url stringlengths 31 38 | title stringlengths 7 229 | abstract stringlengths 44 2.87k | text stringlengths 319 2.51M | meta dict |
|---|---|---|---|---|
https://arxiv.org/abs/2002.03448 | Kelly Criterion: From a Simple Random Walk to Lévy Processes | The original Kelly criterion provides a strategy to maximize the long-term growth of winnings in a sequence of simple Bernoulli bets with an edge, that is, when the expected return on each bet is positive. The objective of this work is to consider more general models of returns and the continuous time, or high frequenc... | \section{Introduction}
Consider repeatedly engaging in a game of chance where one side has an edge
and seeks to optimize the betting in a way that ensures maximal long-term
growth rate of the overall wealth. This problem was posed and analyzed
by John Kelly \cite{Kelly56} at Bell Labs in 1956; the solution was
impl... | {
"timestamp": "2020-02-11T02:18:16",
"yymm": "2002",
"arxiv_id": "2002.03448",
"language": "en",
"url": "https://arxiv.org/abs/2002.03448",
"abstract": "The original Kelly criterion provides a strategy to maximize the long-term growth of winnings in a sequence of simple Bernoulli bets with an edge, that is... |
https://arxiv.org/abs/2209.11307 | A combinatorial bound on the number of distinct eigenvalues of a graph | The smallest possible number of distinct eigenvalues of a graph $G$, denoted by $q(G)$, has a combinatorial bound in terms of unique shortest paths in the graph. In particular, $q(G)$ is bounded below by $k$, where $k$ is the number of vertices of a unique shortest path joining any pair of vertices in $G$. Thus, if $n$... | \section{Motivation and background}
The Inverse Eigenvalue Problem for a Graph (IEPG) starts with a pattern of
zero and nonzero constraints
for a real symmetric matrix, described by a graph $G$ on $n$ vertices, and asks what spectra are possible
for the set of $n \times n$ matrices $\mathcal{S}(G)$ that exhibit this p... | {
"timestamp": "2022-09-26T02:02:24",
"yymm": "2209",
"arxiv_id": "2209.11307",
"language": "en",
"url": "https://arxiv.org/abs/2209.11307",
"abstract": "The smallest possible number of distinct eigenvalues of a graph $G$, denoted by $q(G)$, has a combinatorial bound in terms of unique shortest paths in the... |
https://arxiv.org/abs/1209.5360 | Balanced Allocations and Double Hashing | Double hashing has recently found more common usage in schemes that use multiple hash functions. In double hashing, for an item $x$, one generates two hash values $f(x)$ and $g(x)$, and then uses combinations $(f(x) +k g(x)) \bmod n$ for $k=0,1,2,...$ to generate multiple hash values from the initial two. We first perf... | \section{Introduction}
\label{sec:introduction}
The standard balanced allocation paradigm works as follows: suppose
$n$ balls are sequentially placed into $n$ bins, where each ball is
placed in the least loaded of $d$ uniform independent choices of the
bins. Then the maximum load (that is, the maximum number of balls ... | {
"timestamp": "2014-01-30T02:13:43",
"yymm": "1209",
"arxiv_id": "1209.5360",
"language": "en",
"url": "https://arxiv.org/abs/1209.5360",
"abstract": "Double hashing has recently found more common usage in schemes that use multiple hash functions. In double hashing, for an item $x$, one generates two hash ... |
https://arxiv.org/abs/2203.03772 | The product structure of squaregraphs | A squaregraph is a plane graph in which each internal face is a $4$-cycle and each internal vertex has degree at least 4. This paper proves that every squaregraph is isomorphic to a subgraph of the semi-strong product of an outerplanar graph and a path. We generalise this result for infinite squaregraphs, and show that... | \section{Introduction}
\label{Introduction}
\footnotetext[3]{School of Mathematics, Monash University, Melbourne, Australia (\texttt{\{robert.hickingbotham,david.wood\}@monash.edu}). Research of R.H.\ supported by an Australian Government Research Training Program Scholarship. Research of D.W.\ supported by the Aus... | {
"timestamp": "2022-03-09T02:05:40",
"yymm": "2203",
"arxiv_id": "2203.03772",
"language": "en",
"url": "https://arxiv.org/abs/2203.03772",
"abstract": "A squaregraph is a plane graph in which each internal face is a $4$-cycle and each internal vertex has degree at least 4. This paper proves that every squ... |
https://arxiv.org/abs/1310.4112 | Subalgebras of the Fomin-Kirillov algebra | The Fomin-Kirillov algebra $\mathcal E_n$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on $n$ vertices. For any graph $G$ on $n$ vertices, we define $\mathcal E_G$ to be the subalgebra of $\mathcal E_n$ generated by the edges of $G$. We show that these algebras have many p... | \section{Introduction}
The Fomin-Kirillov algebra $\mathcal E_n$ \cite{FK} is a certain noncommutative algebra with generators $x_{ij}$ for $1 \leq i < j \leq n$ that satisfy a simple set of quadratic relations.
\iffalse
\begin{definition}The \emph{Fomin-Kirillov algebra} $\mathcal E_n$ is the quadratic algebra (say... | {
"timestamp": "2014-03-13T01:04:41",
"yymm": "1310",
"arxiv_id": "1310.4112",
"language": "en",
"url": "https://arxiv.org/abs/1310.4112",
"abstract": "The Fomin-Kirillov algebra $\\mathcal E_n$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on $n$ vertices. For ... |
https://arxiv.org/abs/1204.5704 | A variant of Touchard's Catalan number identity | It is well known that the Catalan number C_n counts dissections of a regular (n+2)-gon into triangles. Here we count such dissections by number of triangles that contain two sides of the polygon among their three edges, leading to a combinatorial interpretation of the identity C_n =sum_{1<=k<=n/2} 2^{n-2k} n-choose-2k ... | \section{Introduction}
\vspace*{-5mm}
Consider a regular polygon of $n+2$ sides with one side designated the base. It is a classic result that there are the Catalan number $C_{n}$ ways to insert noncrossing diagonals connecting nonadjacent vertices of the polygon so as to dissect it into triangles (see illustration i... | {
"timestamp": "2013-05-14T02:02:33",
"yymm": "1204",
"arxiv_id": "1204.5704",
"language": "en",
"url": "https://arxiv.org/abs/1204.5704",
"abstract": "It is well known that the Catalan number C_n counts dissections of a regular (n+2)-gon into triangles. Here we count such dissections by number of triangles... |
https://arxiv.org/abs/1710.00442 | Virtual Element Methods on Meshes with Small Edges or Faces | We consider a model Poisson problem in $\R^d$ ($d=2,3$) and establish error estimates for virtual element methods on polygonal or polyhedral meshes that can contain small edges ($d=2$) or small faces ($d=3$). | \section{Introduction}\label{sec:Introduction}
Let ${\Omega}\subset \mathbb{R}^d$ ($d=2,3$) be a bounded polygonal/polyhedral domain
and $f\in L_2({\Omega})$. The Poisson problem
with the homogeneous Dirichlet boundary condition is to find
$u\in H^1_0({\Omega})$ such that
\begin{equation}\label{eq:Poisson}
a(u,v)... | {
"timestamp": "2017-10-03T02:12:36",
"yymm": "1710",
"arxiv_id": "1710.00442",
"language": "en",
"url": "https://arxiv.org/abs/1710.00442",
"abstract": "We consider a model Poisson problem in $\\R^d$ ($d=2,3$) and establish error estimates for virtual element methods on polygonal or polyhedral meshes that ... |
https://arxiv.org/abs/2112.10171 | Newtonian mechanics in a Riemannian manifold | The work done by Isaac Newton more than three hundred years ago, continues being a path to increase our knowledge of Nature. To better understand all the ideas behind it, one of the finest ways is to generalize them to wider situations. In this report we make a review of one of these enlargements, the one that bears th... | \section{Introduction}
Mechanics is an ancient wisdom of humankind. It is not possible to build the Egyptian Pyramids, for example, without some more or less organised intuitive ideas of mechanics. And they were made near five thousand years ago. Archimedes, 287--212 BC, was one of those that best exploited mechani... | {
"timestamp": "2021-12-21T02:20:17",
"yymm": "2112",
"arxiv_id": "2112.10171",
"language": "en",
"url": "https://arxiv.org/abs/2112.10171",
"abstract": "The work done by Isaac Newton more than three hundred years ago, continues being a path to increase our knowledge of Nature. To better understand all the ... |
https://arxiv.org/abs/1203.0690 | Asymptotic normality of integer compositions inside a rectangle | Among all restricted integer compositions with at most $m$ parts, each of which has size at most $l$, choose one uniformly at random. Which integer does this composition represent? In the current note, we show that underlying distribution is, for large $m$ and $l$, approximately normal with mean value $\frac{ml}{2}$. | \section{Introduction}
An integer composition of a nonnegative integer $n$ is, informally, a
way of writing
$n$ as a sum of nonnegative integers $\pi_1,\dotsc,\pi_k$, for
some $k\ge 0$. Let $h_{l,m}(n)$ denote the number of integer
compositions of the nonnegative integer $n$ with at most $m$ parts,
each of which has s... | {
"timestamp": "2012-03-06T02:01:44",
"yymm": "1203",
"arxiv_id": "1203.0690",
"language": "en",
"url": "https://arxiv.org/abs/1203.0690",
"abstract": "Among all restricted integer compositions with at most $m$ parts, each of which has size at most $l$, choose one uniformly at random. Which integer does thi... |
https://arxiv.org/abs/1203.4200 | Residues and Telescopers for Rational Functions | We give necessary and sufficient conditions for the existence of telescopers for rational functions of two variables in the continuous, discrete and q-discrete settings and characterize which operators can occur as telescopers. Using this latter characterization, we reprove results of Furstenberg and Zeilberger concern... | \section{Introduction}\label{SECT:intro}
Residues have played a ubiquitous and important role in mathematics and
their use in combinatorics has had a lasting impact~(e.g., \cite{Flajolet_Sedgewick}).
In this paper we will show how the notion of residue and its generalizations lead
to new results and a recasting of kno... | {
"timestamp": "2012-03-20T01:05:52",
"yymm": "1203",
"arxiv_id": "1203.4200",
"language": "en",
"url": "https://arxiv.org/abs/1203.4200",
"abstract": "We give necessary and sufficient conditions for the existence of telescopers for rational functions of two variables in the continuous, discrete and q-discr... |
https://arxiv.org/abs/1406.1984 | Discrete Hardy-type Inequalities | This paper studies the Hardy-type inequalities on the discrete intervals. The first result is the variational formulas of the optimal constants. Using these formulas, one may obtain an approximating procedure and the known basic estimates of the optimal constants. The second result, which is the main innovation of this... | \section{Introduction}
For given two constants $p$ and $q$ with $1< p\leqslant q< \infty$, two positive sequences $\mathbf{u}$ and $\mathbf{v}$ on a discrete interval $[1, N]:= \{1, 2, \dots, N \}$ with $N\leqslant +\infty$, here is the discrete Hardy-type inequality:
\begin{equation}\label{Hardy}
\left[\sum_{n= 1}^N... | {
"timestamp": "2014-06-24T02:11:17",
"yymm": "1406",
"arxiv_id": "1406.1984",
"language": "en",
"url": "https://arxiv.org/abs/1406.1984",
"abstract": "This paper studies the Hardy-type inequalities on the discrete intervals. The first result is the variational formulas of the optimal constants. Using these... |
https://arxiv.org/abs/math/0701940 | Monochromatic triangles in two-colored plane | We prove that for any partition of the plane into a closed set $C$ and an open set $O$ and for any configuration $T$ of three points, there is a translated and rotated copy of $T$ contained in $C$ or in $O$. Apart from that, we consider partitions of the plane into two sets whose common boundary is a union of piecewise... | \section{Introduction}
Euclidean Ramsey theory addresses the problems of the following kind:
assume that a finite configuration $X$ of points is given; for what values of
$c$ and $d$ is it true that every coloring of the $d$-dimensional Euclidean
space by $c$ colors contains a monochromatic congruent copy ... | {
"timestamp": "2007-01-31T22:00:13",
"yymm": "0701",
"arxiv_id": "math/0701940",
"language": "en",
"url": "https://arxiv.org/abs/math/0701940",
"abstract": "We prove that for any partition of the plane into a closed set $C$ and an open set $O$ and for any configuration $T$ of three points, there is a trans... |
https://arxiv.org/abs/1301.5020 | Generalized cover ideals and the persistence property | Let $I$ be a square-free monomial ideal in $R = k[x_1,\ldots,x_n]$, and consider the sets of associated primes ${\rm Ass}(I^s)$ for all integers $s \geq 1$. Although it is known that the sets of associated primes of powers of $I$ eventually stabilize, there are few results about the power at which this stabilization oc... | \section{Introduction}
Let $I$ be an ideal of the polynomial ring $R = k[x_1,\ldots,x_n]$ with
$k$ a field.
A prime ideal $P \subseteq R$ is an {\it associated prime} of
$I$ if there exists an element $T \in R$ such that $I:\langle T \rangle = P$.
The {\it set of associated primes} of $I$,
denoted ${\rm Ass}(I)$, ... | {
"timestamp": "2013-12-18T02:12:50",
"yymm": "1301",
"arxiv_id": "1301.5020",
"language": "en",
"url": "https://arxiv.org/abs/1301.5020",
"abstract": "Let $I$ be a square-free monomial ideal in $R = k[x_1,\\ldots,x_n]$, and consider the sets of associated primes ${\\rm Ass}(I^s)$ for all integers $s \\geq ... |
https://arxiv.org/abs/2207.04701 | Spectral radius and edge-disjoint spanning trees | The spanning tree packing number of a graph $G$, denoted by $\tau(G)$, is the maximum number of edge-disjoint spanning trees contained in $G$. The study of $\tau(G)$ is one of the classic problems in graph theory. Cioabă and Wong initiated to investigate $\tau(G)$ from spectral perspectives in 2012 and since then, $\ta... | \section{Introduction}
In this paper, we only consider simply graphs unless otherwise stated and $k$ always denotes a positive integer. The study of edge-disjoint spanning trees has been shown to be very important to graphs and has many applications in fault-tolerance networks as well as network reliability~\cite{Cunni... | {
"timestamp": "2022-07-12T02:28:02",
"yymm": "2207",
"arxiv_id": "2207.04701",
"language": "en",
"url": "https://arxiv.org/abs/2207.04701",
"abstract": "The spanning tree packing number of a graph $G$, denoted by $\\tau(G)$, is the maximum number of edge-disjoint spanning trees contained in $G$. The study ... |
https://arxiv.org/abs/1302.2076 | On measures of symmetry and floating bodies | We consider the following measure of symmetry of a convex n-dimensional body K: $\rho(K)$ is the smallest constant for which there is a point x in K such that for partitions of K by an n-1-dimensional hyperplane passing through x the ratio of the volumes of the two parts is at most $\rho(K)$. It is well known that $\rh... | \section{\@startsection {section}{1}{\z@}{-3.5ex plus-1ex minus
-.2ex}{2.3ex plus.2ex}{\reset@font\large\bf}}
\def\subsection{\@startsection{subsection}{2}{\z@}{-3.25ex plus-1ex
minus-.2ex}{-.1em}{\reset@font\large\bf}}
\def\subsubsection{\@startsection{subsubsection}{3}{\z@}{-3.25ex plus
-1ex minus-.... | {
"timestamp": "2013-02-11T02:02:19",
"yymm": "1302",
"arxiv_id": "1302.2076",
"language": "en",
"url": "https://arxiv.org/abs/1302.2076",
"abstract": "We consider the following measure of symmetry of a convex n-dimensional body K: $\\rho(K)$ is the smallest constant for which there is a point x in K such t... |
https://arxiv.org/abs/0912.5000 | Classification of Q-trivial Bott manifolds | A Bott manifold is a closed smooth manifold obtained as the total space of an iterated $\C P^1$-bundle starting with a point, where each $\C P^1$-bundle is the projectivization of a Whitney sum of two complex line bundles. A \emph{$\Q$-trivial Bott manifold} of dimension $2n$ is a Bott manifold whose cohomology ring is... | \section{Introduction}
A Bott tower of height $n$ is a sequence of $\mathbb{C}P^1$-bundles
\begin{equation} \label{eqn:Bott tower}
B_n\stackrel{\pi_n}\longrightarrow B_{n-1} \stackrel{\pi_{n-1}}\longrightarrow
\dots \stackrel{\pi_2}\longrightarrow B_1 \stackrel{\pi_1}\longrightarrow
B_0=\{\text{a point}\},
\end{equatio... | {
"timestamp": "2009-12-27T06:30:51",
"yymm": "0912",
"arxiv_id": "0912.5000",
"language": "en",
"url": "https://arxiv.org/abs/0912.5000",
"abstract": "A Bott manifold is a closed smooth manifold obtained as the total space of an iterated $\\C P^1$-bundle starting with a point, where each $\\C P^1$-bundle i... |
https://arxiv.org/abs/1911.01151 | Successive shortest paths in complete graphs with random edge weights | Consider a complete graph $K_n$ with edge weights drawn independently from a uniform distribution $U(0,1)$. The weight of the shortest (minimum-weight) path $P_1$ between two given vertices is known to be $\ln n / n$, asymptotically. Define a second-shortest path $P_2$ to be the shortest path edge-disjoint from $P_1$, ... | \section{Introduction}
It is a standard problem to find the shortest $s$--$t$\xspace path in a graph,
i.e., the cheapest path $P_1$ between specified vertices $s$ and $t$,
and its cost $X_1$,
where the cost of a path is the sum of the costs of its edges.
We will use the terms ``cost'' and ``weight'' interchangeably,
a... | {
"timestamp": "2020-10-13T02:13:52",
"yymm": "1911",
"arxiv_id": "1911.01151",
"language": "en",
"url": "https://arxiv.org/abs/1911.01151",
"abstract": "Consider a complete graph $K_n$ with edge weights drawn independently from a uniform distribution $U(0,1)$. The weight of the shortest (minimum-weight) pa... |
https://arxiv.org/abs/2204.11084 | Decompositions of functions defined on finite sets in $\mathbb{R}^d$ | A finite subset $M \subset \mathbb{R}^d$ is basic, if for any function $f \colon M \to \mathbb{R}$ there exists a collection of functions $f_1, \ldots, f_d \colon \mathbb{R} \to \mathbb{R}$ such that for each element $(x_1, \ldots, x_d)\in M$ we have $f(x_1, \ldots, x_d) = f_1(x_1) + \ldots + f_d(x_d)$. For certain fin... | \section{Introduction}
The concept of \textit{basic subsets} arises in connection with Hilbert's thirteenth problem on the superposition of continuous functions. The first time it was introduced in an explicit form by Sternfeld in 1989 \cite{Sternfeld1989}. He called a~subset $M \subset \mathbb{R}^d$ \emph{(continuou... | {
"timestamp": "2022-04-26T02:14:19",
"yymm": "2204",
"arxiv_id": "2204.11084",
"language": "en",
"url": "https://arxiv.org/abs/2204.11084",
"abstract": "A finite subset $M \\subset \\mathbb{R}^d$ is basic, if for any function $f \\colon M \\to \\mathbb{R}$ there exists a collection of functions $f_1, \\ldo... |
https://arxiv.org/abs/1201.6035 | How Accurate is inv(A)*b? | Several widely-used textbooks lead the reader to believe that solving a linear system of equations Ax = b by multiplying the vector b by a computed inverse inv(A) is inaccurate. Virtually all other textbooks on numerical analysis and numerical linear algebra advise against using computed inverses without stating whethe... | \section{Introduction}
Can you accurately compute the solution to a linear equation $Ax=b$
by first computing an approximation $V$ to $A^{-1}$ and then multiplying
$b$ by $V$ (\texttt{x=inv(A){*}b} in Matlab)?
Unfortunately, most of the literature provides a misleading answer
to this question. Many textbooks, incl... | {
"timestamp": "2012-01-31T02:02:27",
"yymm": "1201",
"arxiv_id": "1201.6035",
"language": "en",
"url": "https://arxiv.org/abs/1201.6035",
"abstract": "Several widely-used textbooks lead the reader to believe that solving a linear system of equations Ax = b by multiplying the vector b by a computed inverse ... |
https://arxiv.org/abs/1809.10263 | Counting Shellings of Complete Bipartite Graphs and Trees | A shelling of a graph, viewed as an abstract simplicial complex that is pure of dimension 1, is an ordering of its edges such that every edge is adjacent to some other edges appeared previously. In this paper, we focus on complete bipartite graphs and trees. For complete bipartite graphs, we obtain an exact formula for... | \section{Introduction}
In combinatorial topology, shelling of a simplicial complex is a very useful and important notion that has been well-studied.
\begin{definition}\label{def:shellingcomplex}
An (abstract) simplicial complex $\Delta$ is called \textit{pure} if all of its maximal simplicies have the same dimension. ... | {
"timestamp": "2018-09-28T02:04:40",
"yymm": "1809",
"arxiv_id": "1809.10263",
"language": "en",
"url": "https://arxiv.org/abs/1809.10263",
"abstract": "A shelling of a graph, viewed as an abstract simplicial complex that is pure of dimension 1, is an ordering of its edges such that every edge is adjacent ... |
https://arxiv.org/abs/1611.01153 | On Perfectness of Intersection Graph of Ideals of $\mathbb{Z}_n$ | In this paper, we characterize the positive integers $n$ for which intersection graph of ideals of $\mathbb{Z}_n$ is perfect. | \section{Introduction}
The idea of associating graphs to algebraic structures for characterizing the algebraic structures with graphs and vice versa dates back to Bosak \cite{bosak}. Till then, a lot of research, e.g., \cite{graph-ideal,anderson-livingston,badawi,power2,mks-ideal,angsu-comm-alg-1,angsu-lin-mult-alg,ang... | {
"timestamp": "2016-11-07T02:00:10",
"yymm": "1611",
"arxiv_id": "1611.01153",
"language": "en",
"url": "https://arxiv.org/abs/1611.01153",
"abstract": "In this paper, we characterize the positive integers $n$ for which intersection graph of ideals of $\\mathbb{Z}_n$ is perfect.",
"subjects": "General Ma... |
https://arxiv.org/abs/1508.02851 | On interval edge-colorings of bipartite graphs of small order | An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable if it has an interval $t$-coloring for some positive integer $t$. The probl... | \section{Introduction}
In this paper we consider only finite, undirected graphs, without loops and multiple edges. $V(G)$ and $E(G)$ denote the sets of vertices and edges, respectively. The degree of the vertex $v \in V(G)$ is denoted by $d_G(v)$. The concepts and notations not defined here can be found in \cite{West}.... | {
"timestamp": "2015-08-13T02:06:54",
"yymm": "1508",
"arxiv_id": "1508.02851",
"language": "en",
"url": "https://arxiv.org/abs/1508.02851",
"abstract": "An edge-coloring of a graph $G$ with colors $1,\\ldots,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each ver... |
https://arxiv.org/abs/1004.4953 | The Number of Eigenvalues of a Tensor | Eigenvectors of tensors, as studied recently in numerical multilinear algebra, correspond to fixed points of self-maps of a projective space. We determine the number of eigenvectors and eigenvalues of a generic tensor, and we show that the number of normalized eigenvalues of a symmetric tensor is always finite. We als... | \section{Introduction}
In the current numerical analysis literature, considerable interest has arisen in
extending concepts that are familiar from linear algebra to the setting of
multilinear algebra. One such familiar concept is that of an eigenvalue
of a square matrix. Several authors have explored definitions of e... | {
"timestamp": "2010-05-18T02:00:14",
"yymm": "1004",
"arxiv_id": "1004.4953",
"language": "en",
"url": "https://arxiv.org/abs/1004.4953",
"abstract": "Eigenvectors of tensors, as studied recently in numerical multilinear algebra, correspond to fixed points of self-maps of a projective space. We determine ... |
https://arxiv.org/abs/1509.07908 | Helly-type theorems for the diameter | We study versions of Helly's theorem that guarantee that the intersection of a family of convex sets in $R^d$ has a large diameter. This includes colourful, fractional and $(p,q)$ versions of Helly's theorem. In particular, the fractional and $(p,q)$ versions work with conditions where the corresponding Helly theorem d... | \section{Introduction}
Quantitative results in combinatorial geometry have recently caught new interest. Those surrounding Helly's theorem have as aim to show that \textit{given a finite family of convex set in $\mathds{R}^d$, if the intersection of every small subfamily is large, then the intersection of the whole f... | {
"timestamp": "2015-09-29T02:02:47",
"yymm": "1509",
"arxiv_id": "1509.07908",
"language": "en",
"url": "https://arxiv.org/abs/1509.07908",
"abstract": "We study versions of Helly's theorem that guarantee that the intersection of a family of convex sets in $R^d$ has a large diameter. This includes colourfu... |
https://arxiv.org/abs/2209.10045 | New Lower Bounds for Cap Sets | A cap set is a subset of $\mathbb{F}_3^n$ with no solutions to $x+y+z=0$ other than when $x=y=z$. In this paper, we provide a new lower bound on the size of a maximal cap set. Building on a construction of Edel, we use improved computational methods and new theoretical ideas to show that, for large enough $n$, there is... | \section{Introduction}
\begin{definition}\label{def:cap}
A \emph{cap set} is a set $A \subseteq \mathbb{F}_3^n$ with no solutions to $x+y+z=0$ other than when $x=y=z$, or equivalently $A$ has no 3 distinct elements in arithmetic progression.
\end{definition}
In this paper, we prove the following result.
\begin{theorem}... | {
"timestamp": "2022-09-22T02:05:34",
"yymm": "2209",
"arxiv_id": "2209.10045",
"language": "en",
"url": "https://arxiv.org/abs/2209.10045",
"abstract": "A cap set is a subset of $\\mathbb{F}_3^n$ with no solutions to $x+y+z=0$ other than when $x=y=z$. In this paper, we provide a new lower bound on the size... |
https://arxiv.org/abs/2004.06097 | Saturation problems in the Ramsey theory of graphs, posets and point sets | In 1964, Erdős, Hajnal and Moon introduced a saturation version of Turán's classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a $K_r$-free, $n$-vertex graph with the property that the addition of any further edge yields a copy of $K_r$. We consider analogues of thi... |
\section{Introduction}
Extremal problems have a long history in combinatorics originating with the results of Mantel~\cite{Mantel} in 1907 and Turán~\cite{Turan} in 1947 determining the maximum number of edges in a triangle- and $K_r$-free, $n$-vertex graph, respectively. Erdős, Hajnal and Moon~\cite{ehm} investigated... | {
"timestamp": "2020-05-05T02:35:28",
"yymm": "2004",
"arxiv_id": "2004.06097",
"language": "en",
"url": "https://arxiv.org/abs/2004.06097",
"abstract": "In 1964, Erdős, Hajnal and Moon introduced a saturation version of Turán's classical theorem in extremal graph theory. In particular, they determined the ... |
https://arxiv.org/abs/2010.14043 | The Teaching Dimension of Kernel Perceptron | Algorithmic machine teaching has been studied under the linear setting where exact teaching is possible. However, little is known for teaching nonlinear learners. Here, we establish the sample complexity of teaching, aka teaching dimension, for kernelized perceptrons for different families of feature maps. As a warm-up... |
\section{Acknowledgements}
Yuxin Chen is supported by NSF 2040989 and a C3.ai DTI Research Award 049755.
\section{Experimental Evaluation}\label{appendix: experimentals}
In this section, we provide an algorithmic procedure for constructing the $\epsilon$-approximate teaching set, and quantitatively evaluate our theore... | {
"timestamp": "2021-02-26T02:05:28",
"yymm": "2010",
"arxiv_id": "2010.14043",
"language": "en",
"url": "https://arxiv.org/abs/2010.14043",
"abstract": "Algorithmic machine teaching has been studied under the linear setting where exact teaching is possible. However, little is known for teaching nonlinear l... |
https://arxiv.org/abs/1810.11439 | Hardy-Littlewood-Sobolev and Stein-Weiss inequalities on homogeneous Lie groups | In this note we prove the Stein-Weiss inequality on general homogeneous Lie groups. The obtained results extend previously known inequalities. Special properties of homogeneous norms play a key role in our proofs. Also, we give a simple proof of the Hardy-Littlewood-Sobolev inequality on general homogeneous Lie groups. | \section{Introduction}
Historically, in \cite{HL28}, Hardy and Littlewood considered the one dimensional fractional integral operator on $(0,\infty)$ given by
\begin{equation}\label{1Doper}
T_{\lambda}u(x)=\int_{0}^{\infty}\frac{u(y)}{|x-y|^{\lambda}}dy,\,\,\,\,0<\lambda<1,
\end{equation}
and proved the following theo... | {
"timestamp": "2018-10-29T01:14:50",
"yymm": "1810",
"arxiv_id": "1810.11439",
"language": "en",
"url": "https://arxiv.org/abs/1810.11439",
"abstract": "In this note we prove the Stein-Weiss inequality on general homogeneous Lie groups. The obtained results extend previously known inequalities. Special pro... |
https://arxiv.org/abs/math/9805076 | An Introduction to Total Least Squares | The method of ``Total Least Squares'' is proposed as a more natural way (than ordinary least squares) to approximate the data if both the matrix and and the right-hand side are contaminated by ``errors''. In this tutorial note, we give a elementary unified view of ordinary and total least squares problems and their sol... | \section{Introduction\label{par1}}
\setcounter{equation}{0}
This (tutorial) paper grew out of the need to
motivate the usual formulation of a
``Total Least Squares problem'' and to explain the way it is solved
using the ``Singular Value Decomposition''. Although it is an important
generalization of (ordinary) least sq... | {
"timestamp": "1998-05-18T11:48:36",
"yymm": "9805",
"arxiv_id": "math/9805076",
"language": "en",
"url": "https://arxiv.org/abs/math/9805076",
"abstract": "The method of ``Total Least Squares'' is proposed as a more natural way (than ordinary least squares) to approximate the data if both the matrix and a... |
https://arxiv.org/abs/0809.4621 | Explicit constructions of infinite families of MSTD sets | We explicitly construct infinite families of MSTD (more sums than differences) sets. There are enough of these sets to prove that there exists a constant C such that at least C / r^4 of the 2^r subsets of {1,...,r} are MSTD sets; thus our family is significantly denser than previous constructions (whose densities are a... | \section{Introduction}
Given a finite set of integers $A$, we define its sumset $A+A$ and difference set $A-A$ by \bea A + A & \ = \ & \{a_i + a_j: a_i, a_j \in A\} \nonumber\\ A - A & = & \{a_i - a_j: a_i, a_j \in A\}, \eea and let $|X|$ denote the cardinality of $X$. If $|A+A| > |A-A|$, then, following Nathanson, ... | {
"timestamp": "2008-11-22T15:59:14",
"yymm": "0809",
"arxiv_id": "0809.4621",
"language": "en",
"url": "https://arxiv.org/abs/0809.4621",
"abstract": "We explicitly construct infinite families of MSTD (more sums than differences) sets. There are enough of these sets to prove that there exists a constant C ... |
https://arxiv.org/abs/1402.5826 | Depth and Stanley Depth of the Canonical Form of a factor of monomial ideals | In this paper we show that the depth and the Stanley depth of the factor of two monomial ideals is invariant under taking a so called canonical form. It follows easily that the Stanley Conjecture holds for the factor if and only if it holds for its canonical form. In particular, we construct an algorithm which simplifi... | \section{Introduction}
\vskip 1cm
Let $K$ be a field and $S=K[x_1,\ldots,x_n]$ be the polynomial ring over $K$
in $n$ variables. A Stanley decomposition of a graded $S-$module $M$ is a finite family $$\mathcal{D} = (S_i, u_i)_{i \in I}$$ in which $u_i$ are homogeneous elements of $M$ and $S_i$ are graded $K-$algebra... | {
"timestamp": "2014-04-08T02:03:14",
"yymm": "1402",
"arxiv_id": "1402.5826",
"language": "en",
"url": "https://arxiv.org/abs/1402.5826",
"abstract": "In this paper we show that the depth and the Stanley depth of the factor of two monomial ideals is invariant under taking a so called canonical form. It fol... |
https://arxiv.org/abs/1305.6104 | On node distributions for interpolation and spectral methods | A scaled Chebyshev node distribution is studied in this paper. It is proved that the node distribution is optimal for interpolation in $C_M^{s+1}[-1,1]$, the set of $(s+1)$-time differentiable functions whose $(s+1)$-th derivatives are bounded by a constant $M>0$. Node distributions for computing spectral differentiati... | \section{Introduction}
Choosing nodes is important in interpolating a function and solving differential or integral equations by pseudospectral methods.
Given a sufficiently smooth function, if nodes are not suitably chosen, then the interpolation polynomials do not converge to the function as the number of nodes tend... | {
"timestamp": "2013-05-28T02:03:05",
"yymm": "1305",
"arxiv_id": "1305.6104",
"language": "en",
"url": "https://arxiv.org/abs/1305.6104",
"abstract": "A scaled Chebyshev node distribution is studied in this paper. It is proved that the node distribution is optimal for interpolation in $C_M^{s+1}[-1,1]$, th... |
https://arxiv.org/abs/math/0603630 | Sharp bounds for eigenvalues of triangles | We prove that the first eigenvalue of the Dirichlet Laplacian for a triangle in the plane is bounded above by $\pi^2 L^2\over 9A^2$, where $L$ is the perimeter and $A$ is the area of this triangle. We show that the \mbox{constant 9} is optimal and that the optimal constant for the lower bound of the same form is 16. Th... | \section{Introduction}
The purpose of this paper is to prove the following theorem.
\begin{thm}\label{main}
Let $T$ be a triangle in a plane of area $A$ and perimeter $L$. Then the first eigenvalue $\lambda_T$ of
the Dirichlet Laplacian on $T$ satisfies
\begin{gather}
{\pi^2 L^2\over 16A^2}\leq\lambda_T\l... | {
"timestamp": "2006-03-27T19:33:02",
"yymm": "0603",
"arxiv_id": "math/0603630",
"language": "en",
"url": "https://arxiv.org/abs/math/0603630",
"abstract": "We prove that the first eigenvalue of the Dirichlet Laplacian for a triangle in the plane is bounded above by $\\pi^2 L^2\\over 9A^2$, where $L$ is th... |
https://arxiv.org/abs/2009.12450 | Tinkering with Lattices: A New Take on the Erdős Distance Problem | The Erdős distance problem concerns the least number of distinct distances that can be determined by $N$ points in the plane. The integer lattice with $N$ points is known as \textit{near-optimal}, as it spans $\Theta(N/\sqrt{\log(N)})$ distinct distances, the lower bound for a set of $N$ points (Erdős, 1946). The only ... | \section{Introduction}
In 1946, Paul Erd\H{o}s \cite{Er1} proposed the now-famous Erd\H{o}s distinct distances problem: given $N$ points in a plane, what is the minimum number of distinct distances, $f(N)$, they can determine?
He accompanied this question with the first bounds on $f(N)$,
\begin{equation}\sqrt{N-\fra... | {
"timestamp": "2021-02-02T02:20:34",
"yymm": "2009",
"arxiv_id": "2009.12450",
"language": "en",
"url": "https://arxiv.org/abs/2009.12450",
"abstract": "The Erdős distance problem concerns the least number of distinct distances that can be determined by $N$ points in the plane. The integer lattice with $N$... |
https://arxiv.org/abs/1612.06314 | Computing cohomology of configuration spaces | We give a concrete method to explicitly compute the rational cohomology of the unordered configuration spaces of connected, oriented, closed, even-dimensional manifolds of finite type which we have implemented in Sage [S+09]. As an application, we give acomplete computation of the stable and unstable rational cohomolog... | \section{Introduction}
\subsection{Cohomological stability of configuration spaces.}
Given a sequence of topological spaces or groups, $\{X_n\}$, \emph{(rational) cohomological stability} is the property that, for each $i\geq 0$, $$H^i(X_n; \Q) = H^i(X_{n+1}; \Q)$$ for $n \geq f(i)$, where $f(i)$ is some function of ... | {
"timestamp": "2016-12-20T02:13:53",
"yymm": "1612",
"arxiv_id": "1612.06314",
"language": "en",
"url": "https://arxiv.org/abs/1612.06314",
"abstract": "We give a concrete method to explicitly compute the rational cohomology of the unordered configuration spaces of connected, oriented, closed, even-dimensi... |
https://arxiv.org/abs/2209.07698 | Hitting a prime in 2.43 dice rolls (on average) | What is the number of rolls of fair 6-sided dice until the first time the total sum of all rolls is a prime? We compute the expectation and the variance of this random variable up to an additive error of less than 10^{-4}. This is a solution to a puzzle suggested by DasGupta (2017) in the Bulletin of the Institute of M... | \section{Description of the problem}
The following puzzle appears in the Bulletin of the Institute of
Mathematical Statistics \citep{D2017}:
Let $X_1,X_2,\ldots$ be independent uniform random variables
on the integers $1,2,\ldots,6$,
and define $S_n=X_1+\ldots+X_n$ for $n=1,2,\ldots$.
Denote by $\tau$ the discrete tim... | {
"timestamp": "2022-09-19T02:07:05",
"yymm": "2209",
"arxiv_id": "2209.07698",
"language": "en",
"url": "https://arxiv.org/abs/2209.07698",
"abstract": "What is the number of rolls of fair 6-sided dice until the first time the total sum of all rolls is a prime? We compute the expectation and the variance o... |
https://arxiv.org/abs/math/0510367 | A Weierstrass-type theorem for homogeneous polynomials | By the celebrated Weierstrass Theorem the set of algebraic polynomials is dense in the space of continuous functions on a compact set in R^d. In this paper we study the following question: does the density hold if we approximate only by homogeneous polynomials? Since the set of homogeneous polynomials is nonlinear this... | \section{Introduction} The celebrated theorem of Weierstrass on
the density of real algebraic polynomials in the space of real
continuous functions on an interval $[a,b]$ is one of the main
results in analysis. Its generalization for real multivariate
polynomials was given by Picard, subsequently the
Stone-Weierstrass... | {
"timestamp": "2005-10-18T08:26:27",
"yymm": "0510",
"arxiv_id": "math/0510367",
"language": "en",
"url": "https://arxiv.org/abs/math/0510367",
"abstract": "By the celebrated Weierstrass Theorem the set of algebraic polynomials is dense in the space of continuous functions on a compact set in R^d. In this ... |
https://arxiv.org/abs/1604.00592 | The isoperimetric problem in the plane with the sum of two Gaussian densities | We consider the isoperimetric problem for the sum of two Gaussian densities in the line and the plane. We prove that the double Gaussian isoperimetric regions in the line are rays and that if the double Gaussian isoperimetric regions in the plane are half-spaces, then they must be bounded by vertical lines. | \section{Introduction}
Sudakov-Tsirelson and Borell proved independently (see \cite[18.2]{morgan}) that for $\mathbb{R}^n$ endowed with a Gaussian measure, half-spaces bounded by hyperplanes are isoperimetric, i.e., minimize weighted perimeter for given weighted volume. Ca\~{n}ete et al. \cite[Question 6]{canete}, in ... | {
"timestamp": "2016-08-29T02:00:59",
"yymm": "1604",
"arxiv_id": "1604.00592",
"language": "en",
"url": "https://arxiv.org/abs/1604.00592",
"abstract": "We consider the isoperimetric problem for the sum of two Gaussian densities in the line and the plane. We prove that the double Gaussian isoperimetric reg... |
https://arxiv.org/abs/2006.15797 | Asymptotic enumeration of digraphs and bipartite graphs by degree sequence | We provide asymptotic formulae for the numbers of bipartite graphs with given degree sequence, and of loopless digraphs with given in- and out-degree sequences, for a wide range of parameters. Our results cover medium range densities and close the gaps between the results known for the sparse and dense ranges. In the c... | \section{Introduction}
Enumeration of discrete structures with local constraints has attracted the interest of many researchers and has applications in various areas such as coding theory, statistics and neurostatistical analysis. Exact formulae are often hard to derive or infeasible to compute. Asymptotic formulae ar... | {
"timestamp": "2020-06-30T02:26:30",
"yymm": "2006",
"arxiv_id": "2006.15797",
"language": "en",
"url": "https://arxiv.org/abs/2006.15797",
"abstract": "We provide asymptotic formulae for the numbers of bipartite graphs with given degree sequence, and of loopless digraphs with given in- and out-degree sequ... |
https://arxiv.org/abs/1401.4372 | Regular matchstick graphs | A graph G=(V,E) is called a unit-distance graph in the plane if there is an injective embedding of V in the plane such that every pair of adjacent vertices are at unit distance apart. If additionally the corresponding edges are non-crossing and all vertices have the same degree r we talk of a regular matchstick graph. ... | \section{Introduction}
\noindent
One of the possibly best known problems in combinatorial geometry asks how often the same distance can occur among
$n$ points in the plane. Via scaling we can assume that the most frequent distance has length $1$. Given any set $P$ of points in the plane, we can define the so called uni... | {
"timestamp": "2014-01-20T02:08:36",
"yymm": "1401",
"arxiv_id": "1401.4372",
"language": "en",
"url": "https://arxiv.org/abs/1401.4372",
"abstract": "A graph G=(V,E) is called a unit-distance graph in the plane if there is an injective embedding of V in the plane such that every pair of adjacent vertices ... |
https://arxiv.org/abs/2203.01473 | Powers of posinormal Hilbert-space operators | A bounded linear operator $A$ on a Hilbert space $\mathcal{H}$ is posinormal if there exists a positive operator $P$ such that $AA^{*} = A^{*}PA$. We show that if $A$ is posinormal with closed range, then $A^n$ is posinormal and has closed range for all integers $n\ge 1$. Because the collection of posinormal operators ... | \section{Introduction}
A bounded linear operator $A$ on a Hilbert space $\mathcal{H}$ is said to be \textit{posinormal} if there exists a positive operator $P$ such that $AA^{*} = A^{*}PA$. The operator $P$ is called an \textit{interrupter} of $A$. Note that if $A$ has interrupter $I$, then $A$ is normal. Rhaly int... | {
"timestamp": "2022-03-04T02:06:54",
"yymm": "2203",
"arxiv_id": "2203.01473",
"language": "en",
"url": "https://arxiv.org/abs/2203.01473",
"abstract": "A bounded linear operator $A$ on a Hilbert space $\\mathcal{H}$ is posinormal if there exists a positive operator $P$ such that $AA^{*} = A^{*}PA$. We sho... |
https://arxiv.org/abs/1101.4151 | Tilted Sperner Families | Let \cal A be a family of subsets of an n-set such that \cal A does not contain distinct sets A and B with |A\B| = 2|B\A|. How large can \cal A be? Our aim in this note is to determine the maximum size of such an \cal A. This answers a question of Kalai. We also give some related results and conjectures. | \section{Introduction}
A set system $\mathcal A\subseteq \mathcal {P}[n]=\mathcal {P}(\{1,\ldots ,n\})$ is said to be an \emph{antichain} or \emph{Sperner family} if $A\not\subset B$ for all distinct $A,B\in \mathcal A$. Sperner's theorem \cite{sper} says that any antichain $\mathcal A$ has size at most
$\binom {n}{\... | {
"timestamp": "2011-01-24T02:01:43",
"yymm": "1101",
"arxiv_id": "1101.4151",
"language": "en",
"url": "https://arxiv.org/abs/1101.4151",
"abstract": "Let \\cal A be a family of subsets of an n-set such that \\cal A does not contain distinct sets A and B with |A\\B| = 2|B\\A|. How large can \\cal A be? Our... |
https://arxiv.org/abs/2008.07067 | Revisiting Spectral Bundle Methods: Primal-dual (Sub)linear Convergence Rates | The spectral bundle method proposed by Helmberg and Rendl is well established for solving large-scale semidefinite programs (SDP) thanks to its low per iteration computational complexity and strong practical performance. In this paper, we revisit this classic method showing it achieves sublinear convergence rates in te... | \section{Analysis}\label{sec: analysis}
In this section, we present and derive our sublinear convergence guarantees for both Block-Spec and HR-Spec as well as our improved local linear convergence for Block-Spec under extra condition of strict complementarity (defined momentarily).
Key structural lemmas for our proofs ... | {
"timestamp": "2020-08-18T02:22:55",
"yymm": "2008",
"arxiv_id": "2008.07067",
"language": "en",
"url": "https://arxiv.org/abs/2008.07067",
"abstract": "The spectral bundle method proposed by Helmberg and Rendl is well established for solving large-scale semidefinite programs (SDP) thanks to its low per it... |
https://arxiv.org/abs/1710.05731 | Trees and $n$-Good Hypergraphs | Trees fill many extremal roles in graph theory, being minimally connected and serving a critical role in the definition of $n$-good graphs. In this article, we consider the generalization of trees to the setting of $r$-uniform hypergraphs and how one may extend the notion of $n$-good graphs to this setting. We prove nu... | \section{Introduction}
In graph theory, trees play the important role of being minimally connected. The removal of any edge results in a disconnected graph. So, it is no surprise that trees serve as optimal graphs with regard to certain extremal properties, especially in Ramsey theory. Here, one defines the Ramsey ... | {
"timestamp": "2017-10-17T02:16:42",
"yymm": "1710",
"arxiv_id": "1710.05731",
"language": "en",
"url": "https://arxiv.org/abs/1710.05731",
"abstract": "Trees fill many extremal roles in graph theory, being minimally connected and serving a critical role in the definition of $n$-good graphs. In this articl... |
https://arxiv.org/abs/0902.0958 | Randomized Kaczmarz solver for noisy linear systems | The Kaczmarz method is an iterative algorithm for solving systems of linear equations Ax=b. Theoretical convergence rates for this algorithm were largely unknown until recently when work was done on a randomized version of the algorithm. It was proved that for overdetermined systems, the randomized Kaczmarz method conv... | \section{Introduction}
The Kaczmarz method~\cite{K37:Angena} is one of the most popular solvers of overdetermined linear systems and has numerous applications from computer tomography to image processing. It is an iterative method, and so therefore is practical in the realm of very large systems of equations. The al... | {
"timestamp": "2010-03-25T01:00:13",
"yymm": "0902",
"arxiv_id": "0902.0958",
"language": "en",
"url": "https://arxiv.org/abs/0902.0958",
"abstract": "The Kaczmarz method is an iterative algorithm for solving systems of linear equations Ax=b. Theoretical convergence rates for this algorithm were largely un... |
https://arxiv.org/abs/2107.09029 | Conditions for matchability in groups and field extensions | The origins of the notion of matchings in groups spawn from a linear algebra problem proposed by E. K. Wakeford [24] which was tackled in 1996 [10]. In this paper, we first discuss unmatchable subsets in abelian groups. Then we formulate and prove linear analogues of results concerning matchings, along with a conjectur... | \section{Introduction}
Throughout this paper, we may assume that $G$ is an additive abelian group, unless stated otherwise.
Let $B$ be a finite subset of $G$ which does not contain the neutral element. For any subset $A$ in $G$ with the same cardinality as $B$, a {\it matching} from $A$ to $B$ is defined to be a bijec... | {
"timestamp": "2022-03-09T02:06:03",
"yymm": "2107",
"arxiv_id": "2107.09029",
"language": "en",
"url": "https://arxiv.org/abs/2107.09029",
"abstract": "The origins of the notion of matchings in groups spawn from a linear algebra problem proposed by E. K. Wakeford [24] which was tackled in 1996 [10]. In th... |
https://arxiv.org/abs/2106.12019 | Length-Preserving Directions and Some Diophantine Equations | We study directions along which the norms of vectors are preserved under a linear map. In particular, we find families of matrices for which these directions are determined by integer vectors. We consider the two-dimensional case in detail, and also discuss the extension to the three-dimensional case. | \section{ Introduction.}
In the nice Webinar talk ``Eigenpairs in Maple''
of June 25, 2015 \cite{lopez},
Dr.~Robert Lopez discussed how to use {Maple} to
find eigenvalues and eigenvectors (eigenpairs) of a
matrix $A$.
An eigenvector of $A$ is a (nonzero) vector whose
direction is preserved under multiplication b... | {
"timestamp": "2021-06-24T02:01:39",
"yymm": "2106",
"arxiv_id": "2106.12019",
"language": "en",
"url": "https://arxiv.org/abs/2106.12019",
"abstract": "We study directions along which the norms of vectors are preserved under a linear map. In particular, we find families of matrices for which these directi... |
https://arxiv.org/abs/1803.05419 | Generalised Structural CNNs (SCNNs) for time series data with arbitrary graph topology | Deep Learning methods, specifically convolutional neural networks (CNNs), have seen a lot of success in the domain of image-based data, where the data offers a clearly structured topology in the regular lattice of pixels. This 4-neighbourhood topological simplicity makes the application of convolutional masks straightf... | \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": "2018-05-31T02:15:04",
"yymm": "1803",
"arxiv_id": "1803.05419",
"language": "en",
"url": "https://arxiv.org/abs/1803.05419",
"abstract": "Deep Learning methods, specifically convolutional neural networks (CNNs), have seen a lot of success in the domain of image-based data, where the data off... |
https://arxiv.org/abs/2001.10928 | Discrete Trace Theorems and Energy Minimizing Spring Embeddings of Planar Graphs | Tutte's spring embedding theorem states that, for a three-connected planar graph, if the outer face of the graph is fixed as the complement of some convex region in the plane, and all other vertices are placed at the mass center of their neighbors, then this results in a unique embedding, and this embedding is planar. ... | \section{Introduction}
Graph drawing is an area at the intersection of mathematics, computer
science, and more qualitative fields. Despite the extensive literature
in the field, in many ways the concept of what constitutes the optimal drawing of a
graph is heuristic at best, and subjective at worst. For a general revi... | {
"timestamp": "2020-07-10T02:03:11",
"yymm": "2001",
"arxiv_id": "2001.10928",
"language": "en",
"url": "https://arxiv.org/abs/2001.10928",
"abstract": "Tutte's spring embedding theorem states that, for a three-connected planar graph, if the outer face of the graph is fixed as the complement of some convex... |
https://arxiv.org/abs/1612.00284 | The discrete Pompeiu problem on the plane | We say that a finite subset $E$ of the Euclidean plane $\mathbb{R}^2$ has the discrete Pompeiu property with respect to isometries (similarities), if, whenever $f:\mathbb{R}^2\to \mathbb{C}$ is such that the sum of the values of $f$ on any congruent (similar) copy of $E$ is zero, then $f$ is identically zero. We show t... | \section{Introduction}\label{s1}
Let $K$ be a compact subset of the plane having positive
Lebesgue measure. The set $K$ is said to have the
Pompeiu property if the following condition is satisfied:
whenever $f$ is a continuous function defined on the plane, and
the integral of $f$ over every congruent copy of $K$ is ze... | {
"timestamp": "2016-12-02T02:06:20",
"yymm": "1612",
"arxiv_id": "1612.00284",
"language": "en",
"url": "https://arxiv.org/abs/1612.00284",
"abstract": "We say that a finite subset $E$ of the Euclidean plane $\\mathbb{R}^2$ has the discrete Pompeiu property with respect to isometries (similarities), if, wh... |
https://arxiv.org/abs/2003.10112 | Sparsest Piecewise-Linear Regression of One-Dimensional Data | We study the problem of one-dimensional regression of data points with total-variation (TV) regularization (in the sense of measures) on the second derivative, which is known to promote piecewise-linear solutions with few knots. While there are efficient algorithms for determining such adaptive splines, the difficulty ... | \section*{Acknowledgments}
The authors are thankful to Shayan Aziznejad for many discussions related to this work and for his elegant connection between the (\gBLASSO) problem and its discrete counterpart (see~\eqref{eq:optizlambda}). Julien Fageot was supported by the Swiss National Science Foundation (SNSF) under Gra... | {
"timestamp": "2020-08-04T02:34:30",
"yymm": "2003",
"arxiv_id": "2003.10112",
"language": "en",
"url": "https://arxiv.org/abs/2003.10112",
"abstract": "We study the problem of one-dimensional regression of data points with total-variation (TV) regularization (in the sense of measures) on the second deriva... |
https://arxiv.org/abs/1907.09628 | Limit Shape of Subpartition Maximizing Partitions | This is an expository note answering a question posed to us by Richard Stanley, in which we prove a limit shape theorem for partitions of $n$ which maximize the number of subpartitions. The limit shape and the growth rate of the number of subpartitions are explicit. The key ideas are to use large deviations estimates f... | \section{Maximizing the number of subpartitions} \label{sec:1}
Given a partition $\lambda = (\lambda_1 \ge ...\ge \lambda_k)$ of $n$, we can identify it with a $1$-Lipschitz function which is a finite perturbation of $|x|$ by following the Russian convention for drawing it. Specifically, start with the English convent... | {
"timestamp": "2019-07-24T02:04:53",
"yymm": "1907",
"arxiv_id": "1907.09628",
"language": "en",
"url": "https://arxiv.org/abs/1907.09628",
"abstract": "This is an expository note answering a question posed to us by Richard Stanley, in which we prove a limit shape theorem for partitions of $n$ which maximi... |
https://arxiv.org/abs/1510.01758 | Estimating the Number Of Roots of Trinomials over Finite Fields | We show that univariate trinomials $x^n + ax^s + b \in \mathbb{F}_q[x]$ can have at most $\delta \Big\lfloor \frac{1}{2} +\sqrt{\frac{q-1}{\delta}} \Big\rfloor$ distinct roots in $\mathbb{F}_q$, where $\delta = \gcd(n, s, q - 1)$. We also derive explicit trinomials having $\sqrt{q}$ roots in $\mathbb{F}_q$ when $q$ is ... | \section{Introduction}
For univariate polynomial equations defined over a field, it is desirable to obtain general upper bounds on the number of solutions given in simple terms of plainly available information, such as the coefficients, exponents, or number of terms.
The ubiquitous example of this is the degree bound,... | {
"timestamp": "2016-07-26T02:03:53",
"yymm": "1510",
"arxiv_id": "1510.01758",
"language": "en",
"url": "https://arxiv.org/abs/1510.01758",
"abstract": "We show that univariate trinomials $x^n + ax^s + b \\in \\mathbb{F}_q[x]$ can have at most $\\delta \\Big\\lfloor \\frac{1}{2} +\\sqrt{\\frac{q-1}{\\delta... |
https://arxiv.org/abs/0908.3305 | Cycles are determined by their domination polynomials | Let $G$ be a simple graph of order $n$. A dominating set of $G$ is a set $S$ of vertices of $G$ so that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. The domination polynomial of $G$ is the polynomial $D(G,x)=\sum_{i=1}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of ... | \section{Introduction}
\begin{abstract}
Let $G$ be a simple graph of order $n$. A dominating set of $G$
is a set $S$ of vertices of $G$ so that every vertex of $G$ is
either in $S$ or adjacent to a vertex in $S$. The domination
polynomial of $G$ is the polynomial $D(G,x)=\sum_{i=1}^{n}
d(G,i) x^{i}$,
where $d(G,i)$... | {
"timestamp": "2009-08-23T16:40:08",
"yymm": "0908",
"arxiv_id": "0908.3305",
"language": "en",
"url": "https://arxiv.org/abs/0908.3305",
"abstract": "Let $G$ be a simple graph of order $n$. A dominating set of $G$ is a set $S$ of vertices of $G$ so that every vertex of $G$ is either in $S$ or adjacent to ... |
https://arxiv.org/abs/2209.08164 | Solutions of the Variational Equation for an nth Order Boundary Value Problem with an Integral Boundary Condition | In this paper, we discuss differentiation of solutions to the boundary value problem $y^{(n)} = f(x, y, y^{'}, y^{''}, \ldots, y^{(n-1)}), \; a<x<b,\; y^{(i)}(x_j) = y_{ij},\; 0\leq i \leq m_j, \; 1 \leq j \leq k-1$, and $y^{(i)}(x_k) + \int_c^d p y(x)\;dx = y_{ik}, \;0 \leq i \leq m_k,\;\sum_{i=1}^km_i=n$ with respect... | \section{Introduction}
Our concern is characterizing partial derivatives with respect to the boundary data of solutions to the $n$th order nonlocal boundary value problem
\begin{equation}\label{eq1}
y^{(n)} = f\left(x, y, y^{'}, y^{''}, \ldots, y^{(n-1)}\right), \; a<x<b
\end{equation}
satisfying
\begin{equation}\la... | {
"timestamp": "2022-09-20T02:02:13",
"yymm": "2209",
"arxiv_id": "2209.08164",
"language": "en",
"url": "https://arxiv.org/abs/2209.08164",
"abstract": "In this paper, we discuss differentiation of solutions to the boundary value problem $y^{(n)} = f(x, y, y^{'}, y^{''}, \\ldots, y^{(n-1)}), \\; a<x<b,\\; ... |
https://arxiv.org/abs/2112.11997 | Bohr sets in sumsets I: Compact groups | Let $G$ be a compact abelian group and $\phi_1, \phi_2, \phi_3$ be continuous endomorphisms on $G$. Under certain natural assumptions on the $\phi_i$'s, we prove the existence of Bohr sets in the sumset $\phi_1(A) + \phi_2(A) + \phi_3(A)$, where $A$ is either a set of positive Haar measure, or comes from a finite parti... | \section{Introduction and statements of results}
Let $G$ be an abelian topological group. For a finite set $\Lambda$ of characters (i.e. continuous homomorphisms from $G$ to $S^1 := \{ z \in \mathbb{C}: |z|=1\}$) and $\eta > 0$, the set
\begin{equation*} \label{eq:bohr1}
B(\Lambda; \eta) := \{ x \in G : | \gamm... | {
"timestamp": "2022-01-03T02:17:14",
"yymm": "2112",
"arxiv_id": "2112.11997",
"language": "en",
"url": "https://arxiv.org/abs/2112.11997",
"abstract": "Let $G$ be a compact abelian group and $\\phi_1, \\phi_2, \\phi_3$ be continuous endomorphisms on $G$. Under certain natural assumptions on the $\\phi_i$'... |
https://arxiv.org/abs/math/0702432 | Exceptional points for Lebesgue's density theorem on the real line | For a nontrivial measurable set on the real line, there are always exceptional points, where the lower and upper densities of the set are neither zero nor one. We quantify this statement, following work by V. Kolyada, and obtain the unexpected result that there is always a point where the upper and the lower densities ... | \section{Introduction and notation}
\label{intro}
\subsection{Formulation of the problem}
Denote by $\lambda$ the Lebesgue measure on the real line. We will
call a measurable set $S\subset\R$ {\em nontrivial} if neither $S$ nor
$\R\setminus S$ is of measure zero. A point $p\in\R$ is called a {\em
density point} of $... | {
"timestamp": "2007-02-14T23:55:21",
"yymm": "0702",
"arxiv_id": "math/0702432",
"language": "en",
"url": "https://arxiv.org/abs/math/0702432",
"abstract": "For a nontrivial measurable set on the real line, there are always exceptional points, where the lower and upper densities of the set are neither zero... |
https://arxiv.org/abs/1510.00747 | Elementary triangular matrices and inverses of $k$-Hessenberg and triangular matrices | We use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explicit expressions for the inverses of strict $k$-Hessenberg matrices and banded matrices. Our results can be extended to the cases of block triangular and block Hessenbe... | \section{Introduction}
The importance of triangular, Hessenberg, and banded matrices is well-known. Many
problems in linear algebra and matrix theory are solved by some kind of reduction to problems involving such types of matrices. This occurs, for example, with the $LU$ factorizations and the $QR$ algorithms.
I... | {
"timestamp": "2015-10-06T02:02:26",
"yymm": "1510",
"arxiv_id": "1510.00747",
"language": "en",
"url": "https://arxiv.org/abs/1510.00747",
"abstract": "We use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explic... |
https://arxiv.org/abs/1208.2920 | Fooling sets and rank | An $n\times n$ matrix $M$ is called a \textit{fooling-set matrix of size $n$} if its diagonal entries are nonzero and $M_{k,\ell} M_{\ell,k} = 0$ for every $k\ne \ell$. Dietzfelbinger, Hromkovi{č}, and Schnitger (1996) showed that $n \le (\mbox{rk} M)^2$, regardless of over which field the rank is computed, and asked w... | \section{Introduction}
An $n\times n$ matrix~$M$ over a field~$\mathbb{k}$ is called a \textit{fooling-set matrix of size~$n$} if
\begin{subequations}\label{eq:def-fool}
\begin{align}
M_{kk} &\ne 0 &&\text{ for all~$k$ (its diagonal entries are all nonzero), and} \label{eq:def-fool:diag}\\
M_{k,\ell} \, M_{\e... | {
"timestamp": "2014-01-17T02:11:23",
"yymm": "1208",
"arxiv_id": "1208.2920",
"language": "en",
"url": "https://arxiv.org/abs/1208.2920",
"abstract": "An $n\\times n$ matrix $M$ is called a \\textit{fooling-set matrix of size $n$} if its diagonal entries are nonzero and $M_{k,\\ell} M_{\\ell,k} = 0$ for ev... |
https://arxiv.org/abs/1806.10220 | Genus of The Hypercube Graph And Real Moment-Angle Complexes | In this paper we demonstrate a calculation to find the genus of the hypercube graph $Q_n$ using real moment-angle complex $\mathcal{Z}_\mathcal{K}(D^1,S^0)$ where $\mathcal{K}$ is the boundary of an $n$-gon. We also calculate an upper bound for the genus of the quotient graph $Q_n/C_n$, where $C_n$ represents the cycli... | \section*{Introduction}
In graph theory, the hypercube graph is defined as the 1-skeleton of the $n$-dimensional cube. The graph theoretical properties of this graph has been studied extensively by Harary et al in \cite {Harary}. It is well known that this graph has genus $1+(n-4)2^{n-3}$. This fact was proved by Ringe... | {
"timestamp": "2019-04-05T02:04:27",
"yymm": "1806",
"arxiv_id": "1806.10220",
"language": "en",
"url": "https://arxiv.org/abs/1806.10220",
"abstract": "In this paper we demonstrate a calculation to find the genus of the hypercube graph $Q_n$ using real moment-angle complex $\\mathcal{Z}_\\mathcal{K}(D^1,S... |
https://arxiv.org/abs/1711.10112 | Heuristics for the arithmetic of elliptic curves | This is an introduction to a probabilistic model for the arithmetic of elliptic curves, a model developed in a series of articles of the author with Bhargava, Kane, Lenstra, Park, Rains, Voight, and Wood. We discuss the theoretical evidence for the model, and we make predictions about elliptic curves based on correspon... | \section{Introduction}\label{S:introduction}
Let $E$ be an elliptic curve over $\mathbb{Q}$
(see \cite{SilvermanAEC2009} for basic definitions).
Let $E(\mathbb{Q})$ be the set of rational points on $E$.
The group law on $E$ gives $E(\mathbb{Q})$ the structure of an abelian group,
and Mordell proved that $E(\mathbb{Q}... | {
"timestamp": "2017-12-04T02:01:06",
"yymm": "1711",
"arxiv_id": "1711.10112",
"language": "en",
"url": "https://arxiv.org/abs/1711.10112",
"abstract": "This is an introduction to a probabilistic model for the arithmetic of elliptic curves, a model developed in a series of articles of the author with Bharg... |
https://arxiv.org/abs/2004.10038 | On the spectral gap and the diameter of Cayley graphs | We obtain a new bound connecting the first non--trivial eigenvalue of the Laplace operator of a graph and the diameter of the graph, which is effective for graphs with small diameter or for graphs, having the number of maximal paths comparable to the expectation. | \section{Introduction}
\label{sec:introduction}
Expander graphs were first introduced by Bassalygo and Pinsker \cite{BP}, and their existence first proved by Pinsker \cite{Pinsker_expander} (also, see \cite{Margulis}).
The property of a graph of being an expander is significant in many of mathematical and ... | {
"timestamp": "2020-04-22T02:12:45",
"yymm": "2004",
"arxiv_id": "2004.10038",
"language": "en",
"url": "https://arxiv.org/abs/2004.10038",
"abstract": "We obtain a new bound connecting the first non--trivial eigenvalue of the Laplace operator of a graph and the diameter of the graph, which is effective fo... |
https://arxiv.org/abs/2302.03579 | Unshuffling a deck of cards | We investigate the mathematics behind unshuffles, a type of card shuffle closely related to classical perfect shuffles. To perform an unshuffle, deal all the cards alternately into two piles and then stack the one pile on top of the other. There are two ways this stacking can be done (left stack on top or right stack o... | \section{Introduction}
A well known card shuffling technique in the world of magicians is the so-called {\em perfect shuffle} (also known as the {\em faro shuffle}). Not only are perfect shuffles used by magicians to do card tricks, but gamblers have used these shuffles to cheat at card games since the 1800's~\cite{exp... | {
"timestamp": "2023-02-08T02:19:34",
"yymm": "2302",
"arxiv_id": "2302.03579",
"language": "en",
"url": "https://arxiv.org/abs/2302.03579",
"abstract": "We investigate the mathematics behind unshuffles, a type of card shuffle closely related to classical perfect shuffles. To perform an unshuffle, deal all ... |
https://arxiv.org/abs/math/0703505 | A Neumann Type Maximum Principle for the Laplace Operator on Compact Riemannian Manifolds | In this paper we present a proof of a Neumann type maximum principle for the Laplace operator on compact Riemannian manifolds. A key p oint is the simple geometric nature of the constant in the a priori estimate of this maximum principle. In particular, this maximum principle can be applied to manifolds with Ricci curv... | \section{#1} \setcounter{equation}{0}}
\newtheorem{lem}{Lemma}[section]
\begin{document}
\newtheorem{defn}[lem]{Definition}
\newtheorem{theo}[lem]{Theorem}
\newtheorem{cor}[lem]{Corollary}
\newtheorem{prop}[lem]{Proposition}
\newtheorem{rk}[lem]{Remark}
\newtheorem{ex}[lem]{Example}
\newtheorem{note}[lem]{Note}
\... | {
"timestamp": "2007-11-12T00:04:22",
"yymm": "0703",
"arxiv_id": "math/0703505",
"language": "en",
"url": "https://arxiv.org/abs/math/0703505",
"abstract": "In this paper we present a proof of a Neumann type maximum principle for the Laplace operator on compact Riemannian manifolds. A key p oint is the sim... |
https://arxiv.org/abs/0707.2156 | On Hilbert's construction of positive polynomials | In 1888, Hilbert described how to find real polynomials in more than one variable which take only non-negative values but are not a sum of squares of polynomials. His construction was so restrictive that no explicit examples appeared until the late 1960s. We revisit and generalize Hilbert's construction and present man... | \section{History and Overview}
A real polynomial $f(x_1,\dots,x_n)$ is {\it psd} or {\it positive} if
$f(a) \ge
0$ for all $a \in \mathbb R^n$; it is {\it sos} or a {\it sum of
squares} if there exist real polynomials $h_j$ so that $f = \sum h_j^2$.
For forms, we follow the notation
of \cite{CL1} and use $P_{n,m}$ ... | {
"timestamp": "2007-07-14T17:15:54",
"yymm": "0707",
"arxiv_id": "0707.2156",
"language": "en",
"url": "https://arxiv.org/abs/0707.2156",
"abstract": "In 1888, Hilbert described how to find real polynomials in more than one variable which take only non-negative values but are not a sum of squares of polyno... |
https://arxiv.org/abs/2005.01931 | A strategy for Isolator in the Toucher-Isolator game on trees | In the Toucher-Isolator game, introduced recently by Dowden, Kang, Mikalački and Stojaković, Toucher and Isolator alternately claim an edge from a graph such that Toucher aims to touch as many vertices as possible, while Isolator aims to isolate as many vertices as possible, where Toucher plays first. Among trees with ... | \section{Introduction}\label{section-introduction}
A Maker-Breaker game, introduced by Erd\H{o}s and Selfridge~\cite{erdos1973combinatorial} in 1973, is a positional game played on the complete graph $K_n$ with $n$ vertices, by two players: Maker and Breaker, who alternately claim an edge from the (remaining) graph,... | {
"timestamp": "2020-05-07T02:23:51",
"yymm": "2005",
"arxiv_id": "2005.01931",
"language": "en",
"url": "https://arxiv.org/abs/2005.01931",
"abstract": "In the Toucher-Isolator game, introduced recently by Dowden, Kang, Mikalački and Stojaković, Toucher and Isolator alternately claim an edge from a graph s... |
https://arxiv.org/abs/2207.02271 | Maximum size of a triangle-free graph with bounded maximum degree and matching number | Determining the maximum number of edges under degree and matching number constraints have been solved for general graphs by Chvátal and Hanson (1976), and by Balachandran and Khare (2009). It follows from the structure of those extremal graphs that deciding whether this maximum number decreases or not when restricted t... | \section{Introduction}
In extremal graph theory, an important series of problems, including the celebrated Turán's graphs \cite{turan}, investigate the maximization or the minimization of the number of edges in a graph under a given set of constraints. A question of this kind is to determine the maximum number of edge... | {
"timestamp": "2022-07-07T02:01:49",
"yymm": "2207",
"arxiv_id": "2207.02271",
"language": "en",
"url": "https://arxiv.org/abs/2207.02271",
"abstract": "Determining the maximum number of edges under degree and matching number constraints have been solved for general graphs by Chvátal and Hanson (1976), and... |
https://arxiv.org/abs/1601.04091 | Multigrid Methods for Constrained Minimization Problems and Application to Saddle Point Problems | The first order condition of the constrained minimization problem leads to a saddle point problem. A multigrid method using a multiplicative Schwarz smoother for saddle point problems can thus be interpreted as a successive subspace optimization method based on a multilevel decomposition of the constraint space. Conver... | \section{Introduction}
Given a quadratic energy $E(v)$ defined on a Hilbert space $\mathcal V$, we consider the constrained minimization problem:
\begin{equation}\label{intro:main-opt}
\min _{v\in \mathcal K} E(v),
\end{equation}
where $\mathcal K\subset \mathcal V$ is the null space of a linear and bounded operator $... | {
"timestamp": "2016-01-19T02:01:22",
"yymm": "1601",
"arxiv_id": "1601.04091",
"language": "en",
"url": "https://arxiv.org/abs/1601.04091",
"abstract": "The first order condition of the constrained minimization problem leads to a saddle point problem. A multigrid method using a multiplicative Schwarz smoot... |
https://arxiv.org/abs/2208.11413 | Sharp inequalities for Neumann eigenvalues on the sphere | We prove that the second nontrivial Neumann eigenvalue of the Laplace-Beltrami operator on the unit sphere $\mathbb{S}^n \subseteq \mathbb{R}^{n+1}$ is maximized by the union of two disjoint, equal, geodesic balls among all subsets of $\mathbb{S}^n$ of prescribed volume. In fact, the result holds in a stronger version,... | \section{Introduction}
Let $n \ge 2$ and denote by $\Sn$ the unit sphere of dimension $n$ in $\R^{n+1}$. Let $\Om \subseteq \Sn$ be an open, Lipschitz set of measure $m\in (0,|\Sn|)$. The eigenvalues of the Laplace-Beltrami operator with Neumann boundary conditions on $\Om$ are
$$0= \mu_0(\Om) \le \mu_1(\Om) \le \dot... | {
"timestamp": "2022-08-25T02:12:47",
"yymm": "2208",
"arxiv_id": "2208.11413",
"language": "en",
"url": "https://arxiv.org/abs/2208.11413",
"abstract": "We prove that the second nontrivial Neumann eigenvalue of the Laplace-Beltrami operator on the unit sphere $\\mathbb{S}^n \\subseteq \\mathbb{R}^{n+1}$ is... |
https://arxiv.org/abs/1910.07281 | The diameter and radius of radially maximal graphs | A graph is called radially maximal if it is not complete and the addition of any new edge decreases its radius. In 1976 Harary and Thomassen proved that the radius $r$ and diameter $d$ of any radially maximal graph satisfy $r\le d\le 2r-2.$ Dutton, Medidi and Brigham rediscovered this result with a different proof in 1... | \section{Introduction}
We consider finite simple graphs. Denote by $V(G)$ and $E(G)$ the vertex set and edge set of a graph $G$ respectively. The complement of $G$
is denoted by $\bar{G}.$ The radius and diameter of $G$ are denoted by ${\rm rad}(G)$ and ${\rm diam}(G)$ respectively.
{\bf Definition.} A graph $G$ is ... | {
"timestamp": "2019-10-17T02:11:40",
"yymm": "1910",
"arxiv_id": "1910.07281",
"language": "en",
"url": "https://arxiv.org/abs/1910.07281",
"abstract": "A graph is called radially maximal if it is not complete and the addition of any new edge decreases its radius. In 1976 Harary and Thomassen proved that t... |
https://arxiv.org/abs/2301.09101 | Bounds On the order of the Schur multiplier of $p$-groups | In 1956, Green provided a bound on the order of the Schur multiplier of $p$-groups. This bound, given as a function of the order of the group, is the best possible. Since then, the bound has been refined numerous times by adding other inputs to the function, such as, the minimal number of generators of the group and th... | \section{Introduction}
\vspace{.2cm}
The Schur multiplier $M(G)$ of a finite group $G$ is defined as the second cohomology group of $G$ with coefficients in $\mathbb{C}^{*}$. It plays an important role in the theory of extensions of groups. Finding the bounds on the order, exponents, and ranks of the Schur mult... | {
"timestamp": "2023-01-24T02:11:51",
"yymm": "2301",
"arxiv_id": "2301.09101",
"language": "en",
"url": "https://arxiv.org/abs/2301.09101",
"abstract": "In 1956, Green provided a bound on the order of the Schur multiplier of $p$-groups. This bound, given as a function of the order of the group, is the best... |
https://arxiv.org/abs/1608.08834 | On the edge of the stable range | We prove a general homological stability theorem for certain families of groups equipped with product maps, followed by two theorems of a new kind that give information about the last two homology groups outside the stable range. (These last two unstable groups are the "edge" in our title.) Applying our results to auto... | \section{Introduction}
A sequence of groups and inclusions
$
G_1\hookrightarrow G_1\hookrightarrow G_3\hookrightarrow\cdots
$
is said to satisfy \emph{homological stability}
if in each degree $d$ there is an integer $n_d$ such
that the induced map $H_d(G_{n-1})\to H_d(G_n)$
is an isomorphism for $n> n_d$.
Homological... | {
"timestamp": "2016-10-25T02:01:05",
"yymm": "1608",
"arxiv_id": "1608.08834",
"language": "en",
"url": "https://arxiv.org/abs/1608.08834",
"abstract": "We prove a general homological stability theorem for certain families of groups equipped with product maps, followed by two theorems of a new kind that gi... |
https://arxiv.org/abs/1012.3541 | On the polygonal diameter of the interior, resp. exterior, of a simple closed polygon in the plane | We give a tight upper bound on the polygonal diameter of the interior, resp. exterior, of a simple $n$-gon, $n \ge 3$, in the plane as a function of $n$, and describe an $n$-gon $(n \ge 3)$ for which both upper bounds (for the interior and the exterior) are attained \emph{simultaneously}. | \section{Introduction}
The following is well known
\begin{theorem}{\rm (The Jordan theorem)}\label{theo1.1}
Let $f:[0,1] \to \bR^2$ be a simple closed curve in the plane ($f$ is continous, $f(0) = f(1)$ and $f(u) \not= f(v)$ for $0 < u < v \le 1$).
Define $P=_{\rm def}$ {\rm image}$f= \{f(u) : 0 \le u \le 1\}$,... | {
"timestamp": "2010-12-17T02:01:15",
"yymm": "1012",
"arxiv_id": "1012.3541",
"language": "en",
"url": "https://arxiv.org/abs/1012.3541",
"abstract": "We give a tight upper bound on the polygonal diameter of the interior, resp. exterior, of a simple $n$-gon, $n \\ge 3$, in the plane as a function of $n$, a... |
https://arxiv.org/abs/1002.2847 | A variant of the Johnson-Lindenstrauss lemma for circulant matrices | We continue our study of the Johnson-Lindenstrauss lemma and its connection to circulant matrices started in \cite{HV}. We reduce the bound on $k$ from $k=O(\epsilon^{-2}\log^3n)$ proven there to $k=O(\epsilon^{-2}\log^2n)$. Our technique differs essentially from the one used in \cite{HV}. We employ the discrete Fourie... | \section{Introduction}
Let $x^1,\dots,x^n\in {\mathbb R}^d$ be $n$ points in the $d$-dimensional Euclidean space ${\mathbb R}^d$.
The classical Johnson-Lindenstrauss
lemma tells that, for a given $\varepsilon\in(0,\frac 12)$ and a natural number
$k=O(\varepsilon^{-2}\log n)$, there exists a linear map $f:{\mathbb R}^d... | {
"timestamp": "2010-02-15T11:23:19",
"yymm": "1002",
"arxiv_id": "1002.2847",
"language": "en",
"url": "https://arxiv.org/abs/1002.2847",
"abstract": "We continue our study of the Johnson-Lindenstrauss lemma and its connection to circulant matrices started in \\cite{HV}. We reduce the bound on $k$ from $k=... |
https://arxiv.org/abs/2002.11015 | Parabolic frequency on manifolds | We prove monotonicity of a parabolic frequency on manifolds. This is a parabolic analog of Almgren's frequency function. Remarkably we get monotonicity on all manifolds and no curvature assumption is needed. When the manifold is Euclidean space and the drift operator is the Ornstein-Uhlenbeck operator this can been see... | \section{Introduction}
Bounds on growth for functions satisfying a PDE give crucial information and have many consequences. One of the oldest bounds of this type is Hadamard's three circles theorem for holomorphic functions.
For elliptic equations, such as the Laplace equation, Almgren proved the monotonicity of ... | {
"timestamp": "2020-02-26T02:18:04",
"yymm": "2002",
"arxiv_id": "2002.11015",
"language": "en",
"url": "https://arxiv.org/abs/2002.11015",
"abstract": "We prove monotonicity of a parabolic frequency on manifolds. This is a parabolic analog of Almgren's frequency function. Remarkably we get monotonicity on... |
https://arxiv.org/abs/1806.09953 | On the maximum number of odd cycles in graphs without smaller odd cycles | We prove that for each odd integer $k \geq 7$, every graph on $n$ vertices without odd cycles of length less than $k$ contains at most $(n/k)^k$ cycles of length $k$. This generalizes the previous results on the maximum number of pentagons in triangle-free graphs, conjectured by Erdős in 1984, and asymptotically determ... | \section{Introduction}
In 1984, Erd\H{o}s \cite{Erdos} conjectured that every triangle-free graph on $n$ vertices contains at most $(n/5)^5$ cycles of length 5 and the maximum is attained at the balanced blow-up of a~$C_5$. Gy\"{o}ri \cite{Gyori} proved an upper bound within a~factor 1.03 of the optimal. Using flag al... | {
"timestamp": "2018-10-17T02:03:10",
"yymm": "1806",
"arxiv_id": "1806.09953",
"language": "en",
"url": "https://arxiv.org/abs/1806.09953",
"abstract": "We prove that for each odd integer $k \\geq 7$, every graph on $n$ vertices without odd cycles of length less than $k$ contains at most $(n/k)^k$ cycles o... |
https://arxiv.org/abs/1407.5311 | SB-labelings and posets with each interval homotopy equivalent to a sphere or a ball | We introduce a new class of poset edge labelings for locally finite lattices which we call $SB$-labelings. We prove for finite lattices which admit an $SB$-labeling that each open interval has the homotopy type of a ball or of a sphere of some dimension. Natural examples include the weak order, the Tamari lattice, and ... | \section{Introduction}
Anders Bj\"orner and Curtis Greene have raised the following question (personal communication of Bj\"orner; see also \cite{Gr} by Greene).
\begin{qn}
Why are there so many posets with the property that every interval has M\"obius function equalling $0, 1$ or $-1$? Is there a unifying explanat... | {
"timestamp": "2014-07-22T02:08:53",
"yymm": "1407",
"arxiv_id": "1407.5311",
"language": "en",
"url": "https://arxiv.org/abs/1407.5311",
"abstract": "We introduce a new class of poset edge labelings for locally finite lattices which we call $SB$-labelings. We prove for finite lattices which admit an $SB$-... |
https://arxiv.org/abs/0705.4536 | Refined bound for sum-free sets in groups of prime order | Improving upon earlier results of Freiman and the present authors, we show that if $p$ is a sufficiently large prime and $A$ is a sum-free subset of the group of order $p$, such that $n:=|A|>0.318p$, then $A$ is contained in a dilation of the interval $[n,p-n]\pmod p$. | \section{Introduction}
The subset $A$ of an additively written semigroup is called \emph{sum-free}
if there do not exist $a_1,a_2,a_3\in A$ with $a_1+a_2=a_3$; equivalently, if
$A$ is disjoint with its \emph{sumset} $A+A:=\{a_1+a_2\colon a_1,a_2\in A\}$.
Introduced by Schur in 1916 (``the set of positive integers cann... | {
"timestamp": "2007-05-31T10:17:01",
"yymm": "0705",
"arxiv_id": "0705.4536",
"language": "en",
"url": "https://arxiv.org/abs/0705.4536",
"abstract": "Improving upon earlier results of Freiman and the present authors, we show that if $p$ is a sufficiently large prime and $A$ is a sum-free subset of the gro... |
https://arxiv.org/abs/1301.7544 | The random graph | Erdős and Rényi showed the paradoxical result that there is a unique (and highly symmetric) countably infinite random graph. This graph, and its automorphism group, form the subject of the present survey. | \chapter{The Random Graph}
\label{ch32:chap32}
\textbf{Summary.} Erd\H{o}s and R\'{e}nyi showed the paradoxical
result that there is a unique (and highly symmetric) countably
infinite random graph. This graph, and its automorphism group, form
the subject of the present survey.
\section{Introduction
\label{ch32:sec2.... | {
"timestamp": "2013-02-01T02:02:30",
"yymm": "1301",
"arxiv_id": "1301.7544",
"language": "en",
"url": "https://arxiv.org/abs/1301.7544",
"abstract": "Erdős and Rényi showed the paradoxical result that there is a unique (and highly symmetric) countably infinite random graph. This graph, and its automorphis... |
https://arxiv.org/abs/1008.1286 | Subalgebras of Matrix Algebras Generated by Companion Matrices | Let $f,g\in Z[X]$ be monic polynomials of degree $n$ and let $C,D\in M_n(Z)$ be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra $Z< C,D>$ to be a sublattice of finite index in the full integral lattice $M_n(Z)$, in which case we compute the exact value of this index ... | \section{Introduction} About twenty years ago a question of Chatters [C1]
generated a series of articles concerned with the problem of
identifying full matrix rings. We refer the reader to the papers
[A], [AMR], [C2], [LRS], [R] cited in the bibliography for more
details. In particular, very simple presentations of ful... | {
"timestamp": "2010-08-10T02:00:22",
"yymm": "1008",
"arxiv_id": "1008.1286",
"language": "en",
"url": "https://arxiv.org/abs/1008.1286",
"abstract": "Let $f,g\\in Z[X]$ be monic polynomials of degree $n$ and let $C,D\\in M_n(Z)$ be the corresponding companion matrices. We find necessary and sufficient con... |
https://arxiv.org/abs/2105.08391 | A Steiner general position problem in graph theory | Let $G$ be a graph. The Steiner distance of $W\subseteq V(G)$ is the minimum size of a connected subgraph of $G$ containing $W$. Such a subgraph is necessarily a tree called a Steiner $W$-tree. The set $A\subseteq V(G)$ is a $k$-Steiner general position set if $V(T_B)\cap A = B$ holds for every set $B\subseteq A$ of ca... | \section{Introduction}
\label{sec:intro}
In this work, $G = (V(G), E(G))$ denotes a simple graph. The \emph{distance} $d_G(u,v)$ between two vertices $u$ and $v$ of $G$ is the minimum number of edges on a $u,v$-path in $G$. If there is no such path, then we set $d_G(u,v)=\infty$. A $u,v$-path of length $d_G(u,v)$ is c... | {
"timestamp": "2021-05-19T02:14:29",
"yymm": "2105",
"arxiv_id": "2105.08391",
"language": "en",
"url": "https://arxiv.org/abs/2105.08391",
"abstract": "Let $G$ be a graph. The Steiner distance of $W\\subseteq V(G)$ is the minimum size of a connected subgraph of $G$ containing $W$. Such a subgraph is neces... |
https://arxiv.org/abs/1308.5459 | Unseparated pairs and fixed points in random permutations | In a uniform random permutation \Pi of [n] := {1,2,...,n}, the set of elements k in [n-1] such that \Pi(k+1) = \Pi(k) + 1 has the same distribution as the set of fixed points of \Pi that lie in [n-1]. We give three different proofs of this fact using, respectively, an enumeration relying on the inclusion-exclusion prin... | \section{Introduction}
The goal of any procedure for shuffling a deck of $n$ cards
labeled with, say, $[n] := \{1,2,\ldots,n\}$ is to take
the cards in some specified original order, which we may
take to be $(1,2,\ldots,n)$, and re-arrange them randomly
in such a way that all $n!$ possible orders are close to
being e... | {
"timestamp": "2014-04-29T02:10:46",
"yymm": "1308",
"arxiv_id": "1308.5459",
"language": "en",
"url": "https://arxiv.org/abs/1308.5459",
"abstract": "In a uniform random permutation \\Pi of [n] := {1,2,...,n}, the set of elements k in [n-1] such that \\Pi(k+1) = \\Pi(k) + 1 has the same distribution as th... |
https://arxiv.org/abs/2206.08914 | Graphs with Sudoku number $n-1$ | Recently Lau-Jeyaseeli-Shiu-Arumugam introduced the concept of the "Sudoku colourings" of graphs -- partial $\chi(G)$-colourings of $G$ that have a unique extension to a proper $\chi(G)$-colouring of all the vertices. They introduced the Sudoku number of a graph as the minimal number of coloured vertices in a Sudoku co... | \section{Introduction}
All colourings in this note are vertex-colourings.
A vertex colouring of a graph is \emph{proper} if adjacent vertices receive different colours. The chromatic number, $\chi(G)$ is the minimal number of colours in a proper colouring of $G$. A \emph{partial} proper colouring of a graph is a colour... | {
"timestamp": "2022-06-20T02:21:09",
"yymm": "2206",
"arxiv_id": "2206.08914",
"language": "en",
"url": "https://arxiv.org/abs/2206.08914",
"abstract": "Recently Lau-Jeyaseeli-Shiu-Arumugam introduced the concept of the \"Sudoku colourings\" of graphs -- partial $\\chi(G)$-colourings of $G$ that have a uni... |
https://arxiv.org/abs/1703.06316 | On the linear polarization constants of finite dimensional spaces | We study the linear polarization constants of finite dimensional Banach spaces. We obtain the correct asymptotic behaviour of these constants for the spaces $\ell_p^d$: they behave as $\sqrt[p]{d}$ if $1\le p\le 2$ and as $\sqrt{d}$ if $2\le p<\infty$. For $p=\infty$ we get the asymptotic behavior up to a logarithmic f... | \section*{Introduction}
Given a Banach space $X$, its \textit{$n$th linear polarization constant} is defined as the smallest constant $\mathbf c_n(X)$ such that for any set of $n$ linear functionals $\{\psi_j\}_{j=1}^n\subseteq X^*$, we have
\begin{equation}\label{problempolarizacion}
\Vert \psi_1 \Vert \cdots \Vert ... | {
"timestamp": "2017-03-21T01:04:30",
"yymm": "1703",
"arxiv_id": "1703.06316",
"language": "en",
"url": "https://arxiv.org/abs/1703.06316",
"abstract": "We study the linear polarization constants of finite dimensional Banach spaces. We obtain the correct asymptotic behaviour of these constants for the spac... |
https://arxiv.org/abs/1901.00587 | On bounded elementary generation for $SL_n$ over polynomial rings | Let $F[X]$ be the polynomial ring over a finite field $F$. It is shown that, for $n\geq 3$, the special linear group $SL_n(F[X])$ is boundedly generated by the elementary matrices. | \section{Introduction}
The special linear group $\mathrm{SL}_n(\mathbb{Z})$ is generated by the elementary matrices, that is, matrices which differ from the identity by at most one non-zero off-diagonal entry. Far more remarkable is the following fact:
\begin{thm}[Carter - Keller \cite{CK}]\label{thm: ck}
Let $n\ge... | {
"timestamp": "2019-01-04T02:04:05",
"yymm": "1901",
"arxiv_id": "1901.00587",
"language": "en",
"url": "https://arxiv.org/abs/1901.00587",
"abstract": "Let $F[X]$ be the polynomial ring over a finite field $F$. It is shown that, for $n\\geq 3$, the special linear group $SL_n(F[X])$ is boundedly generated ... |
https://arxiv.org/abs/2209.15493 | Rainbow triangles in families of triangles | We prove that a family $\mathcal{T}$ of distinct triangles on $n$ given vertices that does not have a rainbow triangle (that is, three edges, each taken from a different triangle in $\mathcal{T}$, that form together a triangle) must be of size at most $\frac{n^2}{8}$. We also show that this result is sharp and characte... | \section{Introduction}
Let $\mathcal{F}$ be a family of sets. A rainbow set (with respect to $\mathcal{F}$) is a subset $R\subseteq\cup\mathcal{F}$, together with an injection $\sigma:R\to\mathcal{F}$ such that $e\in\sigma(e)$ for all $e\in R$. We view every member of $\mathcal{F}$ as a different color, and every $... | {
"timestamp": "2022-10-14T02:18:51",
"yymm": "2209",
"arxiv_id": "2209.15493",
"language": "en",
"url": "https://arxiv.org/abs/2209.15493",
"abstract": "We prove that a family $\\mathcal{T}$ of distinct triangles on $n$ given vertices that does not have a rainbow triangle (that is, three edges, each taken ... |
https://arxiv.org/abs/1402.5129 | On a Cohen-Lenstra Heuristic for Jacobians of Random Graphs | In this paper, we make specific conjectures about the distribution of Jacobians of random graphs with their canonical duality pairings. Our conjectures are based on a Cohen-Lenstra type heuristic saying that a finite abelian group with duality pairing appears with frequency inversely proportional to the size of the gro... | \section{Introduction}
Jacobians of graphs are often cyclic. A similar phenomenon has been observed in class groups of imaginary quadratic fields, where it is conjecturally explained by the classical Cohen-Lenstra heuristic, in which a group $\Gamma$ appears with frequency proportional to $1/ \# {\operatorname{Aut \;... | {
"timestamp": "2015-04-22T02:11:30",
"yymm": "1402",
"arxiv_id": "1402.5129",
"language": "en",
"url": "https://arxiv.org/abs/1402.5129",
"abstract": "In this paper, we make specific conjectures about the distribution of Jacobians of random graphs with their canonical duality pairings. Our conjectures are ... |
https://arxiv.org/abs/0801.2115 | A study of counts of Bernoulli strings via conditional Poisson processes | We say that a string of length $d$ occurs, in a Bernoulli sequence, if a success is followed by exactly $(d-1)$ failures before the next success. The counts of such $d$-strings are of interest, and in specific independent Bernoulli sequences are known to correspond to asymptotic $d$-cycle counts in random permutations.... | \section{#1}\setcounter{equation}{0}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem... | {
"timestamp": "2008-01-14T17:45:22",
"yymm": "0801",
"arxiv_id": "0801.2115",
"language": "en",
"url": "https://arxiv.org/abs/0801.2115",
"abstract": "We say that a string of length $d$ occurs, in a Bernoulli sequence, if a success is followed by exactly $(d-1)$ failures before the next success. The counts... |
https://arxiv.org/abs/0802.0316 | Fourier series and approximation on hexagonal and triangular domains | Several problems on Fourier series and trigonometric approximation on a hexagon and a triangle are studied. The results include Abel and Cesàro summability of Fourier series, degree of approximation and best approximation by trigonometric functions, both direct and inverse theorems. One of the objective of this study i... | \section{Introduction}
\setcounter{equation}{0}
A theorem of Fuglede \cite{F} states that a set tiles ${\mathbb R}^n$ by lattice translation
if and only if it has an orthonormal basis of exponentials $e^{i {\langle} \a, x\ra}$ with
$\a$ in the dual lattice. Such a set is called a spectral set. The theorem suggests... | {
"timestamp": "2008-02-04T01:04:45",
"yymm": "0802",
"arxiv_id": "0802.0316",
"language": "en",
"url": "https://arxiv.org/abs/0802.0316",
"abstract": "Several problems on Fourier series and trigonometric approximation on a hexagon and a triangle are studied. The results include Abel and Cesàro summability ... |
https://arxiv.org/abs/2108.12034 | Optimal Point Sets Determining Few Distinct Angles | We characterize the largest point sets in the plane which define at most 1, 2, and 3 angles. For $P(k)$ the largest size of a point set admitting at most $k$ angles, we prove $P(2)=5$ and $P(3)=5$. We also provide the general bounds of $k+2 \leq P(k) \leq 6k$, although the upper bound may be improved pending progress t... | \section{Introduction}
\subsection{Background}
In 1946, Erdős introduced the problem of finding asymptotic bounds on the minimum number of distinct distances among sets of $n$ points in the plane \cite{ErOg}. The Erd\H{o}s distance problem, as it has become known, proved infamously difficult and was only finally (essen... | {
"timestamp": "2021-08-30T02:03:10",
"yymm": "2108",
"arxiv_id": "2108.12034",
"language": "en",
"url": "https://arxiv.org/abs/2108.12034",
"abstract": "We characterize the largest point sets in the plane which define at most 1, 2, and 3 angles. For $P(k)$ the largest size of a point set admitting at most ... |
https://arxiv.org/abs/1506.07941 | Two classes of modular $p$-Stanley sequences | Consider a set $A$ with no $p$-term arithmetic progressions for $p$ prime. The $p$-Stanley sequence of a set $A$ is generated by greedily adding successive integers that do not create a $p$-term arithmetic progression. For $p>3$ prime, we give two distinct constructions for $p$-Stanley sequences which have a regular st... | \section{Introduction}
A set is called \textit{$p$-free} if it contains no $p$-term arithmetic progression. The study of $p$-free sets has been of much interest. Szekeres conjectured that for $p$ an odd prime, the maximum number of elements in a $p$-free subset of $\{0,1,\ldots,n-1\}$ grew as $n^{\log_p(p-1)}$ \cite{e... | {
"timestamp": "2015-06-29T02:02:28",
"yymm": "1506",
"arxiv_id": "1506.07941",
"language": "en",
"url": "https://arxiv.org/abs/1506.07941",
"abstract": "Consider a set $A$ with no $p$-term arithmetic progressions for $p$ prime. The $p$-Stanley sequence of a set $A$ is generated by greedily adding successiv... |
https://arxiv.org/abs/math/0609827 | On A. Zygmund differentiation conjecture | Consider $v$ a Lipschitz unit vector field on $R^n$ and $K$ its Lipschitz constant. We show that the maps $S_s:S_s(X) = X + sv(X)$ are invertible for $0\leq |s|<1/K$ and define nonsingular point transformations. We use these properties to prove first the differentiation in L^p norm for $1\le p<\infty.$ Then we show the... | \section{Introduction}
Lebesgue differentiation theorem states that given a function $f\in
L^1(\mathbb{R})$ the averages $\displaystyle \frac{1}{2t}\int_{-t}^t f(x+u) du$
converge a.e. to $f(x)$ when $t$ tends to zero. The differentiation
for functions $F$ defined on $\mathbb{R}^2$ is more subtle. Actually it is a
long... | {
"timestamp": "2006-09-29T00:36:39",
"yymm": "0609",
"arxiv_id": "math/0609827",
"language": "en",
"url": "https://arxiv.org/abs/math/0609827",
"abstract": "Consider $v$ a Lipschitz unit vector field on $R^n$ and $K$ its Lipschitz constant. We show that the maps $S_s:S_s(X) = X + sv(X)$ are invertible for ... |
https://arxiv.org/abs/2108.12514 | A Bisection Method Like Algorithm for Approximating Extrema of a Continuous Function | For a continuous function $f$ defined on a closed and bounded domain, there is at least one maximum and one minimum. First, we introduce some preliminaries which are necessary through the paper. We then present an algorithm, which is similar to the bisection method, to approximate those maximum and minimum values. We a... | \section{Introduction}
For nonlinear equations, there is no general algorithm or method that computes an exact root of a nonlinear equation, unfortunately. However, in science and engineering, one usually encounters a situation in which she needs to compute a root of a nonlinear equation. For this reason, there are sev... | {
"timestamp": "2021-08-31T02:05:03",
"yymm": "2108",
"arxiv_id": "2108.12514",
"language": "en",
"url": "https://arxiv.org/abs/2108.12514",
"abstract": "For a continuous function $f$ defined on a closed and bounded domain, there is at least one maximum and one minimum. First, we introduce some preliminarie... |
https://arxiv.org/abs/1712.06896 | Integrable geodesic flows on tubular sub-manifolds | In this paper we construct a new class of surfaces whose geodesic flow is integrable (in the sense of Liouville). We do so by generalizing the notion of tubes about curves to 3-dimensional manifolds, and using Jacobi fields we derive conditions under which the metric of the generalized tubular sub-manifold admits an ig... | \section{Introduction}
Let $(\mathcal{M}^n,g)$ be a Riemannian manifold. We may write the geodesic equations as a Hamiltonian system on the cotangent bundle $T^*\mathcal{M}$. The integral curves of this vector field constitute a flow, called the geodesic flow. If this Hamiltonian system has $n$ first integrals, funct... | {
"timestamp": "2017-12-20T02:07:46",
"yymm": "1712",
"arxiv_id": "1712.06896",
"language": "en",
"url": "https://arxiv.org/abs/1712.06896",
"abstract": "In this paper we construct a new class of surfaces whose geodesic flow is integrable (in the sense of Liouville). We do so by generalizing the notion of t... |
https://arxiv.org/abs/2003.08190 | Random triangles on flat tori | Inspired by classical puzzles in geometry that ask about probabilities of geometric phenomena, we give an explicit formula for the probability that a random triangle on a flat torus is homotopically trivial. Our main tool for this computation involves reducing the problem to new invariant of measurable sets in the plan... | \section{Introduction}
A classical problem, recorded as problem 58 in a 1893 book of puzzles by Charles Dodgson \cite{Dodgson}, is to determine the probability that a triangle on the plane is obtuse. Since the plane has no finite measure invariant by translation, this problem is not well defined and several answers can... | {
"timestamp": "2020-03-19T01:09:44",
"yymm": "2003",
"arxiv_id": "2003.08190",
"language": "en",
"url": "https://arxiv.org/abs/2003.08190",
"abstract": "Inspired by classical puzzles in geometry that ask about probabilities of geometric phenomena, we give an explicit formula for the probability that a rand... |
https://arxiv.org/abs/1201.1851 | Enumerating Trees | In this note we discuss trees similar to the Calkin-Wilf tree, a binary tree that enumerates all positive rational numbers in a simple way. The original construction of Calkin and Wilf is reformulated in a more algebraic language, and an elementary application of methods from analytic number theory gives restrictions o... | \section{The Calkin-Wilf Tree}
\noindent In \cite{CalkinWilf}, Neil Calkin and Herbert Wilf introduced a remarkably beautiful\footnote{It was considered worthy by the authors of \cite{AignerZiegler09} to be included into their BOOK.} way to enumerate the positive rational numbers, drawing together several observations... | {
"timestamp": "2012-01-10T02:03:53",
"yymm": "1201",
"arxiv_id": "1201.1851",
"language": "en",
"url": "https://arxiv.org/abs/1201.1851",
"abstract": "In this note we discuss trees similar to the Calkin-Wilf tree, a binary tree that enumerates all positive rational numbers in a simple way. The original con... |
https://arxiv.org/abs/1403.0053 | Bootstrapping and Askey-Wilson polynomials | The mixed moments for the Askey-Wilson polynomials are found using a bootstrapping method and connection coefficients. A similar bootstrapping idea on generating functions gives a new Askey-Wilson generating function. An important special case of this hierarchy is a polynomial which satisfies a four term recurrence, an... | \section{Introduction}
The Askey-Wilson polynomials \cite{AskeyWilson}
$p_n(x;a,b,c,d|q)$ are orthogonal polynomials in $x$
which depend upon five parameters: $a$, $b$, $c$, $d$ and $q$.
In \cite[\S2]{BI} Berg and Ismail use a bootstrapping method to prove
orthogonality of Askey-Wilson polynomials by initially star... | {
"timestamp": "2014-03-04T02:01:59",
"yymm": "1403",
"arxiv_id": "1403.0053",
"language": "en",
"url": "https://arxiv.org/abs/1403.0053",
"abstract": "The mixed moments for the Askey-Wilson polynomials are found using a bootstrapping method and connection coefficients. A similar bootstrapping idea on gener... |
https://arxiv.org/abs/1605.03322 | On tiling the integers with $4$-sets of the same gap sequence | Partitioning a set into similar, if not, identical, parts is a fundamental research topic in combinatorics. The question of partitioning the integers in various ways has been considered throughout history. Given a set $\{x_1, \ldots, x_n\}$ of integers where $x_1<\cdots<x_n$, let the {\it gap sequence} of this set be t... | \section{Introduction}
Let $[n]$ denote the set $\{1, \ldots, n\}$ and let $[a, b]$ denote the set $\{a, \ldots, b\}$.
Note that $[1, 0]=\emptyset$.
An $n$-set is a set of size $n$.
Partitioning a set into similar, if not, identical, parts is a fundamental research topic in combinatorics.
In the literature, it is ... | {
"timestamp": "2016-05-12T02:06:48",
"yymm": "1605",
"arxiv_id": "1605.03322",
"language": "en",
"url": "https://arxiv.org/abs/1605.03322",
"abstract": "Partitioning a set into similar, if not, identical, parts is a fundamental research topic in combinatorics. The question of partitioning the integers in v... |
https://arxiv.org/abs/1903.06317 | Limits of Sums for Binomial and Eulerian Numbers and their Associated Distributions | We provide a unified, probabilistic approach using renewal theory to derive some novel limits of sums for the normalized binomial coefficients and for the normalized Eulerian numbers. We also investigate some corresponding results for their associated distributions -- the binomial distributions for the binomial coeffic... | \section{Introduction}\label{sec:introduction}
Start with Pascal's triangle, the binomial coefficients ${n \choose k}$ arranged in a triangular array as in Figure~\ref{fig:pascals triangle}. Now normalize each row to sum to one: for each $n$ divide the $n$th row by $2^n$. After this normalization we can ask: what are t... | {
"timestamp": "2019-03-18T01:08:10",
"yymm": "1903",
"arxiv_id": "1903.06317",
"language": "en",
"url": "https://arxiv.org/abs/1903.06317",
"abstract": "We provide a unified, probabilistic approach using renewal theory to derive some novel limits of sums for the normalized binomial coefficients and for the... |
https://arxiv.org/abs/0805.2707 | Recurrence Formulas for Fibonacci Sums | In this article we present a new recurrence formula for a finite sum involving the Fibonacci sequence. Furthermore, we state an algorithm to compute the sum of a power series related to Fibonacci series, without the use of term-by-term differentiation theorem | \section{Introducion}
\renewcommand{\theequation}{1.\arabic{equation}}
\setcounter{equation}{0}
The Fibonacci sequence is one of the most famous numerical sequences in mathematics. It is defined in a recursive way: the first two terms are given and the following ones are defined as the sum of the two preceding ones. M... | {
"timestamp": "2008-05-18T02:44:43",
"yymm": "0805",
"arxiv_id": "0805.2707",
"language": "en",
"url": "https://arxiv.org/abs/0805.2707",
"abstract": "In this article we present a new recurrence formula for a finite sum involving the Fibonacci sequence. Furthermore, we state an algorithm to compute the sum... |
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