url stringlengths 31 38 | title stringlengths 7 229 | abstract stringlengths 44 2.87k | text stringlengths 319 2.51M | meta dict |
|---|---|---|---|---|
https://arxiv.org/abs/0707.3476 | Sums of products of congruence classes and of arithmetic progressions | Consider the congruence class R_m(a)={a+im:i\in Z} and the infinite arithmetic progression P_m(a)={a+im:i\in N_0}. For positive integers a,b,c,d,m the sum of products set R_m(a)R_m(b)+R_m(c)R_m(d) consists of all integers of the form (a+im)(b+jm)+(c+km)(d+\ell m) for some i,j,k,\ell\in Z. It is proved that if gcd(a,b,c... | \section{Sums of product sets}
Let \ensuremath{\mathbf Z}\ denote the set of integers and $\ensuremath{ \mathbf N }_0$ the set of nonnegative integers. For every prime $p$ and integer $n$, we denote by $\text{ord}_p(n)$ the greatest integer $k$ such that $p^k$ divides $n$.
Let $X$ and $Y$ be sets of integers. These ... | {
"timestamp": "2007-07-24T04:09:31",
"yymm": "0707",
"arxiv_id": "0707.3476",
"language": "en",
"url": "https://arxiv.org/abs/0707.3476",
"abstract": "Consider the congruence class R_m(a)={a+im:i\\in Z} and the infinite arithmetic progression P_m(a)={a+im:i\\in N_0}. For positive integers a,b,c,d,m the sum... |
https://arxiv.org/abs/2301.00194 | Chordal graphs with bounded tree-width | Given $t\geq 2$ and $0\leq k\leq t$, we prove that the number of labelled $k$-connected chordal graphs with $n$ vertices and tree-width at most $t$ is asymptotically $c n^{-5/2} \gamma^n n!$, as $n\to\infty$, for some constants $c,\gamma >0$ depending on $t$ and $k$. Additionally, we show that the number of $i$-cliques... | \section{Introduction}\label{sec:intro}
Tree-width is a fundamental parameter in structural and algorithmic graph theory, as illustrated for instance in \cite{CFKLMPPS15}.
It can be defined in terms of tree-decompositions or equivalently in terms of $k$-trees.
A $k$-tree is defined recursively as either a complete g... | {
"timestamp": "2023-01-03T02:06:29",
"yymm": "2301",
"arxiv_id": "2301.00194",
"language": "en",
"url": "https://arxiv.org/abs/2301.00194",
"abstract": "Given $t\\geq 2$ and $0\\leq k\\leq t$, we prove that the number of labelled $k$-connected chordal graphs with $n$ vertices and tree-width at most $t$ is ... |
https://arxiv.org/abs/1702.01027 | Random Triangles and Polygons in the Plane | We consider the problem of finding the probability that a random triangle is obtuse, which was first raised by Lewis Caroll. Our investigation leads us to a natural correspondence between plane polygons and the Grassmann manifold of 2-planes in real $n$-space proposed by Allen Knutson and Jean-Claude Hausmann. This cor... | \section{Two paths to a construction}
We start by fixing notation. As is usual in triangle geometry; we let $A, B, C$ refer to the vertices of a triangle, $a$, $b$, and $c$ denote the lengths of the corresponding (opposite) sides, and use $\alpha$, $\beta$, $\gamma$ for the corresponding angles.
We now construct a m... | {
"timestamp": "2017-02-06T02:05:46",
"yymm": "1702",
"arxiv_id": "1702.01027",
"language": "en",
"url": "https://arxiv.org/abs/1702.01027",
"abstract": "We consider the problem of finding the probability that a random triangle is obtuse, which was first raised by Lewis Caroll. Our investigation leads us to... |
https://arxiv.org/abs/1910.13273 | $\varepsilon$-strong simulation of the convex minorants of stable processes and meanders | Using marked Dirichlet processes we characterise the law of the convex minorant of the meander for a certain class of Lévy processes, which includes subordinated stable and symmetric Lévy processes. We apply this characterisaiton to construct $\varepsilon$-strong simulation ($\varepsilon$SS) algorithms for the convex m... | \section{Introduction}
\subsection{Setting and motivation}
The universality of stable laws, processes and their path transformations makes them
ubiquitous in probability theory and many areas of statistics and natural and social
sciences (see e.g.~\citep{MR1745764,MR3160562} and the references therein). Brownian
m... | {
"timestamp": "2019-10-30T01:17:36",
"yymm": "1910",
"arxiv_id": "1910.13273",
"language": "en",
"url": "https://arxiv.org/abs/1910.13273",
"abstract": "Using marked Dirichlet processes we characterise the law of the convex minorant of the meander for a certain class of Lévy processes, which includes subor... |
https://arxiv.org/abs/2010.16415 | Geometric Invariants of Plane and Space Curves | The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to distinguish between two different ones. The aim of this note is to generalize the classic... | \section{Introduction and Geometric Motivation}
In geometry, in order to study and understand the local shape of a plane curve in $\mathbb{A}_{\mathbb{R}}^2$, one usually looks at its tangent line and curvature at a particular point. In the case of a space curve in~$\mathbb{A}_{\mathbb{R}}^3$, one has also the notion ... | {
"timestamp": "2021-03-04T02:28:58",
"yymm": "2010",
"arxiv_id": "2010.16415",
"language": "en",
"url": "https://arxiv.org/abs/2010.16415",
"abstract": "The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunate... |
https://arxiv.org/abs/1701.01270 | Graded components of local cohomology modules | Let $A$ be a regular ring containing a field of characteristic zero and let $R = A[X_1,\ldots, X_m]$. Consider $R$ as standard graded with $deg \ A = 0$ and $deg \ X_i = 1$ for all $i$. In this paper we present a comprehensive study of graded components of local cohomology modules $H^i_I(R)$ where $I$ is an \emph{arbit... | \section{Introduction}
Let $S = \bigoplus_{n \geq 0}S_n$ be a standard graded Noetherian ring and let $S_+$ be it's irrelevant ideal. The theory of local cohomology with respect to $S_+$ is particularly satisfactory. It is well-known (cf. \cite[15.1.5]{BS}) that if $M$ is a finitely generated
graded $S$-module then f... | {
"timestamp": "2017-02-16T02:04:20",
"yymm": "1701",
"arxiv_id": "1701.01270",
"language": "en",
"url": "https://arxiv.org/abs/1701.01270",
"abstract": "Let $A$ be a regular ring containing a field of characteristic zero and let $R = A[X_1,\\ldots, X_m]$. Consider $R$ as standard graded with $deg \\ A = 0$... |
https://arxiv.org/abs/2210.05225 | A Formalisation of a Fast Fourier Transform | This notes explains how a standard algorithm that constructs the discrete Fourier transform has been formalised and proved correct in the Coq proof assistant using the SSReflect extension. | \section{Introduction}
Fast Fourier Transforms are key tools in many areas. In this note,
we are going to explain how they have been formalised in a theorem
prover like Coq.
There are many ways to motivate this work. As univariate polynomials
is at the heart of our formalisation, we take, here, one application :
... | {
"timestamp": "2022-10-17T02:12:13",
"yymm": "2210",
"arxiv_id": "2210.05225",
"language": "en",
"url": "https://arxiv.org/abs/2210.05225",
"abstract": "This notes explains how a standard algorithm that constructs the discrete Fourier transform has been formalised and proved correct in the Coq proof assist... |
https://arxiv.org/abs/1801.03908 | Homogeneous length functions on groups | A pseudo-length function defined on an arbitrary group $G = (G,\cdot,e, (\,)^{-1})$ is a map $\ell: G \to [0,+\infty)$ obeying $\ell(e)=0$, the symmetry property $\ell(x^{-1}) = \ell(x)$, and the triangle inequality $\ell(xy) \leqslant \ell(x) + \ell(y)$ for all $x,y \in G$. We consider pseudo-length functions which sa... | \section{Introduction}
Let $G = (G, \cdot, e, (\,)^{-1})$ be a group (written multiplicatively,
with identity element $e$). A \emph{pseudo-length function} on $G$ is a
map $\normsymb : G \to [0,+\infty)$ that obeys the properties
\begin{itemize}
\item $\norm{e} = 0$,
\item $\norm{ x^{-1} } = \norm{x}$,
\item $\norm{ ... | {
"timestamp": "2018-06-15T02:12:29",
"yymm": "1801",
"arxiv_id": "1801.03908",
"language": "en",
"url": "https://arxiv.org/abs/1801.03908",
"abstract": "A pseudo-length function defined on an arbitrary group $G = (G,\\cdot,e, (\\,)^{-1})$ is a map $\\ell: G \\to [0,+\\infty)$ obeying $\\ell(e)=0$, the symm... |
https://arxiv.org/abs/2207.13892 | Topology and chromatic number of random $ε$-distance graphs on spheres | Given $0<\alpha\leq\pi$, ${\epsilon}>0$ and $n$, we define random graphs on the $d$-dimensional sphere by drawing $n$ i.i.d. uniform random points for the vertices, and edges $u {\sim} v$ whenever the geodesic distance between $u$ and $v$ is ${\epsilon}$-close to ${\alpha}$. This model generalizes distance graphs on sp... |
\section{Introduction}
Given a metric space $(Z,\mathbbm{d})$ and a parameter $\alpha\geq0$, the corresponding \textit{distance graph} on $Z$ is obtained by drawing edges $x\sim y$ whenever $x,y\in Z$ and $\dist{x,y}=\alpha$. Distance graphs have been an active topic of research in the last few decades, popularized b... | {
"timestamp": "2022-07-29T02:10:25",
"yymm": "2207",
"arxiv_id": "2207.13892",
"language": "en",
"url": "https://arxiv.org/abs/2207.13892",
"abstract": "Given $0<\\alpha\\leq\\pi$, ${\\epsilon}>0$ and $n$, we define random graphs on the $d$-dimensional sphere by drawing $n$ i.i.d. uniform random points for... |
https://arxiv.org/abs/1505.07549 | On a conjecture of Erdős and Szekeres | Let f(n) denote the smallest positive integer such that every set of $f(n)$ points in general position in the Euclidean plane contains a convex n-gon. In a seminal paper published in 1935, Erdős and Szekeres proved that f(n) exists and provided an upper bound. In 1961, they also proved a lower bound, which they conject... | \section{Introduction}
In 1935, Paul Erd\H{o}s and George Szekeres published a paper titled {\em 'A combinatorial problem in geometry'} [2], in which they published classical results in Ramsey Theory. One of these results was the proof of the existence of a minimum positive integer $f(n)$ such that, for any set of $f(... | {
"timestamp": "2015-05-29T02:07:09",
"yymm": "1505",
"arxiv_id": "1505.07549",
"language": "en",
"url": "https://arxiv.org/abs/1505.07549",
"abstract": "Let f(n) denote the smallest positive integer such that every set of $f(n)$ points in general position in the Euclidean plane contains a convex n-gon. In ... |
https://arxiv.org/abs/2002.10512 | Sharp Constants of Approximation Theory. IV. Asymptotic Relations in General Settings | In this paper we first introduce the unified definition of the sharp constant that includes constants in three major problems of approximation theory, such as, inequalities for approximating elements, approximation of individual elements, and approximation on classes of elements. Second, we find sufficient conditions t... | \section{Introduction}\label{S1}
\setcounter{equation}{0}
\noindent
We continue the study of the sharp constants
of approximation theory
that began in \cite{G1992,G2000,GT2017,G2017,G2018,G2019a,G2019b}.
In this paper we discuss limit relations between sharp constants of approximation theory
in general settings... | {
"timestamp": "2020-02-26T02:02:15",
"yymm": "2002",
"arxiv_id": "2002.10512",
"language": "en",
"url": "https://arxiv.org/abs/2002.10512",
"abstract": "In this paper we first introduce the unified definition of the sharp constant that includes constants in three major problems of approximation theory, suc... |
https://arxiv.org/abs/0808.2316 | A new secant method for unconstrained optimization | We present a gradient-based algorithm for unconstrained minimization derived from iterated linear change of basis. The new method is equivalent to linear conjugate gradient in the case of a quadratic objective function. In the case of exact line search it is a secant method. In practice, it performs comparably to BFGS ... | \section{Iterated linear change of basis}
We consider the problem of minimizing a differentiable function
$f:\R^n\rightarrow \R$ with no constraints on the variables.
We propose the following algorithm for this problem.
We assume
a starting point $\w_{{0}}$ is given. Let $f_{{0}}$ be identified
with $f$.
\begin{tabbi... | {
"timestamp": "2008-08-17T23:52:04",
"yymm": "0808",
"arxiv_id": "0808.2316",
"language": "en",
"url": "https://arxiv.org/abs/0808.2316",
"abstract": "We present a gradient-based algorithm for unconstrained minimization derived from iterated linear change of basis. The new method is equivalent to linear co... |
https://arxiv.org/abs/1802.10212 | Asymptotic behavior of Rényi entropy in the central limit theorem | We explore an asymptotic behavior of Rényi entropy along convolutions in the central limit theorem with respect to the increasing number of i.i.d. summands. In particular, the problem of monotonicity is addressed under suitable moment hypotheses. | \section{{\bf Introduction}}
\setcounter{equation}{0}
\vskip2mm
\noindent
Given a (continuous) random variable $X$ with density $p$, the associated
R\'enyi entropy and R\'enyi entropy power of index $r$ ($1 < r < \infty$)
are defined by
$$
h_r(X) \, = \, -\frac{1}{r-1}\,\log \int_{-\infty}^\infty p(x)^r\,dx, \qquad
N... | {
"timestamp": "2018-03-01T02:03:39",
"yymm": "1802",
"arxiv_id": "1802.10212",
"language": "en",
"url": "https://arxiv.org/abs/1802.10212",
"abstract": "We explore an asymptotic behavior of Rényi entropy along convolutions in the central limit theorem with respect to the increasing number of i.i.d. summand... |
https://arxiv.org/abs/1502.06241 | Mixture models with a prior on the number of components | A natural Bayesian approach for mixture models with an unknown number of components is to take the usual finite mixture model with Dirichlet weights, and put a prior on the number of components---that is, to use a mixture of finite mixtures (MFM). While inference in MFMs can be done with methods such as reversible jump... | \section{Introduction}
\label{section:intro}
Mixture models are used in a wide range of applications, including
population structure \citep{Pritchard_2000},
document modeling \citep{Blei_2003b},
speaker recognition \citep{Reynolds_2000},
computer vision \citep{stauffer1999adaptive},
phylogenetics \citep{pagel2004phy... | {
"timestamp": "2015-02-24T02:12:16",
"yymm": "1502",
"arxiv_id": "1502.06241",
"language": "en",
"url": "https://arxiv.org/abs/1502.06241",
"abstract": "A natural Bayesian approach for mixture models with an unknown number of components is to take the usual finite mixture model with Dirichlet weights, and ... |
https://arxiv.org/abs/2002.05422 | Closing curves by rearranging arcs | In this paper we show how, under surprisingly weak assumptions, one can split a planar curve into three arcs and rearrange them (matching tangent directions) to obtain a closed curve. We also generalize this construction to curves split into $k$ arcs and comment what can be achieved by rearranging arcs for a curve in h... | \section{Introduction.}
In this paper we study the problem of splitting a given planar curve into arcs and rearrange them (matching tangent directions) in order to make the curve closed. The interest of our result lies in the counterintuitive nature of the statement and in the simplicity of the proof. Using an argumen... | {
"timestamp": "2020-08-24T02:15:55",
"yymm": "2002",
"arxiv_id": "2002.05422",
"language": "en",
"url": "https://arxiv.org/abs/2002.05422",
"abstract": "In this paper we show how, under surprisingly weak assumptions, one can split a planar curve into three arcs and rearrange them (matching tangent directio... |
https://arxiv.org/abs/1106.0602 | Numerical investigation of the smallest eigenvalues of the p-Laplace operator on planar domains | The eigenvalue problem for the p-Laplace operator with p>1 on planar domains with the zero Dirichlet boundary condition is considered. The Constrained Descent Method and the Constrained Mountain Pass Algorithm are used in the Sobolev space setting to numerically investigate the dependence of the two smallest eigenvalue... | \section{Introduction}
For a bounded domain $\Omega\subset\mathbb {R}^N$, $N\in\mathbb {N}$ and a parameter
$p\in(1,\infty)$ consider the nonlinear eigenvalue problem
\begin{equation}
\label{eq:evproblem}
\begin{aligned}
-\Delta_p u &= \lambda |u|^{p-2} u &&\text{in }\Omega, \\
u &= 0 &&\text{on }\partial\O... | {
"timestamp": "2011-06-21T02:08:09",
"yymm": "1106",
"arxiv_id": "1106.0602",
"language": "en",
"url": "https://arxiv.org/abs/1106.0602",
"abstract": "The eigenvalue problem for the p-Laplace operator with p>1 on planar domains with the zero Dirichlet boundary condition is considered. The Constrained Desce... |
https://arxiv.org/abs/2008.11543 | The tree search game for two players | We consider a two-player search game on a tree $T$. One vertex (unknown to the players) is randomly selected as the target. The players alternately guess vertices. If a guess $v$ is not the target, then both players are informed in which subtree of $T \smallsetminus v$ the target lies. The winner is the player who gues... | \section{Introduction}
We consider the following competitive variant of traditional binary search:
two players seek an (unknown, uniformly random) element of
the set~$\{1, \dots, n\}$. The players alternately guess elements of the set; if a guess is incorrect, then both players are informed whether the secret numbe... | {
"timestamp": "2022-02-07T02:18:30",
"yymm": "2008",
"arxiv_id": "2008.11543",
"language": "en",
"url": "https://arxiv.org/abs/2008.11543",
"abstract": "We consider a two-player search game on a tree $T$. One vertex (unknown to the players) is randomly selected as the target. The players alternately guess ... |
https://arxiv.org/abs/1712.07930 | Counting Periodic Trajectories of Finsler Billiards | We provide lower bounds on the number of periodic Finsler billiard trajectories inside a quadratically convex smooth closed hypersurface $M$ in a $d$-dimensional Finsler space with possibly irreversible Finsler metric. An example of such a system is a billiard in a sufficiently weak magnetic field. The $r$-periodic Fin... | \section{Introduction}
The study of mathematical billiards goes back to G. D. Birkhoff who wrote in \cite{Birkhoff1927}:
\begin{quote}
... in this problem the formal side, usually so formidable in dynamics, almost completely disappears, and only the interesting qualitative questions need to be considered.
\end{quote... | {
"timestamp": "2017-12-22T02:06:36",
"yymm": "1712",
"arxiv_id": "1712.07930",
"language": "en",
"url": "https://arxiv.org/abs/1712.07930",
"abstract": "We provide lower bounds on the number of periodic Finsler billiard trajectories inside a quadratically convex smooth closed hypersurface $M$ in a $d$-dime... |
https://arxiv.org/abs/1603.05773 | A framework for structured linearizations of matrix polynomials in various bases | We present a framework for the construction of linearizations for scalar and matrix polynomials based on dual bases which, in the case of orthogonal polynomials, can be described by the associated recurrence relations. The framework provides an extension of the classical linearization theory for polynomials expressed i... |
\section{Introduction}
In recent years much interest has been devoted to finding linearizations
for polynomials and matrix polynomials. The Frobenius linearization, i.e.,
the classical companion,
has been the de-facto standard in polynomial eigenvalue problems and
polynomial rootfinding for a long time ... | {
"timestamp": "2016-07-06T02:07:53",
"yymm": "1603",
"arxiv_id": "1603.05773",
"language": "en",
"url": "https://arxiv.org/abs/1603.05773",
"abstract": "We present a framework for the construction of linearizations for scalar and matrix polynomials based on dual bases which, in the case of orthogonal polyn... |
https://arxiv.org/abs/0706.1120 | Comparison Geometry for the Bakry-Emery Ricci Tensor | For Riemannian manifolds with a measure $(M,g, e^{-f} dvol_g)$ we prove mean curvature and volume comparison results when the $\infty$-Bakry-Emery Ricci tensor is bounded from below and $f$ is bounded or $\partial_r f$ is bounded from below, generalizing the classical ones (i.e. when $f$ is constant). This leads to ext... | \section{#1} \setcounter{equation}{0}}
\newcommand{\mbox{gap}}{\mbox{gap}}
\newcommand{\mathrm{Vol}}{\mathrm{Vol}}
\newcommand{\mathrm{diam}}{\mathrm{diam}}
\newcommand{\mathrm{Ric}}{\mathrm{Ric}}
\newcommand{\mathrm{sn}}{\mathrm{sn}}
\newcommand{\mathrm{Hess}}{\mathrm{Hess}}
\newcommand{\mbox{tr}}{\mbox{t... | {
"timestamp": "2007-09-14T23:28:13",
"yymm": "0706",
"arxiv_id": "0706.1120",
"language": "en",
"url": "https://arxiv.org/abs/0706.1120",
"abstract": "For Riemannian manifolds with a measure $(M,g, e^{-f} dvol_g)$ we prove mean curvature and volume comparison results when the $\\infty$-Bakry-Emery Ricci te... |
https://arxiv.org/abs/2302.09054 | Hanging cables and spider threads | It has been known for more than 300 years that the shape of an inelastic hanging cable, chain, or rope of uniform linear mass density is the graph of the hyperbolic cosine, up to scaling and shifting coordinates. But given two points at which the ends of the cable are attached, how exactly should we scale and shift the... | \section{Introduction}
\begin{figure}[h]
\centering
\includegraphics[scale=0.47]{figure_1.pdf}
\caption{Examples of catenaries.}
\label{fig:HANGING_CHAINS}
\end{figure}
The shape of a hanging cable (or chain or rope) is called a {\em catenary}; see Fig.\ \ref{fig:HANGING_CHAINS} for
examples.
In 1691, in three pap... | {
"timestamp": "2023-02-20T02:17:29",
"yymm": "2302",
"arxiv_id": "2302.09054",
"language": "en",
"url": "https://arxiv.org/abs/2302.09054",
"abstract": "It has been known for more than 300 years that the shape of an inelastic hanging cable, chain, or rope of uniform linear mass density is the graph of the ... |
https://arxiv.org/abs/1705.05911 | Four NP-complete problems about generalizations of perfect graphs | We show that the following problems are NP-complete.1. Can the vertex set of a graph be partitioned into two sets such that each set induces a perfect graph?2. Is the difference between the chromatic number and clique number at most $1$ for every induced subgraph of a graph?3. Can the vertex set of every induced subgra... | \section{Introduction}
All graphs considered in this article are finite and simple. Let $G$ be a
graph. The complement $G^c$ of $G$ is the graph with vertex set $V(G)$ and
such that two vertices are adjacent in $G^c$ if and only if they are
non-adjacent in $G$.
For two graphs $H$ and $G$, $H$ is an {\em induced su... | {
"timestamp": "2017-05-18T02:01:41",
"yymm": "1705",
"arxiv_id": "1705.05911",
"language": "en",
"url": "https://arxiv.org/abs/1705.05911",
"abstract": "We show that the following problems are NP-complete.1. Can the vertex set of a graph be partitioned into two sets such that each set induces a perfect gra... |
https://arxiv.org/abs/1711.01003 | The quasi principal rank characteristic sequence | A minor of a matrix is quasi-principal if it is a principal or an almost-principal minor. The quasi principal rank characteristic sequence (qpr-sequence) of an $n\times n$ symmetric matrix is introduced, which is defined as $q_1 q_2 \cdots q_n$, where $q_k$ is $\tt A$, $\tt S$, or $\tt N$, according as all, some but no... | \section{Introduction}\label{sintro}
$\null$
\indent
Motivated by the famous principal minor assignment problem,
which is stated in \cite{HS}, Brualdi et al.\ \cite{P} introduced the principal rank characteristic sequence,
which is defined as follows:
Given an $n \times n$ symmetric matrix $B$ over a
field $F$,
\textit... | {
"timestamp": "2017-11-06T02:05:37",
"yymm": "1711",
"arxiv_id": "1711.01003",
"language": "en",
"url": "https://arxiv.org/abs/1711.01003",
"abstract": "A minor of a matrix is quasi-principal if it is a principal or an almost-principal minor. The quasi principal rank characteristic sequence (qpr-sequence) ... |
https://arxiv.org/abs/1610.00756 | Ahlswede-Khachatrian Theorems: Weighted, Infinite, and Hamming | The seminal complete intersection theorem of Ahlswede and Khachatrian gives the maximum cardinality of a $k$-uniform $t$-intersecting family on $n$ points, and describes all optimal families. We extend this theorem to several other settings: the weighted case, the case of infinitely many points, and the Hamming scheme.... | \section{Introduction} \label{sec:introduction}
\subsection{Background} \label{sec:background}
The Erd\H{o}s--Ko--Rado theorem~\cite{EKR}, a basic result in extremal combinatorics, states that when $k \leq n/2$, a $k$-uniform intersecting family on $n$ points contains at most $\binom{n-1}{k-1}$ sets; and furthermore,... | {
"timestamp": "2016-10-05T02:00:57",
"yymm": "1610",
"arxiv_id": "1610.00756",
"language": "en",
"url": "https://arxiv.org/abs/1610.00756",
"abstract": "The seminal complete intersection theorem of Ahlswede and Khachatrian gives the maximum cardinality of a $k$-uniform $t$-intersecting family on $n$ points... |
https://arxiv.org/abs/1707.04054 | On Convergence Sets of Power Series with Holomorphic Coefficients | We consider convergence sets of formal power series of the form $f(z,t)=\sum_{n=0}^{\infty} f_n(z)t^n$, where $f_n(z)$ are holomorphic functions on a domain $\Omega$ in $\mathbb{C}$. A subset $E$ of $\Omega$ is said to be a convergence set in $\Omega$ if there is a series $f(z,t)$ such that $E$ is exactly the set of po... | \section{Introduction}
The purpose of this article is to describe the convergence sets of formal power
series with holomorphic coefficients. The study of convergence sets comes from generalizations of Hartogs Theorem (see \cite{Lelong 1951,AM,S,Levenberg and Molzon 1988, spallek}).
Our approach is motivated by recen... | {
"timestamp": "2017-07-14T02:04:47",
"yymm": "1707",
"arxiv_id": "1707.04054",
"language": "en",
"url": "https://arxiv.org/abs/1707.04054",
"abstract": "We consider convergence sets of formal power series of the form $f(z,t)=\\sum_{n=0}^{\\infty} f_n(z)t^n$, where $f_n(z)$ are holomorphic functions on a do... |
https://arxiv.org/abs/1603.05523 | Quantitative combinatorial geometry for continuous parameters | We prove variations of Carathéodory's, Helly's and Tverberg's theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among our results, we present continuous quantitative versions of Lovász's colorful Helly theorem, Bárány's colorful Carathéodory's theorem, and t... | \section{Introduction}
Carath\'eodory's, Helly's, and Tverberg's theorems are undoubtedly among the most important theorems in convex geometry (see \cite{Mbook} for an introduction). Many generalizations and extensions, including colorful, fractional, and topological versions of these theorems have been developed bef... | {
"timestamp": "2016-03-21T01:10:37",
"yymm": "1603",
"arxiv_id": "1603.05523",
"language": "en",
"url": "https://arxiv.org/abs/1603.05523",
"abstract": "We prove variations of Carathéodory's, Helly's and Tverberg's theorems where the sets involved are measured according to continuous functions such as the ... |
https://arxiv.org/abs/2001.04501 | Hysteresis and Stabillity | Hysteresis can be defined from a dynamical systems perspective with respect to equilibrium points. Consequently, hysteresis naturally lends itself as a topic to illustrate and extend concepts in a dynamical systems course. A number of examples exhibiting hysteresis, most motivated by applications, are presented. Althou... | \section{Introduction}
\label{intro}
Hysteresis is a common phenomenon found in real-world systems such as in biological fields \cite{Angeli2004,Collings1990,Mayergoyz2019,Noori2014}, magnetism \cite{Chen2019,Chow2014_ACC,Bernstein2005}, ferroelectric materials \cite{Xu2016,Park2019,Wang2009}, circuits \cite{Elwakil20... | {
"timestamp": "2020-09-01T02:21:20",
"yymm": "2001",
"arxiv_id": "2001.04501",
"language": "en",
"url": "https://arxiv.org/abs/2001.04501",
"abstract": "Hysteresis can be defined from a dynamical systems perspective with respect to equilibrium points. Consequently, hysteresis naturally lends itself as a to... |
https://arxiv.org/abs/1811.08386 | The reduction number and degree bound of projective subschemes | In this paper, we prove the degree upper bound of projective subschemes in terms of the reduction number and show that the maximal cases are only arithmetically Cohen-Macaulay subschemes with linear resolution. Furthermore, it can be shown that there are only two types of reduced, irreducible projective varieties with ... | \section{Introduction}
Let $X\subset \ensuremath{\mathbb P}^{n+e}$ be a non-degenerate closed subscheme of dimension $n$ and codimension $e$ defined over an algebraically closed field $k$ of arbitrary characteristic with the ideal sheaf $\mathcal I_X$. Let $S_0=k[x_0, \ldots, x_{n+e}]$ and $R=S_0/I_X$ be the homogeneo... | {
"timestamp": "2019-08-06T02:03:29",
"yymm": "1811",
"arxiv_id": "1811.08386",
"language": "en",
"url": "https://arxiv.org/abs/1811.08386",
"abstract": "In this paper, we prove the degree upper bound of projective subschemes in terms of the reduction number and show that the maximal cases are only arithmet... |
https://arxiv.org/abs/2112.09338 | Binomial expansion for saturated and symbolic powers of sums of ideals | There are two different notions for symbolic powers of ideals existing in the literature, one defined in terms of associated primes, the other in terms of minimal primes. Elaborating on an idea known to Eisenbud, Herzog, Hibi, and Trung, we interpret both notions of symbolic powers as suitable saturations of the ordina... | \section{Introduction} \label{sec.intro}
Let ${\Bbbk}$ be a field, let $A$ and $B$ be Noetherian ${\Bbbk}$-algebras such that $R = A \otimes_{\Bbbk} B$ is also Noetherian. Let $I \subseteq A$ and $J \subseteq B$ be ideals, and let $I+J\subseteq R$ denote the ideal $IR+JR$. The following binomial expansion for the symb... | {
"timestamp": "2021-12-20T02:09:33",
"yymm": "2112",
"arxiv_id": "2112.09338",
"language": "en",
"url": "https://arxiv.org/abs/2112.09338",
"abstract": "There are two different notions for symbolic powers of ideals existing in the literature, one defined in terms of associated primes, the other in terms of... |
https://arxiv.org/abs/1806.02250 | The Erdos conjecture for primitive sets | A subset of the integers larger than 1 is $primitive$ if no member divides another. Erdos proved in 1935 that the sum of $1/(a\log a)$ for $a$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained for the set of prime numbers. In this pape... | \section{Introduction}
A set of positive integers $>1$ is called {\bf primitive} if no element divides any other (for convenience, we exclude the singleton set $\{1\}$). There are
a number of interesting and sometimes unexpected theorems about primitive sets.
After Besicovitch \cite{besicovitch}, we know that the up... | {
"timestamp": "2018-07-03T02:06:13",
"yymm": "1806",
"arxiv_id": "1806.02250",
"language": "en",
"url": "https://arxiv.org/abs/1806.02250",
"abstract": "A subset of the integers larger than 1 is $primitive$ if no member divides another. Erdos proved in 1935 that the sum of $1/(a\\log a)$ for $a$ running ov... |
https://arxiv.org/abs/cs/0606052 | Topology for Distributed Inference on Graphs | Let $N$ local decision makers in a sensor network communicate with their neighbors to reach a decision \emph{consensus}. Communication is local, among neighboring sensors only, through noiseless or noisy links. We study the design of the network topology that optimizes the rate of convergence of the iterative decision ... | \section{Introduction}
\label{introduction}
The paper studies the problem of designing the topology of a graph network. As a motivational application we consider the problem of describing the connectivity graph of a sensor network, i.e., specifying with which sensors should each sensor in the network communicate. We ... | {
"timestamp": "2006-06-12T20:01:54",
"yymm": "0606",
"arxiv_id": "cs/0606052",
"language": "en",
"url": "https://arxiv.org/abs/cs/0606052",
"abstract": "Let $N$ local decision makers in a sensor network communicate with their neighbors to reach a decision \\emph{consensus}. Communication is local, among ne... |
https://arxiv.org/abs/2103.15778 | Properties of Hamiltonian Circuits in Rectangular Grids | We present properties and invariants of Hamiltonian circuits in rectangular grids. It is proved that all circuits on a $2n \times 2n$ chessboard have at least $4n$ turns and at least $2n$ straights if $n$ is even and $2n+2$ straights if $n$ is odd. The minimum number of turns and straights are presented and proved for ... | \section{Introduction}
A Hamiltonian circuit in a rectangular grid, often also called a {\it rook circuit} or Wazir tour \cite{jeliss} is a circuit on an $n \times m$ chessboard that passes through all squares without crossing. Sometimes the term Hamiltonian circuit is also used for a circuit on a chessboard of a knig... | {
"timestamp": "2021-03-30T02:57:48",
"yymm": "2103",
"arxiv_id": "2103.15778",
"language": "en",
"url": "https://arxiv.org/abs/2103.15778",
"abstract": "We present properties and invariants of Hamiltonian circuits in rectangular grids. It is proved that all circuits on a $2n \\times 2n$ chessboard have at ... |
https://arxiv.org/abs/0704.3363 | Topology and Factorization of Polynomials | For any polynomial $P \in \mathbb{C}[X_1,X_2,...,X_n]$, we describe a $\mathbb{C}$-vector space $F(P)$ of solutions of a linear system of equations coming from some algebraic partial differential equations such that the dimension of $F(P)$ is the number of irreducible factors of $P$. Moreover, the knowledge of $F(P)$ g... | \section{Introduction}
Let $K$ be the algebraic closure of a field $k$ and
let $k[X_1,X_2,...,X_n]$ be the polynomial ring in $n$ indeterminates.
The zero set of
a polynomial $P \in k[X_1,X_2,...,X_n]$ of $\deg d >0$
is a hypersurface $V(P)$ in $K^n.$
As the polynomial ring is a factorial ring, we can write
$P=\prod_{... | {
"timestamp": "2008-04-02T11:15:51",
"yymm": "0704",
"arxiv_id": "0704.3363",
"language": "en",
"url": "https://arxiv.org/abs/0704.3363",
"abstract": "For any polynomial $P \\in \\mathbb{C}[X_1,X_2,...,X_n]$, we describe a $\\mathbb{C}$-vector space $F(P)$ of solutions of a linear system of equations comin... |
https://arxiv.org/abs/1912.05919 | Extensions of Hyperfields | We develop a theory of extensions of hyperfields that generalizes the notion of field extensions. Since hyperfields have a multivalued addition, we must consider two kinds of extensions that we call weak hyperfield extensions and strong hyperfield extensions. For quotient hyperfields, we develop a method to construct s... | \section{Introduction} \label{sec:intro}
Let $F$ be a field, and $f\in F[x]$ be an irreducible polynomial. Results from classical field theory tells us that we can always find some field extension $F\subseteq L$ such that $f$ contains a root in $L$. The construction of this field extension is quite simple, we just ta... | {
"timestamp": "2019-12-13T02:11:58",
"yymm": "1912",
"arxiv_id": "1912.05919",
"language": "en",
"url": "https://arxiv.org/abs/1912.05919",
"abstract": "We develop a theory of extensions of hyperfields that generalizes the notion of field extensions. Since hyperfields have a multivalued addition, we must c... |
https://arxiv.org/abs/1202.4473 | The best of both worlds: stochastic and adversarial bandits | We present a new bandit algorithm, SAO (Stochastic and Adversarial Optimal), whose regret is, essentially, optimal both for adversarial rewards and for stochastic rewards. Specifically, SAO combines the square-root worst-case regret of Exp3 (Auer et al., SIAM J. on Computing 2002) and the (poly)logarithmic regret of UC... |
\section{A simplified SAO algorithm for $K=2$ arms}
\label{sec:2-arms}
\newcommand{C_\mathtt{crn}}{C_\mathtt{crn}}
\newcommand{\mathtt{hf}}{\mathtt{hf}}
\newcommand{\tau_*}{\tau_*}
\newcommand{\mathtt{INT}}{\mathtt{INT}}
\newcommand{post-exploration probability space}{post-exploration probability space}
\newcomman... | {
"timestamp": "2012-02-22T02:00:25",
"yymm": "1202",
"arxiv_id": "1202.4473",
"language": "en",
"url": "https://arxiv.org/abs/1202.4473",
"abstract": "We present a new bandit algorithm, SAO (Stochastic and Adversarial Optimal), whose regret is, essentially, optimal both for adversarial rewards and for stoc... |
https://arxiv.org/abs/1108.5627 | Multiplier Sequences for Simple Sets of Polynomials | In this paper we give a new characterization of simple sets of polynomials B with the property that the set of B-multiplier sequences contains all Q-multiplier sequence for every simple set Q. We characterize sequences of real numbers which are multiplier sequences for every simple set Q, and obtain some results toward... | \section{Introduction} In their seminal work \cite{PS} P\'olya and Schur completely characterized all sequences of real numbers $\{\gamma_k\}_{k=0}^{\infty}$ satisfying the following property.
\medskip
\noindent {\bf Property A.} Given any real polynomial
\[
f(x)=\sum_{k=0}^n a_k x^k
\]
with only real zeros, the ... | {
"timestamp": "2011-08-30T02:03:56",
"yymm": "1108",
"arxiv_id": "1108.5627",
"language": "en",
"url": "https://arxiv.org/abs/1108.5627",
"abstract": "In this paper we give a new characterization of simple sets of polynomials B with the property that the set of B-multiplier sequences contains all Q-multipl... |
https://arxiv.org/abs/1902.02967 | Generic reductions for in-place polynomial multiplication | The polynomial multiplication problem has attracted considerable attention since the early days of computer algebra, and several algorithms have been designed to achieve the best possible time complexity. More recently, efforts have been made to improve the space complexity, developing modified versions of a few specif... |
\section{Introduction}
\subsection{Polynomial multiplication}
Polynomial multiplication is a fundamental problem in mathematical
algorithms. It forms the basis (and key bottleneck) for
other fundamental problems such as division with remainder, GCD computation,
evaluation/interpolation, resultants, factorization, an... | {
"timestamp": "2019-02-11T02:08:26",
"yymm": "1902",
"arxiv_id": "1902.02967",
"language": "en",
"url": "https://arxiv.org/abs/1902.02967",
"abstract": "The polynomial multiplication problem has attracted considerable attention since the early days of computer algebra, and several algorithms have been desi... |
https://arxiv.org/abs/1209.5123 | An n-in-a-row Game | The usual $n$-in-a-row game is a positional game in which two player alternately claim points in $\bb{Z}^2$ with the winner being the first player to claim $n$ consecutive points in a line. We consider a variant of the game, suggested by Croft, where the number of points claimed increases by 1 each turn, and so on turn... | \section{Introduction}
A \emph{positional game} is a pair $(X,\cc{F})$ where $X$ is a set and $\cc{F} \subset \bb{P}(X)$. We call $X$ the \emph{board}, and the members $F \in \cc{F}$ are \emph{winning sets}. The game is played by two players, Red and Blue, who alternately claim unclaimed points from the board. The firs... | {
"timestamp": "2012-09-25T02:04:21",
"yymm": "1209",
"arxiv_id": "1209.5123",
"language": "en",
"url": "https://arxiv.org/abs/1209.5123",
"abstract": "The usual $n$-in-a-row game is a positional game in which two player alternately claim points in $\\bb{Z}^2$ with the winner being the first player to claim... |
https://arxiv.org/abs/1901.05915 | Jacobian syzygies and plane curves with maximal global Tjurina numbers | First we give a sharp upper bound for the cardinal $m$ of a minimal set of generators for the module of Jacobian syzygies of a complex projective reduced plane curve $C$. Next we discuss the sharpness of an upper bound, given by A. du Plessis and C.T.C. Wall, for the global Tjurina number of such a curve $C$, in terms ... | \section{Introduction}
Let $S=\mathbb{C}[x,y,z]$ be the polynomial ring in three variables $x,y,z$ with complex coefficients, and let $C:f=0$ be a reduced curve of degree $d$ in the complex projective plane $\mathbb{P}^2$.
We denote by $J_f$ the Jacobian ideal of $f$, i.e. the homogeneous ideal in $S$ spanned by the... | {
"timestamp": "2019-04-30T02:17:04",
"yymm": "1901",
"arxiv_id": "1901.05915",
"language": "en",
"url": "https://arxiv.org/abs/1901.05915",
"abstract": "First we give a sharp upper bound for the cardinal $m$ of a minimal set of generators for the module of Jacobian syzygies of a complex projective reduced ... |
https://arxiv.org/abs/2009.10395 | Mixed-Projection Conic Optimization: A New Paradigm for Modeling Rank Constraints | We propose a framework for modeling and solving low-rank optimization problems to certifiable optimality. We introduce symmetric projection matrices that satisfy $Y^2=Y$, the matrix analog of binary variables that satisfy $z^2=z$, to model rank constraints. By leveraging regularization and strong duality, we prove that... | \section{Introduction}
Many central problems in optimization, machine learning, and control theory are equivalent to optimizing a low-rank matrix over a convex set. For instance, low-rank constraints successfully model notions of minimal complexity, low dimensionality, or orthogonality in a system. However, while rank ... | {
"timestamp": "2021-04-06T02:02:47",
"yymm": "2009",
"arxiv_id": "2009.10395",
"language": "en",
"url": "https://arxiv.org/abs/2009.10395",
"abstract": "We propose a framework for modeling and solving low-rank optimization problems to certifiable optimality. We introduce symmetric projection matrices that ... |
https://arxiv.org/abs/1801.07584 | The Case for Raabe's Test | Among the techniques for determining the convergence of a series, Raabe's Test remains relatively unfamiliar to most mathematicians. We present several results relating to Raabe's Test that do not seem to be widely known, making the case that Raabe's Test should be featured more prominently in undergraduate calculus an... | \section{Raabe's Test}\label{S:intro}
Although Raabe's Test was first introduced in 1832, its importance and interpretation have largely been overlooked. The purpose of this article is to expand the scope of Raabe's Test, illustrating its benefits and situating it within its proper context.
As most readers are prob... | {
"timestamp": "2019-01-29T02:26:27",
"yymm": "1801",
"arxiv_id": "1801.07584",
"language": "en",
"url": "https://arxiv.org/abs/1801.07584",
"abstract": "Among the techniques for determining the convergence of a series, Raabe's Test remains relatively unfamiliar to most mathematicians. We present several re... |
https://arxiv.org/abs/0909.5106 | Analytical Formulae for Two of A. H. Stroud's Quadrature Rules | Analytical formulae for the points and weights of two fifth-order quadrature rules for C_3, the 3-cube, are given. The rules, originally formulated by A. H. Stroud in 1967, are discussed in greater detail in terms of both the setup of the basic equations and the method of obtaining their solutions analytically. The pri... | \section{Introduction}
In 1967, A.~H.~Stroud published an article~\cite{Stroud_1967} on
fifth-degree integration formulas for several symmetric,
$n$-dimensional regions. In the first sentence of Section 2 of that
article, Stroud mentions that ``Unless stated otherwise we assume that
$n\geq 4$.'' He goes on to give a ... | {
"timestamp": "2009-09-28T17:07:14",
"yymm": "0909",
"arxiv_id": "0909.5106",
"language": "en",
"url": "https://arxiv.org/abs/0909.5106",
"abstract": "Analytical formulae for the points and weights of two fifth-order quadrature rules for C_3, the 3-cube, are given. The rules, originally formulated by A. H.... |
https://arxiv.org/abs/1603.08496 | Maximal spectral surfaces of revolution converge to a catenoid | We consider a maximization problem for eigenvalues of the Laplace-Beltrami operator on surfaces of revolution in $\mathbb{R}^3$ with two prescribed boundary components. For every $j$, we show that there is a surface $\Sigma_j$ which maximizes the $j$-th Dirichlet eigenvalue. The maximizing surface has a meridian which ... | \section{Introduction}
On a smoothly bounded planar domain $\Omega$, the Dirichlet eigenvalues form a sequence
\[
0 < \lambda_1(\Omega) \le \lambda_2(\Omega) \le \lambda_3(\Omega) \le \ldots
\]
The relationship between the eigenvalues and the domain $\Omega$ is complicated.
An interesting problem is to find domains w... | {
"timestamp": "2016-09-13T02:10:31",
"yymm": "1603",
"arxiv_id": "1603.08496",
"language": "en",
"url": "https://arxiv.org/abs/1603.08496",
"abstract": "We consider a maximization problem for eigenvalues of the Laplace-Beltrami operator on surfaces of revolution in $\\mathbb{R}^3$ with two prescribed bound... |
https://arxiv.org/abs/1906.04548 | Spring-Electrical Models For Link Prediction | We propose a link prediction algorithm that is based on spring-electrical models. The idea to study these models came from the fact that spring-electrical models have been successfully used for networks visualization. A good network visualization usually implies that nodes similar in terms of network topology, e.g., co... | \section{Introduction}
The \textit{proceedings} are the records of a conference.\footnote{This
is a footnote} ACM seeks
to give these conference by-products a uniform, high-quality
appearance. To do this, ACM has some rigid requirements for the
format of the proceedings documents: there is a specified format
(bala... | {
"timestamp": "2019-06-12T02:14:42",
"yymm": "1906",
"arxiv_id": "1906.04548",
"language": "en",
"url": "https://arxiv.org/abs/1906.04548",
"abstract": "We propose a link prediction algorithm that is based on spring-electrical models. The idea to study these models came from the fact that spring-electrical... |
https://arxiv.org/abs/0808.0381 | The Chow ring of relative Fulton-MacPherson space | Suppose that X is a nonsingular variety and D is a nonsingular proper subvariety. Configuration spaces of distinct and non-distinct n points in X away from D were constructed by the author and B. Kim inarXiv:0806.3819, by using the method of wonderful compactification. In this paper, we give an explicit presentation of... | \section{Introduction}
Let $X$ be a complex connected nonsingular algebraic variety and let $D$ be a smooth divisor.
In \cite{KS}, two generalizations of Fulton--MacPherson spaces were constructed by using the method of wonderful compactifications \cite{Li1}. Two spaces are following:
\begin{enumerate}
\item A comp... | {
"timestamp": "2008-08-04T10:34:09",
"yymm": "0808",
"arxiv_id": "0808.0381",
"language": "en",
"url": "https://arxiv.org/abs/0808.0381",
"abstract": "Suppose that X is a nonsingular variety and D is a nonsingular proper subvariety. Configuration spaces of distinct and non-distinct n points in X away from ... |
https://arxiv.org/abs/1209.6396 | Chernoff-Hoeffding Inequality and Applications | When dealing with modern big data sets, a very common theme is reducing the set through a random process. These generally work by making "many simple estimates" of the full data set, and then judging them as a whole. Perhaps magically, these "many simple estimates" can provide a very accurate and small representation o... | \section{Chernoff-Hoeffding Inequality}
We consider two specific forms of the Chernoff-Hoeffding bound. They are not the strongest form of the bound, but is for many applications asymptotically equivalent, and it also fairly straight-forward to use.
\begin{theorem}
\label{thm:CH1}
Consider a set of $r$ \emph{i... | {
"timestamp": "2013-02-20T02:01:08",
"yymm": "1209",
"arxiv_id": "1209.6396",
"language": "en",
"url": "https://arxiv.org/abs/1209.6396",
"abstract": "When dealing with modern big data sets, a very common theme is reducing the set through a random process. These generally work by making \"many simple estim... |
https://arxiv.org/abs/1702.07406 | On permutations of order dividing a given integer | We give a detailed analysis of the proportion of elements in the symmetric group on $n$ points whose order divides $m$, for $n$ sufficiently large and $m \ge n$ with $m = O(n)$. | \section{Introduction}
The study of orders of elements in finite symmetric groups goes back at
least to the work of Landau~\cite[p. 222]{Landau09} who proved that the
maximum order of an element of the symmetric group $S_n$ on $n$ points
is $e^{(1+o(1))(n\log n)^{1/2}}$.
Erd\H{o}s and Tur\'an took a probabilistic a... | {
"timestamp": "2017-02-27T02:01:28",
"yymm": "1702",
"arxiv_id": "1702.07406",
"language": "en",
"url": "https://arxiv.org/abs/1702.07406",
"abstract": "We give a detailed analysis of the proportion of elements in the symmetric group on $n$ points whose order divides $m$, for $n$ sufficiently large and $m ... |
https://arxiv.org/abs/2010.11756 | Maximum distances in the four-digit Kaprekar process | For natural numbers $x$ and $b$, the classical Kaprekar function is defined as $K_{b} (x) = D-A$, where $D$ is the rearrangement of the base-$b$ digits of $x$ in descending order and $A$ is ascending. The bases $b$ for which $K_b$ has a $4$-digit non-zero fixed point were classified by Hasse and Prichett, and for each ... | \section{Introduction}
In 1949, Dattatreya Ramchandra Kaprekar \cite{kaprekar1949another} introduced the following process. We start with a base-$b$ four-digit number $x$ (allowing this to contain leading zeros such as $x=0309$). Rearrange the digits of $x$ to be in decreasing order and subtract from this the rearran... | {
"timestamp": "2020-10-23T02:23:56",
"yymm": "2010",
"arxiv_id": "2010.11756",
"language": "en",
"url": "https://arxiv.org/abs/2010.11756",
"abstract": "For natural numbers $x$ and $b$, the classical Kaprekar function is defined as $K_{b} (x) = D-A$, where $D$ is the rearrangement of the base-$b$ digits of... |
https://arxiv.org/abs/2107.11601 | Upper bounds on the extremal number of the 4-cycle | We obtain some new upper bounds on the maximum number $f(n)$ of edges in $n$-vertex graphs without containing cycles of length four. This leads to an asymptotically optimal bound on $f(n)$ for a broad range of integers $n$ as well as a disproof of a conjecture of Erdős from 1970s which asserts that $f(n)=\frac12 n^{3/2... | \section{Introduction}
Let $\ex(n,C_4)$ denote the maximum number of edges in an $n$-vertex {\it $C_4$-free} graph.\footnote{Throughout this paper, a graph is called {\it $C_4$-free} if it does not contain a cycle of length four as a subgraph.}
The study of this extremal number can be dated back to Erd\H{o}s \cite{E3... | {
"timestamp": "2021-10-13T02:10:11",
"yymm": "2107",
"arxiv_id": "2107.11601",
"language": "en",
"url": "https://arxiv.org/abs/2107.11601",
"abstract": "We obtain some new upper bounds on the maximum number $f(n)$ of edges in $n$-vertex graphs without containing cycles of length four. This leads to an asym... |
https://arxiv.org/abs/1910.06890 | A characterization of polynomials whose high powers have non-negative coefficients | Let $f \in \mathbb{R}[x]$ be a polynomial with real coefficients. We say that $f$ is eventually non-negative if $f^m$ has non-negative coefficients for all sufficiently large $m \in \mathbb{N}$. In this short note, we give a classification of all eventually non-negative polynomials. This generalizes a theorem of De Ang... | \section{Introduction}
In this short paper we study the following basic problem about iterated convolutions of sequences of real numbers.
\begin{center}
For what sequences $S = (c_0,\ldots,c_d)$ are all ``high'' convolutions $S \ast S \ast \cdots \ast S$ non-negative?
\end{center}
This is the same as asking ``for wh... | {
"timestamp": "2020-03-31T02:27:53",
"yymm": "1910",
"arxiv_id": "1910.06890",
"language": "en",
"url": "https://arxiv.org/abs/1910.06890",
"abstract": "Let $f \\in \\mathbb{R}[x]$ be a polynomial with real coefficients. We say that $f$ is eventually non-negative if $f^m$ has non-negative coefficients for ... |
https://arxiv.org/abs/2009.10139 | Small Quotients of Braid Groups | We prove that the symmetric group $S_n$ is the smallest non-cyclic quotient of the braid group $B_n$ for $n=5,6$ and that the alternating group $A_n$ is the smallest non-trivial quotient of the commutator subgroup $B_n'$ for $n = 5,6,7,8$. We also give an improved lower bound on the order of any non-cyclic quotient of ... | \section{Introduction}
Let $B_n$ denote the braid group on $n$ strands, and $B_n'$ its commutator subgroup. We are be interested in studying quotients of $B_n$. In light of the fact that the abelianization $B_n/B_n'$ is infinite cyclic, it is simple to create `uninteresting' homomorphisms $B_n \to G$ that factor throu... | {
"timestamp": "2020-09-23T02:01:39",
"yymm": "2009",
"arxiv_id": "2009.10139",
"language": "en",
"url": "https://arxiv.org/abs/2009.10139",
"abstract": "We prove that the symmetric group $S_n$ is the smallest non-cyclic quotient of the braid group $B_n$ for $n=5,6$ and that the alternating group $A_n$ is t... |
https://arxiv.org/abs/2007.09959 | A proof for a conjecture on the regularity of binomial edge ideals | In this paper we introduce the concept of clique disjoint edge sets in graphs. Then, for a graph $G$, we define the invariant $\eta(G)$ as the maximum size of a clique disjoint edge set in $G$. We show that the regularity of the binomial edge ideal of $G$ is bounded above by $\eta(G)$. This, in particular, settles a co... | \section{Introduction}\label{introduction}
Let $G$ be a graph on the vertex set $[n]$ and the edge set $E(G)$. Let also $S=\KK[x_1 ,\ldots ,x_n , y_1 , \ldots , y_n]$ be the polynomial ring over a field $\KK$. Then, the \emph{binomial edge ideal} associated to $G$, denoted by $J_G$, is the ideal in $S$ generated by a... | {
"timestamp": "2020-07-21T02:29:48",
"yymm": "2007",
"arxiv_id": "2007.09959",
"language": "en",
"url": "https://arxiv.org/abs/2007.09959",
"abstract": "In this paper we introduce the concept of clique disjoint edge sets in graphs. Then, for a graph $G$, we define the invariant $\\eta(G)$ as the maximum si... |
https://arxiv.org/abs/2201.04499 | Coloring distance graphs on the plane | We consider the coloring of certain distance graphs on the Euclidean plane. Namely, we ask for the minimal number of colors needed to color all points of the plane in such a way that pairs of points at distance in the interval $[1,b]$ get different colors. The classic Hadwiger-Nelson problem is a special case of this q... | \section{Introduction}\label{sec:intro}
How many colors are needed to color the Euclidean plane $\mathbb{R}^2$ so that no pair of points at distance $1$ get the same color? This famous open question is known as the Hadwiger-Nelson problem, named after Hugo Hadwiger and Edward Nelson. It is also often formulated as a... | {
"timestamp": "2022-01-13T02:20:55",
"yymm": "2201",
"arxiv_id": "2201.04499",
"language": "en",
"url": "https://arxiv.org/abs/2201.04499",
"abstract": "We consider the coloring of certain distance graphs on the Euclidean plane. Namely, we ask for the minimal number of colors needed to color all points of ... |
https://arxiv.org/abs/1011.2972 | Static two-grid mixed finite-element approximations to the Navier-Stokes equations | A two-grid scheme based on mixed finite-element approximations to the incompressible Navier-Stokes equations is introduced and analyzed. In the first level the standard mixed finite-element approximation over a coarse mesh is computed. In the second level the approximation is postprocessed by solving a discrete Oseen-t... | \section{Introduction}
\label{sec:1} We consider the incompressible Navier--Stokes
equations
\begin{eqnarray}
\label{onetwo} u_t -\nu \Delta u + (u\cdot\nabla)u + \nabla p
&=& f,\\ \noalign{\vskip6pt plus 3pt minus 1pt} {\rm div}(u)&=&0,\nonumber
\end{eqnarray}
in a bounded domain $\Omega\subset {\mathbb R}^d$ ($d=2,3... | {
"timestamp": "2010-11-15T02:02:35",
"yymm": "1011",
"arxiv_id": "1011.2972",
"language": "en",
"url": "https://arxiv.org/abs/1011.2972",
"abstract": "A two-grid scheme based on mixed finite-element approximations to the incompressible Navier-Stokes equations is introduced and analyzed. In the first level ... |
https://arxiv.org/abs/1902.03505 | Universal optimal configurations for the $p$-frame potentials | Given $d, N\geq 2$ and $p\in (0, \infty]$ we consider a family of functionals, the $p$-frame potentials FP$_{p, N, d}$, defined on the set of all collections of $N$ unit-norm vectors in $\mathbb R^d$. For the special case $p=2$ and $p=\infty$, both the minima and the minimizers of these potentials have been thoroughly ... | \section{Introduction}
A set of vectors $X=\{x_k\}_{k=1}^{N}
\subseteq \mathbb{{\mathbb{R}}}^d$ is a \textit{frame} for $\mathbb{R}^d$ if there exist $0<A\leq B < \infty$ such that
\begin{equation}\label{eq:frineq}
\quad A\nm{x}{}^{2} \leq \sum_{k=1}^{N} | \ip{x}{x_k}|^{2}
\leq B \nm{x}{}^{2} \quad\text{for a... | {
"timestamp": "2019-02-25T02:08:24",
"yymm": "1902",
"arxiv_id": "1902.03505",
"language": "en",
"url": "https://arxiv.org/abs/1902.03505",
"abstract": "Given $d, N\\geq 2$ and $p\\in (0, \\infty]$ we consider a family of functionals, the $p$-frame potentials FP$_{p, N, d}$, defined on the set of all colle... |
https://arxiv.org/abs/1003.4254 | An Exposition of Götze's Estimation of the Rate of Convergence in the Multivariate Central Limit Theorem | We provide an explanation of the main ideas underlying Götze's main result in using Stein's method. We also provide detailed derivations of various intermediate estimates. Curiously, we are led to a different dimensional dependence of the constant than that given Götze's paper. We would like to dedicate this to Charle... | \section{Introduction}
In his article G\"otze\cite{gotze} used Stein's method to provide
an ingenious derivation of the Berry-Esseen type bound for the class of Borel convex subsets of
${\mathbb R}^k$ in the context of the classical multivariate central limit theorem (CLT).
This approach has proved fruitful in deriving... | {
"timestamp": "2010-03-23T01:03:06",
"yymm": "1003",
"arxiv_id": "1003.4254",
"language": "en",
"url": "https://arxiv.org/abs/1003.4254",
"abstract": "We provide an explanation of the main ideas underlying Götze's main result in using Stein's method. We also provide detailed derivations of various interme... |
https://arxiv.org/abs/1411.6594 | Hirzebruch class and Bialynicki-Birula decomposition | Suppose an algebraic torus acts on a complex algebraic variety $X$. Then a great part of information about global invariants of $X$ are encoded in some data localized around the fixed points. The goal of this note is to present a connection between two approaches to localization for $C^*$-action. The homological result... | \section{Homological localization associated to $S^1$--action}
The homological approach to localization was initiated by Borel \cite{Bo} in 50-ties, but a lot of ideas originate from Smith theory \cite{Sm}. Further development is due to Segal \cite{Se}, Quillen \cite{Qu}, Chang-Skjelbred \cite{ChSk}. The theory was sum... | {
"timestamp": "2015-11-24T02:18:05",
"yymm": "1411",
"arxiv_id": "1411.6594",
"language": "en",
"url": "https://arxiv.org/abs/1411.6594",
"abstract": "Suppose an algebraic torus acts on a complex algebraic variety $X$. Then a great part of information about global invariants of $X$ are encoded in some data... |
https://arxiv.org/abs/1905.03173 | The Gerrymandering Jumble: Map Projections Permute Districts' Compactness Scores | In political redistricting, the compactness of a district is used as a quantitative proxy for its fairness. Several well-established, yet competing, notions of geographic compactness are commonly used to evaluate the shapes of regions, including the Polsby-Popper score, the convex hull score, and the Reock score, and t... |
\section{Convex Hull}\label{sec:ch}
We first consider the \textit{convex hull
score}. We briefly recall the definition of a convex set and then
define this score function.
\begin{definition}
A set in $\ensuremath{\mathbb{R}}\xspace^2$ or $\mbb{S}^2$ is \textbf{convex} if every shortest geodesic segment between ... | {
"timestamp": "2019-05-14T02:40:03",
"yymm": "1905",
"arxiv_id": "1905.03173",
"language": "en",
"url": "https://arxiv.org/abs/1905.03173",
"abstract": "In political redistricting, the compactness of a district is used as a quantitative proxy for its fairness. Several well-established, yet competing, notio... |
https://arxiv.org/abs/math/9912125 | Stratified spaces formed by totally positive varieties | By a theorem of A.Björner, for every interval $[u,v]$ in the Bruhat order of a Coxeter group $W$, there exists a stratified space whose strata are labeled by the elements of $[u,v]$, adjacency is described by the Bruhat order, and each closed stratum (resp., the boundary of each stratum) has the homology of a ball (res... | \section{Introduction and main results}
\label{sec:intro}
In a 1984 paper~\cite{bjorner-cw}, A.~Bj\"orner has shown that
every interval
in the Bruhat order of a Coxeter
group~$W$ is the ``face poset'' of some stratified space,
in which each closed stratum (resp., the boundary of each stratum)
has the homology of a b... | {
"timestamp": "1999-12-15T21:59:14",
"yymm": "9912",
"arxiv_id": "math/9912125",
"language": "en",
"url": "https://arxiv.org/abs/math/9912125",
"abstract": "By a theorem of A.Björner, for every interval $[u,v]$ in the Bruhat order of a Coxeter group $W$, there exists a stratified space whose strata are lab... |
https://arxiv.org/abs/2005.04194 | On the periods of abelian varieties | In this expository paper, we review the formula of Chowla and Selberg for the periods of elliptic curves with complex multiplication, and discuss two methods of proof. One uses Kronecker's limit formula and the other uses the geometry of a family of abelian varieties. We discuss a generalization of this formula, which ... | \section{Looking for a thesis}
In my third year of graduate school at Harvard I was still looking for a thesis topic. John Tate had suggested a problem on $p$-adic Galois representations, but I couldn't see how to make any progress on it. Fortunately for me, Neal Koblitz and David Rohrlich had arrived at Cambridge as... | {
"timestamp": "2020-05-11T02:15:59",
"yymm": "2005",
"arxiv_id": "2005.04194",
"language": "en",
"url": "https://arxiv.org/abs/2005.04194",
"abstract": "In this expository paper, we review the formula of Chowla and Selberg for the periods of elliptic curves with complex multiplication, and discuss two meth... |
https://arxiv.org/abs/1803.08658 | Proving a conjecture on chromatic polynomials by counting the number of acyclic orientations | The chromatic polynomial $P(G,x)$ of a graph $G$ of order $n$ can be expressed as $\sum\limits_{i=1}^n(-1)^{n-i}a_{i}x^i$, where $a_i$ is interpreted as the number of broken-cycle free spanning subgraphs of $G$ with exactly $i$ components. The parameter $\epsilon(G)=\sum\limits_{i=1}^n (n-i)a_i/\sum\limits_{i=1}^n a_i$... | \section{#1} }\newcommand {\rebibitem}[1] {\bibitem{#1} \red{[*: #1]}}
\def\relabel {\label} \def\resection {\section}\def\rebibitem {\bibitem}
\begin{document}
\title
{Proof of Lundow and Markstr\"{o}m's conjecture on chromatic polynomials via novel inequalities
}
\date{}
\def \bg {\hspace{0.3 cm}}
... | {
"timestamp": "2018-03-26T02:05:45",
"yymm": "1803",
"arxiv_id": "1803.08658",
"language": "en",
"url": "https://arxiv.org/abs/1803.08658",
"abstract": "The chromatic polynomial $P(G,x)$ of a graph $G$ of order $n$ can be expressed as $\\sum\\limits_{i=1}^n(-1)^{n-i}a_{i}x^i$, where $a_i$ is interpreted as... |
https://arxiv.org/abs/1505.05263 | Bounding minimal solid angles of polytopes | In this article we study the following question: What can be the measure of the minimal solid angle of a simplex in $\mathbb{R}^d$? We show that in dimensions three it is not greater than the solid angle of the regular simplex. And in dimension four the same holds for simplices sufficiently close to the regular simplex... | \section{Introduction}
In~\cite[Question 7]{karasev2015bounds} it was conjectured that every simplex in $\mathbb R^d$ has a solid angle at a vertex, whose $(d-1)$-dimensional spherical measure is at most that of the solid angle of a regular simplex. In this note we confirm this conjecture for dimension $d=3$ and show ... | {
"timestamp": "2016-04-07T02:09:08",
"yymm": "1505",
"arxiv_id": "1505.05263",
"language": "en",
"url": "https://arxiv.org/abs/1505.05263",
"abstract": "In this article we study the following question: What can be the measure of the minimal solid angle of a simplex in $\\mathbb{R}^d$? We show that in dimen... |
https://arxiv.org/abs/1811.03570 | Dynamics and stationary configurations of heterogeneous foams | We consider the variational foam model, where the goal is to minimize the total surface area of a collection of bubbles subject to the constraint that the volume of each bubble is prescribed. We apply sharp interface methods to develop an efficient computational method for this problem. In addition to simulating time d... | \section{Introduction}
We consider the model for a $d$-dimensional foam ($d=2,3$) comprised of $n$ bubbles, $\{\Omega_i \}_{i=1}^n$, each with a prescribed volume, $ \mathcal H^d(\Omega_i) = V_i$, that arrange themselves as to minimize the total surface area,
\begin{equation}
\label{e:min}
\min_{ \mathcal H^d(\Omega... | {
"timestamp": "2018-11-09T02:19:26",
"yymm": "1811",
"arxiv_id": "1811.03570",
"language": "en",
"url": "https://arxiv.org/abs/1811.03570",
"abstract": "We consider the variational foam model, where the goal is to minimize the total surface area of a collection of bubbles subject to the constraint that the... |
https://arxiv.org/abs/1503.08235 | Convergence properties of the randomized extended Gauss-Seidel and Kaczmarz methods | The Kaczmarz and Gauss-Seidel methods both solve a linear system $\bf{X}\bf{\beta} = \bf{y}$ by iteratively refining the solution estimate. Recent interest in these methods has been sparked by a proof of Strohmer and Vershynin which shows the randomized Kaczmarz method converges linearly in expectation to the solution.... | \section{Introduction}
We consider solving a linear system of equations
\begin{equation}\label{eq:syseq}
{\boldsymbol{X}} {\boldsymbol{\beta}} = {\boldsymbol{y}},
\end{equation}
for a (real or complex) $m\times n$ matrix ${\boldsymbol{X}}$, in various problem settings. Recent interest in the topic was reignited when... | {
"timestamp": "2015-08-26T02:11:46",
"yymm": "1503",
"arxiv_id": "1503.08235",
"language": "en",
"url": "https://arxiv.org/abs/1503.08235",
"abstract": "The Kaczmarz and Gauss-Seidel methods both solve a linear system $\\bf{X}\\bf{\\beta} = \\bf{y}$ by iteratively refining the solution estimate. Recent int... |
https://arxiv.org/abs/1907.09120 | Queens in exile: non-attacking queens on infinite chess boards | Number the cells of a (possibly infinite) chessboard in some way with the numbers 0, 1, 2, ... Consider the cells in order, placing a queen in a cell if and only if it would not attack any earlier queen. The problem is to determine the positions of the queens. We study the problem for a doubly-infinite chessboard of si... | \section{Queens in exile}\label{Sec1}
\begin{figure}[!ht]
\centerline{\includegraphics[angle=0, width=5in]{Tenniel.jpg}}
\caption{The great plain of Attak\'{i}a. [John Tenniel, Illustration for Lewis Carroll,
\emph{Through the looking-glass and what Alice found there} (1871).] }
\label{Fig1}
\end{figure}
The rival qu... | {
"timestamp": "2019-07-30T02:15:39",
"yymm": "1907",
"arxiv_id": "1907.09120",
"language": "en",
"url": "https://arxiv.org/abs/1907.09120",
"abstract": "Number the cells of a (possibly infinite) chessboard in some way with the numbers 0, 1, 2, ... Consider the cells in order, placing a queen in a cell if a... |
https://arxiv.org/abs/1407.2860 | Increasing subsequences of random walks | Given a sequence of $n$ real numbers $\{S_i\}_{i\leq n}$, we consider the longest weakly increasing subsequence, namely $i_1<i_2<\dots <i_L$ with $S_{i_k} \leq S_{i_{k+1}}$ and $L$ maximal. When the elements $S_i$ are i.i.d. uniform random variables, Vershik and Kerov, and Logan and Shepp proved that $\mathbb{E} L=(2+o... | \section{Introduction}
For a function $S\colon \mathbb N\to \mathbb R$, its restriction to a subset $A$ of its
domain is denoted $S|_A$. We say that $S|_A$ is {\bf increasing} if
$S(a)\leq S(b)$ for all $a,b\in A$ with $a\leq b$. Define
\[
\operatorname{LIS}(S|_{[0,n)})=\max\{|A|: A\subset [0,n), S|_A \text{ is incre... | {
"timestamp": "2014-12-24T02:09:17",
"yymm": "1407",
"arxiv_id": "1407.2860",
"language": "en",
"url": "https://arxiv.org/abs/1407.2860",
"abstract": "Given a sequence of $n$ real numbers $\\{S_i\\}_{i\\leq n}$, we consider the longest weakly increasing subsequence, namely $i_1<i_2<\\dots <i_L$ with $S_{i_... |
https://arxiv.org/abs/2006.08968 | Constructing abelian extensions with prescribed norms | Given a number field $K$, a finite abelian group $G$ and finitely many elements $\alpha_1,\ldots,\alpha_t\in K$, we construct abelian extensions $L/K$ with Galois group $G$ that realise all of the elements $\alpha_1,\ldots,\alpha_t$ as norms of elements in $L$. In particular, this shows existence of such extensions for... | \section{Introduction}
Attached to each extension $L/K$ of number fields comes the field-theoretic norm map $\norm_{L/K}:L^\times \to K^\times$. Given an extension $L/K$, a classical problem is to study which elements of $K$ are in the image of this norm map. We consider the inverse problem: given elements $\alpha_1,\... | {
"timestamp": "2021-04-13T02:32:55",
"yymm": "2006",
"arxiv_id": "2006.08968",
"language": "en",
"url": "https://arxiv.org/abs/2006.08968",
"abstract": "Given a number field $K$, a finite abelian group $G$ and finitely many elements $\\alpha_1,\\ldots,\\alpha_t\\in K$, we construct abelian extensions $L/K$... |
https://arxiv.org/abs/0804.0867 | Clique percolation | Derenyi, Palla and Vicsek introduced the following dependent percolation model, in the context of finding communities in networks. Starting with a random graph $G$ generated by some rule, form an auxiliary graph $G'$ whose vertices are the $k$-cliques of $G$, in which two vertices are joined if the corresponding clique... | \section{Cliques sharing vertices}\label{sec_cv}
Fix $k\ge 2$ and $1\le \ell\le k-1$. Given a graph
$G$, let $\Gkl$ be the graph whose vertex set is the set of all copies of $K_k$
in $G$, in which two vertices are adjacent if the corresponding copies
of $K_k$ share at least $\ell$ vertices.
Starting from a random grap... | {
"timestamp": "2008-09-19T21:50:01",
"yymm": "0804",
"arxiv_id": "0804.0867",
"language": "en",
"url": "https://arxiv.org/abs/0804.0867",
"abstract": "Derenyi, Palla and Vicsek introduced the following dependent percolation model, in the context of finding communities in networks. Starting with a random gr... |
https://arxiv.org/abs/1803.05033 | Limiting probabilities for vertices of a given rank in rooted trees | We consider two varieties of labeled rooted trees, and the probability that a vertex chosen from all vertices of all trees of a given size uniformly at random has a given rank. We prove that this probability converges to a limit as the tree size goes to infinity. | \section{Introduction}
Let $\cal T$ be a class of rooted labeled trees. If $v$ is a vertex of a tree $T\in \cal T$, then let the \emph{ rank} of $v$ be the number of edges in the
shortest path from $v$ to a leaf of $T$ that is a descendant of $v$. So leaves are of rank 0, neighbors of leaves are of rank 1, and so... | {
"timestamp": "2018-03-15T01:02:28",
"yymm": "1803",
"arxiv_id": "1803.05033",
"language": "en",
"url": "https://arxiv.org/abs/1803.05033",
"abstract": "We consider two varieties of labeled rooted trees, and the probability that a vertex chosen from all vertices of all trees of a given size uniformly at ra... |
https://arxiv.org/abs/2012.12976 | A Plethora of Polynomials: A Toolbox for Counting Problems | A wide variety of problems in combinatorics and discrete optimization depend on counting the set $S$ of integer points in a polytope, or in some more general object constructed via discrete geometry and first-order logic. We take a tour through numerous problems of this type. In particular, we consider families of such... | \section{Introduction.}
\label{sec:intro}
It's no surprise that geometry is a fertile source of polynomial functions. For example, if we take a bounded $d$-dimensional object, $B$, and dilate by a factor of $t$, then $\vol(tB)=\vol(B)t^d$ is a degree $d$ polynomial in $t$. It's more surprising that we can add considera... | {
"timestamp": "2020-12-25T02:02:10",
"yymm": "2012",
"arxiv_id": "2012.12976",
"language": "en",
"url": "https://arxiv.org/abs/2012.12976",
"abstract": "A wide variety of problems in combinatorics and discrete optimization depend on counting the set $S$ of integer points in a polytope, or in some more gene... |
https://arxiv.org/abs/1707.09417 | An Invitation to Polynomiography via Exponential Series | The subject of Polynomiography deals with algorithmic visualization of polynomial equations, having many applications in STEM and art, see [Kal04]. Here we consider the polynomiography of the partial sums of the exponential series. While the exponential function is taught in standard calculus courses, it is unlikely th... | \section{Introduction} Ever since introducing the term \emph{polynomiography} for the visualization of polynomial equations via iteration functions, when encountering certain polynomials I have found it tempting to consider the shape of their \emph{polynomiographs} in the complex plane. The word \emph{polynomiograph... | {
"timestamp": "2017-08-01T02:01:57",
"yymm": "1707",
"arxiv_id": "1707.09417",
"language": "en",
"url": "https://arxiv.org/abs/1707.09417",
"abstract": "The subject of Polynomiography deals with algorithmic visualization of polynomial equations, having many applications in STEM and art, see [Kal04]. Here w... |
https://arxiv.org/abs/2012.05145 | Decomposition of $(2k+1)$-regular graphs containing special spanning $2k$-regular Cayley graphs into paths of length $2k+1$ | A $P_\ell$-decomposition of a graph $G$ is a set of paths with $\ell$ edges in $G$ that cover the edge set of $G$. Favaron, Genest, and Kouider (2010) conjectured that every $(2k+1)$-regular graph that contains a perfect matching admits a $P_{2k+1}$-decomposition. They also verified this conjecture for $5$-regular grap... | \section{Introduction}\label{sec:introduction}
All graphs in this paper are simple, i.e.,
have no loops nor multiple edges.
A \emph{decomposition} of a graph \(G\) is a set \({\mathcal{D}}\)
of edge-disjoint subgraphs of \(G\) that cover its edge set.
If every element of \({\mathcal{D}}\) is isomorphic to a fixe... | {
"timestamp": "2020-12-10T02:25:34",
"yymm": "2012",
"arxiv_id": "2012.05145",
"language": "en",
"url": "https://arxiv.org/abs/2012.05145",
"abstract": "A $P_\\ell$-decomposition of a graph $G$ is a set of paths with $\\ell$ edges in $G$ that cover the edge set of $G$. Favaron, Genest, and Kouider (2010) c... |
https://arxiv.org/abs/2110.09200 | All Graphs with a Failed Zero Forcing Number of Two | Given a graph $G$, the zero-forcing number of $G$, $Z(G)$, is the smallest cardinality of any set $S$ of vertices on which repeated applications of the forcing rule results in all vertices being in $S$. The forcing rule is: if a vertex $v$ is in $S$, and exactly one neighbor $u$ of $v$ is not in $S$, then $u$ is added ... | \section{Introduction}
Given a graph $G$, the zero-forcing number of $G$, $Z(G)$, is the smallest cardinality of any set $S$ of vertices on which repeated applications of the forcing rule results in all vertices being in $S$. The forcing rule is: if a vertex $v$ is in $S$, and exactly one neighbor $u$ of $v$ is not in ... | {
"timestamp": "2021-10-19T02:35:18",
"yymm": "2110",
"arxiv_id": "2110.09200",
"language": "en",
"url": "https://arxiv.org/abs/2110.09200",
"abstract": "Given a graph $G$, the zero-forcing number of $G$, $Z(G)$, is the smallest cardinality of any set $S$ of vertices on which repeated applications of the fo... |
https://arxiv.org/abs/1804.01740 | On large primitive subsets of $\{1,2,\ldots,2n\}$ | A subset of $\{1,2,\ldots,2n\}$ is said to be primitive if it does not contain any pair of elements $(u,v)$ such that $u$ is a divisor of $v$. Let $D(n)$ denote the number of primitive subsets of $\{1,2,\ldots,2n\}$ with $n$ elements. Numerical evidence suggests that $D(n)$ is roughly $(1.32)^n$. We show that for suffi... | \section{\@startsection {section}{1}{\z@}
{-30pt \@plus -1ex \@minus -.2ex}
{2.3ex \@plus.2ex}
{\normalfont\normalsize\bfseries}}
\renewcommand\subsection{\@startsection{subsection}{2}{\z@}
{-3.25ex\@plus -1ex \@minus -.2ex}
{1.5ex \@plus .2ex}
{\normalfont\normalsize\bfseries}}
\renewcommand{\@seccntformat}[1]{\csna... | {
"timestamp": "2018-04-06T02:06:18",
"yymm": "1804",
"arxiv_id": "1804.01740",
"language": "en",
"url": "https://arxiv.org/abs/1804.01740",
"abstract": "A subset of $\\{1,2,\\ldots,2n\\}$ is said to be primitive if it does not contain any pair of elements $(u,v)$ such that $u$ is a divisor of $v$. Let $D(n... |
https://arxiv.org/abs/1106.5904 | Turàn numbers of Multiple Paths and Equibipartite Trees | The Turán number of a graph H, ex(n;H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let P_l denote a path on l vertices, and kP_l denote k vertex-disjoint copies of P_l. We determine ex(n, kP_3) for n appropriately large, answering in the positive a conjecture of Go... | \section{Introduction}
Our notation in this paper is standard (see, e.g., \cite{mgt}). Thus \(G\cup H\) denotes the disjoint union of graphs \(G\) and \(H\), and we write \(G+H\) for the join of \(G\) and \(H\), the graph obtained from \(G\cup H\) by adding edges between all vertices of \(G\) and all vertices of \(H\),... | {
"timestamp": "2011-08-23T02:01:36",
"yymm": "1106",
"arxiv_id": "1106.5904",
"language": "en",
"url": "https://arxiv.org/abs/1106.5904",
"abstract": "The Turán number of a graph H, ex(n;H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let P_l denote a p... |
https://arxiv.org/abs/1911.08418 | Fast Convergence of Fictitious Play for Diagonal Payoff Matrices | Fictitious Play (FP) is a simple and natural dynamic for repeated play in zero-sum games. Proposed by Brown in 1949, FP was shown to converge to a Nash Equilibrium by Robinson in 1951, albeit at a slow rate that may depend on the dimension of the problem. In 1959, Karlin conjectured that FP converges at the more natura... |
\section{Introduction}
In a two-player zero-sum game, we are given a payoff matrix $A \in \R^{n\times m}$, whose $ij$-th entry denotes how much the row player pays the column player when the two players play actions $i$ and $j$ respectively. When each player selects their actions randomly, with the row player samplin... | {
"timestamp": "2020-11-17T02:24:43",
"yymm": "1911",
"arxiv_id": "1911.08418",
"language": "en",
"url": "https://arxiv.org/abs/1911.08418",
"abstract": "Fictitious Play (FP) is a simple and natural dynamic for repeated play in zero-sum games. Proposed by Brown in 1949, FP was shown to converge to a Nash Eq... |
https://arxiv.org/abs/1705.03096 | Partial Domination in Graphs | A set $S\subseteq V$ is a dominating set of $G$ if every vertex in $V - S$ is adjacent to at least one vertex in $S$. The domination number $\gamma(G)$ of $G$ equals the minimum cardinality of a dominating set $S$ in $G$; we say that such a set $S$ is a $\gamma$-set. The single greatest focus of research in domination ... | \section{Introduction}
Let $G=(V,E)$ be a graph with vertex set $V=\{v_1,v_2,...,v_n\}$ and \emph{order} $n = |V|$. The {\em open neighborhood} of a vertex $v$ is the set $N(v) := \{u\: |\: uv \in E\} $ of vertices $u$ that are adjacent to $v$; the \emph{closed neighborhood} of $v$, $N[v]:=N(v)\cup \{v\}.$ A set $S\... | {
"timestamp": "2017-05-10T02:01:55",
"yymm": "1705",
"arxiv_id": "1705.03096",
"language": "en",
"url": "https://arxiv.org/abs/1705.03096",
"abstract": "A set $S\\subseteq V$ is a dominating set of $G$ if every vertex in $V - S$ is adjacent to at least one vertex in $S$. The domination number $\\gamma(G)$ ... |
https://arxiv.org/abs/1407.5598 | Fractional Gaussian fields: a survey | We discuss a family of random fields indexed by a parameter $s\in \mathbb{R}$ which we call the fractional Gaussian fields, given by\[\mathrm{FGF}_s(\mathbb{R}^d)=(-\Delta)^{-s/2} W,\] where $W$ is a white noise on $\mathbb{R}^d$ and $(-\Delta)^{-s/2}$ is the fractional Laplacian. These fields can also be parameterized... | \section{Introduction}\label{sec:Introduction}
\makeatletter{}\begin{figure}[hpt]
\center
\subfigure[White Noise, $s=0$]{\includegraphics[width=0.45\textwidth]{./figures/whitenoise}}
\subfigure[GFF,
$s=1$]{\includegraphics[width=0.45\textwidth]{./figures/gff}} \\
\subfigure[Bi-Laplacian,
$s=2$]{\includegr... | {
"timestamp": "2016-02-08T02:10:26",
"yymm": "1407",
"arxiv_id": "1407.5598",
"language": "en",
"url": "https://arxiv.org/abs/1407.5598",
"abstract": "We discuss a family of random fields indexed by a parameter $s\\in \\mathbb{R}$ which we call the fractional Gaussian fields, given by\\[\\mathrm{FGF}_s(\\m... |
https://arxiv.org/abs/0902.0315 | Angle contraction between geodesics | We consider here a generalization of a well known discrete dynamical system produced by the bisection of reflection angles that are constructed recursively between two lines in the Euclidean plane. It is shown that similar properties of such systems are observed when the plane is replaced by a regular surface in ${\mat... | \section{Introduction}
Let us consider the following simple problem. In the Euclidean plane
take two rays starting at point $V$ and forming an acute angle of
measure $\mu$. Denote the rays by $L_A$ and $L_B$ and take an
arbitrary transversal segment $A_1B_1$ with $A_1\in L_A$, and
$B_1\in L_B$. Keep constructing furth... | {
"timestamp": "2009-02-02T18:00:03",
"yymm": "0902",
"arxiv_id": "0902.0315",
"language": "en",
"url": "https://arxiv.org/abs/0902.0315",
"abstract": "We consider here a generalization of a well known discrete dynamical system produced by the bisection of reflection angles that are constructed recursively ... |
https://arxiv.org/abs/2301.10131 | Random perfect matchings in regular graphs | We prove that in all regular robust expanders $G$ every edge is asymptotically equally likely contained in a uniformly chosen perfect matching $M$. We also show that given any fixed matching or spanning regular graph $N$ in $G$, the random variable $|M\cap E(N)|$ is approximately Poisson distributed. This in particular... |
\section{Introduction}
A remarkable result due to Kahn and Kim~\cite{kahn1998random} says that in \emph{any} $d$-regular graph $G$
the probability that a vertex is contained in a uniformly chosen matching in $G$ is $1-(1+o_d(1))d^{-\frac{1}{2}}$.
This shows that the structure of a $d$-regular graph has essentiall... | {
"timestamp": "2023-01-25T02:15:47",
"yymm": "2301",
"arxiv_id": "2301.10131",
"language": "en",
"url": "https://arxiv.org/abs/2301.10131",
"abstract": "We prove that in all regular robust expanders $G$ every edge is asymptotically equally likely contained in a uniformly chosen perfect matching $M$. We als... |
https://arxiv.org/abs/1803.03602 | Weyl's polarization theorem in positive characteristic | Let $V$ be an $n$-dimensional algebraic representation over an algebraically closed field $K$ of a group $G$. For $m > 0$, we study the invariant rings $K[V^{ m}]^G$ for the diagonal action of $G$ on $V^m$. In characteristic zero, a theorem of Weyl tells us that we can obtain all the invariants in $K[V^m]^G$ by the pro... | \section{Introduction}
Let $K$ be an algebraically closed field. Suppose $V$ is a rational representation of a reductive group $G$. The ring of invariant polynomials $K[V]^G$ is a finitely generated graded subalgebra of the coordinate ring $K[V]$, see \cite{Haboush,Hilbert1,Hilbert2,Nagata}. A long standing theme in in... | {
"timestamp": "2018-11-27T02:33:34",
"yymm": "1803",
"arxiv_id": "1803.03602",
"language": "en",
"url": "https://arxiv.org/abs/1803.03602",
"abstract": "Let $V$ be an $n$-dimensional algebraic representation over an algebraically closed field $K$ of a group $G$. For $m > 0$, we study the invariant rings $K... |
https://arxiv.org/abs/1312.7555 | Cops and Robbers on diameter two graphs | In this short paper we study the game of Cops and Robbers, played on the vertices of some fixed graph $G$ of order $n$. The minimum number of cops required to capture a robber is called the cop number of $G$. We show that the cop number of graphs of diameter 2 is at most $\sqrt{2n}$, improving a recent result of Lu and... | \section{Introduction}
The game of \emph{Cops and Robbers}, introduced independently by Nowakowski and Winkler \cite{nowwi} and Quillot \cite{qui}, is a perfect information game played on a fixed graph $G$. There are two players, a set of $k\geq 1$ cops and the robber. The cops begin the game by occupying any vertic... | {
"timestamp": "2014-09-30T02:10:07",
"yymm": "1312",
"arxiv_id": "1312.7555",
"language": "en",
"url": "https://arxiv.org/abs/1312.7555",
"abstract": "In this short paper we study the game of Cops and Robbers, played on the vertices of some fixed graph $G$ of order $n$. The minimum number of cops required ... |
https://arxiv.org/abs/0902.1321 | Jeu de taquin and a monodromy problem for Wronskians of polynomials | The Wronskian associates to d linearly independent polynomials of degree at most n, a non-zero polynomial of degree at most d(n-d). This can be viewed as giving a flat, finite morphism from the Grassmannian Gr(d,n) to projective space of the same dimension. In this paper, we study the monodromy groupoid of this map. Wh... | \section{Introduction}
\subsection{The Wronski map}
For any non-negative integer $m$,
let $\Fpol{m}$ denote the $(m{+}1)$-dimensional
vector space of polynomials of degree at most $m$ over a field $\mathbb{F}$:
$$\Fpol{m} := \{f(z) \in \mathbb{F}[z] \mid \deg f(z) \leq m\}\,. $$
Throughout, we fix integers $0<d<n$. ... | {
"timestamp": "2009-09-13T17:30:38",
"yymm": "0902",
"arxiv_id": "0902.1321",
"language": "en",
"url": "https://arxiv.org/abs/0902.1321",
"abstract": "The Wronskian associates to d linearly independent polynomials of degree at most n, a non-zero polynomial of degree at most d(n-d). This can be viewed as gi... |
https://arxiv.org/abs/2209.01303 | Acyclic Orientations and the Chromatic Polynomial of Signed Graphs | We present a new correspondence between acyclic orientations and coloring of a signed graph (symmetric graph). Goodall et al. introduced a bivariate chromatic polynomial $\chi_G(k,l)$ that counts the number of signed colorings using colors $0,\pm1,\dots,\pm k$ along with $l-1$ symmetric colors $0_1,\dots,0_{l-1}$. We s... | \section{Introduction}
In 1973, Richard Stanley \cite{Stanl_Acyclicorientations_73} found a simple yet astonishing connection between the chromatic polynomial and acyclic orientations of a graph. Let $\chi_G$ be the chromatic polynomial of $G$. He showed that $\chi_G(-1)$ is, up to a sign, equal to the number of acycl... | {
"timestamp": "2022-09-07T02:03:39",
"yymm": "2209",
"arxiv_id": "2209.01303",
"language": "en",
"url": "https://arxiv.org/abs/2209.01303",
"abstract": "We present a new correspondence between acyclic orientations and coloring of a signed graph (symmetric graph). Goodall et al. introduced a bivariate chrom... |
https://arxiv.org/abs/1902.08348 | Weighted Fekete points on the real line and the unit circle | Weighted Fekete points are defined as those that maximize the weighted version of the Vandermonde determinant over a fixed set. They can also be viewed as the equilibrium distribution of the unit discrete charges in an external electrostatic field. While these points have many applications, they are very difficult to f... | \section{Weighted Fekete points on the real line} \label{FPR}
For a set of points $Z_n=\{z_k\}_{k=1}^n\subset{\mathbb C},\ n\ge 2$, the associated Vandermonde determinant is defined by
\[
V(Z_n):=\prod_{1\le j<k\le n} (z_j-z_k).
\]
Fekete \cite{Fe} introduced the notion of the $n$th diameter for a compact set $E\subse... | {
"timestamp": "2019-02-25T02:08:06",
"yymm": "1902",
"arxiv_id": "1902.08348",
"language": "en",
"url": "https://arxiv.org/abs/1902.08348",
"abstract": "Weighted Fekete points are defined as those that maximize the weighted version of the Vandermonde determinant over a fixed set. They can also be viewed as... |
https://arxiv.org/abs/2106.11863 | Graph coarsening: From scientific computing to machine learning | The general method of graph coarsening or graph reduction has been a remarkably useful and ubiquitous tool in scientific computing and it is now just starting to have a similar impact in machine learning. The goal of this paper is to take a broad look into coarsening techniques that have been successfully deployed in s... | \section{Introduction}\label{sec:introduction}
The idea of `coarsening,' i.e., exploiting a smaller set in place of a
larger or `finer' set has had numerous uses across many disciplines of
science and engineering. The term `coarsening' employed here is
prevalent in scientific computing, where it refers to the usage of ... | {
"timestamp": "2021-06-23T02:25:05",
"yymm": "2106",
"arxiv_id": "2106.11863",
"language": "en",
"url": "https://arxiv.org/abs/2106.11863",
"abstract": "The general method of graph coarsening or graph reduction has been a remarkably useful and ubiquitous tool in scientific computing and it is now just star... |
https://arxiv.org/abs/0911.2077 | Central Binomial Tail Bounds | An alternate form for the binomial tail is presented, which leads to a variety of bounds for the central tail. A few can be weakened into the corresponding Chernoff and Slud bounds, which not only demonstrates the quality of the presented bounds, but also provides alternate proofs for the classical bounds. | \section{Introduction}
Let $B(p,n)$ denote a binomial random variable comprising $n$ flips of
a bias-$p$ coin, and set $\sigma = \sqrt{p(1-p)}$. The classical form of the
central tail, obtained by summing over the possible outcomes, is
\begin{equation}
\label{eq:classical_tail}
\P[B(p,n) \geq n/2] = \sum_{h=\lceil n/2... | {
"timestamp": "2010-04-07T02:01:07",
"yymm": "0911",
"arxiv_id": "0911.2077",
"language": "en",
"url": "https://arxiv.org/abs/0911.2077",
"abstract": "An alternate form for the binomial tail is presented, which leads to a variety of bounds for the central tail. A few can be weakened into the corresponding ... |
https://arxiv.org/abs/1701.01649 | Existence of Some Signed Magic Arrays | We consider the notion of a signed magic array, which is an $m \times n$ rectangular array with the same number of filled cells $s$ in each row and the same number of filled cells $t$ in each column, filled with a certain set of numbers that is symmetric about the number zero, such that every row and column has a zero ... | \section{Introduction}\label{introduction}
A {\em magic rectangle} is defined as an $m \times n$ array whose entries are precisely the integers from $0$ to $mn-1$ wherein the sum of each row is $c$ and the sum of each column is $r$.
A {\em magic square} is a magic rectangle with $m=n$ and $c=r$.
In \cite{sun} it is pr... | {
"timestamp": "2017-01-09T02:06:23",
"yymm": "1701",
"arxiv_id": "1701.01649",
"language": "en",
"url": "https://arxiv.org/abs/1701.01649",
"abstract": "We consider the notion of a signed magic array, which is an $m \\times n$ rectangular array with the same number of filled cells $s$ in each row and the s... |
https://arxiv.org/abs/0912.2044 | Approximation of projections of random vectors | Let $X$ be a $d$-dimensional random vector and $X_\theta$ its projection onto the span of a set of orthonormal vectors $\{\theta_1,...,\theta_k\}$. Conditions on the distribution of $X$ are given such that if $\theta$ is chosen according to Haar measure on the Stiefel manifold, the bounded-Lipschitz distance from $X_\t... | \section{Introduction}
There is a large class of results dealing with random variables (or
measures) defined in terms of a parameter (say, a point on the
sphere), which say that for a large measure of these parameters, the
behavior of the random variable is well-approximated by some model
distribution. Early work in t... | {
"timestamp": "2011-02-16T02:02:44",
"yymm": "0912",
"arxiv_id": "0912.2044",
"language": "en",
"url": "https://arxiv.org/abs/0912.2044",
"abstract": "Let $X$ be a $d$-dimensional random vector and $X_\\theta$ its projection onto the span of a set of orthonormal vectors $\\{\\theta_1,...,\\theta_k\\}$. Con... |
https://arxiv.org/abs/1910.08548 | An introduction to multiple orthogonal polynomials and Hermite-Padé approximation | We present a brief introduction to the theory of multiple orthogonal polynomials on the basis of known results for an important class of measures known as Nikishin systems. For type I and type II multiple orthogonal polynomials with respect to such systems of measures, we describe some of their most relevant properties... | \section{Introduction} \label{sec:HP}
The object of this paper is to provide an introduction to the study of Hermite-Pad\'e approximation, multiple orthogonal polynomials, and some of their asymptotic properties. For the most part, the attention is restricted to the case of multiple orthogonality with respect to an ... | {
"timestamp": "2019-10-22T02:00:16",
"yymm": "1910",
"arxiv_id": "1910.08548",
"language": "en",
"url": "https://arxiv.org/abs/1910.08548",
"abstract": "We present a brief introduction to the theory of multiple orthogonal polynomials on the basis of known results for an important class of measures known as... |
https://arxiv.org/abs/1801.08564 | An Asymptotically Tight Bound on the Number of Relevant Variables in a Bounded Degree Boolean Function | We prove that there is a constant $C\leq 6.614$ such that every Boolean function of degree at most $d$ (as a polynomial over $\mathbb{R}$) is a $C\cdot 2^d$-junta, i.e. it depends on at most $C\cdot 2^d$ variables. This improves the $d\cdot 2^{d-1}$ upper bound of Nisan and Szegedy [Computational Complexity 4 (1994)]. ... | \section{Introduction}
The \emph{degree} of a Boolean function $f:\{0,1\}^n\rightarrow\{0,1\}$, denoted $\deg(f)$, is the minimum degree of a polynomial in ${\mathbb R}[x_1,...,x_n]$ that agrees with $f$ on all inputs from $\{0,1\}^n$. (It is well known that every Boolean function
has a unique representation over th... | {
"timestamp": "2018-11-20T02:41:21",
"yymm": "1801",
"arxiv_id": "1801.08564",
"language": "en",
"url": "https://arxiv.org/abs/1801.08564",
"abstract": "We prove that there is a constant $C\\leq 6.614$ such that every Boolean function of degree at most $d$ (as a polynomial over $\\mathbb{R}$) is a $C\\cdot... |
https://arxiv.org/abs/2001.02789 | Gallai Ramsey number for double stars | Given a graph $G$ and a positive integer $k$, the \emph{Gallai-Ramsey number} is defined to be the minimum number of vertices $n$ such that any $k$-edge coloring of $K_n$ contains either a rainbow (all different colored) copy of $G$ or a monochromatic copy of $G$. In this paper, we obtain general upper and lower bounds... | \section{Introduction}
In this work, we consider only colorings of the edges of graphs. A coloring of a graph $G$ is called \emph{rainbow} if no two edges in $G$ have the same color.
Edge colorings of complete graphs that contain no rainbow triangle have very interesting and somewhat surprising structure. In 1967, Ga... | {
"timestamp": "2020-01-10T02:03:18",
"yymm": "2001",
"arxiv_id": "2001.02789",
"language": "en",
"url": "https://arxiv.org/abs/2001.02789",
"abstract": "Given a graph $G$ and a positive integer $k$, the \\emph{Gallai-Ramsey number} is defined to be the minimum number of vertices $n$ such that any $k$-edge ... |
https://arxiv.org/abs/2101.09830 | A Generalization of QR Factorization To Non-Euclidean Norms | I propose a way to use non-Euclidean norms to formulate a QR-like factorization which can unlock interesting and potentially useful properties of non-Euclidean norms - for example the ability of $l^1$ norm to suppresss outliers or promote sparsity. A classic QR factorization of a matrix $\mathbf{A}$ computes an upper t... | \section{Introduction}
\label{sec:introduction}
The QR factorization allows for highly stable, robust, and efficient matrix factorization with
beneficial properties to many areas of numerical linear algebra. The factorization may be
implemented using only stable operations such as Householder reflectors or Givens rota... | {
"timestamp": "2021-01-26T02:26:17",
"yymm": "2101",
"arxiv_id": "2101.09830",
"language": "en",
"url": "https://arxiv.org/abs/2101.09830",
"abstract": "I propose a way to use non-Euclidean norms to formulate a QR-like factorization which can unlock interesting and potentially useful properties of non-Eucl... |
https://arxiv.org/abs/0907.2336 | Bounding the rational sums of squares over totally real fields | Let K be a totally real Galois number field. C. J. Hillar proved that if f in Q[x_1,...,x_n] is a sum of m squares in K[x_1,...,x_n], then f is a sum of N(m) squares in Q[x_1,...,x_n]. Modifying Hillar's proof, we improve the improve the bound given for N(m), the proof being constructive as well. | \section{Introduction}
In the theory of semidefinite linear programming, there is a question by Sturmfels
\begin{question}[Sturmfels]
{If $f\in{\Bbb Q}[x_1,\ldots,x_n]$ is a sum of squares in ${\Bbb R}[x_1,\ldots,x_n]$, then is $f$ also a sum of squares in ${\Bbb Q}[x_1,\ldots,x_n]$ ?}
\end{question}
Hillar (\cite... | {
"timestamp": "2009-07-14T11:27:59",
"yymm": "0907",
"arxiv_id": "0907.2336",
"language": "en",
"url": "https://arxiv.org/abs/0907.2336",
"abstract": "Let K be a totally real Galois number field. C. J. Hillar proved that if f in Q[x_1,...,x_n] is a sum of m squares in K[x_1,...,x_n], then f is a sum of N(m... |
https://arxiv.org/abs/1710.10663 | List-decodable zero-rate codes | We consider list-decoding in the zero-rate regime for two cases: the binary alphabet and the spherical codes in Euclidean space. Specifically, we study the maximal $\tau \in [0,1]$ for which there exists an arrangement of $M$ balls of relative Hamming radius $\tau$ in the binary hypercube (of arbitrary dimension) with ... | \section{Introduction}\label{sec:intro}
This work concerns list-decoding under \emph{worst-case} errors in
the zero-rate regime. We consider the case of the binary
alphabet in Sections~\ref{sec:intro}-\ref{sec:proof_three} and the case of the unit sphere in Hilbert space in
Section~\ref{sec:hilbert}.
To motivate o... | {
"timestamp": "2018-05-15T02:20:12",
"yymm": "1710",
"arxiv_id": "1710.10663",
"language": "en",
"url": "https://arxiv.org/abs/1710.10663",
"abstract": "We consider list-decoding in the zero-rate regime for two cases: the binary alphabet and the spherical codes in Euclidean space. Specifically, we study th... |
https://arxiv.org/abs/2002.07248 | Tournaments and the Strong Erdős-Hajnal Property | A conjecture of Alon, Pach and Solymosi, which is equivalent to the celebrated Erdős-Hajnal Conjecture, states that for every tournament $S$ there exists $\epsilon(S)>0$ such that if $T$ is an $n$-vertex tournament that does not contains $S$ as a subtournament, then $T$ contains a transitive subtournament on at least $... | \section{introduction}
A {\em tournament} is a complete graph with directions on edges. A tournament is {\em transitive} if it has no directed triangles. For tournaments $S,T$ we say that $T$ is {\em $S$-free} if no subtournament of $T$ is
isomorphic to $S$.
In \cite{APS} a conjecture was made
concerning tournament... | {
"timestamp": "2021-09-15T02:28:07",
"yymm": "2002",
"arxiv_id": "2002.07248",
"language": "en",
"url": "https://arxiv.org/abs/2002.07248",
"abstract": "A conjecture of Alon, Pach and Solymosi, which is equivalent to the celebrated Erdős-Hajnal Conjecture, states that for every tournament $S$ there exists ... |
https://arxiv.org/abs/2105.12239 | Finite sample guarantees for quantile estimation: An application to detector threshold tuning | In threshold-based anomaly detection, we want to tune the threshold of a detector to achieve an acceptable false alarm rate. However, tuning the threshold is often a non-trivial task due to unknown detector output distributions. A detector threshold that provides an acceptable false alarm rate is equivalent to a specif... | \section{Conclusion}
\label{sec:Conclusion}
In this work, we considered the tuning of detector thresholds and pointed out the equivalence of the detector threshold and a specific quantile of the detector output distribution.
We derived three different finite guarantees for the estimation of a quantile.
The first is ... | {
"timestamp": "2022-05-02T02:03:17",
"yymm": "2105",
"arxiv_id": "2105.12239",
"language": "en",
"url": "https://arxiv.org/abs/2105.12239",
"abstract": "In threshold-based anomaly detection, we want to tune the threshold of a detector to achieve an acceptable false alarm rate. However, tuning the threshold... |
https://arxiv.org/abs/2205.13347 | A Formalization of Finite Group Theory | Previous formulations of group theory in ACL2 and Nqthm, based on either "encapsulate" or "defn-sk", have been limited by their failure to provide a path to proof by induction on the order of a group, which is required for most interesting results in this domain beyond Lagrange's Theorem (asserting the divisibility of ... |
\section{Introduction}
Since ACL2 provides only limited support for quantification, modeling group theory in its logic is a challenging problem. A 1990 paper of Yuan
Yu \cite{yu} presents a formal development of finite group theory in Nqthm based on the {\tt defn-sk} macro (surviving in ACL2 as {\tt defun-sk}),
whic... | {
"timestamp": "2022-05-27T02:16:41",
"yymm": "2205",
"arxiv_id": "2205.13347",
"language": "en",
"url": "https://arxiv.org/abs/2205.13347",
"abstract": "Previous formulations of group theory in ACL2 and Nqthm, based on either \"encapsulate\" or \"defn-sk\", have been limited by their failure to provide a p... |
https://arxiv.org/abs/1912.08080 | Petruska's question on planar convex sets | Given $2k-1$ convex sets in $R^2$ such that no point of the plane is covered by more than $k$ of the sets, is it true that there are two among the convex sets whose union contains all $k$-covered points of the plane? This question due to Gy. Petruska has an obvious affirmative answer for $k=1,2,3$; we show here that th... | \section{Introduction
A family $\mathcal F$ of compact convex sets in $\R^2$
is called a ${\mathcal P}(k)$-family if
$|\mathcal F|=2k-1$ and no point of the plane is
contained in more than $k$ members of $\mathcal F$.
A ${\mathcal P}(1)$-family consists of one set,
thus a ${\mathcal P}(1)$-family is trivially co... | {
"timestamp": "2019-12-18T02:17:45",
"yymm": "1912",
"arxiv_id": "1912.08080",
"language": "en",
"url": "https://arxiv.org/abs/1912.08080",
"abstract": "Given $2k-1$ convex sets in $R^2$ such that no point of the plane is covered by more than $k$ of the sets, is it true that there are two among the convex ... |
https://arxiv.org/abs/2109.01080 | Optimization and Sampling Under Continuous Symmetry: Examples and Lie Theory | In the last few years, the notion of symmetry has provided a powerful and essential lens to view several optimization or sampling problems that arise in areas such as theoretical computer science, statistics, machine learning, quantum inference, and privacy. Here, we present two examples of nonconvex problems in optimi... | \section{Introduction}
In the words of Hermann Weyl:
``{\em A thing is symmetrical if there is something you can do to it so that after you have finished doing it, it looks the same as before.}''
\smallskip
Symmetries can be discrete or continuous.
For instance, the polynomial $f(x_1,\ldots, x_n):=x_1^2 + \cdots + ... | {
"timestamp": "2021-09-03T02:22:46",
"yymm": "2109",
"arxiv_id": "2109.01080",
"language": "en",
"url": "https://arxiv.org/abs/2109.01080",
"abstract": "In the last few years, the notion of symmetry has provided a powerful and essential lens to view several optimization or sampling problems that arise in a... |
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