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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/1410.4191	Propagation time for zero forcing on a graph	Zero forcing (also called graph infection) on a simple, undirected graph $G$ is based on the color-change rule: If each vertex of $G$ is colored either white or black, and vertex $v$ is a black vertex with only one white neighbor $w$, then change the color of $w$ to black. A minimum zero forcing set is a set of black v...	\section{Propagation time} All graphs are simple, finite, and undirected. In a graph $G$ where some vertices are colored black and the remaining vertices are colored white, the {\em color change rule} is: If $v$ is black and $w$ is the only white neighbor of $v$, then change the color of $w$ to black; if we apply th...	{ "timestamp": "2014-10-17T02:00:18", "yymm": "1410", "arxiv_id": "1410.4191", "language": "en", "url": "https://arxiv.org/abs/1410.4191", "abstract": "Zero forcing (also called graph infection) on a simple, undirected graph $G$ is based on the color-change rule: If each vertex of $G$ is colored either whit...
https://arxiv.org/abs/1808.06293	Seymour's Second Neighborhood Conjecture for Subsets of Vertices	Seymour conjectured that every oriented simple graph contains a vertex whose second neighborhood is at least as large as its first. In this note, we put forward a conjecture that we prove is actually equivalent: every oriented simple graph contains a subset of vertices $S$ whose second neighborhood is at least as large...	\section{Introduction} Unless otherwise noted, all digraphs in this paper are oriented simple graphs, and thus do not contain loops or two-cycles. We will use $V(D)$ to denote the set of vertices of a digraph $D$. Given a digraph $D$ and vertices $u$ and $v$, let $d(u, v)$ be the length of the shortest directed pa...	{ "timestamp": "2019-04-15T02:20:56", "yymm": "1808", "arxiv_id": "1808.06293", "language": "en", "url": "https://arxiv.org/abs/1808.06293", "abstract": "Seymour conjectured that every oriented simple graph contains a vertex whose second neighborhood is at least as large as its first. In this note, we put f...
https://arxiv.org/abs/1411.3911	Digits of pi: limits to the seeming randomness	The decimal digits of $\pi$ are widely believed to behave like as statistically independent random variables taking the values $0, 1, 2, 3, 4, 5$, $6, 7, 8, 9$ with equal probabilities $1/10$.In this article, first, another similar conjecture is explored - the seemingly almost random behaviour of digits in the base 3 r...	\section{Introduction} \label{intro} The decimal digits of $\pi$ are widely believed to behave like as statistically independent random variables taking the values $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ with equal probabilities $\frac{1}{10}$ (for an overview, see \cite{marsaglia}). In Section \ref{powers} (it reproduces in ...	{ "timestamp": "2014-11-17T02:10:00", "yymm": "1411", "arxiv_id": "1411.3911", "language": "en", "url": "https://arxiv.org/abs/1411.3911", "abstract": "The decimal digits of $\\pi$ are widely believed to behave like as statistically independent random variables taking the values $0, 1, 2, 3, 4, 5$, $6, 7, 8...
https://arxiv.org/abs/2207.10289	A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks	Physics-informed neural networks (PINNs) have shown to be an effective tool for solving forward and inverse problems of partial differential equations (PDEs). PINNs embed the PDEs into the loss of the neural network, and this PDE loss is evaluated at a set of scattered residual points. The distribution of these points ...	\section{Introduction} Physics-informed neural networks (PINNs)~\cite{raissi2019physics} have emerged in recent years and quickly became a powerful tool for solving both forward and inverse problems of partial differential equations (PDEs) via deep neural networks (DNNs)~\cite{raissi2020Hidden,lu2021deepxde,karniadaki...	{ "timestamp": "2022-07-22T02:10:01", "yymm": "2207", "arxiv_id": "2207.10289", "language": "en", "url": "https://arxiv.org/abs/2207.10289", "abstract": "Physics-informed neural networks (PINNs) have shown to be an effective tool for solving forward and inverse problems of partial differential equations (PD...
https://arxiv.org/abs/math/0608085	On a conjecture of Wilf	Let n and k be natural numbers and let S(n,k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum \sum_{j=0}^{n} (-1)^{j} S(n,j) is nonzero for all n>2. We prove this conjecture for all n not congruent to 2 and not congruent to 2944838 modulo 3145728 and discuss applicati...	\section{Introduction} Let $n$ and $k$ be natural numbers. The Stirling numbers $S(n,k)$ of the second kind are given by \[x^n=\sum_{k=0}^{\infty}S(n,k)(x)_k,\] where $(x)_k:=x(x-1)(x-2)\ldots(x-k+1)$ for $k \in \mathbb{N}\setminus \{0\}$ and $(x)_0:=1.$ $S(n,k)$ is the number of ways in which it is possible to partit...	{ "timestamp": "2007-01-26T13:47:41", "yymm": "0608", "arxiv_id": "math/0608085", "language": "en", "url": "https://arxiv.org/abs/math/0608085", "abstract": "Let n and k be natural numbers and let S(n,k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum \\su...
https://arxiv.org/abs/1306.4434	An Introduction to the Discharging Method via Graph Coloring	We provide a "how-to" guide to the use and application of the Discharging Method. Our aim is not to exhaustively survey results proved by this technique, but rather to demystify the technique and facilitate its wider use, using applications in graph coloring as examples. Along the way, we present some new proofs and ne...	\section{Introduction} The Discharging Method has been used in graph theory for more than 100 years. Its most famous application is the proof of the Four Color Theorem, stating that graphs embeddable in the plane have chromatic number at most $4$. Nevetheless, the method remains a mystery to many graph theorists. Our ...	{ "timestamp": "2013-06-20T02:01:10", "yymm": "1306", "arxiv_id": "1306.4434", "language": "en", "url": "https://arxiv.org/abs/1306.4434", "abstract": "We provide a \"how-to\" guide to the use and application of the Discharging Method. Our aim is not to exhaustively survey results proved by this technique, ...
https://arxiv.org/abs/0902.0583	Witness sets	Given a set C of binary n-tuples and c in C, how many bits of c suffice to distinguish it from the other elements in C? We shed new light on this old combinatorial problem and improve on previously known bounds.	\section{Introduction} Let $C\subset\{0,1\}^n$ be a set of distinct binary vectors that we will call a code, and denote by $[n] = \{1,2,...n\}$ the set of coordinate positions. It is standard in coding theory to ask for codes (or sets) $C$ such that every codeword $c\in C$ is as different as possible from all the other...	{ "timestamp": "2009-02-03T15:28:58", "yymm": "0902", "arxiv_id": "0902.0583", "language": "en", "url": "https://arxiv.org/abs/0902.0583", "abstract": "Given a set C of binary n-tuples and c in C, how many bits of c suffice to distinguish it from the other elements in C? We shed new light on this old combin...
https://arxiv.org/abs/2109.06690	Birth and life of the $L^{2}$ boundedness of the Cauchy Integral on Lipschitz graphs	We review various motives for considering the problem of estimating the Cauchy Singular Integral on Lipschitz graphs in the $L^{2}$ norm. We follow the thread that led to the solution and then describe a few of the innumerable applications and ramifications of this fundamental result. We concentrate on its influence in...	\section{Introduction} In 1982 Coifman, McIntosh and Meyer proved that the Cauchy Singular Integral on a Lipschitz graph is $L^2$ bounded with respect to arc length on the curve \cite{CMM}. This is a deep result, simple to state, elegant, direct. In spite of its apparently specialized nature, it lies at the core of su...	{ "timestamp": "2021-09-15T02:25:50", "yymm": "2109", "arxiv_id": "2109.06690", "language": "en", "url": "https://arxiv.org/abs/2109.06690", "abstract": "We review various motives for considering the problem of estimating the Cauchy Singular Integral on Lipschitz graphs in the $L^{2}$ norm. We follow the th...
https://arxiv.org/abs/2210.00816	The extended Frobenius problem for Fibonacci sequences incremented by a Fibonacci number	We study the extended Frobenius problem for sequences of the form $\{f_a+f_n\}_{n\in\mathbb{N}}$, where $\{f_n\}_{n\in\mathbb{N}}$ is the Fibonacci sequence and $f_a$ is a Fibonacci number. As a consequence, we show that the family of numerical semigroups associated to these sequences satisfies the Wilf's conjecture.	\section{Introduction} Let $S\subseteq \mathbb{N}$ be the set generated by the sequence of positive integers $(a_1,\ldots,a_e)$, that is, $S=\langle a_1,\ldots,a_e \rangle = a_1{\mathbb N}+\cdots+a_e{\mathbb N}$. If $\gcd(a_1,\ldots,a_e)=1$, then it is well known that $S$ has a finite complement in $\mathbb{N}$. T...	{ "timestamp": "2022-11-04T01:01:44", "yymm": "2210", "arxiv_id": "2210.00816", "language": "en", "url": "https://arxiv.org/abs/2210.00816", "abstract": "We study the extended Frobenius problem for sequences of the form $\\{f_a+f_n\\}_{n\\in\\mathbb{N}}$, where $\\{f_n\\}_{n\\in\\mathbb{N}}$ is the Fibonacc...
https://arxiv.org/abs/2106.15667	Even sets of nodes and Gauss genus theory	We observe that a lemma used in the study of even sets of nodes on surfaces applies almost verbatim to prove a celebrated formula of Gauss on the 2-torsion of the class group of a quadratic field.	\section{Introduction} It is a great pleasure for me to dedicate this paper to Herb Clemens. I met Herb in Chile in 1972, and this played a decisive role in my mathematical orientation. I visited Herb for one month in Salt Lake City in 1979. I was interested in surfaces at that time; I~gave in particular a talk about...	{ "timestamp": "2021-07-01T02:01:00", "yymm": "2106", "arxiv_id": "2106.15667", "language": "en", "url": "https://arxiv.org/abs/2106.15667", "abstract": "We observe that a lemma used in the study of even sets of nodes on surfaces applies almost verbatim to prove a celebrated formula of Gauss on the 2-torsio...
https://arxiv.org/abs/2202.11595	Induced Disjoint Paths and Connected Subgraphs for $H$-Free Graphs	Paths $P_1,\ldots, P_k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P_i$ and $P_j$ have neither common vertices nor adjacent vertices. The Induced Disjoint Paths problem is to decide if a graph $G$ with $k$ pairs of specified vertices $(s_i,t_i)$ contains $k$ mutually induced paths $P_i$ such that ea...	\section{Introduction}\label{s-intro} The well-known {\sc Disjoint Paths} problem is one of the problems in Karp's list of {\sf NP}-complete problems. It is to decide if a graph has pairwise vertex-disjoint paths $P^1,\ldots,P^k$ where each $P^i$ connects two pre-specified vertices $s_i$ and $t_i$. Its generalization,...	{ "timestamp": "2022-07-19T02:23:36", "yymm": "2202", "arxiv_id": "2202.11595", "language": "en", "url": "https://arxiv.org/abs/2202.11595", "abstract": "Paths $P_1,\\ldots, P_k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P_i$ and $P_j$ have neither common vertices nor adjacent vertices....
https://arxiv.org/abs/2105.13252	The denominators of the Bernoulli numbers	We study the asymptotic density of the set of subscripts of the Bernoulli numbers having a given denominator. We also study the distribution of distinct Bernoulli denominators and some related problems.	\section{Introduction} Long before the proof of Andrew Wiles, it was thought that the path to Fermat's Last Theorem (FLT) led through the Bernoulli numbers. Defined by the series \[ \frac t{e^t-1}=\sum_{n=0}^\infty B_n\frac{t^n}{n!}, \] the Bernoulli numbers $B_n$ are rationals, in lowest terms $N_n/D_n$, and both t...	{ "timestamp": "2021-11-02T01:33:44", "yymm": "2105", "arxiv_id": "2105.13252", "language": "en", "url": "https://arxiv.org/abs/2105.13252", "abstract": "We study the asymptotic density of the set of subscripts of the Bernoulli numbers having a given denominator. We also study the distribution of distinct B...
https://arxiv.org/abs/1307.6766	An integer optimization problem for non-Hamiltonian periodic flows	Let C be the class of compact 2n-dimensional symplectic manifolds M for which the first or (n-1) Chern class vanish. We point out an integer optimization problem to find a lower bound B(n) on the number of equilibrium points of non-Hamiltonian symplectic periodic flows on manifolds M in C. As a consequence, we confirm ...	\part{\fbox{\mbox{\sf\em{#1}}}}} \newenvironment{skproof}{\par\noindent\emph{{Sketch of Proof.}}}{ \unskip\nobreak\hfill\hbox{$\Box$}\par \bigskip} \setlength{\parskip}{0.5ex plus 0.3ex minus 0.2ex} \renewcommand{\mathcal{A}}{\mathbb{A}} \newcommand{\stackrel{\mathrm{s}}{\sim}}{\stackrel{\mathrm{s}}{\sim}} \newc...	{ "timestamp": "2013-07-26T02:06:03", "yymm": "1307", "arxiv_id": "1307.6766", "language": "en", "url": "https://arxiv.org/abs/1307.6766", "abstract": "Let C be the class of compact 2n-dimensional symplectic manifolds M for which the first or (n-1) Chern class vanish. We point out an integer optimization pr...
https://arxiv.org/abs/2105.03094	Fusion frame and its alternative dual in tensor product of Hilbert spaces	We study fusion frame in tensor product of Hilbert spaces and discuss some of its properties. The resolution of the identity operator on a tensor product of Hilbert spaces is being discussed. An alternative dual of a fusion frame in tensor product of Hilbert spaces is being presented.	\section{Introduction} Fusion frame was introduced by P.\,Casazza and G.\,Kutyniok \cite{Kutyniok}.\,They define frames for closed subspaces of a given Hilbert spaces with respect to the orthogonal projections.\;Fusion frame is a natural generalization of the frame theory in Hilbert space and it has so many applicatio...	{ "timestamp": "2021-05-10T02:09:27", "yymm": "2105", "arxiv_id": "2105.03094", "language": "en", "url": "https://arxiv.org/abs/2105.03094", "abstract": "We study fusion frame in tensor product of Hilbert spaces and discuss some of its properties. The resolution of the identity operator on a tensor product ...
https://arxiv.org/abs/1906.07178	Throttling numbers for adversaries on connected graphs	In this paper, we answer two open problems from [Breen et al., Throttling for the game of Cops and Robbers on graphs, Discrete Math., 341 (2018) 2418-2430]. The throttling number $th_c(G)$ of a graph $G$ is the minimum possible value of $k + capt_k(G)$ over all positive integers $k$, where $capt_k(G)$ is the number of ...	\section{Introduction} In the \emph{cop versus robber} game on a graph $G$, a team of cops choose their initial positions at vertices of $G$, followed by the robber. Multiple cops are allowed to occupy the same vertex. In each round, every cop is allowed to stay at their current vertex or move to a neighboring vertex,...	{ "timestamp": "2019-10-23T02:06:46", "yymm": "1906", "arxiv_id": "1906.07178", "language": "en", "url": "https://arxiv.org/abs/1906.07178", "abstract": "In this paper, we answer two open problems from [Breen et al., Throttling for the game of Cops and Robbers on graphs, Discrete Math., 341 (2018) 2418-2430...
https://arxiv.org/abs/math/0512169	Automorphisms of the Weyl algebra	We discuss a conjecture which says that the automorphism group of the Weyl algebra in characteristic zero is canonically isomorphic to the automorphism group of the corresponding Poisson algebra of classical polynomial symbols. Several arguments in favor of this conjecture are presented, all based on the consideration ...	\section{Introduction}\label{sec:conj} This paper is devoted to the following surprising conjecture. \begin{conj}\label{conj:iso} The automorphism group of the Weyl algebra of index $n$ over $\C$ is isomorphic to the group of the polynomial symplectomorphisms of a $2n$-dimensional affine space $$\operatorname{Aut}(...	{ "timestamp": "2005-12-08T10:15:36", "yymm": "0512", "arxiv_id": "math/0512169", "language": "en", "url": "https://arxiv.org/abs/math/0512169", "abstract": "We discuss a conjecture which says that the automorphism group of the Weyl algebra in characteristic zero is canonically isomorphic to the automorphis...
https://arxiv.org/abs/2201.01101	Disproof of a conjecture on the main spectrum of generalized Bethe trees	An eigenvalue of the adjacency matrix of a graph is said to be main if the all-ones vector is not orthogonal to its associated eigenspace. A generalized Bethe tree with $k$ levels is a rooted tree in which vertices at the same level have the same degree. França and Brondani [On the main spectrum of generalized Bethe tr...	\section{Introduction} \label{intro} Let $G$ be a simple graph with vertex set $\{1,\ldots,n\}$. The \emph{adjacency matrix} of $G$ is the $n\times n$ symmetric matrix $A=A(G)=(a_{i,j})$, where $a_{i,j}=1$ if $i$ and $j$ are adjacent; $a_{i,j}=0$ otherwise. We often identify a graph $G$ with its adjacency matrix $A$....	{ "timestamp": "2022-01-05T02:13:03", "yymm": "2201", "arxiv_id": "2201.01101", "language": "en", "url": "https://arxiv.org/abs/2201.01101", "abstract": "An eigenvalue of the adjacency matrix of a graph is said to be main if the all-ones vector is not orthogonal to its associated eigenspace. A generalized B...
https://arxiv.org/abs/1610.02145	Smoothed nonparametric tests and their properties	In this paper we propose new smoothed sign and Wilcoxon's signed rank tests, which are based on a kernel estimator of the underlying distribution function of data. We discuss approximations of $p$-values and asymptotic properties of these tests. The new smoothed tests are equivalent to the ordinary sign and Wilcoxon's ...	\section{Introduction} \label{intro} Let $X_1,X_2,\cdots,X_n$ be independently and identically distributed ({\it i.i.d.}) random variables with a distribution function $F(x-\theta)$, where the associated desity function satisfies $f(-x)=f(x)$, and $\theta$ is an unknown location parameter. Here we consider a test and...	{ "timestamp": "2016-10-24T02:02:14", "yymm": "1610", "arxiv_id": "1610.02145", "language": "en", "url": "https://arxiv.org/abs/1610.02145", "abstract": "In this paper we propose new smoothed sign and Wilcoxon's signed rank tests, which are based on a kernel estimator of the underlying distribution function...
https://arxiv.org/abs/0903.4637	Tarski's plank problem revisited	In the 1930's, Tarski introduced his plank problem at a time when the field Discrete Geometry was about to born. It is quite remarkable that Tarski's question and its variants continue to generate interest in the geometric and analytic aspects of coverings by planks in the present time as well. The paper is a survey ty...	\section{Introduction} Tarski's plank problem has generated a great interest in understanding the geometry of coverings by planks. There have been a good number of results published in connection with the plank problem of Tarski that are surveyed in this paper. The paper is divided into six sections entitled Plank the...	{ "timestamp": "2009-03-26T18:04:01", "yymm": "0903", "arxiv_id": "0903.4637", "language": "en", "url": "https://arxiv.org/abs/0903.4637", "abstract": "In the 1930's, Tarski introduced his plank problem at a time when the field Discrete Geometry was about to born. It is quite remarkable that Tarski's questi...
https://arxiv.org/abs/2207.04035	The Burning Number Conjecture Holds Asymptotically	The burning number $b(G)$ of a graph $G$ is the smallest number of turns required to burn all vertices of a graph if at every turn a new fire is started and existing fires spread to all adjacent vertices. The Burning Number Conjecture of Bonato et al. (2016) postulates that $b(G)\leq \left\lceil\sqrt{n}\right\rceil$ fo...	\section{Introduction} Consider the following process (or one-player game) on a finite, usually connected, graph $G$. At the start, all vertices are said to be unburned. Then, at every step (or turn) of the process, we may choose to burn (start a new fire) at some vertex. Furthermore, at every step, all vertices adj...	{ "timestamp": "2022-07-11T02:16:57", "yymm": "2207", "arxiv_id": "2207.04035", "language": "en", "url": "https://arxiv.org/abs/2207.04035", "abstract": "The burning number $b(G)$ of a graph $G$ is the smallest number of turns required to burn all vertices of a graph if at every turn a new fire is started a...
https://arxiv.org/abs/2109.10233	An Extension to the Gusić-Tadić Specialization Criterion	Let $E/\mathbb Q(t)$ be an elliptic curve and let $t_0 \in \mathbb Q$ be a rational number for which the specialization $E_{t_0}$ is an elliptic curve. In 2015, Gusić and Tadić gave an easy-to-check criterion, based only on a Weierstrass equation for $E/\mathbb Q(t)$, that is sufficient to conclude that the specializat...	\section{Introduction} Let $C$ be a (complete nonsingular) curve defined over a number field $k$ with function field $k(C)$. Let $E/k(C)$ be an elliptic curve defined by the Weierstrass equation $$y^2 = x^3 +A(t)x+B(t), \qquad A(t), B(t) \in k(C).$$ For any $t_0 \in C(k)$ such that the discriminant $4A(t)^3+27B(t)^2$ ...	{ "timestamp": "2021-09-22T02:21:09", "yymm": "2109", "arxiv_id": "2109.10233", "language": "en", "url": "https://arxiv.org/abs/2109.10233", "abstract": "Let $E/\\mathbb Q(t)$ be an elliptic curve and let $t_0 \\in \\mathbb Q$ be a rational number for which the specialization $E_{t_0}$ is an elliptic curve....
https://arxiv.org/abs/2207.08182	Kuramoto Networks with Infinitely Many Stable Equilibria	We prove that the Kuramoto model on a graph can contain infinitely many non-equivalent stable equilibria. More precisely, we prove that for every positive integer d there is a connected graph such that the set of stable equilibria contains a manifold of dimension d. In particular, we solve a conjecture of R. Delabays, ...	\section{Introduction.} Consider a connected graph~$\G$ with vertices~$1,\ldots,n$ and to each vertex~$j$ associate a \emph{phase}~$\theta_j$ in the $1$-dimensional torus~$\T=\R/2\pi\Z$. Let~$\mathbf N(j)$ denote the set of neighbors of~$j$ and consider the coupled dynamical system \begin{equation} \label{eq:main} \do...	{ "timestamp": "2022-07-19T02:20:07", "yymm": "2207", "arxiv_id": "2207.08182", "language": "en", "url": "https://arxiv.org/abs/2207.08182", "abstract": "We prove that the Kuramoto model on a graph can contain infinitely many non-equivalent stable equilibria. More precisely, we prove that for every positive...
https://arxiv.org/abs/2010.09710	Twice is enough for dangerous eigenvalues	We analyze the stability of a class of eigensolvers that target interior eigenvalues with rational filters. We show that subspace iteration with a rational filter is robust even when an eigenvalue is near a filter's pole. These dangerous eigenvalues contribute to large round-off errors in the first iteration, but are s...	\section{Introduction}\label{sec:intro} When combined with shift-and-invert enhancement, subspace iteration and Arnoldi are two classic iterative schemes for computing a few interior eigenvalues of an $n\times n$ matrix $A$. Each method constructs an orthonormal basis for a search subspace by iteratively applying the ...	{ "timestamp": "2020-11-25T02:28:44", "yymm": "2010", "arxiv_id": "2010.09710", "language": "en", "url": "https://arxiv.org/abs/2010.09710", "abstract": "We analyze the stability of a class of eigensolvers that target interior eigenvalues with rational filters. We show that subspace iteration with a rationa...
https://arxiv.org/abs/1402.6294	Frankl-Rödl type theorems for codes and permutations	We give a new proof of the Frankl-Rödl theorem on forbidden intersections, via the probabilistic method of dependent random choice. Our method extends to codes with forbidden distances, where over large alphabets our bound is significantly better than that obtained by Frankl and Rödl. We also apply our bound to a quest...	\section{Introduction} \let\thefootnote\relax\footnote{\emph{2010 Mathematics Subject Classification.} Primary 05D05. Secondary 05D40, 94B65.} A family $\mathcal{A} $ of sets is said to be \emph{$l$-avoiding} if $\|A \cap B\| \neq l$ for all $A, B \in \mathcal{A} $. Erd\H{o}s conjectured (\cite{Erd}) that for any $\eps...	{ "timestamp": "2014-02-26T02:13:37", "yymm": "1402", "arxiv_id": "1402.6294", "language": "en", "url": "https://arxiv.org/abs/1402.6294", "abstract": "We give a new proof of the Frankl-Rödl theorem on forbidden intersections, via the probabilistic method of dependent random choice. Our method extends to co...
https://arxiv.org/abs/1507.06856	On the Stretch Factor of Convex Polyhedra whose Vertices are (Almost) on a Sphere	Let $P$ be a convex polyhedron in $\mathbb{R}^3$. The skeleton of $P$ is the graph whose vertices and edges are the vertices and edges of $P$, respectively. We prove that, if these vertices are on the unit-sphere, the skeleton is a $(0.999 \cdot \pi)$-spanner. If the vertices are very close to this sphere, then the ske...	\section{Introduction} Let $S$ be a finite set of points in Euclidean space and let $G$ be a graph with vertex set $S$. We denote the Euclidean distance between any two points $p$ and $q$ by $\|pq\|$. Let the length of any edge $pq$ in $G$ be equal to $\|pq\|$, and define the length of a path in $G$ to be the sum of th...	{ "timestamp": "2016-09-05T02:04:06", "yymm": "1507", "arxiv_id": "1507.06856", "language": "en", "url": "https://arxiv.org/abs/1507.06856", "abstract": "Let $P$ be a convex polyhedron in $\\mathbb{R}^3$. The skeleton of $P$ is the graph whose vertices and edges are the vertices and edges of $P$, respective...
https://arxiv.org/abs/2011.04907	A Statistical Perspective on Coreset Density Estimation	Coresets have emerged as a powerful tool to summarize data by selecting a small subset of the original observations while retaining most of its information. This approach has led to significant computational speedups but the performance of statistical procedures run on coresets is largely unexplored. In this work, we d...	\section{Introduction} The ever-growing size of datasets that are routinely collected has led practitioners across many fields to contemplate effective data summarization techniques that aim at reducing the size of the data while preserving the information that it contains. While there are many ways to ach...	{ "timestamp": "2020-12-10T02:03:30", "yymm": "2011", "arxiv_id": "2011.04907", "language": "en", "url": "https://arxiv.org/abs/2011.04907", "abstract": "Coresets have emerged as a powerful tool to summarize data by selecting a small subset of the original observations while retaining most of its informatio...
https://arxiv.org/abs/1609.08186	Extremal functions for Morrey's inequality in convex domains	For a bounded domain $\Omega\subset \mathbb{R}^n$ and $p>n$, Morrey's inequality implies that there is $c>0$ such that $$ c\\|u\\|^p_{\infty}\le \int_\Omega\|Du\|^pdx $$ for each $u$ belonging to the Sobolev space $W^{1,p}_0(\Omega)$. We show that the ratio of any two extremal functions is constant provided that $\Omega$ i...	\section{Introduction} Suppose $\Omega\subset \mathbb{R}^n$ is a bounded domain and $p>n$. Morrey's inequality for $W^{1,p}_0(\Omega)$ functions $u$ may be expressed as $$ c\\|u\\|^p_{C^{1-\frac{n}{p}}(\overline\Omega)}\le \int_\Omega\|Du\|^pdx, $$ where $c>0$ is a constant that is independent of $u$. In particular, \be...	{ "timestamp": "2016-09-28T02:00:32", "yymm": "1609", "arxiv_id": "1609.08186", "language": "en", "url": "https://arxiv.org/abs/1609.08186", "abstract": "For a bounded domain $\\Omega\\subset \\mathbb{R}^n$ and $p>n$, Morrey's inequality implies that there is $c>0$ such that $$ c\\\|u\\\|^p_{\\infty}\\le \\in...
https://arxiv.org/abs/2101.03050	Remarks on manifolds with two sided curvature bounds	We discuss folklore statements about distance functions in manifolds with two sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.	\section{Introduction} \subsection{Distance functions in smooth Riemannian manifolds} We discuss slightly generalized versions of some folklore results about distance functions $d_A$ to subsets $A$ of smooth Riemannian manifolds $M$. The results turn out to be local and are proved without any completeness assumption...	{ "timestamp": "2021-01-11T02:16:14", "yymm": "2101", "arxiv_id": "2101.03050", "language": "en", "url": "https://arxiv.org/abs/2101.03050", "abstract": "We discuss folklore statements about distance functions in manifolds with two sided bounded curvature. The topics include regularity, subsets of positive ...
https://arxiv.org/abs/1005.0781	Adjacent q-cycles in permutations	We introduce a new permutation statistic, namely, the number of cycles of length $q$ consisting of consecutive integers, and consider the distribution of this statistic among the permutations of $\{1,2,...,n\}$. We determine explicit formulas, recurrence relations, and ordinary and exponential generating functions. A ...	\section{Introduction} Let $S_n$ denote the set of all permutations of $[n]=\{1,2,\ldots,n\}$. We use the one-line notation for permutations but then write a permutation according to its standard (disjoint) cycle decomposition. For example, \[\pi =432157869=(14)(23)(5)(678)(9)\] is a permutation in $S_9$ decomposed int...	{ "timestamp": "2010-05-14T17:26:12", "yymm": "1005", "arxiv_id": "1005.0781", "language": "en", "url": "https://arxiv.org/abs/1005.0781", "abstract": "We introduce a new permutation statistic, namely, the number of cycles of length $q$ consisting of consecutive integers, and consider the distribution of th...
https://arxiv.org/abs/1107.0189	The Lasso, correlated design, and improved oracle inequalities	We study high-dimensional linear models and the $\ell_1$-penalized least squares estimator, also known as the Lasso estimator. In literature, oracle inequalities have been derived under restricted eigenvalue or compatibility conditions. In this paper, we complement this with entropy conditions which allow one to improv...	\section{Introduction}\label{intro.section} We derive oracle inequalities for the Lasso estimator for various designs. Results in literature are generally based on restricted eigenvalue or compatibility conditions (see Section \ref{definitions.section} for definitions). We refer to \cite{bickel2009sal}, \cite{Bunea:...	{ "timestamp": "2011-07-04T02:02:39", "yymm": "1107", "arxiv_id": "1107.0189", "language": "en", "url": "https://arxiv.org/abs/1107.0189", "abstract": "We study high-dimensional linear models and the $\\ell_1$-penalized least squares estimator, also known as the Lasso estimator. In literature, oracle inequa...
https://arxiv.org/abs/2009.05921	Numerical semigroups, polyhedra, and posets III: minimal presentations and face dimension	This paper is the third in a series of manuscripts that examine the combinatorics of the Kunz polyhedron $P_m$, whose positive integer points are in bijection with numerical semigroups (cofinite subsemigroups of $\mathbb Z_{\ge 0}$) whose smallest positive element is $m$. The faces of $P_m$ are indexed by a family of f...	\section{Introduction \label{sec:intro A \emph{numerical semigroup} is a cofinite subset $S \subseteq \NN$ of the non-negative integers that is closed under addition and contains $0$. Numerical semigroups are often specified using a set of generators $n_0 < \cdots < n_k$, i.e., \[ S = \<n_0, \ldots, n_k\> = \{a_1n_1 ...	{ "timestamp": "2022-05-25T02:01:32", "yymm": "2009", "arxiv_id": "2009.05921", "language": "en", "url": "https://arxiv.org/abs/2009.05921", "abstract": "This paper is the third in a series of manuscripts that examine the combinatorics of the Kunz polyhedron $P_m$, whose positive integer points are in bijec...
https://arxiv.org/abs/0911.5569	Homogeneous Schrödinger operators on half-line	The differential expression $L_m=-\partial_x^2 +(m^2-1/4)x^{-2}$ defines a self-adjoint operator H_m on L^2(0;\infty) in a natural way when $m^2 \geq 1$. We study the dependence of H_m on the parameter m, show that it has a unique holomorphic extension to the half-plane Re(m) > -1, and analyze spectral and scattering p...	\section{Introduction}\label{s:intro} For $m\ge 1$ real the differential operator $L_m=-\partial_x^2+(m^2-1/4)x^{-2}$ with domain $C_\c^\infty=C_\c^\infty(0,\infty)$ is essentially self-adjoint and we denote by $H_m$ its closure. Let $U_\tau$ be the group of dilations on $L^2$, that is $(U_\tau f)(x)=\mathrm{e}^{\ta...	{ "timestamp": "2009-11-30T09:13:22", "yymm": "0911", "arxiv_id": "0911.5569", "language": "en", "url": "https://arxiv.org/abs/0911.5569", "abstract": "The differential expression $L_m=-\\partial_x^2 +(m^2-1/4)x^{-2}$ defines a self-adjoint operator H_m on L^2(0;\\infty) in a natural way when $m^2 \\geq 1$....
https://arxiv.org/abs/1110.4564	Z-matrix equations in max algebra, nonnegative linear algebra and other semirings	We study the max-algebraic analogue of equations involving Z-matrices and M-matrices, with an outlook to a more general algebraic setting. We show that these equations can be solved using the Frobenius trace down method in a way similar to that in non-negative linear algebra, characterizing the solvability in terms of ...	\section{Introduction} \bigskip A $Z$-matrix is a square matrix of the form $\lambda I-A$ where \lambda $ is real and $A$ is an (elementwise) nonnegative matrix. It is called an $M$-matrix if $\lambda \geq \rho \left( A\right) ,$ where $\rho \left( A\right) $ is the Perron root (spectral radius) of $A$ and it is...	{ "timestamp": "2012-06-07T02:04:51", "yymm": "1110", "arxiv_id": "1110.4564", "language": "en", "url": "https://arxiv.org/abs/1110.4564", "abstract": "We study the max-algebraic analogue of equations involving Z-matrices and M-matrices, with an outlook to a more general algebraic setting. We show that thes...
https://arxiv.org/abs/1501.01741	A linear k-fold Cheeger inequality	Given an undirected graph $G$, the classical Cheeger constant, $h_G$, measures the optimal partition of the vertices into 2 parts with relatively few edges between them based upon the sizes of the parts. The well-known Cheeger's inequality states that $2 \lambda_1 \le h_G \le \sqrt {2 \lambda_1}$ where $\lambda_1$ is t...	\section{Introduction}\label{S:intro} Let $G=(V,E)$ be an undirected graph, and let $\mathcal{L} = \mathbf{D}^{-1/2} (\mathbf{I} - \mathbf{A}) \mathbf{D}^{-1/2}$ be the normalized Laplacian of $G$ with eigenvalues $0=\lambda_0 \le \lambda_1 \le \ldots \le \lambda_{n-1}$. It is a basic fact in spectral graph theory tha...	{ "timestamp": "2015-03-02T02:13:00", "yymm": "1501", "arxiv_id": "1501.01741", "language": "en", "url": "https://arxiv.org/abs/1501.01741", "abstract": "Given an undirected graph $G$, the classical Cheeger constant, $h_G$, measures the optimal partition of the vertices into 2 parts with relatively few edge...
https://arxiv.org/abs/1412.0628	On the Hamiltonicity of the $k$-regular graph game	We consider a game played on an initially empty graph where two players alternate drawing an edge between vertices subject to the condition that no degree can exceed $k$. We show that for $k=3$, either player can avoid a Hamilton cycle, and for $k\geq4$, either player can force the resulting graph to be Hamiltonian.	\section{Introduction} The Hamiltonicity of $k$-regular graphs has been studied extensively. Bollob\'{a}s \cite{BB} and Fenner and Frieze \cite{FF} showed that for sufficiently large $k$, almost all random $k$-regular graphs are Hamiltonian. Robinson and Wormald improved this by showing that almost all cubic graphs ar...	{ "timestamp": "2014-12-02T02:25:30", "yymm": "1412", "arxiv_id": "1412.0628", "language": "en", "url": "https://arxiv.org/abs/1412.0628", "abstract": "We consider a game played on an initially empty graph where two players alternate drawing an edge between vertices subject to the condition that no degree c...
https://arxiv.org/abs/2110.11748	The fractional Makai-Hayman inequality	We prove that the first eigenvalue of the fractional Dirichlet-Laplacian of order $s$ on a simply connected set of the plane can be bounded from below in terms of its inradius only. This is valid for $1/2<s<1$ and we show that this condition is sharp, i.\,e. for $0<s\le 1/2$ such a lower bound is not possible. The cons...	\section{Introduction} \subsection{Background} For an open set $\Omega\subset\mathbb{R}^N$, we indicate by $W^{1,2}_0(\Omega)$ the closure of $C^\infty_0(\Omega)$ in the Sobolev space $W^{1,2}(\Omega)$. We then consider the following quantity \[ \lambda_1(\Omega):=\inf_{u\in W^{1,2}_0(\Omega)\setminus\{0\}}\frac{\disp...	{ "timestamp": "2021-10-25T02:18:28", "yymm": "2110", "arxiv_id": "2110.11748", "language": "en", "url": "https://arxiv.org/abs/2110.11748", "abstract": "We prove that the first eigenvalue of the fractional Dirichlet-Laplacian of order $s$ on a simply connected set of the plane can be bounded from below in ...
https://arxiv.org/abs/1212.3392	Quantization of Galois theory, Examples and Observations	If we consider a q-analogue of linear differential equation, Galoois group of the q-analogue difference equation is still a linear algebraic group. Namely, by a quantization of linear differential equation, Galois group is not quantized. We show by Examples that if we consider non-linear equations Galois group is quant...	\section{Introduction} The pursuit of $q$-analogue of hypergeometric functions goes back to the 19th century. Galois group of a $q$-hypergeometric function is not a quantum group but it is a linear algebraic group. This shows that we consider a $q$-deformations of the hypergeometric equation, Galois theory is not ...	{ "timestamp": "2012-12-17T02:00:50", "yymm": "1212", "arxiv_id": "1212.3392", "language": "en", "url": "https://arxiv.org/abs/1212.3392", "abstract": "If we consider a q-analogue of linear differential equation, Galoois group of the q-analogue difference equation is still a linear algebraic group. Namely, ...
https://arxiv.org/abs/quant-ph/0505026	A matrix representation of graphs and its spectrum as a graph invariant	We use the line digraph construction to associate an orthogonal matrix with each graph. From this orthogonal matrix, we derive two further matrices. The spectrum of each of these three matrices is considered as a graph invariant. For the first two cases, we compute the spectrum explicitly and show that it is determined...	\section{Introduction} Graphs are often conveniently represented using matrices, for example, the adjacency matrix, the Laplacian matrix, \emph{etc.} \cite{g}. Many important properties of a graph are encoded in the eigenvalues of the matrix representation. However, eigenvalues generally fail to separate isomorphism c...	{ "timestamp": "2006-04-05T12:03:09", "yymm": "0505", "arxiv_id": "quant-ph/0505026", "language": "en", "url": "https://arxiv.org/abs/quant-ph/0505026", "abstract": "We use the line digraph construction to associate an orthogonal matrix with each graph. From this orthogonal matrix, we derive two further mat...
https://arxiv.org/abs/2002.09677	Kernel interpolation with continuous volume sampling	A fundamental task in kernel methods is to pick nodes and weights, so as to approximate a given function from an RKHS by the weighted sum of kernel translates located at the nodes. This is the crux of kernel density estimation, kernel quadrature, or interpolation from discrete samples. Furthermore, RKHSs offer a conven...	\section{Introduction} \label{sec:introduction} Kernel approximation is a recurrent task in machine learning \citep*{HaTiFr09}[Chapter 5], signal processing \citep{Uns00} or numerical quadrature \citep{Lar72}. Expressed in its general form, we are given a reproducing kernel Hilbert space $\mathcal{F}$ (RKHS; \citealp{...	{ "timestamp": "2020-02-25T02:08:41", "yymm": "2002", "arxiv_id": "2002.09677", "language": "en", "url": "https://arxiv.org/abs/2002.09677", "abstract": "A fundamental task in kernel methods is to pick nodes and weights, so as to approximate a given function from an RKHS by the weighted sum of kernel transl...
https://arxiv.org/abs/2203.02653	Estimating the circumference of a graph in terms of its leaf number	Let $\mathcal{T}$ be the set of spanning trees of $G$ and let $L(T)$ be the number of leaves in a tree $T$. The leaf number $L(G)$ of $G$ is defined as $L(G)=\max\{L(T)\|T\in \mathcal{T}\}$. Let $G$ be a connected graph of order $n$ and minimum degree $\delta$ such that $L(G)\leq 2\delta-1$. We show that the circumferen...	\section{Introduction} We will deal with only finite nontrivial simple graphs. Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The order and size of a graph $G$ are its number of vertices and edges, respectively. The notations $N_G(v)$ and $N_G[v]$ denote the neighborhood and closed neighborhood of $v\in...	{ "timestamp": "2022-03-08T02:08:17", "yymm": "2203", "arxiv_id": "2203.02653", "language": "en", "url": "https://arxiv.org/abs/2203.02653", "abstract": "Let $\\mathcal{T}$ be the set of spanning trees of $G$ and let $L(T)$ be the number of leaves in a tree $T$. The leaf number $L(G)$ of $G$ is defined as $...