url stringlengths 31 38 | title stringlengths 7 229 | abstract stringlengths 44 2.87k | text stringlengths 319 2.51M | meta dict |
|---|---|---|---|---|
https://arxiv.org/abs/1311.6788 | Twisted Analytic Torsion and Adiabatic Limits | We study an analogue of the analytic torsion for elliptic complexes that are graded by $\mathbb{Z}_2$, orignally constructed by Mathai and Wu. Motivated by topological T-duality, Bouwknegt an Mathai study the complex of forms on an odd-dimensional manifold equipped with with the twisted differential $d_H = d+H$, where ... | \section{Twisted Analytic Torsion}
\subsection{Background}
For an elliptic complex $(E,d)$ over an odd-dimensional compact manifold $M$, one can construct a canonical metric $\| \, \|_{RS} $ on the determinant line $\det H(E,d)$ known as the Ray-Singer (RS) metric. This was first done by Quillen \cite{Quillen:1985a}, ... | {
"timestamp": "2013-11-27T02:12:14",
"yymm": "1311",
"arxiv_id": "1311.6788",
"language": "en",
"url": "https://arxiv.org/abs/1311.6788",
"abstract": "We study an analogue of the analytic torsion for elliptic complexes that are graded by $\\mathbb{Z}_2$, orignally constructed by Mathai and Wu. Motivated by... |
https://arxiv.org/abs/1305.4282 | Dynamics of a Continuous Piecewise Affine Map of the Square | We present a one-parameter family of continuous, piecewise affine, area preserving maps of the square, which are inspired by a dynamical system in game theory. Interested in the coexistence of stochastic and (quasi-)periodic behaviour, we investigate invariant annuli separated by invariant circles. For certain paramete... | \section{Introduction}
Piecewise affine maps and piecewise isometries have received a lot of attention, either as simple,
computationally accessible models for complicated dynamical behaviour, or as a class of systems
with their own unique range of dynamical phenomena. For a list of examples, see
\cite{Bullett1986,W... | {
"timestamp": "2013-05-21T02:01:06",
"yymm": "1305",
"arxiv_id": "1305.4282",
"language": "en",
"url": "https://arxiv.org/abs/1305.4282",
"abstract": "We present a one-parameter family of continuous, piecewise affine, area preserving maps of the square, which are inspired by a dynamical system in game theo... |
https://arxiv.org/abs/0812.1883 | An introduction to exotic 4-manifolds | This article intends to provide an introduction to the construction of small exotic 4-manifolds. Some of the necessary background is covered. An exposition is given of J. Park's construction inarXiv:math.GT/0311395of an exotic CP^2#7(-CP^2). This article does not intend to present any new results. It was originally a M... | \section{Introduction}
A manifold $X$ is \emph{exotic} if it is homeomorphic to another manifold $Y$, but is not diffeomorphic to it. Usually, $Y$ is a well-known manifold and then we say ``$X$ is an exotic $Y$''. \newline
A natural question to ask is ``How easy it is to find an exotic manifold?''. Or, more specifica... | {
"timestamp": "2008-12-31T20:10:20",
"yymm": "0812",
"arxiv_id": "0812.1883",
"language": "en",
"url": "https://arxiv.org/abs/0812.1883",
"abstract": "This article intends to provide an introduction to the construction of small exotic 4-manifolds. Some of the necessary background is covered. An exposition ... |
https://arxiv.org/abs/2006.16811 | Path Integral Based Convolution and Pooling for Graph Neural Networks | Graph neural networks (GNNs) extends the functionality of traditional neural networks to graph-structured data. Similar to CNNs, an optimized design of graph convolution and pooling is key to success. Borrowing ideas from physics, we propose a path integral based graph neural networks (PAN) for classification and regre... | \section{Introduction}
The triumph of convolutional neural networks (CNNs) has motivated researchers to develop similar architectures for graph-structured data. The task is challenging due to the absence of regular grids. One notable proposal is to define convolutions in the Fourier space \cite{BrZaSzLe2013,Bronstein_e... | {
"timestamp": "2020-07-09T02:17:39",
"yymm": "2006",
"arxiv_id": "2006.16811",
"language": "en",
"url": "https://arxiv.org/abs/2006.16811",
"abstract": "Graph neural networks (GNNs) extends the functionality of traditional neural networks to graph-structured data. Similar to CNNs, an optimized design of gr... |
https://arxiv.org/abs/1305.5114 | Uniform spanning trees on Sierpinski graphs | We study spanning trees on Sierpinski graphs (i.e., finite approximations to the Sierpinski gasket) that are chosen uniformly at random. We construct a joint probability space for uniform spanning trees on every finite Sierpinski graph and show that this construction gives rise to a multi-type Galton-Watson tree. We de... | \section{Introduction}
\label{section:intro}
The Sierpi\'nski gasket is certainly one of the most famous fractals, and the Sierpi\'n\-ski graphs, which can be seen as finite approximations of the Sierpi\'nski gasket, are among the most thoroughly studied self-similar graphs. The number of spanning trees in the $n$\nob... | {
"timestamp": "2015-01-14T02:11:53",
"yymm": "1305",
"arxiv_id": "1305.5114",
"language": "en",
"url": "https://arxiv.org/abs/1305.5114",
"abstract": "We study spanning trees on Sierpinski graphs (i.e., finite approximations to the Sierpinski gasket) that are chosen uniformly at random. We construct a join... |
https://arxiv.org/abs/0704.2203 | On Abelian Difference Sets with Parameters of 3-dimensional Projective Geometries | A difference set is said to have classical parameters if $ (v,k, \lambda) = (\frac{q^d-1}{q-1}, \frac{q^{d-1}-1}{q-1}, \frac{q^{d-2}-1}{q-1}).$ The case $d=3$ corresponds to planar difference sets. We focus here on the family of abelian difference sets with $d=4$. The only known examples of such difference sets corresp... | \section{Group Rings}
Let $G$ be a finite abelian group of order $v$ and let $\mathbb{Z}G$
denote the integral group ring of $G$.
Given an element $a = \sum a_ig_i \in \mathbb{Z}G$, we set
\[
a^{(-1)} = \sum a_i g_i^{-1}.
\]
Let $D$ be a $k$-subset of $G$, where $k\ge 1$.
By a standard abuse of notation we will use th... | {
"timestamp": "2007-04-17T19:25:41",
"yymm": "0704",
"arxiv_id": "0704.2203",
"language": "en",
"url": "https://arxiv.org/abs/0704.2203",
"abstract": "A difference set is said to have classical parameters if $ (v,k, \\lambda) = (\\frac{q^d-1}{q-1}, \\frac{q^{d-1}-1}{q-1}, \\frac{q^{d-2}-1}{q-1}).$ The case... |
https://arxiv.org/abs/1912.05102 | Dimension and structure of higher-order Voronoi cells on discrete sites | We study the structure of higher-order Voronoi cells on a discrete set of sites in $\mathbb{R}^n$, focussing on the relations between cells of different order, and paying special attention to the ill-posed case when a large number of points lie on a sphere. In particular, we prove that higher order cells of dimension $... | \section{Introduction}
Voronoi cells are used in computer graphics, crystallography, facility location, and have numerous other applications. The first recorded use of a Voronoi cell-like object goes back to Ren\'e Descartes \cite{descartes}, while mathematical foundations were initially developed by Dirichlet \cite... | {
"timestamp": "2019-12-12T02:06:19",
"yymm": "1912",
"arxiv_id": "1912.05102",
"language": "en",
"url": "https://arxiv.org/abs/1912.05102",
"abstract": "We study the structure of higher-order Voronoi cells on a discrete set of sites in $\\mathbb{R}^n$, focussing on the relations between cells of different ... |
https://arxiv.org/abs/2108.11020 | Logarithmic Euler Maruyama Scheme for Multi Dimensional Stochastic Delay Differential Equation | In this paper, we extend the logarithmic Euler-Maruyama scheme for stochastic delay differential equation in one dimension to the part where we propose a scheme for a system of stochastic delay differential equations. We then show that the scheme always maintains positivity subject to initial conditions. We then show t... | \setcounter{equation}{0}\Section{\setcounter{equation}{0}\Section}
\usepackage{graphicx}
\date{}
\title[Logarithmic Euler-Maruyama scheme]{logarithmic Euler-Maruyama scheme for multi-dimensional stochastic delay equations with jumps}
\author[Agrawal]{Nishant Agrawal}
\address{Department of Mathematical and Sta... | {
"timestamp": "2021-09-01T02:06:53",
"yymm": "2108",
"arxiv_id": "2108.11020",
"language": "en",
"url": "https://arxiv.org/abs/2108.11020",
"abstract": "In this paper, we extend the logarithmic Euler-Maruyama scheme for stochastic delay differential equation in one dimension to the part where we propose a ... |
https://arxiv.org/abs/1510.05360 | The $k$-independent graph of a graph | Let $G=(V,E)$ be a simple graph. A set $I\subseteq V$ is an independent set, if no two of its members are adjacent in $G$. The $k$-independent graph of $G$, $I_k (G)$, is defined to be the graph whose vertices correspond to the independent sets of $G$ that have cardinality at most $k$. Two vertices in $I_k(G)$ are adja... | \section{Introduction}
Given a simple graph $G=(V,E)$, a set $I\subseteq V$ is an independent set of $G$, if there is no edge of $G$ between any two vertices of $I$.
A maximal independent set is an independent set that is not a proper subset of any other independent set.
A maximum independent set is an independent set ... | {
"timestamp": "2015-10-22T02:07:00",
"yymm": "1510",
"arxiv_id": "1510.05360",
"language": "en",
"url": "https://arxiv.org/abs/1510.05360",
"abstract": "Let $G=(V,E)$ be a simple graph. A set $I\\subseteq V$ is an independent set, if no two of its members are adjacent in $G$. The $k$-independent graph of $... |
https://arxiv.org/abs/1104.2511 | On the J-anti-invariant cohomology of almost complex 4-manifolds | For a compact almost complex 4-manifold $(M,J)$, we study the subgroups $H^{\pm}_J$ of $H^2(M, \mathbb{R})$ consisting of cohomology classes representable by $J$-invariant, respectively, $J$-anti-invariant 2-forms. If $b^+ =1$, we show that for generic almost complex structures on $M$, the subgroup $H^-_J$ is trivial. ... | \section{Introduction}
For any almost complex manifold $(M, J)$, the last two authors
\cite{LZ} introduced certain subgroups of the de Rham cohomology
groups, naturally defined by the almost complex structure. These
subgroups are interesting almost
complex invariants and there are several works already devoted to
thei... | {
"timestamp": "2011-04-14T02:02:08",
"yymm": "1104",
"arxiv_id": "1104.2511",
"language": "en",
"url": "https://arxiv.org/abs/1104.2511",
"abstract": "For a compact almost complex 4-manifold $(M,J)$, we study the subgroups $H^{\\pm}_J$ of $H^2(M, \\mathbb{R})$ consisting of cohomology classes representable... |
https://arxiv.org/abs/2205.09116 | Exploring the Adjugate Matrix Approach to Quaternion Pose Extraction | Quaternions are important for a wide variety of rotation-related problems in computer graphics, machine vision, and robotics. We study the nontrivial geometry of the relationship between quaternions and rotation matrices by exploiting the adjugate matrix of the characteristic equation of a related eigenvalue problem to... |
\section{Introduction}
\label{intro.sec}
We address the task of understanding whether there are obstacles to using quaternions
to represent orientation space, typical examples being the determination of
the optimal rotation to align two matched 2D or 3D point clouds, or find the pose
of the 2D or 3D point cloud tha... | {
"timestamp": "2022-05-20T02:00:09",
"yymm": "2205",
"arxiv_id": "2205.09116",
"language": "en",
"url": "https://arxiv.org/abs/2205.09116",
"abstract": "Quaternions are important for a wide variety of rotation-related problems in computer graphics, machine vision, and robotics. We study the nontrivial geom... |
https://arxiv.org/abs/1701.07339 | Loci of Points Inspired by Viviani's Theorem | We consider loci of points such that their sum of distances or sum of squared distances to each of the sides of a given triangle is constant. These loci are inspired by Viviani's theorem and its extension. The former locus is a line segment or the whole triangle and the latter locus is an ellipse. | \section*{Constant sum of distances}
Samelson \cite[p. 225]{Sam} gave a proof of Viviani's theorem that uses
vectors and Chen \& Liang \cite[p. 390-391]{CL} used this vector method to
prove a converse: if inside a triangle there is a circular region in which
the sum of the distances from the sides is constant, th... | {
"timestamp": "2017-01-26T02:06:54",
"yymm": "1701",
"arxiv_id": "1701.07339",
"language": "en",
"url": "https://arxiv.org/abs/1701.07339",
"abstract": "We consider loci of points such that their sum of distances or sum of squared distances to each of the sides of a given triangle is constant. These loci a... |
https://arxiv.org/abs/1602.05476 | On concordances in 3-manifolds | We describe an action of the concordance group of knots in the three-sphere on concordances of knots in arbitrary 3-manifolds. As an application we define the notion of almost-concordance between knots. After some basic results, we prove the existence of non-trivial almost-concordance classes in all non-abelian 3-manif... | \section*{Introduction}\label{sec:intro}
A classical and extensively studied feature of knots in the $3$-sphere is the group structure induced by connected sum on concordance classes. Much is known on the concordance group $\mathcal{C}$, and many recent progresses have been made by Heegaard Floer theoretic techniques.... | {
"timestamp": "2018-01-11T02:06:10",
"yymm": "1602",
"arxiv_id": "1602.05476",
"language": "en",
"url": "https://arxiv.org/abs/1602.05476",
"abstract": "We describe an action of the concordance group of knots in the three-sphere on concordances of knots in arbitrary 3-manifolds. As an application we define... |
https://arxiv.org/abs/1506.04629 | The 3-colorability of planar graphs without cycles of length 4, 6 and 9 | In this paper, we prove that planar graphs without cycles of length 4, 6, 9 are 3-colorable. | \section{Introduction}
The well-known Four Color Theorem
states that every planar graph is 4-colorable. On the 3-colorability
of planar graphs, a famous theorem owing to Gr\"{o}tzsch \cite{Grotzsch1959109} states
that every planar graph without cycles of length 3 is 3-colorable.
Therefore, next sufficient conditions t... | {
"timestamp": "2015-06-16T02:17:34",
"yymm": "1506",
"arxiv_id": "1506.04629",
"language": "en",
"url": "https://arxiv.org/abs/1506.04629",
"abstract": "In this paper, we prove that planar graphs without cycles of length 4, 6, 9 are 3-colorable.",
"subjects": "Combinatorics (math.CO)",
"title": "The 3-... |
https://arxiv.org/abs/1405.2073 | Topological Invariants and Fibration Structure of Complete Intersection Calabi-Yau Four-Folds | We investigate the mathematical properties of the class of Calabi-Yau four-folds recently found in [arXiv:1303.1832]. This class consists of 921,497 configuration matrices which correspond to manifolds that are described as complete intersections in products of projective spaces. For each manifold in the list, we compu... | \section{Introduction and review of the classification of CICY four-folds}\seclabel{review}
Calabi-Yau manifolds have played a central role in many aspects of the development of string theory, from phenomenology to formal theory. Several constructions of Calabi-Yau three-folds have seen extensive use in the literature... | {
"timestamp": "2014-09-19T02:09:39",
"yymm": "1405",
"arxiv_id": "1405.2073",
"language": "en",
"url": "https://arxiv.org/abs/1405.2073",
"abstract": "We investigate the mathematical properties of the class of Calabi-Yau four-folds recently found in [arXiv:1303.1832]. This class consists of 921,497 configu... |
https://arxiv.org/abs/2302.00468 | LS-category and topological complexity of several families of fibre bundles | In this paper, we study upper bounds for the topological complexity of the total spaces of some classes of fibre bundles. We calculate a tight upper bound for the topological complexity of an $n$-dimensional Klein bottle. We also compute the exact value of the topological complexity of $3$-dimensional Klein bottle. We ... | \section{Introduction}\label{sec:intro}
Lusternik and Schnirelmann \cite{LS-paper1} introduced a homotopy invariant of a topological space, known as the `LS-category', to study some problems in variational calculus. This invariant has been studied widely since the 1940s, see \cite{Fox, LS-paper2, CLOT}. Two decades la... | {
"timestamp": "2023-02-02T02:15:12",
"yymm": "2302",
"arxiv_id": "2302.00468",
"language": "en",
"url": "https://arxiv.org/abs/2302.00468",
"abstract": "In this paper, we study upper bounds for the topological complexity of the total spaces of some classes of fibre bundles. We calculate a tight upper bound... |
https://arxiv.org/abs/1706.04069 | Fast Inverse Nonlinear Fourier Transform | This paper considers the non-Hermitian Zakharov-Shabat (ZS) scattering problem which forms the basis for defining the SU$(2)$-nonlinear Fourier transform (NFT). The theoretical underpinnings of this generalization of the conventional Fourier transform is quite well established in the Ablowitz-Kaup-Newell-Segur (AKNS) f... | \section*{Notations}
\label{sec:notations}
The set of non-zero positive real numbers ($\field{R}$) is denoted by
$\field{R}_+$. Non-zero positive (negative) integers are denoted by
$\field{Z}_+$ ($\field{Z}_-$). For any complex number $\zeta$,
$\Re(\zeta)$ and $\Im(\zeta)$ refer to the real
and the imaginary parts of ... | {
"timestamp": "2018-05-09T02:12:00",
"yymm": "1706",
"arxiv_id": "1706.04069",
"language": "en",
"url": "https://arxiv.org/abs/1706.04069",
"abstract": "This paper considers the non-Hermitian Zakharov-Shabat (ZS) scattering problem which forms the basis for defining the SU$(2)$-nonlinear Fourier transform ... |
https://arxiv.org/abs/1309.7275 | Superelliptical laws for complex networks | All dynamical systems of biological interest--be they food webs, regulation of genes, or contacts between healthy and infectious individuals--have complex network structure. Wigner's semicircular law and Girko's circular law describe the eigenvalues of systems whose structure is a fully connected network. However, thes... | \section*{Results}
\subsection*{Symmetric Matrices}
\begin{sidewaysfigure}
\includegraphics[width
= \linewidth]{Figures/5000-1-Normal.pdf} \caption{\textit{Probability
density function for the eigenvalues of symmetric matrices whose
underlying structure is a complex network. The network has average
d... | {
"timestamp": "2013-11-08T02:10:26",
"yymm": "1309",
"arxiv_id": "1309.7275",
"language": "en",
"url": "https://arxiv.org/abs/1309.7275",
"abstract": "All dynamical systems of biological interest--be they food webs, regulation of genes, or contacts between healthy and infectious individuals--have complex n... |
https://arxiv.org/abs/1903.01566 | On Additive Divisor Sums and minorants of divisor functions | We establish asymptotic formulae for various correlations involving general divisor functions $d_k(n)$ and partial divisor functions $d_l(n,A)=\sum_{q|n:q\leq n^A}d_{l-1}(q)$, where $A\in[0,1]$ is a parameter and $k,l\in\mathbb{N}$ are fixed. Our results relate the parameter $A$ to the lengths of arithmetic progression... | \section{}
\begin{abstract}
We establish asymptotic formulae for various correlations involving general divisor functions $d_k(n)$ and partial divisor functions $d_l(n,A)=\sum_{q|n:q\leq n^A}d_{l-1}(q)$, where $A\in[0,1]$ is a parameter and $k,l\in\mathbb{N}$ are fixed. Our results relate the parameter $A$ to the leng... | {
"timestamp": "2019-03-06T02:04:21",
"yymm": "1903",
"arxiv_id": "1903.01566",
"language": "en",
"url": "https://arxiv.org/abs/1903.01566",
"abstract": "We establish asymptotic formulae for various correlations involving general divisor functions $d_k(n)$ and partial divisor functions $d_l(n,A)=\\sum_{q|n:... |
https://arxiv.org/abs/2108.05569 | Agnostic Online Learning and Excellent Sets | We use algorithmic methods from online learning to revisit a key idea from the interaction of model theory and combinatorics, the existence of large "indivisible" sets, called "$\epsilon$-excellent," in $k$-edge stable graphs (equivalently, Littlestone classes). These sets arise in the Stable Regularity Lemma, a theore... | \section{Background and motivation}
\section{Overview}
In this section we briefly present three complementary points of view (combinatorics \S 1.1, online learning \S 1.2, model theory \S 1.3) which inform this work, and state our main results in \S 1.4.
The aim is to allow the paper to be readable by people in all t... | {
"timestamp": "2021-09-06T02:09:52",
"yymm": "2108",
"arxiv_id": "2108.05569",
"language": "en",
"url": "https://arxiv.org/abs/2108.05569",
"abstract": "We use algorithmic methods from online learning to revisit a key idea from the interaction of model theory and combinatorics, the existence of large \"ind... |
https://arxiv.org/abs/1806.02399 | Maximum and minimum nullity of a tree degree sequence | The nullity of a graph is the multiplicity of the eigenvalue zero in its adjacency spectrum. In this paper, we give a closed formula for the minimum and maximum nullity among trees with the same degree sequence, using the notion of matching number and annihilation number. Algorithms for constructing such minimum-nullit... | \section{Introduction}
Collatz and Sinogowitz (1957), see \cite{von1957spektren}, first raised the problem of characterizing all singular or nonsingular graphs. This problem is hard to be solved. On one hand, the nullity is relevant to the rank of symmetric matrices described by graphs. On the other hand, the nullity ... | {
"timestamp": "2018-06-08T02:02:30",
"yymm": "1806",
"arxiv_id": "1806.02399",
"language": "en",
"url": "https://arxiv.org/abs/1806.02399",
"abstract": "The nullity of a graph is the multiplicity of the eigenvalue zero in its adjacency spectrum. In this paper, we give a closed formula for the minimum and m... |
https://arxiv.org/abs/1207.0840 | On Rainbow Cycles and Paths | In a properly edge colored graph, a subgraph using every color at most once is called rainbow. In this thesis, we study rainbow cycles and paths in proper edge colorings of complete graphs, and we prove that in every proper edge coloring of K_n, there is a rainbow path on (3/4-o(1))n vertices, improving on the previous... | \chapter{Introduction}
\section{Rainbow cycles and paths}
Consider an edge colored graph $G$.
A subgraph of $G$ is called \emph{rainbow} (or \emph{heterochromatic})
if no two of its edges receive the same color. We are concerned with
rainbow paths and, to a lesser extent, cycles in proper edge colorings of the comple... | {
"timestamp": "2012-07-05T02:01:04",
"yymm": "1207",
"arxiv_id": "1207.0840",
"language": "en",
"url": "https://arxiv.org/abs/1207.0840",
"abstract": "In a properly edge colored graph, a subgraph using every color at most once is called rainbow. In this thesis, we study rainbow cycles and paths in proper e... |
https://arxiv.org/abs/1203.6397 | Max-Sum Diversification, Monotone Submodular Functions and Dynamic Updates | Result diversification is an important aspect in web-based search, document summarization, facility location, portfolio management and other applications. Given a set of ranked results for a set of objects (e.g. web documents, facilities, etc.) with a distance between any pair, the goal is to select a subset $S$ satisf... | \section{The Greedy Algorithm Applied to Diversification with a Matroid
Constraint}
\label{greedy-matroid}
We observe that
for the more general matroid constraint diversification problem,
the greedy algorithm in section \ref{sec:submo} no longer achieves any constant approximation ratio. More specifically, cons... | {
"timestamp": "2014-08-21T02:05:32",
"yymm": "1203",
"arxiv_id": "1203.6397",
"language": "en",
"url": "https://arxiv.org/abs/1203.6397",
"abstract": "Result diversification is an important aspect in web-based search, document summarization, facility location, portfolio management and other applications. G... |
https://arxiv.org/abs/1903.04968 | A quantitative Lovász criterion for Property B | A well known observation of Lovász is that if a hypergraph is not $2$-colorable, then at least one pair of its edges intersect at a single vertex. %This very simple criterion turned out to be extremly useful . In this short paper we consider the quantitative version of Lovász's criterion. That is, we ask how many pairs... | \section{Introduction}\label{sec:intro}
A hypergraph ${\cal H}=(V,E)$ consists of a vertex set $V$ and a set
of edges $E$ where each
$X \in E$ is a subset of $V$. If all edges of ${\cal H}$ have size $n$
then ${\cal H}$
is called an $n$-uniform hypergraph, or $n$-graph for short.
A hypergraph is $2$-colorable... | {
"timestamp": "2019-03-13T01:17:59",
"yymm": "1903",
"arxiv_id": "1903.04968",
"language": "en",
"url": "https://arxiv.org/abs/1903.04968",
"abstract": "A well known observation of Lovász is that if a hypergraph is not $2$-colorable, then at least one pair of its edges intersect at a single vertex. %This v... |
https://arxiv.org/abs/1601.07678 | Extremal Relations Between Shannon Entropy and $\ell_α$-Norm | The paper examines relationships between the Shannon entropy and the $\ell_{\alpha}$-norm for $n$-ary probability vectors, $n \ge 2$. More precisely, we investigate the tight bounds of the $\ell_{\alpha}$-norm with a fixed Shannon entropy, and vice versa. As applications of the results, we derive the tight bounds betwe... | \section{Introduction}
Information measures of random variables are used in several fields.
The Shannon entropy \cite{shannon} is one of the famous measures of uncertainty for a given random variable.
On the studies of information measures, inequalities for information measures are commonly used in many applications.
... | {
"timestamp": "2016-01-29T02:05:58",
"yymm": "1601",
"arxiv_id": "1601.07678",
"language": "en",
"url": "https://arxiv.org/abs/1601.07678",
"abstract": "The paper examines relationships between the Shannon entropy and the $\\ell_{\\alpha}$-norm for $n$-ary probability vectors, $n \\ge 2$. More precisely, w... |
https://arxiv.org/abs/1509.04632 | The Shape of Data and Probability Measures | We introduce the notion of multiscale covariance tensor fields (CTF) associated with Euclidean random variables as a gateway to the shape of their distributions. Multiscale CTFs quantify variation of the data about every point in the data landscape at all spatial scales, unlike the usual covariance tensor that only qua... | \section{Introduction}
\label{S:intro}
Probing, analyzing and visualizing the shape of complex data are
challenges that are magnified by the intricate dependence of their
structural properties, as basic as dimensionality, on location
and scale (cf.\,\cite{ljm09}). As such, resolving and integrating the
geometry and t... | {
"timestamp": "2017-03-01T02:03:07",
"yymm": "1509",
"arxiv_id": "1509.04632",
"language": "en",
"url": "https://arxiv.org/abs/1509.04632",
"abstract": "We introduce the notion of multiscale covariance tensor fields (CTF) associated with Euclidean random variables as a gateway to the shape of their distrib... |
https://arxiv.org/abs/1305.2944 | Linear combinations of frame generators in systems of translates | A finitely generated shift invariant space $V$ is a closed subspace of $L^2(\R^d)$ that is generated by the integer translates of a finite number of functions. A set of frame generators for $V$ is a set of functions whose integer translates form a frame for $V$. In this note we give necessary and sufficient conditions ... | \section{Introduction}
{\it Shift invariant spaces} (SISs) are closed subspaces of $L^2(\mathbb{R}^d)$ that are invariant under integer translations. They play an important role in approximation theory, harmonic analysis, wavelet
theory, sampling and signal processing \cite{AG01, Gro01, HW96, Mal89}. The structure ... | {
"timestamp": "2013-12-13T02:01:04",
"yymm": "1305",
"arxiv_id": "1305.2944",
"language": "en",
"url": "https://arxiv.org/abs/1305.2944",
"abstract": "A finitely generated shift invariant space $V$ is a closed subspace of $L^2(\\R^d)$ that is generated by the integer translates of a finite number of functi... |
https://arxiv.org/abs/2201.09115 | Disproof of a Conjecture by Woodall | In 2001, Woodall conjectured that for every pair of integers $s,t \ge 1$, all graphs without a $K_{s,t}$-minor are $(s+t-1)$-choosable. In this note we refute this conjecture in a strong form: We prove that for every choice of constants $\varepsilon>0$ and $C \ge 1$ there exists $N=N(\varepsilon,C) \in \mathbb{N}$ such... | \section{Introduction}
\paragraph{\textbf{Preliminaries.}} All graphs considered in this paper are loopless and have no parallel edges. Given numbers $s,t \in \mathbb{N}$ we denote by $K_t$ the complete graph of order $t$ and by $K_{s,t}$ the complete bipartite graph with bipartition classes of size $s$ and $t$, respec... | {
"timestamp": "2022-01-25T02:15:07",
"yymm": "2201",
"arxiv_id": "2201.09115",
"language": "en",
"url": "https://arxiv.org/abs/2201.09115",
"abstract": "In 2001, Woodall conjectured that for every pair of integers $s,t \\ge 1$, all graphs without a $K_{s,t}$-minor are $(s+t-1)$-choosable. In this note we r... |
https://arxiv.org/abs/1804.10920 | Partial complementation of graphs | A partial complement of the graph $G$ is a graph obtained from $G$ by complementing all the edges in one of its induced subgraphs. We study the following algorithmic question: for a given graph $G$ and graph class $\mathcal{G}$, is there a partial complement of $G$ which is in $\mathcal{G}$? We show that this problem c... | \section{Introduction}
One of the most important questions in graph theory concerns the efficiency of recognition of a graph class $\mathcal{G}$. For example, how fast we can decide whether a graph is chordal,
2-connected,
triangle-free,
of bounded treewidth,
bipartite,
$3$-colorable,
or excludes some fixed graph as a... | {
"timestamp": "2018-05-01T02:09:14",
"yymm": "1804",
"arxiv_id": "1804.10920",
"language": "en",
"url": "https://arxiv.org/abs/1804.10920",
"abstract": "A partial complement of the graph $G$ is a graph obtained from $G$ by complementing all the edges in one of its induced subgraphs. We study the following ... |
https://arxiv.org/abs/1601.02085 | Approximating Stochastic Evolution Equations with Additive White and Rough Noises | In this paper, we analyze Galerkin approximations for stochastic evolution equations driven by an additive Gaussian noise which is temporally white and spatially fractional with Hurst index less than or equal to $1/2$. First we regularize the noise by the Wong-Zakai approximation and obtain its optimal order of converg... | \section{Introduction}
\label{sec1}
In this paper we consider the Galerkin approximation of the stochastic evolution equation (SEE)
\begin{align}\label{see}
L u(t,x)=b(u(t,x))+\xi(t,x),\quad (t,x)\in I\times \mathcal O,
\end{align}
with either homogenous Dirichlet boundary condition
\begin{align}\label{dbc}
u(t,0)=... | {
"timestamp": "2017-03-21T01:10:32",
"yymm": "1601",
"arxiv_id": "1601.02085",
"language": "en",
"url": "https://arxiv.org/abs/1601.02085",
"abstract": "In this paper, we analyze Galerkin approximations for stochastic evolution equations driven by an additive Gaussian noise which is temporally white and sp... |
https://arxiv.org/abs/1812.02235 | The Hamiltonian Circuit Polytope | The hamiltonian circuit polytope is the convex hull of feasible solutions for the circuit constraint, which provides a succinct formulation of the traveling salesman and other sequencing problems. We study the polytope by establishing its dimension, developing tools for the identification of facets, and using these too... | \section{Introduction.}\label{intro}
\section{Introduction}
The {\em circuit constraint} \cite{Lau78,CasLab97,ShuBer94} requires that a sequence of vertices in
a directed graph define a hamiltonian circuit. Given a directed graph $G$ on vertices $1, \ldots, n$, the constraint is written
\begin{equation}
\mbox{circ... | {
"timestamp": "2018-12-07T02:02:18",
"yymm": "1812",
"arxiv_id": "1812.02235",
"language": "en",
"url": "https://arxiv.org/abs/1812.02235",
"abstract": "The hamiltonian circuit polytope is the convex hull of feasible solutions for the circuit constraint, which provides a succinct formulation of the traveli... |
https://arxiv.org/abs/1205.4603 | The maximal energy of classes of integral circulant graphs | The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs, which can be characterized by their vertex count $n$ and a set $\cal D$ of divisors of $n$ in such a way that they have vertex set $\mathbb{Z}_n$ and edge ... | \section{Introduction}
Integral circulant graphs have attacted much research attention lately, in particular since
more and more people have become aware that they
play a role in quantum physics \cite{SAX}, \cite{BAS4}. A characteristic property of circulant graphs
is that their vertices can be numbered such that any... | {
"timestamp": "2012-05-22T02:05:08",
"yymm": "1205",
"arxiv_id": "1205.4603",
"language": "en",
"url": "https://arxiv.org/abs/1205.4603",
"abstract": "The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd... |
https://arxiv.org/abs/1403.0103 | On vanishing theorems for local systems associated to Laurent polynomials | We prove some vanishing theorems for the cohomology groups of local systems associated to Laurent polynomials. In particular, we extend one of the results of Gelfand-Kapranov-Zelevinsky into various directions. | \section{Introduction}\label{sec:1}
The study of the cohomology groups of
local systems is an important subject in
algebraic geometry, hyperplane arrangements,
topology and hypergeometric
functions of several variables.
Many mathematicians are interested
in the conditions for
which we have their concen... | {
"timestamp": "2014-10-23T02:08:17",
"yymm": "1403",
"arxiv_id": "1403.0103",
"language": "en",
"url": "https://arxiv.org/abs/1403.0103",
"abstract": "We prove some vanishing theorems for the cohomology groups of local systems associated to Laurent polynomials. In particular, we extend one of the results o... |
https://arxiv.org/abs/0704.3564 | The critical temperature for the BCS equation at weak coupling | For the BCS equation with local two-body interaction $\lambda V(x)$, we give a rigorous analysis of the asymptotic behavior of the critical temperature as $\lambda \to 0$. We derive necessary and sufficient conditions on $V(x)$ for the existence of a non-trivial solution for all values of $\lambda>0$. | \section{Introduction}
The BCS model has played a prominent role in condensed matter physics
in the fifty years since its introduction \cite{BCS}. Originally
introduced as a model for electrons displaying superconductivity, it
has recently also been used to describe dilute cold gases of fermionic
atoms in the case of ... | {
"timestamp": "2007-10-24T22:13:38",
"yymm": "0704",
"arxiv_id": "0704.3564",
"language": "en",
"url": "https://arxiv.org/abs/0704.3564",
"abstract": "For the BCS equation with local two-body interaction $\\lambda V(x)$, we give a rigorous analysis of the asymptotic behavior of the critical temperature as ... |
https://arxiv.org/abs/1207.0765 | Maharaja Nim, Wythoff's Queen meets the Knight | New combinatorial games are introduced, of which the most pertinent is Maharaja Nim. The rules extend those of the well-known impartial game of Wythoff Nim in which two players take turn in moving a single Queen of Chess on a large board, attempting to be the first to put her in the lower left corner. Here, in addition... | \section{Maharaja Nim}
We introduce a 2-player combinatorial game called
\emph{Maharaja Nim}, an extension of the well-known game
of Wythoff Nim \cite{Wy}. (The name `Maharaja' is taken from a variation
of Chess, `The Maharaja and the Sepoys', \cite{Fa}.)
Both these games are impartial, that is, the set of option... | {
"timestamp": "2012-07-04T02:06:02",
"yymm": "1207",
"arxiv_id": "1207.0765",
"language": "en",
"url": "https://arxiv.org/abs/1207.0765",
"abstract": "New combinatorial games are introduced, of which the most pertinent is Maharaja Nim. The rules extend those of the well-known impartial game of Wythoff Nim ... |
https://arxiv.org/abs/2110.04716 | Spectral structure of the Neumann-Poincaré operator on thin ellipsoids and flat domains | We investigate the spectral structure of the Neumann-Poincaré operator on thin ellipsoids. Two types of thin ellipsoids are considered: long prolate ellipsoids and flat oblate ellipsoids. We show that the totality of eigenvalues of the Neumann-Poincaré operators on a sequence of the prolate spheroids is densely distrib... | \section{Introduction}
For a bounded domain $\Omega$ with the Lipschitz continuous boundary in $\mathbb{R}^d$, $d=2,3$, the Neumann--Poincar\'e (abbreviated by NP) operator associated with $\partial\Omega$ is the boundary integral operator on $\partial\Omega$ defined by
\begin{equation}
\mathcal{K}_{\partial\Omega} [... | {
"timestamp": "2021-10-12T02:18:17",
"yymm": "2110",
"arxiv_id": "2110.04716",
"language": "en",
"url": "https://arxiv.org/abs/2110.04716",
"abstract": "We investigate the spectral structure of the Neumann-Poincaré operator on thin ellipsoids. Two types of thin ellipsoids are considered: long prolate ellip... |
https://arxiv.org/abs/1011.4490 | On the constant in Burgess' bound for the number of consecutive residues or non-residues | We give an explicit version of a result due to D. Burgess. Let $\chi$ be a non-principal Dirichlet character modulo a prime $p$. We show that the maximum number of consecutive integers for which $\chi$ takes on a particular value is less than $\left\{\frac{\pi e\sqrt{6}}{3}+o(1)\right\}p^{1/4}\log p$, where the $o(1)$ ... | \section{Introduction}\label{S:1}
Let $\chi$ be a non-principal Dirichlet character to the prime modulus $p$.
In 1963, D.~Burgess showed (see~\cite{bib:burgess.1963}) that the maximum number of consecutive integers for which $\chi$ takes
on any particular value is $O(p^{1/4}\log p)$.
This still constitutes the best kn... | {
"timestamp": "2010-11-22T02:02:31",
"yymm": "1011",
"arxiv_id": "1011.4490",
"language": "en",
"url": "https://arxiv.org/abs/1011.4490",
"abstract": "We give an explicit version of a result due to D. Burgess. Let $\\chi$ be a non-principal Dirichlet character modulo a prime $p$. We show that the maximum n... |
https://arxiv.org/abs/0804.3019 | Three Dimensional Corners: A Box Norm Proof | In an additive group (G,+), a three-dimensional corner is the four points g, g+d(1,0,0), g+d(0,1,0), g+d(0,0,1), where g is in G^3, and d is a non-zero element of G. The Ramsey number of interest is R_3(G) the maximal cardinality of a subset of G^3 that does not contain a three-dimensional corner. Furstenberg and Katzn... | \section{Introduction}
For any discrete abelian group $ (G,+)$, we define a \emph{$ d$-dimensional corner}
to be the $ d+1$ points in $ G ^{d}$ given by
\begin{equation*}
g\,,\, g+h(1,0,0, ,\dotsc, 0)\,,\,
g+ h(0,1,0 ,\dotsc, 0)\, ,\dotsc, g+ h(0,0,0 ,\dotsc, 1)\,, \qquad h\in G- \{0\}\,.
\end{equation*}
The R... | {
"timestamp": "2008-04-18T15:41:50",
"yymm": "0804",
"arxiv_id": "0804.3019",
"language": "en",
"url": "https://arxiv.org/abs/0804.3019",
"abstract": "In an additive group (G,+), a three-dimensional corner is the four points g, g+d(1,0,0), g+d(0,1,0), g+d(0,0,1), where g is in G^3, and d is a non-zero elem... |
https://arxiv.org/abs/2210.11961 | Sets of mutually orthogoval projective and affine planes | A pair of planes, both projective or both affine, of the same order and on the same pointset are orthogoval if each line of one plane intersects each line of the other plane in at most two points. In this paper we prove new constructions for sets of mutually orthogoval planes, both projective and affine, and review kno... | \section{Introduction}
In a projective plane of order $q$ an {\em oval} is a set of $q+1$ points, no three of which are collinear. A beautiful theorem, independently published multiple times, states that in $\PG(2,q)$ there exists a set of $n^2+n+1$ ovals which form the blocks of a second projective plane \cites{baker... | {
"timestamp": "2022-10-24T02:13:30",
"yymm": "2210",
"arxiv_id": "2210.11961",
"language": "en",
"url": "https://arxiv.org/abs/2210.11961",
"abstract": "A pair of planes, both projective or both affine, of the same order and on the same pointset are orthogoval if each line of one plane intersects each line... |
https://arxiv.org/abs/0711.3399 | Improved Poincare inequalities with weights | In this paper we prove that if $\Omega\in\mathbb{R}^n$ is a bounded John domain, the following weighted Poincare-type inequality holds: $$ \inf_{a\in \mathbb{R}}\| (f(x)-a) w_1(x) \|_{L^q(\Omega)} \le C \|\nabla f(x) d(x)^\alpha w_2(x) \|_{L^p(\Omega)} $$ where $f$ is a locally Lipschitz function on $\Omega$, $d(x)$ de... | \section*{}
\setcounter{equation}{0}
\title[]{Improved Poincar\'e inequalities with weights}
\author{Irene Drelichman}
\address{Departamento de Matem\'atica, Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires, 1428 Buenos Aires, Argentina} \email{irene@drelichman.com}
\author{Ricardo G. Dur\'an}
... | {
"timestamp": "2007-11-21T16:48:24",
"yymm": "0711",
"arxiv_id": "0711.3399",
"language": "en",
"url": "https://arxiv.org/abs/0711.3399",
"abstract": "In this paper we prove that if $\\Omega\\in\\mathbb{R}^n$ is a bounded John domain, the following weighted Poincare-type inequality holds: $$ \\inf_{a\\in \... |
https://arxiv.org/abs/1405.3133 | Graph Matching: Relax at Your Own Risk | Graph matching---aligning a pair of graphs to minimize their edge disagreements---has received wide-spread attention from both theoretical and applied communities over the past several decades, including combinatorics, computer vision, and connectomics. Its attention can be partially attributed to its computational dif... | \section{Introduction}
\IEEEPARstart{S}{everal} problems related to the isomorphism and matching of graphs have been an important and enjoyable challenge for the scientific community for a long time,
with applications in pattern recognition (see, for example, \cite{PA2,PA1}), computer vision (see, for example, \cit... | {
"timestamp": "2015-01-13T02:06:04",
"yymm": "1405",
"arxiv_id": "1405.3133",
"language": "en",
"url": "https://arxiv.org/abs/1405.3133",
"abstract": "Graph matching---aligning a pair of graphs to minimize their edge disagreements---has received wide-spread attention from both theoretical and applied commu... |
https://arxiv.org/abs/1508.05869 | Numerical Approximation of Fractional Powers of Regularly Accretive Operators | We study the numerical approximation of fractional powers of accretive operators in this paper. Namely, if $A$ is the accretive operator associated with an accretive sesquilinear form $A(\cdot,\cdot)$ defined on a Hilbert space $\mathbb V$ contained in $L^2(\Omega)$, we approximate $A^{-\beta}$ for $\beta\in (0,1)$. Th... | \section{Introduction.}
The mathematical study of integral or nonlocal operators has received much
attention due to their wide range of applications, see for instance \cite{ISI:000175019600004,MR2450437,MR0521262,MR2223347,MR2480109,MR1918950,MR660727,MR1727557,MR1709781}.
Let $\Omega$ be a bounded domain in ${\math... | {
"timestamp": "2016-07-15T02:03:38",
"yymm": "1508",
"arxiv_id": "1508.05869",
"language": "en",
"url": "https://arxiv.org/abs/1508.05869",
"abstract": "We study the numerical approximation of fractional powers of accretive operators in this paper. Namely, if $A$ is the accretive operator associated with a... |
https://arxiv.org/abs/1309.0434 | Carries, group theory, and additive combinatorics | Given a group G and a normal subgroup H we study the problem of choosing coset representatives with few carries. | \section{Introduction}
When numbers are added in the usual way {\em carries} occur along the
route. These carries cause a mess and it is natural to seek ways to
minimize them. This paper proves that {\em balanced arithmetic}
minimizes the proportion of carries. It also positions carries as
{\em cocycles} in group... | {
"timestamp": "2013-09-03T02:12:33",
"yymm": "1309",
"arxiv_id": "1309.0434",
"language": "en",
"url": "https://arxiv.org/abs/1309.0434",
"abstract": "Given a group G and a normal subgroup H we study the problem of choosing coset representatives with few carries.",
"subjects": "Combinatorics (math.CO); G... |
https://arxiv.org/abs/cond-mat/0612188 | Optimum exploration memory and anomalous diffusion in deterministic partially self-avoiding walks in one-dimensional random media | Consider $N$ points randomly distributed along a line segment of unitary length. A walker explores this disordered medium moving according to a partially self-avoiding deterministic walk. The walker, with memory $\mu$, leaves from the leftmost point and moves, at each discrete time step, to the nearest point, which has... | \section{Introduction}
While random walks in regular or disordered media have been thouroughly explored~\cite{fisher:1984}, deterministic walks in regular~\cite{grassberger:92} and disordered media~\cite{bunimovich:2004,boyer_2004,boyer_2005,boyer_2006} have been much less studied.
Here we are concerned with the prop... | {
"timestamp": "2006-12-07T14:54:27",
"yymm": "0612",
"arxiv_id": "cond-mat/0612188",
"language": "en",
"url": "https://arxiv.org/abs/cond-mat/0612188",
"abstract": "Consider $N$ points randomly distributed along a line segment of unitary length. A walker explores this disordered medium moving according to ... |
https://arxiv.org/abs/2202.08522 | Recovering Unbalanced Communities in the Stochastic Block Model With Application to Clustering with a Faulty Oracle | The stochastic block model (SBM) is a fundamental model for studying graph clustering or community detection in networks. It has received great attention in the last decade and the balanced case, i.e., assuming all clusters have large size, has been well studied. However, our understanding of SBM with unbalanced commun... | \section{Introduction}
Graph clustering (or community detection) is a fundamental problem in computer science and has wide applications in almost all domains, including biology, social science and physics. Among others, stochastic block model (SBM) is one of the most basic models for studying graph clustering, offering... | {
"timestamp": "2022-02-18T02:12:43",
"yymm": "2202",
"arxiv_id": "2202.08522",
"language": "en",
"url": "https://arxiv.org/abs/2202.08522",
"abstract": "The stochastic block model (SBM) is a fundamental model for studying graph clustering or community detection in networks. It has received great attention ... |
https://arxiv.org/abs/1009.0855 | Level Sets of the Takagi Function: Local Level Sets | The Takagi function \tau : [0, 1] \to [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y) = {x : \tau(x) = y} of the Takagi function \tau(x) are studied by introducing a notion of local level set into which level sets are partitioned. Local level sets are simple to anal... | \section{Introduction}
The Takagi function $\tau(x)$ is a function
defined on the unit interval $ x\in [0,1]$
which was introduced by Takagi \cite{Takagi} in 1903
as an example of a continuous nondifferentiable function.
It can be defined by
\begin{equation}\label{eq101}
\tau(x) := \sum_{n=0}^{\infty} \frac{\ll 2^n ... | {
"timestamp": "2012-04-02T02:00:15",
"yymm": "1009",
"arxiv_id": "1009.0855",
"language": "en",
"url": "https://arxiv.org/abs/1009.0855",
"abstract": "The Takagi function \\tau : [0, 1] \\to [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y) = {x : \\tau(x... |
https://arxiv.org/abs/1511.01195 | Small scale equidistribution of random eigenbases | We investigate small scale equidistribution of random orthonormal bases of eigenfunctions (i.e. eigenbases) on a compact manifold M. Assume that the group of isometries acts transitively on M and the multiplicity of eigenfrequency tends to infinity at least logarithmically. We prove that, with respect to the natural pr... | \section{Introduction}
Let $(\M,g)$ be a compact and smooth Riemannian manifold of dimension $n$ without boundary. Denote $\Delta=\Delta_g$ the (positive) Laplace-Beltrami operator and $\{e_j\}_{j=0}^\infty$ an orthonormal basis of eigenfunctions (i.e. eigenbasis) of $\Delta$ with eigenvalues $\lambda_j^2$ (counting mu... | {
"timestamp": "2015-11-05T02:05:35",
"yymm": "1511",
"arxiv_id": "1511.01195",
"language": "en",
"url": "https://arxiv.org/abs/1511.01195",
"abstract": "We investigate small scale equidistribution of random orthonormal bases of eigenfunctions (i.e. eigenbases) on a compact manifold M. Assume that the group... |
https://arxiv.org/abs/1005.5495 | Central Swaths (A Generalization of the Central Path) | We develop a natural generalization to the notion of the central path -- a notion that lies at the heart of interior-point methods for convex optimization. The generalization is accomplished via the "derivative cones" of a "hyperbolicity cone," the derivatives being direct and mathematically-appealing relaxations of th... | \section{{\bf Introduction}} \label{s.a}
Let $ {\mathcal E} $ denote a finite-dimensional Euclidean space and let $ p: {\mathcal E} \rightarrow \reals $ be a hyperbolic polynomial, that is, a homogeneous polynomial for which there is a designated direction vector $ e $ satisfying $ p(e) > 0 $ and having the prope... | {
"timestamp": "2012-03-20T01:02:38",
"yymm": "1005",
"arxiv_id": "1005.5495",
"language": "en",
"url": "https://arxiv.org/abs/1005.5495",
"abstract": "We develop a natural generalization to the notion of the central path -- a notion that lies at the heart of interior-point methods for convex optimization. ... |
https://arxiv.org/abs/1505.04214 | Algorithmic Connections Between Active Learning and Stochastic Convex Optimization | Interesting theoretical associations have been established by recent papers between the fields of active learning and stochastic convex optimization due to the common role of feedback in sequential querying mechanisms. In this paper, we continue this thread in two parts by exploiting these relations for the first time ... | \section{Introduction}
The two fields of convex optimization and active learning seem to have evolved quite independently of each other. Recently, \cite{RR09} pointed out their relatedness due to the inherent sequential nature of both fields and the complex role of feedback in taking future actions. Following that, \c... | {
"timestamp": "2015-05-19T02:01:14",
"yymm": "1505",
"arxiv_id": "1505.04214",
"language": "en",
"url": "https://arxiv.org/abs/1505.04214",
"abstract": "Interesting theoretical associations have been established by recent papers between the fields of active learning and stochastic convex optimization due t... |
https://arxiv.org/abs/2002.07357 | Constructions of regular sparse anti-magic squares | Graph labeling is a well-known and intensively investigated problem in graph theory. Sparse anti-magic squares are useful in constructing vertex-magic labeling for graphs. For positive integers $n,d$ and $d<n$, an $n\times n$ array $A$ based on $\{0,1,\cdots,nd\}$ is called \emph{a sparse anti-magic square of order $n$... | \section{Introduction}
Magic squares and their various generalizations have been objects of
interest for many centuries and in many cultures. A lot of work has
been done on the constructions of magic squares, for more details,
the interested reader may refer to \cite{Abe,Ahmed,Andrews,Hand} and
the references therein.... | {
"timestamp": "2020-02-20T02:08:30",
"yymm": "2002",
"arxiv_id": "2002.07357",
"language": "en",
"url": "https://arxiv.org/abs/2002.07357",
"abstract": "Graph labeling is a well-known and intensively investigated problem in graph theory. Sparse anti-magic squares are useful in constructing vertex-magic lab... |
https://arxiv.org/abs/1807.03462 | A note on breaking ties among sample medians | Given samples $x_1,\cdots,x_n$, it is well known that any sample median value (not necessarily unique) minimizes the absolute loss $\sum_{i=1}^n |q-x_i|$. Interestingly, we show that the minimizer of the loss $\sum_{i=1}^n|q-x_i|^{1+\epsilon}$ exhibits a singular perturbation behaviour that provides a unique definition... | \section{Introduction}
Given samples $x_1,\cdots,x_n$, it is well known that
the sample mean $n^{-1}\sum_i x_i$ is the unique minimizer of the empirical squared loss $\mathbb{E}_{n}(\theta-X)^{2}=n^{-1}\sum_{i:x_i\le\theta}(\theta-x_i)^{2}+n^{-1}\sum_{i:x_i>\theta}(x_i-\theta)^{2}$.
This follows from the first order c... | {
"timestamp": "2019-09-04T02:32:55",
"yymm": "1807",
"arxiv_id": "1807.03462",
"language": "en",
"url": "https://arxiv.org/abs/1807.03462",
"abstract": "Given samples $x_1,\\cdots,x_n$, it is well known that any sample median value (not necessarily unique) minimizes the absolute loss $\\sum_{i=1}^n |q-x_i|... |
https://arxiv.org/abs/1002.4361 | A unification of permutation patterns related to Schubert varieties | We obtain new connections between permutation patterns and singularities of Schubert varieties, by giving a new characterization of Gorenstein varieties in terms of so called bivincular patterns. These are generalizations of classical patterns where conditions are placed on the location of an occurrence in a permutatio... | \section{Introduction}
In this paper we exhibit new connections between permutation patterns and singularities
of Schubert varieties $X_\pi$ in the complete flag variety $\Fl(\CC^n)$,
by giving a new characterization of Gorenstein varieties in terms of which \emph{bivincular patterns} the
permutation $\pi$ avoids. Biv... | {
"timestamp": "2011-06-07T02:07:06",
"yymm": "1002",
"arxiv_id": "1002.4361",
"language": "en",
"url": "https://arxiv.org/abs/1002.4361",
"abstract": "We obtain new connections between permutation patterns and singularities of Schubert varieties, by giving a new characterization of Gorenstein varieties in ... |
https://arxiv.org/abs/1303.6078 | The Bishop-Phelps-Bollobás theorem for operators on $L_1(μ)$ | In this paper we show that the Bishop-Phelps-Bollobás theorem holds for $\mathcal{L}(L_1(\mu), L_1(\nu))$ for all measures $\mu$ and $\nu$ and also holds for $\mathcal{L}(L_1(\mu),L_\infty(\nu))$ for every arbitrary measure $\mu$ and every localizable measure $\nu$. Finally, we show that the Bishop-Phelps-Bollobás theo... | \section{Introduction}
The celebrated Bishop-Phelps theorem of 1961 \cite{BP} states that for a Banach space $X$, every element in its dual space $X^*$ can be approximated by ones that attain their norms. Since then, there has been an extensive research to extend this result to bounded linear operators between Banach ... | {
"timestamp": "2013-03-26T01:04:05",
"yymm": "1303",
"arxiv_id": "1303.6078",
"language": "en",
"url": "https://arxiv.org/abs/1303.6078",
"abstract": "In this paper we show that the Bishop-Phelps-Bollobás theorem holds for $\\mathcal{L}(L_1(\\mu), L_1(\\nu))$ for all measures $\\mu$ and $\\nu$ and also hol... |
https://arxiv.org/abs/1903.04284 | Cracking the problem with 33 | Inspired by the Numberphile video "The uncracked problem with 33" by Tim Browning and Brady Haran, we investigate solutions to $x^3+y^3+z^3=k$ for a few small values of $k$. We find the first known solution for $k=33$. | \section{Introduction}
Let $k$ be a positive integer with $k\not\equiv\pm4\pmod{9}$. Then Heath-Brown
\cite{Heath-Brown} has conjectured that there are infinitely many triples
$(x,y,z)\in\mathbb{Z}^3$ such that
\begin{equation}\label{eq:main}
k=x^3+y^3+z^3.
\end{equation}
Various numerical investigations of \eqref{eq:m... | {
"timestamp": "2019-03-19T01:28:19",
"yymm": "1903",
"arxiv_id": "1903.04284",
"language": "en",
"url": "https://arxiv.org/abs/1903.04284",
"abstract": "Inspired by the Numberphile video \"The uncracked problem with 33\" by Tim Browning and Brady Haran, we investigate solutions to $x^3+y^3+z^3=k$ for a few... |
https://arxiv.org/abs/2108.04147 | Discrete multilinear maximal functions and number theory | Many multilinear discrete operators are primed for pointwise decomposition; such decompositions give structural information but also an essentially optimal range of bounds. We study the (continuous) slicing method of Jeong and Lee -- which when debuted instantly gave sharp multilinear operator bounds -- in the discrete... | \section{Introduction}
Studying analytic operators from both a multilinear perspective and a discrete one has been an active area of research. Typically these operators have non-trivial boundedness properties if the underlying surface of integration is curved. A prototypical curved object is the sphere, and maximal ... | {
"timestamp": "2021-08-10T02:37:57",
"yymm": "2108",
"arxiv_id": "2108.04147",
"language": "en",
"url": "https://arxiv.org/abs/2108.04147",
"abstract": "Many multilinear discrete operators are primed for pointwise decomposition; such decompositions give structural information but also an essentially optima... |
https://arxiv.org/abs/2111.00456 | A generalized Cantor theorem in ZF | It is proved in $\mathsf{ZF}$ (without the axiom of choice) that, for all infinite sets $M$, there are no surjections from $\omega\times M$ onto $\mathscr{P}(M)$. | \section{Introduction}\label{s014}
Throughout this paper, we shall work in $\mathsf{ZF}$
(i.e., the Zermelo--Fraenkel set theory without the axiom of choice).
In~\cite{Cantor1892}, Cantor proves that, for all sets~$M$, there are no injections from $\scrP(M)$ into~$M$,
from which it follows easily that, for all sets~$M... | {
"timestamp": "2021-11-02T01:18:36",
"yymm": "2111",
"arxiv_id": "2111.00456",
"language": "en",
"url": "https://arxiv.org/abs/2111.00456",
"abstract": "It is proved in $\\mathsf{ZF}$ (without the axiom of choice) that, for all infinite sets $M$, there are no surjections from $\\omega\\times M$ onto $\\mat... |
https://arxiv.org/abs/2203.14535 | Asymptotic analysis of k-hop connectivity in the 1D unit disk random graph model | We propose an algorithm for the closed-form recursive computation of joint moments and cumulants of all orders for k-hop counts in the 1D unit disk random graph model with Poisson distributed vertices. Our approach uses decompositions of k-hop counts into multiple Poisson stochastic integrals. As a consequence, using t... | \section{Introduction}
We consider the statistics and asymptotic behavior of
$k$-hop connectivity of the one-dimensional
unit disk random connection model with connection radius $r>0$
on a finite interval, see \cite{drory1997}.
Random geometric graphs
have the ability to model physical systems in e.g.
wireless n... | {
"timestamp": "2022-03-29T02:44:05",
"yymm": "2203",
"arxiv_id": "2203.14535",
"language": "en",
"url": "https://arxiv.org/abs/2203.14535",
"abstract": "We propose an algorithm for the closed-form recursive computation of joint moments and cumulants of all orders for k-hop counts in the 1D unit disk random... |
https://arxiv.org/abs/1310.0852 | Hom Quandles | If $A$ is an abelian quandle and $Q$ is a quandle, the hom set $\mathrm{Hom}(Q,A)$ of quandle homomorphisms from $Q$ to $A$ has a natural quandle structure. We exploit this fact to enhance the quandle counting invariant, providing an example of links with the same counting invariant values but distinguished by the hom ... | \section{\large \textbf{Introduction}}
Ronnie Brown said it best when declaring, ``One of the irritations of group theory is that the set Hom($H$,$K$) of homomorphisms between groups $H$ and $K$ does not have a natural group structure." Of course, when $H$ and $K$ are both commutative, we know that Hom($H$, $K$) is al... | {
"timestamp": "2014-03-11T01:01:07",
"yymm": "1310",
"arxiv_id": "1310.0852",
"language": "en",
"url": "https://arxiv.org/abs/1310.0852",
"abstract": "If $A$ is an abelian quandle and $Q$ is a quandle, the hom set $\\mathrm{Hom}(Q,A)$ of quandle homomorphisms from $Q$ to $A$ has a natural quandle structure... |
https://arxiv.org/abs/1502.01944 | Sums of three cubes, II | Estimates are provided for $s$th moments of cubic smooth Weyl sums, when $4\le s\le 8$, by enhancing the author's iterative method that delivers estimates beyond classical convexity. As a consequence, an improved lower bound is presented for the number of integers not exceeding $X$ that are represented as the sum of th... | \section{Introduction} A heuristic application of the Hardy-Littlewood (circle) method
suggests that the set of integers represented as the sum of three cubes of natural
numbers should have positive density. Although intense effort over the past $75$ years
has delivered a reasonable approximation to this expectat... | {
"timestamp": "2015-02-09T02:12:52",
"yymm": "1502",
"arxiv_id": "1502.01944",
"language": "en",
"url": "https://arxiv.org/abs/1502.01944",
"abstract": "Estimates are provided for $s$th moments of cubic smooth Weyl sums, when $4\\le s\\le 8$, by enhancing the author's iterative method that delivers estimat... |
https://arxiv.org/abs/0710.1521 | Algebraic quantum permutation groups | We discuss some algebraic aspects of quantum permutation groups, working over arbitrary fields. If $K$ is any characteristic zero field, we show that there exists a universal cosemisimple Hopf algebra coacting on the diagonal algebra $K^n$: this is a refinement of Wang's universality theorem for the (compact) quantum p... | \section{Introduction}
A remarkable fact, discovered by Wang \cite{wa}, is the existence
of a largest (universal) compact quantum group, denoted ${\mathcal Q}_n$, acting on the set
$[n] =\{1, \ldots ,n\}$, which is infinite if $n\geq 4$.
In view of its universal property, the quantum group ${\mathcal Q}_n$ is called
t... | {
"timestamp": "2007-10-08T13:55:14",
"yymm": "0710",
"arxiv_id": "0710.1521",
"language": "en",
"url": "https://arxiv.org/abs/0710.1521",
"abstract": "We discuss some algebraic aspects of quantum permutation groups, working over arbitrary fields. If $K$ is any characteristic zero field, we show that there ... |
https://arxiv.org/abs/2008.08503 | The Erdős-Ko-Rado theorem for $2$-intersecting families of perfect matchings | A perfect matching in the complete graph on $2k$ vertices is a set of edges such that no two edges have a vertex in common and every vertex is covered exactly once. Two perfect matchings are said to be $t$-intersecting if they have at least $t$ edges in common. The main result in this paper is an extension of the famou... | \section{Introduction and Preliminaries}
In this paper we present two different approaches to establish a version of the Erd\H{o}s-Ko-Rado theorem for $2$-intersecting families of perfect matchings. There are many recent results that verify analogs of the Erd\H{o}s-Ko-Rado theorem. This research area started with Er... | {
"timestamp": "2020-08-20T02:16:19",
"yymm": "2008",
"arxiv_id": "2008.08503",
"language": "en",
"url": "https://arxiv.org/abs/2008.08503",
"abstract": "A perfect matching in the complete graph on $2k$ vertices is a set of edges such that no two edges have a vertex in common and every vertex is covered exa... |
https://arxiv.org/abs/1908.03669 | A Survey of Tuning Parameter Selection for High-dimensional Regression | Penalized (or regularized) regression, as represented by Lasso and its variants, has become a standard technique for analyzing high-dimensional data when the number of variables substantially exceeds the sample size. The performance of penalized regression relies crucially on the choice of the tuning parameter, which d... | \section{Introduction}
High-dimensional data, where the number of covariates/features (e.g., genes)
may be of the same order or substantially exceed the sample size (e.g., number of patients),
have become common in many fields due to the advancement in science and technology. Statistical methods for analyzing
high-d... | {
"timestamp": "2019-08-13T02:04:14",
"yymm": "1908",
"arxiv_id": "1908.03669",
"language": "en",
"url": "https://arxiv.org/abs/1908.03669",
"abstract": "Penalized (or regularized) regression, as represented by Lasso and its variants, has become a standard technique for analyzing high-dimensional data when ... |
https://arxiv.org/abs/math/0608650 | The boundary behavior of holomorphic functions: Global and local results | We develop a new technique for studying the boundary limiting behavior of a holomorphic function on a domain $\Omega$ -- both in one and several complex variables. The approach involves two new localized maximal functions. As a result of this methodology, theorems of Calderón type about local boundary behavior on a set... | \section{Introduction}
The first theorem about the boundary limiting behavior of holomorphic
functions was proved by P. Fatou in his thesis in 1906 [FAT]. He used Fourier
series techniques to show that if $f$ is a bounded, holomorphic
function on the unit disc $D \subseteq {\Bbb C}$ (i.e., $f \in H^\infty(D)$), then
$... | {
"timestamp": "2006-08-26T07:13:18",
"yymm": "0608",
"arxiv_id": "math/0608650",
"language": "en",
"url": "https://arxiv.org/abs/math/0608650",
"abstract": "We develop a new technique for studying the boundary limiting behavior of a holomorphic function on a domain $\\Omega$ -- both in one and several comp... |
https://arxiv.org/abs/1009.6153 | Three Balls Problem Revisited - On the Limitations of Event-Driven Modeling | If a tennis ball is held above a basket ball with their centers vertically aligned, and the balls are released to collide with the floor, the tennis ball may rebound at a surprisingly high speed. We show in this article that the simple textbook explanation of this effect is an oversimplification, even for the limit of ... | \section{Introduction}
Consider a set of two balls made of the same viscoelastic material whose centers
are vertically aligned at positions $z_1$ and $z_2$ as sketched in Fig.
\ref{fig:sketch}.
\begin{figure}
\centering
\includegraphics[width=0.6\columnwidth]{scetch.pdf}
\caption{Sketch of the problem.}
\label{f... | {
"timestamp": "2010-10-01T02:01:57",
"yymm": "1009",
"arxiv_id": "1009.6153",
"language": "en",
"url": "https://arxiv.org/abs/1009.6153",
"abstract": "If a tennis ball is held above a basket ball with their centers vertically aligned, and the balls are released to collide with the floor, the tennis ball ma... |
https://arxiv.org/abs/1410.6537 | Choices, intervals and equidistribution | We give a sufficient condition for a random sequence in [0,1] generated by a $\Psi$-process to be equidistributed. The condition is met by the canonical example -- the $\max$-2 process -- where the $n$th term is whichever of two uniformly placed points falls in the larger gap formed by the previous $n-1$ points. This s... |
\subsubsection*{Acknowledgments}
Much thanks to Elliot Paquette and Pascal Maillard for many useful conversations. The first referee's careful reading and suggestion to generalize to arbitrary $\Psi$ processes are greatly appreciated. Itai Benjamini is the source of a very similar model that spurred this research. To... | {
"timestamp": "2015-09-08T02:05:09",
"yymm": "1410",
"arxiv_id": "1410.6537",
"language": "en",
"url": "https://arxiv.org/abs/1410.6537",
"abstract": "We give a sufficient condition for a random sequence in [0,1] generated by a $\\Psi$-process to be equidistributed. The condition is met by the canonical ex... |
https://arxiv.org/abs/0810.1288 | Stellar disruption by a supermassive black hole: is the light curve really proportional to $t^{-5/3}$? | In this paper we revisit the arguments for the basis of the time evolution of the flares expected to arise when a star is disrupted by a supermassive black hole. We present a simple analytic model relating the lightcurve to the internal density structure of the star. We thus show that the standard lightcurve proportion... | \section{Introduction}
X-ray flares from quiescent (non-AGN) galaxies are often interpreted as
arising from the tidal disruption of stars as they get close to a dormant
supermassive black hole (SMBH) in the centre of the galaxy \citep{komossa99}.
Similar processes also occur on much smaller scales, such as in compact ... | {
"timestamp": "2008-10-07T22:07:41",
"yymm": "0810",
"arxiv_id": "0810.1288",
"language": "en",
"url": "https://arxiv.org/abs/0810.1288",
"abstract": "In this paper we revisit the arguments for the basis of the time evolution of the flares expected to arise when a star is disrupted by a supermassive black ... |
https://arxiv.org/abs/2010.08767 | Bounds on the running maximum of a random walk with small drift | We derive a lower bound for the probability that a random walk with i.i.d.\ increments and small negative drift $\mu$ exceeds the value $x>0$ by time $N$. When the moment generating functions are bounded in an interval around the origin, this probability can be bounded below by $1-O(x|\mu| \log N)$. The approach is ele... | \section{Introduction}
\subsection{Background}
This paper arose from the need of a random walk estimate for the authors' article \cite{busa-sepp-poly} on directed polymers. This estimate is a {\it positive} lower bound on the running maximum of a random walk with a small {\it negative} drift. Importantly, the bou... | {
"timestamp": "2020-11-12T02:22:18",
"yymm": "2010",
"arxiv_id": "2010.08767",
"language": "en",
"url": "https://arxiv.org/abs/2010.08767",
"abstract": "We derive a lower bound for the probability that a random walk with i.i.d.\\ increments and small negative drift $\\mu$ exceeds the value $x>0$ by time $N... |
https://arxiv.org/abs/1011.1506 | The Dehn functions of Out(F_n) and Aut(F_n) | For n > 2, the Dehn functions of Aut(F_n) and Out(F_n) are exponential. Hatcher and Vogtmann proved that they are at most exponential, and the complementary lower bound in the case n=3 was established by Bridson and Vogtmann. Handel and Mosher completed the proof by reducing the lower bound for n>4 to the case n=3. In ... | \section{Introduction}
Dehn functions provide upper bounds on the complexity of the word problem in finitely presented groups. They are examples of filling functions: if a group $G$ acts properly and cocompactly on a simplicial complex $X$, then the Dehn function of $G$ is asymptotically equivalent to the function that... | {
"timestamp": "2011-11-29T02:01:01",
"yymm": "1011",
"arxiv_id": "1011.1506",
"language": "en",
"url": "https://arxiv.org/abs/1011.1506",
"abstract": "For n > 2, the Dehn functions of Aut(F_n) and Out(F_n) are exponential. Hatcher and Vogtmann proved that they are at most exponential, and the complementary... |
https://arxiv.org/abs/0906.0809 | Minimizing the footprint of your laptop (on your bedside table) | I often work on my laptop in bed. When needed, I park the laptop on the bedside table, where the computer has to share the small available space with a lamp, books, notes, and heaven knows what else. It often gets quite squeezy.Being regularly faced with this tricky situation, it finally occurred to me to determine onc... | \section{Introduction} I often work on my laptop in bed. When needed, I park the laptop on the bedside table, where the computer has to share the small available space with a lamp, books, notes, and heaven knows what else. It often gets quite squeezy.
Being regularly faced with this tricky situation, it finally occur... | {
"timestamp": "2009-06-04T03:50:18",
"yymm": "0906",
"arxiv_id": "0906.0809",
"language": "en",
"url": "https://arxiv.org/abs/0906.0809",
"abstract": "I often work on my laptop in bed. When needed, I park the laptop on the bedside table, where the computer has to share the small available space with a lamp... |
https://arxiv.org/abs/1805.10315 | Modular class of even symplectic manifolds | We provide an intrinsic description of the notion of modular class for an even symplectic manifold and study its properties in this coordinate free setting. | \section{Introduction}
The definition of the modular vector field of a Poisson manifold $%
(M;\{\_,\_\})$, is as follows: given a volume element $\eta$ on $M$, the
modular vector field $Z^{M}$ maps each function $f\in C^{\infty}(M)$ into
the divergence with respect to $\eta$ of the hamiltonian vector field
associated ... | {
"timestamp": "2018-05-29T02:00:37",
"yymm": "1805",
"arxiv_id": "1805.10315",
"language": "en",
"url": "https://arxiv.org/abs/1805.10315",
"abstract": "We provide an intrinsic description of the notion of modular class for an even symplectic manifold and study its properties in this coordinate free settin... |
https://arxiv.org/abs/2208.13926 | Regular projections of the link L6n1 | Given a link projection $P$ and a link $L$, it is natural to ask whether it is possible that $P$ is a projection of $L$. Taniyama answered this question for the cases in which $L$ is a prime knot or link with crossing number at most five. Recently, Takimura settled the issue for the knot $6_2$. We answer this question ... | \section{Introduction}\label{sec:intro}
We work in the piecewise linear category, and links are hosted in the $3$-sphere ${\mathbb S}^3$. If we project a link $L$ onto the $2$-sphere ${\mathbb S}$, we obtain a {\em projection} of $L$. We {\em resolve} a link projection by giving over/under information at each crossing... | {
"timestamp": "2022-08-31T02:04:46",
"yymm": "2208",
"arxiv_id": "2208.13926",
"language": "en",
"url": "https://arxiv.org/abs/2208.13926",
"abstract": "Given a link projection $P$ and a link $L$, it is natural to ask whether it is possible that $P$ is a projection of $L$. Taniyama answered this question f... |
https://arxiv.org/abs/1102.3080 | Covering Point Patterns | An encoder observes a point pattern---a finite number of points in the interval $[0,T]$---which is to be described to a reconstructor using bits. Based on these bits, the reconstructor wishes to select a subset of $[0,T]$ that contains all the points in the pattern. It is shown that, if the point pattern is produced by... | \section{Introduction}
An encoder observes a point pattern---a finite number of points in the
interval $[0,T]$---which is to be described to a reconstructor using
bits. Based on these bits, the reconstructor wishes to produce a
covering-set---a subset of $[0,T]$ containing all the points---of
least Lebesgue measure. Th... | {
"timestamp": "2011-02-16T02:01:57",
"yymm": "1102",
"arxiv_id": "1102.3080",
"language": "en",
"url": "https://arxiv.org/abs/1102.3080",
"abstract": "An encoder observes a point pattern---a finite number of points in the interval $[0,T]$---which is to be described to a reconstructor using bits. Based on t... |
https://arxiv.org/abs/2008.01428 | Canonical trace ideal and residue for numerical semigroup rings | For a numerical semigroup ring $K[H]$ we study the trace of its canonical ideal. The colength of this ideal is called the residue of $H$. This invariant measures how far is $H$ from being symmetric, i.e. $K[H]$ from being a Gorenstein ring. We remark that the canonical trace ideal contains the conductor ideal, and we s... | \section*{Introduction}
\label{sec:introd}
Let $(R,\mm, K)$ be a local ring (or a positively graded $K$-algebra) which is Cohen-Macaulay and possesses a canonical module $\omega_R$. In \cite{HHS} the trace ideal of $\omega_R$ is used as a tool to stratify the Cohen-Macaulay rings and to define the class of nearly ... | {
"timestamp": "2020-08-05T02:12:57",
"yymm": "2008",
"arxiv_id": "2008.01428",
"language": "en",
"url": "https://arxiv.org/abs/2008.01428",
"abstract": "For a numerical semigroup ring $K[H]$ we study the trace of its canonical ideal. The colength of this ideal is called the residue of $H$. This invariant m... |
https://arxiv.org/abs/1910.11766 | On the discrepancy of random subsequences of $\{nα\}$ | For irrational $\alpha$, $\{n\alpha\}$ is uniformly distributed mod 1 in the Weyl sense, and the asymptotic behavior of its discrepancy is completely known. In contrast, very few precise results exist for the discrepancy of subsequences $\{n_k \alpha\}$, with the exception of metric results for exponentially growing $(... | \section{Introduction}\label{intro}
An infinite sequence $(x_k)$ of real numbers is called \textit{uniformly distributed mod} 1 if for every pair $a, b$ of
real numbers with $0\le a < b \le 1$ we have
$$\lim_{N\to\infty} \frac{1}{N}\sum_{k=1}^N I_{[a, b)} (\{x_k\})=b-a.$$
Here $\{ \cdot\}$ denotes fractional part... | {
"timestamp": "2019-10-28T01:16:22",
"yymm": "1910",
"arxiv_id": "1910.11766",
"language": "en",
"url": "https://arxiv.org/abs/1910.11766",
"abstract": "For irrational $\\alpha$, $\\{n\\alpha\\}$ is uniformly distributed mod 1 in the Weyl sense, and the asymptotic behavior of its discrepancy is completely ... |
https://arxiv.org/abs/1703.05378 | Non-crossing Monotone Paths and Binary Trees in Edge-ordered Complete Geometric Graphs | An edge-ordered graph is a graph with a total ordering of its edges. A path $P=v_1v_2\ldots v_k$ in an edge-ordered graph is called increasing if $(v_iv_{i+1}) > (v_{i+1}v_{i+2})$ for all $i = 1,\ldots,k-2$; it is called decreasing if $(v_iv_{i+1}) < (v_{i+1}v_{i+2})$ for all $i = 1,\ldots,k-2$. We say that $P$ is mono... | \section{Introduction}
An \emph{edge-ordering} of a graph is a total order of its edges, an \emph{edge-ordered} graph is a graph with an edge-ordering. A path $P=v_1v_2\ldots v_k$ in an edge-ordered graph is called increasing if $(v_iv_{i+1}) > (v_{i+1}v_{i+2})$ for all $i = 1,\ldots,k-2$; it is called decreasing if $... | {
"timestamp": "2017-05-04T02:02:18",
"yymm": "1703",
"arxiv_id": "1703.05378",
"language": "en",
"url": "https://arxiv.org/abs/1703.05378",
"abstract": "An edge-ordered graph is a graph with a total ordering of its edges. A path $P=v_1v_2\\ldots v_k$ in an edge-ordered graph is called increasing if $(v_iv_... |
https://arxiv.org/abs/1508.03516 | An Adaptive Variable Order Quadrature Strategy | In this article we propose a new adaptive numerical quadrature procedure which includes both local subdivision of the integration domain, as well as local variation of the number of quadrature points employed on each subinterval. In this way we aim to account for local smoothness properties of the function to be integr... | \section{Introduction}
Numerical integration methods have witnessed a tremendous development over the last few decades; see, e.g., \cite{Press:07,DahlquistBjorck:08,DavisRabinowitz:07}. In particular, adaptive quadrature rules have nowadays become an integral part of many scientific computing codes. Here, one of the f... | {
"timestamp": "2015-08-17T02:09:31",
"yymm": "1508",
"arxiv_id": "1508.03516",
"language": "en",
"url": "https://arxiv.org/abs/1508.03516",
"abstract": "In this article we propose a new adaptive numerical quadrature procedure which includes both local subdivision of the integration domain, as well as local... |
https://arxiv.org/abs/2012.09317 | On a fractional queueing model with catastrophes | A $M/M/1$ queue with catastrophes is a modified $M/M/1$ queue model for which, according to the times of a Poisson process, catastrophes occur leaving the system empty. In this work, we study a fractional $M/M/1$ queue with catastrophes, which is formulated by considering fractional derivatives in the Kolmogorov's Forw... | \section{Introduction}
Queueing Theory allows the formulation of mathematical models and methods to deal with stochastic aspects in applied sciences. Roughly speaking, the models are stochastic processes, usually Markovian, to represent phenomena in which customers arrive in a random way at a service facility. Upon arr... | {
"timestamp": "2020-12-18T02:06:19",
"yymm": "2012",
"arxiv_id": "2012.09317",
"language": "en",
"url": "https://arxiv.org/abs/2012.09317",
"abstract": "A $M/M/1$ queue with catastrophes is a modified $M/M/1$ queue model for which, according to the times of a Poisson process, catastrophes occur leaving the... |
https://arxiv.org/abs/1005.1424 | On commuting matrices in max algebra and in classical nonnegative algebra | This paper studies commuting matrices in max algebra and nonnegative linear algebra. Our starting point is the existence of a common eigenvector, which directly leads to max analogues of some classical results for complex matrices. We also investigate Frobenius normal forms of commuting matrices, particularly when the ... | \section{Introduction}\label{s:introduction}
The study of commuting complex matrices has a long history. As
observed in~\cite{Dra-51}, Cayley considers what appears to be a
generic case of commuting matrices in his famous
memoir~\cite{Cay-58}. Frobenius~\cite{Fro-78,Fro-96} showed that if
$A_i$, $i = 1,\ldots ,r... | {
"timestamp": "2010-05-11T02:01:15",
"yymm": "1005",
"arxiv_id": "1005.1424",
"language": "en",
"url": "https://arxiv.org/abs/1005.1424",
"abstract": "This paper studies commuting matrices in max algebra and nonnegative linear algebra. Our starting point is the existence of a common eigenvector, which dire... |
https://arxiv.org/abs/2212.03915 | Combinatorial generation via permutation languages. V. Acyclic orientations | In 1993, Savage, Squire, and West described an inductive construction for generating every acyclic orientation of a chordal graph exactly once, flipping one arc at a time. We provide two generalizations of this result. Firstly, we describe Gray codes for acyclic orientations of hypergraphs that satisfy a simple orderin... | \section{Introduction}
In 1953, Frank Gray registered a patent~\cite{gray_1953} for a method to list all binary words of length~$n$ in such a way that any two consecutive words differ in exactly one bit, and he called it the \emph{binary reflected code}.
More generally, a \emph{combinatorial Gray code}~\cite{ruskey_20... | {
"timestamp": "2022-12-09T02:00:35",
"yymm": "2212",
"arxiv_id": "2212.03915",
"language": "en",
"url": "https://arxiv.org/abs/2212.03915",
"abstract": "In 1993, Savage, Squire, and West described an inductive construction for generating every acyclic orientation of a chordal graph exactly once, flipping o... |
https://arxiv.org/abs/1511.05057 | Inverse tensor eigenvalue problem | A tensor $\mathcal T\in \mathbb T(\mathbb C^n,m+1)$, the space of tensors of order $m+1$ and dimension $n$ with complex entries, has $nm^{n-1}$ eigenvalues (counted with algebraic multiplicities). The inverse eigenvalue problem for tensors is a generalization of that for matrices. Namely, given a multiset $S\in \mathbb... | \section{Introduction}\label{sec:intro}
Eigenvalue of a tensor, as a natural generalized notion of the eigenvalue of a square matrix, has been attracting increasingly attention
in fields related to numerical multilinear algebra (see \cite{cs13,cpz08,fo14,lqz13,oo12,nqww07,q14} and references therein), since the indepe... | {
"timestamp": "2016-05-26T02:08:02",
"yymm": "1511",
"arxiv_id": "1511.05057",
"language": "en",
"url": "https://arxiv.org/abs/1511.05057",
"abstract": "A tensor $\\mathcal T\\in \\mathbb T(\\mathbb C^n,m+1)$, the space of tensors of order $m+1$ and dimension $n$ with complex entries, has $nm^{n-1}$ eigenv... |
https://arxiv.org/abs/1712.05098 | The central limit theorem for the number of clusters of the Arratia flow | In this paper we prove the central limit theorem for the number of clusters formed by the particles of the Arratia flow starting from the interval $[0;n]$ as $n\to\infty$ and obtain an estimate of the Berry-Esseen type for the rate of this convergence. | \section{Introduction}
\label{section1}
In this paper we consider the Arratia flow $\{x(u,\cdot),\; u\in\mbR\}$, which is an ordered family of standard Brownian motions starting from every point of the real line such that for any $u,v\in\mbR$ the joint quadratic variation of $x(u,\cdot)$ and $x(v,\cdot)$ is given... | {
"timestamp": "2017-12-15T02:03:53",
"yymm": "1712",
"arxiv_id": "1712.05098",
"language": "en",
"url": "https://arxiv.org/abs/1712.05098",
"abstract": "In this paper we prove the central limit theorem for the number of clusters formed by the particles of the Arratia flow starting from the interval $[0;n]$... |
https://arxiv.org/abs/1911.11810 | Exceptional points of discrete-time random walks in planar domains | Given a sequence of lattice approximations $D_N\subset\mathbb Z^2$ of a bounded continuum domain $D\subset\mathbb R^2$ with the vertices outside $D_N$ fused together into one boundary vertex $\varrho$, we consider discrete-time simple random walks in $D_N\cup\{\varrho\}$ run for a time proportional to the expected cove... | \section{Introduction}
\noindent
This note is a continuation of earlier work by the first two authors who in~\cite{AB} studied various exceptional level sets associated with the local time of random walks in lattice versions~$D_N\subset\mathbb Z^2$ of bounded open domains~$D\subset\mathbb R^2$, at times proportional ... | {
"timestamp": "2019-11-28T02:01:28",
"yymm": "1911",
"arxiv_id": "1911.11810",
"language": "en",
"url": "https://arxiv.org/abs/1911.11810",
"abstract": "Given a sequence of lattice approximations $D_N\\subset\\mathbb Z^2$ of a bounded continuum domain $D\\subset\\mathbb R^2$ with the vertices outside $D_N$... |
https://arxiv.org/abs/1106.4910 | On Projections of Metric Spaces | Let $X$ be a metric space and let $\mu$ be a probability measure on it. Consider a Lipschitz map $T: X \rightarrow \Rn$, with Lipschitz constant $\leq 1$. Then one can ask whether the image $TX$ can have large projections on many directions. For a large class of spaces $X$, we show that there are directions $\phi \in \... | \section{Introduction}
Let $(X,m)$ be a metric space and let $\mu$ be a probability measure on $X$. Consider a
Lipschitz map $T:X \rightarrow \mathbb{R}^n$, with $\LipNorm{T}\leq 1$, where $\mathbb{R}^n$ is taken with the standard inner product $<\cdot,\cdot>$ and the corresponding Euclidean norm $|\cdot|$.
Define a... | {
"timestamp": "2011-06-27T02:01:43",
"yymm": "1106",
"arxiv_id": "1106.4910",
"language": "en",
"url": "https://arxiv.org/abs/1106.4910",
"abstract": "Let $X$ be a metric space and let $\\mu$ be a probability measure on it. Consider a Lipschitz map $T: X \\rightarrow \\Rn$, with Lipschitz constant $\\leq 1... |
https://arxiv.org/abs/2005.01466 | On dominating pair degree conditions for hamiltonicity in balanced bipartite digraphs | We prove several new sufficient conditions for hamiltonicity and bipancyclicity in balanced bipartite digraphs, in terms of sums of degrees over dominating or dominated pairs of vertices. | \section{Introduction}
\label{sec:intro}
This article is concerned with sufficient conditions for hamiltonicity and bipancyclicity in balanced bipartite digraphs. More specifically, we study several Meyniel-type criteria, that is, theorems asserting existence of hamiltonian cycles under certain conditions on the su... | {
"timestamp": "2020-05-05T02:29:23",
"yymm": "2005",
"arxiv_id": "2005.01466",
"language": "en",
"url": "https://arxiv.org/abs/2005.01466",
"abstract": "We prove several new sufficient conditions for hamiltonicity and bipancyclicity in balanced bipartite digraphs, in terms of sums of degrees over dominatin... |
https://arxiv.org/abs/1806.03196 | Approximation of Hermitian Matrices by Positive Semidefinite Matrices using Modified Cholesky Decompositions | A new algorithm to approximate Hermitian matrices by positive semidefinite Hermitian matrices based on modified Cholesky decompositions is presented. In contrast to existing algorithms, this algorithm allows to specify bounds on the diagonal values of the approximation. It has no significant runtime and memory overhead... | \section{Introduction} \label{sec: introduction}
Algorithms for approximating Hermitian matrices by positive semidefinite Hermitian matrices are useful in several areas. In stochastics they are needed to transform nonpositive semidefinite estimations of covariance and correlation matrices to valid estimations \cite{Ro... | {
"timestamp": "2019-12-12T02:18:57",
"yymm": "1806",
"arxiv_id": "1806.03196",
"language": "en",
"url": "https://arxiv.org/abs/1806.03196",
"abstract": "A new algorithm to approximate Hermitian matrices by positive semidefinite Hermitian matrices based on modified Cholesky decompositions is presented. In c... |
https://arxiv.org/abs/math/0702279 | Infinitely Often Dense Bases of Integers with a Prescribed Representation Function | Nathanson constructed asymptotic bases for the integers with a prescribed representation function, then asked how dense they can be. We can easily obtain an upper bound using a simple argument. In this paper, we will see this is indeed the best bound we can get for asymptotic bases for the integers with an arbitrary re... | \section{Introduction}\label{S:intro}
We will use the following notations: For sets $A, B$ of integers, any integer $t$ and a positive integer $h$, we define the \emph{sumset}
$ A + B = \{ a+b : a\in A, b \in B\}$,
the \emph{translation}
$A + t = \{ a + t : a \in A\},$ and the \emph{dilation}
$ h*A = \{ ha : a \in... | {
"timestamp": "2007-04-26T11:09:57",
"yymm": "0702",
"arxiv_id": "math/0702279",
"language": "en",
"url": "https://arxiv.org/abs/math/0702279",
"abstract": "Nathanson constructed asymptotic bases for the integers with a prescribed representation function, then asked how dense they can be. We can easily obt... |
https://arxiv.org/abs/1911.00659 | Jacobi-type algorithm for low rank orthogonal approximation of symmetric tensors and its convergence analysis | In this paper, we propose a Jacobi-type algorithm to solve the low rank orthogonal approximation problem of symmetric tensors. This algorithm includes as a special case the well-known Jacobi CoM2 algorithm for the approximate orthogonal diagonalization problem of symmetric tensors. We first prove the weak convergence o... | \section{Introduction}
As the higher order analogue of vectors and matrices,
in the last two decades,
tensors have been attracting more and more attentions from various fields,
including signal processing, numerical linear algebra and machine learning
\cite{Cichocki15:review,comon2014tensors,Como10:book,kolda2009tens... | {
"timestamp": "2019-11-05T02:07:59",
"yymm": "1911",
"arxiv_id": "1911.00659",
"language": "en",
"url": "https://arxiv.org/abs/1911.00659",
"abstract": "In this paper, we propose a Jacobi-type algorithm to solve the low rank orthogonal approximation problem of symmetric tensors. This algorithm includes as ... |
https://arxiv.org/abs/1605.07233 | An interpolation proof of Ehrhard's inequality | We prove Ehrhard's inequality using interpolation along the Ornstein-Uhlenbeck semi-group. We also provide an improved Jensen inequality for Gaussian variables that might be of independent interest. | \section{Introduction}
In \cite{Ehr}, A. Ehrhard proved the following Brunn-Minkowski like inequality for convex sets $A, B$ in $\mathbb R^{n}$:
\begin{equation}
\label{Ehr}
\Phi^{-1} \left( \gamma_{n} ( \lambda A + (1-\lambda) B ) \right) \geq \lambda \Phi^{-1} (\gamma_{n} (A) ) + (1-\lambda) \Phi^{-1} (\gamma_{n} (... | {
"timestamp": "2016-05-25T02:03:20",
"yymm": "1605",
"arxiv_id": "1605.07233",
"language": "en",
"url": "https://arxiv.org/abs/1605.07233",
"abstract": "We prove Ehrhard's inequality using interpolation along the Ornstein-Uhlenbeck semi-group. We also provide an improved Jensen inequality for Gaussian vari... |
https://arxiv.org/abs/1910.12228 | Ring-theoretic approaches to point-set topology | In this paper, it is shown that a topological space $X$ is compact iff every maximal ideal of the power set ring $\mathcal{P}(X)$ converges to exactly one point of $X$. Then as an application, simple and ring-theoretic proofs are provided for the Tychonoff theorem and Alexander subbase theorem. As another result in thi... | \section{Introduction}
Some of the mathematicians consider the Tychonoff theorem as the single most important result in general topology, (other mathematicians allow it to share this honor with Urysohn's lemma). Tychonoff theorem is a fundamental ingredient in proving various important results in the areas of topolo... | {
"timestamp": "2020-11-05T02:15:42",
"yymm": "1910",
"arxiv_id": "1910.12228",
"language": "en",
"url": "https://arxiv.org/abs/1910.12228",
"abstract": "In this paper, it is shown that a topological space $X$ is compact iff every maximal ideal of the power set ring $\\mathcal{P}(X)$ converges to exactly on... |
https://arxiv.org/abs/1808.09485 | Spijker's example and its extension | Strongly and weakly stable linear multistep methods can behave very differently. The latter class can produce spurious oscillations in some of the cases for which the former class works flawlessly. The main question is if we can find a well defined property which clearly tells the difference between them. There are man... | \section{Introduction}
This paper focuses on the stability and instability of linear multistep methods. When we introduce linear multistep methods it is unavoidable to talk about the root-condition and usually about the two types of it, which divide these methods into two classes: the weakly and strongly stable linear... | {
"timestamp": "2018-08-30T02:01:14",
"yymm": "1808",
"arxiv_id": "1808.09485",
"language": "en",
"url": "https://arxiv.org/abs/1808.09485",
"abstract": "Strongly and weakly stable linear multistep methods can behave very differently. The latter class can produce spurious oscillations in some of the cases f... |
https://arxiv.org/abs/1710.07007 | Quarter-Turn Baxter Permutations | Baxter permutations are known to be in bijection with a wide number of combinatorial objects. Previously, it was shown that each of these objects had a natural involution which was carried equivariantly by the known bijections, and the number of objects fixed under involution was given by Stembridge's $q=-1$ phenomenon... | \section{Background}
\label{intro}
Baxter permutations are a well-studied class of permutations, which have a number of symmetries and nice properties associated to them.
\begin{definition}
We say that a \emph{Baxter permutation} is a permutation that avoids the patterns 3-14-2 and 2-41-3, where an occurrence of the ... | {
"timestamp": "2017-10-20T02:04:25",
"yymm": "1710",
"arxiv_id": "1710.07007",
"language": "en",
"url": "https://arxiv.org/abs/1710.07007",
"abstract": "Baxter permutations are known to be in bijection with a wide number of combinatorial objects. Previously, it was shown that each of these objects had a na... |
https://arxiv.org/abs/1207.3878 | Solving the Ku-Wales conjecture on the eigenvalues of the derangement graph | We give a new recurrence formula for the eigenvalues of the derangement graph. Consequently, we provide a simpler proof of the Alternating Sign Property of the derangement graph. Moreover, we prove that the absolute value of the eigenvalue decreases whenever the corresponding partition decreases in the dominance order.... | \section{Introduction}
Let $G$ be a finite group and $S$ be a subset of $G$. The Cayley graph $\Gamma(G,S)$ is the graph which has the elements of $G$ as its vertices and two vertices $u, v \in G$ are joined by an edge if and only if $uv^{-1} \in S$. We require that $S$ is a nonempty subset of $G$ satisfying the cond... | {
"timestamp": "2012-07-18T02:01:52",
"yymm": "1207",
"arxiv_id": "1207.3878",
"language": "en",
"url": "https://arxiv.org/abs/1207.3878",
"abstract": "We give a new recurrence formula for the eigenvalues of the derangement graph. Consequently, we provide a simpler proof of the Alternating Sign Property of ... |
https://arxiv.org/abs/2105.13301 | Majority Dynamics: The Power of One | Consider $n=\ell+m$ individuals, where $\ell\le m$, with $\ell$ individuals holding an opinion $A$ and $m$ holding an opinion $B$. Suppose that the individuals communicate via an undirected network $G$, and in each time step, each individual updates her opinion according to a majority rule (that is, according to the op... | \section{Introduction}\label{sec:introduction}
Considerable effort has been devoted to understanding exchange of opinions between individuals, seeing as it plays a major role in all types of social interaction. Of course, no simple model can accurately describe the behavior of many actors in complicated situations, so ... | {
"timestamp": "2021-05-28T02:26:52",
"yymm": "2105",
"arxiv_id": "2105.13301",
"language": "en",
"url": "https://arxiv.org/abs/2105.13301",
"abstract": "Consider $n=\\ell+m$ individuals, where $\\ell\\le m$, with $\\ell$ individuals holding an opinion $A$ and $m$ holding an opinion $B$. Suppose that the in... |
https://arxiv.org/abs/2211.14252 | The extremals of Stanley's inequalities for partially ordered sets | Stanley's inequalities for partially ordered sets establish important log-concavity relations for sequences of linear extensions counts. Their extremals however, i.e., the equality cases of these inequalities, were until now poorly understood with even conjectures lacking. In this work, we solve this problem by providi... | \section{Introduction}
\label{sec:intro}
\subsection{Log-concave sequences}
Finite sequences of numbers $\{a_i\}_{i=1}^n$ often serve as a powerful way to encode properties of algebraic, geometric, and combinatorial objects: $a_i$ can stand for the $i$th coefficient of a Schur polynomial, the dimension of the $i$th c... | {
"timestamp": "2022-11-28T02:27:44",
"yymm": "2211",
"arxiv_id": "2211.14252",
"language": "en",
"url": "https://arxiv.org/abs/2211.14252",
"abstract": "Stanley's inequalities for partially ordered sets establish important log-concavity relations for sequences of linear extensions counts. Their extremals h... |
https://arxiv.org/abs/1607.07067 | Classification of Category $\mathcal{J}$ Modules for Divergence Zero Vector Fields on a Torus | We consider a category of modules that admit compatible actions of the commutative algebra of Laurent polynomials and the Lie algebra of divergence zero vector fields on a torus and have a weight decomposition with finite dimensional weight spaces. We classify indecomposable and irreducible modules in this category. | \section{Introduction}
Consider the algebra $A_N=\mathbb{C}[t_1^{\pm1},\dots,t_{N}^{\pm1}]$ and Lie algebra $\text{Der}(A_N)$ of derivations of $A_N$.
The Lie algebra $\text{Der}(A_N)$ may be identified with the Lie algebra of polynomial vector fields on an $N$-dimensional torus (see Section 2).
In \cite{R2} Eswa... | {
"timestamp": "2016-10-12T02:00:35",
"yymm": "1607",
"arxiv_id": "1607.07067",
"language": "en",
"url": "https://arxiv.org/abs/1607.07067",
"abstract": "We consider a category of modules that admit compatible actions of the commutative algebra of Laurent polynomials and the Lie algebra of divergence zero v... |
https://arxiv.org/abs/2107.05670 | A rainbow connectivity threshold for random graph families | Given a family $\mathcal G$ of graphs on a common vertex set $X$, we say that $\mathcal G$ is rainbow connected if for every vertex pair $u,v \in X$, there exists a path from $u$ to $v$ that uses at most one edge from each graph in $\mathcal G$. We consider the case that $\mathcal G$ contains $s$ graphs, each sampled r... | \section{Introduction}
In this paper, we consider random graphs using the \emph{Erd\H{o}s-R\'enyi model}, which are defined as follows. For a positive integer $n$, we consider a set $X$ of $n$ vertices. Then, for some value $0 \leq p \leq 1$, we construct a graph $G$ on $X$ by independently letting each edge $e \in ... | {
"timestamp": "2021-07-15T02:21:57",
"yymm": "2107",
"arxiv_id": "2107.05670",
"language": "en",
"url": "https://arxiv.org/abs/2107.05670",
"abstract": "Given a family $\\mathcal G$ of graphs on a common vertex set $X$, we say that $\\mathcal G$ is rainbow connected if for every vertex pair $u,v \\in X$, t... |
https://arxiv.org/abs/1508.04675 | Independent Sets, Matchings, and Occupancy Fractions | We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of the results of Kahn, Galvin and Tetali, and Zhao showing that a union of copies of $K_{d,d}$ maximizes the number of independent sets and t... | \section{Independent Sets}
Let $G$ be a graph. The independence polynomial of $G$ is
\[P_G(\lambda) = \sum_{I \in \mathcal I} \lambda^{|I|} \]
where $\mathcal I $ is the set of all independent sets of $G$. By convention we consider the empty independent set to be a member of $\mathcal I$. The \textit{hard-core mode... | {
"timestamp": "2017-05-10T02:05:08",
"yymm": "1508",
"arxiv_id": "1508.04675",
"language": "en",
"url": "https://arxiv.org/abs/1508.04675",
"abstract": "We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this the... |
https://arxiv.org/abs/2003.01172 | Volume Above Distance Below | Given a pair of metric tensors $g_1 \ge g_0$ on a Riemannian manifold, $M$, it is well known that $\operatorname{Vol}_1(M) \ge \operatorname{Vol}_0(M)$. Furthermore one has rigidity: the volumes are equal if and only if the metric tensors are the same $g_1=g_0$. Here we prove that if $g_j \ge g_0$ and $\operatorname{Vo... | \section{Introduction}\label{sect:intro}
Over the past few decades a number of geometric stability theorems have been proven where one assumes a lower bound on Ricci curvature and proves the Riemannian manifolds are close in the Gromov-Hausdorff sense. However, without a lower bound on Ricci curvature, there are us... | {
"timestamp": "2021-01-06T02:24:45",
"yymm": "2003",
"arxiv_id": "2003.01172",
"language": "en",
"url": "https://arxiv.org/abs/2003.01172",
"abstract": "Given a pair of metric tensors $g_1 \\ge g_0$ on a Riemannian manifold, $M$, it is well known that $\\operatorname{Vol}_1(M) \\ge \\operatorname{Vol}_0(M)... |
https://arxiv.org/abs/1811.03527 | A Local Limit Theorem for Cliques in G(n,p) | We prove a local limit theorem the number of $r$-cliques in $G(n,p)$ for $p\in(0,1)$ and $r\ge 3$ fixed constants. Our bounds hold in both the $\ell^\infty$ and $\ell^1$ metric. The main work of the paper is an estimate for the characteristic function of this random variable. This is accomplished by introducing a new t... | \section{Introduction}
In 1960 Erd\H{o}s and R\'enyi introduced the study of $G(n,p)$, the random graph on $n$ vertices where each edge is included independently at random with probability $p$. In \cite{ErdosRenyi} they showed, among other results, that the number of cliques of size $r$ in $G(n,p)$ is concentrated ab... | {
"timestamp": "2018-11-09T02:17:54",
"yymm": "1811",
"arxiv_id": "1811.03527",
"language": "en",
"url": "https://arxiv.org/abs/1811.03527",
"abstract": "We prove a local limit theorem the number of $r$-cliques in $G(n,p)$ for $p\\in(0,1)$ and $r\\ge 3$ fixed constants. Our bounds hold in both the $\\ell^\\... |
https://arxiv.org/abs/1411.3869 | Convergence properties of a geometric mesh smoothing algorithm | We describe a simple geometric transformation of triangles which leads to an efficient and effective algorithm to smooth triangle and tetrahedral meshes. Our focus lies on the convergence properties of this algorithm: we prove the effectivity for some planar triangle meshes and further introduce dynamical methods to st... | \section{Introduction}\label{s.first}
\subsection{Preliminary remarks}
The finite element method is the standard instrument to simulate the behavior of solid bodies or fluids in engineering and physics. The first preparatory step of this method is the discretization of the underlying domain into finitely many elements ... | {
"timestamp": "2014-11-18T02:16:33",
"yymm": "1411",
"arxiv_id": "1411.3869",
"language": "en",
"url": "https://arxiv.org/abs/1411.3869",
"abstract": "We describe a simple geometric transformation of triangles which leads to an efficient and effective algorithm to smooth triangle and tetrahedral meshes. Ou... |
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