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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/2001.03082	Obtaining higher-order Galerkin accuracy when the boundary is polygonally approximated	We study two techniques for correcting the geometrical error associated with domain approximation by a polygon. The first was introduced some time ago \cite{bramble1972projection} and leads to a nonsymmetric formulation for Poisson's equation. We introduce a new technique that yields a symmetric formulation and has sim...	\section{Introduction} When a Dirichlet problem on a smooth domain is approximated by a polygon, an error occurs that is suboptimal for quadratic approximation \cite{lrsBIBaa,lrsBIBab,lrsBIBae}. However, this can be corrected by a modification of the variational form \cite{bramble1972projection}. Here we review this...	{ "timestamp": "2020-01-10T02:13:51", "yymm": "2001", "arxiv_id": "2001.03082", "language": "en", "url": "https://arxiv.org/abs/2001.03082", "abstract": "We study two techniques for correcting the geometrical error associated with domain approximation by a polygon. The first was introduced some time ago \\c...
https://arxiv.org/abs/2211.09530	Rainbow even cycles	We prove that every family of (not necessarily distinct) even cycles $D_1, \dotsc, D_{\lfloor 1.2(n-1) \rfloor+1}$ on some fixed $n$-vertex set has a rainbow even cycle (that is, a set of edges from distinct $D_i$'s, forming an even cycle). This resolves an open problem of Aharoni, Briggs, Holzman and Jiang. Moreover, ...	\section{Introduction} \label{sec:intro} Let $\mathcal{F}$ be a set family. A rainbow set with respect to $\mathcal{F}$ is a subset $R$ (without repeated elements) of $\bigcup_{F \in \mathcal{F}} F$ such that there exists an injection $\sigma \colon R \to \mathcal{F}$ with $r \in \sigma(r)$ for all $r \in R$. In other...	{ "timestamp": "2022-11-18T02:12:21", "yymm": "2211", "arxiv_id": "2211.09530", "language": "en", "url": "https://arxiv.org/abs/2211.09530", "abstract": "We prove that every family of (not necessarily distinct) even cycles $D_1, \\dotsc, D_{\\lfloor 1.2(n-1) \\rfloor+1}$ on some fixed $n$-vertex set has a r...
https://arxiv.org/abs/2202.01296	Sidon sets in a union of intervals	We study the maximum size of Sidon sets in unions of integers intervals. If $A\subseteq\mathbb{N}$ is the union of two intervals and if $\left\| A \right\|=n$ (where $\left\| A \right\|$ denotes the cardinality of $A$), we prove that $A$ contains a Sidon set of size at least $0, 876\sqrt{n}$. On the other hand, by using th...	\section{Introduction} A Sidon set of integers is a subset of $\mathbb{N}$ with the property that all sums of two elements are distinct. Working on Fourier series, Simon Sidon \cite{Simon_Sidon} was the first to take an interest in these sets. He sought to bound the size of the largest Sidon set in $\left\llbracket1,n...	{ "timestamp": "2022-02-04T02:04:02", "yymm": "2202", "arxiv_id": "2202.01296", "language": "en", "url": "https://arxiv.org/abs/2202.01296", "abstract": "We study the maximum size of Sidon sets in unions of integers intervals. If $A\\subseteq\\mathbb{N}$ is the union of two intervals and if $\\left\| A \\rig...
https://arxiv.org/abs/1702.04915	Scaling limit of the uniform prudent walk	We study the 2-dimensional uniform prudent self-avoiding walk, which assigns equal probability to all nearest-neighbor self-avoiding paths of a fixed length that respect the prudent condition, namely, the path cannot take any step in the direction of a previously visited site. The uniform prudent walk has been investig...	\section{Introduction} The prudent walk was introduced in \cite{TD87b,TD87a} and \cite{SSK01} as a simplified version of the self-avoiding walk. It has attracted the attention of the combinatorics community in recent years, see e.g., \cite{B10,BI15,DG08}, and also the probability community, see e.g. \cite{BFV10} and \...	{ "timestamp": "2017-09-08T02:05:20", "yymm": "1702", "arxiv_id": "1702.04915", "language": "en", "url": "https://arxiv.org/abs/1702.04915", "abstract": "We study the 2-dimensional uniform prudent self-avoiding walk, which assigns equal probability to all nearest-neighbor self-avoiding paths of a fixed leng...
https://arxiv.org/abs/2211.14807	Universal convex covering problems under translation and discrete rotations	We consider the smallest-area universal covering of planar objects of perimeter 2 (or equivalently closed curves of length 2) allowing translation and discrete rotations. In particular, we show that the solution is an equilateral triangle of height 1 when translation and discrete rotation of $\pi$ are allowed. Our proo...	\section{Introduction} Given a (possibly infinite) set $S$ of planar objects and a group $G$ of geometric transformations, a $G$-covering\xspace $K$ of $S$ is a region such that every object in $S$ can be contained in $K$ by transforming the object with a suitable transformation $g \in G$. Equivalently, every object...	{ "timestamp": "2022-11-29T02:13:12", "yymm": "2211", "arxiv_id": "2211.14807", "language": "en", "url": "https://arxiv.org/abs/2211.14807", "abstract": "We consider the smallest-area universal covering of planar objects of perimeter 2 (or equivalently closed curves of length 2) allowing translation and dis...
https://arxiv.org/abs/1311.5051	Separating path systems	We study separating systems of the edges of a graph where each member of the separating system is a path. We conjecture that every $n$-vertex graph admits a separating path system of size $O(n)$ and prove this in certain interesting special cases. In particular, we establish this conjecture for random graphs and graphs...	\section{Introduction} Given a set $S$, we say that a family $\mathcal{F}$ of subsets of $S$ \emph{separates} a pair of distinct elements $x,y \in S$ if there exists a set $A\in \mathcal{F}$ which contains exactly one of $x$ and $y$. If $\mathcal{F}$ separates all pairs of distinct elements of $S$, we say that $\mathca...	{ "timestamp": "2013-11-21T02:08:46", "yymm": "1311", "arxiv_id": "1311.5051", "language": "en", "url": "https://arxiv.org/abs/1311.5051", "abstract": "We study separating systems of the edges of a graph where each member of the separating system is a path. We conjecture that every $n$-vertex graph admits a...
https://arxiv.org/abs/1106.3622	On the connectivity of visibility graphs	The visibility graph of a finite set of points in the plane has the points as vertices and an edge between two vertices if the line segment between them contains no other points. This paper establishes bounds on the edge- and vertex-connectivity of visibility graphs.Unless all its vertices are collinear, a visibility g...	\section{Introduction} \seclabel{intro} Let $P$ be a finite set of points in the plane. Two distinct points $v$ and $w$ in the plane are \emph{visible} with respect to $P$ if no point in $P$ is in the open line segment $vw$. The \emph{visibility graph} of $P$ has vertex set $P$, where two vertices are adjacent if and ...	{ "timestamp": "2011-06-21T02:01:06", "yymm": "1106", "arxiv_id": "1106.3622", "language": "en", "url": "https://arxiv.org/abs/1106.3622", "abstract": "The visibility graph of a finite set of points in the plane has the points as vertices and an edge between two vertices if the line segment between them con...
https://arxiv.org/abs/1210.3799	Some remarks on the joint distribution of descents and inverse descents	We study the joint distribution of descents and inverse descents over the set of permutations of n letters. Gessel conjectured that the two-variable generating function of this distribution can be expanded in a given basis with nonnegative integer coefficients. We investigate the action of the Eulerian operators that g...	\section{Introduction} Let $\mathfrak{S}_n$ denote the set of permutations of $\{1, \dotsc, n\}$. The number of \emph{descents} in a permutation $\pi = \pi_1\dotso\pi_n$ is defined as $\des(\pi) = \|\{ i : \pi_i > \pi_{i+1}\}\|.$ Our object of study is the two-variable generating function of descents and \emph{inverse...	{ "timestamp": "2012-10-16T02:03:22", "yymm": "1210", "arxiv_id": "1210.3799", "language": "en", "url": "https://arxiv.org/abs/1210.3799", "abstract": "We study the joint distribution of descents and inverse descents over the set of permutations of n letters. Gessel conjectured that the two-variable generat...
https://arxiv.org/abs/1410.3755	3-manifolds Modulo Surgery Triangles	Surgery triangles are an important computational tool in Floer homology. Given a connected oriented surface $\Sigma$, we consider the abelian group $K(\Sigma)$ generated by bordered 3-manifolds with boundary $\Sigma$, modulo the relation that the three manifolds involved in any surgery triangle sum to zero. We show tha...	\section{Introduction} Floer homology theories can be used to define a number of different invariants of closed, oriented $3$-manifolds. Many of these theories satisfy a ``surgery triangle'', which is a relationship between the invariants of different Dehn surgeries $Y_{\alpha}(K)$ on a knot $K$ in a fixed manifold $...	{ "timestamp": "2014-10-31T01:06:55", "yymm": "1410", "arxiv_id": "1410.3755", "language": "en", "url": "https://arxiv.org/abs/1410.3755", "abstract": "Surgery triangles are an important computational tool in Floer homology. Given a connected oriented surface $\\Sigma$, we consider the abelian group $K(\\Si...
https://arxiv.org/abs/2102.09924	A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions	Gradient descent optimization algorithms are the standard ingredients that are used to train artificial neural networks (ANNs). Even though a huge number of numerical simulations indicate that gradient descent optimization methods do indeed convergence in the training of ANNs, until today there is no rigorous theoretic...	\section{Introduction} Gradient descent (GD) optimization schemes are the standard methods for the training of artificial neural networks (ANNs). Although a large number of numerical simulations hint that GD optimization methods do converge in the training of ANNs, in general there is no mathematical analysis in the ...	{ "timestamp": "2021-02-22T02:17:21", "yymm": "2102", "arxiv_id": "2102.09924", "language": "en", "url": "https://arxiv.org/abs/2102.09924", "abstract": "Gradient descent optimization algorithms are the standard ingredients that are used to train artificial neural networks (ANNs). Even though a huge number ...
https://arxiv.org/abs/0712.2182	Optimal codes for correcting a single (wrap-around) burst of errors	In 2007, Martinian and Trott presented codes for correcting a burst of erasures with a minimum decoding delay. Their construction employs [n,k] codes that can correct any burst of erasures (including wrap-around bursts) of length n-k. The raised the question if such [n,k] codes exist for all integers k and n with 1<= k...	\section{Introduction} In \cite{MaTr}, Martinian and Trott present codes for correcting a burst of erasures with a minimum decoding delay. Their construction employs $[n,k]$ codes that can correct any burst of erasures (including wrap-around bursts) of length $n-k$. Examples of such codes are MDS codes and cyclic code...	{ "timestamp": "2007-12-13T17:33:59", "yymm": "0712", "arxiv_id": "0712.2182", "language": "en", "url": "https://arxiv.org/abs/0712.2182", "abstract": "In 2007, Martinian and Trott presented codes for correcting a burst of erasures with a minimum decoding delay. Their construction employs [n,k] codes that c...
https://arxiv.org/abs/1309.0920	Topology of geometric joins	We consider the geometric join of a family of subsets of the Euclidean space. This is a construction frequently used in the (colorful) Carathéodory and Tverberg theorems, and their relatives. We conjecture that when the family has at least $d+1$ sets, where $d$ is the dimension of the space, then the geometric join is ...	\section{Introduction} The purpose of this paper is to introduce the notion of a geometric join, and to study its topological connectedness. The geometric join is a natural object which appears in the proof of the colorful Carath\'eodory theorem~\cite{bar1982} and Tverberg's theorem~\cite{tver1966}; see chapter 8 in...	{ "timestamp": "2015-01-08T02:09:40", "yymm": "1309", "arxiv_id": "1309.0920", "language": "en", "url": "https://arxiv.org/abs/1309.0920", "abstract": "We consider the geometric join of a family of subsets of the Euclidean space. This is a construction frequently used in the (colorful) Carathéodory and Tver...
https://arxiv.org/abs/2006.01718	Proximity in Concave Integer Quadratic Programming	A classic result by Cook, Gerards, Schrijver, and Tardos provides an upper bound of $n \Delta$ on the proximity of optimal solutions of an Integer Linear Programming problem and its standard linear relaxation. In this bound, $n$ is the number of variables and $\Delta$ denotes the maximum of the absolute values of the s...	\section{Introduction} The relationship between an Integer Linear Programming problem and its standard linear relaxation plays a crucial role in many theoretical and computational aspects of the field, including perfect formulations, cutting planes, and branch-and-bound. Proximity results study one of the most fun...	{ "timestamp": "2021-04-16T02:03:49", "yymm": "2006", "arxiv_id": "2006.01718", "language": "en", "url": "https://arxiv.org/abs/2006.01718", "abstract": "A classic result by Cook, Gerards, Schrijver, and Tardos provides an upper bound of $n \\Delta$ on the proximity of optimal solutions of an Integer Linear...
https://arxiv.org/abs/2206.10220	Linear multistep methods and global Richardson extrapolation	In this work, we study the application the classical Richardson extrapolation (RE) technique to accelerate the convergence of sequences resulting from linear multistep methods (LMMs) for solving initial-value problems of systems of ordinary differential equations numerically. The advantage of the LMM-RE approach is tha...	\section{Introduction} Richardson extrapolation (RE) \cite{richardson1911, richardson1927} is a classical technique to accelerate the convergence of numerical sequences depending on a small parameter, by eliminating the lowest order error term(s) from the corresponding asymptotic expansion. When the sequence is genera...	{ "timestamp": "2022-06-22T02:45:06", "yymm": "2206", "arxiv_id": "2206.10220", "language": "en", "url": "https://arxiv.org/abs/2206.10220", "abstract": "In this work, we study the application the classical Richardson extrapolation (RE) technique to accelerate the convergence of sequences resulting from lin...
https://arxiv.org/abs/2205.00786	Solving PDEs by Variational Physics-Informed Neural Networks: an a posteriori error analysis	We consider the discretization of elliptic boundary-value problems by variational physics-informed neural networks (VPINNs), in which test functions are continuous, piecewise linear functions on a triangulation of the domain. We define an a posteriori error estimator, made of a residual-type term, a loss-function term,...	\section{Introduction} \label{sec1} The possibility of using deep-learning tools for solving complex physical models has attracted the attention of many scientists over the last few years. We have in mind in this paper models that are mathematically described by partial differential equations, supplemented by suitable...	{ "timestamp": "2022-05-03T02:37:59", "yymm": "2205", "arxiv_id": "2205.00786", "language": "en", "url": "https://arxiv.org/abs/2205.00786", "abstract": "We consider the discretization of elliptic boundary-value problems by variational physics-informed neural networks (VPINNs), in which test functions are c...
https://arxiv.org/abs/2011.08364	On Integer Balancing of Digraphs	A weighted digraph is balanced if the sums of the weights of the incoming and of the outgoing edges are equal at each vertex. We show that if these sums are integers, then the edge weights can be integers as well.	\section{Introduction} Let $G = (V, E)$ be a strongly connected digraph on $n$ vertices, with vertex set $V$ and edge set $E$. We use $v_iv_j$ to denote an edge from $v_i$ to $v_j$. The digraph $G$ can have self-arcs. For a vertex $v_i$, let $N^-(v_i):=\{v_j \in V \mid v_iv_j\in E\}$ and $N^+(v_i):=\{v_k \in V\mid v_...	{ "timestamp": "2020-11-20T02:07:34", "yymm": "2011", "arxiv_id": "2011.08364", "language": "en", "url": "https://arxiv.org/abs/2011.08364", "abstract": "A weighted digraph is balanced if the sums of the weights of the incoming and of the outgoing edges are equal at each vertex. We show that if these sums a...
https://arxiv.org/abs/2109.09006	Enumeration of self-reciprocal irreducible monic polynomials with prescribed leading coefficients over a finite field	A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we enumerate self-reciprocal irreducible monic polynomials over a finite field with prescribed leading coefficients. Asymptotic expression with explicit error bound is derived, which is used to show...	\section{ Introduction} In this paper we use our recent results from \cite{GKW21b} to enumerate self-reciprocal irreducible monic polynomials with prescribed leading coefficients. The following is a list of notations which will be used throughout the paper. \begin{itemize} \item ${\mathbb F}_{q}$ denotes the finit...	{ "timestamp": "2021-10-14T02:05:33", "yymm": "2109", "arxiv_id": "2109.09006", "language": "en", "url": "https://arxiv.org/abs/2109.09006", "abstract": "A polynomial is called self-reciprocal (or palindromic) if the sequence of its coefficients is palindromic. In this paper we enumerate self-reciprocal irr...
https://arxiv.org/abs/1703.08834	On connectedness of power graphs of finite groups	The power graph of a group $G$ is the graph whose vertex set is $G$ and two distinct vertices are adjacent if one is a power of the other. This paper investigates the minimal separating sets of power graphs of finite groups. For power graphs of finite cyclic groups, certain minimal separating sets are obtained. Consequ...	\section*{Introduction} Study of graphs associated to algebraic structures has a long history. There are various graphs constructed from groups and semigroups, e.g., Cayley graphs \cite{cayley1878desiderata, budden1985cayley}, intersection graphs \cite{MR3323326, zelinka1975intersection}, and commuting graphs \cite{ba...	{ "timestamp": "2017-03-28T02:07:42", "yymm": "1703", "arxiv_id": "1703.08834", "language": "en", "url": "https://arxiv.org/abs/1703.08834", "abstract": "The power graph of a group $G$ is the graph whose vertex set is $G$ and two distinct vertices are adjacent if one is a power of the other. This paper inve...
https://arxiv.org/abs/0808.2664	Communication-optimal parallel and sequential QR and LU factorizations	We present parallel and sequential dense QR factorization algorithms that are both optimal (up to polylogarithmic factors) in the amount of communication they perform, and just as stable as Householder QR.We prove optimality by extending known lower bounds on communication bandwidth for sequential and parallel matrix m...	\section{Lower Bounds for CAQR} \label{sec:LowerBounds_CAQR} In this section, we review known lower bounds on communication bandwidth for parallel and sequential $\Theta (n^3)$ matrix-matrix multiplication of matrices stored in 2-D layouts, extend some of them to the rectangular case, and then extend them t...	{ "timestamp": "2008-08-19T23:53:43", "yymm": "0808", "arxiv_id": "0808.2664", "language": "en", "url": "https://arxiv.org/abs/0808.2664", "abstract": "We present parallel and sequential dense QR factorization algorithms that are both optimal (up to polylogarithmic factors) in the amount of communication th...
https://arxiv.org/abs/1809.08460	Some notes on the signed bad number in bipartite graphs	In this paper, we deal with the signed bad number and the negative decision number of graphs. We show that two upper bounds concerning these two parameters for bipartite graphs in papers [Discrete Math. Algorithms Appl. 1 (2011), 33--41] and [Australas. J. Combin. 41 (2008), 263--272] are not true as they stand. We cor...	\section{Introduction} \ \ \ Throughout this paper, let $G$ be a finite graph with vertex set $V(G)$ and edge set $E(G)$. We use \cite{we} as a reference for terminology and notation which are not defined here. The {\em open neighborhood} of a vertex $v$ is denoted by $N(v)$, and the {\em closed neighborhood} of $v$ i...	{ "timestamp": "2018-09-25T02:08:10", "yymm": "1809", "arxiv_id": "1809.08460", "language": "en", "url": "https://arxiv.org/abs/1809.08460", "abstract": "In this paper, we deal with the signed bad number and the negative decision number of graphs. We show that two upper bounds concerning these two parameter...
https://arxiv.org/abs/1607.00351	A Comparison of Preconditioned Krylov Subspace Methods for Large-Scale Nonsymmetric Linear Systems	Preconditioned Krylov subspace (KSP) methods are widely used for solving large-scale sparse linear systems arising from numerical solutions of partial differential equations (PDEs). These linear systems are often nonsymmetric due to the nature of the PDEs, boundary or jump conditions, or discretization methods. While i...	\section{Introduction} Preconditioned Krylov subspace (KSP) methods are widely used for solving large sparse linear systems, especially those arising from discretizations of partial differential equations. For most modern applications, these linear systems are nonsymmetric due to various reasons, such as the multiphys...	{ "timestamp": "2016-07-04T02:10:12", "yymm": "1607", "arxiv_id": "1607.00351", "language": "en", "url": "https://arxiv.org/abs/1607.00351", "abstract": "Preconditioned Krylov subspace (KSP) methods are widely used for solving large-scale sparse linear systems arising from numerical solutions of partial dif...
https://arxiv.org/abs/1708.07093	Where is the cone?	Real quadric curves are often referred to as "conic sections," implying that they can be realized as plane sections of circular cones. However, it seems that the details of this equivalence have been partially forgotten by the mathematical community. The definitive analytic treatment was given by Otto Staude in the 188...	\section{Introduction} \label{sec:introduction} A {\em real quadric curve} is defined by an equation of the form \begin{equation} ax^2+bxy+cy^2+dx+ey+f=0 \end{equation} for some real numbers $a,b,c,d,e,f\in\mathbb{R}$. One often sees real quadric curves described as ``conic sections." This terminology suggests that ...	{ "timestamp": "2018-01-12T02:01:43", "yymm": "1708", "arxiv_id": "1708.07093", "language": "en", "url": "https://arxiv.org/abs/1708.07093", "abstract": "Real quadric curves are often referred to as \"conic sections,\" implying that they can be realized as plane sections of circular cones. However, it seems...
https://arxiv.org/abs/1606.04975	Sharp geometric requirements in the Wachspress interpolation error estimate	Geometric conditions on general polygons are given in [9] in order to guarantee the error estimate for interpolants built from generalized barycentric coordinates, and the question about identifying sharp geometric restrictions in this setting is proposed. In this work, we address the question when the construction is ...	\section{Introduction} Many and different conditions on the geometry of finite elements were required in order to guarantee optimal convergence in the interpolation error estimate. Some of them deal with interior angles like the {\it maximum angle condition} (maximum interior angle bounded away from $\pi$) and the {\i...	{ "timestamp": "2017-07-03T02:06:28", "yymm": "1606", "arxiv_id": "1606.04975", "language": "en", "url": "https://arxiv.org/abs/1606.04975", "abstract": "Geometric conditions on general polygons are given in [9] in order to guarantee the error estimate for interpolants built from generalized barycentric coo...
https://arxiv.org/abs/1711.10561	Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations	We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential equations. In this two part treatise, we present our developments in the context of solving two main classes ...	\section{Systematic studies} \subsection{Continuous Time Models} \subsubsection{Example (Burgers' Equation)} As an example, let us consider the Burgers' equation. This equation arises in various areas of applied mathematics, including fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow \cite{basdevan...	{ "timestamp": "2017-11-30T02:01:47", "yymm": "1711", "arxiv_id": "1711.10561", "language": "en", "url": "https://arxiv.org/abs/1711.10561", "abstract": "We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of phy...
https://arxiv.org/abs/2104.15025	The Maximax Minimax Quotient Theorem	We present an optimization problem emerging from optimal control theory and situated at the intersection of fractional programming and linear max-min programming on polytopes. A naïve solution would require solving four nested, possibly nonlinear, optimization problems. Instead, relying on numerous geometric arguments ...	\section{Introduction} The field of fractional programming studies the optimization of a ratio of functions and made its debut in the 1960s with Charnes and Cooper \citep{charnes}. It has since then expanded to more complex and more general problems \citep{phuong}. However, outside of linear fractional programming, ve...	{ "timestamp": "2021-11-19T02:22:45", "yymm": "2104", "arxiv_id": "2104.15025", "language": "en", "url": "https://arxiv.org/abs/2104.15025", "abstract": "We present an optimization problem emerging from optimal control theory and situated at the intersection of fractional programming and linear max-min prog...
https://arxiv.org/abs/1701.02200	Bounding a global red-blue proportion using local conditions	We study the following local-to-global phenomenon: Let $B$ and $R$ be two finite sets of (blue and red) points in the Euclidean plane $\mathbb{R}^2$. Suppose that in each "neighborhood" of a red point, the number of blue points is at least as large as the number of red points. We show that in this case the total number...	\section{Introduction} Consider the following scenario in wireless networks. Suppose we have $n$ clients and $m$ antennas where both are represented as points in the plane (see Figure \ref{fig:hypothesis}). Each client has a wireless device that can communicate with the antennas. Assume also that each client is assoc...	{ "timestamp": "2017-04-21T02:05:47", "yymm": "1701", "arxiv_id": "1701.02200", "language": "en", "url": "https://arxiv.org/abs/1701.02200", "abstract": "We study the following local-to-global phenomenon: Let $B$ and $R$ be two finite sets of (blue and red) points in the Euclidean plane $\\mathbb{R}^2$. Sup...
https://arxiv.org/abs/1809.03926	Constructive regularization of the random matrix norm	We show a simple local norm regularization algorithm that works with high probability. Namely, we prove that if the entries of a $n \times n$ matrix $A$ are i.i.d. symmetrically distributed and have finite second moment, it is enough to zero out a small fraction of the rows and columns of $A$ with largest $L_2$ norms i...	\section{Introduction} \label{introduction} What should we call an \emph{optimal} order of an operator norm of a random $n \times n$ matrix? If we consider a matrix $A$ with independent standard Gaussian entries, then by the classical Bai-Yin law (see, for example, \cite{Tao}) $$ \\|A\\|/\sqrt{n} \to 2 \quad \text{ ...	{ "timestamp": "2018-09-12T02:12:43", "yymm": "1809", "arxiv_id": "1809.03926", "language": "en", "url": "https://arxiv.org/abs/1809.03926", "abstract": "We show a simple local norm regularization algorithm that works with high probability. Namely, we prove that if the entries of a $n \\times n$ matrix $A$ ...
https://arxiv.org/abs/1912.01607	Moments of Student's t-distribution: A Unified Approach	In this note, we derive the closed form formulae for moments of Student's t-distribution in the one dimensional case as well as in higher dimensions through a unified probability framework. Interestingly, the closed form expressions for the moments of Student's t-distribution can be written in terms of the familiar Gam...	\section{Remaining TODO Items} \newpage \section{Introduction} In probability and statistics, the location (e.g., mean), spread (e.g, standard deviation), skewness, and kurtosis play an important role in the modeling of random processes. One often uses the mean and standard deviation to construct confidence in...	{ "timestamp": "2021-03-29T02:09:24", "yymm": "1912", "arxiv_id": "1912.01607", "language": "en", "url": "https://arxiv.org/abs/1912.01607", "abstract": "In this note, we derive the closed form formulae for moments of Student's t-distribution in the one dimensional case as well as in higher dimensions throu...
https://arxiv.org/abs/1710.06916	Switch Functions	We define a switch function to be a function from an interval to $\{1,-1\}$ with a finite number of sign changes. (Special cases are the Walsh functions.) By a topological argument, we prove that, given $n$ real-valued functions, $f_1, \dots, f_n$, in $L^1[0,1]$, there exists a switch function, $\sigma$, with at most $...	\section{Introduction} In this paper, we provide a positive answer to the first (existence) part of a question raised by one of us \cite[end of Sec.~4]{HallTrigEllipt} and also an answer to the second (computation) part of the question in some special cases. \begin{definition} Given a real interval $[a,b]$, we cal...	{ "timestamp": "2018-04-16T02:03:06", "yymm": "1710", "arxiv_id": "1710.06916", "language": "en", "url": "https://arxiv.org/abs/1710.06916", "abstract": "We define a switch function to be a function from an interval to $\\{1,-1\\}$ with a finite number of sign changes. (Special cases are the Walsh functions...
https://arxiv.org/abs/1101.4740	Minimal area ellipses in the hyperbolic plane	We present uniqueness results for enclosing ellipses of minimal area in the hyperbolic plane. Uniqueness can be guaranteed if the minimizers are sought among all ellipses with prescribed axes or center. In the general case, we present a sufficient and easily verifiable criterion on the enclosed set that ensures uniquen...	\section{Introduction and statement of the main result} \label{sec:introduction} By a well-known theorem of convex geometry, a full-dimensional, compact subset $F$ of the Euclidean plane can be enclosed by a unique ellipse $C$ of minimal area. We share the general belief that this is an important but easy result. Th...	{ "timestamp": "2011-07-07T02:01:35", "yymm": "1101", "arxiv_id": "1101.4740", "language": "en", "url": "https://arxiv.org/abs/1101.4740", "abstract": "We present uniqueness results for enclosing ellipses of minimal area in the hyperbolic plane. Uniqueness can be guaranteed if the minimizers are sought amon...
https://arxiv.org/abs/2201.12932	A new proof of the description of the convex hull of space curves with totally positive torsion	We give new proofs of the description convex hulls of space curves $\gamma : [a,b] \mapsto \mathbb{R}^{d}$ having totally positive torsion. These are curves such that all the leading principal minors of $d\times d$ matrix $(\gamma', \gamma'', \ldots, \gamma^{(d)})$ are positive. In particular, we recover parametric rep...	\section{Introduction and a summary of main results} Convex hull of a set $K \subset \mathbb{R}^{d}$ is defined as \begin{align*} \mathrm{conv}(K) = \left\{ \sum_{j=1}^{m} \lambda_{j}x_{j}, \, x_{j} \in K, \, \sum_{j=1}^{m} \lambda_{j}=1, \, \lambda_{j} \geq 0,\, j=1, \ldots, m\; \text{for all}\; m\geq 1\right\}...	{ "timestamp": "2023-01-05T02:12:15", "yymm": "2201", "arxiv_id": "2201.12932", "language": "en", "url": "https://arxiv.org/abs/2201.12932", "abstract": "We give new proofs of the description convex hulls of space curves $\\gamma : [a,b] \\mapsto \\mathbb{R}^{d}$ having totally positive torsion. These are c...
https://arxiv.org/abs/2003.01812	The case for algebraic biology: from research to education	Though it goes without saying that linear algebra is fundamental to mathematical biology, polynomial algebra is less visible. In this article, we will give a brief tour of four diverse biological problems where multivariate polynomials play a central role -- a subfield that is sometimes called "algebraic biology." Name...	\section{Introduction} Nobody would dispute the fundamental role that linear algebra plays in applied fields such as mathematical biology. Systems of linear equations arise both as models of natural phenomena and as approximations of nonlinear models. As such, it is not hard to surmise that systems of nonlinear polyno...	{ "timestamp": "2020-03-05T02:03:58", "yymm": "2003", "arxiv_id": "2003.01812", "language": "en", "url": "https://arxiv.org/abs/2003.01812", "abstract": "Though it goes without saying that linear algebra is fundamental to mathematical biology, polynomial algebra is less visible. In this article, we will giv...
https://arxiv.org/abs/1210.5329	The optimal division between sample and background measurement time for photon counting experiments	Usually, equal time is given to measuring the background and the sample, or even a longer background measurement is taken as it has so few counts. While this seems the right thing to do, the relative error after background subtraction improves when more time is spent counting the measurement with the highest amount of ...	\section{Outline} \emph{Note by the author: It was found after this derivation, that the work presented here had been derived differently but with similar conclusions by Steinhart and Plestil \cite{Steinhart-1993}. We nevertheless believe that this simplified derivation may be of use to some readers.} Usually, equal ...	{ "timestamp": "2012-10-22T02:01:08", "yymm": "1210", "arxiv_id": "1210.5329", "language": "en", "url": "https://arxiv.org/abs/1210.5329", "abstract": "Usually, equal time is given to measuring the background and the sample, or even a longer background measurement is taken as it has so few counts. While thi...
https://arxiv.org/abs/2302.12152	Elastic snap-through instabilities are governed by geometric symmetries	Many elastic structures exhibit rapid shape transitions between two possible equilibrium states: umbrellas become inverted in strong wind and hopper popper toys jump when turned inside-out. This snap-through is a general motif for the storage and rapid release of elastic energy, and it is exploited by many biological a...	\section{Symmetry-breaking of a pitchfork bifurcation: a canonical example}\label{sec:symmetryPitchfork} We review the canonical example of symmetry-breaking in a system with pitchfork bifurcation. It is well-known that starting from a system with pitchfork bifurcation symmetry, the introduction of an additional param...	{ "timestamp": "2023-03-01T02:17:53", "yymm": "2302", "arxiv_id": "2302.12152", "language": "en", "url": "https://arxiv.org/abs/2302.12152", "abstract": "Many elastic structures exhibit rapid shape transitions between two possible equilibrium states: umbrellas become inverted in strong wind and hopper poppe...
https://arxiv.org/abs/2008.01669	Eigenvalues of graph Laplacians via rank-one perturbations	We show how the spectrum of a graph Laplacian changes with respect to a certain type of rank-one perturbation. We apply our finding to give new short proofs of the spectral version of Kirchhoff's Matrix Tree Theorem and known derivations for the characteristic polynomials of the Laplacians for several well known famili...	\section{Introduction} In this paper, we study finite simple graphs, i.e., graphs with finite vertex sets that do not contain loops or multiple edges. We use $V(G)$ and $E(G)$ to denote the vertex set and edge set of a graph $G$, respectively, and we take $V(G) = [n] := \{1,\ldots,n\}$ unless stated otherwise. Recal...	{ "timestamp": "2020-08-05T02:21:05", "yymm": "2008", "arxiv_id": "2008.01669", "language": "en", "url": "https://arxiv.org/abs/2008.01669", "abstract": "We show how the spectrum of a graph Laplacian changes with respect to a certain type of rank-one perturbation. We apply our finding to give new short proo...
https://arxiv.org/abs/1309.5600	A Generalization of Fibonacci Far-Difference Representations and Gaussian Behavior	A natural generalization of base B expansions is Zeckendorf's Theorem: every integer can be uniquely written as a sum of non-consecutive Fibonacci numbers $\{F_n\}$, with $F_{n+1} = F_n + F_{n-1}$ and $F_1=1, F_2=2$. If instead we allow the coefficients of the Fibonacci numbers in the decomposition to be zero or $\pm 1...	\section{Introduction} In this paper we explore signed decompositions of integers by various sequences. After briefly reviewing the literature, we state our results about uniqueness of decomposition, number of summands, and gaps between summands. In the course of our analysis we find a new way to interpret an earlier ...	{ "timestamp": "2014-05-13T02:03:53", "yymm": "1309", "arxiv_id": "1309.5600", "language": "en", "url": "https://arxiv.org/abs/1309.5600", "abstract": "A natural generalization of base B expansions is Zeckendorf's Theorem: every integer can be uniquely written as a sum of non-consecutive Fibonacci numbers $...
https://arxiv.org/abs/2110.02543	A logical approach for temporal and multiplex networks analysis	Many systems generate data as a set of triplets (a, b, c): they may represent that user a called b at time c or that customer a purchased product b in store c. These datasets are traditionally studied as networks with an extra dimension (time or layer), for which the fields of temporal and multiplex networks have exten...	\section{Introduction} \vspace{-10pt} Many systems generate data as a set of triplets $(a, b, c)$: they may represent that user $a$ called $b$ at time $c$ or that customer $a$ purchased product $b$ in store $c$. These datasets are traditionally studied as networks with an extra dimension (time or layer), for which the...	{ "timestamp": "2021-10-07T02:14:42", "yymm": "2110", "arxiv_id": "2110.02543", "language": "en", "url": "https://arxiv.org/abs/2110.02543", "abstract": "Many systems generate data as a set of triplets (a, b, c): they may represent that user a called b at time c or that customer a purchased product b in sto...
https://arxiv.org/abs/1202.3493	Probability calculations under the IAC hypothesis	We show how powerful algorithms recently developed for counting lattice points and computing volumes of convex polyhedra can be used to compute probabilities of a wide variety of events of interest in social choice theory. Several illustrative examples are given.	\section{Introduction} \label{sec:intro} Much research has been undertaken in recent decades with the aim of quantifying the probability of occurrence of certain types of election outcomes for a given voting rule under fixed assumptions on the distribution of voter preferences. Most prominent among these outcomes of i...	{ "timestamp": "2012-02-17T02:00:50", "yymm": "1202", "arxiv_id": "1202.3493", "language": "en", "url": "https://arxiv.org/abs/1202.3493", "abstract": "We show how powerful algorithms recently developed for counting lattice points and computing volumes of convex polyhedra can be used to compute probabilitie...
https://arxiv.org/abs/1812.08485	First-order algorithms converge faster than $O(1/k)$ on convex problems	It is well known that both gradient descent and stochastic coordinate descent achieve a global convergence rate of $O(1/k)$ in the objective value, when applied to a scheme for minimizing a Lipschitz-continuously differentiable, unconstrained convex function. In this work, we improve this rate to $o(1/k)$. We extend th...	\section{Introduction} \label{sec:intro} Consider the unconstrained optimization problem \begin{equation} \min_x \, f(x), \label{eq:f} \end{equation} where $f$ has domain in an inner-product space and is convex and $L$-Lipschitz continuously differentiable for some $L > 0$. We assume throughout that the solution set $\...	{ "timestamp": "2019-05-15T02:07:05", "yymm": "1812", "arxiv_id": "1812.08485", "language": "en", "url": "https://arxiv.org/abs/1812.08485", "abstract": "It is well known that both gradient descent and stochastic coordinate descent achieve a global convergence rate of $O(1/k)$ in the objective value, when a...
https://arxiv.org/abs/1305.3113	Hypergeometric type functions and their symmetries	We give a systematic and unified discussion of various classes of hypergeometric type equations: the hypergeometric equation, the confluent equation, the F_1 equation (equivalent to the Bessel equation), the Gegenbauer equation and the Hermite equation. In particular, we discuss recurrence relations of their solutions,...	\section{Introduction} Following \cite{NU}, we adopt the following terminology. Equations of the form \begin{equation}\left(\sigma(z) \partial_z^2+\tau(z) \partial_z+ \eta\right) f(z)=0,\ \ \ \label{req}\end{equation} where $\sigma$ is a polynomial of degree $\leq2$, \hspace{2.5ex} $\tau$ is a polynomial of degree $\...	{ "timestamp": "2013-08-06T02:06:07", "yymm": "1305", "arxiv_id": "1305.3113", "language": "en", "url": "https://arxiv.org/abs/1305.3113", "abstract": "We give a systematic and unified discussion of various classes of hypergeometric type equations: the hypergeometric equation, the confluent equation, the F_...