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+{"text":"\\section{Introduction}\n\nThe {\\it fast} (asymptotically superlinear or arbitrarily Q-linear) local convergence rate is perhaps computationally the most crucial property of the augmented Lagrangian method.\nThis topic has long been discussed ever since the birth of the augmented Lagrangian method in \\cite{powell}, and proceeded thereafter in \\cite{rockafellar}, \\cite{tre73}, \\cite{ber76}, \\cite{conn}, \\cite{ito}, \\cite{cont93}, \\cite{zhang}, \\cite{sun07}, \\cite{chenjota} and \\cite{cui}, for various problem settings.\nGenerally speaking, it has been observed that if the augmented Lagrangian method achieves a Q-linear convergence rate, the rate should be inversely proportional to the penalty parameter in the algorithm, i.e., the corresponding modulus should tend to zero as the penalty parameter increases to infinity.\nAs a result, both theoretically and practically, large (but not too large to influence the numerical stability) penalty parameters are desirable in implementing the augmented Lagrangian method.\nIn fact, the fast local convergence of the augmented has been exploited in many efficient solvers for large-scale convex optimization problems, e.g., \\cite{zhaoxy}, \\cite{yanglq}, \\cite{chench}, \\cite{lixd}, \\cite{lixd2}, \\cite{lixd3}, \\cite{zhangyj} and \\cite{liu2019}, to name a few.\n\n\nThe main objective of this paper is to investigate whether such a fast local Q-linear convergence is still preserved for the inexact augmented Lagrangian method with a practical relative error criterion proposed by \\cite{eckstein13}.\nThis criterion is of some practical interests since it is checkable whenever the gradient (or the subgradient) of the augmented Lagrangian function is computable.\nMoreover, this criterion is relative in the sense that only {\\it one} tolerance parameter, instead of a summable sequence of nonnegative real numbers, such as that in \\cite{rockafellar}, is used to control the error.\nAs was reported in \\cite{eckstein13}, this approximation criterion improves the augmented Lagrangian method and therefore has significant practical value, but the corresponding algorithm can not be interpreted, at least currently, as an application of the proximal point algorithm (PPA) in \\cite{rockafellar76} or its inexact variants. \nAs a result, the convergence analysis in \\cite{eckstein13} is not based on the convergence properties of the PPAs, which is\nvery different from the convergence analysis for the augmented Lagrangian method in \\cite{rockafellar76}, where the convergence properties, including the fast Q-linear convergence rate, are heavily dependent on those of PPAs.\nConsequently, the convergence properties of the augmented Lagrangian method with the relative error criterion in \\cite{eckstein13} can not be obtained by directly applying some known results.\nTherefore, it is of great interest to know if the algorithm in \\cite{eckstein13} also admits a {\\em fast} (local) convergence and if the modulus is inversely proportional to the penalty parameter.\n\nThe classic augmented Lagrangian method, also known as the method of multipliers, was initiated by \\cite{hestenes69} and \\cite{powell} for solving equality constrained nonlinear programming problems.\nFor general nonlinear programming problems, one may refer to the monograph of \\cite{berbook} and the references therein, where the details for various aspects of the augmented Lagrangian methods were systematically provided.\nIn the context of convex programming, as was illustrated in \\cite{rockafellar}, the classic augmented Lagrangian method can be viewed as a PPA applied to the dual of the given problem.\nMoreover, in the seminal papers \\cite{rockafellar} and \\cite{rockafellar76}, the author has studied both the convergence and the {\\em fast} local convergence rate of the augmented Lagrangian method, even if in the subproblems are approximately solved with a few summable sequences of nonnegative real numbers controlling the errors.\nIn fact, the convergence rate in \\cite{rockafellar} for the augmented Lagrangian method is inherited from the convergence rate of the inexact PPA established in \\cite{rockafellar76}, where the errors are also controlled by a summable sequence of nonnegative real numbers.\n\nAlong a different line, inexact PPAs with relative error criteria, in which summable sequences of real numbers are no longer necessary, also have been well studied in \\cite{solodov99,solodov992,solodov00}.\nMoreover, the corresponding local Q-linear convergence rate was also available in \\cite{solodov992}.\nThere are many concrete instances of applications of these inexact PPAs, but direct applications of them to augmented Lagrangian methods would bring conditions that are not practically verifiable.\nFortunately, in \\cite{eckstein13}, the authors developed a practical relative error criterion for approximately minimizing the subproblems in the augmented Lagrangian method.\nThis criterion is precisely implementable in the sense that all the information needed for checking it can be obtained directly at every candidate approximate solution to the subproblems, instead of using certain values such as the infimum of the objective functions to the subproblems in \\cite{rockafellar}, which are generally not available\\footnote{It wa until recently that practical surrogates of such criteria has been developed for a class of convex composite conic programming problems in \\cite{cui}.}.\nIn \\cite{eckstein13}, an informative discussion on the design of this relative error criterion for the augmented Lagrangian method was provided, and the comparisons of this criterion to those of \\cite{solodov99,solodov992,solodov00} for inexact PPAs are also elaborated.\n\n\nTo be consistent with, and even more general\\footnote{In \\cite{eckstein13}, ${\\mathcal X}\\equiv{\\mathds R}^n$, and the objective function $f$ is assumed to be either a continuous differentiable convex function, or the sum of a continuous differentiable convex function and the indicator function of a closed convex box in ${\\mathds R}^n$. However, the algorithm in \\cite{eckstein13} and all the results established in \\cite{eckstein13} and \\cite{alves} are valid for problem \\eqref{prob}, with the corresponding gradients and normal cones being replaced by subgradients.} than, that of \\cite{eckstein13}, the problem setting of this paper is as follows\n\\(\n\\label{prob}\n\\begin{array}{cl}\n\\displaystyle\\min_{x\\in{\\mathcal X}} &f(x)\\\\[2mm]\n\\mbox{s.t.}& h(x)=0, \\\\[2mm]\n&g(x) \\le 0,\n\\end{array}\n\\)\nwhere ${\\mathcal X}$ is a finite dimensional real Hilbert space endowed with an inner product $\\langle\\cdot,\\cdot\\rangle$ and $\\|\\cdot\\|$ is the corresponding norm,\n$f:{\\mathcal X}\\to(-\\infty,+\\infty]$ is a closed proper convex function, $h:{\\mathcal X}\\to{\\mathds R}^{m_1}$ is an affine mapping,\nand $g:{\\mathcal X}\\to{\\mathds R}^{m_2}$ is a nonlinear mapping, i.e., $g(x)=(g_1(x);\\ldots; g_{m_2}(x))$, with each $g_i:{\\mathcal X}\\to(-\\infty,\\infty)$, $i=1,\\ldots,m_2$ being a continuously differentiable convex function.\n\nIn fact, the original convergence results and the corresponding convergence analysis are not sufficient for analyzing the convergence rate of the algorithm in \\cite{eckstein13}. However, fortunately, the convergence analysis has been further improved in \\cite{alves}, where a novel technique for treating F\\'ejer monotone sequence in product spaces was developed.\nThis partially paves the way for studying the convergence rate of the algorithm in \\cite{eckstein13}.\n\n\nMotivated by the above expositions and the fact that allowing the subproblems being solved approximately would further contribute to the efficiency of the augmented Lagrangian method,\nwe are interested in further exploring the convergence properties of the augmented Lagrangian method with the relative error criterion developed in \\cite{eckstein13} for solving problem \\eqref{prob}.\nIn this paper, we will show that, under a mild local error bound condition, this algorithm also admits a {\\em fast} Q-linear local convergence in the sense that the convergence rate of the dual sequence is Q-linear and the modulus of this rate tends to zero if the penalty parameter increases to infinity.\nBesides, we show that such a local error bound condition is also sufficient to guarantee the convergence of the distance from the primal sequence, generated by the algorithm, to the optimal solution set of problem \\eqref{prob}.\nHere, we should emphasize that neither \\cite{eckstein13} nor \\cite{alves} has established the convergence of the primal sequence.\n\nThis remaining parts of this paper are organized as follows.\nIn Sect. \\ref{sec:pre}, we provide the notation and preliminaries that will be used throughout this paper. In Sect. \\ref{sec:algo}, we present the inexact augmented Lagrangian method for solving problem \\eqref{prob} with the practical relative error criterion developed in \\cite{eckstein13}. Also, we will summarize the corresponding convergence properties from \\cite{eckstein13} and \\cite{alves}.\nHere, we also present some useful equalities and inequalities for further use.\nIn Sect. \\ref{sec:conv}, we introduce a local error bound condition and establish the fast local convergence of the algorithm presented in Sect. \\ref{sec:algo}, as well as the global convergence of the primal sequence.\nWe conclude this paper in Sect. \\ref{sec:conclusion}.\n\n\\subsection*{Notation}\nLet ${\\mathcal H}$ be a finite dimensional real Hilbert space endowed with the inner product $\\langle\\cdot,\\cdot\\rangle$.\nWe use $\\|\\cdot\\|$ to denote the norm induced by this inner product.\nFor any given $z\\in{\\mathcal H}$, we use ${\\mathds B}_\\epsilon(z)$ to denote the closed ball centring at $z$ with the radius $\\epsilon\\ge 0$, i.e.,\n$$\n{\\mathds B}_\\epsilon(z):=\\{z'\\in{\\mathcal H}\\mid \\|z-z'\\|\\le \\epsilon\\}.\n$$\nLet $C\\subset{\\mathcal H}$ be a nonempty closed convex set. The projection of a vector $z\\in{\\mathcal H}$ onto the set $C$ is defined by\n\\[\n\\Pi_C(z):=\\argmin_{z'\\in C}\\left\\{\\frac{1}{2}\\|z'-z\\|^2\\right\\},\n\\]\nwhile the distance from $z\\in{\\mathcal H}$ to the set $C$ is defined by\n\\[\n\\dist (z,C):=\\|z-\\Pi_C(z)\\|.\n\\]\n\nLet $\\theta:{\\mathcal H}\\to(-\\infty,+\\infty]$ be an arbitrary closed proper convex function, we use $\\dom\\, \\theta$ to denote its effective domain, i.e.,\n$$\n\\dom\\, \\theta:=\\{z\\in{\\mathcal H}\\mid \\theta(z)<+\\infty\\}$$\nand $\\partial\\theta$ to denote its subdifferential mapping, i.e.,\n$$\n\\partial \\theta(z):=\\{\n\\gamma \\in{\\mathcal H} \\mid \\theta(z')-\\theta(z)\\ge\\langle \\gamma, z'-z\\rangle,\\ \\forall z'\\in{\\mathcal H}\n\\},\\quad\\forall z\\in{\\mathcal H}.\n$$\nMoreover, if $\\theta$ is continuously differentiable at $z\\in{\\mathcal H}$, one has $\\partial\\theta(z)=\\{\\nabla\\theta(z)\\}$, where $\\nabla\\theta(z)$ is the gradient of $\\theta$ at $z$.\n\nIf ${\\mathcal H}\\equiv{\\mathds R}^l$, i.e., the $l$-dimensional real vector space, we use the dot product as the inner product, i.e.,  $\\langle z,z'\\rangle:=z^Tz'$, so that $\\|\\cdot\\|$ is the conventional $\\ell_2$-norm. In this case we define\n$$\\max\\{z,z'\\}:=\\{\\bar z\\mid \\bar z_i=\\max\\{z_i,z'_i\\}, i=1\\ldots,l\\},\\quad\\forall z,z'\\in{\\mathcal H}, \n$$ i.e., the component-wise maximum between $z$ and $z'$. The component-wise minimum is also defined in the same fashion. \n\n\n\\section{Preliminaries}\n\\label{sec:pre}\nGenerally, the notation and definitions used in this paper are the same as those used in \\cite{eckstein13} and \\cite{alves}. In fact, they are consistent with those in \\cite{rocbook,rocbookcg}.\n\nThe Lagrangian function ${\\mathcal L}:{\\mathcal X}\\times{\\mathds R}^{m_1}\\times{\\mathds R}^{m_2}\\to[-\\infty,\\infty]$ of problem \\eqref{prob} is defined by\n\\[\n{\\mathcal L}(x;\\lambda,\\mu):=\\left\\{\n\\begin{array}{ll}\nf(x)+\\langle \\lambda, h(x)\\rangle+\\langle \\mu, g(x)\\rangle,\\quad &\\mbox{if }\\mu\\ge 0,\\\\[2mm]\n-\\infty, & \\mbox{otherwise}.\n\\end{array}\n\\right.\n\\]\nThen, the dual of problem \\eqref{prob} is given by\n\\(\n\\label{dual}\n\\max_{\\lambda\\in{\\mathds R}^{m_1},\\mu\\in{\\mathds R}^{m_2}}\\left\\{ d(\\lambda,\\mu):=\\inf_{x\\in{\\mathcal X}}{\\mathcal L}(x;\\lambda,\\mu)\\right\\},\n\\)\nwhere $d(\\cdot)$ is called the dual objective function.\nHence, we call $x\\in{\\mathcal X}$ as the primal variable and call $(\\lambda,\\mu)\\in{\\mathds R}^{m_1+m_2}$ as the dual variable.\nNote that the Lagrangian function ${\\mathcal L}$ is convex in $x$ and concave in $(\\lambda,\\mu)$.\nMoreover, for this Lagrangian function,\nthe subdifferential mapping\\footnote{For concave-convex functions, c.f. \\cite[p. 374]{rockafellar}.}\n$\\partial{\\mathcal L}:{\\mathcal X}\\times{\\mathds R}^{m_1}\\times{\\mathds R}^{m_2}\\to{\\mathcal X}\\times{\\mathds R}^{m_1}\\times{\\mathds R}^{m_2}$ is defined as follows\n\\begin{equation*}\n\\begin{array}{lll}\n&(y;u,v)\\in\\partial {\\mathcal L}(x,\\lambda,\\mu)\n\\\\[2mm]\n&\\Leftrightarrow\n\\left\\{\n\\begin{array}{ll}\n{\\mathcal L}(x';\\lambda,\\mu)\\ge {\\mathcal L}(x,\\lambda,\\mu)+\\langle y, x'-x\\rangle, &\\forall x'\\in{\\mathcal X},\n\\\\[2mm]\n{\\mathcal L}(x;\\lambda',\\mu')\\le {\\mathcal L}(x,\\lambda,\\mu)-\\langle u,\\lambda'-\\lambda\\rangle-\\langle v,\\mu'-\\mu\\rangle,\\quad &\\forall (\\lambda',\\mu')\\in{\\mathds R}^{m_1+m_2}.\n\\end{array}\n\\right.\n\\end{array}\n\\end{equation*}\nBased on the above definition, it is easy to verify that the subdifferential mapping\n$\\partial{\\mathcal L}$ is a maximal monotone operator.\nMoreover, if $(x^*,\\lambda^*,\\mu^*)\\in{\\mathcal X}\\times{\\mathds R}^{m_1}\\times{\\mathds R}^{m_2}$ satisfies\n$0\\in\\partial{\\mathcal L}(x^*,\\lambda^*,\\mu^*)$, it holds that $x^*$ is an optimal solution to problem \\eqref{prob} and $(\\lambda^*,\\mu^*)$ is an optimal solution to problem \\eqref{dual}.\nIn this case, $(x^*,\\lambda^*,\\mu^*)$ is called as a saddle point of the Lagrangian function ${\\mathcal L}$, and it holds that $f(x^*)=d(\\lambda^*,\\mu^*)$.\n\nSince $g$ and $h$ are continuously differentiable, we define\n$$\n\\nabla h(x):=\\big(\n\\nabla h_1(x),\\cdots,\n\\nabla h_{m_1}(x)\n\\big)\n\\quad\n\\mbox{and}\n\\quad\n\\nabla g(x):=\\big(\n\\nabla g_1(x),\\cdots,\\nabla g_{m_2}(x)\n\\big).\n$$\nThen, the Karush-Kuhn-Tucker (KKT) system of problem \\eqref{prob} is given by\n\\(\n\\label{kkt}\\left\\{\n\\begin{array}{l}\n0\\in\\partial_x {\\mathcal L}(x,\\lambda,\\mu):=\\partial f(x)+\\nabla h(x) \\lambda+ \\nabla g(x) \\mu,\\\\[2mm]\nh(x)=0,\\\\[2mm]\ng(x)\\le 0,\\ \\mu\\ge0,\\ \\langle\\mu,g(x)\\rangle=0.\n\\end{array}\n\\right.\n\\)\nIf the solution set to the KKT system \\eqref{kkt} is nonempty, from \\cite[Theorem 30.4 \\& Corollary 30.5.1]{rocbook} one knows that a vector\n$(x^*, \\lambda^*,\\mu^*)\\in{\\mathcal X}\\times{\\mathds R}^{m_1}\\times{\\mathds R}^{m_2}$ is a solution to the KKT system \\eqref{kkt} if and only if $x^*$ is an optimal solution to problem \\eqref{prob} and $(\\lambda^*, \\mu^*)$ is an optimal solution to problem \\eqref{dual}.\nMoreover, the solution set to the KKT system \\eqref{kkt} can be written as $X^*\\times P^*$ with $X^*$ being the solution set to problem \\eqref{prob} and $P^*$ being the solution set to problem \\eqref{dual}.\nIn this case, the optimal values of problem \\eqref{prob} and problem \\eqref{dual}\nare equal, and the solution set to the KKT system \\eqref{kkt} is exactly the set of saddle points to the Lagrangian function ${\\mathcal L}$.\n\n\n\n\n\\section{An Inexact Augmented Larangian Method}\n\n\n\n\n\n\n\n\n\\label{sec:algo}\nIn this section, we present the inexact augmented Lagrangian method with the practical relative error criterion of \\cite{eckstein13}, and summarize its convergence properties from \\cite{eckstein13} and \\cite{alves}. Some equalities and inequalities, which are useful to study its convergence rate, are also prepared in this section.\n\nWe dfine the closed convex cone\n$${\\mathcal K}={\\mathds R}^{m_2}_+:=\\{v\\in{\\mathds R}^{m_2}:v\\ge 0\\}.\n$$\nLet $c>0$ be a given real number. \nThe augmented Lagrangian function of problem \\eqref{prob} is defined (with $c$ being the penalty parameter) by\n\\[\n\\begin{array}{l}\n\\displaystyle\n{\\mathcal L}_c(x,\\lambda,\\mu):\n\\displaystyle =f(x)+\\langle\\lambda, h(x)\\rangle+\\frac{c}{2}\\|h(x)\\|^2+\\frac{1}{2c}\\big(\\|\\Pi_{\\cal K}(\\mu+cg(x))\\|^2-\\|\\mu\\|^2\\big)\\\\[4mm]\n\\displaystyle =f(x)+\\sum_{i=1}^{m_1}\\big(\\lambda_ih_i(x)+\\frac{c}{2}(h_i(x))^2\\big)\n+\\frac{1}{2c}\\sum_{i=1}^{m_2}\\Big(\\big(\\max\\{0,\\mu_i+cg_i(x)\\}\\big)^2-\\mu_i^2\\Big),\\quad\n\\\\[4mm]\n\\hfill \\forall x\\in{\\mathcal X}, \\lambda\\in{\\mathds R}^{m_1}, \\mu\\in{\\mathds R}^{m_2}.\n\\end{array}\n\\]\n\nWhen applied to solving problem \\eqref{prob}, the augmented Lagrangian method with the practical relative error criterion proposed in \\cite[(11)-(15)]{eckstein13} can be described as the following Algorithm.\n\n\n\n\\begin{center}\n\\fbox{\\begin{minipage}{.97\\textwidth}\n\\begin{algo}\n\\label{alg:alm}\n{\\bf An inexact augmented Lagrangian method with a practical relative error criterion for solving problem \\eqref{prob}.}\n\\end{algo}\nLet $\\sigma\\in[0,1)$ and let $\\{c_k\\}$ be a sequence of positive real numbers such that $\\inf_{k\\ge 1}\\{c_k\\}>0$.\nChoose $\\lambda^0\\in{\\mathds R}^{m_1}$, $\\mu^0\\in{\\mathds R}^{m_2}_+$ and $w^0\\in{\\mathcal X}$. For $k=1, 2, \\ldots$,\n\\begin{enumerate}\n\\item[\\bf 1.] find $x^{k}\\in{\\mathcal X}$ and $y^k\\in{\\mathcal X}$ such that\n\\(\n\\label{conditionsub}\ny^{k}\\in\\partial_x {\\mathcal L}_{c_k}(x^{k},\\lambda^{k-1},\\mu^{k-1}),\n\\)\nand\n\\begin{equation}\n\\label{condition}\n\\frac{2}{c_k}\n\\Big|\\langle w^{k-1}-x^{k},y^{k}\\rangle\\Big|\n+\n\\|y^k\\|^2\\le \\sigma\n\\left(\n\\|h(x^k)\\|^2\n+\\left\\|\\min\\left\\{\\frac{1}{c_k}\\mu^{k-1},-g(x^k)\\right\\}\\right\\|^2\n\\right);\n\\end{equation}\n\\item[\\bf 2.] set\n\\begin{eqnarray*}\n&&\\lambda^k:=\\lambda^{k-1}+c_kh(x^k),\n\\\\[2mm]\n&&\\mu^k:=\\max\\{0,\\mu^{k-1}+c_kg(x^k)\\},\n\\\\[2mm]\n&&w^k:=w^{k-1}-c_k y^k.\n\\end{eqnarray*}\n\\end{enumerate}\n\\end{minipage}}\n\\end{center}\n\n\\begin{remark}\nIn the above algorithm, each $x^k$ is an approximate solution to the corresponding subproblem with $y^k$ being a subgradient of the corresponding objective function at $x^k$, i.e., $x^k$ approximately solves the problem\n$$\n\\min_{x}{\\mathcal L}_{c_k}(x; \\lambda^{k-1},\\mu^{k-1}).\n$$\nIf this subproblem admits a solution, say $\\tilde x^k$, one can let $y^k=0$ so that \\eqref{condition} is satisfied.\nTherefore, one can always find the sequence $\\{(x^k,y^k)\\}$ via Algorithm \\ref{alg:alm} if the subproblems are well-defined.\n\n\n\n\n\\end{remark}\n\nFor the convenience of further discussions, we denote $m:=m_1+m_2$ and define the sequences $\\{p^k\\}$ and $\\{u^k\\}$ in ${\\mathds R}^m$ via\n\\(\n\\label{def:pu}\np^k:=(\\lambda^k,\\mu^k)\n\\quad\\mbox{and}\\quad\nu^k:=\\frac{1}{c_k}(p^{k-1}-p^k).\n\\)\nThe following theorem is an immediate extension of \\cite[Proposition 1]{eckstein13}.\n\\begin{theorem}\n\\label{prop:main}\nAssume that the solution set to the KKT system \\eqref{kkt} of problem \\eqref{prob} is non-empty.\nSuppose that the infinite sequences generated by Algorithm \\ref{alg:alm} are well-defined. Then,\n\\begin{enumerate}\n\\item[(a)] The sequences $\\{p^k\\}$ and $\\{w^k\\}$ are bounded;\n\\item[(b)] $u^k\\to 0$ and $y^k\\to0$ as $k\\to\\infty$;\n\\item[(c)] Any accumulation point of the sequence $\\{x^k\\}$ is a solutions to problem \\eqref{prob}, and any accumulation point of $\\{p^k\\}$ is a solution to the dual problem \\eqref{dual}.\n\n\n\n\\end{enumerate}\n\\end{theorem}\n\n\nAs can be observed from Theorem \\ref{prop:main}, the convergence results in \\cite{eckstein13} show that the sequence $\\{p^k\\}$ is bounded and each accumulation point of this sequence is a solution to the dual problem \\eqref{dual}.\nTo ensure the full convergence of the sequence $\\{p^k\\}$, in \\cite[Proposition 2]{eckstein13}, the authors introduced the additional criterion to \\eqref{condition} that\n$$\nc_k\\|y^k\\|\\le\\xi\\|p^{k-1}-p^k\\|^2,\n$$\nwhere $\\xi\\ge 0$ is a given constant. As was commented in \\cite{eckstein13}, despite $\\xi$ can be arbitrarily large, this additional criterion seems stringent. Fortunately, in \\cite{alves},\nthe authors showed that it is not necessary to use this extra criterion to guarantee the convergence of the sequence $\\{p^k\\}$,\nthanks to their insightful investigation of F\\'ejer monotone sequences in product spaces.\nThe following theorem directly comes from \\cite[Proposition 2]{alves}.\n\\begin{theorem}\n\\label{theo:main}\nSuppose that all the assumptions and conditions in Proposition \\ref{prop:main} hold.\nThen, the whole sequences $\\{p^k\\}$ converges to a solution to problem \\eqref{dual}.\n\\end{theorem}\n\n\nAs can be seen from Proposition \\ref{prop:main} and Theorem \\ref{theo:main}, the global convergence of Algorithm \\ref{alg:alm} has been well established in the sense that the sequence of the generated dual variables is globally convergent,\nwhich is quite similar to the global convergence properties of the (inexact) augmented Lagrangian method established in \\cite{rockafellar}.\nIn fact, Proposition \\ref{prop:main}, together with Theorem \\ref{theo:main}, lays the foundation for further investigating the convergence rate of Algorithm \\ref{alg:alm} in the next section.\n\nFinally, we should emphasize that, referring to the primal sequence $\\{x^k\\}$, the best possible result in both \\cite{eckstein13} and \\cite{alves} is that any accumulation point of this sequence is a solutions to problem \\eqref{prob}.\n\n\n\n\n\n\\section{Convergence Rate Analysis}\n\\label{sec:conv}\nThis section establishes the local convergence rate of Algorithm \\ref{alg:alm} for solving problem \\eqref{prob}, under a mild error bound condition.\nWe first discuss this error bound condition and then given the main result of this paper.\n\n\\subsection{An error bound condition}\nLet $T:{\\mathcal H}\\to{\\mathcal H}$ be a maximal monotone mapping with $T^{-1}$ being its inverse mapping\\footnote{One may refer to \\cite[Chapter 12]{va} for more information about (maximal) monotone mappings.}.\nFor solving the general inclusion problem of finding $z\\in{\\mathcal H}$ such that\n\\(\n\\label{inclu}\n0\\in T(z),\n\\)\nthe (inexact) PPA in \\cite{rockafellar76} takes the following iteration scheme\n\\(\n\\label{algppa}\nz^{k}\\approx (I+c_k T)^{-1}(z^{k-1}),\\quad k=1,2,\\ldots,\n\\)\nwhere $c_k>0$, and $z^0\\in{\\mathcal H}$ is the given initial point.\nIn \\cite{rockafellar76},\nthe convergence rate of this (inexact) PPA has been analyzed under a local error bound condition in which the solution to \\eqref{inclu} should be a singleton. In \\cite{luque}, the author extended the convergence rate analysis of \\cite{rockafellar76} for the PPA to the case that the solution set of \\eqref{inclu} is not necessarily a singleton.\nThe following definition introduces an error bound condition, which has been used, in the name of a growth condition of maximal monotone operators, in \\cite{luque} for the convergence rate analysis of PPA.\n\n\n\\begin{definition}[\\cite{robinson}]\n\nThe mapping $T^{-1}$ is called locally upper Lipschitz continuous at the origin if $T^{-1}(0)$ is nonempty and there exist constants $\\epsilon>0$ and $\\kappa>0$ such that\n\\(\n\\label{growth}\n\\dist(z,T^{-1}(0))\\le\\kappa\\|\\beta\\|,\\quad\n\\forall \\beta\\in{\\mathds B}_{\\epsilon}(0)\\ \\ \\mbox{and}\\ \\ \\forall z\\in T^{-1}(\\beta).\n\\)\n\\end{definition}\n\nThe augmented Lagrangian method and its inexact version in \\cite{rockafellar} can be explained as an application of the PPA to a certain maximal monotone operator\\footnote{In particular, for problem \\eqref{prob} considered in this paper, this maximal monotone operator is given by $\\partial d$  with $d$ being defined in \\eqref{dual}. Here the subdifferential mapping is in the concave sense, c.f. \\cite[pp. 307--308]{rocbook}.}.\nTherefore, the convergence rate analysis in \\cite{rockafellar} for the augmented Lagrangian method is heavily dependent on the convergence rate analysis in \\cite{rockafellar76} for the PPA.\nHowever, Algorithm \\ref{alg:alm} can not be explained as a certain variant of the PPA, at least currently.\nTherefore, its convergence rate should be studied via a totally different approach.\nIn this paper, to analyze the convergence rate of Algorithm \\ref{alg:alm}, we make the following assumption, which is an error bound condition on problem \\eqref{prob}.\n\n\n\n\n\n\n\\begin{assumption}\n\\label{ass:growth}\nThe mapping $(\\partial{\\mathcal L})^{-1}$ is locally upper Lipschitz continuous at the origin.\n\\end{assumption}\nBefore analyzing the local Q-linear convergence rate of Algorithm \\ref{alg:alm}, we make the following comment to Assumption \\ref{ass:growth}.\n\\begin{remark}\nAssumption \\ref{ass:growth} is not a very restrictive condition. On the one hand, it contains the case that $(\\partial{\\mathcal L})^{-1}$ is locally Lipschitz continuous at $0$, which was used extensively in {\\rm \\cite{rockafellar,rockafellar76}} for deriving the Q-linear convergence rate for PPA and augmented Lagrangian methods.\nNote that even the stronger condition than Assumption \\ref{ass:growth} that $(\\partial{\\mathcal L})^{-1}$ is locally Lipschitz continuous at $0$ can be satisfied, at least heuristically, for ``most'' convex optimization problems in the form of \\eqref{prob} with $f$ being the sum of a twice continuously differentiable convex function plus an indicator function of a closed convex set and $g$ and $h$ being twice continuously differentiable \\cite[p. 105, Remark]{rockafellar}.\nOne may also refer to {\\rm\\cite[Proposition 4]{rockafellar}} and the discussions after this proposition for more information. In addition, we mention that Assumption \\ref{ass:growth} is more general in the sense that $(\\partial{\\mathcal L})^{-1}(0)$ is not necessarily to be a singleton.\n\\end{remark}\n\n\n\n\n\n\\subsection{The convergence rate}\nNow we present the main result of this paper.\n\\begin{theorem}\n\\label{thm:main}\nSuppose that the solution set $X^*\\times P^*$ to the KKT system \\eqref{kkt} of problem \\eqref{prob} is nonempty and the infinite sequences generated by Algorithm \\ref{alg:alm} are well-defined. Suppose that Assumption \\ref{ass:growth} holds (with the parameters $\\kappa>0$, $\\epsilon>0$ such that \\eqref{growth} holds for $T=\\partial{\\mathcal L}$).\nThen, the following statements hold:\n\\begin{enumerate}\n\\item[(a)] for any sufficiently large $k$, it holds that\n\\(\n\\label{result1}\n\\dist(x^k,X^*)\\le\n\\frac{\\kappa(1+\\sqrt{\\sigma})}{c_k}\\|p^{k-1}-p^k\\|,\n\\)\nso that\n\\(\n\\label{pconv}\n\\lim_{k\\to\\infty}\\dist(x^k,X^*)=0;\n\\)\n\\item[(b)]\nif,\nadditionally, one has that\n\\(\n\\label{cond:c}\n\\liminf_{k\\to\\infty}c_k>2\\kappa(\\sigma+\\sqrt{\\sigma}),\n\\)\nthen, for any sufficiently large $k$\n\\(\n\\label{resmain}\n\\dist(p^{k},P^*)\\le \\rho_k \\dist(p^{k-1},P^*)\n\\)\nwith\n$$\n\\rho_k:\n=\\frac{\\kappa\\sqrt{1+\\sigma}}\n{\\sqrt{c_k^2-2\\kappa(\\sigma+\\sqrt{\\sigma}){c_k}+\\kappa^2(1+\\sigma)}}\n$$\nand\n\\(\n\\label{limrho}\n\\limsup_{k\\to\\infty}\\{\\rho_k\\}\n<1.\n\\)\nThis means that the local rate of convergence for Algorithm \\ref{alg:alm} is Q-linear and the modulus of the convergence rate tends to zero if the sequence $\\{c_k\\}$ increases to infinity.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\n{\\it (a)}\nAccording to \\eqref{def:pu} and \\cite[eq. (24)]{eckstein13}, one has that\n$$\n\\begin{array}{l}\n\\big(y^k,\\frac{1}{c_k}(p^{k-1}-p^k)\\big)\n=\n(y^k,u^k)\\in\\partial {\\mathcal L}(x^k,p^k).\n\\end{array}\n$$\nFrom \\cite[eq. (38)]{eckstein13} we know that, for any $p^*\\in P^*$, it holds that\n\\(\n\\label{eq:p}\n\\|p^k-p^*\\|=\\|p^{k-1}-p^*\\|^2-2c_k\\langle u^k,p^k-p^*\\rangle-\\|p^{k-1}-p^k\\|^2.\n\\)\nSince $(0,0)\\in\\partial{\\mathcal L}(x^*,p^*)$ and $\\partial{\\mathcal L}$ is maximally monotone, it holds that for any $x^*\\in X^*$ and $p^*\\in P^*$,\n\\(\n\\label{ineq:px}\n\\langle y^k, x^k-x^*\\rangle+\\langle u^k, p^k-p^*\\rangle\\ge 0.\n\\)\nDefine the sequences $\\{\\bar x^k\\}$ in ${\\mathcal X}$ and $\\{\\bar p^k\\}$ in ${\\mathds R}^{m}$ via\n\\[\n\\bar x^k:=\\Pi_{X^*}(x^k)\n\\quad\\mbox{and}\\quad\n\\bar p^k:=\\Pi_{P^*}(p^k).\n\\]\nCombining \\eqref{eq:p} and \\eqref{ineq:px} together implies that, for any $p^*\\in P^*$,\n\\(\n\\label{ineq:pin1}\n\\begin{array}{ll}\n&\\|p^{k-1}-p^*\\|^2\n-\\|p^{k}-p^*\\|^2\\\\[2mm]\n&=\\|p^{k}-p^{k-1}\\|^2+2c_k\n\\langle u^k,p^k-p^*\\rangle\\\\[2mm]\n&\\ge\\|p^{k}-p^{k-1}\\|^2-2c_k\\langle x^k-\\bar x^k,y^k\\rangle\n\\\\[2mm]\n&\\ge\\|p^{k}-p^{k-1}\\|^2-2c_k\\|x^k-\\bar x^k\\|\\|y^k\\|.\n\\end{array}\n\\)\nSince Assumption \\ref{ass:growth} holds with $\\kappa>0$ and $\\epsilon>0$,\nfrom \\eqref{growth} we know that\n$$\n\\dist\\Big((x,p),(X^*,P^*)\\Big)\\le\\kappa\\big\\|(y,u)\\big\\|,\\quad\n\\forall (y,u)\\in\\partial {\\mathcal L}(x,p) \\mbox{ with }\n\\big\\|(y,u)\\big\\|\\le\\epsilon.\n$$\nFrom Proposition \\ref{prop:main}(b) and \\eqref{ineq:yp1} we know that $\\|(y^k,u^k)\\|\\le\\epsilon$ if $k$ is sufficiently large. Then, there exists a positive integer $k_0$ such that for any $k\\ge k_0$,\n\\(\n\\label{growthbound}\n\\dist\\left((x^{k},p^{k}),(X^*,P^*)\\right)\n\\le\n\\kappa\\left\\|(y^k,u^k)\\right\\|.\n\\)\nThis, together with \\eqref{def:pu}, implies that\n\\[\n\\|x^k-\\bar x^k\\|^2\n\\le\n\\kappa^2\\|y^k\\|^2+\\frac{\\kappa^2}{c_k^2}\\|p^{k-1}-p^k\\|^2.\n\\]\nHence, it holds that, for any $k\\ge k_0$,\n\\(\n\\label{ineq:qq}\n\\|x^k-\\bar x^k\\|\n\\le\n\\kappa\\|y^k\\|+\\frac{\\kappa}{c_k}\\|p^{k-1}-p^k\\|.\n\\)\nNote that \\eqref{condition} can be equivalently written as\n\\(\n\\label{ineq:yp1}\n{2}{c_k}\n\\big|\\langle w^{k-1}-x^{k},y^{k}\\rangle\\big|\n+\nc_k^2\\|y^k\\|^2\\le \\sigma\\|p^{k-1}-p^k\\|^2,\n\\)\nwhich implies that\n\\(\n\\label{ineq:yp2}\n\\|y^k\\|\\le \\frac{\\sqrt{\\sigma}}{c_k}\\|p^{k-1}-p^k\\|.\n\\)\nAs a result, by combining \\eqref{ineq:qq} and \\eqref{ineq:yp2} together one gets\n\\[\n\\|x^k-\\bar x^k\\|\n\\le\n\\frac{\\kappa(1+\\sqrt{\\sigma})}{c_k}\\|p^{k-1}-p^k\\|,\n\\]\nso that \\eqref{result1} is established. Moreover, since $\\inf_{k\\ge 0}\\{c_k\\}>0$,\nthe above inequality, together with \\eqref{def:pu} and Proposition \\ref{prop:main}(b), implies \\eqref{pconv}.\n\n\\medskip\n{\\it (b)}\nOne can get from \\eqref{ineq:qq} that\n\\(\n\\label{ineq:xp1}\n\\|x^k-\\bar x^k\\|\\|y^k\\|\\le\n\\kappa\\|y^k\\|^2+\\frac{\\kappa}{c_k}\\|y^k\\|\\|p^{k-1}-p^k\\|.\n\\)\nBy taking \\eqref{ineq:xp1} into \\eqref{ineq:pin1}, one has that\n\\(\n\\label{ineq:main1}\n\\begin{array}{ll}\n&\\|p^{k-1}-p^*\\|^2\n-\\|p^{k}- p^*\\|^2\n\\\\[2mm]\n&\\displaystyle\n\\ge\\|p^{k}-p^{k-1}\\|^2-2c_k\\left(\\kappa\\|y^k\\|^2+\\frac{\\kappa}{c_k}\\|y^k\\|\\|p^{k-1}-p^k\\|\\right)\n\\\\[4mm]\n&\\displaystyle\n=\\|p^{k}-p^{k-1}\\|^2-2c_k\n\\kappa\\|y^k\\|^2-2\\kappa\\|y^k\\|\\|p^{k-1}-p^k\\|.\n\\end{array}\n\\)\nThen, by combining \\eqref{ineq:yp2} and \\eqref{ineq:main1} together we can get that for any $p^*\\in P^*$,\n\\(\n\\label{ineq:pp2}\n\\begin{array}{llll}\n&\\displaystyle\n\\|p^{k-1}-p^*\\|^2\n-\\|p^{k}-p^*\\|^2\n\\\\[3mm]\n&\\displaystyle\n\\ge\\|p^{k}-p^{k-1}\\|^2-2\n\\kappa\\frac{\\sigma}{c_k}\\|p^{k-1}-p^k\\|^2-2\\kappa\\frac{\\sqrt{\\sigma}}{c_k}\\|p^{k-1}-p^k\\|^2\n\\\\[3mm]\n&\\displaystyle\n=\\left(\n1-\\frac{2\\kappa(\\sigma+\\sqrt{\\sigma})}{c_k}\\right)\n\\|p^{k}-p^{k-1}\\|^2.\n\\end{array}\n\\)\nOn the other hand, as \\eqref{growthbound} holds for $k\\ge k_0$, from \\eqref{ineq:yp2} one has that for any $k\\ge k_0$,\n\\(\n\\label{ineq:pp3}\n\\begin{array}{lllll}\n\\|p^{k}-\\bar p^k\\|^2\n&\\displaystyle\n\\le\n\\kappa^2\\|y^k\\|^2+\\frac{\\kappa^2}{c_k^2}\\|p^{k-1}-p^k\\|^2\n\\\\[3mm]\n&\\displaystyle\\le\n\\frac{\\kappa^2\\sigma}{c^2_k}\\|p^{k-1}-p^k\\|^2\n+\\frac{\\kappa^2}{c_k^2}\\|p^{k-1}-p^k\\|^2\n\\\\[3mm]\n&\\displaystyle\n=\\frac{(1+\\sigma)\\kappa^2}{c_k^2}\\|p^{k-1}-p^k\\|^2.\n\\end{array}\n\\)\nBy combining \\eqref{ineq:pp2} and \\eqref{ineq:pp3} together, one has that for any $k\\ge k_0$,\n\\(\n\\label{ineq:final}\n\\begin{array}{lll}\n&\\displaystyle\n\\|p^{k-1}-\\bar p^{k-1}\\|^2\n-\\|p^{k}-\\bar p^{k}\\|^2\n\\\\[3mm]\n&\\displaystyle\n\\ge\\|p^{k-1}-\\bar p^{k-1}\\|^2\n-\\|p^{k}-\\bar p^{k-1}\\|^2\n\\\\[3mm]\n&\\displaystyle\n\\ge\\frac{c_k^2-{2\\kappa(\\sigma+\\sqrt{\\sigma})}{c_k}}\n{\\kappa^2(1+\\sigma)}\n\\|p^{k}-\\bar p^k\\|^2.\n\\end{array}\n\\)\nAccording to \\eqref{cond:c}, it is easy to see that there exists a certain positive integer $k_0'$ such that for all $k\\ge k'_0$, it holds that $c_k>2\\kappa(\\sigma+\\sqrt{\\sigma})$,\nso that\n$$\n\\frac{c_k^2-{2\\kappa(\\sigma+\\sqrt{\\sigma})}{c_k}}\n{\\kappa^2(1+\\sigma)}>0.\n$$\nHence, from \\eqref{ineq:final} one can get that for any $k\\ge\\max\\{k_0,k'_0\\}$,\n\\[\n\\frac{\\|p^{k}-\\bar p^k\\|^2}{\\|p^{k-1}-\\bar p^{k-1}\\|^2}\n\\le\n\\frac{{\\kappa^2(1+\\sigma)}}{{\\kappa^2(1+\\sigma)}\n+{c_k^2}-{2\\kappa(\\sigma+\\sqrt{\\sigma})}{c_k}}.\n\\]\nThis proves \\eqref{resmain}.\nFinally, denote\n\\(\n\\delta:=\\liminf_{k\\to\\infty}\\{c_k\\}-2\\kappa(\\sigma+\\sqrt{\\sigma})>0.\n\\)\nSince $c_k>2\\kappa(\\sigma+\\sqrt{\\sigma})$ for $k>k_0'$, one has in this case that\n$$\nc_k^2-2\\kappa(\\sigma+\\sqrt{\\sigma}){c_k}>\\delta c_k>2\\delta\\kappa(\\sigma+\\sqrt{\\sigma}),\n$$\nso that\n$$\n\\rho_k=\n\\frac{\\kappa\\sqrt{1+\\sigma}}\n{\\sqrt{c_k^2-2\\kappa(\\sigma+\\sqrt{\\sigma}){c_k}+\\kappa^2(1+\\sigma)}}\n<\n\\frac{\\kappa\\sqrt{1+\\sigma}}\n{\\sqrt{2\\delta\\kappa(\\sigma+\\sqrt{\\sigma})+\\kappa^2(1+\\sigma)}}<1.\n$$\nThis proves \\eqref{limrho} and hence completes the proof of the theorem.\n\\end{proof}\n\nBefore concluding this paper, we should make some comments on the results established above.\n\\begin{remark}\nApparently,\nThe result on the rate of convergence established in Theorem \\ref{thm:main} depends on the condition \\eqref{cond:c}.\nThis can partially be explained by the fact that due to the subproblems of Algorithm \\ref{alg:alm} are solved inexactly, one can not always guarantee the sequence $\\{p^k\\}$ is F\\'ejer monotone to $P^*$ even if $k$ is sufficiently large.\nHowever, this is not something ``bad'' since\n\\eqref{cond:c} is very likely to be satisfied.\nThe reason is that, if $c_k$ increases to $+\\infty$, \\eqref{cond:c} must be satisfied when $k$ is sufficiently large and $\\rho_k$ converges to $0$, no matter what the values of $\\kappa$ and $\\sigma$ are.\nAs a result, large (but not too large to cause the numerical issues when solving the subproblems) penalty parameters are preferred when implementing Algorithm \\ref{alg:alm} as they can both bring better modulus of the convergence rate and enhance the likelihood that \\eqref{cond:c} holds.\n\\end{remark}\n\n\n\n\\begin{remark}\nIt was established in {\\rm\\cite{eckstein13}} that any accumulation point of the primal sequence $\\{x^k\\}$ is a solution to problem \\eqref{prob}, but this sequence is not guaranteed to be bounded.\nAs a result, the convergence properties of this sequence has not been explored.\nMoreover,\neven the sequence $\\{w^k\\}$ is bounded, it is just a sequence of auxiliary variables so that the importance of its convergence properties is not apparent.\nHence, the convergence result of the sequence $\\{x^k\\}$ established in Theorem \\ref{thm:main} (a), to some extent, makes real progress.\n\\end{remark}\n\n\n\n\\begin{remark}\nRecently, in \\cite[Proposition 1]{cui}, the authors have improved the convergence rate analysis of the PPA \\eqref{algppa} for solving \\eqref{inclu} with the error tolerance criteria in \\cite{rockafellar76} by relaxing the upper Lipschitz continuous of $T^{-1}$ at the origin to\nthe condition that $T^{-1}$ is calm\\footnote{\nSuch a calmness assumption is weaker than the upper Lipschitz continuous property of $T^{-1}$, and one may refer to \\cite[Section 2.1]{cui} for details.} at the origin for $z^{\\infty}$ with a certain positive modulus, where $z^{\\infty}$ is the limit of the sequence ${z^k}$ generated by the PPA (which always exists due to the global convergence property of the PPA).\nAs can be seen from Proposition \\ref{prop:main} and Theorem \\ref{theo:main}, the primal sequence generated by Algorithm \\ref{alg:alm} is not guaranteed to be convergent in general. Hence, it is impracticable to impose similar calmness assumptions on $(\\partial {\\mathcal L})^{-1}$ instead of using Assumption \\ref{ass:growth}.\n\n\n\\end{remark}\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusion}\nIn this paper, we have analyzed the fast Q-linear convergence rate of the inexact augmented Lagrangian method with a practical relative error criterion, under a mild local error bound condition.\nThe results established in this paper imply that if the penalty parameter increases to infinity, the modulus of the convergence rate would tend to zero, which constitutes an asymptotically superlinear convergence rate.\nBesides, without introducing any additional condition, the distance from the primal sequence to the solution set of the primal problem is guaranteed to vanish.\nThe results here can serve as the guideline for choosing the penalty parameter when implementing the inexact augmented Lagrangian method of \\cite{eckstein13}.\n\n\n\n\n\n\n\\section*{Acknowledgments}\nThe authors would like to thank Prof. Defeng Sun at the Hong Kong Polytechnic University for bringing the paper of \\cite{eckstein13} to their attentions.\n \n\n\n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRecent developments in technology make it possible to perform for \nthe first time large-area sensitive blind surveys at 21-cm to search\nfor extragalactic H{\\sc i}. This opens many scientific opportunities. Most \nimportantly, the selection effects through the 21-cm window will be quite different \nfrom the well known optical selection effects which may presently bias \nour understanding of galaxy populations (e.g. Impey \\& Bothun 1997, \nDisney 1998). For instance, low surface brightness (LSB) galaxies are \ndimmer than the sky and nearby gas-rich dwarf galaxies show a morphology\nconfusingly similar to background giant galaxies which make both galaxy\ntypes very difficult to  identify optically.  But they may well become\n``visible'' in a 21-cm survey if  they    have  a significant amount \nof H{\\sc i}. LSB galaxies appear to be common - with approximately equal numbers of \ngalaxies, of a given size, in each one-magnitude bin of blue central \nsurface brightness $\\mu_{B,0}$     \nmeasured between $21.5<\\mu_{B,0}<26.5$ (Turner \net al.~1993; McGaugh 1996). Considering the relative narrowness of \nthe bins at fainter magnitudes this is a remarkable result. The \ncosmological significance of LSB galaxies remains controversial\nbecause of uncertain  optical selection effects at faint luminosities \n(see Impey \\& Bothun 1997, Disney 1998 \n   and references therein). Likewise there is \nserious disagreement about the relative number of faint dwarf \ngalaxies. Magnitude-limited surveys tend to find a flat faint-end slope \n(Schechter, 1976, \n   parameter $\\alpha \\sim 1$) to the optical luminosity function \n(e.g. Loveday et al.~1992, Lin et al.~1996). This would suggest that the \ntotal light contribution from dwarfs in the universe is insignificant, \nwhereas surveys within clusters of galaxies (e.g.  \n   de Propris et al. 1995;  Driver \\& Phillipps 1996;  \nTrentham 1997;  Phillipps, \n Parker, Schwartzenberg \\& Jones 1998) find steep faint end slopes with $\\alpha>2$, suggesting a  \ndominant  role for dwarfs in the determination of the light content of \nthe universe. Dwarf galaxies may also contain appreciable amounts of \ndark matter (e.g. Freeman 1997), implying that they may add even more to the \noverall mass in the universe than to the light. Finally, any extragalactic survey \nin which every object is automatically tagged with its velocity-distance \noffers rare opportunities for statistical studies. \n\nBlind 21-cm surveys have a long history but their ambitions \nhave been severely curtailed by technological limitations. \nSingle dish radio telescopes large enough to reach high sensitivity \nhave had beam areas too small to survey large areas of sky, while \ninterferometric surveys are still confined to relatively narrow \nvelocity windows by the computer power required to carry out \nthe necessary cross-correlations between every interferometer \npair. Nevertheless,  in recent times \n  blind surveys have been carried out in \nthe galaxy clusters Hydra by McMahon  et al.~(1993) and Hercules\nby Dickey (1997); in voids by Krumm \\& Bosch (1984) and \nWeinberg et al. (1991);  behind the Galactic plane by Kerr \\& \nHenning (1987); and in the field by Henning (1995). \n Latterly, narrow zenith-strip scans have been done at Arecibo  \n  (see later)  by \nSpitzak \\& Schneider (1998) and Zwaan et al.~(1997).\n\nThe most recent 21-cm survey is our  H{\\sc i} Parkes All-Sky Survey \n(H{\\sc i}PASS ) which started early in 1997 and is expected to be \ncomplete within $\\sim$3 years. This project takes advantage of the \nnew multibeam instrument at the 64m radio telescope at \nParkes (Australia) equipped with 13 separate beams (26 receivers) and \nenough correlator power in each beam to survey 1024 velocity channels \nsimultaneously. The scientific aims of the project include a  \n study  of  the Galactic H{\\sc i} and  of the galaxy content of the local \nuniverse. For this purpose     we scan  \n  the complete southern sky in the \nvelocity range $-1200$ km\\,s$^{-1}$  $<v_{\\odot}<12,700$ km\\,s$^{-1}$ . \n\nFor the purpose of testing the performance of H{\\sc i}PASS   the present \nlimited study was carried out using  the \n   first set of H{\\sc i}PASS    data completed\nto full sensitivity. The selected area covers a substantial fraction of the nearby \nCentaurus A group of galaxies. The main reasons for this choice are twofold:  \n   (1) \nthe Cen\\,A group is among the closest galaxy groups to the Local Group\nat $\\sim 3.5$\\,Mpc (Hui et al.~1993; Sandage et al.~1994) and \ncontains the richest population of   different morphological galaxy types (de \nVaucouleurs 1979), including the peculiar giant elliptical NGC 5128, the \nlarge disk galaxies M 83 and NGC 4945, the S0 galaxy NGC 5102, and the active \nstar forming galaxy NGC 5253. All main group members appear disturbed in \ntheir morphology. It has been suggested (Graham 1979; van Gorkom et al.~1990) \nthat the recent accretion of a population of gas-rich dwarfs could be the \nreason.  The group lies in the Supergalactic Plane with its southern \nextension running down to low galactic latitudes making optical studies \nchallenging because of high extinction. (2) The group has recently been the \n subject of a careful optical survey (C\\^ot\\'e et al.~1997)  enabling \n comparison between contemporary optical \n searches for gas-rich type (irregular) \n galaxies and   our ``blind'' H{\\sc i} \n  survey.   \n\n\n Most \n   earlier  H{\\sc i} studies of groups of galaxies have been done based on  \n  previous  \noptical identifications of candidate members.  In the M~81 group,  \nvan Driel et al.\\ (1998) and Huchtmeier \\& Skillman (1998) have independently \ndone 21~cm searches for optically selected dwarf candidates.  Neither group \ndetected any new members. \n Huchtmeier, Karachentsev \\& Karachentseva (1997) followed up on  \npotential dwarfs found on the Second Palomar Sky Survey (POSS II), and found \n1 new candidate of the IC~342\/Maffei group, and three new dwarf members each \nin the NGC~6946 and NGC~672\/IC~1727 groups.   \nMaia \\&  Willmer (1998) have studied the neutral \nhydrogen content of a number of ``loose groups'', but they also used  \npointed observations of optical candidates. \n\n\n   Several H{\\sc i} searches of  \ncompact groups of galaxies have been done (Oosterloo \\& Iovino 1997; Williams \n\\& van Gorkom 1997; Williams, McMahon \\& \n  van Gorkom 1991), however these groups are usually \nfar enough away that both the spatial resolution and the sensitivity preclude \ndetecting dwarf galaxies.   However, a \n   few nearby groups have been surveyed \n blind at 21~cm, and they include \nM~81, CVnI and  NGC~1023, in which Lo \\&  Sargent (1979) found 4 \n previously uncatalogued members, and a part of  the \n Sculptor   group, studied by \n Haynes \\& Roberts (1979) who found  uncatalogued H{\\sc i} \nobjects thought to be  group members.   \n \n\n\nThe present paper is organised as follows: \\S 2 describes the 21-cm survey;  \nthe observations with results are given in \\S 3 and \\S 4. The optical\nfollow-up on the newly detected galaxies and their optical properties are \npresented in \\S 5 and \\S 6. In \\S 7 we derive the H{\\sc i} Mass Function\nfor the Cen\\,A group. Finally, we discuss detection limits and future projects \nin \\S 8.   \n\n\\section{Survey}\n\n   The Cen\\,A group  covers approximately 1000 square degrees of sky down to and perhaps \n through  the galactic plane. Since it is also projected  against \n the Supergalactic  plane, the area is rich in galaxies and therefore \n well suited to test the performance of H{\\sc i}PASS .  \n\n  The comparison optical search for dwarf irregulars in the area by \nC\\^ot\\'e et \nal.~(1997; hereafter C97)  covered 900 square degrees \nbetween   \n  $12^{h}30^m< \\alpha <15^{h}00^m$ \n   and $-50^{\\circ}< \\delta <-20^{\\circ}$ whereas we have  surveyed a \n smaller  area of 600 square degrees, chosen because it fits \n into the H{\\sc i}PASS   survey grid. C97 found 18 H{\\sc i} rich dIrr's \n(2 new)  of which 14 were in our search-area.  Adding \n AM 1321-304  found by  Matthews \\& Gallagher (1996) and the six well known \n bright group members NGC 5128, M 83, NGC 4945, NGC 5102, \n NGC 5253 and ESO 270-G017  means there were in all 21 \n suspected group members in our 600 square degree \n search area prior to our survey, with another 7 \n outside (see Figure 1). Our search was  confined to \n the velocity range 200 $< v_{\\odot} <$ 1000  km\\,s$^{-1}$ , cut off at \n the lower end because of the possibility of \n confusion with High Velocity Clouds which are numerous in \n the Cen\\,A group direction (Wakker, 1991). One \n group member - CEN 5 ($v_{\\odot}$=160  km\\,s$^{-1}$ ) \n with a velocity lower than 200  km\\,s$^{-1}$  has been previously claimed  (C97) and there remains \n the possibility  that other group members in our data  remain excluded by the 200  km\\,s$^{-1}$  \n restriction. \n\n\n  If we exclude CEN 5 from the statistics  then  there were in our search \n area, prior to this survey, 18 suspected group members with H{\\sc i} \ndetections and two with H$\\alpha$ (NGC 5206 and \n ESO 272-G025).  \n\n\n\n\n\\section{ Observations and Data Reduction}\nThe 21-cm observations for this pilot study were carried out in February and \nMarch 1997 as part of the  H{\\sc i}PASS  survey which is ongoing  at the 64m \nsingle dish Parkes Radio Telescope. This data set comprises the first \ncompleted area observed to full sensitivity with H{\\sc i}PASS . The instrument is a \nmultibeam system consisting of 13 beams, each with 2 orthogonal polarisations. The \nscanning rate across the sky is 1 degree\/minute at constant right ascension. \nThe data from the 26 correlators are stored every 5 seconds. Each correlator has \n1024 channels tuned to a velocity range between $-1200$  km\\,s$^{-1}$  to $12,700$  km\\,s$^{-1}$ , \n corresponding  to a channel separation of $13.2$ km\\,s$^{-1}$ . The \n   system temperature, $T_{\\rm sys}$, is $\\sim 21$ K and the \n  scanning pattern is \norganised so \n   that every point in the sky passes through several beams during a scan, \nand through several different scans separated by days or weeks of time depending on \nthe observing schedule. This allows for the rejection of time-dependent interference. \nThe total integration time per point on the sky is 450 seconds. The 5$\\sigma$ sensitivity of the \nH{\\sc i}PASS  survey is measured at 70 mJy per channel per beam. For more details \nabout the multibeam system we refer to Staveley-Smith et al.~(1996).  \nThe observed spectra were smoothed on-line by applying a 25\\% Tukey filter, to  \nreduce ``ringing'' caused by the strong Galactic signal entering through spectral \nside lobes. This process accounts for a loss in velocity resolution of 37\\%. \nConsequently the actual velocity resolution is $18.2$ km\\,s$^{-1}$ , or $35$ km\\,s$^{-1}$  after \nHanning smoothing. A description of the on-line data reduction at the telescope \nwhich involves bandpass correction can be found in Barnes et al.~(1998). \n Three-dimensional data cubes ($\\alpha$, $\\delta$, $v_{\\odot}$)\nwere produced by putting individual spectra onto a pixel grid of \n$4^{\\prime} \\times 4^{\\prime}$. Due to the lack of an automatic source finder \nat this early stage of the H{\\sc i}PASS  survey, these cubes were visually searched \nfor H{\\sc i} sources by moving through the velocity range $0<v_{\\odot}<1000$  km\\,s$^{-1}$ ,   \nwhich covers the velocity distribution of the Cen\\,A group,  and corresponds \n   to \n1000 km\\,s$^{-1}$ \/35 km\\,s$^{-1}$ =28 independent velocity resolution elements at a given \nspatial pixel on the sky. Given that there are 10,000  \n  such  pixels in this survey, we searched \nthrough 280,000 independent ``cubicles''  (i.e. 3-dimensional pixels). \n\n  The 1$\\sigma$ sensitivity of \nthe H{\\sc i}PASS  survey  was   measured at  \n   14 mJy\/channel. Assuming  noise  normally distributed  about \n  zero, we expect \n statistically to find $\\sim$ 0.1  spurious positive 5-sigma detections in \nthe data. However, 27 objects with velocities $v_{\\odot}>200$ km\\,s$^{-1}$     \nwere found at this detection threshold (plus CEN 5). \n Among others we recovered  main Cen\\,A group \nmembers such as NGC 4945, NGC 5102, M 83, NGC 5237, and NGC 5253. NGC 5128 remained undetected \nbecause  the H{\\sc i} emission was swamped by  \n  strong continuum emission from that galaxy.  \n   \n\nMany more H{\\sc i} sources have been identified with velocities $v_\\odot < 200$ km\\,s$^{-1}$ .\nThese are most likely not galaxies but galactic high velocity clouds,  as \nthere is a clear overlap in velocity space between nearby galaxies and the system of \nHVCs of our Galaxy towards  the Cen\\,A group direction (Wakker 1991). \nBecause of the number dominance of high velocity clouds,  and because only one Cen\\,A \ngroup galaxy is known below $200$ km\\,s$^{-1}$  to date (CEN 5, see Table 1),  \n   we \nrestricted our study and optical follow-up program (see \\S 5) \n   to   detections with \n$v_\\odot>200$  km\\,s$^{-1}$ .   \n\n\n\\section{H{\\sc i} properties}\n\nWe determined the H{\\sc i} fluxes of our  detections by making zero-th \nmoment maps. A MIRIAD (Sault et al.~1995) gauss fitting routine yielded an estimate \nof the object's coordinates  $(\\alpha,\\delta)$ (cols.~[2] and [3] in Table \n 1).   \nPositional accuracy should equal the half-power beamwidth ($14^{\\prime}.8$) divided by \nthe signal-to-noise ratio, which corresponds to $< 3^{\\prime}$ (3.2 kpc at the \ndistance of Cen\\,A) for all sources with a 5$\\sigma$ detection or better. A \nspectrum was extracted from the data cube at the sky position from which the \nheliocentric radial velocity, $v_{\\odot}$ (col.~[4]), was measured at the mid-point \nat 50\\% of the peak of emission, and the total flux was determined (col.~[5]). \nFurthermore, we measured the velocity widths $\\Delta V_{50}$ (cols.~[6]) and \n$\\Delta V_{20}$ (cols.~[7]) at 50\\% and 20\\% of the peak, respectively. The \nH{\\sc i} mass (col.~[8]) was computed by employing the standard formula: \n\n$$\n\\frac {M_{\\rm H{\\sc i}}}{M_{\\odot}} = 2.356 \\times 10^{5} (D[{\\rm Mpc}])^{2} \n\\int{S_{\\nu}dV} \n$$\n\n\\noindent\nwhere D is 3.5 Mpc, our assumed distance to the \n   Cen\\,A   group. \n\n\nThe examination of the Hanning smoothed \n spectra (Figure 2) reveal that none of the detections are \nmarginal. All sources are found to have signals above 3.9 Jy km\\,s$^{-1}$ . \n  The coordinates of the H{\\sc i} detections were cross-correlated with \nthe NASA Extragalactic Database and compared with UK Schmidt films to find optical \ncounterparts. Each source could be associated unambiguously with  a galaxy \n(col.~[1] of Table 1 \n ) which is located within the positional \n   uncertainty \nof 3$^{\\prime}$ radius. In some cases more accurate position informations were  \n  acquired with the Compact Array of the Australia Telescope (see Staveley-Smith et al.~1999)  where   the identifications were \n always confirmed.  \n\nAmong the 28 detection we found ten new Cen\\,A group members. \n  Five of these  galaxies were \npreviously catalogued but not known to be associated with  the group, \n while the  other five\nwere previously uncatalogued galaxies and are identified  according to the H{\\sc i}PASS  \nsurvey nomenclature (HIPASS 1321-31 has recently been detected as \n 131820.5-311605 by  Karachentseva \\& Karachentsev  1998). \n\nThe most gas-rich new group member is ESO 174-G?001 with a H{\\sc i} mass of \n$M_{H{\\sc i}}=2.1\\times 10^{8}$ M$_{\\odot}$,  \n   which is larger than the amount of \nH{\\sc i} observed for instance in NGC5253 or NGC5408. It is a low surface brightness \ngalaxy close to the Galactic plane ($b=8^{\\circ}$) which may explain why its identification as a \ngalaxy was previously uncertain (Lauberts 1982). CEN 5 was previously optically \nidentified and classified as a group member candidate by C97, and so \n we include the spectrum for completeness, although it is excluded from the \n statistics by its low velocity ($<$ 200  km\\,s$^{-1}$ ).   \n  Looking at the spectrum (Fig. 2), we see two peaks. However, \n the   higher velocity   signal is from the   nearby  \n   galaxy  NGC 4945,   the complete profile of \n   which  can   be seen    in Figure 2.  \n\n\n\nFor completeness reasons we also list in Table 1\n  the H{\\sc i} limits \nNGC 5206, ESO 272-G025, and NGC 5128 which remained undetected in this study: NGC 5206\nis an early-type galaxy and ESO 272-G025 was classified as peculiar (Lauberts, 1982).   These  two galaxies  were also not detected in  H{\\sc i} emission \n  by C97, to a more sensitive detection limit.  \nSince the Multibeam failed to detect the powerful radio galaxy NGC 5128, \n the listed H{\\sc i} mass for this galaxy \nis taken   from de Vaucouleurs et al. (1991),  derived from absorbing gas. \n\nIn Figure 3 and Figure 4 we compared the total fluxes and velocity-widths \n($\\Delta$V$_{20}$) with the corresponding data measured by C97. \n   In   general we found\ngood agreement,  showing H{\\sc i}PASS is achieving \n the expected sensitivity,   although for three galaxies we measured   larger integrated fluxes, suggesting that there may be a 21-cm signal \n outside C97's chosen search areas.  \n\n\n\\section{Optical properties}\n\nWe attempted to get CCD images in the optical photometric bands $B$ and $R$ for \nthose galaxies without reliable photometry in the literature. The optical data\nwas acquired as part of an extensive optical follow-up program of H{\\sc i}PASS \ngalaxies. This program employs three telescopes: (i) the 40-inch telescope at \nthe Siding Spring Observatory, (ii) the 1m Swope Telescope at Las Campanas Observatory,\nand (iii) the Curtis Schmidt telescope at CTIO. \n We used $2048^{2}$ CCDs with a $\\sim$ 24$^{\\prime}$ field of \nview and  a pixel size of $0.7^{\\prime\\prime}$, except for the \n Siding Spring  observations, where \n a 800$^{2}$  CCD was used with a $\\sim$ 8$^{\\prime}$ field of \nview and the same pixel size.  For each galaxy a series\n   of 6 exposures of 300 \nseconds each were taken.   The optical data were  \n de-biased and flattened \n with the standard STARLINK CCDPACK (Draper 1993) \n reduction techniques,   then calibrated using \n  Landolt (1992) standards.   \n The seeing was typically \n  1$^{\\prime\\prime}$.  \nIn the case of \\hipass1337-39, we had to rely on the image from the \nDigitised Sky Survey, DSS\\footnotemark  (a   similar   method   to that  \n    employed by   Spitzak \\& \n Schneider, 1998)    to calibrate   an  image for which  we  \n  had no  calibrated   CCD \n  data.  \n\\footnotetext{The Digitized Sky Surveys  \n  were produced at the Space Telescope Science Institute under U.S.  \nGovernment grant NAG W-2166.}\n\n\nFigure 5 shows $B$-band images of the newly identified Cen\\,A group galaxies \nto illustrate the variety in optical morphology. Previously catalogued \ngalaxies (single *) \nare high surface brightness objects and thus likely to be visually identified as \nbackground galaxies. The uncatalogued galaxies  \n  (**),   on the other hand, are faint, low \nsurface brightness galaxies with small angular size and very hard to see.  \n\nIn Table 2 \n  we list photometric parameters of the 10 new group members \n(col.[1]) drawn  from various sources - either   The Third Reference \n Catalogue of Bright Galaxies, RC3 (de Vaucouleurs et al. 1991), the \nDSS, or from our own CCD data (if available). The coordinates (cols.~[2] \nand [3]) are the optical positions either quoted in the literature or measured by us at the \ncentroid of the  optical  image. When CCD data were available, \nthe galaxy image was cleaned by removing superimposed foreground stars. Afterwards, \nthe total apparent magnitude $m_B$ (col.~[4]) was determined by  circular aperture \nphotometry and fitting a growth curve.  \nOtherwise the total apparent $B$ magnitude was \n  taken from the literature. Where \nwe could fit an exponential profile $\\mu(r)=\\mu_0^{exp}+ 1.086(r\/r_0) $ to \nour own optical data we also quote the extrapolated central surface brightness \n$\\mu_{0}$ (col.~[6]) and the exponential scale length $r_0$\n   (cols.~[7] and [8]). \n   In some cases the galaxy was too faint, or too obscured by  stars, \n for a reliable profile fit to be made. \n\n\n\\section{Results} \nTo summarise our results, a shallow   integration  H{\\sc i} survey of the inner 600 square degrees of \nthe Cen\\,A group and the velocity range $200  <v_{\\odot}<1000$ km\\,s$^{-1}$    \n  revealed 28 H{\\sc i} sources \nwhich all have been \n   identified as Cen\\,A group galaxies. All detections have integrated \nflux larger than 3.9 Jy km\\,s$^{-1}$ .  \n Eighteen   objects coincide with previously known group members.  A  \n  further   \n 10 are \n  suggested as new group members.  Of these 10,  5 coincide with previously catalogued \ngalaxies and 5 coincide with previously uncatalogued galaxies. Three main group \nmembers were not detected at our survey sensitivity of 10$^{7}$ M$_{\\odot}$ of H{\\sc i}, assuming an H{\\sc i} velocity width of   \n   50 km\\,s$^{-1}$ , but one (NGC 5128) is \n known as an  H{\\sc i} source  from absorption \n studies.    All new galaxies  detected in this study have optical \n  counterparts.  \n \n\nTable 3 \n   lists various intrinsic parameters for the new group members. \nThe optical radius $R_{opt}$ (col.~[2]) has been estimated by eye on the DSS images.  \nThe model-dependent total $B$ magnitudes,  m$_{T}$ (col.~[4]),  have been calculated for \neach galaxy using the parameters of the exponential profile, $\\mu_{0}^{exp}$ and $r_0$ \n(cols.~[6] and [7], Table~\\ref{photomone}),    using \n\n\n$$\nm_{T}  =  \\mu_{0}^{exp} - 2.5 log(2 \\pi r_0^2)\n$$\n\n\n  However, this  depends on an extrapolation to infinity, and so \n the model magnitudes should be \ncompared with the m$_{B}$ values in Table 2\n   obtained from aperture \nphotometry. The galactic extinction $A_B$ for each galaxy, given in col.(4), \n has been obtained from the dust maps of Schlegel et al.~(1998), wherein dust is \nmapped from the energy it emits in the far infra-red as seen in the high  spatial \nresolution observations made with IRAS and DIRBE. In the case of the highly obscured \nESO174-G?001 we have checked that estimate by measuring the colours of neighbouring background \nelliptical galaxies in our frames, and find excellent agreement ($\\pm$ 0.1 magnitudes). \n\nAssuming a mean distance modulus of $(m-M)=27.8$ for the Cen\\,A group \n  (Saha et al. 1995),  and \ncorrecting for the extinction, we converted the absolute magnitude M$_{B}$ (col.~[5]) \nand determined M$_{\\rm H{\\sc i}}$\/L$_{\\rm B}$ (col.~[6]).  The mass M$_{\\star}$, quoted in \ncol.~(7) is arrived at by simply assuming a value of the stellar mass-to-light ratio \nM$_{\\star}$\/L$_{\\rm B}$ = 1  for such gas-rich dwarf galaxies - a result suggested \nby previous authors (e.g. Staveley-Smith et al.~1990 and de Blok, McGaugh \\&  \nvan der Hulst, 1996). Finally, \nthe ratio between indicative dynamical mass  M$_{\\rm i}$ $\\equiv$ R$_{opt}$($\\Delta$V$_{50,corr})^{2}$\/G \n(Roberts 1978) and photometric mass M$_{\\star}$ is given in col.~(8).   \n   $R_{opt}$   is the quoted \noptical radius   and $\\Delta V_{50}$ is the \nFWHM, corrected for  inclination if the minor and major \n axes of the galaxy could be reliably determined.  \n\n\nAlthough M$_{\\rm i}$  may be a crude estimator of the true dynamical mass, it should provide \na safe lower limit because H{\\sc i} radii are usually  larger than the optical radii. For \ninstance, Bosma (1978) suggests H{\\sc i} disks extend out to 1.5 $\\times$ the optical Holmberg \nradius, while for gas-rich galaxies, the ratio of H{\\sc i} to optical diameter can be much \ngreater (e.g. van Zee et al.~1995).  \n\nThe newly found group members fall in two main categories. Of the five previously \ncatalogued galaxies, ESO 321-G014 (Figure 5)  and ESO 269-G058 \n  (Figure 5)  are not in the C97 area: ESO 321-G014  \nis obviously a dwarf irregular, while ESO 269-G058  is a highly irregular object with obvious \nH{\\sc ii} regions and strong dust lanes. For the latter galaxy,  an incorrect redshift (1853  km\\,s$^{-1}$ ) is \npublished in the literature (de Vaucouleurs et al.~1991). \n Our velocity \nfor IC 4247 (418  km\\,s$^{-1}$ ), Figure 5,  is rather  different \n  from  the value of 274  km\\,s$^{-1}$    published by da Costa et \nal.~(1987), and now places it firmly in the Cen~A group. \n\nThe five previously uncatalogued galaxies exhibit a quite different morphology. They are faint \n($\\sim$ 17$^{m}$),  low surface brightness, inconspicuous objects picked up in a blind survey \nsolely because they are anomalously rich in H{\\sc i}. The survey wouldn't \npick up objects as faint as m$_{B}$=17 unless they had M$_{\\rm H{\\sc i}}$\/L$_{\\rm B}$ \nratios of 4 or more. Indeed, if we  look at their  \n  ratios of   M$_{\\rm H{\\sc i}}$\/L$_{\\rm B}$   \n{\\bf uncorrected} for Galactic extinction we find they have values ranging  \nbetween 4 and 10, with a median of 5.5. Higher values  are \nknown in the literature (e.g. Lequeux \\& Viallefond 1980; Bergvall \\& \nJ\\\"ors\\\"ater 1988; Brinks \\& Klein 1988; van Zee, Haynes \\& \n Giovanelli, 1995)  but are very rare. If such objects have a \nstellar mass-to-light ratio M$_{\\star}$\/L$_{\\rm B}$~=~1,  then they contain significantly \nmore mass in the form of gas than of stars, and it will be interesting to investigate what \ntheir evolutionary status may be. \n\n\n\n\\section {The H{\\sc i} Mass Function}\n\n\n   There is considerable controversy  over the faint-end slope of the  galaxy \n Luminosity Function (see \\S 1) and thus added interest in the low-mass \n end of the  H{\\sc i} Mass Function (H{\\sc i}MF).  \n  The   H{\\sc i}MF   has been recently investigated  by three \n different  groups (Spitzak \\& Schneider 1998; Zwaan et al.~1997; and Kilborn, Staveley-Smith \\& Webster  1998).  The first two groups  used \n deep pencil beam samples observed with Arecibo, whereas Kilborn \nused H{\\sc i}PASS  objects to study a flux-limited sample. Each of these  surveys  contain  between 50 \n and 100 objects  but,  because they are flux-limited, \n they  contain  \n  very few low mass objects, and are therefore  not suited to \n determining  the low mass slope. Our survey, on the \n   other hand, being principally distance-limited  is much better suited \n to determining  that low-mass \n slope, though it lacks the higher masses needed to determine \n $M^{\\star}_{H{\\sc i}}$ well in the usual Schechter function:\n\n\\begin{displaymath}\n\\Phi(M_{\\rm H{\\sc i}}\/M^{\\star}_{\\rm H{\\sc i}}) = \n \\Phi_{\\star} \n(M_{\\rm H{\\sc i}}\/M^{\\star}_{\\rm H{\\sc i}})^{-\\alpha} e^{-(M_{\\rm H{\\sc i}}\/M^{\\star}_{\\rm H{\\sc i}})}  \n\\end{displaymath}\n\n\n\n   In Figure 6  we plot the H{\\sc i} mass function   from the 27 detections \n  of our survey  which have accurate  H{\\sc i} masses. \n \n\n  In order   not to over-interpret  the data  we do not    attempt \n  to find a fit  for \n  $M^{\\star}_{H{\\sc i}}$, only to find   the faint-end slope $\\alpha$. To \n do this we make a weighted linear least-squares fit of \n log $\\rho$ against log $M_{H{\\sc i}}$ (where $\\rho$ is the number of sources per log$_{10}$ $M_{HI}$ bin) for sources with $M_{H{\\sc i}} < 10^9 \n M_\\odot$  (Figure 6)  and find : \n\n log $\\rho$  = (-0.30 $\\pm$ 0.15) log $M_{H{\\sc i}}$ + (3.21 $\\pm$ 1.20)\n\n \\noindent Hence the Schechter  faint-end slope is \n\n $\\alpha$ = 1.30 $\\pm$ 0.15\n\n \\noindent where the   formal error on the slope  (see Bevington, 1969)  \n   arises from assuming  purely Poisson  statistics \n for  the numbers in each $\\rho$-bin in \n Figure 6.    Since the most likely systematic error  is the loss \n from our lowest mass bin of sources  which did not make the first cut, and \n therefore  were not re-measured  more accurately, we \n can ask how $\\alpha$  would change if we added \n 50\\% more sources to the 12 already in \n that bin. In fact $\\alpha$ rises \n to only 1.45 $\\pm$ 0.15.  Similarly the formal error on the slope allows \n$\\alpha = 1.45$ at 1 standard deviation.\n Our data therefore does not rule out a faint-end slope of 1.5 but we\n can exclude a slope as steep as 1.7 at the 95\\% level.   Table 4 \n contains a comparison  of  our Cen~A results  with those from the three other \n H{\\sc i}MF surveys mentioned above.  \n\n\\section{Discussion}\n\nIt is encouraging that the present comparatively shallow blind H{\\sc i} survey  already   increases    \nthe number of known galaxies of the Cen\\,A group by 50\\%, i.e. from 20 to 30 in our search area. \nIt is in particular interesting that this happened after the group was \n  so  carefully studied \nin the optical (C97). \n\n\nA glance at the spectra  (Figure 2) confirms that \nnone of our objects are marginal detections.   All have \nsignals above 3.9 Jy km\\,s$^{-1}$ , whereas the formal 5$\\sigma$ noise limit for \n  optimally smoothed  sources  that are   \n50  km\\,s$^{-1}$  broad is closer to 1.5 Jy km\\,s$^{-1}$ . The dispersion in \nthe velocity of the less significant peaks in our data map     \nmimic the velocity distribution of the Cen\\,A group   galaxies, \nsuggesting that many of them will also turn out to be real members \nof the group.  \n\nWe find only one low surface brightness galaxy with an \n H{\\sc i} mass of more than \n   10$^{8}$ M$_{\\odot}$ - ESO174-G?001  \nwhich contains 2.8  $\\times$ 10$^{8}$ M$_{\\odot}$ of H{\\sc i}. It is 3 \narc minutes across  and its measured  central  surface brightness \n is  24.5 $\\mu_B$, which is indeed low. However, it is the \ngalaxy closest to the Galactic plane ($b=8^{\\circ}$) and turns out to \nhave 2.1 magnitudes of  \n  foreground  blue extinction. When this is allowed for, the central \nsurface brightness rises to 22.4 $\\mu_B$ - which is not very low; \nM$_{\\rm H{\\sc i}}$\/L$_{\\rm B}$ drops to 0.8 and the absolute magnitude,  \nM$_{B}=-15.6$.  \n   ESO174-G?001  was missed by previous studies simply because \n of the large extinction towards it.   Despite its comparatively \nlarge size, ESO174-G?001 is still much smaller than the instrument beam.  If there \nwere a giant LSB galaxy in the group, i.e. one with a diameter larger than \nthe projected beam (15 kpc), we could only have detected it, on \nsignal to noise grounds, if its projected mass were more than 10$^{7}$ M$_{\\odot}$ \nof H{\\sc i}  per  beam; thus it would need to  have a column density, \nN$_{H{\\sc i}}$, of at least 10$^{19}$  atoms cm$^{-2}$. In reality, our \nautomatic baseline fitting procedure would at present almost certainly \n``remove'' the signal from the data. Irrespective of telescope size, \n for sources that fill the beam  \nthere is a lowest column density \n\n\\begin{equation}\n N_{\\rm H{\\sc i}} > 10^{18} \\cdot T_{sys}{\\sqrt{\\frac{\\Delta V [kms^{-1}]}{t_{obs} [s]}}}\\hbox{cm}^{-2}\n\\end{equation}\n\n\n\\noindent  \ndetectable by any radio telescope, set by receiver noise, where \n$\\Delta$V  is the projected linewidth in  km\\,s$^{-1}$  (e.g. Disney \\& Banks \n1997). \nExtreme objects, close to this column density limit, are best found when they \nare matched to the beam. For example, an  L$_{\\star}$ galaxy ten times the \nradius of our own and containing the same amount of H{\\sc i} would \nhave an N$_{H{\\sc i}} < 10^{19}$ atoms cm$^{-2}$ and would be \nfound only in integrations significantly longer than H{\\sc i}PASS , and \nmost probably out near 6000 km\\,s$^{-1}$ .  So the results from the present\nH{\\sc i} survey do not rule out the existence of extended LSB galaxies \nin the Cen\\,A group.  The present study \n is being followed by a much deeper H{\\sc i} survey in  the same region, \nusing the same instrument to address this issue. \n\nAlthough our survey has increased the number of known Cen\\,A group \nmembers by 50\\%, the new members add only 6\\% to the total \nH{\\sc i}, and 4\\% to its light. The H{\\sc i} mass function has \na slope at the low mass end of  1.33 $\\pm$ 0.15, which is unlikely to be\nsteeper than 1.5 due to either statistical or systematic effects.\n\n The five newly \ncatalogued dwarfs have indicative mass to stellar mass ratios \n(col.[8], Table~3) between 40 and 80  (mean 63  $\\pm$ 16). \nSuch ratios probably underestimate the dynamical masses because they \nassume the H{\\sc i} radii (R$_{H{\\sc i}}$) are equal to the optical \nradii (R$_{opt}$). However, H{\\sc i} maps of similar objects by other authors \ngenerally show that R$_{H{\\sc i}} >$ R$_{opt}$  - the most \nstriking example being the VLA observations of H{\\sc i}1225+01, \nChengalur, Giovanelli \\& Haynes 1995  (for comparison, the five \n  new group members that are \n  previously catalogued \n  have a mean ratio of 14). The high ratios add to the \naccumulating evidence (e.g. Kormendy 1985) that H{\\sc i}-rich \ndwarfs have their dynamics controlled by dark matter.\n\nThis survey of the Cen\\,A group was aimed chiefly at validating \nthe performance of the wider H{\\sc i}PASS  survey of the southern sky - \nwhich will be over 90\\% complete by the time this goes to press. It \nconfirms that performance matches expectation, and that large area \nblind H{\\sc i} surveys will open a new window onto the extragalactic \nsky. As to the Cen\\,A group itself, we expect to find further new group\nmembers when we run the data through the automatic galaxy-finding \nalgorithm now under development. We have also completed a much deeper \nsurvey (t$_{obs}$=12 $\\times$ H{\\sc i}PASS ) of a limited region \n(8$^{\\circ}$ x 4$^{\\circ}$)  in the area, to  investigate the \nperformance and promise of long multibeam integrations (Banks 1998). \n\n\\acknowledgements\n\nThe development of the Parkes Multibeam Receiver was supported by \nthe Australia Telescope National Facility, the Universities of \nMelbourne, Western Sydney, Sydney, Cardiff and  Manchester \n and by  Mount \nStromlo Observatory,  PPARC and the ARC. The receiver and hardware \nwere constructed by the staffs of the CSIRO Division of Telecommunications \nand Industrial Physics with assistance from Jodrell Bank. The Multibeam \nsoftware was developed by the AIPS++ team which included staff and students \nof the ATNF and University of Melbourne. \nThe authors are very grateful to \n an anonymous referee  whose comments  have led to \n significant  changes in the \n presentation of the results.  GDB acknowledges the \n support of a PPARC postgraduate research award.  \n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDirichlet p-branes are p+1-dimensional hypersurfaces on which\nsuperstrings can begin and end (see~\\cite{Pol96,wati} for a review).\nThe low energy dynamics of an ensemble of N parallel Dp-branes can be\ndescribed by the U(N) supersymmetric gauge theory obtained by\ndimensional reduction of ten dimensional supersymmetric Yang-Mills\ntheory to the p+1-dimensional world-volume of the brane.\\cite{witt2}\nThe Yang-Mills theory gives an accurate perturbative representation of\nthe Dp-brane dynamics when the separations between the branes is\nlarge.\\cite{gab,kp,dkps} It represents a truncation of the full string\nspectrum to the lowest energy modes.  The full string theoretical\ninteractions between a pair of Dp-branes is computed by considering\nthe annulus diagram shown in fig.~\\ref{f:annulus}.  The short distance\nasymptotics of this diagram are dominated by the open string sector\nwhose lowest modes are the fields of ten dimensional supersymmetric\nYang-Mills theory.  On the other hand, long distance asymptotics are\nmost conveniently described by the dual description of this diagram as\na closed string exchange and the relevant field theoretical modes are\nthose of ten dimensional supergravity.  That these are also\nrepresented by the dimensionally reduced super Yang-Mills theory is a\nresult of supersymmetry and the fact that, for fixed Dp-brane\npositions, the ground state is a BPS state.  At zero temperature,\nbecause of supersymmetry, the interaction potential between a pair of\nstatic D0-branes vanishes independently of their separation.  Their\neffective action has been computed in an expansion in their\nvelocities, divided by powers of the separation and is known to\nbe~\\cite{b,dkps}\n\\begin{equation}\nS_{\\rm eff}(T=0)= \\int dt \\left( \n\\frac{1}{2g_s\\sqrt{\\alpha'}} \\sum_{\\alpha=1}^2 ( {\\dot {\\vec q}}\\,{}^\\alpha \n) ^2 - \n\\frac{15}{16}\\left( \\alpha'\\right)^3\n\\frac{\\vert {\\dot{\\vec q}}\\,{}^1-{\\dot{\\vec q}}\\,{}^2\n\\vert^4}{\\vert {\\vec q}\\,{}^1-{\\vec q}\\,{}^2 \\vert^7}+\\ldots \\right) \n\\label{T=0}\n\\end{equation}\nThis result agrees with the effective potential for the interaction \nof D0-branes in ten dimensional supergravity. Note that, for weak string \ncoupling, the D0-brane is very heavy.\n\n\\begin{figure}\n\\vspace{-0.3in} \n\\hspace{1.5in}\n\\psfig{figure=cylinder.eps,height=1.5in} \n\\vspace{-0.3in} \n\\caption[x]{\\footnotesize Annulus diagram for D0-brane interactions.\nThe bold lines represent world-lines of the D0-branes, separated\nby the distance $L$, which go along the periodic temporal direction. \nThey bound the string world-sheet.} \n\\label{f:annulus} \n\\end{figure} \n\nIn this paper, we shall consider the description of D0-brane\ninteractions in type IIA superstring theory using matrices.  Even at\nvery low temperatures, non-BPS states are important to the leading\ntemperature dependence.\nWe perform 1-loop computation of the effective interaction between\nstatic D0-branes in the matrix theory at finite temperature and\ncompare with the known superstring computations.  We show that the\nresults of the two computations are similar in the low temperature\nlimit but an extra integration over \n the temporal component of the gauge\nfield, is present in the matrix\ntheory. At finite temperature, because the Euclidean \ntime is compact, the  temporal gauge field  can not be removed  \nby a gauge transformation.  {\\it This integration\nis needed in order to describe correctly thermodynamics of $D0$-branes\nboth in the matrix and superstring theories}.\n\nThe paper is organized as follows.  In section~2 we discuss the\nformulation of the matrix theory at finite temperature. In section~3\nwe perform one loop computation of the effective interaction between\nstatic D0-branes at finite temperature and show that it is \nattractive, and short-ranged. In section~4 we compare this result with\nthe superstring computations and discuss the conditions under which\nthe two computations agree. Section~5 is devoted to the discussion of\nour results and, in particular, the origin of the divergence of the\nclassical thermal partition function of D0-branes which is cured by\nquantum statistics.\n                              \n\n\n\\section{Matrix theory at finite temperature}\n\nWe shall consider the matrix theory description~\\cite{bfss} of the effective\ndynamics of D0-branes in a type-IIA superstring theory which is\nderived by the reduction of ten dimensional supersymmertic \nYang-Mills theory which has the action\n\\begin{equation}\nS_{\\rm YM}[A,\\theta]=\\frac{1}{g_{YM}^2}\\int d\\tau {\\rm TR}\\left(\n\\frac{1}{4}F_{\\mu\\nu}^2+\\frac{i}{2}\\theta\\gamma_\\mu D_\\mu\\theta\\right)\n\\end{equation}\nto zero spatial dimension: $A_\\mu=A_\\mu(\\tau)$, $\\theta=\\theta(\\tau)$.\n\nThe thermal partition function of this theory is given by\n\\begin{equation}\nZ_{\\rm YM}=\\int[dA(\\tau)][d\\theta(\\tau)]\\exp\\left( -S_{\\rm YM}[A,\\theta]\\right)\n\\label{thermal}\n\\end{equation}\nwhere $S_{\\rm YM}$ is the Euclidean action and the time coordinate is\nperiodic. The bosonic and fermionic coordinates have periodic and\nanti-periodic boundary conditions,\n\\begin{eqnarray}\nA_\\mu(\\tau+\\beta)= A_\\mu(\\tau),   \\\\\n\\theta(\\tau+\\beta)=-\\theta(\\tau),\\\\\n\\beta=1\/k_BT ,\n\\label{b.c}\n\\end{eqnarray}\nwhere $T$ is the temperature and $k_B$ is Boltzmann's constant.  Gauge\nfixing will be necessary and will involve introducing ghost fields\nwhich will have periodic boundary conditions.\n\nThe representation (\\ref{thermal}) of the  thermal partition \nfunction can be derived in the standard way starting from the known\nHamiltonian of the matrix theory~\\cite{bfss} and representing\nthe  thermal partition function \n\\begin{equation}\nZ_{\\rm YM}= {\\rm tr}\\, e^{-\\beta H}  \n\\end{equation}\nvia the path integral. The trace is calculated here over all states\n obeying Gauss's law which is taken care by the integration\nover $A_0$ in (\\ref{thermal}). This representation of the  matrix theory\nat finite temperature have been already discussed \\cite{OZ98,MOP98,Sat98},\nbut the temperature induced interaction between $D0$-branes described\nbelow was never identified. \n\nIn matrix theory, the diagonal components of the gauge fields,\n$\\vec{a}^\\alpha\\equiv \\vec{A}^{\\alpha\\alpha}$, are interpreted as the\nposition coordinates of the $\\alpha$-th D0-brane and they should be\ntreated as collective variables. Static configurations play a special\nrole since they satisfy classical equations of motion \nwith the periodic boundary conditions and dominate\nthe path integral as $g^2_{\\rm YM}\\rightarrow 0$. Notice that there\nare no such static zero modes for fermionic components since they would not\nsatisfy the antiperiodic boundary conditions.\n\\footnote{This is a difference between our computation at finite temperature \nand computations of the Witten index for the matrix theory where\nfermions obey periodic boundary conditions.}\nThis is an important\ndifference from the  zero temperature case and a manifestation of the fact\nthat supersymmetry is explicitly broken at non-zero temperature.\n\nIn the following, we will construct\nan effective action for these coordinates by integrating the\noff-diagonal gauge fields, the fermionic variables and the ghosts,\n\\begin{equation}\nS_{\\rm eff}[ \\vec a^\\alpha]\\equiv -\\ln \\int[d a^\\alpha_0]\n\\prod_{\\alpha\\neq\\beta}[dA^{\\alpha\\beta}_\\mu][d\\theta][d{\\rm ghost}]\\exp\\left(\n-S_{\\rm YM}-S_{\\rm gf}-S_{\\rm gh}\\right).\n\\end{equation}\nGenerally, this integration can only be done in the a simultaneous\nloop expansion and expansion in the number of derivatives of the\ncoordinates $\\vec a^\\alpha$.  Such an expansion is accurate in the\nlimit where $\\left| \\vec a^\\alpha-\\vec a^\\beta\\right|$ are large for\neach pair of D0-branes and where the velocities are small.  Since\nthese variables are periodic in Euclidean time, small velocities are\nonly possible at low temperatures.  The remaining dynamical problem\nthen defines the statistical mechanics of a gas of D0-branes,\n\\begin{equation}\nZ_{\\rm YM}=\\int\\prod_{\\tau,\\alpha}\n[d \\vec a^\\alpha(\\tau)]\\exp\\left( -S_{\\rm eff}[\\vec a^\\alpha]\n\\right).\n\\label{effec}\n\\end{equation}\nWe expect that the zero temperature limit of $S_{\\rm eff}$ reduces to\n(\\ref{T=0}).  \n\nWe shall find several subtleties with this formulation.  If the\neffective D0-brane action is to reproduce the results of a string\ntheoretical computation, the integration over $a_0^\\alpha$ must be\nperformed in both cases.   \n\nThe effective action is a symmetric functional of the position\nvariables $\\vec a^\\alpha(\\tau)$.  Only the configuration of these\ncoordinates needs to be periodic.  Therefore the individual position\nshould be periodic up to a permutation. The variables in the path\nintegral (\\ref{effec}) should therefore be periodic up to a\npermutation and the integral should be summed over the permutations.\n\n\\section{One loop computation}\n\nWe will compute the effective action $S_{\\rm eff}$ in a simultaneous \nexpansion in the number of loops and in powers of time derivatives \nof the D0-brane positions.\n\nWe decompose the gauge field into diagonal and off-diagonal parts,\n\\begin{equation}\nA_\\mu^{\\alpha\\beta}=a_\\mu^{\\alpha}\\delta^{\\alpha\\beta}+g_{\\rm YM}\n\\bar A^{\\alpha\\beta}_\\mu\n\\end{equation}\nwhere $\\bar A_{\\mu}^{\\alpha\\alpha}=0$ so that the curvature is\n\\begin{equation}\nF_{\\mu\\nu}^{\\alpha\\beta}=\\delta^{\\alpha\\beta}f^\\alpha_{\\mu\\nu}+\ng_{\\rm YM}D_\\mu^{\\alpha\\beta}\n\\bar A^{\\alpha\\beta}_\\nu-g_{\\rm YM}D_\\nu^{\\alpha\\beta}\n\\bar A_\\mu^{\\alpha\\beta}- ig^2_{\\rm YM}\n\\left[ \\bar A_\\mu, \\bar A_\\nu \\right]^{\\alpha\\beta}\n\\end{equation}\nwhere\n\\begin{equation}\nf_{\\mu\\nu}^\\alpha= \\partial_\\mu a^\\alpha_\\nu-\\partial_\\nu a^\\alpha_\\mu\n\\end{equation}\nand\n\\begin{equation}\nD_\\mu^{\\alpha\\beta}=\\partial_\\mu-i\\left(a^\\alpha_\\mu-a^\\beta_\\mu\\right).\n\\end{equation}\nIn the Yang-Mills term in the action, we keep all orders of the\ndiagonal parts of the gauge field and expand up to second order in the\noff-diagonal components,\n\\begin{equation}\n\\frac{1}{g_{\\rm YM}^2}{\\rm TR}\\left( F_{\\mu\\nu}^2 \n\\right)=\\sum_\\alpha \\frac{1}{g_{\\rm YM}^2}\\left(f^\\alpha_{\\mu\\nu}\\right)^2\n+2\\sum_{\\alpha\\beta}\\bar A_{\\mu}^{\\beta\\alpha}\\left(\n\\delta_{\\mu\\nu}{\\buildrel \\leftarrow\\over D}_\\lambda^{\\beta\\alpha}\n\\vec D_\\lambda^{\\alpha\\beta}-{\\buildrel \\leftarrow\\over\nD}_\\mu^{\\beta\\alpha}\\vec D_\\nu^{\\alpha\\beta}+2i\\left(\nf^\\alpha_{\\mu\\nu}- f^\\beta_{\\mu\\nu} \\right)\\right)\\bar\nA_\\nu^{\\alpha\\beta}+\\ldots\n\\end{equation}\nWe will fix the gauge\n\\begin{equation}\nD_\\mu^{\\alpha\\beta}\\bar A_\\mu^{\\alpha\\beta}=0.\n\\end{equation} \nThis entails adding the Fadeev-Popov ghost term to the action\n\\begin{equation}\nS_{\\rm gh}=\\int\\sum_{\\alpha\\beta} \\left\\{\\bar c^{\\alpha\\beta}\\left( \n-D^{\\alpha\\beta}_\\mu\\right)^2c^{\\beta\\alpha}+i\ng_{\\rm YM}\\bar c^{\\beta\\alpha}D_\\mu^{\\alpha\\beta}\n\\left[\\bar A_\\mu,c\\right]\\right\\}\n\\end{equation}\nThere is a residual gauge invariance under the abelian transformation,\n\\begin{eqnarray}\n\\bar A^{\\alpha\\beta}_\\mu\\rightarrow \\bar A^{\\alpha\\beta}_\\mu \ne^{i(\\chi^\\alpha -\\chi^\\beta)},\n\\nonumber \\\\\na_\\mu^\\alpha\\rightarrow a_\\mu^\\alpha+\\partial_\\mu\\chi^\\alpha.\n\\end{eqnarray}\nWe shall use this gauge freedom to set the additional condition\n\\begin{equation}\n\\partial_0 a^{\\alpha}_0=0\n\\end{equation}\nand to fix the constant%\n\\footnote{These constants are related to the eigenvalues of the\nholonomy \n$$\n{\\rm P} \\exp{\\left(i \\int_0^\\beta d\\tau A_0(\\tau)\\right)}= \n\\Omega^\\dagger\\;{\\rm diag}\\; \\left(\ne^{i\\beta a_0^1},\\ldots,e^{i\\beta a_0^{\\rm N}}\\right) \\Omega\n$$\nknown as the Polyakov loop winding along the compact Euclidean time.\nIt can not be made trivial by the gauge transformation if $T\\neq 0$.}\n\\begin{equation}\n-\\pi\/\\beta < a_0^\\alpha\\leq\\pi\/\\beta.\n\\end{equation}\n\nThe ghost for this gauge fixing condition decouples.\n\nKeeping terms up to quadratic order in $\\bar A, c,\\bar c, \\theta$, the\naction is\n\\begin{eqnarray}\nS=\\int\\left\\{\\frac{1}{4g_{\\rm YM}^2}\\sum_\\alpha \\left( \nf^\\alpha_{\\mu\\nu}\\right)^2 +\\frac{1}{2}\n\\sum_{\\alpha\\beta}\\bar A_{\\mu}^{\\beta\\alpha}\n\\left( -\\delta_{\\mu\\nu} D_\\lambda^{\\beta\\alpha} \nD_\\lambda^{\\alpha\\beta}+D_\\mu^{\\beta\\alpha}\nD_\\nu^{\\alpha\\beta}+2i\\left( f^\\alpha_{\\mu\\nu}- f^\\beta_{\\mu\\nu}\n\\right)\\right)\\bar A_\\nu^{\\alpha\\beta}\n\\right.\\nonumber\\\\\n\\left.\n+\\sum_{\\alpha\\beta}\\bar\nc^{\\beta\\alpha}(D_\\mu^{\\alpha\\beta})^2c^{\\alpha\\beta}\n+\\frac{i}{2}\\theta^{\\beta\\alpha}\\gamma_\\mu D_\\mu^{\\alpha\\beta}\n\\theta^{\\alpha\\beta}\n\\right\\}.\n\\end{eqnarray}\nThe effective action obtained by integrating over $\\bar A,\\bar\nc,c,\\theta$ is\n\\begin{eqnarray}\nS_{\\rm eff}=\\int\\sum_\\alpha\\frac{1}{4g_{\\rm YM}^2}(f^\\alpha_{\\mu\\nu})^2\n+\\sum_{\\alpha\\neq\\beta}\\left\\{ \\frac{1}{2}{\\rm TR}\\ln\\left( -\\delta_{\\mu\\nu}(D^{\\alpha\\beta}_\\mu)^2 +2i(f_{\\mu\\nu}^\\alpha-f_{\\mu\\nu}^\\beta)\\right)\n\\right. \\nonumber \\\\  \\left.\n-{\\rm TR}\\ln\\left( -(D_\\mu^{\\alpha\\beta})^2\\right)-\\frac{1}{2}{\\rm TR}\\ln\n\\left( i\\gamma_\\mu D^{\\alpha\\beta}_\\mu\\right) \\right\\}.\n\\label{seff}\n\\end{eqnarray}\n\n\\subsection{Leading order in time derivatives}\n\nWe will evaluate the determinants on the right-hand-side of\n(\\ref{seff}) in an expansion in powers of the derivatives of $\\vec\na(\\tau)$.  The leading order term can be found by setting $\\vec a={\\rm\nconst.}$. In this case, $f^\\alpha_{\\mu\\nu}=0$ and \n(here we retain the tree-level term with time derivatives)\n\\begin{equation}\nS_{\\rm eff}=8\\sum_{\\alpha <\\beta}\\left\\{{\\rm\nTR}_B\\ln\\left( -(D^{\\alpha\\beta}_\\mu)^2 \\right) -{\\rm TR}_F\\ln\\left(\n-(D^{\\alpha\\beta}_\\mu)^2 \\right)\\right\\}\n\\end{equation}\nwhere the subscript $B$ denotes contributions from the gauge fields\nand ghosts, whereas $F$ denotes those from the adjoint fermions.  The\ndeterminants should be evaluated with periodic boundary conditions for\nbosons and anti-periodic boundary conditions for fermions.  (Note\nthat, because of supersymmetry, if both bosons and fermions had\nidentical boundary conditions the determinants would cancel.  This\nwould give the well-known result that the lowest energy state is a BPS\nstate whose energy does not depend on the relative separation of the\nD0-branes.)  \nThe boundary conditions are taken into account by introducing Matsubara \nfrequencies, so that\n\\begin{equation}\ne^{-S_{\\rm eff}}=e^{-S_{0}} \\beta^{\\rm N}\n\\int \\frac{da^\\alpha_0}{2\\pi} \\prod_{\\alpha<\\beta}\n\\prod_{n=-\\infty}^\\infty\n\\left( \\frac{ \\left(\\frac{2\\pi n}{\\beta}\n+\\frac{\\pi}{\\beta}+a_0^\\alpha-a_0^\\beta\\right)^2+\\vert\\vec \na_\\alpha-\\vec a_\\beta\\vert^2}{\\left(  \n\\frac{2\\pi n}{\\beta}+a_0^\\alpha-a_0^\\beta\\right)^2 +\\vert\\vec \na_\\alpha-\\vec a_\\beta\\vert^2}\n\\right)^8\n\\end{equation}\nUsing the formula\n\\begin{equation}\n\\prod_{n=-\\infty}^\\infty\n\\left( \\frac{2\\pi n}{\\beta}+\\omega\\right)=\\sin\\left( \\frac{\\beta\\omega}\n{2}\\right)\n\\label{21}\n\\end{equation}\nwe obtain the result\n\\begin{equation}\ne^{-S_{\\rm eff}}=e^{-S_{0}} \\beta^{\\rm N}\n\\int \\frac{da^\\alpha_0}{2\\pi} \\prod_{\\alpha<\\beta}\n\\left(  \\frac{ \\cosh \\beta\\vert\\vec a^\\alpha-\\vec a^\\beta\\vert+\n\\cos \\beta \\left(a_0^\\alpha- a_0^\\beta\\right)}\n{\\cosh\\beta\\vert\\vec a^\\alpha-\\vec a^\\beta\\vert - \n\\cos\\beta \\left( a_0^\\alpha-a_0^\\beta\\right) }\\right)^8\n\\label{23}\n\\end{equation}\nIn order to find the effective action for $\\vec a^\\alpha$, \nit is now necessary to \nintegrate the temporal gauge fields $a_0^\\alpha$ over the domain \n$(-\\pi\/\\beta,\\pi\/\\beta]$.  This integration implements the projection \nonto the gauge invariant eigenstates of the matrix theory Hamiltonian.\n\nIn the case where there is a single pair of D0-branes, N=2, the integration\nover $a_0^\\alpha$ in (\\ref{23}) can be done explicitly to obtain\nthe effective action\n\\begin{equation}\nS_{\\rm eff}=\\int_0^\\beta d\\tau\\left\\{\n\\sum_1^2\\frac{(\\dot{\\vec a}{}^\\alpha)^2}{2g_{\\rm YM}^2} \n -\\frac{1}{\\beta}\\ln\\left(\n\\frac{ P(z) }{  (1-z^2)^{15} }\n\\right)\\right\\}\n\\label{effac}\n\\end{equation}\nwhere \n\\begin{eqnarray}\nP(z)= 1+241z^2+12649z^4+254009z^6+2434901z^8+12456773z^{10}\n+36119181z^{12}\\nonumber \\\\*+61178589z^{14}+6191459z^{16}\n+36109171z^{18}\n+12462779z^{20}+2432171z^{22}\\nonumber\\\\* \n+254919z^{24}+12439z^{26}+271z^{28}\n-z^{30},\n\\end{eqnarray}\n$z=\\exp(-\\beta\\vert {\\vec a}^1-{\\vec a}^2\\vert)$ and we have\nincluded the tree level term, which gives the non-relativistic kinetic\nenergies of the D0-branes.\n\nThe effective action has the\nlow temperature expansion\n\\begin{eqnarray}\nS_{\\rm eff}=\\int_0^\\beta d\\tau\\left( \\frac{1}{2 g_{\\rm YM}^2}\\sum_\\alpha \n\\left(\\dot{\\vec a}^\\alpha\\right)^2-\\frac{1}{\\beta}\\left( 256 e^{-2\\beta\\vert \n\\vec a^1-\\vec a^2\\vert}-16384 e^{-4\\beta\\vert \\vec a^1-a^2\n\\vert} \\right.\\right. \\nonumber \\\\*   \\left.\\left.\n+\\frac{5614336}{3}e^{-6\\beta\\vert \\vec a^1-\\vec a^2\\vert}+\n\\ldots\\right)\\right)\n\\label{finalSeff}\n\\end{eqnarray}\nWe shall compare in the following section this result with\nthe superstring computation of the effective interaction between D0-branes. \n\n\\section{String theoretical interactions}\n\nThe effective interactions of D0-branes \nin superstring theory is given by computing the annulus diagram \nshown in fig.~\\ref{f:annulus}. \nThis was done in ref.~\\cite{green} (and in~\\cite{VM96} for Dp-branes).  \nThe result of summing over physical (GSO projected) superstring states gives \nthe free energy\n\\begin{equation}\nF[L,\\beta,\\nu]=\\frac{8}{\\sqrt{\\pi \\alpha'}}\\int_0^\\infty \\frac{dl}{l^{3\/2}}\\,\ne^{ -L^2l\/4\\pi^2\\alpha'}\n\\Theta_2 \\left(\\nu \\left \\vert \\frac{i\\beta^2}{\\pi \\alpha' l} \\right. \\right)\n\\prod_{n=1}^\\infty\\left( \\frac{1+e^{-nl}}{1-e^{-nl}}\\right)^8\n\\label{GreenF}\n\\end{equation}\nwhere\n\\begin{equation} \n\\Theta_2 \\left(\\nu \\left \\vert iz \\right. \\right)\n=\\sum_{q=-\\infty}^\\infty\ne^{ -\\pi z(2q+1)^2\/4+i\\pi(2q+1)\\nu},\n\\end{equation}\n$L$ is the brane separation \nand $\\nu$ is a parameter\nwhich weights the winding numbers of strings around the periodic time \ndirection. An extra factor of 2 accounting for the exchange of the\ntwo ends of the superstring~\\cite{Pol96} ending on each of the two \nD0-branes is inserted in~(\\ref{GreenF}). \n\nThe product in the integrand represents the sum over string states, \nwith requisite degeneracies,\n\\begin{equation}\n8\\prod_{n=1}^\\infty\\left( \\frac{1+e^{-nl}}{1-e^{-nl}}\\right)^8\n=\\sum_{N=0}^{\\infty} d_N e^{-Nl}\n\\label{degeneracy}\n\\end{equation}\nwhere $d_N$ is the degeneracy of the \neither superstring state at level $N$.  For \nthe lowest few levels, $d_0=8$ and $E_0=L\/2\\pi\\alpha'$.\n\nInserting (\\ref{degeneracy}) in (\\ref{GreenF}) and integrating over $l$,\nthe free energy has the form\n\\begin{equation}\nF(\\beta,L,\\nu)=\\frac{2}{\\beta}\\sum_{N=0}^\\infty d_N~\n\\ln\\left| \\frac{ 1-e^{-\\beta E_N+i\\pi\\nu}}{ 1+e^{-\\beta E_N+i\\pi\\nu} }\n\\right|\n\\label{Fsuperstring}\n\\end{equation}\nwhere the string energies are given by the formula\n\\begin{equation}\nE_N=\\frac{L}{2\\pi\\alpha'}\\sqrt{ 1+\\frac{4\\pi^2\\alpha'N}{L^2}}.\n\\label{Esuperstring}\n\\end{equation}\nThis results in the partition function\n\\begin{equation}\nZ_{\\rm str}(\\beta,L,\\nu)\\equiv e^{-\\beta F}= \\prod_{N=0}^\\infty \\left|\n\\frac{1+e^{-\\beta E_N+i\\pi \\nu}}{1-e^{-\\beta E_N+i\\pi \\nu}}\n\\right|^{2d_N}.\n\\label{Zsuperstring}\n\\end{equation}\nThe physical meaning of the last formula is obvious:\nthe partition function equals the ratio of the Fermi and\nBose distributions with the power being twice the degeneracy of states\nand $i\\nu$ playing the role of a chemical potential. The factor of 2\nin the exponent $2d_N$ in (\\ref{Zsuperstring}) \nis due to the interchange of the superstring ends as is already\nmentioned. It will provide the agreement with the matrix theory \ncomputation.\n\nIn order to compare with the Yang-Mills computation, we should first re-scale \nthe coordinates so that the mass of the D0-brane appears in the kinetic term\nas in (\\ref{T=0}).  \nThe mass is given by the formula\n\\begin{equation}\nM=\\frac{1}{g_s\\sqrt{\\alpha'}}\n\\end{equation}\nand the Yang-Mills coupling $g_{\\rm YM}$ is related to the string coupling \n$g_s$ by the equation\n\\begin{equation}\ng_{\\rm YM}^2=\\frac{g_s}{4\\pi^2(\\alpha')^{3\/2}}.\n\\end{equation}\nThe physical coordinate of the $\\alpha$-th D0-brane is identified with\n\\begin{equation}\n\\vec q^\\alpha= 2\\pi\\alpha'\\vec a^\\alpha .\n\\label{qvsa}\n\\end{equation}\nTaking N=2 in (\\ref{23}) and identifying\n$L=2\\pi\\alpha' \\vert \\vec a^1-\\vec a^2 \\vert$, we see that the \nintegrand in (\\ref{23}) coincides with (\\ref{Zsuperstring})\ntruncated to the massless modes ($N=0$) provided \n$\\nu=\\beta(a_0^1-a_0^2)$.\n\nIt is clear that the integral over $a_0^\\alpha$ is responsible for \nthe ``mismatch'' of  the effective \nactions between the string theory computation \nand matrix theory computation of the free energy. \nIn the string theory done in the spirit of ref.~\\cite{green}, \nthe parameter $\\nu$ appears in the same place as \n$\\beta(a_0^1-a_0^2)\/\\pi$ but is not integrated. It is associated\nwith the interaction of the ends of the open string with an Abelian\ngauge field background ${A}_{\\mu}(\\tau,\\vec x)$: \n\\begin{equation}\nS_{\\rm int}=\\int dx^\\mu {A}_\\mu.\n\\end{equation}\nIf the ends are separated by the distance $L$, e.g.\\ along the\nfirst spatial axis, then\n\\begin{equation}\n\\nu=\\int_0^\\beta d\\tau \\left( {A}_0(\\tau,0,\\ldots)-\n{A}_0(\\tau,L,\\ldots)\\right)\n\\end{equation}\nsince $\\dot x_\\mu (\\tau)=(1,\\vec 0)$ on the boundaries.\nThe matrix theory automatically takes into account the\nintegration over the background field while in the string theory\ncalculation of ref.~\\cite{green} the background field is fixed.\nThis is just a reflection of the fact that matrix theory is \nan effective low-energy theory of $D$-branes, while the \nolder string theory did not treat the boundaries as dynamical \nobjects. However, it is interesting to notice how close some \nof the earlier string papers came to such a description simply \nby the requirement of consistency \\cite{green1}. Further, \nin the context of matrix theory it is natural to take the \nexponential of the free energy (\\ref{Fsuperstring}) as in\neq.\\ (\\ref{Zsuperstring}), and only integrete over $\\nu$\nafterwards, a procedure not entirely obvious in \na string theory where the boundaries are not dynamical objects.\nThis will be discussed further in the next section.\n\nAn exact coincidence between the matrix theory and superstring results\nis possible only when the higher stringy modes are suppressed.\nUsually, the truncation of the string spectrum to get\nYang-Mills theory is valid for small $\\alpha'$, that is when we are interested\nin temperatures which are much smaller than $1\/\\sqrt{\\alpha'}$.  In fact, the \ncondition in our case  \nis a little different than this once the length $L$ appears as a parameter in \nthe spectrum (\\ref{Esuperstring}).  \nThen, the spectrum can be truncated at the first level only when\n\\begin{equation}\n\\frac{1}{\\beta}\\equiv k_BT\\ll \n\\sqrt{ \\left(  \\frac{L}{2\\pi\\alpha'}\\right)^2+\\frac{1}{\\alpha'} }-\\frac{L}\n{2\\pi\\alpha'}\n\\label{trunk}\n\\end{equation}\nwhich is the energy gap between the first two levels.\nIf the temperature is small, this condition is always satisfied unless\nthe length $L$ is not too large. In other words the truncation of the\nspectrum to the lightest modes is valid for $\\beta \\gg L$\n(or $TL\\ll 1$).\n\nIt is also interesting to discuss what happens in the opposite\nlimit $L \\gg \\beta$ where the interaction between D0-branes\nis mediated by the lightest closed string modes.\nThe superstring free energy can be evaluated in this limit by the\nstandard modular transformation which relates the annulus diagram\nfor an open string with a cylinder diagram for a closed string.\nIntroducing the new integration variable $s=2\\pi^2\/l$, we \nrewrite (\\ref{GreenF}) as~\\cite{green}\n\\begin{equation}\n F[L,\\beta,\\nu]= \\frac{8 \\pi^4}{\\sqrt{2 \\pi \\alpha'}} \n\\int_0^\\infty \\frac{ds}{s^{9\/2}}\\;\ne^{s}\\,e^{-L^2\/2s\\alpha'}\n\\Theta_2 \\left(\\nu \\left \\vert \\frac{i\\beta^2 s}{2\\pi^3 \\alpha' } \n\\right. \\right)\n\\prod_{n=1}^\\infty\\left( \\frac{1-e^{-(2n+1)s}}{1-e^{-2ns}}\\right)^8.\n\\label{viathetas}\n\\end{equation}\nIn the limit where the brane separation is large the integration over $s$ is \nconcentrated in the region of large $s\\sim L^{2}$.\nSubstituting the large-$z$ asymptotics\n\\begin{equation}\n\\Theta_2 \\left(\\nu \\left \\vert i z  \\right. \\right)\n\\rightarrow\n2\\cos{(\\pi\\nu)}e^{-\\pi z\/4\n\\end{equation}\nand evaluating the saddle-point integral, we get\n\\begin{equation}\n F[L,\\beta,\\nu] \\propto \\cos{(\\pi\\nu)} \n\\frac{\\left(\\beta^2-8\\pi^2\\alpha'\\right)^{3\/2}}{L^4}\ne^{-L\\sqrt{\\beta^2-8\\pi^2\\alpha'}\/2\\pi \\alpha'}.\n\\end{equation}\n\nExponentiating and integrating over $\\nu$, we have\n\\begin{equation}\n\\int_{-1}^{1} {d\\nu} Z_{\\rm str}(\\beta,L,\\nu) \n\\propto \\frac{\\beta^2 \\left(\\beta^2-8\\pi^2\\alpha'\\right)^{3}}{L^8}\ne^{-L\\sqrt{\\beta^2-8\\pi^2\\alpha'}\/\\pi \\alpha'}.\n\\end{equation}\nTaking into account (\\ref{qvsa}) the exponent at the low temperatures is\nthe same as in (\\ref{finalSeff}) but the pre-exponential differs.\nThe dependence of the pre-exponential on $L$ in the superstring case \nemerges because the splitting between energy states in\n(\\ref{Esuperstring}) is of order $1\/L$ and the truncation condition\n(\\ref{trunk}) is no longer satisfied when $L$ is large.\nHigher stringy modes are then not separated \nby a gap and the continuum spectrum \nresults in the $L$-dependence of the preexponential.\nAs usual, the limits of $L\\rightarrow\\infty$ and $T\\rightarrow 0$ are not \ninterchangeable in the superstring theory.\n\n\n\\section{Discussion}\n\n\nOur main results concern D0-brane dynamics at finite temperatures. \nWe have computed the 1-loop effective action for the interaction\nof static D0-branes in the matrix theory at finite temperature and\ncompared it with the analogous superstring computation.\nWe have seen that an extra integration over the eigenvalues\nof the holonomy along the compactified Euclidean time is present\nin the matrix theory. The two computations agrees in the \nlow temperature limit provided the superstring thermal partition\nfunction is integrated over the Abelian gauge fields $a_0$'s living on\nD0-branes. \n\nThe integration over $a_0$'s is of course natural in the context \nof the Yang-Mills theory, where it expresses that only \ngauge-invariant states should contribute to ${\\rm TR}\\,e^{-\\beta H}$.\nBut it is also natural from the point\nof view of the D0-brane physics. It can be seen as follows. \nSuppose we make a T-duality transformation, which interchanges the Neumann\nand Dirichlet boundary conditions,\nalong the compactified Euclidean time direction. Then $a_0$'s become\nthe coordinates of D-instantons on the dual circle. The integration\nover $a_0$'s becomes now the integration over the positions of D-instantons.\nThe partition function should involve such an integration over\nthe collective coordinates and since they are collective coordinates\nthe integration appears in front of the exponential of the effective \naction, not in the action itself. Viewed in terms of D0-branes and \nopen strings, we have a gas of D0-branes with open strings between \nthem. The individual strings might have a winding number $q$ (more \nprecisely $2q+1$ in the case of superstrings), describing the \nwinding around the finite-temperature space-time cylinder. The energy \nof such states are $\\propto \\beta q\/2\\pi \\alpha'$. However, the \n$q$'s satisfy $\\sum q=0$ as a result of the integration over $a_0$.\nPhysically this constraint is most easily understood  by going to \nthe closed string channel where we have closed string boundary state \non the dual circle with radius $\\tilde{\\beta} = 4\\pi^2 \\alpha'\/\\beta$\nlocalized at the point $(\\nu \\tilde \\beta, \\vec q ) $. Passing to the momentum\nrepresentation, we write\n\\begin{equation}\n\\Big\\vert B,\\vec q, \\nu \\Big\\rangle =\n\\sum_{q=-\\infty}^\\infty e^{-2i\\pi\\nu q} \\, \\Big\\vert B,\\vec q, \np_0=2\\pi q \/\\tilde{\\beta} \\Big\\rangle .\n\\end{equation}\nHere the temporal momentum is quantized as \n$p_0=2\\pi q \/\\tilde{\\beta}=q\\beta\/2\\pi\\alpha'$, which \nlead to the same energy as the above mentioned open string states.\nIn this representation $\\sum q=0$ simply expresses momentum conservation\nin the thermal direction.\n \n\nThe effective static potential between two D0-branes emerges because \nsupersymmetry\nis broken by finite temperature. This effect of breaking supersymmetry\nis somewhat analogous to the velocity effects at zero temperature\nwhere the matrix theory and superstring computations agree to\nthe leading order of the velocity expansion~\\cite{dkps}.\nWe have thus shown that the leading term in a low temperature expansion \nis correctly reproduced by the matrix theory.\nThe discrepancy between the matrix theory and superstring computations, \nwhich we have observed in the limit of large distances $L T\\gg 1$,\ndoes not contradict this statement since temperature the limits of large \ndistances and small temperatures are not interchangeable.\n\nAn interesting feature of the effective static potential between\nD0-branes is that it is logarithmic and attractive at short distances.\nIn the matrix theory, the singularity of the computed 1-loop potential\noccurs when the distance between the D0-branes vanishes and the \nSU(N) symmetry which is broken by finite distances is restored.\nThe integration over the off-diagonal components can no longer be treated\nin the 1-loop approximation! \nIn the superstring theory, the singularity is exactly the same as in the\nmatrix theory since it is determined only by the massless bosonic modes\n(the NS sector in the superstring theory). Its origin is {\\em not}\\\/ due \nto the presence of massless photon states in the spectrum.\nPutting $\\nu_1=\\nu_2$ in the above D-instanton picture on the dual\ntemporal circle, we see that the mass of the lowest states,\nassociated with the winding numbers $2q+1=\\pm 1$ is \n$\\tilde \\beta{} \/2\\pi\\alpha'$. The divergence at $L=0$ emerges, in\nthis picture, after summing over all the open string states since\nno single state has such a divergence. It shows up only\nat finite temperature where the winding number $q$ exists. \n\n\n\n\nIt is important to notice that the computed partition functions take\ninto account only thermal fluctuations of superstring stretched\nbetween D0-branes but not the fluctuations of D0-branes themselves.  \nTo calculate the thermal partition function of D0-branes,\na further path integration over their periodic trajectories $\\vec a(\\tau)$ \nis to be performed as in (\\ref{effec}). One might think that \nclassical statistics is applicable to this problem since the D0-branes\nare very heavy as $g^2_{\\rm YM}\\rightarrow0$ so that one could restrict\nhimself by the static approximation. \nThis is however not the case\ndue to the singularity of the effective static potential at small distances.\nThe integral over the D0-brane positions $\\vec{a}$'s is divergent\nwhen the two positions coincide.\n\nHowever, this singularity is only in the classical partition function.  \nThe path integral over the periodic trajectories $\\vec a(\\tau)$ \nthat we actually have to do can not diverge since \nthe 2-body quantum mechanical problem has a well-defined spectrum. \n The path integral can then be evaluated as\n$\\sum_n \\exp(-\\beta E_n)$\nwhere $E_n$ are in the spectrum of the operator\n$ H=P^2\/M+V_{\\rm eff}$. There certainly should not be the bound state energy  \neigenvalue at negative infinity for this\nquantum mechanical problem which implies the convergence of the\npath integral. These issues which are related to thermodynamics of\nD0-branes will be considered in a separate publication. \n\nLet us finally discuss when the 1-loop appoximation\nthat we have done is applicable.\nThe loop expansion in Yang-Mills theory computation \nis valid only in the limit where\n$$\ng_{\\rm YM}^2\/\\vert\\vec a\\vert^3\\sim\ng_s \\left(\\frac{ \\sqrt{\\alpha'}}{L}\\right)^3\n$$ \nis small.  This is due to the fact that the distance $L$ plays the role of a \nHiggs mass which cuts off the infrared divergences of the loop expansion in \nthe 0+1~-dimensional gauge theory.\nThus, the perturbative Yang-Mills theory computation is good when \n\\begin{equation}\ng_s^{1\/3}\\sqrt{\\alpha'} \\ll L.\n\\end{equation}   \nThis can be satisfied if either the string coupling is small or if the \nD0-brane \nseparation is large compared with the string length scale. In the latter case,\nthe truncation of the spectrum to the lightest modes is still valid when\n\\begin{equation}\nk_BT\\ll \\frac{1}{L} \\ll\\frac{1}{g_s^{1\/3}\\sqrt{\\alpha'}}\n\\end{equation} \nNote that the first inequality which is independent of both the string \nscale and the string coupling is the one already discussed in\nthe previous section. In this case, the temperature must be less than \nthe inverse distance between D0-branes.   \nIn the case where the string coupling $g_s$ is small, the criterion for \nvalidity of truncation of  the spectrum becomes\n\\begin{equation}\nk_BT<1\/\\sqrt{\\alpha'}\n\\end{equation}\nthat is the usual one.\n\n\n\\section*{Acknowlegments}\n\nWe are gratefull to P.~Di~Vecchia, G.\\ Ferretti, A.~Gorsky, M.~Green, \nM.~Krogh,  \nN.~Nekrasov, N.~Ohta, P.~Olesen, L.~Thorlacius and K.~Zarembo \nfor very useful discussions. \nJ.A. and Y.M. acknowledge the support \nfrom MaPhySto financed by the Danish National Research Foundation\nand from INTAS under the grant 96--0524.  \nThe work by Y.M. is supported in part by the grants\nCRDF 96--RP1--253 and RFFI 97--02--17927.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDistributed algorithms are an important part of solving many applied optimization problems \\cite{shalev2014understanding,mcdonald2010distributed,mcmahan2017communication}. They help to parallelize the computation process and make it faster. In this paper, we focus on the distributed methods for the saddle point problem:\n\\begin{equation}\n    \\label{SPP}\n    \\min_{x \\in \\mathcal{X}} \\max_{y \\in \\mathcal{Y}} f(x,y) := \\frac{1}{M} \\sum\\limits_{m=1}^{M} f_m(x,y).\n\\end{equation}\nIn this formulation of the problem, the original function $f$ is divided into $M$ parts, each of part $f_m$ is stored on its own local device. Therefore, only the device with the number $m$ knows information about $f_m$. Accordingly, in order to obtain complete information about the function $f$, it is necessary to establish a communication process between devices. This process can be organized in two ways: centralized and decentralized. In a centralized approach, communication takes place via a central server, i.e. all devices can send some information about their local $f_m$ function to the central server, the server collects information from the devices and does some additional calculations, and then can send new information or request to the devices. Then the process continues. With this approach, one can easily write centralized gradient descent for distributed sum minimization: $\\min_{x} g(x) := \\frac{1}{M} \\sum_{m=1}^{M} g_m(x)$. All devices compute local gradients in the same current point and then send these gradients to the server, in turn, the server averages the gradients and makes a gradient descent step, thereby obtaining a new current point, which it sends to the devices. Centralized methods for \\eqref{SPP} are discussed in detail, for example, in \\cite{beznosikov2020local}.  However, centralized approach has several problems, e.g. synchronization drawback or high requirements to the server. Possible approach to deal with these drawbacks is to use decentralized architecture \\cite{bertsekas1989parallel}.  In this case, there is no longer any server, and the devices are connected into a certain communication network and workers are able to communicate only with their neighbors and communications are simultaneous. The most popular and frequently used communication methods are the gossip protocol \\cite{kempe2003gossip,boyd2006randomized,nedic2009distributed} and accelerated gossip protocol \\cite{scaman2017optimal,ye2020multi}. In the gossip protocol, nodes iteratively exchange data with their immediate neighbors using a communication matrix and in this way the information diffuses over the network. The rate of convergence depends on the ratio $\\chi$ of maximal and minimal non-zero eigenvalues of the communication matrix, which is typically proportional to diameter of the graph squared. Accelerated consensus can be achieved i.e. by Chebyshev acceleration \\cite{scaman2017optimal} and improves the dependence from $\\chi$ to $\\sqrt\\chi$, which is optimal. However, the non-accelerated variant is more robust, e.g. it can be applied to time-varying (wireless) communication networks.\n\n\n\n\n\\subsection{Our contribution}\n\nIn particular, our contribution can be briefly described as follows\n\n\\textit{Lower bounds.} We present lower bounds for decentralized smooth strongly-convex-strongly-concave and convex-concave saddle-point problems on the time-varying networks. The lower bounds are derived under the assumption that the network is always a connected graph.\n\n\\textit{Optimal algorithm.} The paper constructs an optimal algorithm that meets the lower bounds. The analysis of the algorithm is carried out for smooth strongly-convex-strongly-concave and convex-concave saddle-point problems\n\nSee our results in the column \"time-varying\" of Table 1.\n\n\\subsection{Related works} \n\nOur work is one of the first dedicated to decentralized saddle problems over time-varying networks. Among other works, we can highlight the following paper \\cite{beznosikov2021decentralized}.\nThis work looks at a more general time-varying setting and suggests a new method. The upper bounds for their method are worse than for our method. We also mention papers on related topics:\n\n\\textit{Decentralized saddle point problems.}\n\nThe next work is devoted to centralized and decentralized distributed saddle problems \\cite{beznosikov2020local}. It carries out lower bounds and optimal algorithms in the case when the communication network is constant (non-time-varying). See Table 1 for comparison our results for time-varying topology and results from \\cite{beznosikov2020local} for constant network.\nAlso note the following works devoted to decentralized min-max problems:  in deterministic case \\cite{liu2019decentralizedprox,9304470,rogozin2021decentralized}, in stochastic case \\cite{liu2019decentralized}.\n\n\\textit{Minimization on time-varying networks.}\n\nDecentralized methods are built upon combining iterations of classical first-order methods with communication steps. In the case of time-varying networks, a non-accelerated communication procedure is employed. Paper \\cite{nedic2009distributed} can be named as an initial work on decentralized sub-gradient methods, and \\cite{nedic2017achieving} proposed DIGing -- the first first-order minimization algorithm with linear convergence over time-varying networks. After that, PANDA, which is a dual method capable of working over time-varying graphs, was proposed in \\cite{maros2018}. Analysis of DIGing and PANDA assumes that the underlying network is B-connected, that is, the union of B consequent networks is connected, while the network is allowed to be disconnected at some steps. Considering the time-varying graphs which stay connected at each iteration, decentralized Nesterov method \\cite{rogozin2019optimal} has an accelerated rate under the condition that graph changes happen rarely enough, ADOM \\cite{kovalev2021adom} and ADOM+ \\cite{kovalev2021lower} are first-order optimization methods which achieve lower complexity bounds \\cite{kovalev2021lower}. APM-C \\cite{rogozin2019projected}, Acc-GT \\cite{li2021accelerated} are accelerated methods over time-varying graphs, as well. The mentioned results are devoted to minimization algorithms and can be generalized to saddle-point problems. In this paper we generalize lower bounds of \\cite{kovalev2021lower} to min-max problems and obtain an algorithm which reaches them up to a logarithmic factor.\n\n\\renewcommand{\\arraystretch}{1.5}\n\\renewcommand{\\tabcolsep}{10pt} \n\\begin{table}[h!]\n\\vspace{-0.3cm}\n\\begin{center}\n\\begin{tabular}{ccc}\n\\hline\n\\multicolumn{1}{c}{} & \\multicolumn{1}{c}{\\textbf{time-varying network}} & \\multicolumn{1}{c}{\\textbf{constant network \\cite{beznosikov2020local}}} \\\\ \\hline\n\\multicolumn{1}{c}{}   & \\multicolumn{2}{c}{{\\tt lower}} \\\\ \\hline\n{\\tt sc} & {$\\Omega\\left( R_0^2 \\exp\\left( - \\frac{\\mu K}{256L \\chi} \\right)\\right)$} & {$\\Omega\\left( R_0^2 \\exp\\left( - \\frac{\\mu K}{128L \\sqrt{\\chi}} \\right)\\right)$} \\\\ \\hline\n{\\tt c} & {$\\Omega\\left(\\frac{L D^2 \\chi}{K}\\right)$} & {$\\Omega\\left(\\frac{L D^2 \\sqrt{\\chi}}{K}\\right)$} \\\\ \\hline\n\\multicolumn{1}{c}{}   & \\multicolumn{2}{c}{{\\tt upper}} \\\\ \\hline\n{\\tt sc} & {$\\mathcal{y}{\\overline{\\lambda} O}\\left( R_0^2 \\exp\\left( - \\frac{\\mu K}{8L \\chi} \\right)\\right)$} & {$\\mathcal{y}{\\overline{\\lambda} O}\\left( R_0^2 \\exp\\left( - \\frac{\\mu K}{8L \\sqrt{\\chi}} \\right)\\right)$} \\\\ \\hline\n{\\tt c} & {$\\mathcal{y}{\\overline{\\lambda} O} \\left(\\frac{L D^2 \\chi}{K}\\right)$} & {$\\mathcal{y}{\\overline{\\lambda} O}\\left(\\frac{L D^2 \\sqrt{\\chi}}{K} \\right)$} \\\\ \\hline\n\\end{tabular}\n\\vspace{0.4cm}\n\\caption{Lower and upper bounds for distributed smooth stochastic strongly-convex--strongly-concave ({\\tt sc}) or convex-concave ({\\tt c}) saddle-point problems in centralized and decentralized cases. Notation:\n$L$ -- smothness constant of $f$, $\\mu$ -- strongly-convex-strongly-concave constant, $R^2_0 = \\|x_0 - x^* \\|^2_2 + \\|y_0 - y^* \\|^2_2$, $D$ -- diameter of optimization set, $\\chi$ -- condition number of communication graph (in time-varying case maximum of all graphs), $K$ -- number of communication rounds. In the case of upper bounds in the convex-concave case, the convergence is in terms of the \"saddle-point residual\", in the rest -- in terms of the (squared) distance to the solution.}\n\\end{center}\n\\label{table1}\n\\end{table}\n\n\\section{Preliminaries}\n\nWe use $\\langle z,u \\rangle := \\sum_{i=1}^d z_i u_i$ to denote standard inner product of $z,u\\in\\mathbb R^d$. It induces $\\ell_2$-norm in $\\mathbb R^d$ in the following way $\\|z\\| := \\sqrt{\\langle z, z \\rangle}$. We also introduce the following notation $\\text{proj}_{\\mathcal{Z}}(z) = \\min_{u \\in \\mathcal{Z}}\\| u - z\\|$ -- the Euclidean projection onto $\\mathcal{Z}$.\n\nWe work with the problem \\eqref{SPP}, where the sets $\\mathcal{X} \\subseteq \\mathbb{R}^{n_x}$ and $\\mathcal{Y} \\subseteq \\mathbb{R}^{n_y}$ are convex sets. Additionally, we introduce the set $\\mathcal{Z} = \\mathcal{X} \\times \\mathcal{Y}$, $z = (x,y)$ and the operator $F$:\n\\begin{equation}\n\\label{opSP}\n    F_m(z) = F_m(x,y) = \\begin{pmatrix}\n\\nabla_x f_m(x,y)\\\\\n-\\nabla_y f_m(x,y)\n\\end{pmatrix}.\n\\end{equation}\nThis notation is needed for shortness. \n\n\\textbf{Problem setting.} Next, we introduce the following assumptions:\n\n\\textit{Assumption 1(g).}\n $f(x,y)$ is $L$ - smooth, if for all $z_1, z_2 \\in \\mathcal{Z}$\n    \\begin{eqnarray}\\label{as1g}\n    \\|F(z_1) - F(z_2)\\| \\leq L\\|z_1-z_2\\|.\\end{eqnarray}\n    \n\\textit{Assumption 1(l).}\n For all $m$, $f_m(x,y)$ is Lipschitz continuous with constant $L_{\\max}$, it holds that for all $z_1, z_2 \\in \\mathcal{Z}$\n    \\begin{eqnarray}\\label{as1l}\n    \\|F_m(z_1) - F_m(z_2)\\| \\leq L_{\\max}\\|z_1-z_2\\|.\\end{eqnarray}\n    \n\\textit{Assumption 2(s).}\n$f(x,y)$ is strongly-convex-strongly-concave with constant $\\mu$, if for all $z_1, z_2 \\in \\mathcal{Z}$\n    \\begin{eqnarray} \\label{as2g}\\langle F(z_1) - F(z_2), z_1 - z_2 \\rangle \\geq \\mu\\|z_1-z_2\\|^2.\\end{eqnarray}\n    \n\\textit{Assumption 2.}\n$f(x,y)$ is convex-concave, if $f(x,y)$ is strongly-convex-strongly-concave with $0$.\n\n\\textit{Assumption 3.} $\\mathcal{Z}$ -- compact bounded, i.e. for all $z, z'\\in \\mathcal{Z}$\n    \\begin{eqnarray} \\label{as5}\n    \\| z - z'\\| \\leq D.\n    \\end{eqnarray}\nAll assumptions are standard in the literature.\n\n\\textbf{Network setting.} In each moment of time (iteration) $t$, the communication network  is modeled  as a connected, undirected graph graph $\\mathcal{G}(t) \\triangleq (\\mathcal{V}, \\mathcal{E}(t))$, where $\\mathcal{V} := \\{1,\\ldots, M\\}$ denotes  the vertex    set--the set of devices (does not change in time) and  $\\mathcal{E}(t) := \\{(i,j) \\, |\\, i,j \\in \\mathcal{V} \\}$ represents the set of edges--the communication links at the moment $t$;  $(i,j) \\in \\mathcal{E} (t)$ iff there exists a communication link between devices $i$ and $j$ in moment $t$.  \n\nAs mentioned earlier, the gossip protocol is the most popular communication procedures in decentralized\nsetting. This approach uses a certain matrix $W$. Local vectors during communications are \"weighted\" by multiplication of a vector with $W$. The convergence of decentralized algorithms is determined by the properties of this matrix. Therefore, we introduce it:\n\n\\textbf{Assumption 4.} We call a matrix $W(t)$ a gossip matrix at the moment $t$ if it satisfies the following conditions: 1) $W(t)$ is an $M \\times M$ symmetric, 2) $W(t)$ is positive semi-definite, 3) the kernel of $W(t)$ is the set of constant vectors, 4) $W(t)$ is defined on the edges of the network at the moment $t$: $W_{ij}(t) \\neq 0$ only if $i=j$ or $(i,j) \\in \\mathcal{E}(t)$.\n\nLet $\\lambda_1(W(t)) \\geq \\ldots \\geq \\lambda_M(W(t)) = 0$ the spectrum of $W(t)$, and let define condition number $\\chi = \\max_t \\chi(W(t)) = \\frac{\\lambda_1(W(t))}{\\lambda_{M-1}(W(t))}$. Note that in practice we will use not the matrix $W(t)$, but $y}{\\overline{\\lambda} W(t) = I - \\frac{W(t)}{\\lambda_1(W(t))}$. It is these matrices that are used in consensus algorithms \\cite{boyd2006randomized}. To describe the convergence, we introduce $\\rho = \\max_t \\lambda_2(y}{\\overline{\\lambda} W(t)) = \\max_t \\left[1 - \\frac{\\lambda_{M-1}(W(t))}{\\lambda_1(W(t))}\\right] = \\max_t \\left[1 - \\frac{1}{\\chi(W(t))}\\right] = 1 - \\frac{1}{\\max_t \\chi(W(t))} = 1 - \\frac{1}{\\chi}$.\n\n\\section{Main part}\n\nWe divide our contribution into two main parts, first we discuss lower bounds for decentralized saddle point problems over time-varying graphs. In the second part, we present an algorithm that achieves the lower bounds (up to logarithmic factors and numerical constants).\n\n\\subsection{Lower bounds}\n\nBefore presenting lower bounds, we must restrict the class of algorithms for which our lower bounds are valid. For this we introduce the following back-box procedure.\n\n\\begin{definition} \\label{proc}\n    Each device $m$ has its own local memories $\\mathcal{M}^x_{m}$ and $\\mathcal{M}^y_{m}$ for the $x$- and $y$-variables, respectively--with  initialization $\\mathcal{M}_{m}^x = \\mathcal{M}_{m}^y= \\{0\\}$.    $\\mathcal{M}_{m}^x$ and $\\mathcal{M}_{m}^x$ are updated as follows:\\\\\n   \n    $\\bullet$ \\textbf{Local computation:} Each device $m$ computes and adds to its $\\mathcal{M}^x_{m}$ and $\\mathcal{M}^y_{m}$ a finite number of points $x,y$, each satisfying \n    \\begin{equation}\\begin{aligned}\\label{eq:oracle-opt-step}\n        x \\in \\text{span} \\big\\{x'~,~\\nabla_x f_m(x'',y'')\\big\\},\\quad \n        y \\in \\text{span} \\big\\{y'~,~\\nabla_y f_m(x'',y'')\\big\\}, \n    \\end{aligned}\\end{equation}\n    for given $x', x'' \\in \\mathcal{M}^x_{m}$ and  $y', y'' \\in \\mathcal{M}^y_{m}$.\n\n    $\\bullet$ \\textbf{Communication:} Based upon communication round among neighbouring nodes at the moment $t$,   $\\mathcal{M}^x_{m}$ and $\\mathcal{M}^y_{m}$ are updated according to\n    \\begin{equation}\\label{eq:oracle-comm}\n        \\mathcal{M}^x_{m} := \\text{span}\\left\\{\\bigcup_{(i,m) \\in \\mathcal{E}(t)} \\mathcal{M}^x_{i} \\right\\}, \\quad \n        \\mathcal{M}^{y}_{m} := \\text{span}\\left\\{\\bigcup_{(i,m) \\in \\mathcal{E}(t)} \\mathcal{M}^y_{i} \\right\\}.\n    \\end{equation}\n\n    $\\bullet$ \\textbf{Output:} \n    The final global output at the current moment of time is calculated as: \n    \\begin{align*}\n        x \\in \\text{span}\\left\\{\\bigcup_{m=1}^M \\mathcal{M}^x_{m} \\right\\},~~y \\in \\text{span}\\left\\{\\bigcup_{m=1}^M \\mathcal{M}^y_{m} \\right\\}.\n    \\end{align*}\n\\end{definition} \n\nThis definition includes all algorithms capable of making local gradient updates, as well as exchanging information with neighbors.\nNotice that the proposed oracle builds on   \\cite{scaman2017optimal}  for minimization problems over networks. \n\n\\begin{theorem}\\label{th-LB-distributed}\nFor any $L$, $\\mu$ and $\\chi \\leq 1$, there exist a  saddle point problem in the form (\\ref{SPP}) with $\\mathcal{Z}=\\mathcal{R}^{2d}$(where $d$ is sufficiently\nlarge), and  local functions $f_m$ being  $L$-smooth, $\\mu$-strongly-convex-strongly-concave, and a gossip matrices $W(t)$ over the connected (at each moment) graph  $\\mathcal{G} (t)$,  satisfying Assumption 4 and  with $\\chi$, such that any decentralized algorithm  satisfying Definition~\\ref{proc} and using the gossip matrices $W(t)$  produces    the following estimate on the global output $z=(x,y)$ after $K$ communication rounds\n\\begin{equation*}\n    \\|z^{K} - z^*\\|^2 = \\Omega\\left(\\exp\\left( - \\frac{256\\mu}{L - \\mu} \\cdot \\frac{K}{\\chi} \\right) \\| y^*\\|^2\\right).\n\\end{equation*}\n\\end{theorem}\n\n\\begin{corollary}\nIn the setting of Theorem~\\ref{th-LB-distributed}, the number of communication rounds required to obtain a $\\varepsilon$-solution is lower bounded by\\vspace{-0.2cm}\n\\begin{equation*}\n    \\Omega\\left( \\chi \\frac{L}{\\mu} \\cdot  \\log \\left(\\frac{\\| y^*\\|^2}{\\varepsilon}\\right)\\right).\n\\end{equation*}\nAdditionally, we can get a lower bound for the number of local calculations on each of the devices:\n\\begin{equation*}\n    \\Omega\\left( \\frac{L}{\\mu} \\cdot  \\log \\left(\\frac{\\| y^*\\|^2}{\\varepsilon}\\right)\\right).\n\\end{equation*}\n\\end{corollary}\n\nAlso we want to find lower bounds for the case of  (non strongly) convex-concave problems, one can use  regularization and consider the following objective function \n\\begin{align*}\n    f(x,y) + \\frac{\\varepsilon}{4D^2} \\cdot \\|x - x^0 \\|^2 - \\frac{\\varepsilon}{4D^2}\\cdot \\|y - y^0 \\|^2,\n\\end{align*}\nwhich is   strongly-convex-strongly-concave with constant  $\\mu = \\frac{\\varepsilon}{4D^2}$, where $\\varepsilon$ is a precision of the solution and $D$ is the  diameter of the  sets $\\mathcal{X}$ and $\\mathcal{Y}$. The resulting new SPP problem is solved to $\\varepsilon\/2$-precision in order to guarantee an accuracy $\\varepsilon$ in computing the solution of  the original problem. Therefore, we can easily deduce the lower bounds for convex-concave case\n\\begin{equation*}\n\\label{lower_cc}\n    \\Omega\\left( \\chi \\frac{L D^2}{\\varepsilon} \\right)~\\text{communic. rounds}\\quad \\text{and}\\quad  \\Omega\\left(\\frac{L D^2}{\\varepsilon}\\right) ~\\text{local computations}.\n\\end{equation*}\nSee Table 1 to compare with lower bounds for constant networks. The full proof of Theorem 1 one can find in the full version of our paper [for reviewers: here will be the link to the full version in arxiv.org, but we did not publish the paper before review].\n\n\\subsection{Optimal algorithm}\n\nIn this part, we present an Algorithm that achieves lower bounds (up to logarithmic terms). Our Algorithm uses an auxiliary procedure for communication. This is a classic procedure - Gossip Algorithm.\n\n\\begin{algorithm} [th]\n\t\\caption{Gossip Algorithm ({\\tt Gossip})}\n\t\\label{alg3}\n\t\\begin{algorithmic}\n\\State\n\\noindent {\\bf Parameters:} Vectors $z_1, ..., z_M$, communic. rounds $H$.\n\\State \\noindent {\\bf Initialization:}\nConstruct matrix $\\textbf{z}$ with rows $z^T_1, ..., z^T_M$.\n\\State Choose $\\textbf{z}^0 = \\textbf{z}$.\n\\For {$h=0,1, 2, \\ldots, H$ } \n\\State $\\textbf{z}^{h+1} = y}{\\overline{\\lambda} W (h) \\cdot  \\textbf{z}^{h}$\n\\EndFor\n\\State\n\\noindent {\\bf Output:} rows $z_1, ..., z_M$ of  $\\textbf{z}^{H+1}$ .\n\t\\end{algorithmic}\n\\end{algorithm}\n\nThe essence of the {\\tt Gossip} is very simple. Initially, there are vectors $z_1$ and $z_M$, which are stored on their devices. Our goal is to get a vector close to the $\\bar z = \\frac{1}{M} \\sum_{m=1}^M z_m$ vector on all devices. At each iteration, each device exchange local vectors with its neighbors, and then modify its local vector by averaging local vector and vectors of neighbors with weights from the matrix $W (h)$.\n\nWe are now ready to present our main algorithm. It is based on the classical method for smooth saddle point problems - Extra Step Method (Mirror Prox) \\cite{Nemirovski2004,juditsky2008solving}. With the right choice of $H$, we can achieve averaging of all vectors with good accuracy. In particular, we can assume that $z^k_1 \\approx \\ldots \\approx z^k_M$. For more details about the choice of $H$ and a detailed analysis of the algorithm (taking into account that in the general $z^k_1 \\neq \\ldots \\neq z^k_M$), see in the full version of the paper [for reviewers: here will be the link to the full version in arxiv.org, but we did not publish the paper before review].\n\n\n\\begin{algorithm} [th]\n\t\\caption{Time-Varying Decentralized Extra Step Method ({\\tt TVDESM})}\n\t\\label{alg2}\n\t\\begin{algorithmic}\n\\State\n\\noindent {\\bf Parameters:}  Stepsize $\\gamma \\leq \\frac{1}{4L}$, number of {\\tt Gossip} steps $H$.\\\\\n\\noindent {\\bf Initialization:} Choose  $(x^0,y^0)=z^0\\in \\mathcal{Z}$, $z^0_m = z^0$.\n\\For {$k=0,1, 2, \\ldots, $ } \n \\State Each machine $m$ compute  \\hspace{0.1cm} $\\hat z_m^{k+1\/2} = z_m^{k} - \\gamma \\cdot F_m(z^k_m)$\n \\State  Communication: \\hspace{0.1cm} $y}{\\overline{\\lambda} z^{k+1\/2}_1, \\ldots,  y}{\\overline{\\lambda} z^{k+1\/2}_M$ ={\\tt Gossip}$(\\hat z^{k+1\/2}_1, \\ldots, \\hat z^{k+1\/2}_M, H)$\n\\State Each machine $m$ compute $z^{k+1\/2}_m = \\text{proj}_{\\mathcal{Z}}(y}{\\overline{\\lambda} z^{k+1\/2}_m)$,\n\\State Each machine $m$ compute \\hspace{0.1cm} $\\hat z_m^{k+1} = z_m^{k} - \\gamma \\cdot F_m(z^{k+1\/2}_m)$\n\\State Communication: \\hspace{0.1cm} $y}{\\overline{\\lambda} z^{k+1}_1, \\ldots,  y}{\\overline{\\lambda} z^{k+1}_M$ ={\\tt Gossip}$(\\hat z^{k+1}_1, \\ldots, \\hat z^{k+1}_M, H)$\n\\State Each machine $m$ compute \\hspace{0.1cm} $z^{k+1}_m = \\text{proj}_{\\mathcal{Z}}(y}{\\overline{\\lambda} z^{k+1}_m)$\n\\EndFor\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{theorem} \nLet $\\{ z_m^k\\}^K_{k \\geq 0}$ denote the iterates of Algorithm~\\ref{alg2} for solving problem \\eqref{SPP} after $K$ communication rounds. Let Assumptions 1(g,l) and 4 be satisfied. Then, if $\\gamma \\leq \\frac{1}{4L}$, we have the following estimates in\n\n$\\bullet$ $\\mu$-strongly-convex--strongly-concave case (Assumption 2(s)):\n\\begin{eqnarray*}\n \\mathbb E[\\| \\bar z^{k+1}\\!-\\!z^* \\|^2]\\!=\\!\\mathcal{y}{\\overline{\\lambda} O}\\left( \\| z^{0} - z^* \\|^2 \\exp\\left( -\\frac{\\mu K}{8L\\sqrt{\\chi}} \\right) \\right) ,\n \\end{eqnarray*}\n\n$\\bullet$ convex--concave case (Assumption 2 and 3):\n\\begin{equation*}\n    \\mathbb E[\\text{gap}(\\bar z^{k+1}_{avg})] = \\mathcal{y}{\\overline{\\lambda} O}\\left(\\frac{L \\Omega_z^2 \\chi}{K} \\right),\n\\end{equation*}\n\nwhere $\\bar z^{t} = \\frac{1}{M} \\sum\\limits_{m=1}^M z_m^{t}$, $\\bar z^{k+1}_{avg} = \\frac{1}{M(k+1)} \\sum\\limits_{t=0}^k \\sum\\limits_{m=1}^M z_m^{t+1\/2}$ and $$\\text{gap}(z) = \\max_{y'\\in \\mathcal{Y}} f(x, y') -  \\min_{x'\\in \\mathcal{X}} f(x', y).$$\n\\end{theorem}\n\n\\begin{corollary}\nIn the setting of Theorem~2, the number of communication rounds required for Algorithm 2 to obtain a $\\varepsilon$-solution is upper bounded by\\vspace{-0.2cm}\n\\begin{equation*}\n    \\mathcal{y}{\\overline{\\lambda} O}\\left( \\chi \\frac{L}{\\mu} \\cdot  \\log \\left(\\frac{\\| z^0 - z^*\\|^2}{\\varepsilon}\\right)\\right)\n\\end{equation*}\nin $\\mu$-strongly-convex--strongly-concave case and \n\\begin{equation*}\n\\mathcal{y}{\\overline{\\lambda} O}\\left( \\chi \\frac{L D^2}{\\varepsilon} \\right)\n\\end{equation*}\nin convex-concave case.\nAdditionally, one can obtain upper bounds for the number of local calculations on each of the devices:\n\\begin{equation*}\n    \\mathcal{O}\\left( \\frac{L}{\\mu} \\cdot  \\log \\left(\\frac{\\| z^0 - z^*\\|^2}{\\varepsilon}\\right)\\right)\n\\end{equation*}\nin $\\mu$-strongly-convex--strongly-concave case and \n\\begin{equation*}\n\\mathcal{O}\\left(\\frac{L D^2}{\\varepsilon} \\right)\n\\end{equation*}\nin convex-concave case.\n\\end{corollary}\n\n\\section{Conclusion}\n\nIn conclusion, we briefly summarize the contributions of this paper and discuss the directions for future work. Our findings consist of two parts: lower bounds and optimal (up to a logarithmic factor) algorithms.\n\nFirst, we derived the lower bounds for the classes of convex-concave and strongly-convex-strongly-concave min-max problems over time-varying graphs. The graph is assumed to be connected at each communication round. However, we studied only one class of time-varying networks. Other classes are connected to different assumptions on the network structure. In particular, in B-connected networks \\cite{nedic2017achieving} the graph can be disconnected at some times, but the union of any B consequent graphs must be connected. Yet another possible assumption is the randomly changing graph with a contraction property of $W$ in expectation \\cite{koloskova2020unified}. Developing lower bounds for min-max problems for these two classes is an open question in decentralized optimization.\n\nSecond, we proposed a near-optimal algorithm with a gossip subroutine resulting in squared logarithmic factor. Developing an algorithm without an additional logarithmic factor would close the gap in theory and result in a more practical algorithm with less parameters to fine-tune. Possible directions for developing such an algorithm are generalizations of dual-based approaches for minimization \\cite{kovalev2021adom,maros2018} and gradient-tracking \\cite{nedic2017achieving,maros2018}.\n\nFinally, the comparison of our algorithm to existing works requires additional numerical experiments, which is left for future work.\n\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWe are concerned in this paper with the varieties $W^r_d(C)$ of special linear series on a general curve of fixed genus $g$ and gonality $k$ over an algebraically closed field $K$ with mild restrictions on the characteristic. See Situation \\ref{sit_main} for the necessary hypotheses.\n\nFor a general curve $C$ of genus $g$, we have $k = \\lfloor \\frac{g+3}{2} \\rfloor$ and the Brill-Noether theorem \\cite{gh} says that $\\dim W^r_d(C)$ is equal to the Brill-Noether number\n$$\n\\rho_g(d,r) = g - (r+1)(g-d+r),\n$$\n\nunless $\\rho_g(d,r) < 0$, in which case $W^r_d(C) = \\emptyset$.\n\nOur objective is to compute $\\dim W^r_d(C)$ for non-generic values of $k$. We propose a modification $\\overline{\\rho}_{g,k}(d,r)$ of the Brill-Noether number, incorporating the value of $k$, and prove the following results. By convention, a negative estimate on $\\dim W^r_d(C)$ means that $W^r_d(C)$ is empty.\n\n\\begin{thm} \\label{thm_main} In Situation \\ref{sit_main},\n$\\dim W^r_d(C) \\leq \\overline{\\rho}_{g,k}(d,r)$.\n\\end{thm}\nIn characteristic $0$, we may combine Theorem \\ref{thm_main} with a lower bound from results of Coppens and Martens \\cite{cm02}, to obtain the following sharp results.\n\\begin{thm} \\label{thm_eq}\nIn Situation \\ref{sit_main}, if $\\chr K=0$ and either $k \\leq 5$ or $k \\geq \\frac15 g + 2$, then $\\dim W^r_d(C) = \\overline{\\rho}_{g,k}(d,r)$.\n\\end{thm}\n\n\\begin{thm} \\label{thm_howgeneral}\nIn Situation \\ref{sit_main}, if $\\chr K = 0$ and  $\\rho_g(d,r) \\geq 0$, then $\\dim W^r_d(C) = \\rho_g(d,r)$ if and only if $r=0$, $g-d+r = 1$, or $g - k \\leq d - 2r$.\n\\end{thm}\n\nAll three theorems use the following notation and hypotheses.\n\n\\begin{sit} \\label{sit_main}\nFix nonnegative integers $g,k,r,d$ such that $2 \\leq k \\leq~\\frac{g+3}{2}$ and $g-d+r > 0$. Let $K$ be an algebraically closed field, and $C$ be a general $k$-gonal curve of genus $g$ over $K$. Also assume that \n\\begin{itemize}\n\\item If $k$ is odd, then $\\chr K \\neq 2$,\n\\item If $k=4$ or $10$, then $\\chr K \\neq 3$, and\n\\item If $k=6$, then $\\chr K \\neq 5$.\n\\end{itemize}\n\\end{sit}\n\nThe hypothesis $g-d+r > 0$ is harmless, since $g-d+r \\leq 0$ would imply automatically that $W^r_d(C) = \\Pic^d(C)$. The peculiar restrictions on the characteristic of $K$ arise in our proof when we must construct a \\emph{tame} morphism of metrized complexes with certain properties. The characteristic $0$ assumption in Theorems \\ref{thm_eq} and \\ref{thm_howgeneral} is included because these make use of constructions from \\cite{cm99} and \\cite{cm02}, which assume this hypothesis.\n\n\n\n\\subsection{The estimate $\\overline{\\rho}_{g,k}(d,r)$} The estimate $\\overline{\\rho}_{g,k}(d,r)$ we refer to above is the following.\n\n\\begin{defn} \\label{def_ovr}\nLet $r'$ denote the minimum of $r$ and $g-d+r-1$. Define\n$$\n\\overline{\\rho}_{g,k}(d,r) = \\max_{\\ell \\in \\{0,1,2,\\cdots,r'\\}} \\left( \\rho_g(d,r-\\ell) - \\ell k \\right).\n$$\n\\end{defn}\n\n\\begin{rem} \\label{rem_whatsell}\nThe expression being maximized in this definition is a quadratic function in $\\ell$, which takes its maximum (among all real numbers $\\ell$) at $\\ell_0 = \\frac12(g-d+2r+1-k)$. It is therefore straightforward to write a closed-form expression for $\\overline{\\rho}_{g,k}(d,r)$; see Remark \\ref{rem_threecases} and the equation preceding it.\n\\end{rem}\n\nNote that, by Riemann-Roch, $W^r_d(C) \\cong W^{g-d+r-1}_{2g-2-d}(C)$. This explains the apparently strange appearance of $r'$ in Definition \\ref{def_ovr}: it ensures that $\\overline{\\rho}_{g,k}(d,r)$ is invariant under this duality.\n\nWe will relate $\\overline{\\rho}_{g,k}(d,r)$ to $\\dim W^r_d(C)$ by specializing to a metric graph $\\Gamma_{g,k,\\ell}$ very similar to the metric graphs used in the tropical proof of the Brill-Noether Theorem \\cite{cdpr}, but chosen with \\textit{special} edge lengths. Making use of our description in \\cite{pfl} of the Brill-Noether loci of such chains, we will see (Corollary \\ref{cor_tropdim}) that $\\dim W^r_d(\\Gamma_{g,k,\\ell})$ is equal to $\\overline{\\rho}_{g,k}(d,r)$. The tropical lifting results of \\cite{abbr}, combined with semicontinuity results for tropicalization, give Theorem \\ref{thm_main}.\n\n\\subsection{The lower bound $\\underline{\\rho}_{g,k}(d,r)$} \nTo deduce Theorems \\ref{thm_eq} and \\ref{thm_howgeneral} from Theorem \\ref{thm_main}, we require a lower bound on $\\dim W^r_d(C)$.\n\nCoppens and Martens provide constructions in \\cite{cm99} and \\cite{cm02} that establish lower bounds on $\\dim W^r_d(C)$ in Situation \\ref{sit_main}, with an additional characteristic $0$ hypothesis. Their results imply the existence of irreducible components of $W^r_d(C)$ of dimension $\\rho_g(d,r-\\ell) - \\ell k$ for certain values of $\\ell$ in $\\{0,1,\\cdots,r'\\}$. From their results, one can deduce the following lower bound on $\\dim W^r_d(C)$, which coincides with the ``optimistic guess'' at the end of \\cite{cm99} (they observe in \\cite[Remark on p. 39]{cm02} that this optimistic guess is too small in general, though it is correct for $k \\leq 5$).\n\n\\begin{defn}\nLet $r' = \\min(r,g-d+r-1)$. If $r' \\geq 1$, let\n$$\n\\underline{\\rho}_{g,k}(d,r) = \\max_{\\ell \\in \\{0,1,r'-1,r'\\}} \\left( \\rho_g(d,r-\\ell) - \\ell k \\right).\n$$\nIf $r' = 0$, let $\\underline{\\rho}_{g,k}(d,r) = \\rho_g(d,r)$ (i.e. only allow $\\ell = 0$).\n\\end{defn}\n\n\\begin{thm}[\\cite{cm02}] \\label{thm_cm}\nIn Situation \\ref{sit_main}, if $\\chr K = 0$ then \n$\\dim W^r_d(C) \\geq \\underline{\\rho}_{g,k}(d,r)$.\n\\end{thm}\n\nCoppens and Martens do not state Theorem \\ref{thm_cm} in the form that we have stated it here, but Theorem \\ref{thm_cm} can be rapidly deduced from the main theorem in \\cite{cm02}; the argument is provided in Section \\ref{sec_proofs}.\n\n\\begin{rem} \\label{rem_ab}\nWe will see in our analysis that it is useful to consider instead the quantites $g - \\overline{\\rho}_{g,k}(d,r)$ $g - \\underline{\\rho}_{g,k}(d,r)$ (which are bounds on the \\textit{codimension} of $W^r_d(C)$ in $\\Pic^d(C)$). For fixed $k$, both of these quantities are functions of the following two variables.\n\\begin{eqnarray*}\na &=& r+1\\\\\nb &=& g-d+r\n\\end{eqnarray*}\nObserve that, for \\emph{any} smooth curve $C$ of genus $g$, $W^r_d(C) \\neq \\Pic^d(C)$ if and only if both $a$ and $b$ are positive. It is useful to use these variables, rather than $r$ and $d$, for visualization purposes, such as in Figures \\ref{fig_g20} and \\ref{fig_disagree}; by the previous remark, the region of interest is the first quadrant. The maximizing value $\\ell_0$ mentioned in Remark \\ref{rem_whatsell} is easier to understand in these variables as well: it is $\\ell_0 = \\frac12( a+b-k)$.\n\\end{rem}\n\n\\begin{eg}\nConsider the case $g=20$ in characteristic $0$. Since $\\frac15 g + 2 = 6$, Theorem \\ref{thm_eq} shows that $\\dim W^r_d(C) = \\overline{\\rho}_{20,k}(d,r)$ in all cases. We can visualize how the existence of specific types of linear series varies with the gonality by plotting, for each possible gonality $k$, all points $(r+1,20-d+r)$ such that $W^r_d(C) \\neq \\emptyset$ for a general $k$-gonal curve of genus $20$. These plots are shown in Figure \\ref{fig_g20}. We show only the points with both coordinates positive, since all other points correspond to values of $r,d$ for which $W^r_d(C) = \\Pic^d(C)$.\n\n\\begin{figure}\n\\begin{tabular}{ccccc}\n\\includegraphics{boxplot20-2.png}&\n\\includegraphics{boxplot20-3.png}&\n\\includegraphics{boxplot20-4.png}&\n\\includegraphics{boxplot20-5.png}&\n\\includegraphics{boxplot20-6.png}\\\\\n$k=2$ & $k=3$ & $k=4$ & $k=5$ & $k=6$\\\\\n\\includegraphics{boxplot20-7.png}&\n\\includegraphics{boxplot20-8.png}&\n\\includegraphics{boxplot20-9.png}&\n\\includegraphics{boxplot20-10.png}&\n\\includegraphics{boxplot20-11.png}\\\\\n$k=7$ & $k=8$ & $k=9$ & $k=10$ & $k=11$\n\\end{tabular}\n\\caption{\nThese plots show, for genus $20$ and every possible gonality $k$, the points $(g-d+r,r+1)$ such that $W^r_d(C)$ is nonempty for a general $k$-gonal curve $C$ in characteristic $0$.\n} \\label{fig_g20}\n\\end{figure}\n\\end{eg}\n\nWe consider the form of the results proved in \\cite{cm02}, together with the analysis of the tropical version of the problem, as evidence for the following conjecture. Note in particular that we believe that the hypotheses on the characteristic of the field should be superfluous, although they are needed for technical reasons in our arguments.\n\n\\begin{conj}\nIf $g,k,r,d$ are nonnegative integers with $g-d+r > 0$ and $2 \\leq k \\leq \\frac{g+3}{2}$, and $C$ is a general $k$-gonal curve of genus $g$ over an algebraically closed field, then $\\dim W^r_d(C) = \\overline{\\rho}_{g,k}(d,r)$.\n\\end{conj}\n\nBeyond the dimension itself, one can ask about the dimensions of \\textit{all} irreducible components of $W^r_d(C)$. The form of the upper bound, and the results of \\cite{cm02}, suggest the following question.\n\n\\begin{qu}\nDoes every irreducible component of $W^r_d(C)$ have dimension equal to $\\rho_g(d,r-\\ell) - \\ell k$ for some integer $\\ell$?\n\\end{qu}\n\n\\subsection{The gap between $\\underline{\\rho}_{g,k}(d,r)$ and $\\overline{\\rho}_{g,k}(d,r)$} \\label{ss_gap}\n\nThe definitions of $\\overline{\\rho}_{g,k}(d,r)$ and $\\underline{\\rho}_{g,k}(d,r)$, along with Remark \\ref{rem_whatsell}, reveal that there are four cases where we can immediately conclude that $\\underline{\\rho}_{g,k}(d,r) = \\overline{\\rho}_{g,k}(d,r)$ and compute $\\dim W^r_d(C)$ (for curves over characteristic $0$ fields). These are the cases where the maximum in Definition \\ref{def_ovr} occurs at $\\ell = 0,1,r'-1,r'$, respectively (note that these cases may overlap in situations where the maximum occurs for two integers $\\ell$, or when $r' \\leq 2$). Here $r' = \\min(r,g-d+r-1)$, as in Definition \\ref{def_ovr}.\n\nThe issue of comparing $\\underline{\\rho}_{g,k}(d,r)$ to $\\overline{\\rho}_{g,k}(d,r)$ therefore is confined to the situation where the maximum in Definition \\ref{def_ovr} occurs for a value $\\ell$ such that $2 \\leq \\ell \\leq r'-2$. In the variables $(a,b) = (r+1,g-d+r)$, this is equivalent to the inequalities $2 \\leq \\frac{a+b-k}{2} \\leq \\min(a-3,b-3)$; equivalently $a+b \\geq 4+k$ and $|a-b| \\leq k-6$ (where we assume $a,b > 0$, as is implied by Situation \\ref{sit_main}). See Figure \\ref{fig_disagree}.\n\n\\begin{figure}\n\\begin{tikzpicture}[scale=0.2]\n\\draw[<->,thick] (0,20) -- (0,0) -- (20,0);\n\\draw[<->,fill=lightgray] (17,20) -- (5,8) -- (8,5) -- (20,17);\n\\end{tikzpicture}\n\\caption{The region in which $\\underline{\\rho}_{g,k}(d,r) < \\overline{\\rho}_{g,k}(d,r)$, where the coordinates are $a = r+1$ and $b = g-d+r$. It is bounded by the lines $a+b = 4+k$ and $a-b = \\pm (k-6)$, and empty for $k \\leq 5$.} \\label{fig_disagree}\n\\end{figure}\n\n\nOur proof of Theorem \\ref{thm_eq} proceeds by showing that as long as $k \\geq \\frac15 g + 2$ or $k \\leq 5$, \\textit{all} $d,r$ corresponding to points in this region give $\\overline{\\rho}_{g,k}(d,r) < 0$. The following example shows that even for smaller $k$, the ambiguous cases are nonetheless quite sparse, even in the worst case.\n\n\\begin{eg}\nConsider the case $g=1000$. For each possible value of the gonality $k$, we can enumerate all possible pairs of positive integers $r,d$ with $d \\leq g-1$ and $\\overline{\\rho}_{g,k}(d,r) \\geq 0$ (we will restrict to $d \\leq g-1$, since the other cases are symmetric via Riemann-Roch), and count the proportion of these for which $\\underline{\\rho}_{g,k}(d,r) < \\overline{\\rho}_{g,k}(d,r)$. The largest such proportion is $0.042$, for gonality $k=40$; in that case there are $13123$ possible pairs where $\\overline{\\rho}_{g,k}(d,r) \\geq 0$, of which $552$ satisfy $\\overline{\\rho}_{g,k}(d,r) < \\underline{\\rho}_{g,k}(d,r)$. Of those $552$ cases, there are only $69$ cases where $\\underline{\\rho}_{g,k}(d,r) < 0$. In all other cases there is no ambiguity about whether $W^r_d(C)$ is nonempty, but only about its exact dimension.\n\\end{eg}\n\n\\subsection{Outline of the paper}\n\nIn Section \\ref{sec_lifting}, we define the metric graphs $\\Gamma_{g,k,\\ell}$ where we formulate the tropical analog we will study, and prove that these graphs can be lifted to algebraic curves using results of \\cite{abbr}. We also cite the necessary statements of our \\cite{pfl} that translate the computation of $\\dim W^r_d(\\Gamma_{g,k,\\ell})$ into the analysis of certain \\textit{displacement tableaux}. This analysis is performed in Section \\ref{sec_disp}, which is purely combinatorial. In section \\ref{sec_proofs} we deduce the theorems stated in the introduction.\n\n\\begin{rem}\nAn alternative to the tropical approach that we use in Section \\ref{sec_lifting} would be to instead consider limit linear series on a chain of elliptic curves, where the two attachment points on each elliptic curve differ by $k$-torsion. Such an approach would reduce to the same analysis of displacement tableaux.\n\\end{rem}\n\n\\section{Lifting metric graphs} \\label{sec_lifting}\n\nIn this section, we construct the metric graphs which will be the basis of our argument, and prove the necessary lifting results to algebraic curves.\n\n\\begin{defn}\nLet $\\Gamma_{g,k,\\ell}$ denote the metric graph formed by connecting $g$ cycles in a chain as in Figure \\ref{fig_chain}, with edge lengths as follows.\n\\begin{itemize}\n\\item The length of the top (clockwise) edge from $w_i$ to $w_{i+1}$ is $\\ell$.\n\\item The length of the bottom (counterclockwise) edge from $w_i$ to $w_{i+1}$ is $k-\\ell$.\n\\end{itemize}\n\\end{defn}\n\n\\begin{figure}\n\\begin{tikzpicture}[scale=2]\n\\draw (0,0) arc (105:75:1.931);\n\\draw (1,0) arc (0:-180:0.5);\n\\draw (1,0) arc (105:75:1.931);\n\\draw (2,0) arc (0:-180:0.5);\n\\draw (2,0) arc (105:75:1.931);\n\\draw (3,0) arc (0:-180:0.5);\n\\draw (3.5,0) node {$\\cdots$};\n\\draw (4,0) arc (105:75:1.931);\n\\draw (5,0) arc (0:-180:0.5);\n\n\\foreach \\x in {0,1,2,4} {\n\\draw (\\x+0.5,0.05) node[above] {$\\ell$};\n\\draw (\\x+0.5,-0.5) node[below] {$k - \\ell$};\n\\draw (\\x+0.5,-2) node[below] {$\\ell(k-\\ell)$};\n}\n\n\\draw[->,ultra thick] (2.5,-1) -- (2.5,-1.5);\n\\draw (2.5,-1.25) node[right] {degree $k$};\n\\draw (0,-2) -- (3.25,-2);\n\\draw[fill] (0,-2) circle[radius=0.05] node[above] {$u_0$};\n\\draw[fill] (1,-2) circle[radius=0.05] node[above] {$u_1$};\n\\draw[fill] (2,-2) circle[radius=0.05] node[above] {$u_2$};\n\\draw[fill] (3,-2) circle[radius=0.05] node[above] {$u_3$};\n\\draw (3.5,-2) node {$\\cdots$};\n\\draw (3.75,-2) -- (5,-2);\n\\draw[fill] (4,-2) circle[radius=0.05] node[above] {$u_{g-1}$};\n\\draw[fill] (5,-2) circle[radius=0.05] node[above] {$u_g$};\n\n\\draw[fill] (0,0) circle[radius=0.05] node[above] {$w_0$};\n\\draw[fill] (1,0) circle[radius=0.05] node[above] {$w_1$};\n\\draw[fill] (2,0) circle[radius=0.05] node[above] {$w_2$};\n\\draw[fill] (3,0) circle[radius=0.05] node[above] {$w_3$};\n\\draw[fill] (4,0) circle[radius=0.05] node[above] {$w_{g-1}$};\n\\draw[fill] (5,0) circle[radius=0.05] node[above] {$w_g$};\n\\end{tikzpicture}\n\\caption{The chain of cycles $\\Gamma_{g,k,\\ell}$ and the harmonic morphism to an interval.} \\label{fig_chain}\n\\end{figure}\n\n\\begin{defn}\nLet $k,\\ell$ be positive integers with $\\ell < k$, and let $p$ be either a prime number or $0$. Call the triple $(p,k,\\ell)$ \\emph{admissible} if $\\gcd(\\ell,k) = \\gcd(\\ell,p) = \\gcd(k-\\ell,p) = 1$.\n\\end{defn}\n\nThis notion of admissibility underlies the restrictions on the characteristic of the field in Situation \\ref{sit_main}.\n\n\\begin{lemma} \\label{lem_lexists}\nLet $p,k$ be integers, with $k \\geq 2$ and either $p=0$ or $p$ prime. There exists an integer $\\ell \\in \\{1,2,\\cdots,k-1\\}$ such that $(p,k,\\ell)$ is admissible if and only if \n$$\n(p,k) \\not\\in \\{(2,k):\\ k \\mbox{ odd}\\} \\cup \\{(3,4),(3,10),(5,6)\\}.\n$$\n\\end{lemma}\n\\begin{proof}\nAssume that $(p,k)$ are chosen, not belonging to any of these four cases stated in the lemma. Let $\\ell$ be an integer, chosen as follows.\n\\begin{enumerate}\n\\item If $k \\not\\equiv 1 \\pmod{p}$, let $\\ell = 1$ (this case includes the case $p=0$).\n\\item If $k \\equiv 1 \\pmod{p}$ and $k$ is odd, let $\\ell = 2$.\n\\item If $k \\equiv 1 \\pmod{p}$ and $k$ is even, choose $\\ell$ according to the following grid. Observe that our assumptions imply, in this case, that $p$ is an odd prime.\n\\begin{center}\n{\n\\setlength\\extrarowheight{5pt}\n$\\begin{array}{|l |l l l |}\\hline\n& p=3 & p=5 & p > 5 \\\\\\hline\nk \\equiv 0 \\pmod{4} & \\ell = \\frac12 k - 3 & \\ell = \\frac12 k - 1 & \\ell = \\frac12 k - 1\\\\\nk \\equiv 2 \\pmod{4} & \\ell = \\frac12 k - 6 & \\ell = \\frac12 k - 4 & \\ell = \\frac12 k - 2\\\\\\hline\n\\end{array}$\n}\n\\end{center}\n\\end{enumerate}\nIn cases (1) and (2), it follows immediately that $(p,k,\\ell)$ is admissible (note in case (2) that our assumption has ruled out $p=2$). In case $3$, we must verify that $\\ell \\not\\equiv 0,1 \\pmod{p}$, $\\gcd(\\ell,k) = 1$, and $\\ell \\geq 1$. This is a routine verification in each of the six sub-cases.\n\nConversely, it is straightforward to check that no such $\\ell$ exists in the four cases mentioned in the statement.\n\\end{proof}\n\n\\begin{lemma} \\label{lem_lift}\nLet $K$ be a complete algebraically closed non-Archimedean field of characteristic $p$ (possibly $p=0$), with nontrivial valuation and residue field of the same characteristic. Let $\\ell$ be a positive integer such that $(p,k,\\ell)$ is admissible. There exists a smooth projective curve $C$ over $K$ with the following properties:\n\\begin{enumerate}\n\\item The genus of $C$ is $g$;\n\\item There is a degree $k$ map $C \\rightarrow \\textbf{P}^1$;\n\\item The minimal skeleton of $C$ is isometric to $\\Gamma_{g,k,\\ell}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWe apply the results of \\cite{abbr} on lifting harmonic morphisms of metrized complexes to morphisms of algebraic curves.\n\nLet $I$ denote a metric graph consisting of vertices $u_0,u_1,\\cdots,u_g$ joined (in that order) into a path, with all edge lengths equal to $\\ell(k-\\ell)$ (see Figure \\ref{fig_chain}).\n\nDefine a piecewise-linear map $\\pi: \\Gamma_{g,k,\\ell} \\rightarrow I$ by $\\pi(w_i) = u_i$, with $\\pi$ linear between these vertices. This map of metric graphs has integer slopes, with expansion factor $k-\\ell$ on the top (clockwise) edges $w_i w_{i+1}$, and expansion factor $\\ell$ on the bottom (counterclockwise) edges $w_i w_{i+1}$. Since $(p,k,\\ell)$ is admissible, all expansion factors are positive integers, none divisible by the characteristic of $K$, such that at any vertex of $\\Gamma_{g,k,\\ell}$, the expansion factors along any outward rays mapping to the leftward ray in $I$ add up to $k$ (and likewise for outward rays mapping to the rightward ray in $I$). Therefore $\\pi$ is a \\textit{finite harmonic} map of metric graphs, of degree $k$ \\cite[Definitions 2.4,\\ 2.6]{abbr}.\n\nThis harmonic map of metric graphs gives rise to harmonic morphism of \\textit{metrized complexes} \\cite[Definition 2.19]{abbr} as follows: to each vertex $v$ in either $\\Gamma_{g,k,\\ell}$ or $I$ we associate a curve $C_v$ isomorphic to $\\textbf{P}^1$ over the residue field of $K$; for each vertex $w_i$ of $\\Gamma_{g,k,\\ell}$ we associate a degree $k$ morphism $C_{w_i} \\rightarrow C_{u_i}$ defined by a rational function with divisor $(k-\\ell) \\cdot A + \\ell \\cdot B - (k-\\ell) \\cdot C - \\ell \\cdot D$, where $A,B,C,D$ denote any four rational points of $C_{w_i}$. We identify the two points $A,B$ of $C_{w_i}$ with the two tangent directions from $w_i$ that point to the left (if they exist), and we identify with $C,D$ the tangent directions that points to the right (if they exist). We identify the rational point $(0)$ of $C_{u_i}$ with the leftward tangent direction in $I$, and the rational point $(\\infty)$ with the rightward tangent direction.\n\nThe morphism $\\pi'$ is \\textit{tame} \\cite[Definition 2.21]{abbr}, since none of the expansion factors of $\\pi$ are divisible by the characteristic of the residue field of $K$. Therefore \\cite[Proposition 7.15]{abbr} implies the existence of a degree $k$ morphism $C \\rightarrow \\textbf{P}^1$ of smooth projective curves over $K$ specializing to the morphism $\\pi'$ of metrized complexes. In particular, the minimal skeleton of $C$ is isometric to $\\Gamma_{g,k,\\ell}$, and $C$ has a degree $k$ map to $\\textbf{P}^1$, as desired.\n\\end{proof}\n\nLemma \\ref{lem_lift} makes it possible to deduce results about algebraic curves from the tropical Brill-Noether theory of the graph $\\Gamma_{g,k,\\ell}$, i.e. of the loci $W^r_d(\\Gamma_{g,k,\\ell})$, which can be defined in terms of the Baker-Norine rank of divisors on metric graphs (see \\cite{gk} or \\cite{mz}). Our \\cite{pfl} develops the tools necessary for this analysis, applicable to any chain of cycles with any edge lengths. For convenience, we summarize the needed facts in the context necessary for this paper.\n\nThe structure of the tropical Brill-Noether loci $W^r_d(\\Gamma_{g,k,\\ell})$ depends on the \\textit{torsion profile} of the metric graph $\\Gamma_{g,k,\\ell}$, which is defined as the tuple $m_2, m_3, \\cdots, m_{g-1}$, where $m_i$ is the generator of the arithmetic progression $\\{m \\in \\textbf{Z}: m \\cdot w_{i-1} \\sim m \\cdot w_i \\}$ \\cite[Definition 1.9]{pfl} (in \\cite{pfl}, the torsion profile also includes a final number $m_g$, but this number is irrelevant in the context of this paper). For the metric graph $\\Gamma_{g,k,\\ell}$, where $\\ell$ is chosen relatively prime to $k$ (which is part of the definition of $(p,k,\\ell)$ being admissible), all of the numbers $m_i$ are equal to $k$. We therefore make the following definition, which is a simplified version of \\cite[Definition 2.1]{pfl} for the case where all of the $m_i$ equal the same number $k$.\n\n\\begin{defn} \\label{def_disp}\nLet $a,b$ be positive integers, and let $\\lambda$ be the set $\\{1,2,\\cdots,b\\} \\times \\{1,2,\\cdots,a\\}$. A \\emph{$k$-uniform displacement tableau} on $\\lambda$ is a function $t: \\lambda \\rightarrow \\textbf{Z}_{>0}$ subject to the following two conditions.\n\\begin{enumerate}\n\\item $t(x+1,y) > t(x,y)$ and $t(x,y+1) > t(x,y)$ whenever both sides of the inequality are defined.\n\\item If $t(x,y) = t(x',y')$, then $x-y \\equiv x'-y' \\pmod{k}$.\n\\end{enumerate}\nThe elements of $\\lambda$ will be called ``boxes'' and the integer $t(x,y)$ will be called the ``label'' of the box $(x,y)$.\n\\end{defn}\n\n\\begin{eg}\nThe following is a $3$-uniform displacement tableau, with $a = b = 3$. The labels $3$ and $5$ are repeated, which is permitted since the occurrences are in boxes of lattice distance $k=3$ from each other.\n$$\n\\young(567,345,123)\n$$\n\\end{eg}\n\n\\begin{lemma} \\label{l_td}\nFix integers $r,d$, and let \n$$\n\\lambda = \\{1,2,\\cdots,g-d+r\\} \\times \\{1,2,\\cdots,r+1\\}.$$\nSuppose that $\\ell$ is relatively prime to $k$. The locus $W^r_d(\\Gamma_{g,k,\\ell})$ is nonempty if and only if there exists a $k$-uniform displacement tableau on $\\lambda$ with at most $g$ distinct labels. If it is nonempty, then its dimension is equal to $g - | t(\\lambda) |$, where $t$ has the fewest possible number of distinct labels in such a tableau.\n\\end{lemma}\n\n\\begin{proof}\nIf such a tableau exists, then we may assume without loss of generality that the labels used are all chosen from $\\{1,2,\\cdots, g\\}$, by applying an order-preserving function to the labels used. The result is now \\cite[Corollary 3.7]{pfl}, applied to the case where $\\lambda$ is rectangular and all the torsion orders are equal to $k$.\n\\end{proof}\n\n\\begin{cor} \\label{cor_punchline}\nLet $C$ be a curve of the form guaranteed to exist by Lemma \\ref{lem_lift}. Then the dimension of $W^r_d(C)$ is bounded above by the maximum number of omitted labels from $\\{1,2,\\cdots,g\\}$ in a $k$-uniform displacement tableau on $\\{1,2,\\cdots,g-d+r\\} \\times \\{ 1,2,\\cdots, r+1 \\}$.\n\\end{cor}\n\n\\begin{proof}\nThis follows from \\cite[Proposition 5.1]{pfl} applied to the case of a rectangular partition, which states that the dimensions of the Brill-Noether loci of $C$ are bounded above by the dimensions of the Brill-Noether loci of $\\Gamma_{g,k,\\ell}$.\n\\end{proof}\n\nWe have therefore reduced our task to the analysis of $k$-uniform displacement tableaux, which we undertake in the following section.\n\n\\section{Uniform displacement tableaux} \\label{sec_disp}\n\nOur purpose in this section is to compute the minimum number of symbols needed to form a $k$-uniform displacement tableau on a rectangular partition. The answer is given by the following function.\n\n\\begin{defn} \\label{def_cd}\nLet $a,b,k$ be positive integers with $k \\geq 2$. Let $\\cd(a,b,k)$ denote the fewest number of distinct symbols used in a $k$-uniform displacement tableau on an $a \\times b$ rectangular partition. Define $\\delta(a,b,k)$ by\n$$\n\\delta(a,b,k) = \\min_{\\ell \\in \\{0,1,\\cdots,\\min(a,b)-1\\}} \\left( (a-\\ell)(b-\\ell) + k \\ell \\right).\n$$\n\\end{defn}\n\nThe name $\\cd$ is chosen since this quantity will be used to bound the codimensions of Brill-Noether loci. Indeed, note that by definition, $$\\overline{\\rho}_{g,k}(d,r) = g - \\delta(r+1,g-d+r,k).$$\n\nObserve that $\\cd(a,b,k) = \\cd(b,a,k)$ and $\\delta(a,b,k) = \\delta(b,a,k)$. Therefore it suffices to consider the case $a \\leq b$, which will simplify some statements.\n\n\\begin{rem} \\label{rem_threecases}\nThe function being minimized in the definition of $\\delta(a,b,k)$ is a quadratic function in $\\ell$, which achieves its minimum (for real values of $\\ell$) at $\\ell = \\frac12 (a+b-k)$. If $\\ell$ is constrained to integer values, the minimum is attained by both the floor or the ceiling of $\\frac12 (a+b-k)$. An elementary calculation shows that if $a \\leq b$, then\n$$\n\\delta(a,b,k) = \\begin{cases}\n(k-1)(a-1) + b & \\mbox{if $k \\leq b-a+3$},\\\\\nab - \\left\\lfloor \\left( \\frac{a+b-k}{2} \\right)^2 \\right\\rfloor & \\mbox{if $b-a+1 \\leq k \\leq a+b+1$},\\\\\nab & \\mbox{if $k \\geq a+b-1.$}\n\\end{cases}\n$$\nThe overlap of the hypotheses for these cases reflects the fact that there are two minimizing values of $\\ell$ when $a+b-k$ is odd. If $a \\geq b$, we obtain similar formulas by symmetry.\n\\end{rem}\n\n\\begin{prop} \\label{p_cd}\nFor any positive integers $a,b,k$ with $k \\geq 2$, $$\\cd(a,b,k) = \\delta(a,b,k).$$\n\\end{prop}\n\nWe prove Proposition \\ref{p_cd} by proving inequalities in both directions. In fact, only the first inequality is needed for this paper's main results, but we present the second inequality to emphasize that our result is the strongest possible using this method.\n\n\\begin{lemma} \\label{l_cdlower}\nFor any positive integers $a,b,k$ with $k \\geq 2$, $$\\cd(a,b,k) \\geq \\delta(a,b,k).$$\n\\end{lemma}\n\n\\begin{proof} \nWe assume without loss of generality that $a \\leq b$.\n\nLet $\\lambda$ denote the rectangular partition $\\{1,2,\\cdots,b\\} \\times \\{1,2,\\cdots,a\\}$, and let $t: \\lambda \\rightarrow \\textbf{Z}_{>0}$ be any $k$-uniform displacement tableau on $\\lambda$. \nWe consider three cases separately (corresponding to the three cases of Remark \\ref{rem_threecases}). In each case, we will give a subset $S$ of the boxes of $\\lambda$, with $\\delta(a,b,k)$ elements, such that each box must have a different value in $t$. This will imply the statement of the lemma.\n\nThroughout, we will say that a box $(x',y')$ \\emph{dominates} another box $(x,y)$ if $x' \\geq x$ and $y' \\geq y$. If so, then $(x',y')$ and $(x,y)$ cannot have the same label.\n\n\n\\textit{Case 1:} $k \\geq a+b-1$. For any box $(x,y) \\in \\lambda$, $1-a \\leq x-y \\leq b-1$. Therefore, if $(x,y),(x',y')$ are two boxes with $x-y \\equiv x'-y' \\pmod{k}$, then $x-y = x'-y'$ and one box dominates the other. Therefore we take $S$ to be all of $\\lambda$; all boxes must have distinct labels, so $\\cd(a,b,k) = \\delta(a,b,k) = ab$.\n\n\\textit{Case 2:} $k \\leq b-a+2$. Define $S \\subseteq \\lambda$ to be the disjoint union $S_1 \\cup S_2$, where\n\\begin{eqnarray*}\nS_1 &=& \\left\\{ (x,y):\\ 1 \\leq y \\leq a-1 \\textrm{ and } y \\leq x \\leq y+k-1 \\right\\} \\\\\nS_2 &=&  \\left\\{ (x,a):\\ a \\leq x \\leq b \\right\\}.\n\\end{eqnarray*}\nAn example is shown in Figure \\ref{fig_s}. Any box $(x,y)$ in $S_1$ satisfies $0 \\leq x-y \\leq k-1$, hence no two boxes in $S_1$ can have the same label, by the same reasoning as in case $1$. Any two boxes in $S_2$ must have different labels since one dominates the other. Finally, if $(x,y) \\in S_1$ and $(x',a) \\in S_2$ satisfy $x-y \\equiv x'-a \\pmod{k}$, then $(x',a)$ either dominates or is equal to $(x-y+a, a)$ (since $x' \\geq a$, $x' \\equiv x-y + a \\pmod{k}$, and $0 \\leq x-y \\leq k-1$), which dominates $(x,y)$. Therefore no box of $S_1$ can have the same label as a box of $S_2$. It follows that the boxes of $S$ all have distinct labels in $t$. The size of $S$ is $k(a-1) + b-a+1 = (k-1)(a-1) + b = \\delta(a,b,k)$, so $\\cd(a,b,k) \\geq \\delta(a,b,k)$.\n\n\\begin{figure}\n\\begin{tabular}{cc}\n\\begin{tikzpicture}[scale=0.5]\n\\draw (0,0) rectangle (8,3);\n\\draw[fill=lightgray] (0,0) -- (4,0) -- (4,1) -- (5,1) -- (5,2) -- (8,2) -- (8,3) -- (2,3) -- (2,2) -- (1,2) -- (1,1) -- (0,1) -- cycle;\n\\foreach \\x in {1,2,...,7} {\\draw (\\x,0) -- (\\x,3);}\n\\foreach \\y in {1,2} {\\draw (0,\\y) -- (8,\\y);}\n\\end{tikzpicture}\n&\n\\begin{tikzpicture}[scale=0.5]\n\\draw (0,0) rectangle (6,5);\n\\draw[fill=lightgray] (0,0) -- (3,0) -- (3,1) -- (4,1) -- (4,2) -- (5,2) -- (5,3) -- (6,3) -- (6,5) -- (3,5) -- (3,4) -- (2,4) -- (2,3) -- (1,3) -- (1,2) -- (0,2) -- cycle;\n\\foreach \\x in {1,2,...,5} {\\draw (\\x,0) -- (\\x,5);}\n\\foreach \\y in {1,2,3,4} {\\draw (0,\\y) -- (6,\\y);}\n\\end{tikzpicture}\n\\\\\n$a=3,\\ b=8,\\ k=4$ (case 2)\n& \n$a=5,\\ b=6,\\ k=4$ (case 3)\n\\end{tabular}\n\\caption{Examples of the regions $S$ is cases $2$ and $3$ of the proof of Lemma \\ref{l_cdlower}.} \\label{fig_s}\n\\end{figure}\n\n\\textit{Case 3:} $b-a+3 \\leq k \\leq a+b-2$. Define\n$$\nS = \\left\\{ (x,y) \\in \\lambda:\\ - \\left\\lceil \\frac{k-1-(b-a)}{2} \\right\\rceil \\leq x-y \\leq \\left\\lfloor \\frac{k-1+(b-a)}{2} \\right\\rfloor \\right\\}.\n$$\n\nSince $(b-a) \\leq k-3$, the lower bound on $x-y$ in this definition is indeed negative. An example is shown in Figure \\ref{fig_s}.\n\nObserve that the difference between the upper bound and the lower bound for $x-y$ in the definition of $S$ is exactly $k-1$. Therefore any two boxes in $S$ with values of $x-y$ that are congruent modulo $k$ must in fact have equal values of $x-y$, hence one dominates the other. It follows that $\\cd(a,b,k) \\geq |S|$. It remains to calculate the size of the set $S$.\n\nThe complement of $S$ can be written as the union of two disjoint parts.\n\\begin{eqnarray*}\nT_1 &=& \\left\\{ (x,y) \\in \\lambda:\\ y \\geq x + 1 + \\left\\lceil \\frac{k-1-(b-a)}{2} \\right\\rceil \\right\\}\\\\\nT_2 &=& \\left\\{ (x,y) \\in \\lambda:\\ x \\geq y + 1 + \\left\\lfloor \\frac{k-1+(b-a)}{2} \\right\\rfloor \\right\\}\n\\end{eqnarray*}\n\nBoth $T_1$ and $T_2$ are triangular regions in the upper left and lower right corners of $\\lambda$. By determining the side lengths of these two regions, it follows that\n\n\\begin{eqnarray*}\n|T_1| &=& \\binom{ \\lfloor (a+b-k+1)\/2 \\rfloor }{2},\\\\\n|T_2| &=& \\binom{ \\lceil (a+b -k+1)\/2 \\rceil }{2}.\n\\end{eqnarray*}\n\nLet $z = \\frac{a+b-k+1}{2}$ and $\\varepsilon = z - \\lfloor z \\rfloor$. Then\n\n\\begin{eqnarray*}\n|T_1| + |T_2| &=& \\frac12 (z - \\varepsilon)(z - 1 - \\varepsilon) + \\frac12 (z + \\varepsilon)(z-1+\\varepsilon)\\\\\n&=& z(z-1) + \\varepsilon^2\\\\\n&=& \\left(z-\\frac12\\right)^2 + \\varepsilon^2 - \\frac14\\\\\n&=& \\left \\lfloor \\left( \\frac{a+b-k}{2} \\right)^2 \\right\\rfloor\n\\end{eqnarray*}\n\nTherefore $|S| = ab - \\left \\lfloor \\left( \\frac{a+b-k}{2} \\right)^2 \\right\\rfloor = \\delta(a,b,k)$ (by the formula in Remark \\ref{rem_threecases}). Since all the boxes of $S$ must have distinct labels, it follows that $\\cd(a,b,k) \\geq \\delta(a,b,k)$.\n\\end{proof}\n\n\\begin{lemma} \\label{l_cdupper}\nFor any positive integers $a,b,k$ with $k \\geq 2$,\n$$\n\\cd(a,b,k) \\leq \\delta(a,b,k).\n$$\n\\end{lemma}\n\\begin{proof}\nWe assume without loss of generality that $a \\leq b$. The case $a=1$ can be checked directly (both $\\cd(1,b,k)$ and $\\delta(1,b,k)$ are equal to $b$), so we also assume that $a \\geq 2$.\n\nWe give an explicit construction for $k$-uniform displacement tableaux with exactly $\\delta(a,b,k)$ symbols.\n\n\\textit{Case 1:} $k \\geq a+b-1$. In this case, $\\delta(a,b,k) = ab$, so we can take $t$ to be any tableau with all distinct symbols.\n\n\\textit{Case 2:} $k \\leq a+b-2$. Define $\\ell = \\min \\left( a-1,\\ \\left\\lceil \\frac{a+b-k}{2} \\right\\rceil \\right)$; this value is chosen so that $\\delta(a,b,k) = (a-\\ell)(b-\\ell) + k \\ell$. Note that our assumptions imply that $\\ell \\geq 1$. We will construct a tableau $t$ on $\\lambda$ with exactly $(a-\\ell)(b-\\ell) + k \\ell$ distinct symbols, which will establish that $\\cd(a,b,k) \\leq \\delta(a,b,k)$.\n\n\\textit{Definition of $t$}. Informally, $t$ will be defined by first dividing the partition $\\lambda$ into strips of width $1$ and height $a-\\ell$ (with the top row of strips truncated to a smaller height), and filling each strip with consecutive numbers from bottom to top. In each row of strips, we fill the strips in order, from left to right. In order to minimize the distinct symbols used, we begin filling each row of strips before the previous row is complete: the first strip of one row is filled simultaneously with the $(k+\\ell-a+1)$th strip of the row below.\n\nFormally, define first a function $t: \\lambda \\rightarrow \\textbf{Z}_{>0}$ as follows.\n\n\\begin{eqnarray*}\nt(x,y) &=& (a-\\ell) \\cdot \\left[ (x-1) + q (k+\\ell-a) \\right] + r,\\\\\n\\mbox{where } q &=& \\left\\lfloor \\frac{y-1}{a-\\ell} \\right\\rfloor\\\\\n\\mbox{and } r &=& y - q \\cdot (a-\\ell).\n\\end{eqnarray*}\n\nAn example is shown in Figure \\ref{fig_construction}.\n\n\\begin{figure}\n\\begin{tikzpicture}[scale=0.7]\n\\draw (0,0) -- (7,0) -- (7,7) -- (0,7) -- (0,0);\n\\draw (1,0) -- (1,7);\n\\draw (2,0) -- (2,7);\n\\draw (3,0) -- (3,7);\n\\draw (4,0) -- (4,7);\n\\draw (5,0) -- (5,7);\n\\draw (6,0) -- (6,7);\n\\draw (0,3) -- (7,3);\n\\draw (0,6) -- (7,6);\n\\draw[fill=lightgray] (0,6) rectangle (1,7);\n\\draw[fill=lightgray] (3,3) rectangle (4,6);\n\\draw[fill=lightgray] (6,0) rectangle (7,3);\n\\draw (0.500000,0.500000) node {$1$};\n\\draw (0.500000,1.500000) node {$2$};\n\\draw (0.500000,2.500000) node {$3$};\n\\draw (0.500000,3.500000) node {$10$};\n\\draw (0.500000,4.500000) node {$11$};\n\\draw (0.500000,5.500000) node {$12$};\n\\draw (0.500000,6.500000) node {$19$};\n\\draw (1.500000,0.500000) node {$4$};\n\\draw (1.500000,1.500000) node {$5$};\n\\draw (1.500000,2.500000) node {$6$};\n\\draw (1.500000,3.500000) node {$13$};\n\\draw (1.500000,4.500000) node {$14$};\n\\draw (1.500000,5.500000) node {$15$};\n\\draw (1.500000,6.500000) node {$22$};\n\\draw (2.500000,0.500000) node {$7$};\n\\draw (2.500000,1.500000) node {$8$};\n\\draw (2.500000,2.500000) node {$9$};\n\\draw (2.500000,3.500000) node {$16$};\n\\draw (2.500000,4.500000) node {$17$};\n\\draw (2.500000,5.500000) node {$18$};\n\\draw (2.500000,6.500000) node {$25$};\n\\draw (3.500000,0.500000) node {$10$};\n\\draw (3.500000,1.500000) node {$11$};\n\\draw (3.500000,2.500000) node {$12$};\n\\draw (3.500000,3.500000) node {$19$};\n\\draw (3.500000,4.500000) node {$20$};\n\\draw (3.500000,5.500000) node {$21$};\n\\draw (3.500000,6.500000) node {$28$};\n\\draw (4.500000,0.500000) node {$13$};\n\\draw (4.500000,1.500000) node {$14$};\n\\draw (4.500000,2.500000) node {$15$};\n\\draw (4.500000,3.500000) node {$22$};\n\\draw (4.500000,4.500000) node {$23$};\n\\draw (4.500000,5.500000) node {$24$};\n\\draw (4.500000,6.500000) node {$31$};\n\\draw (5.500000,0.500000) node {$16$};\n\\draw (5.500000,1.500000) node {$17$};\n\\draw (5.500000,2.500000) node {$18$};\n\\draw (5.500000,3.500000) node {$25$};\n\\draw (5.500000,4.500000) node {$26$};\n\\draw (5.500000,5.500000) node {$27$};\n\\draw (5.500000,6.500000) node {$34$};\n\\draw (6.500000,0.500000) node {$19$};\n\\draw (6.500000,1.500000) node {$20$};\n\\draw (6.500000,2.500000) node {$21$};\n\\draw (6.500000,3.500000) node {$28$};\n\\draw (6.500000,4.500000) node {$29$};\n\\draw (6.500000,5.500000) node {$30$};\n\\draw (6.500000,6.500000) node {$37$};\n\\end{tikzpicture}\n\\caption{The construction in the proof of lemma \\ref{l_cdupper}, in the case $a=7,\\ b=7,\\ k=6$. The tableau is constructed by filling $3 \\times 1$ strips in sequence. Although the labels of the tableau go up to $37$, there are four values that do not occur ($32,33,35,36$), so there are exactly $33$ distinct labels.} \\label{fig_construction}\n\\end{figure}\n\n\\textit{Validity of $t$}. We first verify that this function does in fact give a $k$-uniform displacement tableau. Clearly $t(x+1,y) > t(x,y)$. The value $t(x,y+1)-t(x,y)$ is either $1$ (if $y \\not \\equiv 0 \\pmod{a-\\ell}$), or $(a-\\ell)(k+\\ell-a) - (a-\\ell-1)$ otherwise. The latter quantity is equal to $(a - \\ell)( k + \\ell - a -1) + 1$, which is positive since $k + \\ell -a - 1 \\geq 0$ and $\\ell \\leq a-1$ (these follow from our choice of the number $\\ell$). Hence $t(x,y)$ is strictly increasing in rows and columns; it remains to show that any two boxes $(x,y)$ with the same label in $t$ have values of $x-y$ that are congruent modulo $k$.\n\nSince $1 \\leq r \\leq a - \\ell$, two boxes satisfy $t(x,y) = t(x',y')$ if and only if $r = r'$ and $(x-1) + q(k+\\ell-a) = (x'-1) + q'(k+\\ell-a)$ (where $q',r'$ are obtained from $y'$ in the same manner as $q,r$ are obtained from $y$). The latter equation implies that \n$$\nx - q (a - \\ell) \\equiv x' - q'(a-\\ell) \\pmod{k},\n$$\nand subtracting $r = r'$ from both sides gives $x-y \\equiv x' - y' \\pmod{k}$. So $t$ is indeed a $k$-uniform displacement tableau.\n\n\\textit{Number of symbols in $t$.} We now enumerate the number of distinct symbols that occur in $t$ on $\\lambda$. The primary difficulty in this computation is that, as we have constructed it, the symbols of $t$ are not consecutive; some symbols are skipped.\n\nEvery value $t(x,y)$ for $(x,y) \\in \\lambda$ has the form $(a - \\ell) B + r$, where $1 \\leq r \\leq a - \\ell$; clearly two such values are equal if and only if they arise from equal values of $B$ and $r$, so it suffices to enumerate the number of distinct pairs $(B,r)$ which occur.\n\nThe number $B$ is equal to $(x-1) + q(k + \\ell - a)$. The number $q$ depends on $y$ alone; its possible values are the integers $0$ through $\\left\\lfloor \\frac{a-1}{a - \\ell} \\right\\rfloor$, inclusive. Note that the largest possible value of $q$ is at least $1$, since our assumptions imply that $\\ell \\geq 1$. For each possible value of $q$ in $\\{0,1,\\cdots, \\left\\lfloor \\frac{a-1}{a - \\ell} \\right\\rfloor - 1\\}$, the possible values of $y$ giving this value of $q$ give all possible values of $r$ (from $1$ to $a - \\ell$). Since $x-1$ takes all values from $0$ to $b-1$ inclusive, this shows that all pairs $(B,r)$ satisfying the following inequalities will occur.\n\n\\begin{eqnarray*}\n0 &\\leq B \\leq& b-1 + \\left( \\left\\lfloor \\frac{a-1}{a - \\ell} \\right\\rfloor - 1 \\right) (k + \\ell -a)\\\\\n1 &\\leq r \\leq& a - \\ell\n\\end{eqnarray*}\n\nThis accounts for \n\\begin{equation} \\label{monster1}\n(a - \\ell) \\left( b + \\left( \\left\\lfloor \\frac{a-1}{a - \\ell} \\right\\rfloor - 1 \\right) (k + \\ell -a) \\right) \n\\end{equation}\ndistinct values of $t(x,y)$.\n\nAll of the remaining values of $B$ must occur for the largest possible value of $q$, namely $q = \\left\\lfloor \\frac{a-1}{a-\\ell} \\right\\rfloor$. For this value of $q$, the possible values of $r$ range from $1$ to $a - \\left\\lfloor \\frac{a-1}{a-\\ell} \\right\\rfloor (a - \\ell)$, inclusive. For this value of $q$, the value of $B$ is distinct from those listed above if and only if $x$ is taken to be one of the $(k+\\ell-a)$ largest values from the range $\\{1,2,\\cdots,b\\}$ (here we use the fact that $k+\\ell-a \\leq b$, which follows from the assumption $a+b-k \\geq 2$ and the choice of $\\ell$). Therefore the number of values of $t(x,y)$ not accounted for is precisely\n\n\\begin{equation} \\label{monster2}\n\\left( a - \\left\\lfloor \\frac{a-1}{a-\\ell} \\right\\rfloor (a - \\ell) \\right) (k + \\ell - a).\n\\end{equation}\n\nAdding the quantities (\\ref{monster1}) and (\\ref{monster2}) and rearranging terms gives the quantity $(a-\\ell)(b-\\ell) + k \\ell$, which is therefore the number of distinct symbols in $t$. It follows that $\\cd(a,b,k) \\leq (a-\\ell)(b-\\ell) + k \\ell$, as desired.\n\\end{proof}\n\nLemmas \\ref{l_cdlower} and \\ref{l_cdupper} together prove Proposition \\ref{p_cd}.\n\n\\begin{cor} \\label{cor_tropdim}\nIf $k$ and $\\ell$ are relatively prime, then the dimension of $W^r_d(\\Gamma_{g,k,\\ell})$ is equal to $\\overline{\\rho}_{g,k}(d,r)$.\n\\end{cor}\n\\begin{proof}\nBy Riemann-Roch, we may assume without loss of generality that $d \\leq g-1$. Then $\\dim W^r_d(\\Gamma_{g,k,\\ell}) = g - \\cd(r+1,g-d+r,k)$ by Lemma \\ref{l_td}, which is $g- \\delta(r+1,g-d+r,k)$ by Proposition \\ref{p_cd}. By definition, this is equal to $\\overline{\\rho}_{g,k}(d,r)$.\n\\end{proof}\n\n\\section{Proofs of main theorems} \\label{sec_proofs}\n\nWe can now combine our results from the previous two sections to prove the theorems from the introduction. We assume that Situation \\ref{sit_main} holds throughout, i.e. integers $g,k,r,d$, a field $K$, and a general $k$-gonal curve $C$ are fixed, with the characteristic of $K$ meeting our mild restrictions. Before each proof we briefly recall the theorem statement.\n\nOur main result is Theorem \\ref{thm_main}, which asserts that in Situation \\ref{sit_main}, $\\overline{\\rho}_{g,k}(d,r)$ is an upper bound for $\\dim W^r_d(C)$.\n\n\\begin{proof}[Proof of theorem \\ref{thm_main}]\nWe first prove that the result holds when $K$ is a complete algebraically closed non-Archimedean field with nontrivial valuation. By Lemma \\ref{lem_lexists}, there exists an integer $\\ell$ such that $(\\chr K, k, \\ell)$ is admissible. By Lemma \\ref{lem_lift} and Corollary \\ref{cor_punchline}, there exists a smooth projective curve $C'$ over $K$ such that for all $r,d$,\n$$\n\\dim W^r_d(C') \\leq g - \\cd(g-d+r,r+1,k),\n$$\nwhere $\\cd$ is the function from Definition \\ref{def_cd}. Proposition \\ref{p_cd}, together with the fact that $\\overline{\\rho}_{g,k}(d,r) = g - \\delta(r+1,g-d+r,k)$, gives the inequality $\\dim W^r_d(C') \\leq \\overline{\\rho}_{g,k}(d,r)$.\n\nIn particular, letting $r=1$, these bounds show that $W^1_d(C')$ is empty for all $d < k$ (the inequalities $d \\leq k-1$ and $k \\leq \\frac{g+3}{2}$ imply that $\\overline{\\rho}_{g,k}(d,1) = d-k \\leq -1$). So the gonality of $C'$ is at least $k$. By construction, $C'$ does have a degree $k$ line bundle with two linearly independent sections, so the gonality of $C'$ is exactly $k$. By the irreducibility of the $k$-gonal locus and upper semicontinuity, it follows that these upper bounds on dimension hold on a dense open subset of the set of $k$-gonal curves over the field $K$. Hence $\\dim W^r_d(C) \\leq \\overline{\\rho}_{g,k}(d,r)$ for a \\textit{general} $k$-gonal curve $C$ of genus $g$ over $K$.\n\n\\textit{Extension to arbitrary algebraically closed fields.} So far we have deduced the theorem only for a specific type of field. To obtain the result for all fields, observe that the coarse moduli space of curves of genus $g$ is a scheme of finite type over $\\Spec \\textbf{Z}$, and the locus of $k$-gonal curves for which the theorem holds is a locally closed subscheme whose image in $\\Spec \\textbf{Z}$ includes the generic point and $\\Spec \\textbf{F}_p$ for all $p$ not excluded by Situation \\ref{sit_main}. Therefore it has points in every algebraically closed field of characteristic not excluded by Situation \\ref{sit_main}.\n\\end{proof}\n\nTheorem \\ref{thm_cm} states that, under a characteristic $0$ hypothesis, $\\underline{\\rho}_{g,k}(d,r)$ is a lower bound for $\\dim W^r_d(C)$; it follows from the main theorem of \\cite{cm02} by an elementary argument. We give this elementary argument below for completeness. First, we restate the main theorem of \\cite{cm02} in the notation of this paper.\\\\\n\n\\noindent\\textbf{Main theorem of \\cite{cm02}}. \\textit{\nLet $C$ be a general $k$-gonal curve of genus $g$ over the complex numbers, and let $r,d$ be positive integers. Let $\\ell \\in \\{0,1,\\cdots,r\\}$ be an integer satisfying the following hypotheses.\n\\begin{enumerate}\n\\item $\\ell \\geq r- k$,\n\\item $r+1 - \\ell$ divides either $r$ or $r+1$, and\n\\item $\\rho_g(d,r-\\ell) - \\ell k \\geq \\max(0,\\rho_g(d,r))$.\n\\end{enumerate}\nThen $W^r_d(C)$ has an irreducible component of dimension $\\rho_g(d,r-\\ell) - \\ell k$.\n}\n\n\\begin{proof}[Proof of Theorem \\ref{thm_cm}]\nNote first of all that to prove this results for all algebraically closed fields of characteristic $0$, it suffices to verify the result for the field $K = \\textbf{C}$ of complex numbers, by standard arguments (e.g. as in the last paragraph of the proof of Theorem \\ref{thm_main}). We use the notation of Situation \\ref{sit_main}. By Riemann-Roch, we may assume without loss of generality that $d \\leq g-1$, so that $r' = r$ in the notation of Definition \\ref{def_ovr}. Assume also that $\\underline{\\rho}_{g,k}(d,r) \\geq 0$ and $\\underline{\\rho}_{g,k}(d,r) > \\rho_g(d,r)$, since otherwise the desired result is either vacuous or follows from $\\dim W^r_d(C) \\geq \\rho_g(d,r)$. Note that this implies $r > 0$.\n\nLet $\\ell$ be whichever integer $\\ell \\in \\{0,1,r-1,r\\}$ is closest to $\\ell_0 = \\frac{g-d+2r - k + 1}{2}$, where we choose the largest value in case of a tie. In this case $\\rho_g(d,r-\\ell) - \\ell k = \\underline{\\rho}_{g,k}(d,r)$. So it suffices to verify hypotheses (1), (2), and (3) in the theorem statement above.\n\nHypothesis (3) follows from our assumptions. Hypothesis (2) holds for all $\\ell \\in \\{0,1,r-1,r\\}$ for elementary reasons.  It remains to verify hypothesis (1). If $\\ell \\in \\{r-1,r\\}$ then hypothesis (1) follows from $k \\geq 2$. We are assuming that $\\underline{\\rho}_{g,k}(d,r) > \\rho_g(d,r)$, hence $\\ell \\neq 0$; it remains to consider the case $\\ell = 1$. We may assume that $r \\geq 3$, otherwise either $r=0$, $\\ell = 1 = r-1$ or $\\ell = 1 = r$. Since $\\ell = 1$ must be closer to $\\ell_0$ than $r-1$ (which is strictly greater than $1$), it follows that $\\ell_0 < \\frac12 r$. From the definition of $\\ell_0$, this implies that $r < k + d - g - 1$. We are assuming that $d < g$, hence $r < k - 1$. So certainly $1 \\geq r - k$ in this case, and hypothesis (1) holds.\n\nTherefore all three hypotheses hold, and the main theorem of \\cite{cm02} implies that $W^r_d(C)$ has a component of dimension $\\underline{\\rho}_{g,k}(d,r)$.\n\\end{proof}\n\nTheorem \\ref{thm_eq} asserts that, in characteristic $0$, when $k \\leq 5$ or $k \\geq \\frac15g+2$, the dimension of $W^r_d(C)$ is exactly $\\overline{\\rho}_{g,k}(d,r)$ (unless this number is negative, in which case $W^r_d(C)$ is empty). The proof is elementary, given Theorems \\ref{thm_main} and \\ref{thm_cm}.\n\n\\begin{proof}[Proof of Theorem \\ref{thm_eq}]\nWe must show that if $k \\leq 5$ or $k \\geq \\frac15 g + 2$, then either $\\underline{\\rho}_{g,k}(d,r) = \\overline{\\rho}_{g,k}(d,r)$ or $\\overline{\\rho}_{g,k}(d,r) < 0$. We assume without loss of generality that $d \\leq g-1$.\n\nSuppose that $\\underline{\\rho}_{g,k}(d,r) \\neq \\overline{\\rho}_{g,k}(d,r)$. By the discussion in \\ref{ss_gap},\n$$\n2 \\leq \\frac{g-d+2r-k+1}{2} \\leq r-2.\n$$\n\n\nDefine two auxiliary variables, as follows.\n\\begin{eqnarray*}\n\\ell_0 &=& \\frac{g-d+2r-k+1}{2}\\\\\n\\delta &=& \\frac{g-d-1}{2}\n\\end{eqnarray*}\n\nWe deduce from the inequalities above that $\\ell_0 \\geq 2$ and $\\delta \\leq \\frac{k}{2} - 3$.  An elementary (but tedious) rearrangement of terms shows that, for any integer $\\ell$,\n$$\n\\rho_g(d,r-\\ell) -k \\ell = g - (\\ell - \\ell_0)^2 - \\left( \\frac{k}{2} \\right)^2 - k \\ell_0 + \\delta^2.\n$$\nThe maximum value occurs at $\\ell = \\lfloor \\ell_0 \\rfloor$, hence\n\\begin{eqnarray*}\n\\overline{\\rho}_{g,k}(d,r) &=& \\rho_g(d,r-\\lfloor \\ell_0 \\rfloor) - k \\lfloor \\ell_0 \\rfloor\\\\\n&\\leq& g - (\\lfloor \\ell_0 \\rfloor - \\ell_0)^2 - \\left( \\frac{k}{2} \\right)^2 - k \\ell_0 + \\delta^2\\\\\n&\\leq& g - 0 - \\left( \\frac{k}{2} \\right)^2 - k \\cdot 2 + \\left( \\frac{k}{2} - 3 \\right)^2\\\\\n&\\leq& g - 5k + 9.\n\\end{eqnarray*}\nIn the case $k > \\frac15 g + 2$, it follows that $\\overline{\\rho}_{g,k}(d,r) \\leq -1$. In the case $k \\leq 5$, the inequality $\\delta \\leq \\frac{k}{2} -3$ would imply that $\\delta < 0$, which contradicts the assumption that $d \\leq g-1$. Therefore we conclude that in either case, $\\underline{\\rho}_{g,k}(d,r) = \\overline{\\rho}_{g,k}(d,r)$ unless $\\overline{\\rho}_{g,k}(d,r) < 0$. The result now follows from Theorems \\ref{thm_main} and \\ref{thm_cm}.\n\\end{proof}\n\nTheorem \\ref{thm_howgeneral} characterizes, in characteristic $0$, those values of $d,r$ for which a general $k$-gonal curve of genus $g$ satisfies $\\dim W^r_d(C) = \\rho_g(d,r)$ (i.e. the same as for a general curve of genus $g$). This occurs if and only if $r=0,\\ g-d+r = 1,$ or $g-k \\leq d-2r$. \n\n\\begin{proof}[Proof of Theorem \\ref{thm_howgeneral}]\nThe $r=0$ case (and dually, the $g-d+r = 1$ case) follow from the fact that the $d$th symmetric product of $C$ surjects onto $W^0_d(C)$, so assume that $r \\geq 1$ and $g-d+r \\geq 2$. Let $\\ell_0$ be as in Section \\ref{ss_gap}. Then $\\ell_0 \\leq \\frac12$ is equivalent to $g-k \\leq d-2r$; the discussion in Section \\ref{ss_gap} shows that in this case $\\dim W^r_d(C) = \\rho_g(d,r)$. For the converse, suppose that $g-k > d - 2r$; equivalently, $\\ell_0 \\geq 1$. Then $\\rho_g(d,r-1) - k > \\rho_g(d,r)$, hence $\\underline{\\rho}_{g,k}(d,r) > \\rho_g(d,r)$. By Theorem \\ref{thm_cm}, $\\dim W^r_d(C) > \\rho_g(d,r)$, as desired.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}