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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/1309.3354	Laurent polynomials, Eulerian numbers, and Bernstein's theorem	Erman, Smith, and Várilly-Alvarado showed that the expected number of doubly monic Laurent polynomials $f(z) = z^{-m} + a_{-m+1}z^{-m+1} + \cdots + a_{n-1}z^{n-1} + z^n$ whose first $m+n-1$ powers have vanishing constant term is the Eulerian number $\brac{m+n-1}{m-1}$, as well as a more refined result about sparse Laur...	\section{Introduction} Fix positive integers $m$ and $n$, and let $f(z) = z^{-m} + a_{-m+1}z^{-m+1} + \cdots + a_{n-1}z^{n-1} + z^n$ be a doubly monic Laurent polynomial (with complex coefficients). Denote by $\llbracket f^k \rrbracket$, for all positive integers $k$, the constant coefficient of the $k$th power of $f(...	{ "timestamp": "2013-09-16T02:02:16", "yymm": "1309", "arxiv_id": "1309.3354", "language": "en", "url": "https://arxiv.org/abs/1309.3354", "abstract": "Erman, Smith, and Várilly-Alvarado showed that the expected number of doubly monic Laurent polynomials $f(z) = z^{-m} + a_{-m+1}z^{-m+1} + \\cdots + a_{n-1}...
https://arxiv.org/abs/1807.06527	On monotonicity of Ramanujan function for binomial random variables	For a binomial random variable $\xi$ with parameters $n$ and $b/n$, it is well known that the median equals $b$ when $b$ is an integer. In 1968, Jogdeo and Samuels studied the behaviour of the relative difference between ${\sf P}(\xi=b)$ and $1/2-{\sf P}(\xi<b)$. They proved its monotonicity in $n$ and posed a question...	\section{Introduction} Given a non-negative integer random variable $\xi$, we call {\it the median of $\xi$} $$ \mu(\xi):=\min\left\{m\in\mathbb{Z}_+:\,{\sf P}(\xi\leq m)\geq\frac{1}{2}\right\}. $$ Consider a Poisson random variable $\eta_b$ with a positive integer parameter $b$. From the general result of Choi~\ci...	{ "timestamp": "2020-05-06T02:20:12", "yymm": "1807", "arxiv_id": "1807.06527", "language": "en", "url": "https://arxiv.org/abs/1807.06527", "abstract": "For a binomial random variable $\\xi$ with parameters $n$ and $b/n$, it is well known that the median equals $b$ when $b$ is an integer. In 1968, Jogdeo a...
https://arxiv.org/abs/1103.2041	Random sum-free subsets of Abelian groups	We characterize the structure of maximum-size sum-free subsets of a random subset of an Abelian group $G$. In particular, we determine the threshold $p_c \approx \sqrt{\log n / n}$ above which, with high probability as $\|G\| \to \infty$, each such subset is contained in a maximum-size sum-free subset of $G$, whenever $q...	\section{Introduction} One of the most important developments in Combinatorics over the past twenty years has been the introduction and proof of various `random analogues' of well-known theorems in Extremal Graph Theory, Ramsey Theory, and Additive Combinatorics. Such questions were first introduced for graphs by Baba...	{ "timestamp": "2012-11-19T02:01:42", "yymm": "1103", "arxiv_id": "1103.2041", "language": "en", "url": "https://arxiv.org/abs/1103.2041", "abstract": "We characterize the structure of maximum-size sum-free subsets of a random subset of an Abelian group $G$. In particular, we determine the threshold $p_c \\...
https://arxiv.org/abs/1610.03528	Explicit Hilbert Irreducibility	Let $P(T,X)$ be an irreducible polynomial in two variables with rational coefficients. It follows from Hilbert's Irreducibility Theorem that for most rational numbers $t$ the specialized polynomial $P(t,X)$ is irreducible and has the same Galois group as $P$. We discuss here a method for obtaining an explicit descripti...	\section{Introduction} Let $P\in \mathbf{Q}[T,X]$ be an irreducible polynomial in two variables with rational coefficients. Regarding $P$ as an element of the ring $\mathbf{Q}(T)[X]$, let $G$ be the Galois group of $P$, i.e., the Galois group of a splitting field for $P$ over $\mathbf{Q}(T)$. For any rational number $...	{ "timestamp": "2016-10-13T02:00:49", "yymm": "1610", "arxiv_id": "1610.03528", "language": "en", "url": "https://arxiv.org/abs/1610.03528", "abstract": "Let $P(T,X)$ be an irreducible polynomial in two variables with rational coefficients. It follows from Hilbert's Irreducibility Theorem that for most rati...
https://arxiv.org/abs/1602.07668	Analysis of the mean squared derivative cost function	In this paper, we investigate the mean squared derivative cost functions that arise in various applications such as in motor control, biometrics and optimal transport theory. We provide qualitative properties, explicit analytical formulas and computational algorithms for the cost functions. We also perform numerical si...	\section{Introduction} \subsection{The mean squared derivative cost function} This paper is concerned with the analysis of \textit{the mean squared derivative cost function} defined as follows: \begin{equation} \label{eq: Ch cost} C_{n,h}(x_{0},x_{1},\ldots,x_{n-1};y_{0},y_{1},\ldots,y_{n-1}):=\inf\limits_{\xi}\int_0...	{ "timestamp": "2017-03-01T02:06:51", "yymm": "1602", "arxiv_id": "1602.07668", "language": "en", "url": "https://arxiv.org/abs/1602.07668", "abstract": "In this paper, we investigate the mean squared derivative cost functions that arise in various applications such as in motor control, biometrics and optim...
https://arxiv.org/abs/1708.02436	Subsets of posets minimising the number of chains	A well-known theorem of Sperner describes the largest collections of subsets of an $n$-element set none of which contains another set from the collection. Generalising this result, Erdős characterised the largest families of subsets of an $n$-element set that do not contain a chain of sets $A_1 \subset \dotsc \subset A...	\section{Introduction} \label{sec:introduction} The classical theorem of Sperner~\cite{Sp28} describes the largest families of subsets of a finite set none of which contains another set from the family. Originally a result in extremal set theory, it reached a broader mathematical audience as the key lemma in Erd\H{o}s...	{ "timestamp": "2017-08-09T02:06:10", "yymm": "1708", "arxiv_id": "1708.02436", "language": "en", "url": "https://arxiv.org/abs/1708.02436", "abstract": "A well-known theorem of Sperner describes the largest collections of subsets of an $n$-element set none of which contains another set from the collection....
https://arxiv.org/abs/2302.01770	A conjecture related to the nilpotency of groups with isomorphic non-commuting graphs	In this work we discuss whether the non-commuting graph of a finite group can determine its nilpotency. More precisely, Abdollahi, Akbari and Maimani conjectured that if $G$ and $H$ are finite groups with isomorphic non-commuting graphs and $G$ is nilpotent, then $H$ must be nilpotent as well (Conjecture 2). We pose a ...	\section{Introduction} Given a finite group $G$, one can consider the graph whose vertices are (some) elements of $G$ and whose edges reflect a certain structure property of $G$. Such a technique has proven to be a valuable tool to study certain aspects of finite groups: we suggest Cameron's survey the see for example ...	{ "timestamp": "2023-02-07T02:32:47", "yymm": "2302", "arxiv_id": "2302.01770", "language": "en", "url": "https://arxiv.org/abs/2302.01770", "abstract": "In this work we discuss whether the non-commuting graph of a finite group can determine its nilpotency. More precisely, Abdollahi, Akbari and Maimani conj...
https://arxiv.org/abs/1711.07029	Universal Cycles of Restricted Words	A connected digraph in which the in-degree of any vertex equals its out-degree is Eulerian, this baseline result is used as the basis of existence proofs for universal cycles (also known as generalized deBruijn cycles or U-cycles) of several combinatorial objects. We extend the body of known results by presenting new r...	\section{Introduction}A universal cycle, or U-Cycle, is a cyclic ordering of a set of objects ${\mathcal C}$, each represented as a string of length $k$. The ordering requires that object $b = b_0b_1\ldots b_{k-1}$ follow object $a = a_0a_1\ldots a_{k-1}$ only if $a_1a_2\ldots a_{k-1} = b_0b_1\ldots b_{k-2}$. U-cycles ...	{ "timestamp": "2017-11-21T02:12:00", "yymm": "1711", "arxiv_id": "1711.07029", "language": "en", "url": "https://arxiv.org/abs/1711.07029", "abstract": "A connected digraph in which the in-degree of any vertex equals its out-degree is Eulerian, this baseline result is used as the basis of existence proofs ...
https://arxiv.org/abs/1607.02092	A Delayed Yule Process	In now classic work, David Kendall (1966) recognized that the Yule process and Poisson process could be related by a (random) time change. Furthermore, he showed that the Yule population size rescaled by its mean has an almost sure exponentially distributed limit as $t\to \infty$. In this note we introduce a class of c...	\section{Introduction} The {\it basic Yule process} $Y = \{Y_t:t\ge 0\}$ is a continuous time branching process starting from a single progenitor in which a particle survives for a mean one, exponentially distributed time before being replaced by two offspring independently evolving in the same manner. $Y_t$ repr...	{ "timestamp": "2016-07-08T02:11:51", "yymm": "1607", "arxiv_id": "1607.02092", "language": "en", "url": "https://arxiv.org/abs/1607.02092", "abstract": "In now classic work, David Kendall (1966) recognized that the Yule process and Poisson process could be related by a (random) time change. Furthermore, he...
https://arxiv.org/abs/math/0702316	Matroids with nine elements	We describe the computation of a catalogue containing all matroids with up to nine elements, and present some fundamental data arising from this cataogue. Our computation confirms and extends the results obtained in the 1960s by Blackburn, Crapo and Higgs. The matroids and associated data are stored in an online databa...	\section{Introduction} In the late 1960s, Blackburn, Crapo \& Higgs published a technical report describing the results of a computer search for all simple matroids on up to eight elements (although the resulting paper \cite{MR0419270} did not appear until 1973). In both the report and the paper they said \begin{q...	{ "timestamp": "2007-02-12T11:27:26", "yymm": "0702", "arxiv_id": "math/0702316", "language": "en", "url": "https://arxiv.org/abs/math/0702316", "abstract": "We describe the computation of a catalogue containing all matroids with up to nine elements, and present some fundamental data arising from this catao...
https://arxiv.org/abs/1603.06697	On the exponent of the automorphism group of a compact Riemann surface	Let $X$ be a compact Riemann surface of genus $g\geq 2$, and let $Aut(X)$ be its group of automorphims. We show that the exponent of $Aut(X)$ is bounded by $42(g-1)$. We also determine explicitly the infinitely many values of $g$ for which this bound is reached and the corresponding groups. Finally we discuss related q...	\subsection*{1. Introduction} \noindent Throughout the paper $X$ will be a compact Riemann surface of genus $g\geq 2$. We write $Aut(X)$ for the full group of conformal automorphisms of $X$. \par The order of a group $G$ is denoted by $\|G\|$, the cyclic group of order $n$ by $C_n$, and the neutral element in a gro...	{ "timestamp": "2016-10-04T02:07:56", "yymm": "1603", "arxiv_id": "1603.06697", "language": "en", "url": "https://arxiv.org/abs/1603.06697", "abstract": "Let $X$ be a compact Riemann surface of genus $g\\geq 2$, and let $Aut(X)$ be its group of automorphims. We show that the exponent of $Aut(X)$ is bounded ...
https://arxiv.org/abs/0806.4372	The 1-fixed-endpoint Path Cover Problem is Polynomial on Interval Graph	We consider a variant of the path cover problem, namely, the $k$-fixed-endpoint path cover problem, or kPC for short, on interval graphs. Given a graph $G$ and a subset $\mathcal{T}$ of $k$ vertices of $V(G)$, a $k$-fixed-endpoint path cover of $G$ with respect to $\mathcal{T}$ is a set of vertex-disjoint paths $\mathc...	\section{Introduction} {\bf Framework--Motivation.} A well studied problem with numerous practical applications in graph theory is to find a minimum number of vertex-disjoint paths of a graph $G$ that cover the vertices of $G$. This problem, also known as the path cover problem (PC), finds application in the fields of ...	{ "timestamp": "2008-06-26T20:13:32", "yymm": "0806", "arxiv_id": "0806.4372", "language": "en", "url": "https://arxiv.org/abs/0806.4372", "abstract": "We consider a variant of the path cover problem, namely, the $k$-fixed-endpoint path cover problem, or kPC for short, on interval graphs. Given a graph $G$ ...
https://arxiv.org/abs/math/0510054	Euler and the pentagonal number theorem	``In this paper we give the history of Leonhard Euler's work on the pentagonal number theorem, and his applications of the pentagonal number theorem to the divisor function, partition function and divergent series. We have attempted to give an exhaustive review of all of Euler's correspondence and publications about th...	\section{Introduction} \label{section:introduction} The pentagonal numbers are those numbers of the form $\frac{n(3n-1)}{2}$ for $n$ a positive integer. They represent the number of distinct points which may be arranged to form superimposed regular pentagons with the same number of equally spaced points on the sides of...	{ "timestamp": "2006-08-17T15:30:11", "yymm": "0510", "arxiv_id": "math/0510054", "language": "en", "url": "https://arxiv.org/abs/math/0510054", "abstract": "``In this paper we give the history of Leonhard Euler's work on the pentagonal number theorem, and his applications of the pentagonal number theorem t...
https://arxiv.org/abs/1605.06481	The Sphere Covering Inequality and Its Applications	In this paper, we show that the total area of two distinct surfaces with Gaussian curvature equal to 1, which are also conformal to the Euclidean unit disk with the same conformal factor on the boundary, must be at least 4{\pi}. In other words, the areas of these surfaces must cover the whole unit sphere after a proper...	\section{Introduction} A large number of important second order nonlinear elliptic equations involve exponential nonlinearities. These equations arise, for example, in the study of Gaussian curvature of surfaces with metrics conformal to Euclidean metric ( \cite{CLiu-MR1209959}, \cite{CY2-MR925123}, \cite{CY1-...	{ "timestamp": "2016-10-28T02:09:22", "yymm": "1605", "arxiv_id": "1605.06481", "language": "en", "url": "https://arxiv.org/abs/1605.06481", "abstract": "In this paper, we show that the total area of two distinct surfaces with Gaussian curvature equal to 1, which are also conformal to the Euclidean unit dis...
https://arxiv.org/abs/2207.06679	Learning to Prove Trigonometric Identities	Automatic theorem proving with deep learning methods has attracted attentions recently. In this paper, we construct an automatic proof system for trigonometric identities. We define the normalized form of trigonometric identities, design a set of rules for the proof and put forward a method which can generate theoretic...	\section{Synthetic dataset} \subsection{Synthetic identities} The full details of generating identities are as follows: \begin{enumerate} \item Randomly generate $n$ elements in the form of $\sin(Ax + B)$ or $\cos(Ax + B)$. Multiply these $n$ elements and a coefficient $C$ to get a term in the form of $C*\prod_...	{ "timestamp": "2022-07-15T02:08:54", "yymm": "2207", "arxiv_id": "2207.06679", "language": "en", "url": "https://arxiv.org/abs/2207.06679", "abstract": "Automatic theorem proving with deep learning methods has attracted attentions recently. In this paper, we construct an automatic proof system for trigonom...
https://arxiv.org/abs/1005.3750	Rectangle Free Coloring of Grids	A two-dimensional \emph{grid} is a set $\Gnm = [n]\times[m]$. A grid $\Gnm$ is \emph{$c$-colorable} if there is a function $\chi_{n,m}: \Gnm \to [c]$ such that there are no rectangles with all four corners the same color. We address the following question: for which values of $n$ and $m$ is $\Gnm$ $c$-colorable? This p...	\section{Introduction} A two-dimensional \emph{grid} is a set $\Gnm = [n]\times[m]$ where $[t] = \{1, \ldots, t\}$. A \emph{rectangle} of $\Gnm$ is a subset of the form $\{(a,b),(a+c_1,b),(a+c_1,b+c_2),(a,b+c_2)\}$ for some constants $c_1$ and $c_2$. A grid $\Gnm$ is \emph{$c$-colorable} if there is a functio...	{ "timestamp": "2010-05-21T02:01:55", "yymm": "1005", "arxiv_id": "1005.3750", "language": "en", "url": "https://arxiv.org/abs/1005.3750", "abstract": "A two-dimensional \\emph{grid} is a set $\\Gnm = [n]\\times[m]$. A grid $\\Gnm$ is \\emph{$c$-colorable} if there is a function $\\chi_{n,m}: \\Gnm \\to [c]...
https://arxiv.org/abs/1806.09027	Joint similarity for commuting families of power bounded matrices	An example due to Pisier shows that two commuting, completely polynomially bounded Hilbert space operators may not be simultaneously similar to contractions. Thus, while each operator is individually similar to a contraction, the pair is not jointly similar to a pair of commuting contractions. We show that this phenome...	\section{Introduction} The classical von Neumann inequality \cite{vN1951} states that for a contractive linear operator $T$ acting on a Hilbert space, we always have that \[ \\|f(T)\\|\leq \sup_{z\in {\mathbb{D}}}\|f(z)\| \] for every polynomial $f$, where ${\mathbb{D}}$ denotes the open unit disc in the complex plane. Th...	{ "timestamp": "2018-06-26T02:07:12", "yymm": "1806", "arxiv_id": "1806.09027", "language": "en", "url": "https://arxiv.org/abs/1806.09027", "abstract": "An example due to Pisier shows that two commuting, completely polynomially bounded Hilbert space operators may not be simultaneously similar to contractio...
https://arxiv.org/abs/2112.08283	Guarantees for existence of a best canonical polyadic approximation of a noisy low-rank tensor	The canonical polyadic decomposition (CPD) of a low rank tensor plays a major role in data analysis and signal processing by allowing for unique recovery of underlying factors. However, it is well known that the low rank CPD approximation problem is ill-posed. That is, a tensor may fail to have a best rank $R$ CPD appr...	\section{Introduction} Tensors, i.e. multiindexed arrays, are natural generalizations of matrices and are commonplace in fields such as machine learning and signal processing where data sets often have inherent higher-order structure \cite{Setal17,Cetal15}. Tensor decompositions are powerful tools which can be ...	{ "timestamp": "2021-12-16T02:25:49", "yymm": "2112", "arxiv_id": "2112.08283", "language": "en", "url": "https://arxiv.org/abs/2112.08283", "abstract": "The canonical polyadic decomposition (CPD) of a low rank tensor plays a major role in data analysis and signal processing by allowing for unique recovery ...
https://arxiv.org/abs/2109.10222	Extremal Uniquely Resolvable Multisets	For positive integers $n$ and $m$, consider a multiset of non-empty subsets of $[m]$ such that there is a \textit{unique} partition of these subsets into $n$ partitions of $[m]$. We study the maximum possible size $g(n,m)$ of such a multiset. We focus on the regime $n \leq 2^{m-1}-1$ and show that $g(n,m) \geq \Omega(\...	\section{Introduction} In this paper, we study the maximum size of a multiset that admits a certain kind of unique partition. This problem was originally motivated by attempting to count the minimum number of rules required to guarantee the existence of a unique solution for a logic based problem called the zebra puzz...	{ "timestamp": "2022-09-02T02:00:45", "yymm": "2109", "arxiv_id": "2109.10222", "language": "en", "url": "https://arxiv.org/abs/2109.10222", "abstract": "For positive integers $n$ and $m$, consider a multiset of non-empty subsets of $[m]$ such that there is a \\textit{unique} partition of these subsets into...
https://arxiv.org/abs/2110.08319	A Proof of the Optimal Leapfrogging Conjecture	Suppose we place checkers in the lower left corner of a Go board and wish to move them to the upper right corner in as few moves as possible, where the pieces move as in the game of Chinese checkers. Auslander, Benjamin, and Wilkerson in 1993 generalized this game for integer lattices and defined a measure of speed for...	\section{Introduction} Suppose we have some checkers placed in the lower left corner of a Go board, and we wish to move them to the upper right corner in as few moves as possible. There are no opponent pieces present, and the pieces move as they would in the game of Chinese Checkers, where for one move, a piece may ei...	{ "timestamp": "2021-10-19T02:02:07", "yymm": "2110", "arxiv_id": "2110.08319", "language": "en", "url": "https://arxiv.org/abs/2110.08319", "abstract": "Suppose we place checkers in the lower left corner of a Go board and wish to move them to the upper right corner in as few moves as possible, where the pi...
https://arxiv.org/abs/1912.01253	Tropical convex hulls of polyhedral sets	In this paper we focus on the tropical convex hull of convex sets and polyhedral complexes. We give a vertex description of the tropical convex hull of a line segment and a ray. %in \RR^{n+1}/\RR\mathbf{1}.Next we show that tropical convex hull and ordinary convex hull commute in two dimensions and characterize tropica...	\section*{Introduction} \label{sec:intro} Tropical convexity is the analog of classical convexity in the {\em tropical semiring} $(\mathbb{R},\oplus,\odot)$ where $a\oplus b=\min(a,b),$ and $a\odot b = a+b.$ The goal of this paper is to explore the interplay between tropical convexity and its classical counterpart. O...	{ "timestamp": "2020-09-08T02:06:41", "yymm": "1912", "arxiv_id": "1912.01253", "language": "en", "url": "https://arxiv.org/abs/1912.01253", "abstract": "In this paper we focus on the tropical convex hull of convex sets and polyhedral complexes. We give a vertex description of the tropical convex hull of a ...
https://arxiv.org/abs/1409.0488	Generalizing Zeckendorf's Theorem: The Kentucky Sequence	By Zeckendorf's theorem, an equivalent definition of the Fibonacci sequence (appropriately normalized) is that it is the unique sequence of increasing integers such that every positive number can be written uniquely as a sum of non-adjacent elements; this is called a legal decomposition. Previous work examined the dist...	\section{Introduction} One of the standard definitions of the Fibonacci numbers $\{F_n\}$ is that it is the unique sequence satisfying the recurrence $F_{n+1} = F_n + F_{n-1}$ with initial conditions $F_1 = 1$, $F_2 = 2$. An interesting and equivalent definition is that it is the unique increasing sequence of pos...	{ "timestamp": "2014-09-02T02:14:14", "yymm": "1409", "arxiv_id": "1409.0488", "language": "en", "url": "https://arxiv.org/abs/1409.0488", "abstract": "By Zeckendorf's theorem, an equivalent definition of the Fibonacci sequence (appropriately normalized) is that it is the unique sequence of increasing integ...
https://arxiv.org/abs/1702.01718	Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow	We show how to view the standard Follow-the-Leader (FtL) model as a numerical method to compute numerically the solution of the Lighthill--Whitham--Richards (LWR) model for traffic flow. As a result we offer a simple proof that FtL models converge to the LWR model for traffic flow when traffic becomes dense. The proof ...	\section{Introduction} \label{sec:intro} There are two paradigms in the mathematical modeling of traffic flow. One is based on an individual modeling of each vehicle with the dynamics governed by the distance between adjacent vehicles. The other is based on the assumption of dense traffic where the vehicles are represe...	{ "timestamp": "2017-09-22T02:09:39", "yymm": "1702", "arxiv_id": "1702.01718", "language": "en", "url": "https://arxiv.org/abs/1702.01718", "abstract": "We show how to view the standard Follow-the-Leader (FtL) model as a numerical method to compute numerically the solution of the Lighthill--Whitham--Richar...
https://arxiv.org/abs/1001.0559	Julius and Julia: Mastering the art of the Schwarz lemma	This article discusses classical versions of the Schwarz lemma at the boundary of the unit disk in the complex plane. The exposition includes commentary on the history, the mathematics, and the applications.	\section{Introduction.} Despite teaching complex analysis for a quarter century, I still didn't know Jack about the Schwarz lemma. That's what I found out after a graduate student buttonholed me at the door of the printer room. ``Oh, professor, I noticed while studying for the qualifying exam that the book of Bak and ...	{ "timestamp": "2010-01-04T20:42:31", "yymm": "1001", "arxiv_id": "1001.0559", "language": "en", "url": "https://arxiv.org/abs/1001.0559", "abstract": "This article discusses classical versions of the Schwarz lemma at the boundary of the unit disk in the complex plane. The exposition includes commentary on ...
https://arxiv.org/abs/1709.03638	Quantitative representation stability over linear groups	We introduce a technique for proving quantitative representation stability theorems for sequences of representations of certain finite linear groups over a field of characteristic zero. In particular, we prove a vanishing result for higher syzygies of VIC- and SI-modules, which can be thought of as a weaker version of ...	\section{Introduction} Putman--Sam \cite{PS} introduced techniques for proving \emph{representation stability} results in the sense of Church--Ellenberg--Farb \cite{CEF} for sequences of representations of several families of finite linear groups. They applied their tools to prove stability results for the homolog...	{ "timestamp": "2018-10-05T02:11:18", "yymm": "1709", "arxiv_id": "1709.03638", "language": "en", "url": "https://arxiv.org/abs/1709.03638", "abstract": "We introduce a technique for proving quantitative representation stability theorems for sequences of representations of certain finite linear groups over ...
https://arxiv.org/abs/1706.08253	Lebesgue and gaussian measure of unions of basic semi-algebraic sets	Given a finite Borel measure $\mu$ on R n and basic semi-algebraic sets $\Omega$\_i $\subset$ R n , i = 1,. .. , p, we provide a systematic numerical scheme to approximate as closely as desired $\mu$(\cup\_i $\Omega$\_i), when all moments of $\mu$ are available (and finite). More precisely , we provide a hierarchy of s...	\section*{Abstract} Given a finite Borel measure $\mu$ on $\mathbb{R}^n$ and basic semi-algebraic sets $\mathbf{\Omega}_i\subset\mathbb{R}^n$, $i=1,\ldots,p$, we provide a systematic numerical scheme to approximate as closely as desired $\mu(\bigcup_i\mathbf{\Omega}_i)$, when all moments of $\mu$ are available (and f...	{ "timestamp": "2017-06-27T02:09:56", "yymm": "1706", "arxiv_id": "1706.08253", "language": "en", "url": "https://arxiv.org/abs/1706.08253", "abstract": "Given a finite Borel measure $\\mu$ on R n and basic semi-algebraic sets $\\Omega$\\_i $\\subset$ R n , i = 1,. .. , p, we provide a systematic numerical ...
https://arxiv.org/abs/2107.14067	Polynomial approximation avoiding values in sets II	We prove some results on when functions on compact sets $K \subset \mathbb C$ can be approximated by polynomials avoiding values in given sets. We also prove some higher dimensional analogues. In particular we prove that a continuous function from a compact set $K \subset \mathbb R^n$ without interior points to $\mathb...	\section{Introduction} In \cite{AndRos}, \cite{Rousu} the following version of Lavrent\'ev's theorem was proved \begin{thm} \label{TH1} Let $A \subset \C $ be any countable set, let $K \subset \C$ be a compact set with connected complement and without interior points, and let $f$ be a continuous function on $K$....	{ "timestamp": "2021-08-17T02:21:15", "yymm": "2107", "arxiv_id": "2107.14067", "language": "en", "url": "https://arxiv.org/abs/2107.14067", "abstract": "We prove some results on when functions on compact sets $K \\subset \\mathbb C$ can be approximated by polynomials avoiding values in given sets. We also ...
https://arxiv.org/abs/math/0606356	$f$-Vectors of Barycentric Subdivisions	For a simplicial complex or more generally Boolean cell complex $\Delta$ we study the behavior of the $f$- and $h$-vector under barycentric subdivision. We show that if $\Delta$ has a non-negative $h$-vector then the $h$-polynomial of its barycentric subdivision has only simple and real zeros. As a consequence this imp...	\section{Introduction} \label{introduction} This work is concerned with the effect of barycentric subdivision on the enumerative structure of a simplicial complex. More precisely, we study the behavior of the $f$- and $h$-vector of a simplicial complex, or more generally, Boolean cell complex, under barycentric subdi...	{ "timestamp": "2006-06-15T13:44:05", "yymm": "0606", "arxiv_id": "math/0606356", "language": "en", "url": "https://arxiv.org/abs/math/0606356", "abstract": "For a simplicial complex or more generally Boolean cell complex $\\Delta$ we study the behavior of the $f$- and $h$-vector under barycentric subdivisi...
https://arxiv.org/abs/2111.00497	An Algorithm taking Kirby diagrams to Trisection diagrams	We present an algorithm taking a Kirby diagram of a closed oriented $4$-manifold to a trisection diagram of the same manifold. This algorithm provides us with a large number of examples for trisection diagrams of closed oriented $4$-manifolds since many Kirby-diagrammatic descriptions of closed oriented $4$-manifolds a...	\section{Introduction} \label{section:introduction} Kirby diagrams are a classical way of diagrammatically describing closed $4$-manifolds that has been hugely successful and that is well understood. Their usefulness lies in the fact that they fully describe a closed $4$-manifold using only a framed link and pairs of ...	{ "timestamp": "2021-11-02T01:19:49", "yymm": "2111", "arxiv_id": "2111.00497", "language": "en", "url": "https://arxiv.org/abs/2111.00497", "abstract": "We present an algorithm taking a Kirby diagram of a closed oriented $4$-manifold to a trisection diagram of the same manifold. This algorithm provides us ...
https://arxiv.org/abs/2102.08181	On Greedily Packing Anchored Rectangles	Consider a set P of points in the unit square U, one of them being the origin. For each point p in P you may draw a rectangle in U with its lower-left corner in p. What is the maximum area such rectangles can cover without overlapping each other? Freedman [1969] posed this problem in 1969, asking whether one can always...	\section{Introduction}% \label{sec:introduction} The \LLARPlong/ (\LLARP/) problem considers a finite set $P \subseteq \cU \coloneqq \intco{0, 1}^2$ with $(0, 0) \in P$ of \emph{input points}. The goal is to find a set of non-empty, interior-disjoint rectangles $\intoo{r_p}_{p \in P}$ with $p$ being the lower-left co...	{ "timestamp": "2021-02-17T02:22:48", "yymm": "2102", "arxiv_id": "2102.08181", "language": "en", "url": "https://arxiv.org/abs/2102.08181", "abstract": "Consider a set P of points in the unit square U, one of them being the origin. For each point p in P you may draw a rectangle in U with its lower-left cor...
https://arxiv.org/abs/1612.01212	Counting numerical semigroups by genus and even gaps	Let $n_g$ be the number of numerical semigroups of genus $g$. We present an approach to compute $n_g$ by using even gaps, and the question: Is it true that $n_{g+1}>n_g$? is investigated. Let $N_\gamma(g)$ be the number of numerical semigroups of genus $g$ whose number of even gaps equals $\gamma$. We show that $N_\gam...	\section{Introduction}\label{s1} A {\em numerical semigroup} $S$ is a submonoid of the set of nonnegative integers $\mathbb N_0$, equipped with the usual addition, such that $G(S):=\mathbb N_0\setminus S$, the set of {\em gaps} of $S$, is finite. The number of elements $g=g(S)$ of $G(S)$ is called the {\em genus} ...	{ "timestamp": "2017-08-15T02:05:48", "yymm": "1612", "arxiv_id": "1612.01212", "language": "en", "url": "https://arxiv.org/abs/1612.01212", "abstract": "Let $n_g$ be the number of numerical semigroups of genus $g$. We present an approach to compute $n_g$ by using even gaps, and the question: Is it true tha...
https://arxiv.org/abs/1010.2014	Recurrence and Polya number of general one-dimensional random walks	The recurrence properties of random walks can be characterized by Pólya number, i.e., the probability that the walker has returned to the origin at least once. In this paper, we consider recurrence properties for a general 1D random walk on a line, in which at each time step the walker can move to the left or right wit...	\section{The method of creative telescoping (MCT)} The method of creative telescoping, also known as Zeilberger's algorithm~\cite{rn14,rn15,rn16}, is a powerful tool for solving problem involving definite integration and summation of hypergeometric function. Suppose we are given a certain holonomic function of two vari...	{ "timestamp": "2010-10-12T02:02:54", "yymm": "1010", "arxiv_id": "1010.2014", "language": "en", "url": "https://arxiv.org/abs/1010.2014", "abstract": "The recurrence properties of random walks can be characterized by Pólya number, i.e., the probability that the walker has returned to the origin at least on...
https://arxiv.org/abs/0909.4966	Pattern avoidance and RSK-like algorithms for alternating permutations and Young tableaux	We define a class L_{n, k} of permutations that generalizes alternating (up-down) permutations and give bijective proofs of certain pattern-avoidance results for this class. As a special case of our results, we give two bijections between the set A_{2n}(1234) of alternating permutations of length 2n with no four-term i...	\section{Introduction}\label{sec:intro} A classical problem asks for the number of permutations that avoid a certain permutation pattern. This problem has received a great deal of attention (see e.g., \cite{simion, 1342, 1234}) and has led to a number of interesting variations including the enumeration of special cla...	{ "timestamp": "2010-03-24T01:00:36", "yymm": "0909", "arxiv_id": "0909.4966", "language": "en", "url": "https://arxiv.org/abs/0909.4966", "abstract": "We define a class L_{n, k} of permutations that generalizes alternating (up-down) permutations and give bijective proofs of certain pattern-avoidance result...
https://arxiv.org/abs/0903.1795	A Parameter-Uniform Finite Difference Method for Multiscale Singularly Perturbed Linear Dynamical Systems	A system of singularly perturbed ordinary differential equations of first order with given initial conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These parameters are assumed to be distinct and they determine the different scales in the solution to this problem....	\section{Introduction} We consider the initial value problem for the singularly perturbed system of linear first order differential equations \begin{eqnarray}\label{IVP} E\vec{u}^{\prime}(t)+A(t)\vec{u}(t)=\vec{f}(t), \;\; t \in (0,T], \;\; \vec{u}(0)\;\; \mathrm{given}. \end{eqnarray} Here $\vec{u}$ is a column $n$-v...	{ "timestamp": "2009-03-10T16:22:42", "yymm": "0903", "arxiv_id": "0903.1795", "language": "en", "url": "https://arxiv.org/abs/0903.1795", "abstract": "A system of singularly perturbed ordinary differential equations of first order with given initial conditions is considered. The leading term of each equati...
https://arxiv.org/abs/1806.00900	Algorithmic Regularization in Learning Deep Homogeneous Models: Layers are Automatically Balanced	We study the implicit regularization imposed by gradient descent for learning multi-layer homogeneous functions including feed-forward fully connected and convolutional deep neural networks with linear, ReLU or Leaky ReLU activation. We rigorously prove that gradient flow (i.e. gradient descent with infinitesimal step ...	\subsection{Fully Connected Neural Networks} \label{subsec:conserved-fully-connected} We first formally define a fully connected feed-forward neural network with $N$ ($N\ge2$) layers. Let $\mathbf{W}^{(h)} \in \mathbb{R}^{n_h \times n_{h-1}}$ be the weight matrix in the $h$-th layer, and define $\vect{w} = ( \mathb...	{ "timestamp": "2018-11-01T01:14:46", "yymm": "1806", "arxiv_id": "1806.00900", "language": "en", "url": "https://arxiv.org/abs/1806.00900", "abstract": "We study the implicit regularization imposed by gradient descent for learning multi-layer homogeneous functions including feed-forward fully connected and...
https://arxiv.org/abs/1703.07870	General Heuristics for Nonconvex Quadratically Constrained Quadratic Programming	We introduce the Suggest-and-Improve framework for general nonconvex quadratically constrained quadratic programs (QCQPs). Using this framework, we generalize a number of known methods and provide heuristics to get approximate solutions to QCQPs for which no specialized methods are available. We also introduce an open-...	\section{Splitting quadratic forms}\label{s-split} In this appendix, we explore various ways of splitting an indefinite matrix $P$ as a difference $P_+ - P_-$ of two positive semidefinite matrices, as in~\eqref{quadratic-split}. The motivation for this discussion is that the performance of convex-concave procedure can ...	{ "timestamp": "2017-05-18T02:00:56", "yymm": "1703", "arxiv_id": "1703.07870", "language": "en", "url": "https://arxiv.org/abs/1703.07870", "abstract": "We introduce the Suggest-and-Improve framework for general nonconvex quadratically constrained quadratic programs (QCQPs). Using this framework, we genera...
https://arxiv.org/abs/0706.1062	Power-law distributions in empirical data	Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution -- the pa...	\section{Introduction} Many empirical quantities cluster around a typical value. The speeds of cars on a highway, the weights of apples in a store, air pressure, sea level, the temperature in New York at noon on Midsummer's Day. All of these things vary somewhat, but their distributions place a negligible amount of ...	{ "timestamp": "2009-02-02T18:49:44", "yymm": "0706", "arxiv_id": "0706.1062", "language": "en", "url": "https://arxiv.org/abs/0706.1062", "abstract": "Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phen...
https://arxiv.org/abs/2108.12680	Avoiding unwanted results in locally linear embedding: A new understanding of regularization	We demonstrate that locally linear embedding (LLE) inherently admits some unwanted results when no regularization is used, even for cases in which regularization is not supposed to be needed in the original algorithm. The existence of one special type of result, which we call ``projection pattern'', is mathematically p...	\section{Introduction} Let $\mathcal{X}=\{x_i\}_{i=1}^N$ be a collection of points in some high dimensional space $\mathbb{R}^D$. The general goal of manifold learning (or nonlinear dimensionality reduction) is to find for $\mathcal{X}$ a representation $\mathcal{Y}=\{y_i\}_{i=1}^N$ in some lower dimensional $\mat...	{ "timestamp": "2021-08-31T02:13:24", "yymm": "2108", "arxiv_id": "2108.12680", "language": "en", "url": "https://arxiv.org/abs/2108.12680", "abstract": "We demonstrate that locally linear embedding (LLE) inherently admits some unwanted results when no regularization is used, even for cases in which regular...
https://arxiv.org/abs/2109.07524	Exact and Approximation Algorithms for Many-To-Many Point Matching in the Plane	Given two sets $S$ and $T$ of points in the plane, of total size $n$, a {many-to-many} matching between $S$ and $T$ is a set of pairs $(p,q)$ such that $p\in S$, $q\in T$ and for each $r\in S\cup T$, $r$ appears in at least one such pair. The {cost of a pair} $(p,q)$ is the (Euclidean) distance between $p$ and $q$. In ...	\section{Introduction} Let $G=(V=S\cup T, E)$ be a simple bipartite graph where each edge has a non-negative real weight, and no vertex is isolated. The \emph{many-to-many} matching problem on $G$ is to find a subset of edges $E'\subseteq E$ of minimum total weight such that for each vertex $v\in V$ there is an edge ...	{ "timestamp": "2021-09-17T02:01:24", "yymm": "2109", "arxiv_id": "2109.07524", "language": "en", "url": "https://arxiv.org/abs/2109.07524", "abstract": "Given two sets $S$ and $T$ of points in the plane, of total size $n$, a {many-to-many} matching between $S$ and $T$ is a set of pairs $(p,q)$ such that $p...
https://arxiv.org/abs/1804.03752	Conjectured lower bound for the clique number of a graph	It is well known that $n/(n - \mu)$, where $\mu$ is the spectral radius of a graph with $n$ vertices, is a lower bound for the clique number. We conjecture that $\mu$ can be replaced in this bound with $\sqrt{s^+}$, where $s^+$ is the sum of the squares of the positive eigenvalues. We prove this conjecture for various ...	\section{Introduction} Let $G$ be a graph, with no isolated vertices, with $n$ vertices, edge set $E$ with $\|E\| = m$, average degree $d$, chromatic number $\chi(G)$ and clique number $\omega(G)$. We also let $A$ denote the adjacency matrix of $G$ and let $\mu = \mu_1 \ge \ldots \ge \mu_n$ denote the eigenvalues of $...	{ "timestamp": "2018-08-10T02:01:27", "yymm": "1804", "arxiv_id": "1804.03752", "language": "en", "url": "https://arxiv.org/abs/1804.03752", "abstract": "It is well known that $n/(n - \\mu)$, where $\\mu$ is the spectral radius of a graph with $n$ vertices, is a lower bound for the clique number. We conject...