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url stringlengths 31 38	title stringlengths 7 229	abstract stringlengths 44 2.87k	text stringlengths 319 2.51M	meta dict
https://arxiv.org/abs/1607.04115	The problème des ménages revisited	We present an alternative proof to the Touchard-Kaplansky formula for the problème des ménages, which, we believe, is simpler than the extant ones and is in the spirit of the elegant original proof by Kaplansky (1943). About the latter proof, Bogart and Doyle (1986) argued that despite its cleverness, suffered from opt...	\section{Introduction} \label{intro:sec} \bigskip\noindent The {\em probl\`{e}me des m\'{e}nages} (the couples problem) asks to count the number of ways that $n$ man-woman couples can sit around a circular table so that no one sits next to her partner or someone of the same gender. The problem was first stated i...	{ "timestamp": "2016-07-26T02:09:29", "yymm": "1607", "arxiv_id": "1607.04115", "language": "en", "url": "https://arxiv.org/abs/1607.04115", "abstract": "We present an alternative proof to the Touchard-Kaplansky formula for the problème des ménages, which, we believe, is simpler than the extant ones and is ...
https://arxiv.org/abs/1112.4327	On the optimal convergence rate of a Robin-Robin domain decomposition method	In this work, we solve a long-standing open problem: Is it true that the convergence rate of the Lions' Robin-Robin nonoverlapping domain decomposition(DD) method can be constant, independent of the mesh size $h$? We closed this twenty-year old problem with a positive answer. Our theory is also verified by numerical te...	\section{Introduction} \def\frac{\partial u}{\partial \b n }{\frac{\partial u}{\partial \b n }} \def\frac{\partial w}{\partial \b n }{\frac{\partial w}{\partial \b n }} Domain decomposition (DD) methods are important tools for solving partial differential equations, especially by parallel computers. In this p...	{ "timestamp": "2012-07-11T02:04:18", "yymm": "1112", "arxiv_id": "1112.4327", "language": "en", "url": "https://arxiv.org/abs/1112.4327", "abstract": "In this work, we solve a long-standing open problem: Is it true that the convergence rate of the Lions' Robin-Robin nonoverlapping domain decomposition(DD) ...
https://arxiv.org/abs/2105.14368	Fit without fear: remarkable mathematical phenomena of deep learning through the prism of interpolation	In the past decade the mathematical theory of machine learning has lagged far behind the triumphs of deep neural networks on practical challenges. However, the gap between theory and practice is gradually starting to close. In this paper I will attempt to assemble some pieces of the remarkable and still incomplete math...	\section{Preface} In recent years we have witnessed triumphs of Machine Learning in practical challenges from machine translation to playing chess to protein folding. These successes rely on advances in designing and training complex neural network architectures and on availability of extensive datasets. Yet, whi...	{ "timestamp": "2021-06-01T02:15:44", "yymm": "2105", "arxiv_id": "2105.14368", "language": "en", "url": "https://arxiv.org/abs/2105.14368", "abstract": "In the past decade the mathematical theory of machine learning has lagged far behind the triumphs of deep neural networks on practical challenges. However...
https://arxiv.org/abs/2012.15517	Minimization of the sum under product constraints	We systematically explore a class of constrained optimization problems with linear objective function and constraints that are linear combinations of logarithms of the optimization variables. Such problems can be viewed as a generalization of the inequality between the arithmetic and geometric means. The existence and ...	\section{Introduction} \label{sec:intro} \addtocounter{subsection}{10} \newcounter{ssecaux} \setcounter{ssecaux}{0} \renewcommand{\thesubsection}{\arabic{ssecaux}} \refstepcounter{ssecaux} \subsection{The class of optimization problems} This work is concerned with optimization of a linear objective function \beq{su...	{ "timestamp": "2021-01-01T02:33:40", "yymm": "2012", "arxiv_id": "2012.15517", "language": "en", "url": "https://arxiv.org/abs/2012.15517", "abstract": "We systematically explore a class of constrained optimization problems with linear objective function and constraints that are linear combinations of loga...
https://arxiv.org/abs/2106.03576	A Generalised Continuous Primitive Integral and Some of Its Applications	Using the Laplace derivative a Perron type integral, the Laplace integral, is defined. Moreover, it is shown that this integral includes Perron integral and to show that the inclusion is proper, an example of a function is constructed, which is Laplace integrable but not Perron integrable. Properties of integrals such ...	\section{Introduction} \par In integral calculus, there are two fundamental concepts of integration, one is the Riemann integral, and the other is Newton integral, which are not comparable. Even the Lebesgue integral does not contain the Newton integral. This phenomenon leads to the problem of defining an integral whi...	{ "timestamp": "2021-06-08T02:42:28", "yymm": "2106", "arxiv_id": "2106.03576", "language": "en", "url": "https://arxiv.org/abs/2106.03576", "abstract": "Using the Laplace derivative a Perron type integral, the Laplace integral, is defined. Moreover, it is shown that this integral includes Perron integral a...
https://arxiv.org/abs/math/0610684	Gaussian binomials and the number of sublattices	The purpose of this short communication is to make some observations on the connections between various existing formulas of counting the number of sublattices of a fixed index in an $n$-dimensional lattice and their connection with the Gaussian binomials.	\section{Existing formulas} There are various ways of determining the number of sublattices of a fixed index in a lattice, they can be found in Cassels (1971), Baake (1997), and Gruber (1997). To determine the number $f_{n}(m)$ (notation as in Baake (1997)) of sublattices of index $m$ in an $n$-dimensional lattice is t...	{ "timestamp": "2006-10-23T16:40:29", "yymm": "0610", "arxiv_id": "math/0610684", "language": "en", "url": "https://arxiv.org/abs/math/0610684", "abstract": "The purpose of this short communication is to make some observations on the connections between various existing formulas of counting the number of su...
https://arxiv.org/abs/2101.12290	Independent Hyperplanes in Oriented Paving Matroids	In 1993, Csima and Sawyer proved that in a non-pencil arrangement of n pseudolines, there are at least $\frac{6}{13}n$ simple points of intersection. Since pseudoline arrangements are the topological representations of reorientation classes of oriented matroids of rank $3$, in this paper, we will use this result to pro...	\section{Introduction} \label{sec:intro} In 1893 Sylvester asked in \cite{sylvester} the following question: ``Prove that it is not possible to arrange any finite number of real points so that a right line through every two of them shall pass through a third, unless they all lie in the same right line." This questi...	{ "timestamp": "2021-02-01T02:03:00", "yymm": "2101", "arxiv_id": "2101.12290", "language": "en", "url": "https://arxiv.org/abs/2101.12290", "abstract": "In 1993, Csima and Sawyer proved that in a non-pencil arrangement of n pseudolines, there are at least $\\frac{6}{13}n$ simple points of intersection. Sin...
https://arxiv.org/abs/1607.00519	Randomized isoperimetric inequalities	We discuss isoperimetric inequalities for convex sets. These include the classical isoperimetric inequality and that of Brunn-Minkowski, Blaschke-Santalo, Busemann-Petty and their various extensions. We show that many such inequalities admit stronger randomized forms in the following sense: for natural families of asso...	\section{Introduction} The focus of this paper is stochastic forms of isoperimetric inequalities for convex sets. To set the stage, we begin with two examples. Among the most fundamental isoperimetric inequalities is the Brunn-Minkowski inequality for the volume $V_n$ of convex bodies $K,L\subseteq \mathbb R^n$, \beg...	{ "timestamp": "2016-07-05T02:04:52", "yymm": "1607", "arxiv_id": "1607.00519", "language": "en", "url": "https://arxiv.org/abs/1607.00519", "abstract": "We discuss isoperimetric inequalities for convex sets. These include the classical isoperimetric inequality and that of Brunn-Minkowski, Blaschke-Santalo,...
https://arxiv.org/abs/1910.08791	A correction of the historiographical record on the probability integral	We correct a common (but mistaken) attribution of the evaluation of the probability integral, usually attributed to Poisson, Gauss, or Laplace.	\section{Common but mistaken attributions} The \textit{probability integral} is the following one: \begin{equation} \int_\mathbb{R} e^{-x^2}\,dx=\sqrt{\pi} \tag{E} \end{equation} Some modern authors attribute the probability integral~(E) to \textit{Poisson} \cite[p. 132]{Gnedenko}. However, Gauss himself (in an impo...	{ "timestamp": "2019-10-22T02:08:35", "yymm": "1910", "arxiv_id": "1910.08791", "language": "en", "url": "https://arxiv.org/abs/1910.08791", "abstract": "We correct a common (but mistaken) attribution of the evaluation of the probability integral, usually attributed to Poisson, Gauss, or Laplace.", "subje...
https://arxiv.org/abs/2109.09000	Smaller Gershgorin disks for multiple eigenvalues for complex matrices	Extending an earlier result for real matrices we show that multiple eigenvalues of a complex matrix lie in a reduced Gershgorin disk. One consequence is a slightly better estimate in the real case. Another one is a geometric application. Further results of a similar type are given for normal and almost symmetric matric...	\section{Introduction} Gershgorin's classic result \cite{GE} has been a main tool to estimate the eigenvalues of an $n \times n$ (complex) matrix $A=(a_{ij})_{ij=1}^n$ for the last 90 years. It says that if $\lambda$ is an eigenvalue of $A$, then there is $i \in [n]:=\{1,\ldots,n\}$ such that \[ \|a_{ii}-\lambda\| \leqs...	{ "timestamp": "2021-09-21T02:15:05", "yymm": "2109", "arxiv_id": "2109.09000", "language": "en", "url": "https://arxiv.org/abs/2109.09000", "abstract": "Extending an earlier result for real matrices we show that multiple eigenvalues of a complex matrix lie in a reduced Gershgorin disk. One consequence is a...
https://arxiv.org/abs/1803.11122	The sieving phenomenon for finite groups	The cyclic sieving phenomenon is a well-studied occurrence in combinatorics appearing when a cyclic group acts on a finite set. In this paper, we demonstrate a natural extension of this theory to finite abelian groups. We also present a similar result for dihedral groups and suggest approaches for natural generalizatio...	\section{Introduction} The \textit{cyclic sieving phenomenon} (CSP) refers to the existence of a polynomial with nonnegative integer coefficients which, when the appropriate roots of unity are plugged in, give the number of fixed points of a cyclic action on a finite set. Several surveys have been written on this topi...	{ "timestamp": "2018-03-30T02:11:01", "yymm": "1803", "arxiv_id": "1803.11122", "language": "en", "url": "https://arxiv.org/abs/1803.11122", "abstract": "The cyclic sieving phenomenon is a well-studied occurrence in combinatorics appearing when a cyclic group acts on a finite set. In this paper, we demonstr...
https://arxiv.org/abs/2107.12031	Defective Ramsey Numbers and Defective Cocolorings in Some Subclasses of Perfect Graphs	In this paper, we investigate a variant of Ramsey numbers called defective Ramsey numbers where cliques and independent sets are generalized to $k$-dense and $k$-sparse sets, both commonly called $k$-defective sets. We focus on the computation of defective Ramsey numbers restricted to some subclasses of perfect graphs....	\section{Introduction} Ramsey Theory deals with the existence of some unavoidable structures as the number of vertices in a graph grows. In (classical) Ramsey numbers, we are interested in the minimum order of a graph which guarantees the existence of a clique or an independent set of given sizes. It is well-known that...	{ "timestamp": "2021-07-27T02:29:18", "yymm": "2107", "arxiv_id": "2107.12031", "language": "en", "url": "https://arxiv.org/abs/2107.12031", "abstract": "In this paper, we investigate a variant of Ramsey numbers called defective Ramsey numbers where cliques and independent sets are generalized to $k$-dense ...
https://arxiv.org/abs/1004.4311	The Pentagram Integrals on Inscribed Polygons	The pentagram map is a natural iteration on projective equivalence classes of (twisted) n-gons in the projective plane. It was recently proved ([OST]) that the pentagram map is completely integrable, with the complete set of Poisson commuting integrals given by the polynomials O1,...,O[n/2],On and E1,...,E[n/2],En, pre...	\section{Introduction} \label{intro} The pentagram map is a geometric iteration defined on polygons. This map is defined in practically any field, but it is most easily described for polygons in the real projective plane. Geometrically, the pentagram map carries the polygon $P$ to the polygon $Q$, as shown in Figure...	{ "timestamp": "2010-04-27T02:00:55", "yymm": "1004", "arxiv_id": "1004.4311", "language": "en", "url": "https://arxiv.org/abs/1004.4311", "abstract": "The pentagram map is a natural iteration on projective equivalence classes of (twisted) n-gons in the projective plane. It was recently proved ([OST]) that ...
https://arxiv.org/abs/math/0509025	Counting the Positive Rationals: A Brief Survey	We discuss some examples that illustrate the countability of the positive rational numbers and related sets. Techniques include radix representations, Godel numbering, the fundamental theorem of arithmetic, continued fractions, Egyptian fractions, and the sequence of ratios of successive hyperbinary representation numb...	\section{Introduction}\label{sect:Intro} Ginsberg~\cite{Gins} gave an injective mapping of the rational numbers ${\mathbf Q}$ into the positive integers ${\mathbf Z}^+$ by interpreting the fraction $-a/b$ as a positive integer written in base 12, with the division slash and the minus sign as symbols for 10 and 11, resp...	{ "timestamp": "2005-09-01T20:45:19", "yymm": "0509", "arxiv_id": "math/0509025", "language": "en", "url": "https://arxiv.org/abs/math/0509025", "abstract": "We discuss some examples that illustrate the countability of the positive rational numbers and related sets. Techniques include radix representations,...
https://arxiv.org/abs/2104.13505	Clique number of Xor products of Kneser graphs	In this article we investigate a problem in graph theory, which has an equivalent reformulation in extremal set theory similar to the problems researched in "A general 2-part Erdős-Ko-Rado theorem" by Gyula O.H. Katona, who proposed our problem as well. In the graph theoretic form we examine the clique number of the Xo...	\section{\@startsection{section}{1}% \z@{.7\linespacing\@plus\linespacing}{.5\linespacing}% {\normalfont\large\scshape\centering\bfseries}} } \makeatother \parindent 15pt \begin{document} \mainsectionstyle \title{Clique number of Xor products of Kneser graphs} \author{ András Imolay \and Anett Kocsis \and Ádám ...	{ "timestamp": "2021-05-25T02:42:07", "yymm": "2104", "arxiv_id": "2104.13505", "language": "en", "url": "https://arxiv.org/abs/2104.13505", "abstract": "In this article we investigate a problem in graph theory, which has an equivalent reformulation in extremal set theory similar to the problems researched ...
https://arxiv.org/abs/1503.01243	A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights	We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method. This ODE exhibits approximate equivalence to Nesterov's scheme and thus can serve as a tool for analysis. We show that the continuous time ODE allows for a better understanding of Nesterov's schem...	\section{Restarting} \label{sec:accelerate} The example discussed in Section \ref{sec:low-friction} demonstrates that Nesterov's scheme and its generalizations \eqref{eq:nesterov_general} are not capable of fully exploiting strong convexity. That is, this example suggests evidence that $O(1/\mathtt{poly}(k))$ is the be...	{ "timestamp": "2015-10-29T01:02:16", "yymm": "1503", "arxiv_id": "1503.01243", "language": "en", "url": "https://arxiv.org/abs/1503.01243", "abstract": "We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method. This ODE exhibits approximate ...
https://arxiv.org/abs/2102.04576	Statistical Enumeration of Groups by Double Cosets	Let $H$ and $K$ be subgroups of a finite group $G$. Pick $g \in G$ uniformly at random. We study the distribution induced on double cosets. Three examples are treated in detail: 1) $H = K = $ the Borel subgroup in $GL_n(\mathbb{F}_q)$. This leads to new theorems for Mallows measure on permutations and new insights into...	\section{Introduction} Let $G$ be a finite group. Pick $g \in G$ uniformly at random. What does $g$ `look like'? This ill-posed question can be sharpened in a variety of ways; this is the subject of `probabilistic group theory' initiated by Erd\H{o}s and Turan \cite{ErdosTuranI}, \cite{ErdosTuranII}, \cite{ErdosTuranII...	{ "timestamp": "2021-03-09T02:01:18", "yymm": "2102", "arxiv_id": "2102.04576", "language": "en", "url": "https://arxiv.org/abs/2102.04576", "abstract": "Let $H$ and $K$ be subgroups of a finite group $G$. Pick $g \\in G$ uniformly at random. We study the distribution induced on double cosets. Three example...
https://arxiv.org/abs/1610.08389	Stability results for graphs with a critical edge	The classical stability theorem of Erdős and Simonovits states that, for any fixed graph with chromatic number $k+1 \ge 3$, the following holds: every $n$-vertex graph that is $H$-free and has within $o(n^2)$ of the maximal possible number of edges can be made into the $k$-partite Turán graph by adding and deleting $o(...	\section{Introduction} \tikzstyle{vertex} = [fill,shape=circle,node distance=80pt] \tikzstyle{vertex2} = [draw,shape=circle,node distance=80pt,text width={width("$M^1_1$")},align=center] \tikzstyle{hex} = [draw,regular polygon, regular polygon sides = 6] \tikzstyle{switch} = [fill,shape=circle,node distance=80pt] \tikz...	{ "timestamp": "2018-07-11T02:06:01", "yymm": "1610", "arxiv_id": "1610.08389", "language": "en", "url": "https://arxiv.org/abs/1610.08389", "abstract": "The classical stability theorem of Erdős and Simonovits states that, for any fixed graph with chromatic number $k+1 \\ge 3$, the following holds: every $n...
https://arxiv.org/abs/2107.06739	On the Lebesgue measure of the boundary of the evoluted set	The evoluted set is the set of configurations reached from an initial set via a fixed flow for all times in a fixed interval. We find conditions on the initial set and on the flow ensuring that the evoluted set has negligible boundary (i.e. its Lebesgue measure is zero). We also provide several counterexample showing t...	\section{Introduction} The study of the attainable set from a point is a crucial problem in control theory, starting from the classical orbit, Rashevsky-Chow and Krener theorems, see \cite{agra,jurdj,krener}. If the initial state is not precisely identified, but lies in a given set, the problem gets even more comp...	{ "timestamp": "2021-07-15T02:20:40", "yymm": "2107", "arxiv_id": "2107.06739", "language": "en", "url": "https://arxiv.org/abs/2107.06739", "abstract": "The evoluted set is the set of configurations reached from an initial set via a fixed flow for all times in a fixed interval. We find conditions on the in...
https://arxiv.org/abs/2102.03541	On generalized Minkowski arrangements	The concept of a Minkowski arrangement was introduced by Fejes Tóth in 1965 as a family of centrally symmetric convex bodies with the property that no member of the family contains the center of any other member in its interior. This notion was generalized by Fejes Tóth in 1967, who called a family of centrally symmetr...	\section{Introduction}\label{sec:intro} The notion of a \emph{Minkowski arrangement} of convex bodies was introduced by L. Fejes T\'oth in \cite{FTL}, who defined it as a family $\mathcal{F}$ of centrally symmetric convex bodies in the $d$-dimensional Euclidean space $\Re^d$, with the property that no member of $\math...	{ "timestamp": "2021-12-14T02:28:13", "yymm": "2102", "arxiv_id": "2102.03541", "language": "en", "url": "https://arxiv.org/abs/2102.03541", "abstract": "The concept of a Minkowski arrangement was introduced by Fejes Tóth in 1965 as a family of centrally symmetric convex bodies with the property that no mem...
https://arxiv.org/abs/2106.13372	Graphs with Many Hamiltonian Paths	A graph is \emph{hamiltonian-connected} if every pair of vertices can be connected by a hamiltonian path, and it is \emph{hamiltonian} if it contains a hamiltonian cycle. Every hamiltonian-connected graph is hamiltonian, however we also construct families of nonhamiltonian graphs with `many' hamiltonian paths, where 'm...	\section{Introduction} A graph is \emph{hamiltonian} if it admits a hamiltonian cycle, and \emph{homogeneously traceable} if every vertex of $G$ is the starting vertex of a hamiltonian path. If every pair of vertices in $G$ is connected with a hamiltonian path, then $G$ is \emph{hamiltonian-connected}. This class of gr...	{ "timestamp": "2022-06-03T02:02:50", "yymm": "2106", "arxiv_id": "2106.13372", "language": "en", "url": "https://arxiv.org/abs/2106.13372", "abstract": "A graph is \\emph{hamiltonian-connected} if every pair of vertices can be connected by a hamiltonian path, and it is \\emph{hamiltonian} if it contains a ...
https://arxiv.org/abs/2007.16081	Proportions of vanishing elements in finite groups	In this paper, we study the proportion of vanishing elements of finite groups. We show that the proportion of vanishing elements of every finite non-abelian group is bounded below by $1/2$ and classify all finite groups whose proportions of vanishing elements attain this bound. For symmetric groups of degree at least $...	\section{Introduction} Let $G$ be a finite group. An element $g\in G$ is called a \emph{vanishing element} of $G$ if there exists an irreducible complex character $\chi$ of $G$ such that $\chi(g)=0$. In this case, $g$ is said to be a \emph{zero} of $\chi$. Let $\van(G)$ denote the set of all vanishing elements of $G$...	{ "timestamp": "2021-01-18T02:14:14", "yymm": "2007", "arxiv_id": "2007.16081", "language": "en", "url": "https://arxiv.org/abs/2007.16081", "abstract": "In this paper, we study the proportion of vanishing elements of finite groups. We show that the proportion of vanishing elements of every finite non-abeli...
https://arxiv.org/abs/1811.05430	On the Mean Order of Connected Induced Subgraphs of Block Graphs	The average order of the connected induced subgraphs of a graph $G$ is called the mean connected induced subgraph (CIS) order of $G$. This is an extension of the mean subtree order of a tree, first studied by Jamison. In this article, we demonstrate that among all connected block graphs of order $n$, the path $P_n$ has...	\section{Introduction} Jamison~\cite{Jamison1983} initiated the study of the mean subtree order of a tree. A number of extensions of this mean to other (connected) graphs have recently been considered: \begin{itemize} \item the mean order of the sub-$k$-trees of a $k$-tree~\cite{StephensOellermann2018}, \item the mea...	{ "timestamp": "2018-11-14T02:17:54", "yymm": "1811", "arxiv_id": "1811.05430", "language": "en", "url": "https://arxiv.org/abs/1811.05430", "abstract": "The average order of the connected induced subgraphs of a graph $G$ is called the mean connected induced subgraph (CIS) order of $G$. This is an extension...
https://arxiv.org/abs/1706.02132	Newton correction methods for computing real eigenpairs of symmetric tensors	Real eigenpairs of symmetric tensors play an important role in multiple applications. In this paper we propose and analyze a fast iterative Newton-based method to compute real eigenpairs of symmetric tensors. We derive sufficient conditions for a real eigenpair to be a stable fixed point for our method, and prove that ...	\section{Discussion and summary}\label{sec:discussion} In this paper we developed and analyzed a Newton-based iterative approach to compute real eigenpairs of symmetric tensors. We now briefly discuss three important issues: its runtime, its ability to find all tensor eigenpairs, and its optimization point of view. \...	{ "timestamp": "2018-03-06T02:10:37", "yymm": "1706", "arxiv_id": "1706.02132", "language": "en", "url": "https://arxiv.org/abs/1706.02132", "abstract": "Real eigenpairs of symmetric tensors play an important role in multiple applications. In this paper we propose and analyze a fast iterative Newton-based m...
https://arxiv.org/abs/1201.3593	Path Following in the Exact Penalty Method of Convex Programming	Classical penalty methods solve a sequence of unconstrained problems that put greater and greater stress on meeting the constraints. In the limit as the penalty constant tends to $\infty$, one recovers the constrained solution. In the exact penalty method, squared penalties are replaced by absolute value penalties, and...	\section{Introduction} Penalties and barriers are both potent devices for solving constrained optimization problems \citep{BoydVandenberghe04Book,Forsgren02interior,LuenbergerYe08Book,NocedalWright06Book,Ruszczynski06Book,Zangwill67Penalty}. The general idea is to replace hard constraints by penalties or barriers and ...	{ "timestamp": "2012-01-18T02:03:52", "yymm": "1201", "arxiv_id": "1201.3593", "language": "en", "url": "https://arxiv.org/abs/1201.3593", "abstract": "Classical penalty methods solve a sequence of unconstrained problems that put greater and greater stress on meeting the constraints. In the limit as the pen...
https://arxiv.org/abs/2108.08804	Helly-type Problems	In this paper, we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems, and explain some of its main themes and goals.	\section{Helly, Carath\'eodory, and Radon theorems} In this paper, we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carath\'eodory, and Tverberg. Helly's theorem~\cite{Helly:1923wr} asserts that for a family $\{K_1,K_2,\ldots, K_n\}$ of convex...	{ "timestamp": "2021-08-20T02:22:21", "yymm": "2108", "arxiv_id": "2108.08804", "language": "en", "url": "https://arxiv.org/abs/2108.08804", "abstract": "In this paper, we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and...
https://arxiv.org/abs/1910.03775	Approximate semi-amenability of Banach algebras	In recent work of the authors the notion of a derivation being approximately semi-inner arose as a tool for investigating (approximate) amenability questions for Banach algebras. Here we investigate this property in its own right, together with the consequent one of approximately semi-amenability. Under certain hypothe...	\section{Introduction} \label{Introduction} The concept of amenability for a Banach algebra, introduced by Johnson in \cite{John}, has proved to be of enormous importance in Banach algebra theory. In \cite{GhaLoy}, and subsequently in \cite{GhaLoyZh}, several modifications of this notion were introduced, in particul...	{ "timestamp": "2019-10-10T02:06:49", "yymm": "1910", "arxiv_id": "1910.03775", "language": "en", "url": "https://arxiv.org/abs/1910.03775", "abstract": "In recent work of the authors the notion of a derivation being approximately semi-inner arose as a tool for investigating (approximate) amenability questi...
https://arxiv.org/abs/1903.05951	Metrics which turn tilings into binary perfect codes	In this work, we consider tilings of the Hamming cube and look for metrics which turn the tilings into a perfect code. We consider the family of metrics which are determined by a weight and are compatible with the support of vectors (TS-metrics). We determine which of the tilings with small tiles or high rank can be a ...	\section{Introduction} The study of perfect codes is an important topic in coding theory, since it satisfies an optimality condition: the coincidence between the packing and covering radii. Finding perfect codes is a difficult issue. For the Hamming metric, there is a complete characterization of its parameters, which...	{ "timestamp": "2019-05-09T02:19:01", "yymm": "1903", "arxiv_id": "1903.05951", "language": "en", "url": "https://arxiv.org/abs/1903.05951", "abstract": "In this work, we consider tilings of the Hamming cube and look for metrics which turn the tilings into a perfect code. We consider the family of metrics w...
https://arxiv.org/abs/2106.07331	Vertex separators, chordality and virtually free groups	In this paper we consider some results obtained for graphs using minimal vertex separators and generalized chordality and translate them to the context of Geometric Group Theory. Using these new tools, we are able to give two new characterizations for a group to be virtually free. Furthermore, we prove that the Baumsla...	\section{Introduction} In \cite{M2} the second author studied some relations between vertex separator sets, certain chordality properties that generalize being chordal and conditions for a graph to be quasi-isometric to a tree. Some of these ideas can be easily translated to the language of Geometric Group...	{ "timestamp": "2021-06-15T02:36:51", "yymm": "2106", "arxiv_id": "2106.07331", "language": "en", "url": "https://arxiv.org/abs/2106.07331", "abstract": "In this paper we consider some results obtained for graphs using minimal vertex separators and generalized chordality and translate them to the context of...
https://arxiv.org/abs/1912.00417	A graph inequality on the common neighbourhood	In this note we prove a graph inequality based on the sizes of the common neighbourhoods. We also characterize the extremal graphs that achieve the equality.The result was first discovered as a consequence of the classical Forster's theorem in electric networks. We also present a short combinatorial proof that was insp...	\section{Introduction} A (simple) graph $G$ is a pair $G=(V, E)$ where $V$ is the vertex set and $E \subseteq \binom{V}{2}$ is the edge set. For any pair of distinct vertices $u$ and $v$, we denote $\{u, v\}$ by $uv$. For a vertex $u$, the (open) neighbourhood is defined as $N(u) = \{v \in V : uv \in E\}$. In particul...	{ "timestamp": "2019-12-03T02:15:23", "yymm": "1912", "arxiv_id": "1912.00417", "language": "en", "url": "https://arxiv.org/abs/1912.00417", "abstract": "In this note we prove a graph inequality based on the sizes of the common neighbourhoods. We also characterize the extremal graphs that achieve the equali...
https://arxiv.org/abs/2006.12294	Connecting Graph Convolutional Networks and Graph-Regularized PCA	Graph convolution operator of the GCN model is originally motivated from a localized first-order approximation of spectral graph convolutions. This work stands on a different view; establishing a \textit{mathematical connection between graph convolution and graph-regularized PCA} (GPCA). Based on this connection, GCN a...	\subsection{Graph Convolution} Consider a node-attributed input graph $G=(V,E,X)$ with $\|V\|=n$ nodes and $\|E\|=m$ edges, where $X\in \mathbb{R}^{n\times d}$ denotes the feature matrix with $d$ features. Similar to other neural networks stacked with repeated layers, GCN \cite{kipf2017semi} contains multiple graph conv...	{ "timestamp": "2021-03-03T02:39:09", "yymm": "2006", "arxiv_id": "2006.12294", "language": "en", "url": "https://arxiv.org/abs/2006.12294", "abstract": "Graph convolution operator of the GCN model is originally motivated from a localized first-order approximation of spectral graph convolutions. This work s...
https://arxiv.org/abs/1706.00738	Contractive inequalities for Hardy spaces	We state and discuss several interrelated results, conjectures, and questions regarding contractive inequalities for classical $H^p$ spaces of the unit disc. We study both coefficient estimates in terms of weighted $\ell^2$ sums and the Riesz projection viewed as a map from $L^q$ to $H^p$ with $q\ge p$. Some numerical ...	\section{Introduction} This paper deals with certain contractive inequalities for the classical Hardy spaces $H^p$ of the unit disc $\mathbb{D}$, where as usual $f$ belongs to $H^p$ for $0<p<\infty$ if $f$ is analytic in $\mathbb{D}$ and \[ \\| f\\|_{H^p}:=\sup_{0<r<1} \left(\int_{0}^{2\pi} \|f(re^{i\theta})\|^p \frac{d\...	{ "timestamp": "2018-05-28T02:05:04", "yymm": "1706", "arxiv_id": "1706.00738", "language": "en", "url": "https://arxiv.org/abs/1706.00738", "abstract": "We state and discuss several interrelated results, conjectures, and questions regarding contractive inequalities for classical $H^p$ spaces of the unit di...
https://arxiv.org/abs/math/0612620	Congruence and Uniqueness of Certain Markoff Numbers	By making use of only simple facts about congruence, we first show that every even Markoff number is congruent to 2 modulo 32, and then, generalizing an earlier result of Baragar, establish the uniqueness for those Markoff numbers c where one of 3c - 2 and 3c + 2 is a prime power, 4 times a prime power, or 8 times a pr...	\section{Introduction}\label{s:intro} \noindent It is A. A. Markoff who first studied the Diophantine equation---now known as the {\it Markoff equation} \begin{eqnarray}\label{eqn:markoff} a^2+b^2+c^2=3abc \end{eqnarray} in late 1870s in his famous work \cite{markoff1880ma} on the minima of real, indefinite, binary qu...	{ "timestamp": "2006-12-29T20:25:27", "yymm": "0612", "arxiv_id": "math/0612620", "language": "en", "url": "https://arxiv.org/abs/math/0612620", "abstract": "By making use of only simple facts about congruence, we first show that every even Markoff number is congruent to 2 modulo 32, and then, generalizing ...
https://arxiv.org/abs/2302.08169	The Commuting Algebra	Let $KQ$ be a path algebra, where $Q$ is a finite quiver and $K$ is a field. We study $KQ/C$ where $C$ is the two-sided ideal in $KQ$ generated by all differences of parallel paths in $Q$. We show that $KQ/C$ is always finite dimensional and its global dimension is finite. Furthermore, we prove that $KQ/C$ is Morita eq...	\section{Introduction} In the study of the representation theory of finite dimensional $K$-algebras with $K$ a field, the algebras one encounters often are of the form $KQ/I$, where $I$ is an admissible ideal; that is $J^n\subseteq I\subseteq J^2$, for some positive integer $n$, and $J$ is the ideal in $KQ$...	{ "timestamp": "2023-02-22T02:07:22", "yymm": "2302", "arxiv_id": "2302.08169", "language": "en", "url": "https://arxiv.org/abs/2302.08169", "abstract": "Let $KQ$ be a path algebra, where $Q$ is a finite quiver and $K$ is a field. We study $KQ/C$ where $C$ is the two-sided ideal in $KQ$ generated by all dif...
https://arxiv.org/abs/1701.02793	Equidistribution of Neumann data mass on triangles	In this paper we study the behaviour of the Neumann data of Dirichlet eigenfunctions on triangles. We prove that the $L^2$ norm of the (semi-classical) Neumann data on each side is equal to the length of the side divided by the area of the triangle. The novel feature of this result is that it is {\it not} an asymptotic...	\section{Introduction} Given a compact surface or manifold with boundary, it is an interesting question to consider restrictions of eigenfunctions to hypersurfaces; either the Dirichlet data or Neumann data (or both, the Cauchy data) can be considered. Perhaps the simplest question is to consider boundary values. ...	{ "timestamp": "2017-01-12T02:01:41", "yymm": "1701", "arxiv_id": "1701.02793", "language": "en", "url": "https://arxiv.org/abs/1701.02793", "abstract": "In this paper we study the behaviour of the Neumann data of Dirichlet eigenfunctions on triangles. We prove that the $L^2$ norm of the (semi-classical) Ne...
https://arxiv.org/abs/1907.09784	Connecting optimization with spectral analysis of tri-diagonal matrices	We show that the global minimum (resp. maximum) of a continuous function on a compact set can be approximated from above (resp. from below) by computing the smallest (rest. largest) eigenvalue of a hierarchy of (r x r) tri-diagonal univariate moment matrices of increasing size. Equivalently it reduces to computing the ...	\section{Introduction} \label{section-1} The goal of this paper is show that the global minimum (resp. maximum) of a continuous function $f$ on a compact set $\mathbf{\Omega}\subset\mathbb{R}^n$ can be approximated as closely as desired from above (resp. from below) by the smallest eigenvalues $(\tau^\ell_r)_{r\in\mat...	{ "timestamp": "2020-03-17T01:26:48", "yymm": "1907", "arxiv_id": "1907.09784", "language": "en", "url": "https://arxiv.org/abs/1907.09784", "abstract": "We show that the global minimum (resp. maximum) of a continuous function on a compact set can be approximated from above (resp. from below) by computing t...
https://arxiv.org/abs/1002.4060	On the three-rowed skew standard Young tableaux	Let $\mathcal{T}_3$ be the three-rowed strip. Recently Regev conjectured that the number of standard Young tableaux with $n-3$ entries in the "skew three-rowed strip" $\mathcal{T}_3 / (2,1,0)$ is $m_{n-1}-m_{n-3}$, a difference of two Motzkin numbers. This conjecture, together with hundreds of similar identities, were ...	\section{Introduction} The enumeration of standard Young tableaux (SYTs) is a fundamental problem in combinatorics and representation theory. For example, it is known that the number of SYTs of a given shape $\lambda \vdash n$ is counted by the hook-length formula~\cite{Frame_54}. However, the problem of counting ...	{ "timestamp": "2010-05-13T02:00:28", "yymm": "1002", "arxiv_id": "1002.4060", "language": "en", "url": "https://arxiv.org/abs/1002.4060", "abstract": "Let $\\mathcal{T}_3$ be the three-rowed strip. Recently Regev conjectured that the number of standard Young tableaux with $n-3$ entries in the \"skew three-...
https://arxiv.org/abs/2011.12007	Index of a matrix, complex logarithms, and multidimensional Fresnel integrals	We critically discuss the problem of finding the $\lambda$-index $\mathcal{N}(\lambda)\in [0,1,\ldots,N]$ of a real symmetric matrix $\mathbf{M}$, defined as the number of eigenvalues smaller than $\lambda$, using the entries of $\mathbf{M}$ as only input. We show that a widely used formula $$ \mathcal{N}(\lambda)=\lim...	\section{The question} Consider a $N\times N$ real symmetric matrix $\bm M = (M_{ij})$, whose eigenvalues (all real) are $\lambda_1\leq\lambda_2\leq\ldots\leq \lambda_N$. Is there a way to count how many eigenvalues of $\bm M$ fall below a threshold $\lambda$, using as only input\footnote{In particular, explicit diagon...	{ "timestamp": "2020-11-25T02:17:42", "yymm": "2011", "arxiv_id": "2011.12007", "language": "en", "url": "https://arxiv.org/abs/2011.12007", "abstract": "We critically discuss the problem of finding the $\\lambda$-index $\\mathcal{N}(\\lambda)\\in [0,1,\\ldots,N]$ of a real symmetric matrix $\\mathbf{M}$, d...
https://arxiv.org/abs/1906.05224	Estimation of the Shapley value by ergodic sampling	The idea of approximating the Shapley value of an n-person game by Monte Carlo simulation was first suggested by Mann and Shapley (1960) and they also introduced four different heuristical methods to reduce the estimation error. Since 1960, several statistical methods have been developed to reduce the standard deviatio...	\section{Introduction} Since 1953 the Shapley value \citep{shapley1953value} is a fundamental solution concept of cooperative games with transferable utility (TU). As it is the sum of $2^{n-1}$ terms (where $n$ is the number of players) the exact value cannot be calculated in polynomial time of the number of players, ...	{ "timestamp": "2019-06-13T02:17:08", "yymm": "1906", "arxiv_id": "1906.05224", "language": "en", "url": "https://arxiv.org/abs/1906.05224", "abstract": "The idea of approximating the Shapley value of an n-person game by Monte Carlo simulation was first suggested by Mann and Shapley (1960) and they also int...
https://arxiv.org/abs/0805.3167	Smooth analysis of the condition number and the least singular value	Let $\a$ be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $\a$ and $M$ be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value of the matrix $M...	\section{Previous results} \label{section:previous} Let us first discuss the gaussian case. Improving results of Kostlan and Oceanu \cite{Smale}, Edelman \cite{Edel} computed the limiting distribution of $\sqrt n s_{n} (N_{n}) $ when $N_{n} $ is gaussian. His result implies \begin{theorem} \label{theorem:Edel} Th...	{ "timestamp": "2009-08-11T00:40:19", "yymm": "0805", "arxiv_id": "0805.3167", "language": "en", "url": "https://arxiv.org/abs/0805.3167", "abstract": "Let $\\a$ be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $\\...