url stringlengths 31 38 | title stringlengths 7 229 | abstract stringlengths 44 2.87k | text stringlengths 319 2.51M | meta dict |
|---|---|---|---|---|
https://arxiv.org/abs/0710.2387 | Characterizing arbitrarily slow convergence in the method of alternating projections | In 1997, Bauschke, Borwein, and Lewis have stated a trichotomy theorem that characterizes when the convergence of the method of alternating projections can be arbitrarily slow. However, there are two errors in their proof of this theorem. In this note, we show that although one of the errors is critical, the theorem it... | \section{Introduction}\label{S: intro}
For the notation and basic Hilbert space results necessary to read this paper, the book \cite{deu;01} is a good source, especially chapter 9.
Let $H$ be a (real or complex) Hilbert space with inner product $\langle x, y \rangle$ and norm $\|x\|=\sqrt{\langle x, x\... | {
"timestamp": "2007-10-12T07:31:40",
"yymm": "0710",
"arxiv_id": "0710.2387",
"language": "en",
"url": "https://arxiv.org/abs/0710.2387",
"abstract": "In 1997, Bauschke, Borwein, and Lewis have stated a trichotomy theorem that characterizes when the convergence of the method of alternating projections can ... |
https://arxiv.org/abs/1808.03230 | Does Hamiltonian Monte Carlo mix faster than a random walk on multimodal densities? | Hamiltonian Monte Carlo (HMC) is a very popular and generic collection of Markov chain Monte Carlo (MCMC) algorithms. One explanation for the popularity of HMC algorithms is their excellent performance as the dimension $d$ of the target becomes large: under conditions that are satisfied for many common statistical mode... | \section{Introduction}
Markov chain Monte Carlo (MCMC) algorithms, and in particular the Hamiltonian Monte Carlo (HMC) algorithms, are workhorses in many scientific fields including physics \cite{Hybrid_MCMC}, statistics and machine learning \cite{NUTS,Riemannian_HMC2, MCMC_Application_Machine_Learning,welling2011baye... | {
"timestamp": "2018-09-05T02:32:46",
"yymm": "1808",
"arxiv_id": "1808.03230",
"language": "en",
"url": "https://arxiv.org/abs/1808.03230",
"abstract": "Hamiltonian Monte Carlo (HMC) is a very popular and generic collection of Markov chain Monte Carlo (MCMC) algorithms. One explanation for the popularity o... |
https://arxiv.org/abs/1410.7791 | Hölder stability for Serrin's overdetermined problem | In a bounded domain $\Omega$, we consider a positive solution of the problem $\Delta u+f(u)=0$ in $\Omega$, $u=0$ on $\partial\Omega$, where $f:\mathbb{R}\to\mathbb{R}$ is a locally Lipschitz continuous function. Under sufficient conditions on $\Omega$ (for instance, if $\Omega$ is convex), we show that $\partial\Omega... | \section[Introduction]{Introduction}
Serrin's overdetermined problem has been the object of many investigations.
In its classical form, it involves a sufficiently smooth bounded domain $\Omega$ in ${\mathbb R^N}$ and a
classical solution of the set of equations:
\begin{eqnarray}
&\Delta u + f(u) = 0 \ \mbox{ and } \
u\... | {
"timestamp": "2015-06-22T02:06:55",
"yymm": "1410",
"arxiv_id": "1410.7791",
"language": "en",
"url": "https://arxiv.org/abs/1410.7791",
"abstract": "In a bounded domain $\\Omega$, we consider a positive solution of the problem $\\Delta u+f(u)=0$ in $\\Omega$, $u=0$ on $\\partial\\Omega$, where $f:\\mathb... |
https://arxiv.org/abs/1101.5638 | Lower Bound for Convex Hull Area and Universal Cover Problems | In this paper, we provide a lower bound for an area of the convex hull of points and a rectangle in a plane. We then apply this estimate to establish a lower bound for a universal cover problem. We showed that a convex universal cover for a unit length curve has area at least 0.232239. In addition, we show that a conve... | \section{Introduction}
One of the open classical problems in discrete geometry is the Moser's Worm problem, which originally asked for ``the smallest cover for any unit-length curve''. In the other words, the question asks for a minimal universal cover for any curve of unit length -- also called unit worm. Although it... | {
"timestamp": "2011-02-01T02:00:18",
"yymm": "1101",
"arxiv_id": "1101.5638",
"language": "en",
"url": "https://arxiv.org/abs/1101.5638",
"abstract": "In this paper, we provide a lower bound for an area of the convex hull of points and a rectangle in a plane. We then apply this estimate to establish a lowe... |
https://arxiv.org/abs/1711.04504 | Tilings with noncongruent triangles | We solve a problem of R. Nandakumar by proving that there is no tiling of the plane with pairwise noncongruent triangles of equal area and equal perimeter. We also show that no convex polygon with more than three sides can be tiled with finitely many triangles such that no pair of them share a full side. | \section{Introduction}
In his blog, R. Nandakumar~\cite{Na06, Na14} raised several interesting questions on tilings. They have triggered a lot of research in geometry and topology. In particular, he and Ramana Rao~\cite{NaR12} conjectured that for every natural number $n$, any plane convex body can be partitioned into ... | {
"timestamp": "2018-04-12T02:09:06",
"yymm": "1711",
"arxiv_id": "1711.04504",
"language": "en",
"url": "https://arxiv.org/abs/1711.04504",
"abstract": "We solve a problem of R. Nandakumar by proving that there is no tiling of the plane with pairwise noncongruent triangles of equal area and equal perimeter... |
https://arxiv.org/abs/1404.7254 | On Hodge Theory of Singular Plane Curves | The dimensions of the graded quotients of the cohomology of a plane curve complement with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed in detail. | \section{Introduction}
The Hodge theory of the complement of projective hypersurfaces have received a lot of attention, see for instance Griffiths \cite{G} in the smooth case, Dimca-Saito \cite{DS} and Sernesi \cite{Se} in the singular case. In this paper we consider the case of plane curves and continue the study init... | {
"timestamp": "2015-04-08T02:05:10",
"yymm": "1404",
"arxiv_id": "1404.7254",
"language": "en",
"url": "https://arxiv.org/abs/1404.7254",
"abstract": "The dimensions of the graded quotients of the cohomology of a plane curve complement with respect to the Hodge filtration are described in terms of simple g... |
https://arxiv.org/abs/1711.04909 | An Optimal Convergence Rate for the Gaussian Regularized Shannon Sampling Series | We consider the reconstruction of a bandlimited function from its finite localized sample data. Truncating the classical Shannon sampling series results in an unsatisfactory convergence rate due to the slow decay of the sinc function. To overcome this drawback, a simple and highly effective method, called the Gaussian ... | \section{Introduction}
The classical Shannon sampling theorem \cite{Jerri, Kotelnikov33,Nyquist28, Shannon,Unser,Whittaker} states that any bandlimited function with bandwidth $\pi$ can be completely reconstructed by its infinite samples at integers. In practice, however, we can only sum over finite sample data ``nea... | {
"timestamp": "2018-11-07T02:19:58",
"yymm": "1711",
"arxiv_id": "1711.04909",
"language": "en",
"url": "https://arxiv.org/abs/1711.04909",
"abstract": "We consider the reconstruction of a bandlimited function from its finite localized sample data. Truncating the classical Shannon sampling series results i... |
https://arxiv.org/abs/2110.02264 | Generic Generalized Diagonal Matrices | Generalized diagonal matrices are matrices that have two ladders of entries that are zero in the upper right and bottom left corners. The minors of generic generalized diagonal matrices have square-free initial ideals. We give a description of the facets of their Stanley-Reisner complex. With this description, we chara... | \section{Introduction}
The initial ideals of determinantal ideals are square-free, hence a combinatorial approach, using Stanley-Reisner complexes, can be applied to study them. Herzog and Trung in \cite{Herzog1992} gave a description of the Stanley-Reisner complexes of generic determinantal ideals. They use this desc... | {
"timestamp": "2022-06-06T02:15:13",
"yymm": "2110",
"arxiv_id": "2110.02264",
"language": "en",
"url": "https://arxiv.org/abs/2110.02264",
"abstract": "Generalized diagonal matrices are matrices that have two ladders of entries that are zero in the upper right and bottom left corners. The minors of generi... |
https://arxiv.org/abs/math/0610498 | Bounds on changes in Ritz values for a perturbed invariant subspace of a Hermitian matrix | The Rayleigh-Ritz method is widely used for eigenvalue approximation. Given a matrix $X$ with columns that form an orthonormal basis for a subspace $\X$, and a Hermitian matrix $A$, the eigenvalues of $X^HAX$ are called Ritz values of $A$ with respect to $\X$. If the subspace $\X$ is $A$-invariant then the Ritz values ... | \section{Introduction} \label{sec:1intro}
Eigenvalue problems appear in many applications.
For example eigenvalues represent the
frequencies of vibration in mechanical vibrations,
while the energy levels of a system are the
eigenvalues of the Hamiltonian
in quantum mechanics.
Eigenvalue problems are used today in thes... | {
"timestamp": "2008-02-04T03:09:54",
"yymm": "0610",
"arxiv_id": "math/0610498",
"language": "en",
"url": "https://arxiv.org/abs/math/0610498",
"abstract": "The Rayleigh-Ritz method is widely used for eigenvalue approximation. Given a matrix $X$ with columns that form an orthonormal basis for a subspace $\... |
https://arxiv.org/abs/0906.2809 | Sandpile groups and spanning trees of directed line graphs | We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph G and its directed line graph LG. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when G is regular of degree k, we ... | \section{Introduction}
Let $G=(V,E)$ be a finite directed graph, which may have loops and multiple edges. Each edge $e \in E$ is directed from its source vertex $\mathtt{s}(e)$ to its target vertex $\mathtt{t}(e)$. The \emph{directed line graph} $\line G = (E,E_2)$ has as vertices the edges of $G$, and as edges the s... | {
"timestamp": "2010-04-08T02:00:16",
"yymm": "0906",
"arxiv_id": "0906.2809",
"language": "en",
"url": "https://arxiv.org/abs/0906.2809",
"abstract": "We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph G and its directed line graph LG. The sandpile group is an abelian... |
https://arxiv.org/abs/1311.5718 | On floors and ceilings of the k-Catalan arrangement | The set of dominant regions of the $k$-Catalan arrangement of a crystallographic root system $\Phi$ is a well-studied object enumerated by the Fuß-Catalan number $Cat^{(k)}(\Phi)$. It is natural to refine this enumeration by considering floors and ceilings of dominant regions. A conjecture of Armstrong states that coun... | \section{Introduction}
Let $\Phi$ be a crystallographic root system of rank $n$ with simple system $S$, positive system $\Phi^+$, and ambient vector space $V$. For background on root systems see \cite{humphreys90reflection}. For $k$ a positive integer, we define the \emph{$k$-Catalan arrangement} of $\Phi$ as the hyper... | {
"timestamp": "2014-11-06T02:10:21",
"yymm": "1311",
"arxiv_id": "1311.5718",
"language": "en",
"url": "https://arxiv.org/abs/1311.5718",
"abstract": "The set of dominant regions of the $k$-Catalan arrangement of a crystallographic root system $\\Phi$ is a well-studied object enumerated by the Fuß-Catalan ... |
https://arxiv.org/abs/0710.2216 | The dying rabbit problem revisited | In this paper we study a generalization of the Fibonacci sequence in which rabbits are mortal and take more that two months to become mature. In particular we give a general recurrence relation for these sequences (improving the work in the paper Hoggatt, V. E., Jr.; Lind, D. A. "The dying rabbit problem". Fibonacci Qu... | \section{Introduction}
Fibonacci numbers arose in the answer to a problem proposed by Leonardo de Pisa who asked for the number of rabbits at the $n^{th}$ month if there is one pair of rabbits at the $0^{th}$ month which becomes mature one month later and that breeds another pair in each of the succeeding months, and i... | {
"timestamp": "2007-10-11T13:16:43",
"yymm": "0710",
"arxiv_id": "0710.2216",
"language": "en",
"url": "https://arxiv.org/abs/0710.2216",
"abstract": "In this paper we study a generalization of the Fibonacci sequence in which rabbits are mortal and take more that two months to become mature. In particular ... |
https://arxiv.org/abs/1810.08742 | Some remarks on the correspondence between elliptic curves and four points in the Riemann sphere | In this paper we relate some classical normal forms for complex elliptic curves in terms of 4-point sets in the Riemann sphere. Our main result is an alternative proof that every elliptic curve is isomorphic as a Riemann surface to one in the Hesse normal form. In this setting, we give an alternative proof of the equiv... | \section*{Introduction} A complex elliptic curve is by definition a
compact Riemann surface of genus 1. By the \emph{uniformization
theorem}, every elliptic curve is conformally equivalent to an
algebraic curve given by a cubic polynomial in the form
\begin{equation}\label{Weierstrass-intro}
E\colon \ y^2=4x^3-g_... | {
"timestamp": "2018-10-23T02:04:24",
"yymm": "1810",
"arxiv_id": "1810.08742",
"language": "en",
"url": "https://arxiv.org/abs/1810.08742",
"abstract": "In this paper we relate some classical normal forms for complex elliptic curves in terms of 4-point sets in the Riemann sphere. Our main result is an alte... |
https://arxiv.org/abs/1712.00887 | Regularity of Edge Ideals and Their Powers | We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\text{reg} I(G)$ and the asymptotic linear function $\text{reg} I(G)^q$, for $q \geq 1,$ in terms of combinatorial data of the given graph $G.$ | \section{Introduction}
Monomial ideals are classical objects that live at the crossroad of three areas in mathematics: algebra, combinatorics and topology. Investigating monomial ideals has led to many important results in these areas. The new construction of edge ideals of (hyper)graphs has again resurrected much int... | {
"timestamp": "2018-03-13T01:14:18",
"yymm": "1712",
"arxiv_id": "1712.00887",
"language": "en",
"url": "https://arxiv.org/abs/1712.00887",
"abstract": "We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of $\\te... |
https://arxiv.org/abs/2208.07229 | Proof of a conjecture on the determinant of walk matrix of rooted product with a path | The walk matrix of an $n$-vertex graph $G$ with adjacency matrix $A$, denoted by $W(G)$, is $[e,Ae,\ldots,A^{n-1}e]$, where $e$ is the all-ones vector. Let $G\circ P_m$ be the rooted product of $G$ and a rooted path $P_m$ (taking an endvertex as the root), i.e., $G\circ P_m$ is a graph obtained from $G$ and $n$ copies ... | \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 $A=(a_{i,j})$, where $a_{i,j}=1$ if $i$ and $j$ are adjacent; $a_{i,j}=0$ otherwise. For a graph $G$, the \emph{walk matrix} of $G$ is
\begin{equatio... | {
"timestamp": "2022-08-16T02:29:59",
"yymm": "2208",
"arxiv_id": "2208.07229",
"language": "en",
"url": "https://arxiv.org/abs/2208.07229",
"abstract": "The walk matrix of an $n$-vertex graph $G$ with adjacency matrix $A$, denoted by $W(G)$, is $[e,Ae,\\ldots,A^{n-1}e]$, where $e$ is the all-ones vector. L... |
https://arxiv.org/abs/2106.14755 | Counting Divisions of a $2\times n$ Rectangular Grid | Consider a $2\times n$ rectangular grid composed of $1\times 1$ squares. Cutting only along the edges between squares, how many ways are there to divide the board into $k$ pieces? Building off the work of Durham and Richmond, who found the closed-form solutions for the number of divisions into 2 and 3 pieces, we prove ... |
\section{Algorithm for Counting Divisions of $n\times m$ Board into $k$ Pieces} \label{sec:algorithm}
While $d_k^m(n)$ has directly proven explicit formulas for small values for $m$ and $k$, computational methods are much more useful to collect data about greater values. We used those data to fit polynomials that pre... | {
"timestamp": "2021-07-23T02:02:23",
"yymm": "2106",
"arxiv_id": "2106.14755",
"language": "en",
"url": "https://arxiv.org/abs/2106.14755",
"abstract": "Consider a $2\\times n$ rectangular grid composed of $1\\times 1$ squares. Cutting only along the edges between squares, how many ways are there to divide... |
https://arxiv.org/abs/1809.05858 | The Method of Alternating Projections | The method of alternating projections involves orthogonally projecting an element of a Hilbert space onto a collection of closed subspaces. It is known that the resulting sequence always converges in norm if the projections are taken periodically, or even quasiperiodically. We present proofs of such well known results,... | \section{Introduction} \label{introduction}
The method of alternating projections has been widely studied in mathematics. Interesting not only for its rich theory, it also has many wide-reaching applications, for instance to the iterative solution of large linear systems, in the theory of partial differential equations... | {
"timestamp": "2018-09-18T02:10:36",
"yymm": "1809",
"arxiv_id": "1809.05858",
"language": "en",
"url": "https://arxiv.org/abs/1809.05858",
"abstract": "The method of alternating projections involves orthogonally projecting an element of a Hilbert space onto a collection of closed subspaces. It is known th... |
https://arxiv.org/abs/1404.5886 | Log-concavity and strong log-concavity: a review | We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on $\mathbb{R}$ under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efr... | \section{Introduction: log-concavity}
\label{sec:intro}
Log-concave distributions and various properties related to log-concavity
play an increasingly important role in probability, statistics, optimization
theory, econometrics and other areas of applied mathematics. In view of
these developments, the basic propertie... | {
"timestamp": "2014-04-24T02:11:47",
"yymm": "1404",
"arxiv_id": "1404.5886",
"language": "en",
"url": "https://arxiv.org/abs/1404.5886",
"abstract": "We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-... |
https://arxiv.org/abs/1102.2141 | The Turán number of $F_{3,3}$ | Let $F_{3,3}$ be the 3-graph on 6 vertices, labelled abcxyz, and 10 edges, one of which is abc, and the other 9 of which are all triples that contain 1 vertex from abc and 2 vertices from xyz. We show that for all $n \ge 6$, the maximum number of edges in an $F_{3,3}$-free 3-graph on $n$ vertices is $\binom{n}{3} - \bi... | \section{Introduction}
The {\em Tur\'an number} $\mbox{ex}(n,F)$ is the maximum number of edges
in an $F$-free $r$-graph on $n$ vertices.%
\footnote{
An {\em $r$-graph} (or {\em $r$-uniform hypergraph}) $G$ consists of a vertex set
and an edge set, each edge being some $r$-set of vertices.
We say $G$ is {\em $F$-free... | {
"timestamp": "2011-02-11T02:02:16",
"yymm": "1102",
"arxiv_id": "1102.2141",
"language": "en",
"url": "https://arxiv.org/abs/1102.2141",
"abstract": "Let $F_{3,3}$ be the 3-graph on 6 vertices, labelled abcxyz, and 10 edges, one of which is abc, and the other 9 of which are all triples that contain 1 vert... |
https://arxiv.org/abs/1908.08910 | Counting pop-stacked permutations in polynomial time | Permutations in the image of the pop-stack operator are said to be pop-stacked. We give a polynomial-time algorithm to count pop-stacked permutations up to a fixed length and we use it to compute the first 1000 terms of the corresponding counting sequence. Only the first 16 terms had previously been computed. With the ... | \section{Pop-stacked permutations}
The abstract data type known as a \emph{stack} has two operations:
\emph{push} adds an element at the top of the stack; \emph{pop} removes
the top element from the stack. A \emph{pop-stack} is a variation of
this introduced by Avis and Newborn~\cite{avis1981pop} in which the pop
oper... | {
"timestamp": "2019-08-26T02:13:19",
"yymm": "1908",
"arxiv_id": "1908.08910",
"language": "en",
"url": "https://arxiv.org/abs/1908.08910",
"abstract": "Permutations in the image of the pop-stack operator are said to be pop-stacked. We give a polynomial-time algorithm to count pop-stacked permutations up t... |
https://arxiv.org/abs/1111.2450 | The Bernstein-Orlicz norm and deviation inequalities | We introduce two new concepts designed for the study of empirical processes. First, we introduce a new Orlicz norm which we call the Bernstein-Orlicz norm. This new norm interpolates sub-Gaussian and sub-exponential tail behavior. In particular, we show how this norm can be used to simplify the derivation of deviation ... | \section{Introduction}\label{introduction.section}
We introduce a new Orlicz norm which we name the Bernstein-Orlicz norm.
It interpolates sub-Gaussian and
sub-exponential tail behavior. With this new norm, we apply the
usual techniques based on Orlicz norms. In particular, we derive
deviation
inequalities for suprema ... | {
"timestamp": "2011-11-22T02:02:37",
"yymm": "1111",
"arxiv_id": "1111.2450",
"language": "en",
"url": "https://arxiv.org/abs/1111.2450",
"abstract": "We introduce two new concepts designed for the study of empirical processes. First, we introduce a new Orlicz norm which we call the Bernstein-Orlicz norm. ... |
https://arxiv.org/abs/0711.1400 | Polynomials associated with Partitions: Polynomials associated with Partitions: Their Asymptotics and Zeros | Let $p_n$ be the number of partitions of an integer $n$. For each of the partition statistics of counting their parts, ranks, or cranks, there is a natural family of integer polynomials. We investigate their asymptotics and the limiting behavior of their zeros as sets and densities. | \section{Introduction}\label{section:introduction}
The purpose of this paper is to survey
several natural polynomial families associated with integer partitions
focusing on
their asymptotics and the limiting behavior
of their zeros. Our principal families are
\begin{enumerate}
\item
Taylor polynomials of the analy... | {
"timestamp": "2007-11-09T03:16:41",
"yymm": "0711",
"arxiv_id": "0711.1400",
"language": "en",
"url": "https://arxiv.org/abs/0711.1400",
"abstract": "Let $p_n$ be the number of partitions of an integer $n$. For each of the partition statistics of counting their parts, ranks, or cranks, there is a natural ... |
https://arxiv.org/abs/1810.03050 | Leaky Roots and Stable Gauss-Lucas Theorems | Let $p:\mathbb{C} \rightarrow \mathbb{C}$ be a polynomial. The Gauss-Lucas theorem states that its critical points, $p'(z) = 0$, are contained in the convex hull of its roots. A recent quantitative version Totik shows that if almost all roots are contained in a bounded convex domain $K \subset \mathbb{C}$, then almost ... | \section{Introduction and result}
\subsection{Introduction.} The Gauss-Lucas Theorem, stated by Gauss \cite{gauss} in 1836 and proved by Lucas \cite{lucas} in 1879, says that if $p_n:\mathbb{C} \rightarrow \mathbb{C}$ is a polynomial of degree $n$, then the $n-1$ zeroes of $p_n'$ lie inside the convex hull of the $n$ z... | {
"timestamp": "2019-01-23T02:18:48",
"yymm": "1810",
"arxiv_id": "1810.03050",
"language": "en",
"url": "https://arxiv.org/abs/1810.03050",
"abstract": "Let $p:\\mathbb{C} \\rightarrow \\mathbb{C}$ be a polynomial. The Gauss-Lucas theorem states that its critical points, $p'(z) = 0$, are contained in the c... |
https://arxiv.org/abs/1804.07104 | From a Consequence of Bertrand's Postulate to Hamilton Cycles | A consequence of Bertrand's postulate, proved by L. Greenfield and S. Greenfield in 1998, assures that the set of integers $\{1,2,\cdots, 2n\}$ can be partitioned into pairs so that the sum of each pair is a prime number for any positive integer $n$. Cutting through it from the angle of Graph Theory, this paper provide... | \section{Introduction}
Primes are the collection that has been extensively studied in Number Theory. This paper is motivated from a result, by Greenfield and Greenfield in 1998 \cite{greenfield}, concerning prime numbers.
\begin{thm}\label{greenfield}\cite{greenfield}\\
The set of integers $\{1, 2, 3, \cdots, 2n\... | {
"timestamp": "2018-04-20T02:07:37",
"yymm": "1804",
"arxiv_id": "1804.07104",
"language": "en",
"url": "https://arxiv.org/abs/1804.07104",
"abstract": "A consequence of Bertrand's postulate, proved by L. Greenfield and S. Greenfield in 1998, assures that the set of integers $\\{1,2,\\cdots, 2n\\}$ can be ... |
https://arxiv.org/abs/1912.02717 | Investigating transversals as generating sets for groups | In [3] is was shown that for any group $G$ whose rank (i.e., minimal number of generators) is at most 3, and any finite index subgroup $H\leq G$ with index $[G:H]\geq rank(G)$, one can always find a left-right transversal of $H$ which generates $G$. In this paper we extend this result to groups of rank at most 4. We al... | \section{Introduction}
Given a group $G$ and a subgroup $H \leq G$ of finite index, one can consider sets $S \subset G$ which are a \emph{left} (resp. \textit{right}) \textit{ transversals} of $H$ in $G$. That is, a complete set of left (resp. right) coset representatives. It then follows that we can consider \em... | {
"timestamp": "2019-12-06T02:23:30",
"yymm": "1912",
"arxiv_id": "1912.02717",
"language": "en",
"url": "https://arxiv.org/abs/1912.02717",
"abstract": "In [3] is was shown that for any group $G$ whose rank (i.e., minimal number of generators) is at most 3, and any finite index subgroup $H\\leq G$ with ind... |
https://arxiv.org/abs/1306.0943 | Extremal Problems for Subset Divisors | Let $A$ be a set of $n$ positive integers. We say that a subset $B$ of $A$ is a divisor of $A$, if the sum of the elements in $B$ divides the sum of the elements in $A$. We are interested in the following extremal problem. For each $n$, what is the maximum number of divisors a set of $n$ positive integers can have? We ... | \section{Introduction}
Let $A$ be a finite set of positive integers and let $B$ be a subset of $A$. We say that $B$ is a \emph{divisor} of $A$, if the sum of the elements in $B$ divides the sum of the elements in $A$. We are interested in the number of divisors a set of positive integers can have. Toward that end, ... | {
"timestamp": "2014-09-22T02:12:59",
"yymm": "1306",
"arxiv_id": "1306.0943",
"language": "en",
"url": "https://arxiv.org/abs/1306.0943",
"abstract": "Let $A$ be a set of $n$ positive integers. We say that a subset $B$ of $A$ is a divisor of $A$, if the sum of the elements in $B$ divides the sum of the ele... |
https://arxiv.org/abs/1105.5178 | The peak sidelobe level of random binary sequences | Let $A_n=(a_0,a_1,\dots,a_{n-1})$ be drawn uniformly at random from $\{-1,+1\}^n$ and define \[ M(A_n)=\max_{0<u<n}\,\Bigg|\sum_{j=0}^{n-u-1}a_ja_{j+u}\Bigg|\quad\text{for $n>1$}. \] It is proved that $M(A_n)/\sqrt{n\log n}$ converges in probability to $\sqrt{2}$. This settles a problem first studied by Moon and Moser ... | \section{Introduction}
Consider a binary sequence $A=(a_0,a_1,\dots,a_{n-1})$ of length $n$, namely an element of $\{-1,+1\}^n$. Define the \emph{aperiodic autocorrelation} at shift $u$ of $A$ to be
\[
C_u(A):=\sum_{j=0}^{n-u-1}a_ja_{j+u}\quad \mbox{for $u\in\{0,1,\dots,n-1\}$}
\]
and define the \emph{peak sidelobe ... | {
"timestamp": "2011-05-27T02:00:51",
"yymm": "1105",
"arxiv_id": "1105.5178",
"language": "en",
"url": "https://arxiv.org/abs/1105.5178",
"abstract": "Let $A_n=(a_0,a_1,\\dots,a_{n-1})$ be drawn uniformly at random from $\\{-1,+1\\}^n$ and define \\[ M(A_n)=\\max_{0<u<n}\\,\\Bigg|\\sum_{j=0}^{n-u-1}a_ja_{j... |
https://arxiv.org/abs/0911.0563 | Judicious partitions of 3-uniform hypergraphs | The vertices of any graph with $m$ edges can be partitioned into two parts so that each part meets at least $\frac{2m}{3}$ edges. Bollobás and Thomason conjectured that the vertices of any $r$-uniform graph may be likewise partitioned into $r$ classes such that each part meets at least $cm$ edges, with $c=\frac{r}{2r-1... | \section{Introduction}
Given a graph $G$, it is easy to find a bipartition $V(G)=V_1\cup V_2$
such that at least half of the edges in $G$ join $V_1$ to $V_2$. It is
only slightly less trivial to find a bipartition $V_1\cup V_2$ such that
each of $V_1$ and $V_2$ meets at least $2/3$ of the edges;
equivalently, each cla... | {
"timestamp": "2009-11-03T13:07:39",
"yymm": "0911",
"arxiv_id": "0911.0563",
"language": "en",
"url": "https://arxiv.org/abs/0911.0563",
"abstract": "The vertices of any graph with $m$ edges can be partitioned into two parts so that each part meets at least $\\frac{2m}{3}$ edges. Bollobás and Thomason con... |
https://arxiv.org/abs/1804.09455 | Nonnegative Polynomials and Circuit Polynomials | The concept of sums of nonnegative circuit polynomials (SONC) was recently introduced as a new certificate of nonnegativity especially for sparse polynomials. In this paper, we explore the relationship between nonnegative polynomials and SONC polynomials. As a first result, we provide sufficient conditions for nonnegat... | \section{Introduction}
A real polynomial $f\in{\mathbb{R}}[{\mathbf{x}}]={\mathbb{R}}[x_1,\ldots,x_n]$ is called a {\em nonnegative polynomial} if its evaluation on every real point is nonnegative. All of nonnegative polynomials form a convex cone, denoted by PSD. Certifying nonnegativity of polynomials is a central pr... | {
"timestamp": "2020-01-08T02:16:44",
"yymm": "1804",
"arxiv_id": "1804.09455",
"language": "en",
"url": "https://arxiv.org/abs/1804.09455",
"abstract": "The concept of sums of nonnegative circuit polynomials (SONC) was recently introduced as a new certificate of nonnegativity especially for sparse polynomi... |
https://arxiv.org/abs/1409.7890 | Using Brouwer's fixed point theorem | Brouwer's fixed point theorem from 1911 is a basic result in topology - with a wealth of combinatorial and geometric consequences. In these lecture notes we present some of them, related to the game of HEX and to the piercing of multiple intervals. We also sketch stronger theorems, due to Oliver and others, and explain... | \section{The Game of HEX and the Brouwer Fixed Point Theorem}
Let's start with a game: ``HEX'' is a board game for two players,
invented by the ingenious Danish poet, designer and engineer Piet
Hein in 1942 \cite{Gardner}, and rediscovered in 1948 by the mathematician
John Nash \cite{Milnor-nash} who got a Nobel pri... | {
"timestamp": "2014-09-30T02:11:03",
"yymm": "1409",
"arxiv_id": "1409.7890",
"language": "en",
"url": "https://arxiv.org/abs/1409.7890",
"abstract": "Brouwer's fixed point theorem from 1911 is a basic result in topology - with a wealth of combinatorial and geometric consequences. In these lecture notes we... |
https://arxiv.org/abs/1603.00317 | Finite element approximation for the fractional eigenvalue problem | The purpose of this work is to study a finite element method for finding solutions to the eigenvalue problem for the fractional Laplacian. We prove that the discrete eigenvalue problem converges to the continuous one and we show the order of such convergence. Finally, we perform some numerical experiments and compare o... | \section{Introduction and Main Results}
Anomalous diffusion phenomena are ubiquitous in nature \cite{Klafter, MetzlerKlafter}, and
the study of nonlocal operators has been an active area of research in different branches of
mathematics. Such operators arise in applications as image processing~\cite{BuadesColl, Gat... | {
"timestamp": "2016-03-02T02:13:01",
"yymm": "1603",
"arxiv_id": "1603.00317",
"language": "en",
"url": "https://arxiv.org/abs/1603.00317",
"abstract": "The purpose of this work is to study a finite element method for finding solutions to the eigenvalue problem for the fractional Laplacian. We prove that t... |
https://arxiv.org/abs/1712.02499 | Connecting the q-Multiplicative Convolution and the Finite Difference Convolution | In a recent paper, Brändén, Krasikov, and Shapiro consider root location preservation properties of finite difference operators. To this end, the authors describe a natural polynomial convolution operator and conjecture that it preserves root mesh properties. We prove this conjecture using two methods. The first develo... | \section{Introduction}
Let $\mathbb{C}[x]$ denote the space of polynomials with complex coefficients, and let $\mathbb{C}_n[x]$ denote the subspace of polynomials in $\mathbb{C}[x]$ of degree at most $n$. (We make the analogous definitions for $\mathbb{R}[x]$ and $\mathbb{R}_n[x]$.) The Walsh additive (\cite{walsh}) ... | {
"timestamp": "2017-12-08T02:05:20",
"yymm": "1712",
"arxiv_id": "1712.02499",
"language": "en",
"url": "https://arxiv.org/abs/1712.02499",
"abstract": "In a recent paper, Brändén, Krasikov, and Shapiro consider root location preservation properties of finite difference operators. To this end, the authors ... |
https://arxiv.org/abs/2011.12218 | Tverberg's theorem, disks, and Hamiltonian cycles | For a finite set $S$ of points in the plane and a graph with vertices on $S$ consider the disks with diameters induced by the edges. We show that for any odd set $S$ there exists a Hamiltonian cycle for which these disks share a point, and for an even set $S$ there exists a Hamiltonian path with the same property. We d... | \section{Introduction}
In 1966, Helge Tverberg proved that \textit{for any set of $(r-1)(d+1)+1$ points in $\mathds{R}^d$ there exists a partition of them into $r$ parts whose convex hulls intersect} \cite{Tverberg:1966tb}. We call these partitions \emph{Tverberg partitions}. Among the variations and generalizations... | {
"timestamp": "2020-11-30T02:02:23",
"yymm": "2011",
"arxiv_id": "2011.12218",
"language": "en",
"url": "https://arxiv.org/abs/2011.12218",
"abstract": "For a finite set $S$ of points in the plane and a graph with vertices on $S$ consider the disks with diameters induced by the edges. We show that for any ... |
https://arxiv.org/abs/2009.05000 | Primes in short intervals: Heuristics and calculations | We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length $y$ around $x$, where $y\ll (\log x)^2$. In particular we conjecture that the maximum grows surprisingly slowly as $y$ ranges from $\log x$ to $(\log x)^2$. We will show that our conjectures are som... | \section{Introduction}
We are interested in estimating the maximum and minimum number of primes in a length $y$ sub-interval of $(x,2x]$, denoted by
\[
M(x,y):=\max_{X\in (x,2x]} \pi(X+y)-\pi(X) \text{ and } m(x,y):=\min_{X\in (x,2x]} \pi(X+y)-\pi(X) ,
\]
respectively, so that
\[
m(x,y)\leq \pi(X+y)-\pi(X) \leq M(x... | {
"timestamp": "2021-05-05T02:04:47",
"yymm": "2009",
"arxiv_id": "2009.05000",
"language": "en",
"url": "https://arxiv.org/abs/2009.05000",
"abstract": "We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length $y$ around $x$, where $y\\ll (\\... |
https://arxiv.org/abs/1902.00051 | Shape Analysis, Lebesgue Integration and Absolute Continuity Connections | As shape analysis of the form presented in Srivastava and Klassen's textbook 'Functional and Shape Data Analysis' is intricately related to Lebesgue integration and absolute continuity, it is advantageous to have a good grasp of the latter two notions. Accordingly, in these notes we review basic concepts and results ab... | \section{\large Introduction}
The concepts of Lebesgue integration and absolute continuity play a major
role in the theory of shape analysis or more generally in the theory of
functional data analysis of the form presented in~\cite{srivastava}. In
fact, well-known connections between Lebesgue integration and absolute
c... | {
"timestamp": "2019-07-01T02:03:42",
"yymm": "1902",
"arxiv_id": "1902.00051",
"language": "en",
"url": "https://arxiv.org/abs/1902.00051",
"abstract": "As shape analysis of the form presented in Srivastava and Klassen's textbook 'Functional and Shape Data Analysis' is intricately related to Lebesgue integ... |
https://arxiv.org/abs/math/0607686 | The Modulo 1 Central Limit Theorem and Benford's Law for Products | We derive a necessary and sufficient condition for the sum of M independent continuous random variables modulo 1 to converge to the uniform distribution in L^1([0,1]), and discuss generalizations to discrete random variables. A consequence is that if X_1, ..., X_M are independent continuous random variables with densit... | \section{Introduction}
We investigate necessary and sufficient conditions for the
distribution of a sum of random variables modulo $1$ to converge to
the uniform distribution. This topic has been fruitfully studied by
many previous researchers. Our purpose here is to provide an
elementary proof of prior results, and e... | {
"timestamp": "2007-11-20T15:22:31",
"yymm": "0607",
"arxiv_id": "math/0607686",
"language": "en",
"url": "https://arxiv.org/abs/math/0607686",
"abstract": "We derive a necessary and sufficient condition for the sum of M independent continuous random variables modulo 1 to converge to the uniform distributi... |
https://arxiv.org/abs/0802.3874 | On low rank perturbation of matrices | The article is devoted to different aspects of the question "What can be done with a matrix by low rank perturbation?" It is proved that one can change a geometrically simple spectrum drastically by a rank 1 permutation, but the situation is quite different if one restricts oneself to normal matrices. Also, the Jordan ... | \section{Introduction}
The article is devoted to different aspects of the question: "What
can be done with a complex-valued matrix by a low rank
perturbation?"\footnote{The authors would like to thank
V.S.Savchenko, J. Moro and F. Dopico
for useful comments and references.
The work was partially supported by CONACyT ... | {
"timestamp": "2008-09-02T02:36:20",
"yymm": "0802",
"arxiv_id": "0802.3874",
"language": "en",
"url": "https://arxiv.org/abs/0802.3874",
"abstract": "The article is devoted to different aspects of the question \"What can be done with a matrix by low rank perturbation?\" It is proved that one can change a ... |
https://arxiv.org/abs/1904.10507 | Fekete's lemma for componentwise subadditive functions of two or more real variables | We prove an analogue of Fekete's subadditivity lemma for functions of several real variables which are subadditive in each variable taken singularly. This extends both the classical case for subadditive functions of one real variable, and a result in a previous paper by the author. While doing so, we prove that the fun... | \section{Introduction}
A real-valued function $f$ defined on a semigroup $(S,\cdot)$ is
\emph{subadditive} if
\begin{equation}
\label{eq:subadd}
f(x \cdot y) \leq f(x) + f(y)
\end{equation}
for every $x,y\in{S}$. Examples of subadditive functions include the
absolute value of a complex number; the ceiling of a rea... | {
"timestamp": "2021-02-04T02:12:55",
"yymm": "1904",
"arxiv_id": "1904.10507",
"language": "en",
"url": "https://arxiv.org/abs/1904.10507",
"abstract": "We prove an analogue of Fekete's subadditivity lemma for functions of several real variables which are subadditive in each variable taken singularly. This... |
https://arxiv.org/abs/2103.08498 | On the nilradical of a Leibniz algebra | The purpose of this short note is to correct an error which appears in the literature concerning Leibniz algebras $L$: namely, that $N(L/I)=N(L)/I$ where $N(L)$ is the nilradical of $L$ and $I$ is the Leibniz kernel. | \section{Introduction}
\medskip
An algebra $L$ over a field $F$ is called a {\em Leibniz algebra} if, for every $x,y,z \in L$, we have
\[ [x,[y,z]]=[[x,y],z]-[[x,z],y]
\]
In other word,s the right multiplication operator $R_x : L \rightarrow L : y\mapsto [y,x]$ is a derivation of $L$. As a result, such algebras... | {
"timestamp": "2021-03-16T01:38:35",
"yymm": "2103",
"arxiv_id": "2103.08498",
"language": "en",
"url": "https://arxiv.org/abs/2103.08498",
"abstract": "The purpose of this short note is to correct an error which appears in the literature concerning Leibniz algebras $L$: namely, that $N(L/I)=N(L)/I$ where ... |
https://arxiv.org/abs/2006.14228 | Primitive point packing | A point in the $d$-dimensional integer lattice $\mathbb{Z}^d$ is primitive when its coordinates are relatively prime. Two primitive points are multiples of one another when they are opposite, and for this reason, we consider half of the primitive points within the lattice, the ones whose first non-zero coordinate is po... | \section{Introduction}\label{PPP.sec.0}
Lattice polytopes appear in many branches of mathematics, as for instance in algebraic geometry where they are associated with certain toric varieties. It is noteworthy that, in relation with the study of these and similar objects, methods from combinatorics and algebraic geomet... | {
"timestamp": "2020-06-26T02:09:36",
"yymm": "2006",
"arxiv_id": "2006.14228",
"language": "en",
"url": "https://arxiv.org/abs/2006.14228",
"abstract": "A point in the $d$-dimensional integer lattice $\\mathbb{Z}^d$ is primitive when its coordinates are relatively prime. Two primitive points are multiples ... |
https://arxiv.org/abs/2206.06981 | Generalized graph splines and the Universal Difference Property | We study the generalized graph splines introduced by Gilbert, Tymoczko, and Viel and focus on an attribute known as the Universal Difference Property (UDP). We prove that paths, trees, and cycles satisfy UDP. We explore UDP on graphs pasted at a single vertex and use Prüfer domains to illustrate that not every edge lab... | \section{Introduction}\label{intro}
Splines are perhaps best known for their usage in analysis and for their applications in finding approximate solutions to differential equations, but splines also appear in a variety of other contexts including geometry and topology. To unify these various notions of splines, Gilbe... | {
"timestamp": "2022-06-15T02:22:52",
"yymm": "2206",
"arxiv_id": "2206.06981",
"language": "en",
"url": "https://arxiv.org/abs/2206.06981",
"abstract": "We study the generalized graph splines introduced by Gilbert, Tymoczko, and Viel and focus on an attribute known as the Universal Difference Property (UDP... |
https://arxiv.org/abs/1311.3874 | An Algorithm to Solve the Equal-Sum-Product Problem | A recursive algorithm is constructed which finds all solutions to a class of Diophantine equations connected to the problem of determining ordered n-tuples of positive integers satisfying the property that their sum is equal to their product. An examination of the use of Binary Search Trees in implementing the algorith... | \section{Introduction}
Suppose we are asked to consider the following three arithmetic identities
\[
2+2=4\mbox{ ,}\hspace{1.7cm}1+2+3=6\mbox{ ,}\hspace{1.7cm}1+1+2+2+2=8\mbox{ .}
\]
What can we say is a feature common to each of the three identities? Looking at the second equality we might first think that we are deal... | {
"timestamp": "2013-11-18T02:09:43",
"yymm": "1311",
"arxiv_id": "1311.3874",
"language": "en",
"url": "https://arxiv.org/abs/1311.3874",
"abstract": "A recursive algorithm is constructed which finds all solutions to a class of Diophantine equations connected to the problem of determining ordered n-tuples ... |
https://arxiv.org/abs/2111.02635 | The 3x+1 Problem: An Overview | This paper is an overview and survey of work on the 3x+1 problem, also called the Collatz problem, and generalizations of it. It gives a history of the problem. It addresses two questions: (1) What can mathematics currently say about this problem? (as of 2010). (2) How can this problem be hard, when it is so easy to st... | \section{Introduction}
The $3x+1$ problem concerns
the following innocent seeming arithmetic procedure applied to integers: If an integer $x$ is odd
then ``multiply by three and add one", while if it is even then ``divide by two".
This operation is described by the {\em Collatz function}
$$
C(x) =
\left\{
\beg... | {
"timestamp": "2021-11-05T01:09:01",
"yymm": "2111",
"arxiv_id": "2111.02635",
"language": "en",
"url": "https://arxiv.org/abs/2111.02635",
"abstract": "This paper is an overview and survey of work on the 3x+1 problem, also called the Collatz problem, and generalizations of it. It gives a history of the pr... |
https://arxiv.org/abs/1810.02532 | Sharp error bounds for Ritz vectors and approximate singular vectors | We derive sharp bounds for the accuracy of approximate eigenvectors (Ritz vectors) obtained by the Rayleigh-Ritz process for symmetric eigenvalue problems. Using information that is available or easy to estimate, our bounds improve the classical Davis-Kahan $\sin\theta$ theorem by a factor that can be arbitrarily large... | \section{Introduction}
It is well known that the eigenvector
corresponding to
a near-multiple eigenvalue is ill-conditioned.
Specifically,
the classical Davis-Kahan theory~\cite{daviskahan} implies that the condition number of eigenvectors of symmetric or Hermitian matrices is $1/{\rm gap}$, where ${\rm gap}$ is the ... | {
"timestamp": "2020-01-01T02:11:31",
"yymm": "1810",
"arxiv_id": "1810.02532",
"language": "en",
"url": "https://arxiv.org/abs/1810.02532",
"abstract": "We derive sharp bounds for the accuracy of approximate eigenvectors (Ritz vectors) obtained by the Rayleigh-Ritz process for symmetric eigenvalue problems... |
https://arxiv.org/abs/1704.07022 | Note on the union-closed sets conjecture | The union-closed sets conjecture states that if a family of sets $\mathcal{A} \neq \{\emptyset\}$ is union-closed, then there is an element which belongs to at least half the sets in $\mathcal{A}$. In 2001, D. Reimer showed that the average set size of a union-closed family, $\mathcal{A}$, is at least $\frac{1}{2} \log... | \section{Introduction}
Given the set $[n]= \{1, \ldots, n\}$ and a family ${\mathcal A} \subseteq 2^{[n]}$ we say ${\mathcal A}$ is union-closed if for $A, B \in {\mathcal A}$ we have $A \cup B \in {\mathcal A}$. The Union-Closed Sets Conjecture, due to P. Frankl \cite{Rival}, states that if ${\mathcal A} \subseteq 2^... | {
"timestamp": "2017-04-25T02:09:08",
"yymm": "1704",
"arxiv_id": "1704.07022",
"language": "en",
"url": "https://arxiv.org/abs/1704.07022",
"abstract": "The union-closed sets conjecture states that if a family of sets $\\mathcal{A} \\neq \\{\\emptyset\\}$ is union-closed, then there is an element which bel... |
https://arxiv.org/abs/1203.2295 | Techniques for Solving Sudoku Puzzles | Solving Sudoku puzzles is one of the most popular pastimes in the world. Puzzles range in difficulty from easy to very challenging; the hardest puzzles tend to have the most empty cells. The current paper explains and compares three algorithms for solving Sudoku puzzles. Backtracking, simulated annealing, and alternati... | \section{Introduction}
As all good mathematical scientists know, a broad community has contributed to the invention of modern algorithms. Computer scientists, applied mathematicians, statisticians, economists, and physicists, to name just a few, have made lasting contributions. Exposing students to a variety of perspe... | {
"timestamp": "2013-05-17T02:02:22",
"yymm": "1203",
"arxiv_id": "1203.2295",
"language": "en",
"url": "https://arxiv.org/abs/1203.2295",
"abstract": "Solving Sudoku puzzles is one of the most popular pastimes in the world. Puzzles range in difficulty from easy to very challenging; the hardest puzzles tend... |
https://arxiv.org/abs/1901.06096 | Repeated minimizers of $p$-frame energies | For a collection of $N$ unit vectors $\mathbf{X}=\{x_i\}_{i=1}^N$, define the $p$-frame energy of $\mathbf{X}$ as the quantity $\sum_{i\neq j} |\langle x_i,x_j \rangle|^p$. In this paper, we connect the problem of minimizing this value to another optimization problem, so giving new lower bounds for such energies. In pa... | \section{Introduction}\label{sec:intro}
Let $\mathbf{A}=A_{i,j}$ be an $N\times N$ real matrix of rank less than or equal to $d$, and with ones along the diagonal. The $p$-frame energy of matrix $\mathbf{A}$ is denoted
$$E_p(\mathbf{A})=\sum\limits_{i\neq j} |A_{ij}|^p.$$
An interesting question is what the optimiz... | {
"timestamp": "2019-08-13T02:17:10",
"yymm": "1901",
"arxiv_id": "1901.06096",
"language": "en",
"url": "https://arxiv.org/abs/1901.06096",
"abstract": "For a collection of $N$ unit vectors $\\mathbf{X}=\\{x_i\\}_{i=1}^N$, define the $p$-frame energy of $\\mathbf{X}$ as the quantity $\\sum_{i\\neq j} |\\la... |
https://arxiv.org/abs/1902.09450 | Asymptotic complements in the integers | Let $W\subseteq \mathbb{Z}$ be a non-empty subset of the integers. A nonempty set $C\subseteq \mathbb{Z}$ is said to be an asymptotic complement to $W$ if $W+C$ contains almost all the integers except a set of finite size. $C$ is said to be a minimal asymptotic complement if $C$ is an asymptotic complement, but $C\setm... | \section{Introduction}
\subsection{Background and Motivation}
Let $(G,\cdot)$ be a group where $\cdot$ denotes its group operation. If $A, B$ are two nonempty subsets of $G$, then we define the product set $A\cdot B$ as
$$A\cdot B := \{a\cdot b \,|\, a\in A, b\in B\}.$$
If $A$ contains only one element $a$, then $A\c... | {
"timestamp": "2019-12-11T02:19:55",
"yymm": "1902",
"arxiv_id": "1902.09450",
"language": "en",
"url": "https://arxiv.org/abs/1902.09450",
"abstract": "Let $W\\subseteq \\mathbb{Z}$ be a non-empty subset of the integers. A nonempty set $C\\subseteq \\mathbb{Z}$ is said to be an asymptotic complement to $W... |
https://arxiv.org/abs/1302.6597 | Independence of l-adic representations of geometric Galois groups | Let k be an algebraically closed field of arbitrary characteristic,let K/k be a finitely generated field extension and let X be a separated scheme of finite type over K. For each prime ell, the absolute Galois group of K acts on the ell-adic etale cohomology modules of X. We prove that this family of representations va... | \section{Introduction}
Let $G$ be a profinite group and $L$ a set of prime numbers. For every $\ell\in L$ let $G_\ell$ be a profinite
group and $\rho_\ell: G\to G_\ell$ a homomorphism. Denote by
$$\rho: G\to \prod\limits_{\ell\in L} G_\ell$$
the homomorphism induced by the $\rho_\ell$. Following the notation in \cite{b... | {
"timestamp": "2013-02-28T02:00:13",
"yymm": "1302",
"arxiv_id": "1302.6597",
"language": "en",
"url": "https://arxiv.org/abs/1302.6597",
"abstract": "Let k be an algebraically closed field of arbitrary characteristic,let K/k be a finitely generated field extension and let X be a separated scheme of finite... |
https://arxiv.org/abs/1412.0380 | Fixed points for weak contractions in metric type spaces | In this article, we prove some fixed point theorems in metric type spaces. This article is just a generalization some results previously proved in \cite{niyi-gaba}. In particular, we give some coupled common fixed points theorems under weak contractions. These results extend well known similar results existing in the l... | \section*{\small Keyword: Metric Type Spaces; Fixed Point.}
\end{abstract}
\rule{5.75in}{.01in}
\section{Introduction}
The Banach contraction principle is a fundamental result in fixed point theory. Due to its importance, several authors have obtained many interesting extensions and generalizations, from sin... | {
"timestamp": "2015-05-12T02:08:58",
"yymm": "1412",
"arxiv_id": "1412.0380",
"language": "en",
"url": "https://arxiv.org/abs/1412.0380",
"abstract": "In this article, we prove some fixed point theorems in metric type spaces. This article is just a generalization some results previously proved in \\cite{ni... |
https://arxiv.org/abs/1603.03526 | Cycles in graphs of fixed girth with large size | Consider a family of graphs having a fixed girth and a large size. We give an optimal lower asymptotic bound on the number of even cycles of any constant length, as the order of the graphs tends to infinity. | \section{Introduction}
All graphs we consider in this article are simple graphs. We denote the size of $G$ by $e(G)$ and the order of $G$ by $v(G)$. A $j$-path in $G$ is a path of length $j$ in $G$. A $j$-cycle in $G$ is a cycle of length $j$ in $G$, and it is called an even cycle if $j$ is even. The {\em girth} of a ... | {
"timestamp": "2016-03-14T01:04:57",
"yymm": "1603",
"arxiv_id": "1603.03526",
"language": "en",
"url": "https://arxiv.org/abs/1603.03526",
"abstract": "Consider a family of graphs having a fixed girth and a large size. We give an optimal lower asymptotic bound on the number of even cycles of any constant ... |
https://arxiv.org/abs/2202.04722 | On the stability of unevenly spaced samples for interpolation and quadrature | Unevenly spaced samples from a periodic function are common in signal processing and can often be viewed as a perturbed equally spaced grid. In this paper, we analyze how the uneven distribution of the samples impacts the quality of interpolation and quadrature. Starting with equally spaced nodes on $[-\pi,\pi)$ with g... | \section{Introduction}\label{sec:introduction}
In signal processing, function approximation, and econometrics, unevenly spaced time series data naturally occur. For example, natural disasters occur at irregular time intervals~\cite{quan}, observational astronomy takes measurements of celestial bodies at times determine... | {
"timestamp": "2022-02-11T02:03:16",
"yymm": "2202",
"arxiv_id": "2202.04722",
"language": "en",
"url": "https://arxiv.org/abs/2202.04722",
"abstract": "Unevenly spaced samples from a periodic function are common in signal processing and can often be viewed as a perturbed equally spaced grid. In this paper... |
https://arxiv.org/abs/1507.00827 | Estimating the number of communities in networks by spectral methods | Community detection is a fundamental problem in network analysis with many methods available to estimate communities. Most of these methods assume that the number of communities is known, which is often not the case in practice. We study a simple and very fast method for estimating the number of communities based on th... | \section{Introduction}
\input{introduction}
\section{Preliminaries}
\input{preliminaries}
\section{Spectral estimates of the number of communities}
\input{methods}
\section{Consistency}
\input{consistency}
\section{Numerical results}\label{sec: simulation}
\input{simulation}
\section{Discussion}\lab... | {
"timestamp": "2015-07-06T02:04:54",
"yymm": "1507",
"arxiv_id": "1507.00827",
"language": "en",
"url": "https://arxiv.org/abs/1507.00827",
"abstract": "Community detection is a fundamental problem in network analysis with many methods available to estimate communities. Most of these methods assume that th... |
https://arxiv.org/abs/1001.1406 | Some experiments with integral Apollonian circle packings | Bounded Apollonian circle packings (ACP's) are constructed by repeatedly inscribing circles into the triangular interstices of a configuration of four mutually tangent circles, one of which is internally tangent to the other three. If the original four circles have integer curvature, all of the circles in the packing w... | \section{Introduction}\label{intro}
Start with four mutually tangent circles, one of them internally tangent to the other three as in Fig.~\ref{circles}. One can inscribe into each of the curvilinear triangles in this picture a unique circle (the uniqueness follows from an old theorem of Apollonius of Perga circa 200... | {
"timestamp": "2010-01-12T01:44:00",
"yymm": "1001",
"arxiv_id": "1001.1406",
"language": "en",
"url": "https://arxiv.org/abs/1001.1406",
"abstract": "Bounded Apollonian circle packings (ACP's) are constructed by repeatedly inscribing circles into the triangular interstices of a configuration of four mutua... |
https://arxiv.org/abs/1203.4066 | On the last digit and the last non-zero digit of $n^n$ in base $b$ | In this paper we study the sequences defined by the last and the last non-zero digits of $n^n$ in base $b$. For the sequence given by the last digits of $n^n$ in base $b$, we prove its periodicity using different techniques than those used by W. Sierpinski and R. Hampel. In the case of the sequence given by the last no... | \section{Introduction}
The study of the last digit of the elements in a sequence is a recurrent topic in Number Theory. In this sense, one of the most studied sequences is, of course, the Fibonacci sequence which was already studied by Lagrange
observing that the last digit of the Fibonacci sequence repeats with peri... | {
"timestamp": "2012-03-20T01:03:46",
"yymm": "1203",
"arxiv_id": "1203.4066",
"language": "en",
"url": "https://arxiv.org/abs/1203.4066",
"abstract": "In this paper we study the sequences defined by the last and the last non-zero digits of $n^n$ in base $b$. For the sequence given by the last digits of $n^... |
https://arxiv.org/abs/1310.1181 | On the expectation of normalized Brownian functionals up to first hitting times | Let B be a Brownian motion and T its first hitting time of the level 1. For U a uniform random variable independent of B, we study in depth the distribution of T^{-1/2}B_{UT}, that is the rescaled Brownian motion sampled at uniform time. In particular, we show that this variable is centered. | \section{Introduction}\label{intro}
In this paper, we study the expectations of the random variables $A_a^{(m)}$ and $\tilde A_a^{(m)}$ defined for $a>0$ and $m\geq 0$ by
$$A_a^{(m)}=\frac{1}{T_a^{1+m/2}}\int_0^{T_a}|B_s|^m\text{sgn}(B_s) ds,~~\tilde A_a^{(m)}=\frac{1}{T_a^{1+m/2}}\int_0^{T_a}|B_s|^mds,$$
where $B$ is... | {
"timestamp": "2013-10-07T02:04:51",
"yymm": "1310",
"arxiv_id": "1310.1181",
"language": "en",
"url": "https://arxiv.org/abs/1310.1181",
"abstract": "Let B be a Brownian motion and T its first hitting time of the level 1. For U a uniform random variable independent of B, we study in depth the distribution... |
https://arxiv.org/abs/2208.14334 | Doubly Sequenceable Groups | Given a sequence ${\bf g}: g_0,\ldots, g_{m}$, in a finite group $G$ with $g_0=1_G$, let ${\bf \bar g}: \bar g_0,\ldots, \bar g_{m}$, be the sequence defined by $\bar g_0=1_G$ and $\bar g_i=g_{i-1}^{-1}g_i$ for $1\leq i \leq m$. We say that $G$ is doubly sequenceable if there exists a sequence ${\bf g}$ in $G$ such tha... | \section{Introduction}
Let $G$ be a finite group with $n$ elements and ${\bf g}: g_0,\ldots, g_{kn-1},$ be a sequence of elements of $G$ such that $g_0=1_G$, where $1_G$ is the identity element of $G$. The sequence of consecutive quotients ${\bf \bar g}$ is defined by ${\bar g}_0=1_G$ and ${\bar g}_i=g_{i-1}^{-1}g_{i}... | {
"timestamp": "2022-08-31T02:20:53",
"yymm": "2208",
"arxiv_id": "2208.14334",
"language": "en",
"url": "https://arxiv.org/abs/2208.14334",
"abstract": "Given a sequence ${\\bf g}: g_0,\\ldots, g_{m}$, in a finite group $G$ with $g_0=1_G$, let ${\\bf \\bar g}: \\bar g_0,\\ldots, \\bar g_{m}$, be the sequen... |
https://arxiv.org/abs/2103.07644 | A Jordan Curve Theorem for 2-dimensional Tilings | The classical Jordan curve theorem for digital curves asserts that the Jordan curve theorem remains valid in the Khalimsky plane. Since the Khalimsky plane is a quotient space of $\mathbb R^2$ induced by a tiling of squares, it is natural to ask for which other tilings of the plane it is possible to obtain a similar re... | \section{Introduction}
The Jordan curve theorem asserts that a simple closed curve divides the plane into two connected components, one of these components is bounded whereas the other one is not. This theorem was proved by Camille Jordan in 1887 in his book \textit{Cours d'analyse} \cite{Jordan}.
During the decade ... | {
"timestamp": "2021-03-16T01:07:33",
"yymm": "2103",
"arxiv_id": "2103.07644",
"language": "en",
"url": "https://arxiv.org/abs/2103.07644",
"abstract": "The classical Jordan curve theorem for digital curves asserts that the Jordan curve theorem remains valid in the Khalimsky plane. Since the Khalimsky plan... |
https://arxiv.org/abs/1704.05494 | The pinnacle set of a permutation | The peak set of a permutation records the indices of its peaks. These sets have been studied in a variety of contexts, including recent work by Billey, Burdzy, and Sagan, which enumerated permutations with prescribed peak sets. In this article, we look at a natural analogue of the peak set of a permutation, instead rec... | \section{Introduction}\label{sec:intro}
Let $S_n$ denote the set of permutations of $[n] = \{1,2,\ldots, n\}$, which we will always write as words, $w = w(1)w(2)\cdots w(n)$. An \emph{ascent} of a permutation $w$ is an index $i$ such that $w(i)< w(i+1)$, while a \emph{descent} is an index $i$ such that $w(i) > w(i+1)$... | {
"timestamp": "2017-04-20T02:00:48",
"yymm": "1704",
"arxiv_id": "1704.05494",
"language": "en",
"url": "https://arxiv.org/abs/1704.05494",
"abstract": "The peak set of a permutation records the indices of its peaks. These sets have been studied in a variety of contexts, including recent work by Billey, Bu... |
https://arxiv.org/abs/2111.10624 | Spectral perturbation by rank one matrices | Let $A$ be a matrix of size $n \times n$ over an algebraically closed field $F$ and $q(t)$ a monic polynomial of degree $n$. In this article, we describe the necessary and sufficient conditions of $q(t)$ so that there exists a rank one matrix $B$ such that the characteristic polynomial of $A+B$ is $q(t)$. | \section{Introduction and main results}
Let $A$ be an $n \times n$ matrix over an algebraically closed field $F$. The eigenspectrum of matrices of the form $A+B$ where $B$ is a low rank matrix has been studied extensively in the literature (for example, see \cite{[Bau]}, \cite{[Kato]}, \cite{[Lidskii]}, \cite{[MMRR]}... | {
"timestamp": "2021-11-23T02:12:11",
"yymm": "2111",
"arxiv_id": "2111.10624",
"language": "en",
"url": "https://arxiv.org/abs/2111.10624",
"abstract": "Let $A$ be a matrix of size $n \\times n$ over an algebraically closed field $F$ and $q(t)$ a monic polynomial of degree $n$. In this article, we describe... |
https://arxiv.org/abs/2111.10331 | Maximum arrangements of nonattacking kings on the $2n\times 2n$ chessboard | To count the number of maximum independent arrangements of $n^2$ kings on a $2n\times 2n$ chessboard, we build a $2^n \times (n+1)$ matrix whose entries are independent arrangements of $n$ kings on $2\times 2n$ rectangles. Utilizing upper and lower bound functions dependent of the entries of the matrix, we recursively ... | \section{Introduction}
The problem of finding and counting the number of independent, also called nonattacking, arrangements of pieces on a chessboard is a long-established problem in mathematics and computer science. Many variations have been studied including modifications of traditional pieces and of board size an... | {
"timestamp": "2021-11-22T02:20:09",
"yymm": "2111",
"arxiv_id": "2111.10331",
"language": "en",
"url": "https://arxiv.org/abs/2111.10331",
"abstract": "To count the number of maximum independent arrangements of $n^2$ kings on a $2n\\times 2n$ chessboard, we build a $2^n \\times (n+1)$ matrix whose entries... |
https://arxiv.org/abs/2103.04483 | An Amazing Prime Heuristic | Dickson conjectured that a set of polynomials will take on infinitely many simultaneous prime values. Later others, such as Hardy and Littlewood, gave estimates for the number of these primes. In this article we look at this conjecture, develop a simple heuristic and rederive these classic estimates. We then apply them... | \section{Introduction}
\addtocontents{toc}{\vspace{3pt}}
The record for the largest known twin prime is constantly
changing. For example, in October of 2000, David Underbakke
found the record primes:
$$83475759\cdot 2^{64955}\pm 1.$$
The very next day Giovanni La Barbera found the new record primes:
$$1693965\cdot... | {
"timestamp": "2021-03-09T02:29:03",
"yymm": "2103",
"arxiv_id": "2103.04483",
"language": "en",
"url": "https://arxiv.org/abs/2103.04483",
"abstract": "Dickson conjectured that a set of polynomials will take on infinitely many simultaneous prime values. Later others, such as Hardy and Littlewood, gave est... |
https://arxiv.org/abs/1712.02315 | Pair Correlations in Uniform Countable Sets | We determine the pair correlations of countable sets $T \subset \mathbb{R}^n$ satisfying natural equidistribution conditions. The pair correlations are computed as the volume of a certain region in $\mathbb{R}^{2n}$, which can be expressed in terms of the incomplete Beta function. For $n=2$ and $n=3$ we give closed for... | \section{Introduction}
In many areas of geometry it is important to determine the spatial statistics of various discrete sets of points in $n$-dimensions. We study an interesting property of the statistics of integer lattice points $\mathbb{Z}^{n}$ (and subsets of the lattice points), demonstrating an interesting rela... | {
"timestamp": "2017-12-07T02:09:31",
"yymm": "1712",
"arxiv_id": "1712.02315",
"language": "en",
"url": "https://arxiv.org/abs/1712.02315",
"abstract": "We determine the pair correlations of countable sets $T \\subset \\mathbb{R}^n$ satisfying natural equidistribution conditions. The pair correlations are ... |
https://arxiv.org/abs/1803.07744 | Passivity and Evolutionary Game Dynamics | This paper investigates an energy conservation and dissipation -- passivity -- aspect of dynamic models in evolutionary game theory. We define a notion of passivity using the state-space representation of the models, and we devise systematic methods to examine passivity and to identify properties of passive dynamic mod... | \section{Introduction} \label{sec_introduction}
Of central interest in evolutionary game theory \cite{weibull1995_mit, hofbauer2003_ams} is the study of strategic interactions among players in large populations. Each player engaged in a game chooses a strategy among a finite set of options and repeatedly revises its st... | {
"timestamp": "2018-03-22T01:05:33",
"yymm": "1803",
"arxiv_id": "1803.07744",
"language": "en",
"url": "https://arxiv.org/abs/1803.07744",
"abstract": "This paper investigates an energy conservation and dissipation -- passivity -- aspect of dynamic models in evolutionary game theory. We define a notion of... |
https://arxiv.org/abs/1807.01450 | Ramsey theory for hypergroups | In this paper, Ramsey theory for discrete hypergroups is introduced with emphasis on polynomial hypergroups, discrete orbit hypergroups and hypergroup deformations of semigroups. In this context, new notions of Ramsey principle for hypergroups and $\alpha$-Ramsey hypergroup, $0 \leq \alpha<1,$ are defined and studied. | \section{Introduction} Ramsey theory \cite{Ramsey1}, now a well-developed branch of combinatorics, has a long history dating back to 1892 starting with David Hilbert \cite{Hilbert}. For a discrete semigroup $(S, \cdot),$ the algebra structure of Stone-$\check{\mbox{C}}$ech compactification $\beta S$ of $S$ has been uti... | {
"timestamp": "2018-10-11T02:01:14",
"yymm": "1807",
"arxiv_id": "1807.01450",
"language": "en",
"url": "https://arxiv.org/abs/1807.01450",
"abstract": "In this paper, Ramsey theory for discrete hypergroups is introduced with emphasis on polynomial hypergroups, discrete orbit hypergroups and hypergroup def... |
https://arxiv.org/abs/1512.02086 | Hypercube Unfoldings that Tile R^3 and R^2 | We show that the hypercube has a face-unfolding that tiles space, and that unfolding has an edge-unfolding that tiles the plane. So the hypercube is a "dimension-descending tiler." We also show that the hypercube cross unfolding made famous by Dali tiles space, but we leave open the question of whether or not it has an... | \section{Introduction}
\seclab{Introduction}
The cube in ${\mathbb{R}}^3$ has $11$ distinct (incongruent)
edge-unfoldings\footnote{
An \emph{edge-unfolding} cuts along edges.
}
to $6$-square
planar polyominoes, each of which tiles the plane~\cite{Etudes}.
A single tile (a \emph{prototile}) that tiles the plane with c... | {
"timestamp": "2015-12-09T02:09:48",
"yymm": "1512",
"arxiv_id": "1512.02086",
"language": "en",
"url": "https://arxiv.org/abs/1512.02086",
"abstract": "We show that the hypercube has a face-unfolding that tiles space, and that unfolding has an edge-unfolding that tiles the plane. So the hypercube is a \"d... |
https://arxiv.org/abs/1810.05317 | Independence Equivalence Classes of Paths and Cycles | The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size. Two graphs are said to be \textit{independence equivalent} if they have equivalent independence polynomials. We extend previous work by showing that independence equivalence class of every odd path has s... | \section*{References}}
\begin{document}
\tikzset{bignode/.style={minimum size=3em,}}
\maketitle
\begin{abstract}
The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size. Two graphs are said to be \textit{independence equivalent} if they have equivalent indep... | {
"timestamp": "2018-10-15T02:05:15",
"yymm": "1810",
"arxiv_id": "1810.05317",
"language": "en",
"url": "https://arxiv.org/abs/1810.05317",
"abstract": "The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size. Two graphs are said to be \\textit{in... |
https://arxiv.org/abs/1507.01613 | A note on the clique number of complete $k$-partite graphs | In this note, we show that a complete $k$-partite graph is the only graph with clique number $k$ among all degree-equivalent simple graphs. This result gives a lower bound on the clique number, which is sharper than existing bounds on a large family of graphs. | \section{Preliminaries}
We first recall select graph theoretic notions used in the sequel; see \cite{bondy} for further details. All graphs considered in this note are simple graphs.
Let $G=(V,E)$ be a graph. The number of vertices and edges in $G$ are denoted by $n$ and $m$, respectively. The \emph{neighborhood} in... | {
"timestamp": "2015-07-08T02:00:45",
"yymm": "1507",
"arxiv_id": "1507.01613",
"language": "en",
"url": "https://arxiv.org/abs/1507.01613",
"abstract": "In this note, we show that a complete $k$-partite graph is the only graph with clique number $k$ among all degree-equivalent simple graphs. This result gi... |
https://arxiv.org/abs/1106.4753 | New approaches to plactic monoid via Gröbner-Shirshov bases | We present the plactic algebra on an arbitrary alphabet set $A$ by row generators and column generators respectively. We give Gröbner-Shirshov bases for such presentations. In the case of column generators, a finite Gröbner-Shirshov basis is given if $A$ is finite. From the Composition-Diamond lemma for associative alg... | \section{Introduction}\label{Intro}
Let $A=\{1,2,\dots,n\}$ with $1<2<\dots<n$. Then we call
$$
Pl(A):=sgp\langle A|\Omega\rangle=A^*/\equiv
$$
a plactic monoid on the alphabet set $A$, see \cite{M.L}, where
$A^*$ is the free monoid generated by $A$, $\equiv$ is the
congruence of $A^*$ generated by the Knuth relations... | {
"timestamp": "2011-06-24T02:03:37",
"yymm": "1106",
"arxiv_id": "1106.4753",
"language": "en",
"url": "https://arxiv.org/abs/1106.4753",
"abstract": "We present the plactic algebra on an arbitrary alphabet set $A$ by row generators and column generators respectively. We give Gröbner-Shirshov bases for suc... |
https://arxiv.org/abs/2202.00762 | Extending FEniCS to Work in Higher Dimensions Using Tensor Product Finite Elements | We present a method to extend the finite element library FEniCS to solve problems with domains in dimensions above three by constructing tensor product finite elements. This methodology only requires that the high dimensional domain is structured as a Cartesian product of two lower dimensional subdomains. In this study... | \section{Introduction}
In the last decade, open source libraries with high level APIs that automate the process of solving partial differential equations (PDEs) using the finite element method (FEM) have become valuable tools in computational research. Some prominent libraries of this kind include Firedrake~\cite{rath... | {
"timestamp": "2022-02-03T02:04:18",
"yymm": "2202",
"arxiv_id": "2202.00762",
"language": "en",
"url": "https://arxiv.org/abs/2202.00762",
"abstract": "We present a method to extend the finite element library FEniCS to solve problems with domains in dimensions above three by constructing tensor product fi... |
https://arxiv.org/abs/2211.09989 | A simple construction of infinite finitely generated torsion groups | The goal of this note is to provide yet another proof of the following theorem of Golod: there exists an infinite finitely generated group $G$ such that every element of $G$ has finite order. Our proof is based on the Nielsen-Schreier index formula and is simple enough to be included in a standard group theory course. | \section{Introduction}
Recall that a group $G$ is said to be \emph{torsion} (or \emph{periodic}) if every element of $G$ has finite order. Obviously, every finite group is torsion. Infinite torsion groups can be constructed as direct products of finite groups; note, however, that these groups are not finitely generated... | {
"timestamp": "2022-12-05T02:02:02",
"yymm": "2211",
"arxiv_id": "2211.09989",
"language": "en",
"url": "https://arxiv.org/abs/2211.09989",
"abstract": "The goal of this note is to provide yet another proof of the following theorem of Golod: there exists an infinite finitely generated group $G$ such that e... |
https://arxiv.org/abs/2005.03250 | Cohomological dimension of ideals defining Veronese subrings | Given a standard graded polynomial ring over a commutative Noetherian ring $A$, we prove that the cohomological dimension and the height of the ideals defining any of its Veronese subrings are equal. This result is due to Ogus when $A$ is a field of characteristic zero, and follows from a result of Peskine and Szpiro w... | \section{Introduction}
Throughout this paper, all rings are assumed to be commutative, Noetherian, and with an identity element.
Let $T = \mathbb{Z} [x_1,x_2,\ldots,x_k]$ be the standard graded polynomial ring in $n$ indeterminants over the integers. Consider a minimal minimal presentation of its $n$-th Veronese subr... | {
"timestamp": "2020-05-08T02:06:00",
"yymm": "2005",
"arxiv_id": "2005.03250",
"language": "en",
"url": "https://arxiv.org/abs/2005.03250",
"abstract": "Given a standard graded polynomial ring over a commutative Noetherian ring $A$, we prove that the cohomological dimension and the height of the ideals def... |
https://arxiv.org/abs/2001.05767 | Universal arrays | A word on $q$ symbols is a sequence of letters from a fixed alphabet of size $q$. For an integer $k\ge 1$, we say that a word $w$ is $k$-universal if, given an arbitrary word of length $k$, one can obtain it by removing entries from $w$. It is easily seen that the minimum length of a $k$-universal word on $q$ symbols i... | \section{Introduction}
A \emph{universal} mathematical structure is one which contains all possible substructures of a particular form. Famous examples of universal structures are De Bruijn sequences~\cite{DeBruijn}, which are periodic words that contain, exactly once, every possible word of a fixed size as a sub... | {
"timestamp": "2021-08-24T02:05:05",
"yymm": "2001",
"arxiv_id": "2001.05767",
"language": "en",
"url": "https://arxiv.org/abs/2001.05767",
"abstract": "A word on $q$ symbols is a sequence of letters from a fixed alphabet of size $q$. For an integer $k\\ge 1$, we say that a word $w$ is $k$-universal if, gi... |
https://arxiv.org/abs/2205.02798 | Effective poset inequalities | We explore inequalities on linear extensions of posets and make them effective in different ways. First, we study the Björner--Wachs inequality and generalize it to inequalities on order polynomials and their $q$-analogues via direct injections and FKG inequalities. Second, we give an injective proof of the Sidorenko i... | \section{Introduction}
\medskip
\subsection{Foreword}\label{ss:intro-foreword}
There are two schools of thought on what to do when an \defng{interesting
combinatorial inequality} is established. The first approach would
be to treat it as a tool to prove a desired result. The inequality
can still be sharpened or gen... | {
"timestamp": "2022-05-25T02:07:33",
"yymm": "2205",
"arxiv_id": "2205.02798",
"language": "en",
"url": "https://arxiv.org/abs/2205.02798",
"abstract": "We explore inequalities on linear extensions of posets and make them effective in different ways. First, we study the Björner--Wachs inequality and genera... |
https://arxiv.org/abs/2207.02189 | Accelerating Hamiltonian Monte Carlo via Chebyshev Integration Time | Hamiltonian Monte Carlo (HMC) is a popular method in sampling. While there are quite a few works of studying this method on various aspects, an interesting question is how to choose its integration time to achieve acceleration. In this work, we consider accelerating the process of sampling from a distribution $\pi(x) \... | \section{Introduction}
Markov chain Monte Carlo (MCMC) algorithms are fundamental techniques for sampling from probability distributions, which is a task that naturally arises in statistics \citep{Duane87,Duane1987216}, optimization \citep{FKM05,DBW12,JGNKJ17}, machine learning and others \citep{Wetal20,SM08,KF09,WT1... | {
"timestamp": "2022-07-06T02:21:57",
"yymm": "2207",
"arxiv_id": "2207.02189",
"language": "en",
"url": "https://arxiv.org/abs/2207.02189",
"abstract": "Hamiltonian Monte Carlo (HMC) is a popular method in sampling. While there are quite a few works of studying this method on various aspects, an interestin... |
https://arxiv.org/abs/0707.2885 | Sylvester's Minorant Criterion, Lagrange-Beltrami Identity, and Nonnegative Definiteness | We consider the characterizations of positive definite as well as nonnegative definite quadratic forms in terms of the principal minors of the associated symmetric matrix. We briefly review some of the known proofs, including a classical approach via the Lagrange-Beltrami identity. For quadratic forms in up to 3 variab... | \section{Introduction}
\label{sec:intro}
Let $A=(a_{ij})$ be an $n\times n$ real symmetric matrix and
$$
Q(\mathbf{x})=Q(x_1, \dots, x_n):= \mathbf{x}A\mathbf{x}^T = \sum_{i=1}^n \sum_{j=1}^n a_{ij} \, x_i x_j \,
$$
be the corresponding (real) quadratic form in $n$ variables. Recall that the matrix $A$ or the form $... | {
"timestamp": "2007-07-19T14:27:22",
"yymm": "0707",
"arxiv_id": "0707.2885",
"language": "en",
"url": "https://arxiv.org/abs/0707.2885",
"abstract": "We consider the characterizations of positive definite as well as nonnegative definite quadratic forms in terms of the principal minors of the associated sy... |
https://arxiv.org/abs/1809.09036 | Combinatorial interpretations of Lucas analogues of binomial coefficients and Catalan numbers | The Lucas sequence is a sequence of polynomials in s, and t defined recursively by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. On specialization of s and t one can recover the Fibonacci numbers, the nonnegative integers, and the q-integers [n]_q. Given a quantity which is expressed in terms of products and quotient... | \section{Introduction}
Let $s$ and $t$ be two indeterminants. The corresponding {\em Lucas sequence} is defined inductively by letting $\{0\}=0$, $\{1\}=1$, and
$$
\{n\}=s\{n-1\}+t\{n-2\}
$$
for $n\ge2$. For example
$$
\{2\}=s, \{3\}=s^2+t, \{4\}=s^3+2st,
$$
and so forth. Clearly when $s=t=1$ one recovers the Fibo... | {
"timestamp": "2018-09-25T02:25:17",
"yymm": "1809",
"arxiv_id": "1809.09036",
"language": "en",
"url": "https://arxiv.org/abs/1809.09036",
"abstract": "The Lucas sequence is a sequence of polynomials in s, and t defined recursively by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. On specialization of s ... |
https://arxiv.org/abs/1601.00609 | Small drift limit theorems for random walks | We show analogs of the classical arcsine theorem for the occupation time of a random walk in $(-\infty,0)$ in the case of a small positive drift. To study the asymptotic behavior of the total time spent in $(-\infty,0)$ we consider parametrized classes of random walks, where the convergence of the parameter to zero imp... | \section{Introduction}
For the classical symmetric random walk with $\pm 1$ steps it is well known that the three random variables ``time spent on the positive axis", ``position of the first maximum"
and ``last exit from zero'' are identically distributed and (suitably normalized) asymptotically arcsine-distributed.
... | {
"timestamp": "2016-05-31T02:06:15",
"yymm": "1601",
"arxiv_id": "1601.00609",
"language": "en",
"url": "https://arxiv.org/abs/1601.00609",
"abstract": "We show analogs of the classical arcsine theorem for the occupation time of a random walk in $(-\\infty,0)$ in the case of a small positive drift. To stud... |
https://arxiv.org/abs/1908.04065 | On nilpotent generators of the symplectic Lie algebra | Let $\mathfrak{sp}_{2n}(\mathbb {K})$ be the symplectic Lie algebra over an algebraically closed field of characteristic zero. We prove that for any nonzero nilpotent element $X \in \mathfrak{sp}_{2n}(\mathbb {K})$ there exists a nilpotent element $Y \in \mathfrak{sp}_{2n}(\mathbb {K})$ such that $X$ and $Y$ generate $... | \section{Introduction}
It is an important problem to find a minimal generating set of a given algebra. This problem was studied actively for semisimple Lie algebras. In 1951, Kuranishi \cite{MK} observed that any semisimple Lie algebra over a field of characteristic zero can be generated by two elements. Twenty-five ... | {
"timestamp": "2019-08-13T02:18:47",
"yymm": "1908",
"arxiv_id": "1908.04065",
"language": "en",
"url": "https://arxiv.org/abs/1908.04065",
"abstract": "Let $\\mathfrak{sp}_{2n}(\\mathbb {K})$ be the symplectic Lie algebra over an algebraically closed field of characteristic zero. We prove that for any non... |
https://arxiv.org/abs/0704.2716 | Constructing a quadrilateral inside another one | Connect each vertex of a convex quadrilateral Q to the midpoint of the next (proceeding counterclockwise) side. The four connecting lines create an interior quadrilateral I. We study the ratio area(I)/area(Q). We also determine what happens to area(I)/area(Q) when the four midpoints are replaced by points which divide ... | \section{\label{s:1}The quadrilateral ratio problem}
The description of Project 54 in 101 Project Ideas for the Geometer's
Sketchpad \cite{Key} reads (in part):
\qquad\parbox[1in]{4.5in}{On the Units panel of Preferences, set
Scalar Precision to hundredths. Construct a generic quadrilateral
and the midpoints of the s... | {
"timestamp": "2007-09-17T17:24:20",
"yymm": "0704",
"arxiv_id": "0704.2716",
"language": "en",
"url": "https://arxiv.org/abs/0704.2716",
"abstract": "Connect each vertex of a convex quadrilateral Q to the midpoint of the next (proceeding counterclockwise) side. The four connecting lines create an interior... |
https://arxiv.org/abs/1004.2445 | The Cauchy-Schlomilch transformation | The Cauchy-Schlömilch transformation states that for a function $f$ and $a, \, b > 0$, the integral of $f(x^{2})$ and $af((ax-bx^{-1})^{2}$ over the interval $[0, \infty)$ are the same. This elementary result is used to evaluate many non-elementary definite integrals, most of which cannot be obtained by symbolic pack... | \section{Introduction} \label{intro}
\setcounter{equation}{0}
The problem of analytic evaluations of definite integrals has been of
interest to scientists for a long time. The central
question can be stated as follows: \\
\begin{center}
{\em given a class of
functions} $\mathfrak{F}$ {\em and an
interval} $[a,b... | {
"timestamp": "2010-04-15T02:01:34",
"yymm": "1004",
"arxiv_id": "1004.2445",
"language": "en",
"url": "https://arxiv.org/abs/1004.2445",
"abstract": "The Cauchy-Schlömilch transformation states that for a function $f$ and $a, \\, b > 0$, the integral of $f(x^{2})$ and $af((ax-bx^{-1})^{2}$ over the interv... |
https://arxiv.org/abs/1506.08962 | Product of positive semi-definite matrices | It is known that every complex square matrix with nonnegative determinant is the product of positive semi-definite matrices. There are characterizations of matrices that require two or five positive semi-definite matrices in the product. However, the characterizations of matrices that require three or four positive sem... | \section{Introduction}
Let $M_n$ be the set of $n\times n$ complex matrices.
In \cite{Wu}, the author
showed that a matrix in $M_n$ with nonnegative determinant can always be
written as the product of five or fewer positive semi-definite
matrices. This is an extension to the result in \cite{B} asserting that
every mat... | {
"timestamp": "2015-09-29T02:10:18",
"yymm": "1506",
"arxiv_id": "1506.08962",
"language": "en",
"url": "https://arxiv.org/abs/1506.08962",
"abstract": "It is known that every complex square matrix with nonnegative determinant is the product of positive semi-definite matrices. There are characterizations o... |
https://arxiv.org/abs/2006.16797 | Confirming the Labels of Coins in One Weighing | There are $n$ bags with coins that look the same. Each bag has an infinite number of coins and all coins in the same bag weigh the same amount. Coins in different bags weigh 1, 2, 3, and so on to $n$ grams exactly. There is a unique label from the set 1 through $n$ attached to each bag that is supposed to correspond to... | \section{Introduction}\label{sec:intro}
Coin puzzles have fascinated mathematicians for a long time. Guy and Nowakowsky summarized the most famous coin problem in their paper \cite{GN}. We are interested in the particular famous coin-weighing puzzle below. This puzzle appeared on the 2000 Streamline Olympiad for 8th g... | {
"timestamp": "2020-07-01T02:20:02",
"yymm": "2006",
"arxiv_id": "2006.16797",
"language": "en",
"url": "https://arxiv.org/abs/2006.16797",
"abstract": "There are $n$ bags with coins that look the same. Each bag has an infinite number of coins and all coins in the same bag weigh the same amount. Coins in d... |
https://arxiv.org/abs/1805.08340 | Reducing Parameter Space for Neural Network Training | For neural networks (NNs) with rectified linear unit (ReLU) or binary activation functions, we show that their training can be accomplished in a reduced parameter space. Specifically, the weights in each neuron can be trained on the unit sphere, as opposed to the entire space, and the threshold can be trained in a boun... | \section{Summary} \label{sec:summary}
In this paper we presented a set of constraints on multi-layer
feedforward NNs with ReLU and binary activation functions. The weights
in each neuron are constrained on the unit sphere, as opposed to the
entire space. This effectively reduces the number of parameters in
weigh... | {
"timestamp": "2020-01-30T02:15:10",
"yymm": "1805",
"arxiv_id": "1805.08340",
"language": "en",
"url": "https://arxiv.org/abs/1805.08340",
"abstract": "For neural networks (NNs) with rectified linear unit (ReLU) or binary activation functions, we show that their training can be accomplished in a reduced p... |
https://arxiv.org/abs/1102.3394 | Poincare Analyticity and the Complete Variational Equations | According to a theorem of Poincare, the solutions to differential equations are analytic functions of (and therefore have Taylor expansions in) the initial conditions and various parameters providing the right sides of the differential equations are analytic in the variables, the time, and the parameters. We describe h... | \section*{Abstract}
According to a theorem of Poincar\'{e}, the solutions to differential equations are analytic functions of (and therefore have Taylor expansions in) the initial conditions and various parameters providing the right sides of the differential equations are analytic in the variables, the time, and the ... | {
"timestamp": "2011-02-17T02:02:31",
"yymm": "1102",
"arxiv_id": "1102.3394",
"language": "en",
"url": "https://arxiv.org/abs/1102.3394",
"abstract": "According to a theorem of Poincare, the solutions to differential equations are analytic functions of (and therefore have Taylor expansions in) the initial ... |
https://arxiv.org/abs/1810.06275 | Functional limit theorems for random walks | We survey some geometrical properties of trajectories of $d$-dimensional random walks via the application of functional limit theorems. We focus on the functional law of large numbers and functional central limit theorem (Donsker's theorem). For the latter, we survey the underlying weak convergence theory, drawing heav... | \section{Introduction}
The early days of limit theorems in the form of a \lq law of averages\rq~saw slow progress. The first direct study was the theorem of Bernoulli \cite{bernoulli} on the sums of binary random variables, but this was only stated in 1713 over a century after comments of Cardano in his work on dice g... | {
"timestamp": "2018-10-16T02:19:38",
"yymm": "1810",
"arxiv_id": "1810.06275",
"language": "en",
"url": "https://arxiv.org/abs/1810.06275",
"abstract": "We survey some geometrical properties of trajectories of $d$-dimensional random walks via the application of functional limit theorems. We focus on the fu... |
https://arxiv.org/abs/1706.05975 | The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg | We discuss five discrete results: the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carathéodory, Helly, and Tverberg from combinatorial geometry. We explore their connections and emphasize their broad impact in application areas such as game theory, graph theory, mathematical optimizatio... | \section*{Acknowledgments}
J. A. De Loera was partially supported by the LabEx
Bezout (ANR-10-LABX-58). He is grateful to the labex BEZOUT and the CERMICS
research center at \'Ecole National des Ponts et Chauss\'ees for the support
received, and the enjoyable and welcoming environment in which the
topics in this pape... | {
"timestamp": "2018-10-09T02:20:39",
"yymm": "1706",
"arxiv_id": "1706.05975",
"language": "en",
"url": "https://arxiv.org/abs/1706.05975",
"abstract": "We discuss five discrete results: the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carathéodory, Helly, and Tverberg from ... |
https://arxiv.org/abs/2009.05922 | An algorithm for finding minimal generating sets of finite groups | In this article, we study connections between components of the Cayley graph $\mathrm{Cay}(G,A)$, where $A$ is an arbitrary subset of a group $G$, and cosets of the subgroup of $G$ generated by $A$. In particular, we show how to construct generating sets of $G$ if $\mathrm{Cay}(G,A)$ has finitely many components. Furth... | \section{Introduction}\label{sec: introdcution}
The problem of determining (minimal) generating sets of groups has been studied widely; see, for instance, \cite{MR3814347, MR2361465, MR572868, MR3073659, MR2023255, MR1289999, MR643291}. Although the existence of a generating set of a certain group is known, it might be... | {
"timestamp": "2021-04-20T02:08:48",
"yymm": "2009",
"arxiv_id": "2009.05922",
"language": "en",
"url": "https://arxiv.org/abs/2009.05922",
"abstract": "In this article, we study connections between components of the Cayley graph $\\mathrm{Cay}(G,A)$, where $A$ is an arbitrary subset of a group $G$, and co... |
https://arxiv.org/abs/0912.0239 | On k-crossings and k-nestings of permutations | We introduce k-crossings and k-nestings of permutations. We show that the crossing number and the nesting number of permutations have a symmetric joint distribution. As a corollary, the number of k-noncrossing permutations is equal to the number of k-nonnesting permutations. We also provide some enumerative results for... | \section{Introduction}
Nestings and crossings are equidistributed in many combinatorial objects, such as matchings, set partitions, permutations, and embedded labelled graphs~\cite{Chetal07, Corteel07, deMi07}. More surprising is the symmetric joint distribution of the crossing and nesting numbers: A set of $k$~arcs f... | {
"timestamp": "2009-12-01T20:47:28",
"yymm": "0912",
"arxiv_id": "0912.0239",
"language": "en",
"url": "https://arxiv.org/abs/0912.0239",
"abstract": "We introduce k-crossings and k-nestings of permutations. We show that the crossing number and the nesting number of permutations have a symmetric joint dist... |
https://arxiv.org/abs/1404.6550 | A note on coloring vertex-transitive graphs | We prove bounds on the chromatic number $\chi$ of a vertex-transitive graph in terms of its clique number $\omega$ and maximum degree $\Delta$. We conjecture that every vertex-transitive graph satisfies $\chi \le \max \left\{\omega, \left\lceil\frac{5\Delta + 3}{6}\right\rceil\right\}$ and we prove results supporting t... | \section{Introduction}
Many results and conjectures in the graph coloring literature have the
form: \emph{if the chromatic number $\chi$ of a graph is close to its maximum
degree $\Delta$,
then the graph contains a big clique, i.e., $\omega$ is large}
(\cite{brooks1941colouring,
borodin1977upper, reed1999strengthening,... | {
"timestamp": "2014-04-29T02:00:57",
"yymm": "1404",
"arxiv_id": "1404.6550",
"language": "en",
"url": "https://arxiv.org/abs/1404.6550",
"abstract": "We prove bounds on the chromatic number $\\chi$ of a vertex-transitive graph in terms of its clique number $\\omega$ and maximum degree $\\Delta$. We conjec... |
https://arxiv.org/abs/2002.05670 | Experimental Design in Two-Sided Platforms: An Analysis of Bias | We develop an analytical framework to study experimental design in two-sided marketplaces. Many of these experiments exhibit interference, where an intervention applied to one market participant influences the behavior of another participant. This interference leads to biased estimates of the treatment effect of the in... | \subsection{Estimation with the $\ensuremath{\mathsf{TSR}}$ design}
\label{ssec:TSR_est}
The preceding sections reveal that each of the naive $\ensuremath{\mathsf{LR}}$ and $\ensuremath{\mathsf{CR}}$ estimators has its virtues, depending on market balance conditions. In this section, we explore whether we can develop... | {
"timestamp": "2021-09-28T02:27:22",
"yymm": "2002",
"arxiv_id": "2002.05670",
"language": "en",
"url": "https://arxiv.org/abs/2002.05670",
"abstract": "We develop an analytical framework to study experimental design in two-sided marketplaces. Many of these experiments exhibit interference, where an interv... |
https://arxiv.org/abs/1802.08015 | Spanned lines and Langer's inequality | We collect some results in combinatorial geometry that follow from an inequality of Langer in algebraic geometry. Langer's inequality gives a lower bound on the number of incidences between a point set and its spanned lines, and was recently used by Han to improve the constant in the weak Dirac conjecture. Here we obse... | \section{Introduction}
We consider the following standard notions from combinatorial geometry.
For a point set $P$ in a space $\mathbb R^d$ or $\mathbb C^d$, we write $L(P)$ for the set of lines spanned by $P$,
and we set $n = |L(P)|$.
We write $\ell_i$ for the number of lines in $L(P)$ containing exactly $i$ points o... | {
"timestamp": "2018-02-23T02:08:25",
"yymm": "1802",
"arxiv_id": "1802.08015",
"language": "en",
"url": "https://arxiv.org/abs/1802.08015",
"abstract": "We collect some results in combinatorial geometry that follow from an inequality of Langer in algebraic geometry. Langer's inequality gives a lower bound ... |
https://arxiv.org/abs/1503.04749 | The height of multiple edge plane trees | Multi-edge trees as introduced in a recent paper of Dziemiańczuk are plane trees where multiple edges are allowed. We first show that $d$-ary multi-edge trees where the out-degrees are bounded by $d$ are in bijection with classical $d$-ary trees. This allows us to analyse parameters such as the height.The main part of ... | \section{Introduction}
Dziemia{\'n}czuk~\cite{Dziemianczuk:2014:enumer-raney} has introduced a tree model based on plane (=planar) trees~\cite[p.~31]{Flajolet-Sedgewick:ta:analy},
which are enumerated by Catalan numbers. Instead of connecting two vertices by
one edge, in his multi-edge model, two vertices can be con... | {
"timestamp": "2015-03-17T01:19:29",
"yymm": "1503",
"arxiv_id": "1503.04749",
"language": "en",
"url": "https://arxiv.org/abs/1503.04749",
"abstract": "Multi-edge trees as introduced in a recent paper of Dziemiańczuk are plane trees where multiple edges are allowed. We first show that $d$-ary multi-edge t... |
https://arxiv.org/abs/1701.03203 | Stability of the Heisenberg Product on Symmetric Functions | The Heisenberg product is an associative product defined on symmetric functions which interpolates between the usual product and the Kronecker product. In 1938, Murnaghan discovered that the Kronecker product of two Schur functions stabilizes. We prove an analogous result for the Heisenberg product of Schur functions. | \section{Introduction}
\label{intro}
Aguiar, Ferrer Santos, and Moreira introduced a new product on symmetric functions (also on representations of symmetric group) \cite{M-2015}. Unlike the usual product and the Kronecker product, the terms appearing in Heisenberg product of two Schur functions have different degrees.... | {
"timestamp": "2017-01-13T02:02:19",
"yymm": "1701",
"arxiv_id": "1701.03203",
"language": "en",
"url": "https://arxiv.org/abs/1701.03203",
"abstract": "The Heisenberg product is an associative product defined on symmetric functions which interpolates between the usual product and the Kronecker product. In... |
https://arxiv.org/abs/1207.3468 | Minimal Convex Decompositions | Let $P$ be a set of $n$ points on the plane in general position. We say that a set $\Gamma$ of convex polygons with vertices in $P$ is a convex decomposition of $P$ if: Union of all elements in $\Gamma$ is the convex hull of $P,$ every element in $\Gamma$ is empty, and for any two different elements of $\Gamma$ their i... | \section{Introduction}
Let $P_n$ denote a set of $n$ points on the plane in general position. We denote as $Conv(P_n)$ the convex
hull of $P_n$ and $c$ the number of its vertices, and given a polygon $\alpha$ we denote as $\alpha^o$ its interior. We say that a set
$\Gamma=\{\gamma_1,\gamma_2,...,\gamma_k\}$ of $k$ c... | {
"timestamp": "2012-07-17T02:01:32",
"yymm": "1207",
"arxiv_id": "1207.3468",
"language": "en",
"url": "https://arxiv.org/abs/1207.3468",
"abstract": "Let $P$ be a set of $n$ points on the plane in general position. We say that a set $\\Gamma$ of convex polygons with vertices in $P$ is a convex decompositi... |
https://arxiv.org/abs/0906.2795 | Descent sets of cyclic permutations | We present a bijection between cyclic permutations of {1,2,...,n+1} and permutations of {1,2,...,n} that preserves the descent set of the first n entries and the set of weak excedances. This non-trivial bijection involves a Foata-like transformation on the cyclic notation of the permutation, followed by certain conjuga... | \section{Introduction}\label{sec:intro}
\subsection{Permutations, cycles, and descents}
Let $[n]=\{1,2,\dots,n\}$, and let $\S_n$ denote the set of permutations of $[n]$. We will use both the one-line notation of $\pi\in\S_n$
as $\pi=\pi(1)\pi(2)\dots\pi(n)$ and its decomposition as a a product
of cycles of the form... | {
"timestamp": "2009-06-15T22:25:51",
"yymm": "0906",
"arxiv_id": "0906.2795",
"language": "en",
"url": "https://arxiv.org/abs/0906.2795",
"abstract": "We present a bijection between cyclic permutations of {1,2,...,n+1} and permutations of {1,2,...,n} that preserves the descent set of the first n entries an... |
https://arxiv.org/abs/1310.5924 | The symplectic geometry of closed equilateral random walks in 3-space | A closed equilateral random walk in 3-space is a selection of unit length vectors giving the steps of the walk conditioned on the assumption that the sum of the vectors is zero. The sample space of such walks with $n$ edges is the $(2n-3)$-dimensional Riemannian manifold of equilateral closed polygons in $\mathbb{R}^3$... | \section{Introduction}
In this paper, we consider the classical model of a random walk in $\mathbb{R}^3$-- the walker chooses each step uniformly from the unit sphere. Some of the first results in the theory of these random walks are based on the observation that if a point is distributed uniformly on the surface of a... | {
"timestamp": "2013-10-23T02:07:12",
"yymm": "1310",
"arxiv_id": "1310.5924",
"language": "en",
"url": "https://arxiv.org/abs/1310.5924",
"abstract": "A closed equilateral random walk in 3-space is a selection of unit length vectors giving the steps of the walk conditioned on the assumption that the sum of... |
https://arxiv.org/abs/1407.5516 | A DEIM Induced CUR Factorization | We derive a CUR matrix factorization based on the Discrete Empirical Interpolation Method (DEIM). For a given matrix $A$, such a factorization provides a low rank approximate decomposition of the form $A \approx C U R$, where $C$ and $R$ are subsets of the columns and rows of $A$, and $U$ is constructed to make $CUR$ a... |
\section{Conclusions}
\vspace*{-5pt}
\noindent
The Discrete Empirical Interpolation Method (DEIM) is an index selection procedure that gives simple, deterministic CUR factorizations of the matrix ${\bf A}$.\ \
Since DEIM utilizes (approximate) singular vectors, we propose an effective one-pass incremental approxi... | {
"timestamp": "2015-09-22T02:00:52",
"yymm": "1407",
"arxiv_id": "1407.5516",
"language": "en",
"url": "https://arxiv.org/abs/1407.5516",
"abstract": "We derive a CUR matrix factorization based on the Discrete Empirical Interpolation Method (DEIM). For a given matrix $A$, such a factorization provides a lo... |
https://arxiv.org/abs/1712.06587 | Solving satisfiability using inclusion-exclusion | Using Maple, we implement a SAT solver based on the principle of inclusion-exclusion and the Bonferroni inequalities. Using randomly generated input, we investigate the performance of our solver as a function of the number of variables and number of clauses. We also test it against Maple's built-in tautology procedure.... | \section{Introduction to SAT}
First, some terminology. A \emph{Boolean variable} is a variable which can take on values in $\{true, false\}$, or, equivalently, $\{0,1\}$ (e.g. $x$). A \emph{literal} is a Boolean variable or its negation (e.g. $\lnot x$). \emph{Disjunction} means ``or" ($\lor$) and conjunction means ... | {
"timestamp": "2017-12-20T02:00:10",
"yymm": "1712",
"arxiv_id": "1712.06587",
"language": "en",
"url": "https://arxiv.org/abs/1712.06587",
"abstract": "Using Maple, we implement a SAT solver based on the principle of inclusion-exclusion and the Bonferroni inequalities. Using randomly generated input, we i... |
https://arxiv.org/abs/1308.1427 | Regions of Stability for a Linear Differential Equation with Two Rationally Dependent Delays | Stability analysis is performed for a linear differential equation with two delays. Geometric arguments show that when the two delays are rationally dependent, then the region of stability increases. When the ratio has the form 1/n, this study finds the asymptotic shape and size of the stability region. For example, a ... | \section*{Regions of Stability for a Linear Differential Equation with
Two Rationally Dependent Delays}
\end{center}
\begin{center}
Joseph M. Mahaffy \\
\begin{small}
Nonlinear Dynamical Systems Group \\
Department of Mathematics \\
San Diego State University \\ San Diego, CA 92182, USA \\
\end{sm... | {
"timestamp": "2013-08-08T02:00:43",
"yymm": "1308",
"arxiv_id": "1308.1427",
"language": "en",
"url": "https://arxiv.org/abs/1308.1427",
"abstract": "Stability analysis is performed for a linear differential equation with two delays. Geometric arguments show that when the two delays are rationally depende... |
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