url stringlengths 31 38 | title stringlengths 7 229 | abstract stringlengths 44 2.87k | text stringlengths 319 2.51M | meta dict |
|---|---|---|---|---|
https://arxiv.org/abs/1110.2956 | Brauer spaces for commutative rings and structured ring spectra | Using an analogy between the Brauer groups in algebra and the Whitehead groups in topology, we first use methods of algebraic K-theory to give a natural definition of Brauer spectra for commutative rings, such that their homotopy groups are given by the Brauer group, the Picard group and the group of units. Then, in th... | \section*{Introduction}
Let~$K$ be a field, and let us consider all finite-dimensional associative~$K$-algebras~$A$ up to isomorphism. For many purposes, a much coarser equivalence relation than isomorphism is appropriate: Morita equivalence. Recall that two such algebras~$A$ and~$B$ are called {\it Morita equi\-valen... | {
"timestamp": "2013-07-31T02:03:17",
"yymm": "1110",
"arxiv_id": "1110.2956",
"language": "en",
"url": "https://arxiv.org/abs/1110.2956",
"abstract": "Using an analogy between the Brauer groups in algebra and the Whitehead groups in topology, we first use methods of algebraic K-theory to give a natural def... |
https://arxiv.org/abs/1809.06946 | Adding a point to configurations in closed balls | We answer the question of when a new point can be added in a continuous way to configurations of $n$ distinct points in a closed ball of arbitrary dimension. We show that this is possible given an ordered configuration of $n$ points if and only if $n \neq 1$. On the other hand, when the points are not ordered and the d... | \section{Introduction}
Let $\DD^m$ be the closed ball of dimension $m$, with $m \geq 1$. This paper answers the following basic question:
\begin{quotation}
\emph{Is there a continuous rule that adds a new distinct point to every configuration of $n$ distinct points in $\DD^m$?}
\end{quotation}
The challenge he... | {
"timestamp": "2019-05-09T02:03:48",
"yymm": "1809",
"arxiv_id": "1809.06946",
"language": "en",
"url": "https://arxiv.org/abs/1809.06946",
"abstract": "We answer the question of when a new point can be added in a continuous way to configurations of $n$ distinct points in a closed ball of arbitrary dimensi... |
https://arxiv.org/abs/1711.09858 | Geometric Bounds for Favard Length | Given a set in the plane, the average length of its projections over all directions is called Favard length. This quantity measures the size of a set, and is closely related to metric and geometric properties of the set such as rectifiability, Hausdorff dimension, and analytic capacity. In this paper, we develop new ge... | \section{Introduction}
Given a set $E$ in the plane, its Favard length is the average
$$\operatorname{Fav}(E) = \int_0^{2\pi} |\pi_{\theta} E|\, d\theta$$
where $\pi_{\theta}$ is orthogonal projection onto a line $L_{\theta}$ through the origin at angle $\theta$ to the positive $x$-axis, and $|.|$ is the length measure... | {
"timestamp": "2017-11-28T02:20:31",
"yymm": "1711",
"arxiv_id": "1711.09858",
"language": "en",
"url": "https://arxiv.org/abs/1711.09858",
"abstract": "Given a set in the plane, the average length of its projections over all directions is called Favard length. This quantity measures the size of a set, and... |
https://arxiv.org/abs/1207.5381 | On the connectivity of manifold graphs | This paper is concerned with lower bounds for the connectivity of graphs (one-dimensional skeleta) of triangulations of compact manifolds. We introduce a structural invariant b_M for simplicial d-manifolds M taking values in the range 0 <= b_M <= d-1. The main result is that b_M influences connectivity in the following... | \section{Introduction}
Consider a pure $d$-dimensional polyhedral complex $\Delta$.
The {\em graph ${\mathcal{G}}(\Delta)$}, or {\em $1$-skeleton}, of $\Delta$ is the undirected simple graph that has the
vertices of $\Delta$ as nodes and the one-dimensional faces of $\Delta$ as edges.
The study of graph-theoretic c... | {
"timestamp": "2013-10-23T02:08:31",
"yymm": "1207",
"arxiv_id": "1207.5381",
"language": "en",
"url": "https://arxiv.org/abs/1207.5381",
"abstract": "This paper is concerned with lower bounds for the connectivity of graphs (one-dimensional skeleta) of triangulations of compact manifolds. We introduce a st... |
https://arxiv.org/abs/1910.02977 | Colored Multipermutations and a Combinatorial Generalization of Worpitzky's Identity | Worpitzky's identity expresses $n^p$ in terms of the Eulerian numbers and binomial coefficients: $$n^p = \sum_{i=0}^{p-1} \genfrac<>{0pt}{}{p}{i} \binom{n+i}{p}.$$Pita-Ruiz recently defined numbers $A_{a,b,r}(p,i)$ implicitly to satisfy a generalized Worpitzky identity $$\binom{an+b}{r}^p = \sum_{i=0}^{rp} A_{a,b,r}(p,... | \section{Generalized Eulerian Numbers}
The famous \emph{Eulerian numbers} $\displaystyle\eu{p}{i}$ appear first in Euler's 1755 manuscript \emph{Institutiones calculi differentialis}~\cite{euler:foundations} and are now known to count a variety of combinatorial objects. The Eulerian numbers can be defined by the recur... | {
"timestamp": "2019-10-09T02:00:30",
"yymm": "1910",
"arxiv_id": "1910.02977",
"language": "en",
"url": "https://arxiv.org/abs/1910.02977",
"abstract": "Worpitzky's identity expresses $n^p$ in terms of the Eulerian numbers and binomial coefficients: $$n^p = \\sum_{i=0}^{p-1} \\genfrac<>{0pt}{}{p}{i} \\bino... |
https://arxiv.org/abs/1012.3932 | Balanced Interval Coloring | We consider the discrepancy problem of coloring $n$ intervals with $k$ colors such that at each point on the line, the maximal difference between the number of intervals of any two colors is minimal. Somewhat surprisingly, a coloring with maximal difference at most one always exists. Furthermore, we give an algorithm w... | \section{Introduction}
In this paper, we consider the following load balancing problem: We
are given a set $\mathcal{I} = \{ I_1, \dots, I_n \}$ of tasks, where
each task is represented by an interval $I = [\ell, r] \in
\mathcal{I}$ with starttime~$\ell$ and endtime $r$. Furthermore, we
are given $k$ servers and have ... | {
"timestamp": "2010-12-20T02:01:57",
"yymm": "1012",
"arxiv_id": "1012.3932",
"language": "en",
"url": "https://arxiv.org/abs/1012.3932",
"abstract": "We consider the discrepancy problem of coloring $n$ intervals with $k$ colors such that at each point on the line, the maximal difference between the number... |
https://arxiv.org/abs/1901.05834 | Multicolour bipartite Ramsey number of paths | The $k$-colour bipartite Ramsey number of a bipartite graph $H$ is the least integer $N$ for which every $k$-edge-coloured complete bipartite graph $K_{N,N}$ contains a monochromatic copy of $H$. The study of bipartite Ramsey numbers was initiated over 40 years ago by Faudree and Schelp and, independently, by Gyárfás a... | \section{Introduction}
Ramsey theory refers to a large body of mathematical results, which roughly say that any sufficiently large structure is guaranteed to have a large well-organised substructure. For example, the celebrated theorem of Ramsey \cite{ramsey1929problem} says that for any fixed graph $H$, every $k$-edg... | {
"timestamp": "2019-09-18T02:19:47",
"yymm": "1901",
"arxiv_id": "1901.05834",
"language": "en",
"url": "https://arxiv.org/abs/1901.05834",
"abstract": "The $k$-colour bipartite Ramsey number of a bipartite graph $H$ is the least integer $N$ for which every $k$-edge-coloured complete bipartite graph $K_{N,... |
https://arxiv.org/abs/1907.12668 | Perspectives on CUR Decompositions | This note discusses an interesting matrix factorization called the CUR Decomposition. We illustrate various viewpoints of this method by comparing and contrasting them in different situations. Additionally, we offer a new characterization of CUR decompositions which synergizes these viewpoints and shows that they are i... | \section{Introduction}
In many data analysis applications, two key tools are dimensionality reduction and compression, in which data obtained as vectors in a high dimensional Euclidean space are approximated in a basis or frame which spans a much lower dimensional space (reduction) or a sketch of the total data ma... | {
"timestamp": "2019-09-05T02:01:54",
"yymm": "1907",
"arxiv_id": "1907.12668",
"language": "en",
"url": "https://arxiv.org/abs/1907.12668",
"abstract": "This note discusses an interesting matrix factorization called the CUR Decomposition. We illustrate various viewpoints of this method by comparing and con... |
https://arxiv.org/abs/1308.4438 | On varieties of commuting nilpotent matrices | Let $N(d,n)$ be the variety of all $d$-tuples of commuting nilpotent $n\times n$ matrices. It is well-known that $N(d,n)$ is irreducible if $d=2$, if $n\le 3$ or if $d=3$ and $n=4$. On the other hand $N(3,n)$ is known to be reducible for $n\ge 13$. We study in this paper the reducibility of $N(d,n)$ for various values ... | \section{Introduction}
Let $C(d,n)$ denote the set of all $d$-tuples of commuting $n\times n$ matrices over an algebraically closed field $\mathbb{F}$ and let $N(d,n)$ be its subset consisting of all $d$-tuples of {\em nilpotent} commuting matrices. Both sets are defined by polynomial equations in the entries of matric... | {
"timestamp": "2014-03-28T01:00:56",
"yymm": "1308",
"arxiv_id": "1308.4438",
"language": "en",
"url": "https://arxiv.org/abs/1308.4438",
"abstract": "Let $N(d,n)$ be the variety of all $d$-tuples of commuting nilpotent $n\\times n$ matrices. It is well-known that $N(d,n)$ is irreducible if $d=2$, if $n\\l... |
https://arxiv.org/abs/2108.02055 | Recovery of Sobolev functions restricted to iid sampling | We study $L_q$-approximation and integration for functions from the Sobolev space $W^s_p(\Omega)$ and compare optimal randomized (Monte Carlo) algorithms with algorithms that can only use iid sample points, uniformly distributed on the domain. The main result is that we obtain the same optimal rate of convergence if we... | \section{Introduction and main results}
Let $\Omega \subset \ensuremath{\mathbb{R}}^d$ be open and bounded.
We assume that $\Omega$ satisfies an interior cone condition.
We study the problem of approximating
a function $f$ from the Sobolev space $W_p^s(\Omega)$
in the $L_q(\Omega)$-norm based on function values $f(x_... | {
"timestamp": "2021-08-05T02:19:33",
"yymm": "2108",
"arxiv_id": "2108.02055",
"language": "en",
"url": "https://arxiv.org/abs/2108.02055",
"abstract": "We study $L_q$-approximation and integration for functions from the Sobolev space $W^s_p(\\Omega)$ and compare optimal randomized (Monte Carlo) algorithms... |
https://arxiv.org/abs/2203.00068 | Perturbation of invariant subspaces for ill-conditioned eigensystem | Given a diagonalizable matrix $A$, we study the stability of its invariant subspaces when its matrix of eigenvectors is ill-conditioned. Let $\mathcal{X}_1$ be some invariant subspace of $A$ and $X_1$ be the matrix storing the right eigenvectors that spanned $\mathcal{X}_1$. It is generally believed that when the condi... | \section{Introduction}\label{sec:intro}
Let $A\in\mathbb{C}^{n,n}$ be a diagonalizable matrix. An invariant subspace $\mathcal{X}$ of $A$ is one that satisfies
\[
A\mathcal{X}\subseteq \mathcal{X}.
\]
When a small perturbation is added to $A$, its invariant subspace $\mathcal{X}$ will be perturbed accordingly. The goa... | {
"timestamp": "2022-03-02T02:03:37",
"yymm": "2203",
"arxiv_id": "2203.00068",
"language": "en",
"url": "https://arxiv.org/abs/2203.00068",
"abstract": "Given a diagonalizable matrix $A$, we study the stability of its invariant subspaces when its matrix of eigenvectors is ill-conditioned. Let $\\mathcal{X}... |
https://arxiv.org/abs/2001.00653 | On the Number of Independent Sets in Uniform, Regular, Linear Hypergraphs | We study the problems of bounding the number weak and strong independent sets in $r$-uniform, $d$-regular, $n$-vertex linear hypergraphs with no cross-edges. In the case of weak independent sets, we provide an upper bound that is tight up to the first order term for all (fixed) $r\ge 3$, with $d$ and $n$ going to infin... | \section{Introduction}
A classic result in the extremal theory of bounded-degree graphs is the result of Jeff Kahn~\cite{Kahn2001} that a disjoint union of copies of the complete $d$-regular bipartite graph ($K_{d,d}$) maximizes the number of independent sets over all $d$-regular bipartite graphs on the same number ... | {
"timestamp": "2021-07-06T02:15:07",
"yymm": "2001",
"arxiv_id": "2001.00653",
"language": "en",
"url": "https://arxiv.org/abs/2001.00653",
"abstract": "We study the problems of bounding the number weak and strong independent sets in $r$-uniform, $d$-regular, $n$-vertex linear hypergraphs with no cross-edg... |
https://arxiv.org/abs/1509.08166 | Finite Element Methods for Interface Problems: Robust and Local Optimal A Priori Error Estimates | For elliptic interface problems in two- and three-dimensions, this paper establishes a priori error estimates for Crouzeix-Raviart nonconforming, Raviart-Thomas mixed, and discontinuous Galerkin finite element approximations. These estimates are robust with respect to the diffusion coefficient and optimal with respect ... |
\section{Introduction}\label{intro}
\setcounter{equation}{0}
As a prototype of problems with interface singularities, this paper studies {\it a priori}
error estimates of various finite element methods for the following
interface problem (i.e., the diffusion problem with discontinuous coefficients):
\begin{equation}... | {
"timestamp": "2015-10-26T01:05:36",
"yymm": "1509",
"arxiv_id": "1509.08166",
"language": "en",
"url": "https://arxiv.org/abs/1509.08166",
"abstract": "For elliptic interface problems in two- and three-dimensions, this paper establishes a priori error estimates for Crouzeix-Raviart nonconforming, Raviart-... |
https://arxiv.org/abs/2301.04763 | Behaviors of pairs of dimensions and depths of edge ideals | Edge ideals of finite simple graphs $G$ on $n$ vertices are the ideals $I(G)$ of the polynomial ring $S$ in $n$ variables generated by the quadratic monomials associated with the edges of $G$. In this paper, we consider the possible pairs of dimensions and depths of $S/I(G)$ for connected graphs with a fixed number of ... | \section{Introduction}
Monomial ideals are one of the most well-studied objects in the area of combinatorial commutative algebra.
Especially, edge ideals of graphs are of particular interest in this area.
On the other hand, dimensions and depths are fundamental invariants on graded (or local) rings.
The goal o... | {
"timestamp": "2023-01-18T02:22:12",
"yymm": "2301",
"arxiv_id": "2301.04763",
"language": "en",
"url": "https://arxiv.org/abs/2301.04763",
"abstract": "Edge ideals of finite simple graphs $G$ on $n$ vertices are the ideals $I(G)$ of the polynomial ring $S$ in $n$ variables generated by the quadratic monom... |
https://arxiv.org/abs/2009.09322 | The geometry of random tournaments | A tournament is an orientation of a graph. Each edge represents a match, directed towards the winner. The score sequence lists the number of wins by each team. Landau (1953) characterized score sequences of the complete graph. Moon (1963) showed that the same conditions are necessary and sufficient for mean score seque... | \section{Introduction}\label{S_intro}
Let $G=(V,E)$ be a graph on $V=[n]$.
A {\it tournament} on $G$ is an orientation of
$E$. Intuitively, for each edge $(i,j)\in E$, teams
$i$ and $j$ play a match, and then this edge
is directed towards the winner.
The {\it score sequence} $s=(s_1,\ldots,s_n)$
lists the number of ... | {
"timestamp": "2020-09-22T02:12:56",
"yymm": "2009",
"arxiv_id": "2009.09322",
"language": "en",
"url": "https://arxiv.org/abs/2009.09322",
"abstract": "A tournament is an orientation of a graph. Each edge represents a match, directed towards the winner. The score sequence lists the number of wins by each ... |
https://arxiv.org/abs/2012.12859 | Limit Theorems for Fréchet Mean Sets | For $1\le p \le \infty$, the Fréchet $p$-mean of a probability measure on a metric space is an important notion of central tendency that generalizes the usual notions in the real line of mean ($p=2$) and median ($p=1$). In this work we prove a collection of limit theorems for Fréchet means and related objects, which, i... | \section{Introduction}
For $1 \le p < \infty$ and a metric space $(X,d)$, denote by $\mathcal{P}_p(X)$ the set of Borel probability measures $\mu$ on $X$ such that $\int_X d^p(x,y) \, d\mu(y) < \infty$ for all $x \in X$.
\begin{definition}
For $1 \le p < \infty$, the \textit{Fr\'echet $p$-mean} of a probability measur... | {
"timestamp": "2020-12-24T02:23:33",
"yymm": "2012",
"arxiv_id": "2012.12859",
"language": "en",
"url": "https://arxiv.org/abs/2012.12859",
"abstract": "For $1\\le p \\le \\infty$, the Fréchet $p$-mean of a probability measure on a metric space is an important notion of central tendency that generalizes th... |
https://arxiv.org/abs/2009.07137 | Gaussian vectors with Markov property | We demonstrate the parallel between the properties of Gaussian vectors and the Euclidean geometry. In particular we study the Markov property and give various equivalent Euclidean and probabilistic characterizations. We also give a simple Euclidean proof of the conditional maximality of the differential entropy for the... | \section{Introduction}\label{int}
It is a math folklore that the study of
Gaussian vectors is in some respects part of Euclidean geometry.
In this note we demonstrate this parallel by exploring
the Markov property in a Gaussian vector.
We say that a random vector $X= (X_1,X_2,\dots,X_n)$ is {\it Markov}
if for ever... | {
"timestamp": "2020-09-16T02:20:17",
"yymm": "2009",
"arxiv_id": "2009.07137",
"language": "en",
"url": "https://arxiv.org/abs/2009.07137",
"abstract": "We demonstrate the parallel between the properties of Gaussian vectors and the Euclidean geometry. In particular we study the Markov property and give var... |
https://arxiv.org/abs/1705.03842 | On the linear independence of shifted powers | We call shifted power a polynomial of the form $(x-a)^e$. The main goal of this paper is to obtain broadly applicable criteria ensuring that the elements of a finite family $F$ of shifted powers are linearly independent or, failing that, to give a lower bound on the dimension of the space of polynomials spanned by $F$.... | \section{Introduction}
In this article, we
consider families of univariate polynomials of the
form: $$F = \{ (x-a_i)^{e_i} \;:\: 1 \leq i \leq s \},$$
where $e_i \in \mathbb{N}$ and the $a_i$ belong to a field $\mathbb{K}$ of characteristic~0.
An element of $F$ will be called a \emph{shifted power}
(polynomials of th... | {
"timestamp": "2017-10-23T02:08:18",
"yymm": "1705",
"arxiv_id": "1705.03842",
"language": "en",
"url": "https://arxiv.org/abs/1705.03842",
"abstract": "We call shifted power a polynomial of the form $(x-a)^e$. The main goal of this paper is to obtain broadly applicable criteria ensuring that the elements ... |
https://arxiv.org/abs/1307.7819 | Complex versus real orthogonal polynomials of two variables | Orthogonal polynomials of two real variables can often be represented in complex variables. We explore the connection between the two types of representations and study the structural relations of complex orthogonal polynomials. The complex Hermite orthogonal polynomials and the disk polynomials are used as illustratin... | \section{Introduction}
\setcounter{equation}{0}
For a real valued weight function $W(x,y)$ defined on a domain $\Omega \subset {\mathbb R}^2$, orthogonal polynomials
of two variables with respect to $W$ are usually defined as polynomials that are orthogonal with respect to the inner
product
\begin{equation} \labe... | {
"timestamp": "2013-07-31T02:02:47",
"yymm": "1307",
"arxiv_id": "1307.7819",
"language": "en",
"url": "https://arxiv.org/abs/1307.7819",
"abstract": "Orthogonal polynomials of two real variables can often be represented in complex variables. We explore the connection between the two types of representatio... |
https://arxiv.org/abs/2206.09411 | A Stirling-type formula for the distribution of the length of longest increasing subsequences | The discrete distribution of the length of longest increasing subsequences in random permutations of $n$ integers is deeply related to random matrix theory. In a seminal work, Baik, Deift and Johansson provided an asymptotics in terms of the distribution of the scaled largest level of the large matrix limit of GUE. As ... | \section{Introduction}
As witnessed by a number of outstanding surveys and monographs (see, e.g., \cite{MR1694204, MR3468920, MR3468738, MR2334203}), a surprisingly rich topic in combinatorics and probability theory, deeply related to representation theory and to random matrix theory, is the study of the lengths $L_n... | {
"timestamp": "2022-07-05T02:24:37",
"yymm": "2206",
"arxiv_id": "2206.09411",
"language": "en",
"url": "https://arxiv.org/abs/2206.09411",
"abstract": "The discrete distribution of the length of longest increasing subsequences in random permutations of $n$ integers is deeply related to random matrix theor... |
https://arxiv.org/abs/1309.7295 | Linear extensions of orders invariant under abelian group actions | Let G be an abelian group acting on a set X, and suppose that no element of G has any finite orbit of size greater than one. We show that every partial order on X invariant under $G$ extends to a linear order on X also invariant under G. We then discuss extensions to linear preorders when the orbit condition is not met... | \section{Linear orders}
Szpilrajn's Theorem~\cite{Szpilrajn} (proved independently by a number of others) says that given the Axiom of Choice, any partial order can be extended to a linear order, where $\le^*$ extends $\le$ provided that $x\le y$ implies $x\le^* y$. There has been much work on what properties of the p... | {
"timestamp": "2013-09-30T02:08:19",
"yymm": "1309",
"arxiv_id": "1309.7295",
"language": "en",
"url": "https://arxiv.org/abs/1309.7295",
"abstract": "Let G be an abelian group acting on a set X, and suppose that no element of G has any finite orbit of size greater than one. We show that every partial orde... |
https://arxiv.org/abs/1702.08510 | Orthogonal Polynomials and Sharp Estimates for the Schrödinger Equation | In this paper we study sharp estimates for the Schrödinger operator via the framework of orthogonal polynomials. We use spherical harmonics and Gegenbauer polynomials to prove a new weighted inequality for the Schrödinger equation that is maximized by radial functions. We use Hermite and Laguerre polynomial expansions ... | \section{Introduction}
Let $2\leq p,q \leq \infty$. The Strichartz estimate for the Schr\"odinger equation (see \cite[Theorem 2.3]{Tao}) states that there exists a constant $C$ such that
\es{\label{St-est}
\|\|e^{it\Delta}f(\boldsymbol{x})\|_{L^p(\d\boldsymbol{x})}\|_{L^q(\d t)} \leq C \|f(\boldsymbol{x})\|_{L^2(\d\bol... | {
"timestamp": "2017-08-28T02:01:51",
"yymm": "1702",
"arxiv_id": "1702.08510",
"language": "en",
"url": "https://arxiv.org/abs/1702.08510",
"abstract": "In this paper we study sharp estimates for the Schrödinger operator via the framework of orthogonal polynomials. We use spherical harmonics and Gegenbauer... |
https://arxiv.org/abs/1903.11620 | Characterising bimodal collections of sets in finite groups | A collection of disjoint subsets ${\cal A}=\{A_1,A_2,\dotsc,A_m\}$ of a finite abelian group is said to have the \emph{bimodal} property if, for any non-zero group element $\delta$, either $\delta$ never occurs as a difference between an element of $A_i$ and an element of some other set $A_j$, or else for every element... | \section{Introduction}\label{sec:background}
Let $G$ be a finite abelian group of order $n$, written additively, with identity $0$. Let ${\cal A}=\{A_1,A_2,\dotsc,A_m\}$ be a collection of disjoint subsets of $G$. Then $\cal A$ is said to have the {\em bimodal property} if, for any non-identity element $\delta$ of $G... | {
"timestamp": "2019-03-29T01:00:23",
"yymm": "1903",
"arxiv_id": "1903.11620",
"language": "en",
"url": "https://arxiv.org/abs/1903.11620",
"abstract": "A collection of disjoint subsets ${\\cal A}=\\{A_1,A_2,\\dotsc,A_m\\}$ of a finite abelian group is said to have the \\emph{bimodal} property if, for any ... |
https://arxiv.org/abs/1503.00298 | Jamison sequences in countably infinite discrete abelian groups | We extend the definition of Jamison sequences in the context of topological abelian groups. Then we study such sequences when the abelian group is discrete and countably infinite. An arithmetical characterization of such sequences is obtained, extending the result of Badea and Grivaux about Jamison sequences of integer... | \section{Introduction}
To begin, let us recall the original definition of Jamison sequences \cite{BadeaGrivaux1} in the context of bounded linear operators (and $C_0$-semigroups) and some important results around these sequences.
\subsection{Integer Jamison sequences, Jamison sequences for semigroups}
The subject o... | {
"timestamp": "2015-03-03T02:12:59",
"yymm": "1503",
"arxiv_id": "1503.00298",
"language": "en",
"url": "https://arxiv.org/abs/1503.00298",
"abstract": "We extend the definition of Jamison sequences in the context of topological abelian groups. Then we study such sequences when the abelian group is discret... |
https://arxiv.org/abs/1011.1716 | Least Squares Ranking on Graphs | Given a set of alternatives to be ranked, and some pairwise comparison data, ranking is a least squares computation on a graph. The vertices are the alternatives, and the edge values comprise the comparison data. The basic idea is very simple and old: come up with values on vertices such that their differences match th... |
\section{Introduction} \label{sec:intro}
This paper is about ranking of items, of which some pairs have been
compared. The formulation we use (and which we did not invent) leads
to a least squares computation on graphs, and a deeper analysis
requires a second least squares solution. The topology of the graph
plays ... | {
"timestamp": "2011-09-07T02:02:33",
"yymm": "1011",
"arxiv_id": "1011.1716",
"language": "en",
"url": "https://arxiv.org/abs/1011.1716",
"abstract": "Given a set of alternatives to be ranked, and some pairwise comparison data, ranking is a least squares computation on a graph. The vertices are the alterna... |
https://arxiv.org/abs/2209.07010 | Computing Galois groups of Fano problems | A Fano problem consists of enumerating linear spaces of a fixed dimension on a variety, generalizing the classical problem of 27 lines on a cubic surface. Those Fano problems with finitely many linear spaces have an associated Galois group that acts on these linear spaces and controls the complexity of computing them i... | \section{Introduction}
The classical problem of 27 lines on a smooth cubic surface in $\mathbb{P}^3$ is one of the first examples of a Fano problem: enumerating $r$--planes lying on a variety $X$. The family of $r$--planes on $X$ is a subscheme of the Grassmanian called the Fano scheme of $X$. Debarre and Manivel studi... | {
"timestamp": "2022-09-16T02:06:32",
"yymm": "2209",
"arxiv_id": "2209.07010",
"language": "en",
"url": "https://arxiv.org/abs/2209.07010",
"abstract": "A Fano problem consists of enumerating linear spaces of a fixed dimension on a variety, generalizing the classical problem of 27 lines on a cubic surface.... |
https://arxiv.org/abs/1604.03004 | Aliquot sequences with small starting values | We describe the results of the computation of aliquot sequences with small starting values. In particular all sequences with starting values less than a million have been computed until either termination occurred (at 1 or a cycle), or an entry of 100 decimal digits was encountered. All dependencies were recorded, and ... | \section{Introduction}
\noindent
{\it Aliquot sequences} arise from iterating the sum-of-proper-divisors function
$$s(n)=\sum_{\genfrac{}{}{0pt}{}{d\vert n}{d<n}}d,$$
assigning to an integer $n>1$ the sum of its {\it aliquot} divisors
(that is, excluding $n$ itself).
Iteration is denoted exponentially, so $s^k$ is sho... | {
"timestamp": "2016-04-12T02:19:12",
"yymm": "1604",
"arxiv_id": "1604.03004",
"language": "en",
"url": "https://arxiv.org/abs/1604.03004",
"abstract": "We describe the results of the computation of aliquot sequences with small starting values. In particular all sequences with starting values less than a m... |
https://arxiv.org/abs/1408.3518 | On Augmentation Algorithms for Linear and Integer-Linear Programming: From Edmonds-Karp to Bland and Beyond | Motivated by Bland's linear-programming generalization of the renowned Edmonds-Karp efficient refinement of the Ford-Fulkerson maximum-flow algorithm, we discuss three closely-related natural augmentation rules for linear and integer-linear optimization. In several nice situations, we show that polynomially-many augmen... | \section{Introduction.}\label{intro}
\section{Introduction.}
We consider a general framework for solving linear programs (LPs) and integer-linear programs (ILPs) of the form
\begin{equation} \label{themilp}
\min\set{\,\vecc^\T\vex\ :\ A\vex=\veb,\ \ve 0\leq\vex\leq\veu,\ \vex\in X\,},
\end{equation}
where $A\in{\ma... | {
"timestamp": "2015-01-16T02:15:50",
"yymm": "1408",
"arxiv_id": "1408.3518",
"language": "en",
"url": "https://arxiv.org/abs/1408.3518",
"abstract": "Motivated by Bland's linear-programming generalization of the renowned Edmonds-Karp efficient refinement of the Ford-Fulkerson maximum-flow algorithm, we di... |
https://arxiv.org/abs/1608.04079 | Twisted Centralizer Codes | Given an $n\times n$ matrix $A$ over a field $F$ and a scalar $a\in F$, we consider the linear codes $C(A,a):=\{B\in F^{n\times n}\mid \,AB=aBA\}$ of length $n^2$. We call $C(A,a)$ a twisted centralizer code. We investigate properties of these codes including their dimensions, minimum distances, parity-check matrices, ... | \section{Introduction}
Denote the $n\times n$ matrices over a field $F$ by $F^{n\times n}$.
Fix a matrix $A\in F^{n\times n}$ and a scalar $a\in F$. As we are motivated by applications to coding theory we focus on
the case where $F$ is a finite field $\mathbb{F}_q$ of order $q.$ The \emph{centralizer of $A$, twisted by... | {
"timestamp": "2017-03-14T01:07:27",
"yymm": "1608",
"arxiv_id": "1608.04079",
"language": "en",
"url": "https://arxiv.org/abs/1608.04079",
"abstract": "Given an $n\\times n$ matrix $A$ over a field $F$ and a scalar $a\\in F$, we consider the linear codes $C(A,a):=\\{B\\in F^{n\\times n}\\mid \\,AB=aBA\\}$... |
https://arxiv.org/abs/1812.03214 | On the lengths of $t$-based confidence intervals | Given $n=mk$ $iid$ samples from $N(\theta,\sigma^2)$ with $\theta$ and $\sigma^2$ unknown, we have two ways to construct $t$-based confidence intervals for $\theta$. The traditional method is to treat these $n$ samples as $n$ groups and calculate the intervals. The second, and less frequently used, method is to divide ... | \section{Introduction}\label{sec:intro}
In this paper, we consider $t$-based confidence intervals for mean parameter $\theta$ under normal cases. Suppose we have a group of variables $X_1,\cdots,X_n$, which are $iid$ $N(\theta,\sigma^2)$ random variables, where $\theta$ and $\sigma^2$ are both unknown, and the confide... | {
"timestamp": "2018-12-11T02:00:55",
"yymm": "1812",
"arxiv_id": "1812.03214",
"language": "en",
"url": "https://arxiv.org/abs/1812.03214",
"abstract": "Given $n=mk$ $iid$ samples from $N(\\theta,\\sigma^2)$ with $\\theta$ and $\\sigma^2$ unknown, we have two ways to construct $t$-based confidence interval... |
https://arxiv.org/abs/2207.00051 | Reflective numerical semigroups | We define a reflective numerical semigroup of genus $g$ as a numerical semigroup that has a certain reflective symmetry when viewed within $\mathbb{Z}$ as an array with $g$ columns. Equivalently, a reflective numerical semigroup has one gap in each residue class modulo $g$. In this paper, we give an explicit descriptio... |
\section{Introduction}
Let $\N_0$ denote the set of non-negative integers, a monoid under addition, and let $\N$ denote the set of positive integers. A numerical semigroup $S$ is a submonoid of $\N_0$ with finite complement. We denote the complement by $\HH(S)=\N_0\setminus S$. Elements of $\HH(S)$ are called gaps of ... | {
"timestamp": "2022-07-04T02:01:16",
"yymm": "2207",
"arxiv_id": "2207.00051",
"language": "en",
"url": "https://arxiv.org/abs/2207.00051",
"abstract": "We define a reflective numerical semigroup of genus $g$ as a numerical semigroup that has a certain reflective symmetry when viewed within $\\mathbb{Z}$ a... |
https://arxiv.org/abs/1912.09538 | Maximal edge colorings of graphs | For a graph $G$ of order $n$ a maximal edge coloring is a proper edge coloring with $\chi'(K_n)$ colors such that adding any edge to $G$ in any color makes it improper. Meszka and Tyniec proved that for some values of the number of edges there are no graphs with a maximal edge coloring, while for some other values, the... | \section{Introduction}
The problems associated with maximality of a family of some objects are widely considered in Combinatorics.
When we have a subset of the space of objects that satisfies some fixed properties we say it is \textit{maximal} when adding to this set any new element from the space results in violatin... | {
"timestamp": "2019-12-23T02:02:00",
"yymm": "1912",
"arxiv_id": "1912.09538",
"language": "en",
"url": "https://arxiv.org/abs/1912.09538",
"abstract": "For a graph $G$ of order $n$ a maximal edge coloring is a proper edge coloring with $\\chi'(K_n)$ colors such that adding any edge to $G$ in any color mak... |
https://arxiv.org/abs/dg-ga/9510008 | Theorem on six vertices of a plane curve via the Sturm theory | We discuss the theorem on the existence of six points on a convex closed plane curve in which the curve has a contact of order six with the osculating conic. (This is the ``projective version'' of the well known four vertices theorem for a curve in the Euclidean plane.) We obtain this classical fact as a corollary of s... | \section{introduction}
The well known classical theorem states that a convex curve
on the Euclidean plane has
at least four vertices (critical points of its curvature). This theorem
has been frequently discussed in mathematical literature (see
\cite{arn1,tab}). Beautiful applications of this theorem to symplectic
geo... | {
"timestamp": "1995-10-26T15:27:26",
"yymm": "9510",
"arxiv_id": "dg-ga/9510008",
"language": "en",
"url": "https://arxiv.org/abs/dg-ga/9510008",
"abstract": "We discuss the theorem on the existence of six points on a convex closed plane curve in which the curve has a contact of order six with the osculati... |
https://arxiv.org/abs/1012.4017 | A Note on Solid Coloring of Pure Simplicial Complexes | We establish a simple generalization of a known result in the plane. The simplices in any pure simplicial complex in R^d may be colored with d+1 colors so that no two simplices that share a (d-1)-facet have the same color. In R^2 this says that any planar map all of whose faces are triangles may be 3-colored, and in R^... | \section{Introduction}
\seclab{Introduction}
The famous 4-color theorem says that the regions of any planar map
may be colored with four colors such that no two regions that share a
positive-length border receive the same color.
A lesser-known special case is that if all the regions are triangles,
three colors suffice.... | {
"timestamp": "2010-12-21T02:00:10",
"yymm": "1012",
"arxiv_id": "1012.4017",
"language": "en",
"url": "https://arxiv.org/abs/1012.4017",
"abstract": "We establish a simple generalization of a known result in the plane. The simplices in any pure simplicial complex in R^d may be colored with d+1 colors so t... |
https://arxiv.org/abs/1902.09302 | Configuration Models of Random Hypergraphs | Many empirical networks are intrinsically polyadic, with interactions occurring within groups of agents of arbitrary size. There are, however, few flexible null models that can support statistical inference for such polyadic networks. We define a class of null random hypergraphs that hold constant both the node degree ... | \subsection*{Outline}
We begin in \Cref{sec:lit_review} with a survey of the landscape of null models for relational data, including the dyadic configuration model, random hypergraphs, and random simplicial complexes.
In \Cref{sec:model_def}, we define stub- and vertex-labeled configuration models of random hyperg... | {
"timestamp": "2019-12-17T02:00:51",
"yymm": "1902",
"arxiv_id": "1902.09302",
"language": "en",
"url": "https://arxiv.org/abs/1902.09302",
"abstract": "Many empirical networks are intrinsically polyadic, with interactions occurring within groups of agents of arbitrary size. There are, however, few flexibl... |
https://arxiv.org/abs/1605.06360 | Eigenvalues of subgraphs of the cube | We consider the problem of maximising the largest eigenvalue of subgraphs of the hypercube $Q_d$ of a given order. We believe that in most cases, Hamming balls are maximisers, and our results support this belief. We show that the Hamming balls of radius $o(d)$ have largest eigenvalue that is within $1 + o(1)$ of the ma... | \section{Introduction}
In the last few decades much research has been done on spectra of graphs,
i.e.~the eigenvalues of the adjacency matrices of graphs; see Finck and
Grohmann~\cite{FG}, Hoffman~\cite{hof1, hof2}, Nosal~\cite{nos},
Cvetkovi\'c, Doob and Sachs~\cite{CDS}, Neumaier~\cite{Neum}, Brigha... | {
"timestamp": "2016-05-23T02:10:33",
"yymm": "1605",
"arxiv_id": "1605.06360",
"language": "en",
"url": "https://arxiv.org/abs/1605.06360",
"abstract": "We consider the problem of maximising the largest eigenvalue of subgraphs of the hypercube $Q_d$ of a given order. We believe that in most cases, Hamming ... |
https://arxiv.org/abs/1702.03114 | Locality of the heat kernel on metric measure spaces | We will discuss what it means for a general heat kernel on a metric measure space to be local. We show that the Wiener measure associated to Brownian motion is local. Next we show that locality of the Wiener measure plus a suitable decay bound of the heat kernel implies locality of the heat kernel. We define a class of... | \section{Introduction and motivation}
\label{sec:intro}
\nocite{AGS13}
This article is about the well-known statement: `The heat kernel is
local'. We will discuss what this means exactly and when it
holds. This statement first appeared around 1950, for example in the
works of~\cite{MinakshisundaramPleijel49}. They st... | {
"timestamp": "2017-11-08T02:06:24",
"yymm": "1702",
"arxiv_id": "1702.03114",
"language": "en",
"url": "https://arxiv.org/abs/1702.03114",
"abstract": "We will discuss what it means for a general heat kernel on a metric measure space to be local. We show that the Wiener measure associated to Brownian moti... |
https://arxiv.org/abs/2301.06131 | Volume Product | Our purpose here is to give an overview of known results and open questions concerning the volume product ${\mathcal P}(K)=\min_{z\in K}{\rm vol}(K){\rm vol}((K-z)^*)$ of a convex body $K$ in ${\mathbb R}^n$. We present a number of upper and lower bounds for ${\mathcal P}(K)$, in particular, we discuss the Mahler's con... | \section{Introduction}
More or less attached to the name of Kurt Mahler (1903-1988), there are at least two celebrated unsolved problems:
\begin{itemize}
\item Lehmer's problem on algebraic numbers
\item The lower bound for the volume product of convex bodies.
\end{itemize}
The celebrity of these two problems co... | {
"timestamp": "2023-01-18T02:13:38",
"yymm": "2301",
"arxiv_id": "2301.06131",
"language": "en",
"url": "https://arxiv.org/abs/2301.06131",
"abstract": "Our purpose here is to give an overview of known results and open questions concerning the volume product ${\\mathcal P}(K)=\\min_{z\\in K}{\\rm vol}(K){\... |
https://arxiv.org/abs/1806.11359 | A note on polynomial sequences modulo integers | We study the uniform distribution of the polynomial sequence $\lambda(P)=(\lfloor P(k) \rfloor )_{k\geq 1}$ modulo integers, where $P(x)$ is a polynomial with real coefficients. In the nonlinear case, we show that $\lambda(P)$ is uniformly distributed in $\mathbb{Z}$ if and only if $P(x)$ has at least one irrational co... | \section{Introduction}
A sequence $(r_k)_{k\geq 1}$ of real numbers is said to be u.d.\ mod 1 if for all $0\leq a<b <1$,
$$\lim_{N \rightarrow \infty}\dfrac{1}{N} \#\{k \in \{1,\ldots, N\}: a\leq \{r_k\} \leq b\}=b-a,$$
where $\{r_k\}$ denotes the fractional part of $r_k$.
An integer sequence $(a_k)_{k\geq 1}$ is s... | {
"timestamp": "2018-12-18T02:31:20",
"yymm": "1806",
"arxiv_id": "1806.11359",
"language": "en",
"url": "https://arxiv.org/abs/1806.11359",
"abstract": "We study the uniform distribution of the polynomial sequence $\\lambda(P)=(\\lfloor P(k) \\rfloor )_{k\\geq 1}$ modulo integers, where $P(x)$ is a polynom... |
https://arxiv.org/abs/0805.0419 | Normal and Anomalous Diffusion: A Tutorial | The purpose of this tutorial is to introduce the main concepts behind normal and anomalous diffusion. Starting from simple, but well known experiments, a series of mathematical modeling tools are introduced, and the relation between them is made clear. First, we show how Brownian motion can be understood in terms of a ... | \section{Introduction}
\label{intro}
The art of doing research in physics usually starts with the observation of
a natural
phenomenon. Then follows a qualitative idea on "How the phenomenon can be
interpreted",
and one proceeds with the construction of a model equation or a simulation,
with the aim that it resembles... | {
"timestamp": "2008-05-05T20:33:49",
"yymm": "0805",
"arxiv_id": "0805.0419",
"language": "en",
"url": "https://arxiv.org/abs/0805.0419",
"abstract": "The purpose of this tutorial is to introduce the main concepts behind normal and anomalous diffusion. Starting from simple, but well known experiments, a se... |
https://arxiv.org/abs/2106.10233 | True-pairs of Real Linear Operators and Factorization of Real Polynomials | A linear operator on a finite dimensional nonzero real vector space may not have an eigenvalue. We define a related notion of a true-pair of a linear operator, and then show that each linear operator on a finite dimensional nonzero real vector space has a true-pair. This is usually proved by using the Fundamental theor... | \section{Introduction}
Let $V$ be a real vector space of dimension $n\geq 1$. Let $T:V\to V$ be a linear operator. Recall that a real number $\lambda$ is called an eigenvalue of $T$ if there exists a nonzero vector $u\in V$ such that $Tu=\lambda u$. In this case, any such vector $u$ is called an eigenvector. The polyno... | {
"timestamp": "2021-06-21T02:22:20",
"yymm": "2106",
"arxiv_id": "2106.10233",
"language": "en",
"url": "https://arxiv.org/abs/2106.10233",
"abstract": "A linear operator on a finite dimensional nonzero real vector space may not have an eigenvalue. We define a related notion of a true-pair of a linear oper... |
https://arxiv.org/abs/1503.01865 | Hyperbolic geometry in the work of Johann Heinrich Lambert | The memoir Theorie der Parallellinien (1766) by Johann Heinrich Lambert is one of the founding texts of hyperbolic geometry, even though its author's aim was, like many of his pre-decessors', to prove that such a geometry does not exist. In fact, Lambert developed his theory with the hope of finding a contradiction in ... | \section{Introduction}
The first published treatise on hyperbolic geometry is Lobachevsky's \emph{Elements of geometry} \cite{Loba-Elements}, printed in installments in the \emph{Kazan Messenger} in the years 1829-1830. Before that article, Lobachevsky wrote a memoir on the same subject, which he presented on the 1... | {
"timestamp": "2015-03-09T01:05:55",
"yymm": "1503",
"arxiv_id": "1503.01865",
"language": "en",
"url": "https://arxiv.org/abs/1503.01865",
"abstract": "The memoir Theorie der Parallellinien (1766) by Johann Heinrich Lambert is one of the founding texts of hyperbolic geometry, even though its author's aim ... |
https://arxiv.org/abs/2003.09618 | Backward Error Measures for Roots of Polynomials | We analyze different measures for the backward error of a set of numerical approximations for the roots of a polynomial. We focus mainly on the element-wise mixed backward error introduced by Mastronardi and Van Dooren, and the tropical backward error introduced by Tisseur and Van Barel. We show that these measures are... |
\section{Introduction}
In this article we analyze the problem of measuring the backward error for a set of approximations for the roots of a polynomial with complex coefficients. For a general introduction to the notion of backward error analysis, the reader can consult for instance \cite[Section 1.5]{Higham:2002}. C... | {
"timestamp": "2020-03-24T01:06:31",
"yymm": "2003",
"arxiv_id": "2003.09618",
"language": "en",
"url": "https://arxiv.org/abs/2003.09618",
"abstract": "We analyze different measures for the backward error of a set of numerical approximations for the roots of a polynomial. We focus mainly on the element-wi... |
https://arxiv.org/abs/2010.01155 | Representational aspects of depth and conditioning in normalizing flows | Normalizing flows are among the most popular paradigms in generative modeling, especially for images, primarily because we can efficiently evaluate the likelihood of a data point. This is desirable both for evaluating the fit of a model, and for ease of training, as maximizing the likelihood can be done by gradient des... |
\section{Appendix}
\section{Conclusion}
In this paper, we initiated a theoretical study of representation aspects of flow models, providing separations with non-invertible models like GANs, and studied the limitations of a popular architecture based on affine coupling blocks.
Given the popularity of both families... | {
"timestamp": "2021-06-29T02:06:09",
"yymm": "2010",
"arxiv_id": "2010.01155",
"language": "en",
"url": "https://arxiv.org/abs/2010.01155",
"abstract": "Normalizing flows are among the most popular paradigms in generative modeling, especially for images, primarily because we can efficiently evaluate the li... |
https://arxiv.org/abs/1106.5305 | The Theory of the Interleaving Distance on Multidimensional Persistence Modules | In 2009, Chazal et al. introduced $\epsilon$-interleavings of persistence modules. $\epsilon$-interleavings induce a pseudometric $d_I$ on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of $\epsilon$-interleavings and $d_I$ generalize readily to multidimensional persistence mod... | \subsection*{Acknowledgments}
The first version of this paper \cite{lesnick2011optimalityV2} was written while I was a graduate student at Stanford.
My discussions with my Ph.D. adviser Gunnar Carlsson catalyzed the research presented here in several key ways. In our many conversations about multidimensional persis... | {
"timestamp": "2013-02-06T02:04:13",
"yymm": "1106",
"arxiv_id": "1106.5305",
"language": "en",
"url": "https://arxiv.org/abs/1106.5305",
"abstract": "In 2009, Chazal et al. introduced $\\epsilon$-interleavings of persistence modules. $\\epsilon$-interleavings induce a pseudometric $d_I$ on (isomorphism cl... |
https://arxiv.org/abs/2109.13688 | Square roots of some classical operators | In this paper we give complete descriptions of the set of square roots of certain classical operators, often providing specific formulas. The classical operators included in this discussion are the square of the unilateral shift, the Volterra operator, certain compressed shifts, the unilateral shift plus its adjoint, t... | \section{Introduction}
Let $\mathcal{H}$ be a Hilbert space, and let $\mathcal{B}(\mathcal{H})$ denote the space of all bounded operators on $\mathcal{H}$. We say that $A \in \mathcal{B}(\mathcal{H})$ has a square root in $\mathcal{B}(\mathcal{H})$ if there is $B \in \mathcal{B}(\mathcal{H})$ such that $A=B^2$. In the ... | {
"timestamp": "2021-09-29T02:18:29",
"yymm": "2109",
"arxiv_id": "2109.13688",
"language": "en",
"url": "https://arxiv.org/abs/2109.13688",
"abstract": "In this paper we give complete descriptions of the set of square roots of certain classical operators, often providing specific formulas. The classical op... |
https://arxiv.org/abs/1904.09046 | Direct Synthesis of Iterative Algorithms With Bounds on Achievable Worst-Case Convergence Rate | Iterative first-order methods such as gradient descent and its variants are widely used for solving optimization and machine learning problems. There has been recent interest in analytic or numerically efficient methods for computing worst-case performance bounds for such algorithms, for example over the class of stron... | \section{Introduction}\label{sec:intro}
First-order methods are a widely used class of iterative algorithms for solving optimization problems of the form
$\min_{x\in\R^n} f(x)$,
where $f$ is continuously differentiable and we only have access to first-order (gradient) measurements of $f$.
An example of a first-order m... | {
"timestamp": "2019-04-22T02:07:00",
"yymm": "1904",
"arxiv_id": "1904.09046",
"language": "en",
"url": "https://arxiv.org/abs/1904.09046",
"abstract": "Iterative first-order methods such as gradient descent and its variants are widely used for solving optimization and machine learning problems. There has ... |
https://arxiv.org/abs/1403.4741 | Diameter 2 Cayley Graphs of Dihedral Groups | We consider the degree-diameter problem for Cayley graphs of dihedral groups. We find upper and lower bounds on the maximum number of vertices of such a graph with diameter 2 and degree $d$. We completely determine the asymptotic behaviour of this class of graphs by showing that both limits are asymptotically $d^2/2$. | \section{Introduction}
The \emph{degree-diameter problem} seeks to determine the largest possible number $n(d,k)$ of vertices of a graph of maximum
degree $d$ and diameter $k$. The survey \cite{miller2005moore} summarises current known results for the general case and also for various restricted problems
where we co... | {
"timestamp": "2015-02-17T02:09:32",
"yymm": "1403",
"arxiv_id": "1403.4741",
"language": "en",
"url": "https://arxiv.org/abs/1403.4741",
"abstract": "We consider the degree-diameter problem for Cayley graphs of dihedral groups. We find upper and lower bounds on the maximum number of vertices of such a gra... |
https://arxiv.org/abs/0811.1310 | Classification theorems for sumsets modulo a prime | Let $\Z/pZ$ be the finite field of prime order $p$ and $A$ be a subsequence of $\Z/pZ$. We prove several classification results about the following questions: (1) When can one represent zero as a sum of some elements of $A$ ? (2) When can one represent every element of $\Z/pZ$ as a sum of some elements of $A$ ? (3) Whe... | \section{Introduction.}
\noindent Let $G$ be an additive group and $A$ be a sequence of (not
necessarily different) elements of $G$. We denote by $S_A$ the
collection of partial sums of $A$
$$
\sum(A) := \left \{ \sum_{x\in B}x \ |\ \emptyset \neq B\subset A,
|B| < \infty \right \}.
$$
\noindent For a positive integ... | {
"timestamp": "2009-01-27T05:46:35",
"yymm": "0811",
"arxiv_id": "0811.1310",
"language": "en",
"url": "https://arxiv.org/abs/0811.1310",
"abstract": "Let $\\Z/pZ$ be the finite field of prime order $p$ and $A$ be a subsequence of $\\Z/pZ$. We prove several classification results about the following questi... |
https://arxiv.org/abs/2103.04707 | Iterating the RSK Bijection | We investigate the dynamics of the well-known RSK bijection on permutations when iterated on various reading words of the recording tableau. In the setting of the ordinary (row) reading word, we show that there is exactly one fixed point per partition shape, and that it is always reached within two steps from any start... | \section{Introduction}
A \textbf{permutation} of $1,2,\ldots,n$ is a rearrangement of the numbers $1,2,3,\ldots,n$ in some order. A \textbf{standard Young tableau}, or SYT, is a way of writing the numbers $1,2,\ldots,n$ in the unit squares in the first quadrant such that the rows are left-aligned and increasing from ... | {
"timestamp": "2021-03-09T02:38:19",
"yymm": "2103",
"arxiv_id": "2103.04707",
"language": "en",
"url": "https://arxiv.org/abs/2103.04707",
"abstract": "We investigate the dynamics of the well-known RSK bijection on permutations when iterated on various reading words of the recording tableau. In the settin... |
https://arxiv.org/abs/2012.07302 | Torsion of digraphs and path complexes | We define the notions of Reidemeister torsion and analytic torsion for directed graphs by means of the path homology theory introduced by the authors in \cite{Grigoryan-Lin-Muranov-Yau2013, Grigoryan-Lin-Muranov-Yau2014, Grigoryan-Lin-Muranov-Yau2015, Grigoryan-Lin-Muranov-Yau2020}. We prove the identity of the two not... |
\section{Introduction}
\label{S:intr} \setcounter{equation}{0}Let $M$ be a compact oriented
Riemannian manifold. Assume that $M$ is triangulated by a simplicial complex
$K$. Let $\rho $ be a acyclic representation of $\pi _{1}(K)$ by orthogonal
matrices, i.e., the twisted cohomology group $H^{p}(K;\rho )$ is tr... | {
"timestamp": "2020-12-15T02:31:17",
"yymm": "2012",
"arxiv_id": "2012.07302",
"language": "en",
"url": "https://arxiv.org/abs/2012.07302",
"abstract": "We define the notions of Reidemeister torsion and analytic torsion for directed graphs by means of the path homology theory introduced by the authors in \... |
https://arxiv.org/abs/1202.5484 | Which finitely generated Abelian groups admit isomorphic Cayley graphs? | We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality. The proof uses only elementary arguments and is... | \section{Introduction}
Cayley graphs allow us to view groups as combinatorial and geometric
objects. For instance, one of the main objectives of geometric group
theory is to understand the relation between algebraic properties of
finitely generated groups and (large scale) geometric properties
of their Cayley graphs. ... | {
"timestamp": "2012-07-23T02:01:36",
"yymm": "1202",
"arxiv_id": "1202.5484",
"language": "en",
"url": "https://arxiv.org/abs/1202.5484",
"abstract": "We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admi... |
https://arxiv.org/abs/1603.00945 | High-Order, Stable, And Efficient Pseudospectral Method Using Barycentric Gegenbauer Quadratures | The work reported in this article presents a high-order, stable, and efficient Gegenbauer pseudospectral method to solve numerically a wide variety of mathematical models. The proposed numerical scheme exploits the stability and the well-conditioning of the numerical integration operators to produce well-conditioned sy... | \section{Introduction}
\label{int}
The past few decades have seen a conspicuous attention towards the solution of differential problems by working on their integral reformulations; cf. \cite{Elgindy2009,Elgindy2013a,Franccolin2014,Elgindy2012d,Tang2015,Coutsias1996,Greengard1991,Viswanath2015,Driscoll2010,Olver2013,El-... | {
"timestamp": "2016-10-28T02:00:35",
"yymm": "1603",
"arxiv_id": "1603.00945",
"language": "en",
"url": "https://arxiv.org/abs/1603.00945",
"abstract": "The work reported in this article presents a high-order, stable, and efficient Gegenbauer pseudospectral method to solve numerically a wide variety of mat... |
https://arxiv.org/abs/1811.02768 | Some special property of Farey sequence | We discuss some special property of the Farey sequence. We show in each term of the Farey sequence, ratio of the sum of elements in the denominator and the sum of elements in the numerator is exactly two. We also show that the Farey sequence contains a palindromic structure. | \section{Introduction}
Farey sequence is named after British geologist John Farey, Sr., who publised a result in Philosophical Magazine in 1816 about these sequence without giving a proof. Later, Cauchy proved the result conjectured by Farey. Though, Charles Haros proved similar result in 1802 which were not known t... | {
"timestamp": "2018-11-08T02:06:39",
"yymm": "1811",
"arxiv_id": "1811.02768",
"language": "en",
"url": "https://arxiv.org/abs/1811.02768",
"abstract": "We discuss some special property of the Farey sequence. We show in each term of the Farey sequence, ratio of the sum of elements in the denominator and th... |
https://arxiv.org/abs/1802.05953 | Weak Dynamic Coloring of Planar Graphs | The \textit{$k$-weak-dynamic number} of a graph $G$ is the smallest number of colors we need to color the vertices of $G$ in such a way that each vertex $v$ of degree $d(v)$ sees at least $\rm{min}\{k,d(v)\}$ colors on its neighborhood. We use reducible configurations and list coloring of graphs to prove that all plana... | \section{Introduction}
A \textit{proper coloring} of $G$ is a vertex coloring of $G$ in which adjacent vertices receive different colors. The \textit{chromatic number} of $G$, written as $\chi(G)$, is the smallest number of colors needed to find a proper coloring of $G$. For notation and definitions not de... | {
"timestamp": "2018-02-19T02:07:49",
"yymm": "1802",
"arxiv_id": "1802.05953",
"language": "en",
"url": "https://arxiv.org/abs/1802.05953",
"abstract": "The \\textit{$k$-weak-dynamic number} of a graph $G$ is the smallest number of colors we need to color the vertices of $G$ in such a way that each vertex ... |
https://arxiv.org/abs/0906.4027 | Directed Simplices In Higher Order Tournaments | It is well known that a tournament (complete oriented graph) on $n$ vertices has at most ${1/4}\binom{n}{3}$ directed triangles, and that the constant 1/4 is best possible. Motivated by some geometric considerations, our aim in this paper is to consider some `higher order' versions of this statement. For example, if we... | \section{Introduction}
A \textit{tournament} is a complete graph in which each edge is assigned a direction. It is well known (see e.g. \cite{moon}) that there are at most $\frac{1}{4}\binom{n}{3}+O(n^2)$ directed triangles in a tournament on $n$ vertices. The constant $\frac{1}{4}$ is easily seen to be best possible,... | {
"timestamp": "2009-06-22T16:58:41",
"yymm": "0906",
"arxiv_id": "0906.4027",
"language": "en",
"url": "https://arxiv.org/abs/0906.4027",
"abstract": "It is well known that a tournament (complete oriented graph) on $n$ vertices has at most ${1/4}\\binom{n}{3}$ directed triangles, and that the constant 1/4 ... |
https://arxiv.org/abs/1401.2691 | The Location of the First Ascent in a 123-Avoiding Permutation | It is natural to ask, given a permutation with no three-term ascending subsequence, at what index the first ascent occurs. We shall show, using both a recursion and a bijection, that the number of 123-avoiding permutations at which the first ascent occurs at positions $k,k+1$ is given by the $k$-fold Catalan convolutio... | \section{Introduction} For $n\ge0$, the Catalan numbers $C_n$ are given by
\[C_n=\frac{1}{n+1}{{2n}\choose{n}};\] generalizing this fact,
Catalan \cite{cata} proved the $k$-fold Catalan convolution formula
\[C_{n,k}:=\sum_{i_1+\ldots+i_k=n}\prod_{r=1}^kC_{i_r-1}=\frac{k}{2n-k}{{2n-k}\choose{n}}.\]
The theory of patter... | {
"timestamp": "2014-01-14T02:11:09",
"yymm": "1401",
"arxiv_id": "1401.2691",
"language": "en",
"url": "https://arxiv.org/abs/1401.2691",
"abstract": "It is natural to ask, given a permutation with no three-term ascending subsequence, at what index the first ascent occurs. We shall show, using both a recur... |
https://arxiv.org/abs/2210.16406 | Gallai's Conjecture for Complete and "Nearly Complete" Graphs | The famous Gallai's Conjecture states that any connected graph with n vertices has a path decomposition containing at most (n+1)/2 paths. In this note, we explore graphs generated from removing edges from complete graphs. We first provide an explicit construction for a path decomposition of complete graphs that satisfi... | \section{Introduction}\label{sec:definition}
One of the primary areas of study in graph theory is the decomposition of graphs. We say that a set of subgraphs $S = \{ S_1, S_2, \dots S_n \}$ is a \textit{decomposition} of the original graph $G$ if every edge in $G$ is in exactly one of the subgraphs in $S$. Researcher... | {
"timestamp": "2022-11-01T01:03:17",
"yymm": "2210",
"arxiv_id": "2210.16406",
"language": "en",
"url": "https://arxiv.org/abs/2210.16406",
"abstract": "The famous Gallai's Conjecture states that any connected graph with n vertices has a path decomposition containing at most (n+1)/2 paths. In this note, we... |
https://arxiv.org/abs/1912.02297 | A Densest ternary circle packing in the plane | We consider circle packings in the plane with circles of sizes $1$, $r\simeq 0.834$ and $s\simeq 0.651$. These sizes are algebraic numbers which allow a compact packing, that is, a packing in which each hole is formed by three mutually tangent circles. Compact packings are believed to maximize the density when there ar... | \section{Introduction}
A {\em circle packing} in the plane is a set of interior-disjoint circles.
Its {\em density} $\delta$ is defined by
\begin{displaymath}
\delta:=\limsup_{k\to \infty}\frac{\textrm{area in the square $[-k,k]^2$ in the circle interiors}}{\textrm{area of the square $[-k,k]^2$}}.
\end{displaymath}
A ... | {
"timestamp": "2019-12-06T02:04:52",
"yymm": "1912",
"arxiv_id": "1912.02297",
"language": "en",
"url": "https://arxiv.org/abs/1912.02297",
"abstract": "We consider circle packings in the plane with circles of sizes $1$, $r\\simeq 0.834$ and $s\\simeq 0.651$. These sizes are algebraic numbers which allow a... |
https://arxiv.org/abs/2002.09995 | Counting independent sets in regular hypergraphs | Amongst $d$-regular $r$-uniform hypergraphs on $n$ vertices, which ones have the largest number of independent sets? While the analogous problem for graphs (originally raised by Granville) is now well-understood, it is not even clear what the correct general conjecture ought to be; our goal here is propose such a gener... | \section{Introduction}
This paper concerns the hypergraph analogue of an old (and now resolved) graph-theoretic problem of Granville (see~\citep{Alon}). Granville raised the following problem: which $d$-regular graphs on $n$ vertices have the maximum number of independent sets? This problem was also considered by Kahn~... | {
"timestamp": "2020-02-25T02:19:20",
"yymm": "2002",
"arxiv_id": "2002.09995",
"language": "en",
"url": "https://arxiv.org/abs/2002.09995",
"abstract": "Amongst $d$-regular $r$-uniform hypergraphs on $n$ vertices, which ones have the largest number of independent sets? While the analogous problem for graph... |
https://arxiv.org/abs/2103.14284 | On Co-Maximal Subgroup Graph of a Group | The co-maximal subgroup graph $\Gamma(G)$ of a group $G$ is a graph whose vertices are non-trivial proper subgroups of $G$ and two vertices $H$ and $K$ are adjacent if $HK=G$. In this paper, we continue the study of $\Gamma(G)$, especially when $\Gamma(G)$ has isolated vertices. We define a new graph $\Gamma^*(G)$, whi... | \section{Introduction}
The idea of associating graphs with groups, which started from Cayley graphs, is now an important topic of research in modern algebraic graph theory. The most prominent graphs defined on groups in recent years are power graphs \cite{Cameron-Ghosh}, enhanced power graphs \cite{enhanced-power-graph... | {
"timestamp": "2021-04-13T02:20:11",
"yymm": "2103",
"arxiv_id": "2103.14284",
"language": "en",
"url": "https://arxiv.org/abs/2103.14284",
"abstract": "The co-maximal subgroup graph $\\Gamma(G)$ of a group $G$ is a graph whose vertices are non-trivial proper subgroups of $G$ and two vertices $H$ and $K$ a... |
https://arxiv.org/abs/1701.08155 | The Moser's formula for the division of the circle by chords problem revisited | The enumeration of the regions formed when circle is divided by secants drawn from points on the circle is one of the examples where the inductive reasoning fails as was pointed out by Leo Moser in the Mathematical Miscellany in 1949. The formula that gives the right number of regions can be deduced by combinatorics re... | \section{Introduction}
A problem sometimes known as Moser's circle problem asks to determine the number of pieces into which a circle is divided if $m$ points on its circumference are joined by chords with no three internally concurrent.
The number of regions formed inside the circle when it is divided by the chords ... | {
"timestamp": "2017-01-31T02:00:08",
"yymm": "1701",
"arxiv_id": "1701.08155",
"language": "en",
"url": "https://arxiv.org/abs/1701.08155",
"abstract": "The enumeration of the regions formed when circle is divided by secants drawn from points on the circle is one of the examples where the inductive reasoni... |
https://arxiv.org/abs/1307.6233 | Comparing skew Schur functions: a quasisymmetric perspective | Reiner, Shaw and van Willigenburg showed that if two skew Schur functions s_A and s_B are equal, then the skew shapes A and B must have the same "row overlap partitions." Here we show that these row overlap equalities are also implied by a much weaker condition than skew Schur equality: that s_A and s_B have the same s... | \section{Introduction}
For well-documented reasons (see, for example, \cite{Ful97, Ful00, Sag01, ec2}), the Schur functions $s_\lambda$ are often considered to be the most important basis for symmetric functions. Furthermore, skew Schur functions $s_{\lambda/\mu}$ are both a natural generalization of Schur functions ... | {
"timestamp": "2014-01-17T02:12:32",
"yymm": "1307",
"arxiv_id": "1307.6233",
"language": "en",
"url": "https://arxiv.org/abs/1307.6233",
"abstract": "Reiner, Shaw and van Willigenburg showed that if two skew Schur functions s_A and s_B are equal, then the skew shapes A and B must have the same \"row overl... |
https://arxiv.org/abs/1609.08801 | On Notions of Distortion and an Almost Minimum Spanning Tree with Constant Average Distortion | This paper makes two main contributions: The first is the construction of a near-minimum spanning tree with constant average distortion. The second is a general equivalence theorem relating two refined notions of distortion: scaling distortion and prioritized distortion.Minimum Spanning Trees of weighted graphs are fun... | \section{Introduction}
One of the fundamental problems in graph theory is that of constructing a Minimum Spanning Tree (MST) of a given weighted graph $G=(V,E)$. This problem and its variants received much attention, and has found numerous applications. In many of these applications, one may desire not only minimizi... | {
"timestamp": "2016-09-29T02:03:18",
"yymm": "1609",
"arxiv_id": "1609.08801",
"language": "en",
"url": "https://arxiv.org/abs/1609.08801",
"abstract": "This paper makes two main contributions: The first is the construction of a near-minimum spanning tree with constant average distortion. The second is a g... |
https://arxiv.org/abs/0807.1581 | On the Completeness of Gradient Ricci Solitons | A gradient Ricci soliton is a triple $(M,g,f)$ satisfying $R_{ij}+\nabla_i\nabla_j f=\lambda g_{ij}$ for some real number $\lambda$. In this paper, we will show that the completeness of the metric $g$ implies that of the vector field $\nabla f$. | \section*{This is an unnumbered first-level section head}
\section{Introduction}
\begin{definition}
Let $(M,g,X)$ be a smooth Riemannian manifold with $X$ a smooth
vector field. We call $M$ a Ricci soliton if
$Ric+\frac{1}{2}{\mathcal{L}}_X g={\lambda}g$ for some real number
$\lambda$. It is called shrinking when $... | {
"timestamp": "2008-10-22T06:55:13",
"yymm": "0807",
"arxiv_id": "0807.1581",
"language": "en",
"url": "https://arxiv.org/abs/0807.1581",
"abstract": "A gradient Ricci soliton is a triple $(M,g,f)$ satisfying $R_{ij}+\\nabla_i\\nabla_j f=\\lambda g_{ij}$ for some real number $\\lambda$. In this paper, we w... |
https://arxiv.org/abs/2007.10251 | Capacitance matrix revisited | The capacitance matrix relates potentials and charges on a system of conductors. We review and rigorously generalize its properties, block-diagonal structure and inequalities, deduced from the geometry of system of conductors and analytic properties of the permittivity tensor. Furthermore, we discuss alternative choice... | \section{Introduction}
One of the fundamental problems of electrostatics considers the system of charged ideal conductors. In a basic setting one either assigns the potentials on each of the conductors and asks for a total charge on each of them or, vice versa, asks for potentials if the charges are known. Since Maxwe... | {
"timestamp": "2021-03-12T02:25:58",
"yymm": "2007",
"arxiv_id": "2007.10251",
"language": "en",
"url": "https://arxiv.org/abs/2007.10251",
"abstract": "The capacitance matrix relates potentials and charges on a system of conductors. We review and rigorously generalize its properties, block-diagonal struct... |
https://arxiv.org/abs/2008.04937 | Combinatorics of Multicompositions | Integer compositions with certain colored parts were introduced by Andrews in 2007 to address a number-theoretic problem. Integer compositions allowing zero as some parts were introduced by Ouvry and Polychronakos in 2019. We give a bijection between these two varieties of compositions and determine various combinatori... | \section{Introduction}
A composition of a positive integer $n$ is an ordered collection of positive integers whose sum is $n$. For instance, there are four compositions of 3: $1+1+1$, $1+2$, $2+1$, and $3$. We refer to the summands as parts.
The next section begins with the formal definition of multicompositions t... | {
"timestamp": "2020-08-13T02:00:53",
"yymm": "2008",
"arxiv_id": "2008.04937",
"language": "en",
"url": "https://arxiv.org/abs/2008.04937",
"abstract": "Integer compositions with certain colored parts were introduced by Andrews in 2007 to address a number-theoretic problem. Integer compositions allowing ze... |
https://arxiv.org/abs/1510.08211 | On commuting probability of finite rings | The commuting probability of a finite ring $R$, denoted by $\Pr(R)$, is the probability that any two randomly chosen elements of $R$ commute. In this paper, we obtain several bounds for $\Pr(R)$ through a generalization of $\Pr(R)$. Further, we define ${\Z}$-isoclinism between two pairs of rings and show that the gener... | \section{Introduction}
Throughout this paper $S$ denotes a subring of a finite ring $R$.
We define $Z(S, R) = \{s \in S : sr = rs \; \forall \; r \in R\}$. Note that $Z(R, R) = Z(R)$, the center of $R$, and $Z(S, R) = Z(R) \cap S$.
For any two elements $s$ and $r$ of a ring $R$, we write $[s, r]$ to denote the... | {
"timestamp": "2015-10-29T01:07:03",
"yymm": "1510",
"arxiv_id": "1510.08211",
"language": "en",
"url": "https://arxiv.org/abs/1510.08211",
"abstract": "The commuting probability of a finite ring $R$, denoted by $\\Pr(R)$, is the probability that any two randomly chosen elements of $R$ commute. In this pap... |
https://arxiv.org/abs/2206.13409 | Homomesies on permutations -- an analysis of maps and statistics in the FindStat database | In this paper, we perform a systematic study of permutation statistics and bijective maps on permutations in which we identify and prove 122 instances of the homomesy phenomenon. Homomesy occurs when the average value of a statistic is the same on each orbit of a given map. The maps we investigate include the Lehmer co... | \section{Introduction}
Dynamical algebraic combinatorics is the study of objects important in algebra and combinatorics through the lens of dynamics.
In this paper, we focus on permutations, which are fundamental objects in algebra, combinatorics, representation theory, geometry, probability, and many other areas of m... | {
"timestamp": "2022-06-28T02:33:44",
"yymm": "2206",
"arxiv_id": "2206.13409",
"language": "en",
"url": "https://arxiv.org/abs/2206.13409",
"abstract": "In this paper, we perform a systematic study of permutation statistics and bijective maps on permutations in which we identify and prove 122 instances of ... |
https://arxiv.org/abs/1905.08230 | Open problems in wavelet theory | We present a collection of easily stated open problems in wavelet theory and we survey the current status of answering them. This includes a problem of Larson on minimally supported frequency wavelets. We show that it has an affirmative answer for MRA wavelets. | \section{Introduction}
The goal of this paper is twofold. The first goal is to present a collection of open problems on wavelets which have simple formulations. Many of these problems are well-known, such as connectivity of the set of wavelets. Others are less known, but nevertheless deserve a wider dissemination. At... | {
"timestamp": "2019-05-21T02:34:34",
"yymm": "1905",
"arxiv_id": "1905.08230",
"language": "en",
"url": "https://arxiv.org/abs/1905.08230",
"abstract": "We present a collection of easily stated open problems in wavelet theory and we survey the current status of answering them. This includes a problem of La... |
https://arxiv.org/abs/2106.08443 | Reproducing Kernel Hilbert Space, Mercer's Theorem, Eigenfunctions, Nyström Method, and Use of Kernels in Machine Learning: Tutorial and Survey | This is a tutorial and survey paper on kernels, kernel methods, and related fields. We start with reviewing the history of kernels in functional analysis and machine learning. Then, Mercer kernel, Hilbert and Banach spaces, Reproducing Kernel Hilbert Space (RKHS), Mercer's theorem and its proof, frequently used kernels... | \section{Introduction}\label{section_introduction}
\subsection{History of Kernels}
It is 1904 when David Hilbert proposed his work on kernels and defined a definite kernel \cite{hilbert1904grundzuge}, and later, his and Erhard Schmidt's works proposed integral equations such as Fredholm integral equations \cite{hilb... | {
"timestamp": "2021-06-17T02:05:50",
"yymm": "2106",
"arxiv_id": "2106.08443",
"language": "en",
"url": "https://arxiv.org/abs/2106.08443",
"abstract": "This is a tutorial and survey paper on kernels, kernel methods, and related fields. We start with reviewing the history of kernels in functional analysis ... |
https://arxiv.org/abs/1910.12923 | Ensemble Kalman Sampler: mean-field limit and convergence analysis | Ensemble Kalman Sampler (EKS) is a method to find approximately $i.i.d.$ samples from a target distribution. As of today, why the algorithm works and how it converges is mostly unknown. The continuous version of the algorithm is a set of coupled stochastic differential equations (SDEs). In this paper, we prove the well... | \section{Introduction}
Sampling from a target distribution is a core problem in Bayesian statistics, machine learning and data assimilation. It has wide applications in atmospheric science, petroleum engineering, remote sensing and epidemiology in the form of volume computation, and bandit optimization~\cite{FABIAN1981... | {
"timestamp": "2020-08-24T02:14:30",
"yymm": "1910",
"arxiv_id": "1910.12923",
"language": "en",
"url": "https://arxiv.org/abs/1910.12923",
"abstract": "Ensemble Kalman Sampler (EKS) is a method to find approximately $i.i.d.$ samples from a target distribution. As of today, why the algorithm works and how ... |
https://arxiv.org/abs/1603.02807 | Constructions and nonexistence results for suitable sets of permutations | A set of $N$ permutations of $\{1,2,\dots,v\}$ is $(N,v,t)$-suitable if each symbol precedes each subset of $t-1$ others in at least one permutation. The central problems are to determine the smallest $N$ for which such a set exists for given $v$ and $t$, and to determine the largest $v$ for which such a set exists for... | \section{Introduction}
\label{sec:intro}
A set of $N$ permutations of $[v] = \set{1,2,\dots,v}$ is \ii{$(N,v,t)$-suitable} if each symbol precedes each subset of $t-1$ others in at least one permutation; necessarily we must have $t \le \min(v,N)$. We represent such a set as an $N\times v$ array called an \ii{$(N,v,t)... | {
"timestamp": "2016-12-02T02:07:45",
"yymm": "1603",
"arxiv_id": "1603.02807",
"language": "en",
"url": "https://arxiv.org/abs/1603.02807",
"abstract": "A set of $N$ permutations of $\\{1,2,\\dots,v\\}$ is $(N,v,t)$-suitable if each symbol precedes each subset of $t-1$ others in at least one permutation. T... |
https://arxiv.org/abs/1802.03087 | A Note on Intervals in the Hales-Jewett Theorem | The Hales-Jewett theorem for alphabet of size 3 states that whenever the Hales-Jewett cube [3]^n is r-coloured there is a monochromatic line (for n large). Conlon and Kamcev conjectured that, for any n, there is a 2-colouring of [3]^n for which there is no monochromatic line whose active coordinate set is an interval. ... | \section{Introduction}
In order to state the Hales-Jewett theorem we need some notation.
Given positive integers $k$ and $n$ let $[k]^{n}$ be the set of
all words in symbols $\{1,\dots,k\}$ of length $n$. A set $L \subset
[k]^{n}$ is called
a combinatorial line if there exist a nonempty set $S\subset [n]$
and integer... | {
"timestamp": "2018-02-12T02:02:43",
"yymm": "1802",
"arxiv_id": "1802.03087",
"language": "en",
"url": "https://arxiv.org/abs/1802.03087",
"abstract": "The Hales-Jewett theorem for alphabet of size 3 states that whenever the Hales-Jewett cube [3]^n is r-coloured there is a monochromatic line (for n large)... |
https://arxiv.org/abs/0712.1507 | Spectral analysis of metric graphs and related spaces | The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We present the basic concept of metric graphs and natural Laplacians acting on it and explicitly allow infinite graphs. Motivated by the general form of a Laplacian on ... | \section{Introduction}
\label{sec:intro}
A \emph{metric graph} $X$ is by definition a topological graph (i.e.,
a CW~complex of dimension $1$), where each edge $e$ is assigned a
length $\ell_e$. The resulting metric measure space allows to
introduce a family of ordinary differential operators acting on each
edge $e$ co... | {
"timestamp": "2008-02-15T17:28:39",
"yymm": "0712",
"arxiv_id": "0712.1507",
"language": "en",
"url": "https://arxiv.org/abs/0712.1507",
"abstract": "The aim of the present article is to give an overview of spectral theory on metric graphs guided by spectral geometry on discrete graphs and manifolds. We p... |
https://arxiv.org/abs/1202.2595 | The number of bit comparisons used by Quicksort: an average-case analysis | The analyses of many algorithms and data structures (such as digital search trees) for searching and sorting are based on the representation of the keys involved as bit strings and so count the number of bit comparisons. On the other hand, the standard analyses of many other algorithms (such as Quicksort) are performed... | \section{Introduction and summary}
\label{S:intro}
Algorithms for sorting and searching (together with their accompanying
analyses) generally fall into one of two categories: either the algorithm is
regarded as comparing items pairwise irrespective of their internal structure
(and so the analysis focuses on the numbe... | {
"timestamp": "2012-02-14T02:02:15",
"yymm": "1202",
"arxiv_id": "1202.2595",
"language": "en",
"url": "https://arxiv.org/abs/1202.2595",
"abstract": "The analyses of many algorithms and data structures (such as digital search trees) for searching and sorting are based on the representation of the keys inv... |
https://arxiv.org/abs/1512.01187 | On the State Complexity of the Shuffle of Regular Languages | We investigate the shuffle operation on regular languages represented by complete deterministic finite automata. We prove that $f(m,n)=2^{mn-1} + 2^{(m-1)(n-1)}(2^{m-1}-1)(2^{n-1}-1)$ is an upper bound on the state complexity of the shuffle of two regular languages having state complexities $m$ and $n$, respectively. W... | \section{An Upper Bound for the Shuffle Operation}
\label{sec:bound}
The \emph{state complexity of a regular language} $L$~\cite{Yu01} is the number of states in a complete minimal deterministic finite automaton (DFA) recognizing the language; it
will be denoted by $\kappa(L)$.
The \emph{state complexity of an operat... | {
"timestamp": "2016-07-18T02:06:04",
"yymm": "1512",
"arxiv_id": "1512.01187",
"language": "en",
"url": "https://arxiv.org/abs/1512.01187",
"abstract": "We investigate the shuffle operation on regular languages represented by complete deterministic finite automata. We prove that $f(m,n)=2^{mn-1} + 2^{(m-1)... |
https://arxiv.org/abs/1507.01258 | Expected number of real zeros for random orthogonal polynomials | We study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with i.i.d. Gaussian coefficients, have only $(2/\pi + o(1))\log{n}$ expected real zeros in terms of the degree $n$. If the basis is given by the orthonormal p... | \section{Introduction}
The expected number of real zeros ${\mathbb E}[N_n({\mathbb R})]$ for random polynomials of the form $P_n(x)=\sum_{k=0}^{n} c_k x^k,$ where $\{c_k\}_{k=0}^n$ are independent and identically distributed random variables, was studied since the 1930's. In particular, Bloch and P\'olya \cite{BP} gav... | {
"timestamp": "2015-07-07T02:10:19",
"yymm": "1507",
"arxiv_id": "1507.01258",
"language": "en",
"url": "https://arxiv.org/abs/1507.01258",
"abstract": "We study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by mon... |
https://arxiv.org/abs/2301.02628 | Pinnacle sets of signed permutations | Pinnacle sets record the values of the local maxima for a given family of permutations. They were introduced by Davis-Nelson-Petersen-Tenner as a dual concept to that of peaks, previously defined by Billey-Burdzy-Sagan. In recent years pinnacles and admissible pinnacles sets for the type $A$ symmetric group have been w... | \section{Introduction}
The study of permutation statistics is an active subdiscipline of combinatorics. Given a permutation $w=w(1)w(2)\cdots w(n)$, two particularly well-studied statistics are \newword{descents} and \newword{peaks}. Respectively, these statistics refer to indices $i$ such that $w(i)>w(i+1)$, and indi... | {
"timestamp": "2023-01-09T02:13:38",
"yymm": "2301",
"arxiv_id": "2301.02628",
"language": "en",
"url": "https://arxiv.org/abs/2301.02628",
"abstract": "Pinnacle sets record the values of the local maxima for a given family of permutations. They were introduced by Davis-Nelson-Petersen-Tenner as a dual con... |
https://arxiv.org/abs/2006.14070 | Enumeration of Standard Puzzles | We introduce a large family of combinatorial objects, called standard puzzles, defined by very simple rules. We focus on the standard puzzles for which the enumeration problems can be solved by explicit formulas or by classical numbers, such as binomial coefficients, Fibonacci numbers, tangent numbers, Catalan numbers,... | \section{Introduction}
We introduce a large family of combinatorial objects,
called {\it standard puzzles}, defined by very simple rules,
and study their enumeration problems. The topic of the paper may be classified
as belonging to {\it Enumerative Combinatorics}, since several of those
standard puzzles
can be solv... | {
"timestamp": "2020-06-26T02:04:23",
"yymm": "2006",
"arxiv_id": "2006.14070",
"language": "en",
"url": "https://arxiv.org/abs/2006.14070",
"abstract": "We introduce a large family of combinatorial objects, called standard puzzles, defined by very simple rules. We focus on the standard puzzles for which th... |
https://arxiv.org/abs/2209.03247 | Some extensions of Krasnoselskii's fixed point result for real functions | We extend Krasnoselskii's fixed point result to non-self-real functions. We find a new and simple proof for Hillam's result. In our approach, we don't assume the image of the related mapping to be compact or bounded. In this way, we extend Hillam's result to self-mappings on $\mathbb R$. Finally, we present a new proof... | \section{Introduction}\label{sec1}
We start our discussion by presenting the definition of L-Lipschitz functions, Krasnoselskii's theorem, and some generalizations of this result. In these results, the related function is self-mapping. In this manuscript, we extend Theorem \ref{hillam} to non-self mappings that are mo... | {
"timestamp": "2022-09-08T02:19:16",
"yymm": "2209",
"arxiv_id": "2209.03247",
"language": "en",
"url": "https://arxiv.org/abs/2209.03247",
"abstract": "We extend Krasnoselskii's fixed point result to non-self-real functions. We find a new and simple proof for Hillam's result. In our approach, we don't ass... |
https://arxiv.org/abs/1611.09707 | An unconstrained framework for eigenvalue problems | In this paper, we propose an unconstrained framework for eigenvalue problems in both discrete and continuous settings. We begin our discussion to solve a generalized eigenvalue problem $A{\bf x} = \lambda B{\bf x}$ with two $N\times N$ real symmetric matrices $A, B$ via minimizing a proposed functional whose nonzero cr... | \section{Introduction}
\label{intro}
Given an $N\times N$ matrix $A$, the eigenvalue problem of our interest is to find an eigenvalue and its corresponding eigenvector of $A$, that is, to solve $A{\bf x} = \lambda {\bf x}$ for ${\bf x}$ and $\lambda$. This is one of the most fundamental problems in mathematics with ap... | {
"timestamp": "2017-08-01T02:10:16",
"yymm": "1611",
"arxiv_id": "1611.09707",
"language": "en",
"url": "https://arxiv.org/abs/1611.09707",
"abstract": "In this paper, we propose an unconstrained framework for eigenvalue problems in both discrete and continuous settings. We begin our discussion to solve a ... |
https://arxiv.org/abs/1102.5764 | Rational approximations to algebraic Laurent series with coefficients in a finite field | In this paper we give a general upper bound for the irrationality exponent of algebraic Laurent series with coefficients in a finite field. Our proof is based on a method introduced in a different framework by Adamczewski and Cassaigne. It makes use of automata theory and, in our context, of a classical theorem due to ... | \subsection{Morphisms and Cobham's theorem}\label{3.Morphisms and Cobham's theorem}
Let $\mathcal A$ (respectively $\mathcal B$) be a finite alphabet and let $\mathcal{A}^*$ (respectively $\mathcal{B}^*$) be the corresponding free monoid. A morphism $\sigma$ is a map from $\mathcal{A}^*$ to $\mathcal{B}^*$ such that ... | {
"timestamp": "2011-03-01T02:04:29",
"yymm": "1102",
"arxiv_id": "1102.5764",
"language": "en",
"url": "https://arxiv.org/abs/1102.5764",
"abstract": "In this paper we give a general upper bound for the irrationality exponent of algebraic Laurent series with coefficients in a finite field. Our proof is bas... |
https://arxiv.org/abs/1308.6005 | Graphs with Equal Chromatic Symmetric Functions | Stanley [9] introduced the chromatic symmetric function ${\bf X}_G$ associated to a simple graph $G$ as a generalization of the chromatic polynomial of $G$. In this paper we present a novel technique to write ${\bf X}_G$ as a linear combination of chromatic symmetric functions of smaller graphs. We use this technique t... | \section*{Introduction}
In 1995, Stanley \cite{bST} introduced a symmetric function ${\bf X}_G = {\bf X}_G(x_1, x_2, \ldots)$ associated with any simple graph $G$ (see Section 1 for a precise definition) called the \emph{chromatic symmetric function} of $G$. ${\bf X}_G$ has the property that when we specialize the v... | {
"timestamp": "2013-08-29T02:01:14",
"yymm": "1308",
"arxiv_id": "1308.6005",
"language": "en",
"url": "https://arxiv.org/abs/1308.6005",
"abstract": "Stanley [9] introduced the chromatic symmetric function ${\\bf X}_G$ associated to a simple graph $G$ as a generalization of the chromatic polynomial of $G$... |
https://arxiv.org/abs/1603.07737 | The Planar Tree Packing Theorem | Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T1 and T2 on n vertices and have to find a planar graph on n vertices that is the edge-disjoint union of T1 ... |
\section{Introduction}\label{sec:introduction}
The \emph{packing problem} is to find a graph $G$ on $n$ vertices that
contains a given collection $G_1,\ldots, G_k$ of graphs on $n$ vertices
each as edge-disjoint subgraphs. This problem has been studied in a wide
variety of scenarios (see, e.g., \cite{AkiyamaC90,CaroY9... | {
"timestamp": "2016-03-28T02:00:37",
"yymm": "1603",
"arxiv_id": "1603.07737",
"language": "en",
"url": "https://arxiv.org/abs/1603.07737",
"abstract": "Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. ... |
https://arxiv.org/abs/1401.3714 | Testing Equivalence of Polynomials under Shifts | Two polynomials $f, g \in \mathbb{F}[x_1, \ldots, x_n]$ are called shift-equivalent if there exists a vector $(a_1, \ldots, a_n) \in \mathbb{F}^n$ such that the polynomial identity $f(x_1+a_1, \ldots, x_n+a_n) \equiv g(x_1,\ldots,x_n)$ holds. Our main result is a new randomized algorithm that tests whether two given po... | \section{Introduction}
In this paper we address the following problem, which we call {\em Shift Equivalence Testing} (SET). Given two polynomials
$f,g\in {\mathbb{F}}[{\mathbf x}]$ (we use boldface letters to denote vectors), decide whether there exists a shift ${\mathbf a}\in {\mathbb{F}}^n$ such that
$f({\mathbf x}+... | {
"timestamp": "2014-02-20T02:03:49",
"yymm": "1401",
"arxiv_id": "1401.3714",
"language": "en",
"url": "https://arxiv.org/abs/1401.3714",
"abstract": "Two polynomials $f, g \\in \\mathbb{F}[x_1, \\ldots, x_n]$ are called shift-equivalent if there exists a vector $(a_1, \\ldots, a_n) \\in \\mathbb{F}^n$ suc... |
https://arxiv.org/abs/1403.5975 | Monochromatic cycle partitions in local edge colourings | An edge colouring of a graph is said to be an $r$-local colouring if the edges incident to any vertex are coloured with at most $r$ colours. Generalising a result of Bessy and Thomassé, we prove that the vertex set of any $2$-locally coloured complete graph may be partitioned into two disjoint monochromatic cycles of d... | \section{Introduction}
A well-known result of Erd\H{o}s, Gy\'arf\'as and Pyber~\cite{EGP91} says that there exists a constant $c(r)$, depending only on $r$, such that if the edges of the complete graph $K_n$ have been coloured with $r$ colours, then the vertex set of $K_n$ may be partitioned into at most $c(r)$ disjoi... | {
"timestamp": "2015-05-12T02:08:50",
"yymm": "1403",
"arxiv_id": "1403.5975",
"language": "en",
"url": "https://arxiv.org/abs/1403.5975",
"abstract": "An edge colouring of a graph is said to be an $r$-local colouring if the edges incident to any vertex are coloured with at most $r$ colours. Generalising a ... |
https://arxiv.org/abs/1210.2279 | The parbelos, a parabolic analog of the arbelos | The arbelos is a classical geometric shape bounded by three mutually tangent semicircles with collinear diameters. We introduce a parabolic analog, the parbelos. After a review of the parabola, we use theorems of Archimedes and Lambert to demonstrate seven properties of the parbelos, drawing analogies to similar proper... | \section{Introduction: The Arbelos and the Parbelos.}
The \emph{arbelos} or \emph{shoemaker's knife} is a classic figure from Greek geometry bounded by three pairwise tangent semicircles with diameters lying on the same line. (See Figure~\ref{FIG:arbelos}.)
There is a long list of remarkable properties of the arbelos-... | {
"timestamp": "2013-05-07T02:01:10",
"yymm": "1210",
"arxiv_id": "1210.2279",
"language": "en",
"url": "https://arxiv.org/abs/1210.2279",
"abstract": "The arbelos is a classical geometric shape bounded by three mutually tangent semicircles with collinear diameters. We introduce a parabolic analog, the parb... |
https://arxiv.org/abs/1706.02351 | Recognizing difference quotients of real functions | For a real function $f:[0,1]\to\mathbb{R}$, the difference quotient of $f$ is the function of two real variables $\operatorname{DQ}_f(a,b)=\dfrac{f(b)-f(a)}{b-a}$, which we view as defined on the triangle $\mathcal{T}=\{(a,b):0\leq a<b\leq1\}$. In this paper we investigate how to determine whether a given function of t... | \section{Introduction}
Given a function $f(x)$ defined at two distinct points $a,b\in[0,1]$, the difference quotient of $f$ from $a$ to $b$ is the slope of the line segment connecting points $(a,f(a))$ and $(b,f(b))$. This we denote as \[\operatorname{DQ}_f(a,b)=\dfrac{f(b)-f(a)}{b-a}.\]
However, simplification and re... | {
"timestamp": "2017-06-09T02:01:06",
"yymm": "1706",
"arxiv_id": "1706.02351",
"language": "en",
"url": "https://arxiv.org/abs/1706.02351",
"abstract": "For a real function $f:[0,1]\\to\\mathbb{R}$, the difference quotient of $f$ is the function of two real variables $\\operatorname{DQ}_f(a,b)=\\dfrac{f(b)... |
https://arxiv.org/abs/1312.2245 | Edge-disjoint spanning trees and eigenvalues of regular graphs | Partially answering a question of Paul Seymour, we obtain a sufficient eigenvalue condition for the existence of $k$ edge-disjoint spanning trees in a regular graph, when $k\in \{2,3\}$. More precisely, we show that if the second largest eigenvalue of a $d$-regular graph $G$ is less than $d-\frac{2k-1}{d+1}$, then $G$ ... | \section{Introduction}
Our graph notation is standard (see West \cite{West} for undefined terms). The adjacency matrix of a graph $G$ with $n$ vertices has its rows and columns indexed after the vertices of $G$ and the $(u,v)$-entry of $A$ is $1$ if $uv=\{u,v\}$ is an edge of $G$
and $0$ otherwise. If $G$ is undirect... | {
"timestamp": "2013-12-10T02:08:56",
"yymm": "1312",
"arxiv_id": "1312.2245",
"language": "en",
"url": "https://arxiv.org/abs/1312.2245",
"abstract": "Partially answering a question of Paul Seymour, we obtain a sufficient eigenvalue condition for the existence of $k$ edge-disjoint spanning trees in a regul... |
https://arxiv.org/abs/2302.01977 | Construction of Hierarchically Semi-Separable matrix Representation using Adaptive Johnson-Lindenstrauss Sketching | We extend an adaptive partially matrix-free Hierarchically Semi-Separable (HSS) matrix construction algorithm by Gorman et al. [SIAM J. Sci. Comput. 41(5), 2019] which uses Gaussian sketching operators to a broader class of Johnson--Lindenstrauss (JL) sketching operators. We present theoretical work which justifies thi... | \section{Conclusions}
\label{sec:conclusion}
In this paper we extend the adaptive HSS compression algorithm from \cite{gorman2019robust} which required a Gaussian sketching operator to use any Johnson--Lindenstrauss sketching operator. We provide theoretical guarantees that the adaptive stopping criterion holds for all... | {
"timestamp": "2023-02-07T02:01:04",
"yymm": "2302",
"arxiv_id": "2302.01977",
"language": "en",
"url": "https://arxiv.org/abs/2302.01977",
"abstract": "We extend an adaptive partially matrix-free Hierarchically Semi-Separable (HSS) matrix construction algorithm by Gorman et al. [SIAM J. Sci. Comput. 41(5)... |
https://arxiv.org/abs/2202.01943 | PSO-PINN: Physics-Informed Neural Networks Trained with Particle Swarm Optimization | Physics-informed neural networks (PINN) have recently emerged as a promising application of deep learning in a wide range of engineering and scientific problems based on partial differential equation (PDE) models. However, evidence shows that PINN training by gradient descent displays pathologies that often prevent con... |
\section{Experimental Results}
\label{sec:results}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=1\columnwidth]{fig/poisson_train.png}
\hfill
\end{center}
\caption{
\textbf{Poisson Equation} -- First row: solutions in the PSO-PINN ensemble as training progresses. Second ... | {
"timestamp": "2022-10-24T02:03:38",
"yymm": "2202",
"arxiv_id": "2202.01943",
"language": "en",
"url": "https://arxiv.org/abs/2202.01943",
"abstract": "Physics-informed neural networks (PINN) have recently emerged as a promising application of deep learning in a wide range of engineering and scientific pr... |
https://arxiv.org/abs/2211.09304 | Spectral conditions for $k$-extendability and $k$-factors of bipartite graphs | Let $G$ be a connected graph. If $G$ contains a matching of size $k$, and every matching of size $k$ is contained in a perfect matching of $G$, then $G$ is said to be \emph{$k$-extendable}. A $k$-regular spanning subgraph of $G$ is called a \textit{$k$-factor}. In this paper, we provide spectral conditions for a (balan... | \section{Introduction}
Perfect matchings theory, which studies the simplest nontrivial substructures of graphs, is one of the earliest reasearch areas in graph theory. The characterization of perfect matchings was initially given by Frobenius\cite{Frobenius} in 1917, who proved that a bipartite graph of order $n$ has a... | {
"timestamp": "2022-11-18T02:04:58",
"yymm": "2211",
"arxiv_id": "2211.09304",
"language": "en",
"url": "https://arxiv.org/abs/2211.09304",
"abstract": "Let $G$ be a connected graph. If $G$ contains a matching of size $k$, and every matching of size $k$ is contained in a perfect matching of $G$, then $G$ i... |
https://arxiv.org/abs/2002.00999 | Enumerative problems for arborescences and monotone paths on polytope graphs | Every generic linear functional $f$ on a convex polytope $P$ induces an orientation on the graph of $P$. From the resulting directed graph one can define a notion of $f$-arborescence and $f$-monotone path on $P$, as well as a natural graph structure on the vertex set of $f$-monotone paths. These concepts are important ... | \section{On the diameter of monotone path graphs}
\label{sec:diameter}
The main goal of this section is to prove
Theorem~\ref{thm:diam-max}.
The lower bound
of~(\ref{eq:diam-max}) for the maximum diameter follows from
Lemma~\ref{lem:diam-stack}. The upper bound will be deduced from
the following result. Clearly,... | {
"timestamp": "2021-04-26T02:09:08",
"yymm": "2002",
"arxiv_id": "2002.00999",
"language": "en",
"url": "https://arxiv.org/abs/2002.00999",
"abstract": "Every generic linear functional $f$ on a convex polytope $P$ induces an orientation on the graph of $P$. From the resulting directed graph one can define ... |
https://arxiv.org/abs/1807.09030 | Bounding the Number of Minimal Transversals in Tripartite 3-Uniform Hypergraphs | We bound the number of minimal hypergraph transversals that arise in tri-partite 3-uniform hypergraphs, a class commonly found in applications dealing with data. Let H be such a hypergraph on a set of vertices V. We give a lower bound of 1.4977 |V | and an upper bound of 1.5012 |V | . | \section{Introduction}
Hypergraphs are a generalization of graphs, where edges may have arity different than 2.
They have been formalized by Berge in the seventies~\cite{berge1973graphs}.
Formally, a \emph{hypergraph} $\m H$ is a pair $(V,\m E)$, where $V$ is a set of vertices and $\m E$ is a family of subsets of $V$... | {
"timestamp": "2018-07-31T02:16:47",
"yymm": "1807",
"arxiv_id": "1807.09030",
"language": "en",
"url": "https://arxiv.org/abs/1807.09030",
"abstract": "We bound the number of minimal hypergraph transversals that arise in tri-partite 3-uniform hypergraphs, a class commonly found in applications dealing wit... |
https://arxiv.org/abs/1511.00137 | Error Analysis of Finite Differences and the Mapping Parameter in Spectral Differentiation | The Chebyshev points are commonly used for spectral differentiation in non-periodic domains. The rounding error in the Chebyshev approximation to the $n$-the derivative increases at a rate greater than $n^{2m}$ for the $m$-th derivative. The mapping technique of Kosloff and Tal-Ezer (\emph{J. Comp. Physics}, vol. 104 (... | \section{Introduction}
The Chebyshev points $x_{j}=\cos(j\pi/n)$, $n=0,1,\ldots,n$, are
commonly used to discretize the interval $[-1,1]$ when the boundary
conditions are not periodic. The $m$-th derivative $f^{(m)}(x)$
may be approximated as $\sum_{k=0}^{m}f(x_{k})w_{k,m}$ where $w_{k,m}$
are differentiation weights.... | {
"timestamp": "2015-11-03T02:08:58",
"yymm": "1511",
"arxiv_id": "1511.00137",
"language": "en",
"url": "https://arxiv.org/abs/1511.00137",
"abstract": "The Chebyshev points are commonly used for spectral differentiation in non-periodic domains. The rounding error in the Chebyshev approximation to the $n$-... |
https://arxiv.org/abs/1408.1940 | On the classification of Stanley sequences | An integer sequence is said to be 3-free if no three elements form an arithmetic progression. Following the greedy algorithm, the Stanley sequence $S(a_0,a_1,\ldots,a_k)$ is defined to be the 3-free sequence $\{a_n\}$ having initial terms $a_0,a_1,\ldots,a_k$ and with each subsequent term $a_n>a_{n-1}$ chosen minimally... | \section{Introduction}
A set of non-negative integers is \emph{3-free} if no three elements form an arithmetic progression. Given a 3-free set $A$ with elements $a_0<a_1<\cdots<a_k$, we define the \emph{Stanley sequence} $S(A)=\{a_n\}$ according to the greedy algorithm, as follows: Assuming $a_n$ has been defined, let... | {
"timestamp": "2014-08-11T02:10:28",
"yymm": "1408",
"arxiv_id": "1408.1940",
"language": "en",
"url": "https://arxiv.org/abs/1408.1940",
"abstract": "An integer sequence is said to be 3-free if no three elements form an arithmetic progression. Following the greedy algorithm, the Stanley sequence $S(a_0,a_... |
https://arxiv.org/abs/2205.01031 | Auslander-Reiten and Huneke-Wiegand conjectures over quasi-fiber product rings | In this paper we explore consequences of the vanishing of ${\rm Ext}$ for finitely generated modules over a quasi-fiber product ring $R$; that is, $R$ is a local ring such that $R/(\underline x)$ is a non-trivial fiber product ring, for some regular sequence $\underline x$ of $R$. Equivalently, the maximal ideal of $R/... | \section{Introduction}
This article is motivated
by the celebrated Auslander-Reiten Conjecture (ARC)
and the Huneke-Wiegand Conjecture for integral domains (HWC$_d$); see \cite[p. 70]{AR}, \cite{HL}, and
\cite[pp. 473--474]{HW}:
\begin{definition}\label{defarchwc} Let $R$ be a commutative Noetherian local rin... | {
"timestamp": "2022-05-03T02:48:06",
"yymm": "2205",
"arxiv_id": "2205.01031",
"language": "en",
"url": "https://arxiv.org/abs/2205.01031",
"abstract": "In this paper we explore consequences of the vanishing of ${\\rm Ext}$ for finitely generated modules over a quasi-fiber product ring $R$; that is, $R$ is... |
https://arxiv.org/abs/1708.00854 | Average-case reconstruction for the deletion channel: subpolynomially many traces suffice | The deletion channel takes as input a bit string $\mathbf{x} \in \{0,1\}^n$, and deletes each bit independently with probability $q$, yielding a shorter string. The trace reconstruction problem is to recover an unknown string $\mathbf{x}$ from many independent outputs (called "traces") of the deletion channel applied t... |
\section{Introduction}
The \emph{deletion channel} takes as input a bit string $\mathbf{x} \in
\{0,1\}^n$. Each bit of $\mathbf{x}$ is (independently of other bits)
retained with probability $p$ and deleted with probability $q := 1 -
p$. The channel then outputs the concatenation of the retained bits;
such an output... | {
"timestamp": "2017-08-03T02:10:45",
"yymm": "1708",
"arxiv_id": "1708.00854",
"language": "en",
"url": "https://arxiv.org/abs/1708.00854",
"abstract": "The deletion channel takes as input a bit string $\\mathbf{x} \\in \\{0,1\\}^n$, and deletes each bit independently with probability $q$, yielding a short... |
https://arxiv.org/abs/1711.09962 | On positivity of Ehrhart polynomials | Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive if all Ehrhart coefficients are positive (which is not true for all integral polyto... | \subsection{\@startsection{subsection}{2
\renewcommand\subsubsection{\@startsection{subsubsection}{3
\bm{z}@{.5\linespacing\@plus.7\linespacing}{-.5em
{\normalfont\bfseries}}\makeatother
\def\multiset#1#2{\ensuremath{\left(\kern-.3em\left(\genfrac{}{}{0pt}{}{#1}{#2}\right)\kern-.3em\right)}}
\def\Poly{\oper... | {
"timestamp": "2018-09-05T02:01:24",
"yymm": "1711",
"arxiv_id": "1711.09962",
"language": "en",
"url": "https://arxiv.org/abs/1711.09962",
"abstract": "Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficien... |
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