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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/1605.05226	On two-generator subgroups in SL_2(Z), SL_2(Q), and SL_2(R)	We consider what some authors call 'parabolic Möbius subgroups' of matrices over Z, Q, and R and focus on the membership problem in these subgroups and complexity of relevant algorithms.	\section{Introduction: two theorems of Sanov} Denote $A(k) = \left( \begin{array}{cc} 1 & k \\ 0 & 1 \end{array} \right) , \hskip .2cm B(k) = \left( \begin{array}{cc} 1 & 0 \\ k & 1 \end{array} \right).$ In an old paper \cite{Sanov}, I. N. Sanov proved two simple yet remarkable theorems: \begin{theorem}\label{th1...	{ "timestamp": "2017-02-07T02:07:08", "yymm": "1605", "arxiv_id": "1605.05226", "language": "en", "url": "https://arxiv.org/abs/1605.05226", "abstract": "We consider what some authors call 'parabolic Möbius subgroups' of matrices over Z, Q, and R and focus on the membership problem in these subgroups and co...
https://arxiv.org/abs/2011.03400	Apéry Limits: Experiments and Proofs	An important component of Apéry's proof that $\zeta (3)$ is irrational involves representing $\zeta (3)$ as the limit of the quotient of two rational solutions to a three-term recurrence. We present various approaches to such Apéry limits and highlight connections to continued fractions as well as the famous theorems o...	\section{Introduction} A fundamental ingredient of Ap\'ery's groundbreaking proof \cite{apery} of the irrationality of $\zeta (3)$ is the binomial sum \begin{equation} A (n) = \sum_{k = 0}^n \binom{n}{k}^2 \binom{n + k}{k}^2 \label{eq:apery3} \end{equation} and the fact that it satisfies the three-term recurrence \b...	{ "timestamp": "2020-11-09T02:16:14", "yymm": "2011", "arxiv_id": "2011.03400", "language": "en", "url": "https://arxiv.org/abs/2011.03400", "abstract": "An important component of Apéry's proof that $\\zeta (3)$ is irrational involves representing $\\zeta (3)$ as the limit of the quotient of two rational so...
https://arxiv.org/abs/2104.04086	Halperin's conjecture in formal dimensions up to 20	A 1976 conjecture of Halperin on positively elliptic spaces was recently confirmed in formal dimensions up to 16. In this article, we shorten the proof and extend the result up to formal dimension 20. We work with Meier's algebraic characterization of the conjecture, so the proof is elementary in that it involves only ...	\section{Introduction} We consider Artinian complete intersection algebras \[H^ = \Q[x_1,\ldots,x_k] / (u_1,\ldots,u_k)\] with a grading concentrated in even degrees. Examples include the rational cohomology of positively elliptic topological spaces, so for simplicity we refer to these algebras as {\it positively el...	{ "timestamp": "2021-04-12T02:04:21", "yymm": "2104", "arxiv_id": "2104.04086", "language": "en", "url": "https://arxiv.org/abs/2104.04086", "abstract": "A 1976 conjecture of Halperin on positively elliptic spaces was recently confirmed in formal dimensions up to 16. In this article, we shorten the proof an...
https://arxiv.org/abs/0907.0513	The Gift Exchange Problem	The aim of this paper is to solve the "gift exchange" problem: you are one of n players, and there are n wrapped gifts on display; when your turn comes, you can either choose any of the remaining wrapped gifts, or you can "steal" a gift from someone who has already unwrapped it, subject to the restriction that no gift ...	\section{\@startsection {section}{1}{\z@}{-3.5ex plus -1ex minus -.2ex}{2.3ex plus .2ex}{\normalsize\bf}} \makeatother \makeatletter \def\subsection{\@startsection {subsection}{1}{\z@}{-3.5ex plus -1ex minus -.2ex}{2.3ex plus .2ex}{\normalsize\bf}} \makeatother \begin{document} \begin{center} {\large\bf The Gift...	{ "timestamp": "2009-07-03T03:36:07", "yymm": "0907", "arxiv_id": "0907.0513", "language": "en", "url": "https://arxiv.org/abs/0907.0513", "abstract": "The aim of this paper is to solve the \"gift exchange\" problem: you are one of n players, and there are n wrapped gifts on display; when your turn comes, y...
https://arxiv.org/abs/2012.08365	Two generalizations of the Butterfly Theorem	We establish two direct extensions to the Butterfly Theorem on the cyclic quadrilateral along with the proofs using the projective method and analytic geometry of the Cartesian coordinate system.	\section{Introduction} We repeat the Butterfly Theorem expressed with the chord of the circle; see \cite{1,2,3,4,4a,4b}. This is an interesting and important theorem of plane Euclidean geometry. This classic theorem also has a lot of solutions; see \cite{1,2,3,4,4b}. Previously, the first author of this article also c...	{ "timestamp": "2020-12-16T02:21:52", "yymm": "2012", "arxiv_id": "2012.08365", "language": "en", "url": "https://arxiv.org/abs/2012.08365", "abstract": "We establish two direct extensions to the Butterfly Theorem on the cyclic quadrilateral along with the proofs using the projective method and analytic geo...
https://arxiv.org/abs/1207.1258	Matrices that commute with their derivative. On a letter from Schur to Wielandt	We examine when a matrix whose elements are differentiable functions in one variable commutes with its derivative. This problem was discussed in a letter from Issai Schur to Helmut Wielandt written in 1934, which we found in Wielandt's Nachlass. We present this letter and its translation into English. The topic was red...	\section{Introduction} What are the conditions that force a matrix of differentiable functions to commute with its elementwise derivative? This problem, discussed in a letter from I. Schur to H. Wielandt \cite{Sch34}, has been discussed in a large number of papers \cite{Ama54,Asc50,Asc52,BogC59,Die74,Eps63,Eru46,Gof...	{ "timestamp": "2012-11-28T02:00:32", "yymm": "1207", "arxiv_id": "1207.1258", "language": "en", "url": "https://arxiv.org/abs/1207.1258", "abstract": "We examine when a matrix whose elements are differentiable functions in one variable commutes with its derivative. This problem was discussed in a letter fr...
https://arxiv.org/abs/2012.10015	A Gallery of Gaussian Periods	Gaussian periods are certain sums of roots of unity whose study dates back to Gauss's seminal work in algebra and number theory. Recently, large scale plots of Gaussian periods have been revealed to exhibit striking visual patterns, some of which have been explored in the second named author's prior work. In 2020, the ...	\section*{Introduction} Gaussian periods, certain sums of roots of unity introduced by Gauss, have played a key role in several mathematical developments. For example, Gauss employed them in his work on constructibility of regular polygons by unmarked straightedge and compass, as well as in number theory. In the pas...	{ "timestamp": "2020-12-21T02:06:06", "yymm": "2012", "arxiv_id": "2012.10015", "language": "en", "url": "https://arxiv.org/abs/2012.10015", "abstract": "Gaussian periods are certain sums of roots of unity whose study dates back to Gauss's seminal work in algebra and number theory. Recently, large scale plo...
https://arxiv.org/abs/1205.4851	Green's Theorem for Generalized Fractional Derivatives	We study three types of generalized partial fractional operators. An extension of Green's theorem, by considering partial fractional derivatives with more general kernels, is proved. New results are obtained, even in the particular case when the generalized operators are reduced to the standard partial fractional deriv...	\section{Introduction} In 1828, the English mathematician George Green (1793-1841), who up to his forties was working as a baker and a miller, published an essay where he introduced a formula connecting the line integral around a simple closed curve with a double integral. Within years, this result turned out to be us...	{ "timestamp": "2012-10-29T01:00:43", "yymm": "1205", "arxiv_id": "1205.4851", "language": "en", "url": "https://arxiv.org/abs/1205.4851", "abstract": "We study three types of generalized partial fractional operators. An extension of Green's theorem, by considering partial fractional derivatives with more g...
https://arxiv.org/abs/2210.12371	Structure of singular and nonsingular tournament matrices	A tournament is a directed graph resulting from an orientation of the complete graph; so, if $M$ is a tournament's adjacency matrix, then $M + M^T$ is a matrix with $0$s on its diagonal and all other entries equal to $1$. An outstanding question in tournament theory asks to classify the adjacency matrices of tournament...	\section{Introduction} A \underline{tournament matrix} of order $n$ is an $n \times n$ $(0,1)$-matrix $M = [m_{ij}]$ which satisfies $$M + M^T = J_n - I_n,$$ where $J_n$ denotes the $n \times n$ matrix of all 1s and $I_n$ denotes the $n \times n$ identity matrix. A \underline{tournament} of order $n$ is a digraph obta...	{ "timestamp": "2022-10-25T02:07:00", "yymm": "2210", "arxiv_id": "2210.12371", "language": "en", "url": "https://arxiv.org/abs/2210.12371", "abstract": "A tournament is a directed graph resulting from an orientation of the complete graph; so, if $M$ is a tournament's adjacency matrix, then $M + M^T$ is a m...
https://arxiv.org/abs/2105.10618	Tight bounds on the maximal perimeter of convex equilateral small polygons	A small polygon is a polygon that has diameter one. The maximal perimeter of a convex equilateral small polygon with $n=2^s$ sides is not known when $s \ge 4$. In this paper, we construct a family of convex equilateral small $n$-gons, $n=2^s$ and $s \ge 4$, and show that their perimeters are within $O(1/n^4)$ of the ma...	\section{Introduction} The {\em diameter} of a polygon is the largest Euclidean distance between pairs of its vertices. A polygon is said to be {\em small} if its diameter equals one. For an integer $n \ge 3$, the maximal perimeter problem consists in finding a convex small $n$-gon with the longest perimeter. The probl...	{ "timestamp": "2021-06-02T02:20:22", "yymm": "2105", "arxiv_id": "2105.10618", "language": "en", "url": "https://arxiv.org/abs/2105.10618", "abstract": "A small polygon is a polygon that has diameter one. The maximal perimeter of a convex equilateral small polygon with $n=2^s$ sides is not known when $s \\...
https://arxiv.org/abs/cs/9910009	Locked and Unlocked Polygonal Chains in 3D	In this paper, we study movements of simple polygonal chains in 3D. We say that an open, simple polygonal chain can be straightened if it can be continuously reconfigured to a straight sequence of segments in such a manner that both the length of each link and the simplicity of the chain are maintained throughout the m...	\section{Introduction} \seclab{Introduction} A {\em polygonal chain\/} $P=(v_0,v_1,\ldots,v_{n-1})$ is a sequence of consecutively joined segments (or edges) $e_i =v_iv_{i+1}$ of fixed lengths $\ell_i = \|e_i\|$, embedded in space.\footnote{ All index arithmetic throughout the paper is mod $n$. } A chain is {\em clo...	{ "timestamp": "1999-10-08T14:04:18", "yymm": "9910", "arxiv_id": "cs/9910009", "language": "en", "url": "https://arxiv.org/abs/cs/9910009", "abstract": "In this paper, we study movements of simple polygonal chains in 3D. We say that an open, simple polygonal chain can be straightened if it can be continuou...
https://arxiv.org/abs/1904.08002	Quotients of Hurwitz Primes	Quotient sets have attracted the attention of mathematicians in the past three decades. The set of quotients of primes is dense in the positive real numbers and the set of all quotients of Gaussian primes is also dense in the complex plane. Sittinger has proved that the set of quotients of primes in an imaginary quadra...	\section{\baselineskip=17pt} \title{Quotients of Hurwitz Primes} \author{Minghao Pan} \address{Department of Mathematics, University of California, Los Angeles, CA 90095, United States} \email{minghaopan@g.ucla.edu} \author{Wentao Zhang} \address{Shenzhen Middle School, No.18 Shenzhong Street, Luohu, Shenzhen, Guan...	{ "timestamp": "2019-04-18T02:04:18", "yymm": "1904", "arxiv_id": "1904.08002", "language": "en", "url": "https://arxiv.org/abs/1904.08002", "abstract": "Quotient sets have attracted the attention of mathematicians in the past three decades. The set of quotients of primes is dense in the positive real numbe...
https://arxiv.org/abs/1401.1736	Statistical Topology of Three-Dimensional Poisson-Voronoi Cells and Cell Boundary Networks	Voronoi tessellations of Poisson point processes are widely used for modeling many types of physical and biological systems. In this paper, we analyze simulated Poisson-Voronoi structures containing a total of 250,000,000 cells to provide topological and geometrical statistics of this important class of networks. We al...	\section{Introduction} Poisson-Voronoi tessellations are random subdivisions of space that have found applications as models for many physical systems \cite{1992okabe, stoyan1995stochastic}. They have been used to study how galaxies are distributed throughout space \cite{icke1988voronoi, yoshioka1989large} and have ai...	{ "timestamp": "2014-01-09T02:10:06", "yymm": "1401", "arxiv_id": "1401.1736", "language": "en", "url": "https://arxiv.org/abs/1401.1736", "abstract": "Voronoi tessellations of Poisson point processes are widely used for modeling many types of physical and biological systems. In this paper, we analyze simul...
https://arxiv.org/abs/2005.03921	Closed formulas and determinantal expressions for higher-order Bernoulli and Euler polynomials in terms of Stirling numbers	In this paper, applying the Faà di Bruno formula and some properties of Bell polynomials, several closed formulas and determinantal expressions involving Stirling numbers of the second kind for higher-order Bernoulli and Euler polynomials are presented.	\section{Introduction} The classical Bernoulli polynomials $B_{n}(x)$ and Euler polynomials $E_{n}(x)$ are usually defined by means of the following generating functions \[ \dfrac{te^{xt}}{e^{t}-1} {\displaystyle\sum\limits_{n=0}^{\infty}} B_{n}(x)\dfrac{t^{n}}{n!}\text{ }\left( \left\vert t\right\vert <2\pi\rig...	{ "timestamp": "2020-05-11T02:09:12", "yymm": "2005", "arxiv_id": "2005.03921", "language": "en", "url": "https://arxiv.org/abs/2005.03921", "abstract": "In this paper, applying the Faà di Bruno formula and some properties of Bell polynomials, several closed formulas and determinantal expressions involving ...
https://arxiv.org/abs/1608.06905	Pólya's conjecture fails for the fractional Laplacian	The analogue of Pólya's conjecture is shown to fail for the fractional Laplacian (-Delta)^{alpha/2} on an interval in 1-dimension, whenever 0 < alpha < 2. The failure is total: every eigenvalue lies below the corresponding term of the Weyl asymptotic.In 2-dimensions, the fractional Pólya conjecture fails already for th...	\subsection*{\bf Introduction} The Weyl asymptotic for the $n$-th eigenvalue of the Dirichlet Laplacian on a bounded domain of volume $V$ in ${\mathbb R}^d$ says that \[ \lambda_n \sim (nC_d/V)^{2/d} \qquad \text{as $n \to \infty$,} \] where $C_d = (2\pi)^d/\omega_d$ and $\omega_d=$ volume of the unit ball in ${\mathb...	{ "timestamp": "2016-08-25T02:06:15", "yymm": "1608", "arxiv_id": "1608.06905", "language": "en", "url": "https://arxiv.org/abs/1608.06905", "abstract": "The analogue of Pólya's conjecture is shown to fail for the fractional Laplacian (-Delta)^{alpha/2} on an interval in 1-dimension, whenever 0 < alpha < 2....
https://arxiv.org/abs/1502.07805	On Hopital-style rules for monotonicity and oscillation	We point out the connection of the so-called Hôpital-style rules for monotonicity and oscillation to some well-known properties of concave/convex functions. From this standpoint, we are able to generalize the rules under no differentiability requirements and greatly extend their usability. The improved rules can handle...	\section{Introduction and historical remarks} Since the 1990's, many authors have successfully applied the so-called monotone L'H\^opital's\footnote{A well-known anecdote, recounted in some undergraduate textbooks and the Wikipedia, claims that H\^opital might have cheated his teacher Johann Bernoulli to earn the cred...	{ "timestamp": "2015-03-02T02:05:42", "yymm": "1502", "arxiv_id": "1502.07805", "language": "en", "url": "https://arxiv.org/abs/1502.07805", "abstract": "We point out the connection of the so-called Hôpital-style rules for monotonicity and oscillation to some well-known properties of concave/convex function...
https://arxiv.org/abs/1404.2443	Polygons as sections of higher-dimensional polytopes	We show that every heptagon is a section of a $3$-polytope with $6$ vertices. This implies that every $n$-gon with $n\geq 7$ can be obtained as a section of a $(2+\lfloor\frac{n}{7}\rfloor)$-dimensional polytope with at most $\lceil\frac{6n}{7}\rceil$ vertices; and provides a geometric proof of the fact that every nonn...	\section{Introduction} Let $P$ be a (convex) polytope. An \defn{extension} of~$P$ is any polytope~$Q$ such that~$P$ is the image of $Q$ under a linear projection; the \defn{extension complexity} of~$P$, denoted~\defn{$\ec(P)$}, is the minimal number of facets of an extension of~$P$. This concept is relevant in comb...	{ "timestamp": "2015-02-11T02:18:15", "yymm": "1404", "arxiv_id": "1404.2443", "language": "en", "url": "https://arxiv.org/abs/1404.2443", "abstract": "We show that every heptagon is a section of a $3$-polytope with $6$ vertices. This implies that every $n$-gon with $n\\geq 7$ can be obtained as a section o...
https://arxiv.org/abs/2201.12949	The shallow permutations are the unlinked permutations	Diaconis and Graham studied a measure of distance from the identity in the symmetric group called total displacement and showed that it is bounded below by the sum of length and reflection length. They asked for a characterization of the permutations where this bound is an equality; we call these the shallow permutatio...	\section{Introduction} There are many measures for how far a given permutation $w\in S_n$ is from being the identity. The most classical are length and reflection length, which are defined as follows. Let $s_i$ denote the adjacent transposition $s_i=(i\,\,i+1)$ and $t_{ij}$ the transposition $t_{ij}=(i\,\,j)$. Th...	{ "timestamp": "2022-02-01T02:32:50", "yymm": "2201", "arxiv_id": "2201.12949", "language": "en", "url": "https://arxiv.org/abs/2201.12949", "abstract": "Diaconis and Graham studied a measure of distance from the identity in the symmetric group called total displacement and showed that it is bounded below b...
https://arxiv.org/abs/1405.2805	Cross-intersecting families of vectors	Given a sequence of positive integers $p = (p_1, . . ., p_n)$, let $S_p$ denote the family of all sequences of positive integers $x = (x_1,...,x_n)$ such that $x_i \le p_i$ for all $i$. Two families of sequences (or vectors), $A,B \subseteq S_p$, are said to be $r$-cross-intersecting if no matter how we select $x \in A...	\section{Introduction} The Erd\H os-Ko-Rado theorem~\cite{EKR61} states that for $n\geq 2k$, every family of pairwise intersecting $k$-element subsets of an $n$-element set consists of at most ${n-1\choose k-1}$ subsets, as many as the star-like family of all subsets containing a fixed element of the underlying set. ...	{ "timestamp": "2015-02-02T02:09:05", "yymm": "1405", "arxiv_id": "1405.2805", "language": "en", "url": "https://arxiv.org/abs/1405.2805", "abstract": "Given a sequence of positive integers $p = (p_1, . . ., p_n)$, let $S_p$ denote the family of all sequences of positive integers $x = (x_1,...,x_n)$ such th...
https://arxiv.org/abs/1203.5207	Linear extensions of partial orders and Reverse Mathematics	We introduce the notion of \tau-like partial order, where \tau is one of the linear order types \omega, \omega, \omega+\omega, and \zeta. For example, being \omega-like means that every element has finitely many predecessors, while being \zeta-like means that every interval is finite. We consider statements of the fo...	\section{Introduction} Szpilrajn's Theorem (\cite{Szp30}) states that any partial order has a linear extension. This theorem raises many natural questions, where in general we search for properties of the partial order which are preserved by some or all its linear extensions. For example it is well-known that a parti...	{ "timestamp": "2012-04-17T02:03:06", "yymm": "1203", "arxiv_id": "1203.5207", "language": "en", "url": "https://arxiv.org/abs/1203.5207", "abstract": "We introduce the notion of \\tau-like partial order, where \\tau is one of the linear order types \\omega, \\omega, \\omega+\\omega, and \\zeta. For examp...
https://arxiv.org/abs/1810.13418	Sharp error estimates for spline approximation: explicit constants, $n$-widths, and eigenfunction convergence	In this paper we provide a priori error estimates in standard Sobolev (semi-)norms for approximation in spline spaces of maximal smoothness on arbitrary grids. The error estimates are expressed in terms of a power of the maximal grid spacing, an appropriate derivative of the function to be approximated, and an explicit...	\section{Introduction} Splines are piecewise polynomial functions that are glued together in a certain smooth way. When using them in an approximation method, the availability of sharp error estimates is of utmost importance. Depending on the problem to be addressed, one needs to tailor the norm to measure the error, t...	{ "timestamp": "2019-07-09T02:08:12", "yymm": "1810", "arxiv_id": "1810.13418", "language": "en", "url": "https://arxiv.org/abs/1810.13418", "abstract": "In this paper we provide a priori error estimates in standard Sobolev (semi-)norms for approximation in spline spaces of maximal smoothness on arbitrary g...
https://arxiv.org/abs/1805.04759	An Analog of Matrix Tree Theorem for Signless Laplacians	A spanning tree of a graph is a connected subgraph on all vertices with the minimum number of edges. The number of spanning trees in a graph $G$ is given by Matrix Tree Theorem in terms of principal minors of Laplacian matrix of $G$. We show a similar combinatorial interpretation for principal minors of signless Laplac...	\section{Introduction} For a simple graph $G$ on $n$ vertices $1,2,\ldots,n$ and $m$ edges $1,2,\ldots, m$ we define its \emph{degree matrix} $D$, \emph{adjacency matrix} $A$, and \emph{incidence matrix} $N$ as follows: \begin{enumerate} \item $D=[d_{ij}]$ is an $n \times n$ diagonal matrix where $d_{ii}$ is the d...	{ "timestamp": "2018-05-15T02:08:22", "yymm": "1805", "arxiv_id": "1805.04759", "language": "en", "url": "https://arxiv.org/abs/1805.04759", "abstract": "A spanning tree of a graph is a connected subgraph on all vertices with the minimum number of edges. The number of spanning trees in a graph $G$ is given ...
https://arxiv.org/abs/1609.06172	Optimal stretching for lattice points and eigenvalues	We aim to maximize the number of first-quadrant lattice points in a convex domain with respect to reciprocal stretching in the coordinate directions. The optimal domain is shown to be asymptotically balanced, meaning that the stretch factor approaches 1 as the "radius" approaches infinity. In particular, the result imp...	\section{\bf Introduction} Among ellipses of given area centered at the origin and symmetric about both axes, which one encloses the most integer lattice points in the open first quadrant? One might guess the optimal ellipse would be circular, but a non-circular ellipse can enclose more lattice points, as shown in \au...	{ "timestamp": "2017-09-07T02:05:14", "yymm": "1609", "arxiv_id": "1609.06172", "language": "en", "url": "https://arxiv.org/abs/1609.06172", "abstract": "We aim to maximize the number of first-quadrant lattice points in a convex domain with respect to reciprocal stretching in the coordinate directions. The ...
https://arxiv.org/abs/2109.09928	A numerical study of L-convex polyominoes and 201-avoiding ascent sequences	For L-convex polyominoes we give the asymptotics of the generating function coefficients, obtained by analysis of the coefficients derived from the functional equation given by Castiglione et al. \cite{CFMRR7}. For 201-avoiding ascent sequences, we conjecture the solution, obtained from the first 23 coefficients of the...	\section{Introduction} \label{introduction} In \cite{CFMRR7}, Castiglione et al. gave a functional equation for the number of $L$-convex polyominoes. These are defined as polyominoes with the property that any two cells may be joined by an $L$-shaped path. An example is shown in Fig. \ref{Lconvex}, and it can be seen t...	{ "timestamp": "2021-09-30T02:05:50", "yymm": "2109", "arxiv_id": "2109.09928", "language": "en", "url": "https://arxiv.org/abs/2109.09928", "abstract": "For L-convex polyominoes we give the asymptotics of the generating function coefficients, obtained by analysis of the coefficients derived from the functi...
https://arxiv.org/abs/0909.5024	Generalized Sidon sets	We give asymptotic sharp estimates for the cardinality of a set of residue classes with the property that the representation function is bounded by a prescribed number. We then use this to obtain an analogous result for sets of integers, answering an old question of Simon Sidon.	\section{Introduction} A \emph{Sidon set} $A$ in a commutative group is a set with the property that the sums $a_1+a_2$, $a_i\in A$ are all distinct except when they coincide because of commutativity. We consider the case when, instead of that, a bound is imposed on the number of such representations. When this b...	{ "timestamp": "2009-09-28T09:12:44", "yymm": "0909", "arxiv_id": "0909.5024", "language": "en", "url": "https://arxiv.org/abs/0909.5024", "abstract": "We give asymptotic sharp estimates for the cardinality of a set of residue classes with the property that the representation function is bounded by a prescr...
https://arxiv.org/abs/1208.3958	A Note on why Enforcing Discrete Maximum Principles by a simple a Posteriori Cutoff is a Good Idea	Discrete maximum principles in the approximation of partial differential equations are crucial for the preservation of qualitative properties of physical models. In this work we enforce the discrete maximum principle by performing a simple cutoff. We show that for many problems this a posteriori procedure even improves...	\section{Introduction} \label{sec:introduction} Consider a function $u:\Omega\to \setR$ on some bounded domain $\Omega\subset\setR^d$ such that \begin{align}\label{eq:rd} -\Delta u + c \,u \le 0 \end{align} in the variational sense for some nonnegative $c$. Then for $u^+=\max\{0,u\}$, we have the estimate \begin{a...	{ "timestamp": "2012-08-21T02:07:42", "yymm": "1208", "arxiv_id": "1208.3958", "language": "en", "url": "https://arxiv.org/abs/1208.3958", "abstract": "Discrete maximum principles in the approximation of partial differential equations are crucial for the preservation of qualitative properties of physical mo...
https://arxiv.org/abs/2211.07227	Stochastic approximation approaches for CVaR-based variational inequalities	This paper considers variational inequalities (VI) defined by the conditional value-at-risk (CVaR) of uncertain functions and provides three stochastic approximation schemes to solve them. All methods use an empirical estimate of the CVaR at each iteration. The first algorithm constrains the iterates to the feasible se...	\section{Introduction} \label{sec:introduction} This document is a template for \LaTeX. If you are reading a paper or PDF version of this document, please download the electronic file, trans\_jour.tex, from the IEEE Web site at \underline {http://www.ieee.org/authortools/trans\_jour.tex} so you can use it to prepare ...	{ "timestamp": "2022-11-16T02:11:51", "yymm": "2211", "arxiv_id": "2211.07227", "language": "en", "url": "https://arxiv.org/abs/2211.07227", "abstract": "This paper considers variational inequalities (VI) defined by the conditional value-at-risk (CVaR) of uncertain functions and provides three stochastic ap...
https://arxiv.org/abs/1404.3469	Polynomial reconstruction of the matching polynomial	The matching polynomial of a graph is the generating function of the numbers of its matchings with respect to their cardinality. A graph polynomial is polynomial reconstructible, if its value for a graph can be determined from its values for the vertex-deleted subgraphs of the same graph. This note discusses the polyno...	\section{Counterexamples for arbitrary graphs} \label{sec:counterexamples} While it is true that the matching polynomials of graphs with an odd number of vertices or with an pendant edge are polynomial reconstructible, it does not hold for arbitrary graphs. There are graphs which have the same polynomial deck and yet...	{ "timestamp": "2014-04-15T02:10:55", "yymm": "1404", "arxiv_id": "1404.3469", "language": "en", "url": "https://arxiv.org/abs/1404.3469", "abstract": "The matching polynomial of a graph is the generating function of the numbers of its matchings with respect to their cardinality. A graph polynomial is polyn...
https://arxiv.org/abs/1611.06842	Almost tiling of the Boolean lattice with copies of a poset	Let $P$ be a partially ordered set. If the Boolean lattice $(2^{[n]},\subset)$ can be partitioned into copies of $P$ for some positive integer $n$, then $P$ must satisfy the following two trivial conditions:(1) the size of $P$ is a power of $2$,(2) $P$ has a unique maximal and minimal element.Resolving a conjecture of ...	\section{Introduction} The \emph{Boolean lattice} $(2^{[n]},\subset)$ is the power set of $[n]=\{1,...,n\}$ ordered by inclusion. If $P$ and $Q$ are partially ordered sets (posets), a subset $P'\subset Q$ is a \emph{copy} of $P$ if the subposet of $Q$ induced on $P'$ is isomorphic to $P$. A \emph{chain} is a copy of...	{ "timestamp": "2016-11-22T02:12:24", "yymm": "1611", "arxiv_id": "1611.06842", "language": "en", "url": "https://arxiv.org/abs/1611.06842", "abstract": "Let $P$ be a partially ordered set. If the Boolean lattice $(2^{[n]},\\subset)$ can be partitioned into copies of $P$ for some positive integer $n$, then ...
https://arxiv.org/abs/1304.7515	A short note on short pants	It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied and the best upper bounds to date are linear in ge...	\section{Introduction} \label{introduction} A pants decomposition of a hyperbolic surface is a maximal collection of disjoint simple closed geodesics, which as its name indicates, decomposes the surface into three holed spheres or pairs of pants. In the case of closed surfaces of genus $g\geq 2$, a pants decomposition ...	{ "timestamp": "2013-05-23T02:01:14", "yymm": "1304", "arxiv_id": "1304.7515", "language": "en", "url": "https://arxiv.org/abs/1304.7515", "abstract": "It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends ...
https://arxiv.org/abs/2206.13158	Sharp inequalities involving the Cheeger constant of planar convex sets	We are interested in finding sharp bounds for the Cheeger constant via different geometrical quantities, such as the area $\|\cdot\|$, the perimeter $P$, the inradius $r$, the circumradius $R$, the minimal width $w$ and the diameter $d$. In particular, we provide new sharp inequalities between these quantities for planar...	\section{Introduction} Let $\Omega$ be a bounded subset of $\mathbb{R}^2$. The Cheeger constant of $\Omega$, introduced by Jeff Cheeger in \cite{cheger} in connection with the first eigenvalue of the Laplacian, is defined as \begin{equation} \label{chee} h(\Omega):=\inf\left\{\frac{P(E)}{\abs{E}} \, : \, E \,...	{ "timestamp": "2022-06-28T02:26:51", "yymm": "2206", "arxiv_id": "2206.13158", "language": "en", "url": "https://arxiv.org/abs/2206.13158", "abstract": "We are interested in finding sharp bounds for the Cheeger constant via different geometrical quantities, such as the area $\|\\cdot\|$, the perimeter $P$, t...
https://arxiv.org/abs/1801.07238	On a Helly-type question for central symmetry	We study a certain Helly-type question by Konrad Swanepoel. Assume that $X$ is a set of points such that every $k$-subset of $X$ is in centrally symmetric convex position, is it true that $X$ must also be in centrally symmetric convex position? It is easy to see that this is false if $k\le 5$, but it may be true for su...	\section{Introduction} The classical Carathéodory theorem in dimension $2$ can be stated in the following equivalent way: Let $X$ be a set of points in the plane, if any $4$ points from $X$ are in convex positions then $X$ is in convex position. In 2010, Konrad Swanepoel \cite{KS} asked the following Helly-type questio...	{ "timestamp": "2018-01-23T02:17:59", "yymm": "1801", "arxiv_id": "1801.07238", "language": "en", "url": "https://arxiv.org/abs/1801.07238", "abstract": "We study a certain Helly-type question by Konrad Swanepoel. Assume that $X$ is a set of points such that every $k$-subset of $X$ is in centrally symmetric...
https://arxiv.org/abs/1109.4930	Multiset metrics on bounded spaces	We discuss five simple functions on finite multisets of metric spaces. The first four are all metrics iff the underlying space is bounded and are complete metrics iff it is also complete. Two of them, and the fifth function, all generalise the usual Hausdorff metric on subsets. Some possible applications are also consi...	\section{Introduction} Metrics on subsets and multisets (subsets-with-repetition-allowed) of metric spaces have or could have numerous fields of application such as credit rating, pattern or image recognition and synthetic biology. We employ three related models (called $E,F$ and $G$) for the space of multisets on the...	{ "timestamp": "2011-09-23T02:05:52", "yymm": "1109", "arxiv_id": "1109.4930", "language": "en", "url": "https://arxiv.org/abs/1109.4930", "abstract": "We discuss five simple functions on finite multisets of metric spaces. The first four are all metrics iff the underlying space is bounded and are complete m...
https://arxiv.org/abs/1812.11043	Toric degenerations in symplectic geometry	A toric degeneration in algebraic geometry is a process where a given projective variety is being degenerated into a toric one. Then one can obtain information about the original variety via analyzing the toric one, which is a much easier object to study. Harada and Kaveh described how one incorporates a symplectic str...	\section{Introduction} Manifolds and algebraic varieties equipped with a group action are usually better understood as a presence of an action is a sign of certain symmetries. In particular, {\it toric varieties} form a very well understood class of varieties. These are varieties which contain an algebraic torus $T^n_{...	{ "timestamp": "2018-12-31T02:19:02", "yymm": "1812", "arxiv_id": "1812.11043", "language": "en", "url": "https://arxiv.org/abs/1812.11043", "abstract": "A toric degeneration in algebraic geometry is a process where a given projective variety is being degenerated into a toric one. Then one can obtain inform...
https://arxiv.org/abs/0804.1268	k-Wise Independent Random Graphs	We study the k-wise independent relaxation of the usual model G(N,p) of random graphs where, as in this model, N labeled vertices are fixed and each edge is drawn with probability p, however, it is only required that the distribution of any subset of k edges is independent. This relaxation can be relevant in modeling p...	\section{{{\underline{\underline{#1}}}}} \newcommand {\mysubsubsection} [1] {\subsubsection{{{#1}}} \newcommand {\mysection} [1] {\section{{{#1}}} \newcommand {\mysubsection} [1] {\subsection{{{#1}}} \newcommand {\myParagraph} [1] {\subsubsection*{#1}} \newcommand {\SaveSizeParagraph} [1] {\vspace{-1.6mm}\paragraph{#1}...	{ "timestamp": "2008-04-08T15:10:58", "yymm": "0804", "arxiv_id": "0804.1268", "language": "en", "url": "https://arxiv.org/abs/0804.1268", "abstract": "We study the k-wise independent relaxation of the usual model G(N,p) of random graphs where, as in this model, N labeled vertices are fixed and each edge is...
https://arxiv.org/abs/1610.00954	Exact and Positive Controllability of Boundary Control Systems	Using the semigroup approach to abstract boundary control problems we characterize the space of all exactly reachable states. Moreover, we study the situation when the controls of the system are required to be positive. The abstract results are applied to flows in networks with static as well as dynamic boundary condit...	\section{Introduction} This paper is a continuation of \cite{EKNS:08, EKKNS:10} where we introduced a semigroup approach to boundary control problems and applied it to the control of flows in networks. While in these previous works we concentrated on maximal \emph{approximate} controllability, we now focus on the \emph...	{ "timestamp": "2016-10-05T02:04:50", "yymm": "1610", "arxiv_id": "1610.00954", "language": "en", "url": "https://arxiv.org/abs/1610.00954", "abstract": "Using the semigroup approach to abstract boundary control problems we characterize the space of all exactly reachable states. Moreover, we study the situa...
https://arxiv.org/abs/2104.01689	What does a typical metric space look like?	The collection $\mathcal{M}_n$ of all metric spaces on $n$ points whose diameter is at most $2$ can naturally be viewed as a compact convex subset of $\mathbb{R}^{\binom{n}{2}}$, known as the metric polytope. In this paper, we study the metric polytope for large $n$ and show that it is close to the cube $[1,2]^{\binom{...	\section{Introduction} For a positive integer $n$, let $\br{n} :=\{1, \dotsc, n\}$ and let $\binom{\br{n}}{2}$ be the set of all unordered pairs of distinct elements in $\br{n}$. A finite metric space on $n\ge 2$ points can be regarded as an array $(\dist{i}{j})$ with $\{i,j\}\in \binom{\br{n}}{2}$, where $\dist{i}{j}$...	{ "timestamp": "2021-04-06T02:24:49", "yymm": "2104", "arxiv_id": "2104.01689", "language": "en", "url": "https://arxiv.org/abs/2104.01689", "abstract": "The collection $\\mathcal{M}_n$ of all metric spaces on $n$ points whose diameter is at most $2$ can naturally be viewed as a compact convex subset of $\\...
https://arxiv.org/abs/1405.7334	Strong duality in Lasserre's hierarchy for polynomial optimization	A polynomial optimization problem (POP) consists of minimizing a multivariate real polynomial on a semi-algebraic set $K$ described by polynomial inequalities and equations. In its full generality it is a non-convex, multi-extremal, difficult global optimization problem. More than an decade ago, J.~B.~Lasserre proposed...	\section{Introduction} Consider the following polynomial optimization problem (POP) \begin{equation} \label{eq:pop} \begin{array}{ll} \inf_x & f(x) := \sum_\alpha f_{\alpha} x^\alpha \\ \mathrm{s.t.} & g_i(x) := \sum_\alpha g_{i,\alpha} x^\alpha \geq 0, \quad i=1,\ldots,m \end{array} \end{equation} where we use the mul...	{ "timestamp": "2014-05-29T02:10:57", "yymm": "1405", "arxiv_id": "1405.7334", "language": "en", "url": "https://arxiv.org/abs/1405.7334", "abstract": "A polynomial optimization problem (POP) consists of minimizing a multivariate real polynomial on a semi-algebraic set $K$ described by polynomial inequaliti...
https://arxiv.org/abs/2005.08903	Iterative and doubling algorithms for Riccati-type matrix equations: a comparative introduction	We review a family of algorithms for Lyapunov- and Riccati-type equations which are all related to each other by the idea of \emph{doubling}: they construct the iterate $Q_k = X_{2^k}$ of another naturally-arising fixed-point iteration $(X_h)$ via a sort of repeated squaring.The equations we consider are Stein equation...	\section{Introduction} Riccati-type matrix equations are a family of matrix equations that appears very frequently in literature and applications, especially in systems theory. One of the reasons why they are so ubiquitous is that they are equivalent to certain invariant subspace problems; this equivalence connects th...	{ "timestamp": "2020-05-19T02:37:09", "yymm": "2005", "arxiv_id": "2005.08903", "language": "en", "url": "https://arxiv.org/abs/2005.08903", "abstract": "We review a family of algorithms for Lyapunov- and Riccati-type equations which are all related to each other by the idea of \\emph{doubling}: they constr...
https://arxiv.org/abs/math/0611614	Matchings in arbitrary groups	A matching in a group G is a bijection f from a subset A to a subset B in G such that af(a) does not belong to A for all a in A. The group G is said to have the matching property if, for any finite subsets A,B in G of same cardinality with B avoiding 1, there is a matching from A to B.Using tools from additive number t...	\section{Introduction} Let $G$ be a group, written multiplicatively. Given nonempty finite subsets $A,B$ in $G$, a \textit{matching} from $A$ to $B$ is a map $\varphi:A \rightarrow B$ which is bijective and satisfies the condition $$a\varphi(a) \notin A$$ for all $a \in A$. This notion was introduced in \cite{FanLo...	{ "timestamp": "2006-11-20T18:10:23", "yymm": "0611", "arxiv_id": "math/0611614", "language": "en", "url": "https://arxiv.org/abs/math/0611614", "abstract": "A matching in a group G is a bijection f from a subset A to a subset B in G such that af(a) does not belong to A for all a in A. The group G is said t...