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url stringlengths 31 38	title stringlengths 7 229	abstract stringlengths 44 2.87k	text stringlengths 319 2.51M	meta dict
https://arxiv.org/abs/1602.07209	Polytopes and simplexes in p-adic fields	We introduce topological notions of polytopes and simplexes, the latter being expected to play in p-adically closed fields the role played by real simplexes in the classical results of triangulation of semi-algebraic sets over real closed fields. We prove that the faces of every p-adic polytope are polytopes and that t...	\section{Introduction} \label{se:intro} Throughout all this paper we fix a $p$\--adically closed field $(K,v)$. The reader unfamiliar with this notion may restrict to the special case where $K={\rm\bf Q}_p$ or a finite extension of it, and $v$ is its $p$\--adic valuation. We let $R$ denote the valuation ring of $v$,...	{ "timestamp": "2016-11-15T02:12:28", "yymm": "1602", "arxiv_id": "1602.07209", "language": "en", "url": "https://arxiv.org/abs/1602.07209", "abstract": "We introduce topological notions of polytopes and simplexes, the latter being expected to play in p-adically closed fields the role played by real simplex...
https://arxiv.org/abs/1202.1218	A geometrical triumvirate of real random matrices	We present a five-step method for the calculation of eigenvalue correlation functions for various ensembles of real random matrices, based upon the method of (skew-) orthogonal polynomials. This scheme systematises existing methods and also involves some new techniques. The ensembles considered are: the Gaussian Orthog...	\section*{} \pagenumbering{roman} \thispagestyle{empty} \begin{centering} \vspace{60pt}A GEOMETRICAL TRIUMVIRATE OF REAL RANDOM MATRICES \vspace{12pt} Anthony Mays \vspace{280pt}Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy \vspace{50pt}November 2011 \vspace{12pt}Departmen...	{ "timestamp": "2012-02-07T02:03:50", "yymm": "1202", "arxiv_id": "1202.1218", "language": "en", "url": "https://arxiv.org/abs/1202.1218", "abstract": "We present a five-step method for the calculation of eigenvalue correlation functions for various ensembles of real random matrices, based upon the method o...
https://arxiv.org/abs/1611.07820	Local variational study of 2d lattice energies and application to Lennard-Jones type interactions	In this paper, we focus on finite Bravais lattice energies per point in two dimensions. We compute the first and second derivatives of these energies. We prove that the Hessian at the square and the triangular lattice are diagonal and we give simple sufficient conditions for the local minimality of these lattices. Furt...	\section{Introduction} \subsection{Minimization at high and low densities: our previous works} In our previous work with Zhang \cite{Betermin:2014fy}, generalized in \cite{BetTheta15}, we studied some two-dimensional lattice energies among Bravais lattices. More precisely, these energies are defined, for any Bravais l...	{ "timestamp": "2016-11-24T02:06:38", "yymm": "1611", "arxiv_id": "1611.07820", "language": "en", "url": "https://arxiv.org/abs/1611.07820", "abstract": "In this paper, we focus on finite Bravais lattice energies per point in two dimensions. We compute the first and second derivatives of these energies. We ...
https://arxiv.org/abs/2206.01544	Polynomial approximation on $C^2$-domains	We introduce appropriate computable moduli of smoothness to characterize the rate of best approximation by multivariate polynomials on a connected and compact $C^2$-domain $\Omega\subset \mathbb{R}^d$. This new modulus of smoothness is defined via finite differences along the directions of coordinate axes, and along a ...	\subsection{Decomposition of the domain in $\RR^d$} \subjclass[2010]{Primary 41A10, 41A17, 41A27, 41A63;\\Secondary 41A55, 65D32} \keywords{$C^2$-domains, polynomial approximation, modulus of smoothness, Jackson inequality, inverse theorem} \begin{abstract} We introduce appropriate computable moduli of smoot...	{ "timestamp": "2022-06-06T02:13:14", "yymm": "2206", "arxiv_id": "2206.01544", "language": "en", "url": "https://arxiv.org/abs/2206.01544", "abstract": "We introduce appropriate computable moduli of smoothness to characterize the rate of best approximation by multivariate polynomials on a connected and com...
https://arxiv.org/abs/0906.3132	Constructing matrix geometric means	In this paper, we analyze the process of "assembling" new matrix geometric means from existing ones, through function composition or limit processes. We show that for n=4 a new matrix mean exists which is simpler to compute than the existing ones. Moreover, we show that for n>4 the existing proving strategies cannot pr...	\section{Introduction} \paragraph{Literature review} In the last few years, several papers have been devoted to defining a proper way to generalize the concept of geometric mean to $n \geq 3$ Hermitian, positive definite $m \times m$ matrices. A seminal paper by Ando, Li and Mathias \cite{alm} defined the mathematical ...	{ "timestamp": "2010-07-27T02:01:03", "yymm": "0906", "arxiv_id": "0906.3132", "language": "en", "url": "https://arxiv.org/abs/0906.3132", "abstract": "In this paper, we analyze the process of \"assembling\" new matrix geometric means from existing ones, through function composition or limit processes. We s...
https://arxiv.org/abs/1207.5015	Free and Very Free Morphisms into a Fermat Hypersurface	This paper studies the existence of free and very free curves on the degree 5 Fermat hypersurface in P^5 over a field of characteristic 2. We find that such curves exist in degrees 8 and 9 and not in lower degrees.	\section{Introduction} \noindent Any smooth projective Fano variety in characteristic zero is rationally connected and hence contains a very free rational curve. In positive characteristic a smooth projective Fano variety is rationally chain connected. However, it is not known whether such varieties are separably rati...	{ "timestamp": "2012-07-26T02:06:22", "yymm": "1207", "arxiv_id": "1207.5015", "language": "en", "url": "https://arxiv.org/abs/1207.5015", "abstract": "This paper studies the existence of free and very free curves on the degree 5 Fermat hypersurface in P^5 over a field of characteristic 2. We find that such...
https://arxiv.org/abs/1001.3885	Improved Source Coding Exponents via Witsenhausen's Rate	We provide a novel upper-bound on Witsenhausen's rate, the rate required in the zero-error analogue of the Slepian-Wolf problem; our bound is given in terms of a new information-theoretic functional defined on a certain graph. We then use the functional to give a single letter lower-bound on the error exponent for the ...	\section{Introduction} Under consideration is the communication problem depicted in Figure \ref{fig:scsi}; nature produces a sequence $(X_i,Y_i)$ governed by the i.i.d. distribution $P_{XY}$ on alphabet $\mathcal{X} \times \mathcal{Y}$. An encoder, observing the sequence $X^n$, must send a message to a decoder, obse...	{ "timestamp": "2010-01-21T22:30:48", "yymm": "1001", "arxiv_id": "1001.3885", "language": "en", "url": "https://arxiv.org/abs/1001.3885", "abstract": "We provide a novel upper-bound on Witsenhausen's rate, the rate required in the zero-error analogue of the Slepian-Wolf problem; our bound is given in terms...
https://arxiv.org/abs/1406.3233	The Skolem-Abouzaid theorem in the singular case	Let F(X;Y) in Q[X;Y] be a Q-irreducible polynomial. In 1929 Skolem proved the following theorem: "Assume that F(0;0) = 0. Then for every non-zero integer d, the equation F(X;Y) = 0 has only finitely many solutions in integers (X;Y) with gcd(X;Y) = d". Skolem method allows one to bound the solutions explicitly in terms ...	\section{Introduction} Let ${F(X,Y)\in\Q[X,Y]}$ be a $\Q$-irreducible polynomial. In 1929 Skolem~\cite{Sk29} proved the following beautiful theorem: \begin{theorem}[Skolem] \label{tskol} Assume that \begin{equation} \label{e00} F(0,0)=0. \end{equation} Then for every non-zero integer~$d$, the equation ${F(X,Y)=0}$...	{ "timestamp": "2015-01-22T02:15:27", "yymm": "1406", "arxiv_id": "1406.3233", "language": "en", "url": "https://arxiv.org/abs/1406.3233", "abstract": "Let F(X;Y) in Q[X;Y] be a Q-irreducible polynomial. In 1929 Skolem proved the following theorem: \"Assume that F(0;0) = 0. Then for every non-zero integer d...
https://arxiv.org/abs/1312.3311	Stochastic De Giorgi Iteration and Regularity of Stochastic Partial Differential Equation	Under general conditions we show that the solution of a stochastic parabolic partial differential equation of the form \[ \partial_t u = \mathrm{div} (A \nabla u) + f(t,x, u) + g_i (t,x,u) \dot{w}^i_t \] is almost surely Hölder continuous in both space and time variables.	\section{Introduction} Stochastic partial differential equations (SPDEs) arise in many pure and applied sciences. Regularity of solutions is of central importance for theoretical development as well as for numerical simulations. For linear equations, $W^{k,2}$-theory has been well developed (see Pardoux~\cite{Par07} ...	{ "timestamp": "2014-05-14T02:03:58", "yymm": "1312", "arxiv_id": "1312.3311", "language": "en", "url": "https://arxiv.org/abs/1312.3311", "abstract": "Under general conditions we show that the solution of a stochastic parabolic partial differential equation of the form \\[ \\partial_t u = \\mathrm{div} (A ...
https://arxiv.org/abs/1911.05799	Congruences satisfied by eta-quotients	The values of the partition function, and more generally the Fourier coefficients of many modular forms, are known to satisfy certain congruences. Results given by Ahlgren and Ono for the partition function and by Treneer for more general Fourier coefficients state the existence of infinitely many families of congruenc...	\section{Introduction}\label{sec:intro} The partition function $p(n)$ gives the number of non-increasing sequences of positive integers that sum to $n$. Ramanujan first discovered that $p(n)$ satisfies \begin{align*} p(5n+4)&\equiv 0\pmod{5},\\ p(7n+5)&\equiv 0\pmod{7}, \;\mathrm{and}\\ p(11n+6)&\equiv 0\pmod{11}, \en...	{ "timestamp": "2021-02-01T02:13:13", "yymm": "1911", "arxiv_id": "1911.05799", "language": "en", "url": "https://arxiv.org/abs/1911.05799", "abstract": "The values of the partition function, and more generally the Fourier coefficients of many modular forms, are known to satisfy certain congruences. Results...
https://arxiv.org/abs/2005.01576	Group Presentations for Links in Thickened Surfaces	Using a combinatorial argument, we prove the well-known result that the Wirtinger and Dehn presentations of a link in 3-space describe isomorphic groups. The result is not true for links $\ell$ in a thickened surface $S \times [0,1]$. Their precise relationship, as given in the 2012 thesis of R.E. Byrd, is established ...	\section{Introduction} Modern knot theory, which began in the early 1900's, was propelled by the nearly simultaneous publications of two different methods for computing presentations of knot groups, fundamental groups of knot complements. The methods are due to W. Wirtinger and M. Dehn. Both are combinatorial, beginnin...	{ "timestamp": "2020-05-05T02:32:31", "yymm": "2005", "arxiv_id": "2005.01576", "language": "en", "url": "https://arxiv.org/abs/2005.01576", "abstract": "Using a combinatorial argument, we prove the well-known result that the Wirtinger and Dehn presentations of a link in 3-space describe isomorphic groups. ...
https://arxiv.org/abs/0710.2020	Valiron's construction in higher dimension	We consider holomorphic self-maps $\v$ of the unit ball $\B^N$ in $\C^N$ ($N=1,2,3,...$). In the one-dimensional case, when $\v$ has no fixed points in $\D\defeq \B^1$ and is of hyperbolic type, there is a classical renormalization procedure due to Valiron which allows to semi-linearize the map $\phi$, and therefore, i...	\section{Introduction} \subsection{The one-dimensional case} Let $\varphi$ be a holomorphic map on $\mathbb D$ with $\varphi(\mathbb D)\subset\mathbb D$. If $\varphi$ has no fixed points in $\mathbb D$, then by the classical Wolff lemma (see, {\sl e.g.}, \cite{Abate}) there exists a unique point $\tau\in\partial\ma...	{ "timestamp": "2007-10-10T16:13:43", "yymm": "0710", "arxiv_id": "0710.2020", "language": "en", "url": "https://arxiv.org/abs/0710.2020", "abstract": "We consider holomorphic self-maps $\\v$ of the unit ball $\\B^N$ in $\\C^N$ ($N=1,2,3,...$). In the one-dimensional case, when $\\v$ has no fixed points in ...
https://arxiv.org/abs/1907.03635	Distance from the Nucleus to a Uniformly Random Point in the 0-cell and the Typical Cell of the Poisson-Voronoi Tessellation	Consider the distances $\tilde{R}_o$ and $R_o$ from the nucleus to a uniformly random point in the 0-cell and the typical cell, respectively, of the $d$-dimensional Poisson-Voronoi (PV) tessellation. The main objective of this paper is to characterize the exact distributions of $\tilde{R}_o$ and $R_o$. First, using the...	\section{Introduction}\label{sec:Intro} The Poisson point process (PPP) has found many applications in science and engineering due to its useful mathematical properties. Several of these applications specifically focus on the Poisson-Voronoi (PV) tessellation \cite{moller1989random}, which partitions space into disjo...	{ "timestamp": "2019-12-06T02:19:02", "yymm": "1907", "arxiv_id": "1907.03635", "language": "en", "url": "https://arxiv.org/abs/1907.03635", "abstract": "Consider the distances $\\tilde{R}_o$ and $R_o$ from the nucleus to a uniformly random point in the 0-cell and the typical cell, respectively, of the $d$-...
https://arxiv.org/abs/1301.2342	A Linear Time Algorithm for the Feasibility of Pebble Motion on Graphs	Given a connected, undirected, simple graph $G = (V, E)$ and $p \le \|V\|$ pebbles labeled $1,..., p$, a configuration of these $p$ pebbles is an injective map assigning the pebbles to vertices of $G$. Let $S$ and $D$ be two such configurations. From a configuration, pebbles can move on $G$ as follows: In each step, at m...	\section{Introduction} In Sam Loyd's 15-puzzle \cite{Loy59}, a player is asked to arrange square game pieces labeled 1-15, scrambled on a $4 \times 4$ grid, to a shuffled row major ordering, using one empty swap cell: In each step, one of the labeled pieces neighboring the empty cell may be moved to the empty cell (...	{ "timestamp": "2013-01-22T02:02:36", "yymm": "1301", "arxiv_id": "1301.2342", "language": "en", "url": "https://arxiv.org/abs/1301.2342", "abstract": "Given a connected, undirected, simple graph $G = (V, E)$ and $p \\le \|V\|$ pebbles labeled $1,..., p$, a configuration of these $p$ pebbles is an injective m...
https://arxiv.org/abs/2006.11612	Unstable Modules with the Top $k$ Squares	Unstable modules over the Steenrod algebra with only the top $k$ operations are introduced in the language of ringoids. We prove the category of such modules has homological dimension at most $k$. A pratical method, which generalizes the $\Lambda$ complex, to compute the $\mathrm{Ext}$ group from such modules to sphere...	\section*{Introduction} Let $A$ be the Steenrod algebra over the field $\mathbb{F}_{2}$. The purpose of this paper is to investigate the category $\mathcal{U}_{k}$ of unstable left $A$-modules where only the top $k$ Steenrod squares are allowed. In general, on an homogeneous element of degree $n$ the top $k$ Steenrod s...	{ "timestamp": "2020-06-23T02:13:06", "yymm": "2006", "arxiv_id": "2006.11612", "language": "en", "url": "https://arxiv.org/abs/2006.11612", "abstract": "Unstable modules over the Steenrod algebra with only the top $k$ operations are introduced in the language of ringoids. We prove the category of such modu...
https://arxiv.org/abs/1405.5587	Parking functions, Shi arrangements, and mixed graphs	The \emph{Shi arrangement} is the set of all hyperplanes in $\mathbb R^n$ of the form $x_j - x_k = 0$ or $1$ for $1 \le j < k \le n$. Shi observed in 1986 that the number of regions (i.e., connected components of the complement) of this arrangement is $(n+1)^{n-1}$. An unrelated combinatorial concept is that of a \emph...	\section{Introduction} Our goal is to draw (bijective) connections between three seemingly unrelated concepts; their names form the title of our paper, and we start by introducing them one by one. \subsection{Parking Functions} Imagine a one-way street with $n$ parking spots and a cliff at its end. We'll give the f...	{ "timestamp": "2014-09-09T02:11:45", "yymm": "1405", "arxiv_id": "1405.5587", "language": "en", "url": "https://arxiv.org/abs/1405.5587", "abstract": "The \\emph{Shi arrangement} is the set of all hyperplanes in $\\mathbb R^n$ of the form $x_j - x_k = 0$ or $1$ for $1 \\le j < k \\le n$. Shi observed in 19...
https://arxiv.org/abs/1306.3396	Symmetry minimizes the principal eigenvalue: an example for the Pucci's sup operator	We explicitly evaluate the principal eigenvalue of the extremal Pucci's sup--operator for a class of special plane domains, and we prove that, for fixed area, the eigenvalue is minimal for the most symmetric set.	\section{Introduction}\label{intro} In 1951, P\'olya and Szego conjectured: {\em Of all $n$-polygons with the same area, the regular $n$-polygon has the smallest first Dirichlet eigenvalue,} referring to the Dirichlet eigenvalue of the Laplacian. It is very simple to see that among all rectangles of same area, the o...	{ "timestamp": "2013-07-08T02:03:33", "yymm": "1306", "arxiv_id": "1306.3396", "language": "en", "url": "https://arxiv.org/abs/1306.3396", "abstract": "We explicitly evaluate the principal eigenvalue of the extremal Pucci's sup--operator for a class of special plane domains, and we prove that, for fixed are...
https://arxiv.org/abs/1806.07260	The graphs with all but two eigenvalues equal to $2$ or $-1$	In this paper, all graphs whose adjacency matrix has at most two eigenvalues (multiplicities included) different from $2$ and $-1$ are determined. These graphs conclude a class of generalized friendship graphs $F_{t,r,k}, $ which is the graph of $k$ copies of the complete graph $K_t$ meeting in common $r$ vertices such...	\section{ Introduction} All graphs in this paper are simple graphs and all spectrum of a graph are adjacency spectrum. Let $G=(V,E)$ be a graph. The adjacency matrix $A(G)$ (or $A$) of $G$ is an $n\times n$ matrix, whose $(i,j)$-entry is $1$ if vertex $v_{i}$ is adjacent to $v_{j}$ (denote by $v_i \sim v_j$), and is ...	{ "timestamp": "2018-06-20T02:11:23", "yymm": "1806", "arxiv_id": "1806.07260", "language": "en", "url": "https://arxiv.org/abs/1806.07260", "abstract": "In this paper, all graphs whose adjacency matrix has at most two eigenvalues (multiplicities included) different from $2$ and $-1$ are determined. These g...
https://arxiv.org/abs/1805.02201	RealCertify: a Maple package for certifying non-negativity	Let $\mathbb{Q}$ (resp. $\mathbb{R}$) be the field of rational (resp. real) numbers and $X = (X_1, \ldots, X_n)$ be variables. Deciding the non-negativity of polynomials in $\mathbb{Q}[X]$ over $\mathbb{R}^n$ or over semi-algebraic domains defined by polynomial constraints in $\mathbb{Q}[X]$ is a classical algorithmic ...	\section{Introduction} Let ${\mathbb{Q}}$ (resp.~${\mathbb{R}}$) be the field of rational (resp.~real) numbers and $X = (X_1, \ldots, X_n)$ be a sequence of variables. We consider the problem of deciding the non-negativity of $f \in {\mathbb{Q}}[X]$ either over ${\mathbb{R}}^n$ or over a semi-algebraic set $S$ defined...	{ "timestamp": "2018-05-08T02:11:12", "yymm": "1805", "arxiv_id": "1805.02201", "language": "en", "url": "https://arxiv.org/abs/1805.02201", "abstract": "Let $\\mathbb{Q}$ (resp. $\\mathbb{R}$) be the field of rational (resp. real) numbers and $X = (X_1, \\ldots, X_n)$ be variables. Deciding the non-negativ...
https://arxiv.org/abs/1907.00551	Plateau's problem as a singular limit of capillarity problems	Soap films at equilibrium are modeled, rather than as surfaces, as regions of small total volume through the introduction of a capillarity problem with a homotopic spanning condition. This point of view introduces a length scale in the classical Plateau's problem, which is in turn recovered in the vanishing volume limi...	\section{Introduction} \subsection{Overview}\label{section overview} The theory of minimal surfaces with prescribed boundary data provides the basic model for soap films hanging from a wire frame: given an $(n-1)$-dimensional surface $\Gamma\subset\mathbb{R}^{n+1}$ without boundary, one seeks $n$-dimensional surfaces ...	{ "timestamp": "2021-05-05T02:18:23", "yymm": "1907", "arxiv_id": "1907.00551", "language": "en", "url": "https://arxiv.org/abs/1907.00551", "abstract": "Soap films at equilibrium are modeled, rather than as surfaces, as regions of small total volume through the introduction of a capillarity problem with a ...
https://arxiv.org/abs/2207.06074	Optimal Reach Estimation and Metric Learning	We study the estimation of the reach, an ubiquitous regularity parameter in manifold estimation and geometric data analysis. Given an i.i.d. sample over an unknown $d$-dimensional $\mathcal{C}^k$-smooth submanifold of $\mathbb{R}^D$, we provide optimal nonasymptotic bounds for the estimation of its reach. We build upon...	\subsection{Geometric Inference} Topological data analysis and geometric methods now constitute a standard toolbox in statistics and machine learning~\cite{Wasserman18,Chazal21}. In this family of methods, data $\mathbb{X}_n := \{X_1,\dots,X_n\}$ are usually seen as point clouds in high dimension, for which complex st...	{ "timestamp": "2022-07-14T02:12:44", "yymm": "2207", "arxiv_id": "2207.06074", "language": "en", "url": "https://arxiv.org/abs/2207.06074", "abstract": "We study the estimation of the reach, an ubiquitous regularity parameter in manifold estimation and geometric data analysis. Given an i.i.d. sample over a...
https://arxiv.org/abs/1707.04247	On the maximum diameter of path-pairable graphs	A graph is path-pairable if for any pairing of its vertices there exist edge disjoint paths joining the vertices in each pair. We obtain sharp bounds on the maximum possible diameter of path-pairable graphs which either have a given number of edges, or are c- degenerate. Along the way we show that a large family of gra...	\section{Introduction} \emph{Path-pairability} is a graph theoretical notion that emerged from a practical networking problem introduced by Csaba, Faudree, Gy\'arf\'as, Lehel, and Schelp \cite{CS}, and further studied by Faudree, Gy\'arf\'as, and Lehel \cite{mpp,F,pp} and by Kubicka, Kubicki and Lehel \cite{grid}. Give...	{ "timestamp": "2017-07-14T02:07:51", "yymm": "1707", "arxiv_id": "1707.04247", "language": "en", "url": "https://arxiv.org/abs/1707.04247", "abstract": "A graph is path-pairable if for any pairing of its vertices there exist edge disjoint paths joining the vertices in each pair. We obtain sharp bounds on t...
https://arxiv.org/abs/1812.11169	Spherical harmonic d-tensors	Tensor harmonics are a useful mathematical tool for finding solutions to differential equations which transform under a particular representation of the rotation group $\mathrm{SO}(3)$. The aim of this work is to make use of this tool also in the setting of Finsler geometry, or more general geometries on the tangent bu...	\section{Introduction}\label{sec:intro} It is commonly understood that problems in differential geometry and its applications in physics simplify if they exhibit any symmetries, such as spherical or planar symmetries, which are most common to appear in physics via the action of a corresponding symmetry group on some un...	{ "timestamp": "2018-12-31T02:21:06", "yymm": "1812", "arxiv_id": "1812.11169", "language": "en", "url": "https://arxiv.org/abs/1812.11169", "abstract": "Tensor harmonics are a useful mathematical tool for finding solutions to differential equations which transform under a particular representation of the r...
https://arxiv.org/abs/0801.4726	Stochastic extrema as stationary phases of characteristic functions	The paper is dealing with semi-classical asymptotics of a characteristic function for a stochastic process. The main technical tool is provided by the stationary phase method. The extremal range for a stochastic process is defined by limit values of the complex logarithm of the characteristic function. The paper also o...	\section{INTRODUCTION} The extremum for a stochastic process admits transparent numerical presentation in terms of limit set of its characteristic function which is treated as a high frequency integral \cite{Guillemin}, \cite{McClure}, \cite{Maslov}, \cite{Maslov_Fedoriuk}. The proposed concept of stochastic extremum ...	{ "timestamp": "2008-01-30T18:35:40", "yymm": "0801", "arxiv_id": "0801.4726", "language": "en", "url": "https://arxiv.org/abs/0801.4726", "abstract": "The paper is dealing with semi-classical asymptotics of a characteristic function for a stochastic process. The main technical tool is provided by the stati...
https://arxiv.org/abs/1204.6530	Independent sets in hypergraphs	Many important theorems in combinatorics, such as Szemerédi's theorem on arithmetic progressions and the Erdős-Stone Theorem in extremal graph theory, can be phrased as statements about independent sets in uniform hypergraphs. In recent years, an important trend in the area has been to extend such classical results to ...	\section{Introduction} A great many of the central questions in combinatorics fall into the following general framework: Given a finite set $V$ and a collection $\mathcal{H} \subseteq \mathcal{P}(V)$ of \emph{forbidden structures}, what can be said about sets $I \subseteq V$ that do not contain any member of $\mathcal...	{ "timestamp": "2014-03-24T01:07:52", "yymm": "1204", "arxiv_id": "1204.6530", "language": "en", "url": "https://arxiv.org/abs/1204.6530", "abstract": "Many important theorems in combinatorics, such as Szemerédi's theorem on arithmetic progressions and the Erdős-Stone Theorem in extremal graph theory, can b...
https://arxiv.org/abs/1403.7920	Computing the dimension of ideals in group algebras, with an application to coding theory	The problem of computing the dimension of a left/right ideal in a group algebra F[G] of a finite group G over a field F is considered. The ideal dimension is related to the rank of a matrix originating from a regular left/right representation of G; in particular, when F[G] is semisimple, the dimension of a principal id...	\section{Introduction and preliminaries}\label{sect1} Let $\mathcal G=\{g_1,g_2, \ldots,g_n\}$ be a finite multiplicative group of order $n=\|\mathcal G\|$, with neutral element $g_1=1$. Let $\mathbb F$ be a field of characteristic $p$. Finite fields of order $q=p^m$ are denoted as $\mathbb F_q$. The group algebra...	{ "timestamp": "2014-04-01T02:13:40", "yymm": "1403", "arxiv_id": "1403.7920", "language": "en", "url": "https://arxiv.org/abs/1403.7920", "abstract": "The problem of computing the dimension of a left/right ideal in a group algebra F[G] of a finite group G over a field F is considered. The ideal dimension i...
https://arxiv.org/abs/1608.01596	Heat kernel estimates on connected sums of parabolic manifolds	We obtain matching two sided estimates of the heat kernel on a connected sum of parabolic manifolds, each of them satisfying the Li-Yau estimate. The key result is the on-diagonal upper bound of the heat kernel at a central point. Contrary to the nonparabolic case (which was settled in [15]), the on-diagonal behavior o...	\section{Introduction} \label{Introduction} Let $M$ be a Riemannian manifold. The heat kernel $p(t,x,y)$ on $M$ is the minimal positive fundamental solution of the heat equation $\partial _{t}u=\Delta u$ on $M$ where $u=u\left( t,x\right) $, $t>0$, $x\in M$ and $% \Delta $ is the (negative definite) Laplace-Beltrami...	{ "timestamp": "2016-08-05T02:09:59", "yymm": "1608", "arxiv_id": "1608.01596", "language": "en", "url": "https://arxiv.org/abs/1608.01596", "abstract": "We obtain matching two sided estimates of the heat kernel on a connected sum of parabolic manifolds, each of them satisfying the Li-Yau estimate. The key ...
https://arxiv.org/abs/2212.10759	The construction of $ε$-splitting map	For a geodesic ball with non-negative Ricci curvature and almost maximal volume, without using compactness argument, we construct an $\epsilon$-splitting map on a concentric geodesic ball with uniformly small radius. There are two new technical points in our proof. The first one is the way of finding $n$ directional po...	\section{Introduction} For a compact $n$-dimensional Riemannian manifold with $Rc\geq (n- 1)$, if its volume is close to the volume of unit round sphere $\mathbb{S}^{n}$, Colding \cite{Colding-shape} proved that the manifold is Gromov-Hausdorff close to $\mathbb{S}^{n}$. Analogue to the positive Ricci curvature case, ...	{ "timestamp": "2022-12-22T02:06:40", "yymm": "2212", "arxiv_id": "2212.10759", "language": "en", "url": "https://arxiv.org/abs/2212.10759", "abstract": "For a geodesic ball with non-negative Ricci curvature and almost maximal volume, without using compactness argument, we construct an $\\epsilon$-splitting...
https://arxiv.org/abs/1809.06296	Improvements for eigenfunction averages: An application of geodesic beams	Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{\phi_\lambda \}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-\Delta_g\phi_\lambda =\lambda^2 \phi_\lambda$. Given a smooth submanifold $H \subset M$ of codimension $k\geq 1$, we find conditions on the pair $(M,H)$, even when $H=\{x\}$, for which ...	\section{Introduction} On a smooth compact Riemannian manifold without boundary of dimension $n$, $(M,g)$, we consider sequences of Laplace eigenfunctions $\{\phi_\lambda\}$ solving \[ (-\Delta_g-\lambda^2)\phi_\lambda=0,\qquad{\\|\phi_\lambda\\|_{L^2(M)}=1.} \] We study the average oscillatory behavior of $\phi_\lambd...	{ "timestamp": "2019-03-22T01:07:21", "yymm": "1809", "arxiv_id": "1809.06296", "language": "en", "url": "https://arxiv.org/abs/1809.06296", "abstract": "Let $(M,g)$ be a smooth, compact Riemannian manifold and $\\{\\phi_\\lambda \\}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-\\Delta_g\\phi_...
https://arxiv.org/abs/2106.11573	Approximation convergence in the inverse first-passage time problem	The inverse first-passage time problem determines a boundary such that the first-passage time of a Wiener process to this boundary has a given distribution. An approximation which is based on the starting value of the boundary to a smooth boundary by a piecewise linear boundary is given by equating the probability of t...	\section{Introduction} In the theory of stochastic processes, first-passage time related problems have been studied extensively. When specified to a standard Brownian motion $(W_t)_{t \geq 0}$, the related first passage to an upper boundary continuous function $g: \mathbb{R}^+ \rightarrow \mathbb{R}$ satisfying $g(0) \...	{ "timestamp": "2021-06-23T02:12:53", "yymm": "2106", "arxiv_id": "2106.11573", "language": "en", "url": "https://arxiv.org/abs/2106.11573", "abstract": "The inverse first-passage time problem determines a boundary such that the first-passage time of a Wiener process to this boundary has a given distributio...
https://arxiv.org/abs/1610.07232	Symbolic Iterative Solution of Two-Point Boundary Value Problems	In this work we give an efficient method involving symbolic manipulation, Picard iteration, and auxiliary variables for approximating solutions of two-point boundary value problems.	\section{Introduction}\label{s:intro} There exist a variety of numerical methods for approximating solutions of two-point boundary value problems, among them shooting methods, finite difference techniques, power series methods, and variational methods, all of which are described in detail in classical texts (for examp...	{ "timestamp": "2016-10-25T02:07:02", "yymm": "1610", "arxiv_id": "1610.07232", "language": "en", "url": "https://arxiv.org/abs/1610.07232", "abstract": "In this work we give an efficient method involving symbolic manipulation, Picard iteration, and auxiliary variables for approximating solutions of two-poi...
https://arxiv.org/abs/1710.09301	The Loewner Equation for Multiple Hulls	Kager, Nienhuis, and Kadanoff conjectured that the hull generated from the Loewner equation driven by two constant functions with constant weights could be generated by a single rapidly and randomly oscillating function. We prove their conjecture and generalize to multiple continuous driving functions. In the process, ...	\section{Introduction} The Loewner equation is the initial value problem \begin{equation}\label{eqn:LEintro} \frac{\partial}{\partial t} g_t(z) =\frac{2}{g_t(z)-\lambda(t)}, \quad g_0(z)=z. \end{equation} where $\lambda:[0,T]\to\mathbb{R}$ is called the driving function. For $z\in\mathbb{H}$, a solution ex...	{ "timestamp": "2017-10-26T02:09:02", "yymm": "1710", "arxiv_id": "1710.09301", "language": "en", "url": "https://arxiv.org/abs/1710.09301", "abstract": "Kager, Nienhuis, and Kadanoff conjectured that the hull generated from the Loewner equation driven by two constant functions with constant weights could b...
https://arxiv.org/abs/math/0612552	Isomorphisms between Leavitt algebras and their matrix rings	Let $K$ be any field, let $L_n$ denote the Leavitt algebra of type $(1,n-1)$ having coefficients in $K$, and let ${\rm M}_d(L_n)$ denote the ring of $d \times d$ matrices over $L_n$. In our main result, we show that ${\rm M}_d(L_n) \cong L_n$ if and only if $d$ and $n-1$ are coprime. We use this isomorphism to answer a...	\section*{Introduction} Let $K$ be any field, and let $m<n$ be positive integers. The ring $R$ is said to have {\it invariant basis number} (IBN) if no two free left $R$-modules of differing rank over $R$ are isomorphic. On the other hand, $R$ is said to have {\it module type} $(m,n-m)$ in case for every pair of posit...	{ "timestamp": "2008-02-22T14:00:05", "yymm": "0612", "arxiv_id": "math/0612552", "language": "en", "url": "https://arxiv.org/abs/math/0612552", "abstract": "Let $K$ be any field, let $L_n$ denote the Leavitt algebra of type $(1,n-1)$ having coefficients in $K$, and let ${\\rm M}_d(L_n)$ denote the ring of ...
https://arxiv.org/abs/1112.5206	Vanishing of negative $K$-theory in positive characteristic	We show how a theorem of Gabber on alterations can be used to apply work of Cisinski, Suslin, Voevodsky, and Weibel to prove that $K_n(X)[1/p] = 0$ for $n < - \dim X$ where $X$ is a quasi-excellent noetherian scheme, $p$ is a prime that is nilpotent on $X$, and $K_n$ is the $K$-theory of Bass-Thomason-Trobaugh. This gi...	\section{Introduction} In \cite[2.9]{Wei80} Weibel asks if $K_n(X) = 0$ for $n < - \dim X$ for every noetherian scheme $X$ where $K_n$ is the $K$-theory of Bass-Thomason-Trobaugh. This question was answered in the affirmative in \cite{CHSW} for schemes essentially of finite type over a field of characteristic zero. As...	{ "timestamp": "2013-05-24T02:02:42", "yymm": "1112", "arxiv_id": "1112.5206", "language": "en", "url": "https://arxiv.org/abs/1112.5206", "abstract": "We show how a theorem of Gabber on alterations can be used to apply work of Cisinski, Suslin, Voevodsky, and Weibel to prove that $K_n(X)[1/p] = 0$ for $n <...
https://arxiv.org/abs/math/0502402	Topological fundamental groups can distinguish spaces with isomorphic homotopy groups	We exhibit a map f between aspherical spaces X and Y such that f induces an isomorphism on homotopy groups but, with natural topologies, X and Y fail to have homeomorphic fundamental groups. Thus the topological fundamental group has the capacity to distinguish homotopy type when the Whitehead theorem fails.	\section{Introduction} Given CW complexes $X$ and $Y,$ the Whitehead theorem (\cite{hatch}) asserts that a map $f:X\rightarrow Y$ is a homotopy equivalence provided $f$ induces an isomorphism on homotopy groups. However the result can fail in the context of path connected metric spaces. For example the standard Warsaw...	{ "timestamp": "2005-04-19T22:26:04", "yymm": "0502", "arxiv_id": "math/0502402", "language": "en", "url": "https://arxiv.org/abs/math/0502402", "abstract": "We exhibit a map f between aspherical spaces X and Y such that f induces an isomorphism on homotopy groups but, with natural topologies, X and Y fail ...
https://arxiv.org/abs/2301.05414	Higher order first integrals of autonomous non-Riemannian dynamical systems	We consider autonomous holonomic dynamical systems defined by equations of the form $\ddot{q}^{a}=-\Gamma_{bc}^{a}(q) \dot{q}^{b}\dot{q}^{c}$ $-Q^{a}(q)$, where $\Gamma^{a}_{bc}(q)$ are the coefficients of a symmetric (possibly non-metrical) connection and $-Q^{a}(q)$ are the generalized forces. We prove a theorem whic...	\section{Introduction} \label{sec.intro} A first integral (FI) of a second order set of dynamical equations with generalized coordinates $q^{a}$ and generalized velocities $\dot{q}^{a}\equiv \frac{dq^{a}}{dt}$ is a function $I(t,q^{a},\dot{q}^{a})$ satisfying the condition $\frac{dI}{dt}=0$ along the dynamical eq...	{ "timestamp": "2023-01-16T02:06:30", "yymm": "2301", "arxiv_id": "2301.05414", "language": "en", "url": "https://arxiv.org/abs/2301.05414", "abstract": "We consider autonomous holonomic dynamical systems defined by equations of the form $\\ddot{q}^{a}=-\\Gamma_{bc}^{a}(q) \\dot{q}^{b}\\dot{q}^{c}$ $-Q^{a}(...
https://arxiv.org/abs/math/9805098	On Dynamics of Cubic Siegel Polynomials	Motivated by the work of Douady, Ghys, Herman and Shishikura on Siegel quadratic polynomials, we study the one-dimensional slice of the cubic polynomials which have a fixed Siegel disk of rotation number theta, with theta being a given irrational number of Brjuno type. Our main goal is to prove that when theta is of bo...	\section{A Blaschke Parameter Space} \label{sec:blapar} Now we focus on a certain class of degree $5$ Blaschke products. These are the maps $B$ with the following two properties:\\ \begin{enumerate} \item[(i)] $B$ has the form \begin{equation} \label{eqn:blass} B:z\mapsto e^{2 \pi i t} z^3 \left ( \frac{z-p}{1-\ov...	{ "timestamp": "1998-05-22T00:05:37", "yymm": "9805", "arxiv_id": "math/9805098", "language": "en", "url": "https://arxiv.org/abs/math/9805098", "abstract": "Motivated by the work of Douady, Ghys, Herman and Shishikura on Siegel quadratic polynomials, we study the one-dimensional slice of the cubic polynomi...
https://arxiv.org/abs/1109.1857	Some remarks about interpolating sequences in reproducing kernel Hilbert spaces	In this paper we study two separate problems on interpolation. We first give some new equivalences of Stout's Theorem on necessary and sufficient conditions for a sequence of points to be an interpolating sequence on a finite open Riemann surface. We next turn our attention to the question of interpolation for reproduc...	\section*{Notation} \section{Introduction and Statement of Main Results} Recall that a sequence $Z=\{z_j\}\subset\mathbb{D}$ is called an $H^\infty$-\textit{interpolating} sequence if for every $a=\{a_j\}\in\ell^\infty$ there exists a function $f\in H^\infty$ such that $$ f(z_j)=a_j\quad\forall j. $$ Similarly, for t...	{ "timestamp": "2013-04-23T02:02:49", "yymm": "1109", "arxiv_id": "1109.1857", "language": "en", "url": "https://arxiv.org/abs/1109.1857", "abstract": "In this paper we study two separate problems on interpolation. We first give some new equivalences of Stout's Theorem on necessary and sufficient conditions...
https://arxiv.org/abs/math/0604457	On some properties of contracting matrices	The concepts of paracontracting, pseudocontracting and nonexpanding operators have been shown to be useful in proving convergence of asynchronous or parallel iteration algorithms. The purpose of this paper is to give characterizations of these operators when they are linear and finite-dimensional. First we show that ps...	\section{Introduction\label{sec:introduction}} \begin{definition}[\cite{nelson:paracontractive:1987}] Let $\\|\cdot\\|$ be a vector norm in ${\mathbb C}^n$. An $n$ by $n$ matrix $B$ is {\em nonexpansive} with respect to $\\|\cdot\\|$ if \begin{equation}\label{eqn:noncontractive} \forall x \in {\mathbb C}^n, \\|Bx\\|\leq \\|...	{ "timestamp": "2006-04-20T22:33:18", "yymm": "0604", "arxiv_id": "math/0604457", "language": "en", "url": "https://arxiv.org/abs/math/0604457", "abstract": "The concepts of paracontracting, pseudocontracting and nonexpanding operators have been shown to be useful in proving convergence of asynchronous or p...
https://arxiv.org/abs/1702.06068	On a problem of Pethő	In this paper we deal with a problem of Pethő related to existence of quartic algebraic integer $\alpha$ for which $$ \beta=\frac{4\alpha^4}{\alpha^4-1}-\frac{\alpha}{\alpha-1} $$ is a quadratic algebraic number. By studying rational solutions of certain Diophantine system we prove that there are infinitely many $\alph...	\section{introduction} Buchmann and Peth\H{o} \cite{BuPe} found an interesting unit in the number field $K=\mathbb{Q}(\alpha)$ with $\alpha^7-3=0$ it is as follows $$ 10+9\alpha+8\alpha^2+7\alpha^3+6\alpha^4+5\alpha^5+4\alpha^6. $$ That is the coordinates $(x_0,\ldots,x_6)\in\mathbb{Z}^7$ of a solution of the norm form...	{ "timestamp": "2017-03-16T01:05:18", "yymm": "1702", "arxiv_id": "1702.06068", "language": "en", "url": "https://arxiv.org/abs/1702.06068", "abstract": "In this paper we deal with a problem of Pethő related to existence of quartic algebraic integer $\\alpha$ for which $$ \\beta=\\frac{4\\alpha^4}{\\alpha^4...