Subset (6)

default · 38.7k rows

Split (1)

train · 38.7k rows

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.

url stringlengths 31 38	title stringlengths 7 229	abstract stringlengths 44 2.87k	text stringlengths 319 2.51M	meta dict
https://arxiv.org/abs/0912.1436	On the number of zeros of multiplicity r	Let S be a finite subset of a field. For multivariate polynomials the generalized Schwartz-Zippel bound [2], [4] estimates the number of zeros over Sx...xS counted with multiplicity. It does this in terms of the total degree, the number of variables and \|S\|. In the present work we take into account what is the leading ...	\section{Introduction}\label{secintro} In this paper we consider multivariate polynomials over an arbitrary field ${\mathbf{F}}$. Our studies focus on the zeros of given prescribed multiplicity, a concept to be defined more formally below. The definition of multiplicity that we will use relies on the Hasse derivative...	{ "timestamp": "2009-12-21T16:13:22", "yymm": "0912", "arxiv_id": "0912.1436", "language": "en", "url": "https://arxiv.org/abs/0912.1436", "abstract": "Let S be a finite subset of a field. For multivariate polynomials the generalized Schwartz-Zippel bound [2], [4] estimates the number of zeros over Sx...xS ...
https://arxiv.org/abs/1603.07261	A characterization theorem and its applications for d-orthogonality of Sheffer polynomial sets	The purpose of this paper is to find the characterization of the Sheffer polynomial sets satisfying the d-orthogonality conditions. The generating function form of these polynomial sets is given in Theorem 2.2. As applications of the Theorem 2.2, we revisit the d-orthogonal polynomial sets exist in the literature and d...	\section{Introduction} Recently, the generalization of orthogonal polynomials called "$d -orthogonal polynomials" have attracted so much attention from many authors. The well-known properties of orthogonal polynomials such as recurrence relations, Favard theorem, generating function relations and differential equat...	{ "timestamp": "2016-03-24T01:11:17", "yymm": "1603", "arxiv_id": "1603.07261", "language": "en", "url": "https://arxiv.org/abs/1603.07261", "abstract": "The purpose of this paper is to find the characterization of the Sheffer polynomial sets satisfying the d-orthogonality conditions. The generating functio...
https://arxiv.org/abs/2003.08083	Additive Representations of Natural Numbers	Every natural number greater than two may be written as the sum of a prime and a square-free number. We establish several generalisations of this, by placing divisibility conditions on the square-free number.	\section{Introduction} In 1742, Goldbach conjectured that every even integer greater than two is the sum of two primes. In 1966, Chen \cite{Chen66,Chen78} proved that every sufficiently large even integer is the sum of one prime and a product of at most two primes\footnote{Yamada \cite{Yamada2015} has shown that Chen'...	{ "timestamp": "2020-11-12T02:08:47", "yymm": "2003", "arxiv_id": "2003.08083", "language": "en", "url": "https://arxiv.org/abs/2003.08083", "abstract": "Every natural number greater than two may be written as the sum of a prime and a square-free number. We establish several generalisations of this, by plac...
https://arxiv.org/abs/1701.08504	A generalization of the practical numbers	A positive integer $n$ is practical if every $m \leq n$ can be written as a sum of distinct divisors of $n$. One can generalize the concept of practical numbers by applying an arithmetic function $f$ to each of the divisors of $n$ and asking whether all integers in a given interval can be expressed as sums of $f(d)$'s,...	\section{Introduction} Srinivasan first introduced the practical numbers as integers $n$ for which every number between $1$ and $n$ is representable as a sum of distinct divisors of $n$. In her Ph.D. thesis, the second author adapted this concept to study the degrees of divisors of $x^n-1$. Recall that $x^n - 1 = \prod...	{ "timestamp": "2017-03-24T01:02:13", "yymm": "1701", "arxiv_id": "1701.08504", "language": "en", "url": "https://arxiv.org/abs/1701.08504", "abstract": "A positive integer $n$ is practical if every $m \\leq n$ can be written as a sum of distinct divisors of $n$. One can generalize the concept of practical ...
https://arxiv.org/abs/1506.08869	Carries and the arithmetic progression structure of sets	If we want to represent integers in base $m$, we need a set $A$ of digits, which needs to be a complete set of residues modulo $m$. When adding two integers with last digits $a_1, a_2 \in A$, we find the unique $a \in A$ such that $a_1 + a_2 \equiv a$ mod $m$, and call $(a_1 + a_2 -a)/m$ the carry. Carries occur also w...	\section{Introduction} If we want to represent integers in base $m$, we need a set $A$ of digits, which needs to be a complete set of residues modulo $m$. The most popular choices are the integers in $[0, m-1]$ and the integers in $(-m/2, m/2]$. When adding two integers with last digits $a_1, a_2 \in A$, w...	{ "timestamp": "2015-07-01T02:01:33", "yymm": "1506", "arxiv_id": "1506.08869", "language": "en", "url": "https://arxiv.org/abs/1506.08869", "abstract": "If we want to represent integers in base $m$, we need a set $A$ of digits, which needs to be a complete set of residues modulo $m$. When adding two intege...
https://arxiv.org/abs/1602.08698	Equal Sums of Like Powers with Minimum Number of Terms	This paper is concerned with the diophantine system, $\sum_{i=1}^{s_1} x_i^r=\sum_{i=1}^{s_2} y_i^r,\, r=1,\,2,\,\ldots,\,k, $ where $s_1$ and $s_2$ are integers such that the total number of terms on both sides, that is, $s_1+s_2,$ is as small as possible. We define $\beta(k)$ to be the minimum value of $s_1+s_2$ for ...	\section{Introduction} \hspace*{0.25in}The Tarry-Escott problem of degree $k$ consists of finding two distinct sets of integers $\{x_1,\,x_2,\,\ldots,\,x_s\}$ and $\{y_1,\,y_2,\,\ldots,\,y_s\}$ such that \begin{equation} \sum_{i=1}^s x_i^r=\sum_{i=1}^s y_i^r,\quad r=1,\,2,\,\ldots,\,k. \label{tepsk} \end{equat...	{ "timestamp": "2016-03-01T02:11:04", "yymm": "1602", "arxiv_id": "1602.08698", "language": "en", "url": "https://arxiv.org/abs/1602.08698", "abstract": "This paper is concerned with the diophantine system, $\\sum_{i=1}^{s_1} x_i^r=\\sum_{i=1}^{s_2} y_i^r,\\, r=1,\\,2,\\,\\ldots,\\,k, $ where $s_1$ and $s_2...
https://arxiv.org/abs/2204.05250	Revisiting and improving upper bounds for identifying codes	An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C$. These codes have been widely studied for over two decades. We give an improvement over all the best known upper bounds, some of which have stood for over 20 years,...	\section{Introduction} An identifying code of a graph is a subset of its vertices that allows to distinguish all pairs of vertices by means of their neighbourhoods within the identifying code. This concept is related to other similar notions that deal with domination-based identification of the vertices/edges of a gra...	{ "timestamp": "2022-04-12T02:47:25", "yymm": "2204", "arxiv_id": "2204.05250", "language": "en", "url": "https://arxiv.org/abs/2204.05250", "abstract": "An identifying code $C$ of a graph $G$ is a dominating set of $G$ such that any two distinct vertices of $G$ have distinct closed neighbourhoods within $C...
https://arxiv.org/abs/1606.01443	Filters in the partition lattice	Given a filter $\Delta$ in the poset of compositions of $n$, we form the filter $\Pi^{}_{\Delta}$ in the partition lattice. We determine all the reduced homology groups of the order complex of $\Pi^{}_{\Delta}$ as ${\mathfrak S}_{n-1}$-modules in terms of the reduced homology groups of the simplicial complex $\Delta$...	\section{Introduction} \label{section_introduction} In his physics dissertation Sylvester~\cite{Sylvester} considered the even partition lattice, that is, the poset of all set partitions where the blocks have even size. He computed the M\"obius function of this lattice and showed that it equals, up to a sign, the tan...	{ "timestamp": "2016-06-07T02:08:30", "yymm": "1606", "arxiv_id": "1606.01443", "language": "en", "url": "https://arxiv.org/abs/1606.01443", "abstract": "Given a filter $\\Delta$ in the poset of compositions of $n$, we form the filter $\\Pi^{*}_{\\Delta}$ in the partition lattice. We determine all the reduc...
https://arxiv.org/abs/1712.09488	Positive Solutions of p-th Yamabe Type Equations on Infinite Graphs	Let $G=(V,E)$ be a connected infinite and locally finite weighted graph, $\Delta_p$ be the $p$-th discrete graph Laplacian. In this paper, we consider the $p$-th Yamabe type equation $$-\Delta_pu+h\|u\|^{p-2}u=gu^{\alpha-1}$$ on $G$, where $h$ and $g$ are known, $2<\alpha\leq p$. The prototype of this equation comes from...	\section{Introduction} Recently, the investigations of discrete weighted Laplacians and various equations on graphs have attracted much attention (cf. \cite{GLY,GLY2,GLY3,Ge1,Ge2,Ge3,Ge4,ZL}). Grigor'yan, Lin and Yang \cite{GLY3} first studied a Yamabe type equation on a finite graph $G$ as follows \begin{equa...	{ "timestamp": "2018-01-17T02:05:06", "yymm": "1712", "arxiv_id": "1712.09488", "language": "en", "url": "https://arxiv.org/abs/1712.09488", "abstract": "Let $G=(V,E)$ be a connected infinite and locally finite weighted graph, $\\Delta_p$ be the $p$-th discrete graph Laplacian. In this paper, we consider th...
https://arxiv.org/abs/2210.11270	Factorisation in the semiring of finite dynamical systems	Finite dynamical systems (FDSs) are commonly used to model systems with a finite number of states that evolve deterministically and at discrete time steps. Considered up to isomorphism, those correspond to functional graphs. As such, FDSs have a sum and product operation, which correspond to the direct sum and direct p...	\section{Introduction} Finite dynamical systems are commonly used to model systems with a finite number of states that evolve deterministically and at discrete time steps. Multiple models have been proposed for various settings, such as Boolean networks \cite{BoolNets, Goles90}, reaction systems \cite{ReactionSyste...	{ "timestamp": "2022-10-21T02:14:29", "yymm": "2210", "arxiv_id": "2210.11270", "language": "en", "url": "https://arxiv.org/abs/2210.11270", "abstract": "Finite dynamical systems (FDSs) are commonly used to model systems with a finite number of states that evolve deterministically and at discrete time steps...
https://arxiv.org/abs/2009.11069	Towards accelerated rates for distributed optimization over time-varying networks	We study the problem of decentralized optimization over time-varying networks with strongly convex smooth cost functions. In our approach, nodes run a multi-step gossip procedure after making each gradient update, thus ensuring approximate consensus at each iteration, while the outer loop is based on accelerated Nester...	\section{Introduction} In this work, we study a sum-type minimization problem \begin{align}\label{eq:initial_problem} f(x) = \frac{1}{n}\sum_{i=1}^n f_i(x) \to \min_{x\in\mathbb{R}^d}. \end{align} Convex functions $f_i$ are stored separately by nodes in communication network, which is represented by an undirec...	{ "timestamp": "2022-04-21T02:04:37", "yymm": "2009", "arxiv_id": "2009.11069", "language": "en", "url": "https://arxiv.org/abs/2009.11069", "abstract": "We study the problem of decentralized optimization over time-varying networks with strongly convex smooth cost functions. In our approach, nodes run a mul...
https://arxiv.org/abs/1603.04637	On the Randomization of Frolov's Algorithm for Multivariate Integration	We are concerned with the numerical integration of functions from the Sobolev space $H^{r,\text{mix}}([0,1]^d)$ of dominating mixed smoothness $r\in\mathbb{N}$ over the $d$-dimensional unit cube.In 1976, K. K. Frolov introduced a deterministic quadrature rule whose worst case error has the order $n^{-r} \, (\log n)^{(d...	\section{Introduction} Many applications deal with multivariate functions $f$ which are smooth in the sense that certain weak derivatives $\diff^\alpha f$ exist and are square-integrable, functions from a \textit{Sobolev space}. Which derivatives $\diff^\alpha f$ of $f$ are known to be existent and square-integrable ...	{ "timestamp": "2016-03-17T01:09:50", "yymm": "1603", "arxiv_id": "1603.04637", "language": "en", "url": "https://arxiv.org/abs/1603.04637", "abstract": "We are concerned with the numerical integration of functions from the Sobolev space $H^{r,\\text{mix}}([0,1]^d)$ of dominating mixed smoothness $r\\in\\ma...
https://arxiv.org/abs/2110.08492	Asymmetric coloring of locally finite graphs and profinite permutation groups: Tucker's Conjecture confirmed	An asymmetric coloring of a graph is a coloring of its vertices that is not preserved by any non-identity automorphism of the graph. The motion of a graph is the minimal degree of its automorphism group, i.e., the minimum number of elements displaced by any non-identity automorphism. In this paper we confirm Tom Tucker...	\section{Introduction} A graph is \emph{locally finite} if every vertex has finite degree. A graph is \emph{asymmetric} if it has no nontrivial automorphisms. A \emph{coloring} of the vertices of a graph is asymmetric if no non-identity automorphisms of the graph preserves the coloring. This author established in 197...	{ "timestamp": "2021-11-16T02:11:31", "yymm": "2110", "arxiv_id": "2110.08492", "language": "en", "url": "https://arxiv.org/abs/2110.08492", "abstract": "An asymmetric coloring of a graph is a coloring of its vertices that is not preserved by any non-identity automorphism of the graph. The motion of a graph...
https://arxiv.org/abs/1912.02225	Intrinsic Topological Transforms via the Distance Kernel Embedding	Topological transforms are parametrized families of topological invariants, which, by analogy with transforms in signal processing, are much more discriminative than single measurements. The first two topological transforms to be defined were the Persistent Homology Transform and Euler Characteristic Transform, both of...	\section{Introduction} \input{intro} \section{The Distance Kernel} \label{sec:distkernel} \input{distkernel} \section{The Distance Kernel Embedding} \label{sec:distkernelembed} \input{distkernelembed} \section{Finite Approximation of the Distance Kernel Embedding} \label{sec:discretization} \input{dis...	{ "timestamp": "2020-04-01T02:03:23", "yymm": "1912", "arxiv_id": "1912.02225", "language": "en", "url": "https://arxiv.org/abs/1912.02225", "abstract": "Topological transforms are parametrized families of topological invariants, which, by analogy with transforms in signal processing, are much more discrimi...
https://arxiv.org/abs/1809.08554	An explicit solution for a multimarginal mass transportation problem	We construct an explicit solution for the multimarginal transportation problem on the unit cube $[0,1]^3$ with the cost function $xyz$ and one-dimensional uniform projections. We show that the primal problem is concentrated on a set with non-constant local dimension and admits many solutions, whereas the solution to th...	\section{Introduction} \subsection{Notation} Assume we are given $n$ polish spaces $X_1$, $X_2$, \dots, $X_n$, equipped with probability measures $\mu_i$ on $X_i$ and a cost function $c: X_1 \times \dots \times X_n \to \mathbb{R}$. In multimarginal Monge-Kantorovich problem (called primal problem throughout this pape...	{ "timestamp": "2020-02-25T02:10:29", "yymm": "1809", "arxiv_id": "1809.08554", "language": "en", "url": "https://arxiv.org/abs/1809.08554", "abstract": "We construct an explicit solution for the multimarginal transportation problem on the unit cube $[0,1]^3$ with the cost function $xyz$ and one-dimensional...
https://arxiv.org/abs/1607.00854	Lecture Notes on the ARV Algorithm for Sparsest Cut	One of the landmarks in approximation algorithms is the $O(\sqrt{\log n})$-approximation algorithm for the Uniform Sparsest Cut problem by Arora, Rao and Vazirani from 2004. The algorithm is based on a semidefinite program that finds an embedding of the nodes respecting the triangle inequality. Their core argument show...	\section{Introduction} Let $G = (V,E)$ be a complete, undirected graph on $\|V\| = n$ nodes and let $c : E \to \setR_{\geq 0}$ be a cost function on the edges. For a subset $S \subseteq V$ of nodes, let $\delta(S) := \{ \{ i,j\} \in E \mid \|\{ i,j\} \cap S\| = 1\}$ be the induced \emph{cut}. We abbreviate $c(\delta(S)) :...	{ "timestamp": "2016-07-05T02:13:18", "yymm": "1607", "arxiv_id": "1607.00854", "language": "en", "url": "https://arxiv.org/abs/1607.00854", "abstract": "One of the landmarks in approximation algorithms is the $O(\\sqrt{\\log n})$-approximation algorithm for the Uniform Sparsest Cut problem by Arora, Rao an...
https://arxiv.org/abs/2012.07221	Intersecting longest paths in chordal graphs	We consider the size of the smallest set of vertices required to intersect every longest path in a chordal graph. Such sets are known as longest path transversals. We show that if $\omega(G)$ is the clique number of a chordal graph $G$, then there is a transversal of order at most $4\lceil\frac{\omega(G)}{5}\rceil$. We...	\section{Introduction} In this paper we consider the paths of maximum length among all paths in a graph. As longest paths represent the most indirect way one could travel through the graph from some place to another, we will adopt the terminology of Kapoor et al.~\cite{KapoorEtal,JobsonEtal} and refer to them as \emp...	{ "timestamp": "2020-12-15T02:27:31", "yymm": "2012", "arxiv_id": "2012.07221", "language": "en", "url": "https://arxiv.org/abs/2012.07221", "abstract": "We consider the size of the smallest set of vertices required to intersect every longest path in a chordal graph. Such sets are known as longest path tran...
https://arxiv.org/abs/1011.6646	Sparse random graphs: Eigenvalues and Eigenvectors	In this paper we prove the semi-circular law for the eigenvalues of regular random graph $G_{n,d}$ in the case $d\rightarrow \infty$, complementing a previous result of McKay for fixed $d$. We also obtain a upper bound on the infinity norm of eigenvectors of Erdős-Rényi random graph $G(n,p)$, answering a question raise...	\section{Introduction} \subsection{Overview} In this paper, we consider two models of random graphs, the Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ and the random regular graph $G_{n,d}$. Given a real number $p=p(n)$,$0\le p\le 1$, the Erd\H{o}s-R\'{e}nyi graph on a vertex set of size $n$ is obtained by drawing an ...	{ "timestamp": "2010-12-01T02:02:55", "yymm": "1011", "arxiv_id": "1011.6646", "language": "en", "url": "https://arxiv.org/abs/1011.6646", "abstract": "In this paper we prove the semi-circular law for the eigenvalues of regular random graph $G_{n,d}$ in the case $d\\rightarrow \\infty$, complementing a prev...
https://arxiv.org/abs/1609.02181	Geometry and a natural symplectic structure of phase tropical hypersurfaces	First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in $(\mathbb{C}^*)^n$. Next, we prove that complex hyperplanes are diffeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin's decomposition into pairs-of-...	\section{introduction} In this paper we deal with smooth algebraic hypersurfaces in the complex projective space $\mathbb{CP}^n$. So, let $V$ be a smooth hypersurface in $\mathbb{CP}^n$ of degree $d$. Recall that for a fixed degree, generically a hypersurface in the projective space is smooth and transverse to all...	{ "timestamp": "2016-09-09T02:00:39", "yymm": "1609", "arxiv_id": "1609.02181", "language": "en", "url": "https://arxiv.org/abs/1609.02181", "abstract": "First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in $(\\mathbb{C}^*)^n$. Next, we p...
https://arxiv.org/abs/2209.09379	Characterization of Graphs With Failed Skew Zero Forcing Number of 1	Given a graph $G$, the zero forcing number of $G$, $Z(G)$, is the smallest cardinality of any set $S$ of vertices on which repeated applications of the forcing rule results in all vertices being in $S$. The forcing rule is: if a vertex $v$ is in $S$, and exactly one neighbor $u$ of $v$ is not in $S$, then $u$ is added ...	\section{Introduction} Given a graph $G$, the zero forcing number of $G$, $Z(G)$, is the smallest cardinality of any set $S$ of vertices on which repeated applications of the forcing rule results in all vertices being in $S$. The forcing rule is: if a vertex $v$ is in $S$, and exactly one neighbor $u$ of $v$ is not in ...	{ "timestamp": "2022-09-21T02:05:59", "yymm": "2209", "arxiv_id": "2209.09379", "language": "en", "url": "https://arxiv.org/abs/2209.09379", "abstract": "Given a graph $G$, the zero forcing number of $G$, $Z(G)$, is the smallest cardinality of any set $S$ of vertices on which repeated applications of the fo...
https://arxiv.org/abs/1109.2934	On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime	Given a finite family F of linear forms with integer coefficients, and a compact abelian group G, an F-free set in G is a measurable set which does not contain solutions to any equation L(x)=0 for L in F. We denote by d_F(G) the supremum of m(A) over F-free sets A in G, where m is the normalized Haar measure on G. Our ...	\section{Introduction} Much work in arithmetic combinatorics concerns the maximal density that a subset of a finite abelian group can have if it does not contain a non-trivial solution to a given linear equation. Examples include the study of sum-free sets, where the equation to be avoided is $x_1+x_2-x_3=0$, and the ...	{ "timestamp": "2011-09-15T02:00:37", "yymm": "1109", "arxiv_id": "1109.2934", "language": "en", "url": "https://arxiv.org/abs/1109.2934", "abstract": "Given a finite family F of linear forms with integer coefficients, and a compact abelian group G, an F-free set in G is a measurable set which does not cont...
https://arxiv.org/abs/1001.4169	Stolarsky's conjecture and the sum of digits of polynomial values	Let $s_q(n)$ denote the sum of the digits in the $q$-ary expansion of an integer $n$. In 1978, Stolarsky showed that $$ \liminf_{n\to\infty} \frac{s_2(n^2)}{s_2(n)} = 0. $$ He conjectured that, as for $n^2$, this limit infimum should be 0 for higher powers of $n$. We prove and generalize this conjecture showing that fo...	\section{Introduction} Let $q\geq 2$ and denote by $s_q(n)$ the sum of digits in the $q$-ary representation of an integer $n$. In recent years, much effort has been made to get a better understanding of the distribution properties of $s_q$ regarding certain subsequences of the positive integers. We mention the ground-...	{ "timestamp": "2010-01-23T17:27:11", "yymm": "1001", "arxiv_id": "1001.4169", "language": "en", "url": "https://arxiv.org/abs/1001.4169", "abstract": "Let $s_q(n)$ denote the sum of the digits in the $q$-ary expansion of an integer $n$. In 1978, Stolarsky showed that $$ \\liminf_{n\\to\\infty} \\frac{s_2(n...
https://arxiv.org/abs/1205.4416	On the Local-Global Conjecture for integral Apollonian gaskets	We prove that a set of density one satisfies the local-global conjecture for integral Apollonian gaskets. That is, for a fixed integral, primitive Apollonian gasket, almost every (in the sense of density) admissible (passing local obstructions) integer is the curvature of some circle in the gasket.	\section{Introduction} \begin{figure} \includegraphics[width 2in]{pic.pdf} \vskip-1.25in \hskip-2.5in {\Huge -11} \vskip1in \caption{The Apollonian gasket with root quadruple $v_{0}=(-11, 21,24,28)^{t}$.} \label{fig1} \end{figure} \subsection{The Local-Global Conjecture}\ Let $\sG$ be an Apollonian gasket, see Fig...	{ "timestamp": "2013-05-15T02:00:46", "yymm": "1205", "arxiv_id": "1205.4416", "language": "en", "url": "https://arxiv.org/abs/1205.4416", "abstract": "We prove that a set of density one satisfies the local-global conjecture for integral Apollonian gaskets. That is, for a fixed integral, primitive Apollonia...
https://arxiv.org/abs/1607.08865	Law of Iterated Logarithm for random graphs	A milestone in Probability Theory is the law of the iterated logarithm (LIL), proved by Khinchin and independently by Kolmogorov in the 1920s, which asserts that for iid random variables $\{t_i\}_{i=1}^{\infty}$ with mean $0$ and variance $1$$$ \Pr \left[ \limsup_{n\rightarrow \infty} \frac{ \sum_{i=1}^n t_i }{\sigma_n...	\section{Introduction} Let $\{t_i\}_{i=1}^{\infty}$ be an infinite sequence of iid random variables with mean $0$ and variance $1$. Two key results in probability theory are the central limit theorem and the law of the iterated logarithm. The \emph{central limit theorem} (CLT) states that for $X_n:= \sum_{i=1}^n t_i$,...	{ "timestamp": "2017-10-12T02:02:51", "yymm": "1607", "arxiv_id": "1607.08865", "language": "en", "url": "https://arxiv.org/abs/1607.08865", "abstract": "A milestone in Probability Theory is the law of the iterated logarithm (LIL), proved by Khinchin and independently by Kolmogorov in the 1920s, which asser...
https://arxiv.org/abs/1009.4913	Equivalence of concentration inequalities for linear and non-linear functions	We consider a random variable $X$ that takes values in a (possibly infinite-dimensional) topological vector space $\mathcal{X}$. We show that, with respect to an appropriate "normal distance" on $\mathcal{X}$, concentration inequalities for linear and non-linear functions of $X$ are equivalent. This normal distance cor...	\section{Introduction} \label{sec:intro} It is by now almost classical that smooth enough convex functions enjoy good concentration properties; see \emph{e.g.}\ \cite{Ledoux:2001} \cite{Lugosi:2009} \cite{McDiarmid:1998} \cite{MilmanSchechtman:1986} for surveys of the literature. It is also known that convexity can ...	{ "timestamp": "2010-09-27T02:02:24", "yymm": "1009", "arxiv_id": "1009.4913", "language": "en", "url": "https://arxiv.org/abs/1009.4913", "abstract": "We consider a random variable $X$ that takes values in a (possibly infinite-dimensional) topological vector space $\\mathcal{X}$. We show that, with respect...
https://arxiv.org/abs/1905.06366	Equivalence and invariance of the chi and Hoffman constants of a matrix	We show that the following two condition measures of a full column rank matrix $A \in \mathbb{R}^{m\times n}$ are identical: the chi constant and a signed Hoffman constant. This identity is naturally suggested by the evident invariance of the chi constant under sign changes of the rows of $A$. We also show that similar...	\section{Introduction} \label{sec.intro} We show a novel equivalence between the following two condition measures of a matrix that play central roles in numerical linear algebra and in convex optimization: the chi measure~\cite{BenTT90,Diki74,Fors96,Stew89,Todd90} and the Hoffman constant~\cite{Hoff52,guler1995,KlatT9...	{ "timestamp": "2020-05-19T02:33:29", "yymm": "1905", "arxiv_id": "1905.06366", "language": "en", "url": "https://arxiv.org/abs/1905.06366", "abstract": "We show that the following two condition measures of a full column rank matrix $A \\in \\mathbb{R}^{m\\times n}$ are identical: the chi constant and a sig...
https://arxiv.org/abs/1501.01685	Spaces of regular abstract martingales	In \cite{Troitsky:05,Korostenski:08}, the authors introduced and studied the space $\mathcal M_r$ of regular martingales on a vector lattice and the space $M_r$ of bounded regular martingales on a Banach lattice. In this note, we study these two spaces from the vector lattice point of view. We show, in particular, that...	\section{The space of regular martingales on a vector lattice} Let $F$ be a vector lattice. A sequence $(E_n)$ of positive projections on $F$ such that $E_nE_m=E_{n\wedge m}$ is said to be a \term{filtration}. We will try to impose as few additional assumptions on the filtration as possible. A sequence $X=(x_n)$ in ...	{ "timestamp": "2015-01-09T02:04:35", "yymm": "1501", "arxiv_id": "1501.01685", "language": "en", "url": "https://arxiv.org/abs/1501.01685", "abstract": "In \\cite{Troitsky:05,Korostenski:08}, the authors introduced and studied the space $\\mathcal M_r$ of regular martingales on a vector lattice and the spa...
https://arxiv.org/abs/math/0601143	L-functions and higher order modular forms	It is believed that Dirichlet series with a functional equation and Euler product of a particular form are associated to holomorphic newforms on a Hecke congruence group. We perform computer algebra experiments which find that in certain cases one can associate a kind of ``higher order modular form'' to such Dirichlet ...	\section{Introduction} We investigate the relationship between degree-2 $L$-functions and modular forms. We find that degree-2 $L$-functions can be associated to functions on the upper half-plane which have similar properties to ``second order modular forms.'' Since it is conjectured that degree-2 $L$-functions can...	{ "timestamp": "2006-01-07T20:54:52", "yymm": "0601", "arxiv_id": "math/0601143", "language": "en", "url": "https://arxiv.org/abs/math/0601143", "abstract": "It is believed that Dirichlet series with a functional equation and Euler product of a particular form are associated to holomorphic newforms on a Hec...
https://arxiv.org/abs/2008.00584	Optimal rates of convergence and error localization of Gegenbauer projections	Motivated by comparing the convergence behavior of Gegenbauer projections and best approximations, we study the optimal rate of convergence for Gegenbauer projections in the maximum norm. We show that the rate of convergence of Gegenbauer projections is the same as that of best approximations under conditions of the un...	\section{Introduction}\label{sec:introduction} Orthogonal polynomial approximations, such as Gegenbauer as well as Legendre and Chebyshev approximations, play an important role in many branches of numerical analysis, including function approximations and quadrature \cite{davis1984methods,trefethen2013atap}, the re...	{ "timestamp": "2020-08-04T02:26:43", "yymm": "2008", "arxiv_id": "2008.00584", "language": "en", "url": "https://arxiv.org/abs/2008.00584", "abstract": "Motivated by comparing the convergence behavior of Gegenbauer projections and best approximations, we study the optimal rate of convergence for Gegenbauer...
https://arxiv.org/abs/2212.01211	Sometimes Two Irrational Guards are Needed	In the art gallery problem, we are given a closed polygon $P$, with rational coordinates and an integer $k$. We are asked whether it is possible to find a set (of guards) $G$ of size $k$ such that any point $p\in P$ is seen by a point in $G$. We say two points $p$, $q$ see each other if the line segment $pq$ is contain...	\section{Introduction} In the art gallery problem, we are given a closed polygon $P$, on $n$ vertices, with rational coordinates and an integer $k$. We are asked whether it is possible to find a set (of guards) $G$ of size $k$ such that any point $p\in P$ is seen by a point in $G$. We...	{ "timestamp": "2022-12-05T02:13:36", "yymm": "2212", "arxiv_id": "2212.01211", "language": "en", "url": "https://arxiv.org/abs/2212.01211", "abstract": "In the art gallery problem, we are given a closed polygon $P$, with rational coordinates and an integer $k$. We are asked whether it is possible to find a...
https://arxiv.org/abs/1812.00114	The Duality between Ideals of Multilinear Operators and Tensor Norms	We develop the duality theory between ideals of multilinear operators and tensor norms that arises from the geometric approach of $\Sigma$-operators. To this end, we introduce and develop the notions of $\Sigma$-ideals of multilinear operators and $\Sigma$-tensor norms. We establish the foundations of this theory by pr...	\section{Introduction} The theory of operator ideals, mainly developed by Pietsch in \cite{pietsch78}, is a branch of Functional Analysis that provides a systematic framework to study linear operators grouping them according to the so called ideal property. Many examples of operator ideals find applications in a wide...	{ "timestamp": "2018-12-04T02:05:50", "yymm": "1812", "arxiv_id": "1812.00114", "language": "en", "url": "https://arxiv.org/abs/1812.00114", "abstract": "We develop the duality theory between ideals of multilinear operators and tensor norms that arises from the geometric approach of $\\Sigma$-operators. To ...
https://arxiv.org/abs/2105.05732	Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control	In a separable Hilbert space $X$, we study the controlled evolution equation \begin{equation} u'(t)+Au(t)+p(t)Bu(t)=0, \end{equation} where $A\geq-\sigma I$ ($\sigma\geq0$) is a self-adjoint linear operator, $B$ is a bounded linear operator on $X$, and $p\in L^2_{loc}(0,+\infty)$ is a bilinear control.We give suffici...	\section{Introduction} In a separable Hilbert space $X$ consider the nonlinear control system \begin{equation}\label{u} \left\{ \begin{array}{ll} u'(t)+Au(t)+p(t)Bu(t)=0,& t>0\\\\ u(0)=u_0. \end{array}\right. \end{equation} where $A:D(A)\subset X\to X$ is a linear self-adjoint operator on $X$ such that $A\geq-\sigma I$...	{ "timestamp": "2021-05-13T02:20:59", "yymm": "2105", "arxiv_id": "2105.05732", "language": "en", "url": "https://arxiv.org/abs/2105.05732", "abstract": "In a separable Hilbert space $X$, we study the controlled evolution equation \\begin{equation} u'(t)+Au(t)+p(t)Bu(t)=0, \\end{equation} where $A\\geq-\\...
https://arxiv.org/abs/0809.0918	Intersecting random graphs and networks with multiple adjacency constraints: A simple example	When studying networks using random graph models, one is sometimes faced with situations where the notion of adjacency between nodes reflects multiple constraints. Traditional random graph models are insufficient to handle such situations.A simple idea to account for multiple constraints consists in taking the intersec...	\section{Introduction} \label{sec:Introduction} Graphs provide simple and useful representations for networks with the presence of an edge between a pair of nodes marking their ability to communicate with each other. Thus, for some set $V$ of nodes, an undirected graph $G \equiv (V,E)$ with edge set $E$ is defined ...	{ "timestamp": "2008-09-04T23:13:14", "yymm": "0809", "arxiv_id": "0809.0918", "language": "en", "url": "https://arxiv.org/abs/0809.0918", "abstract": "When studying networks using random graph models, one is sometimes faced with situations where the notion of adjacency between nodes reflects multiple const...
https://arxiv.org/abs/2107.14054	Detecting and diagnosing prior and likelihood sensitivity with power-scaling	Determining the sensitivity of the posterior to perturbations of the prior and likelihood is an important part of the Bayesian workflow. We introduce a practical and computationally efficient sensitivity analysis approach using importance sampling to estimate properties of posteriors resulting from power-scaling the pr...	\section{Introduction} \label{sec:intro} Bayesian inference is characterised by the derivation of a posterior from a prior and a likelihood. As the posterior is dependent on the specification of these two components, investigating its sensitivity to perturbations of the prior and likelihood is a critical step in the...	{ "timestamp": "2022-05-06T02:25:01", "yymm": "2107", "arxiv_id": "2107.14054", "language": "en", "url": "https://arxiv.org/abs/2107.14054", "abstract": "Determining the sensitivity of the posterior to perturbations of the prior and likelihood is an important part of the Bayesian workflow. We introduce a pr...
https://arxiv.org/abs/1908.10684	Downlink Analysis for the Typical Cell in Poisson Cellular Networks	Owing to its unparalleled tractability, the Poisson point process (PPP) has emerged as a popular model for the analysis of cellular networks. Considering a stationary point process of users, which is independent of the base station (BS) point process, it is well known that the typical user does not lie in the typical c...	\section{Introduction} \label{sec:Introduction} The previous decade has witnessed a significant growth in research efforts related to the modeling and analysis of cellular networks using stochastic geometry. A vast majority of these works, e.g., \cite{AndBacJ2011,DhiGanJ2012}, rely on the homogeneous PPP model for the...	{ "timestamp": "2019-08-29T02:12:15", "yymm": "1908", "arxiv_id": "1908.10684", "language": "en", "url": "https://arxiv.org/abs/1908.10684", "abstract": "Owing to its unparalleled tractability, the Poisson point process (PPP) has emerged as a popular model for the analysis of cellular networks. Considering ...
https://arxiv.org/abs/1912.11063	Does scrambling equal chaos?	Focusing on semiclassical systems, we show that the parametrically long exponential growth of out-of-time order correlators (OTOCs), also known as scrambling, does not necessitate chaos. Indeed, scrambling can simply result from the presence of unstable fixed points in phase space, even in a classically integrable mode...	\section{Saddle-dominated scrambling in Dicke model}\label{sec:dicke} \begin{figure} \centering \includegraphics[width=.8\columnwidth]{dicke.pdf} \caption{(a) The OTOC in the quantum Dicke model ($N = 40$) with $\hat{O} = \hat{p}$, for a microcanonical ensemble of 40 eigenstates around $H = -1$, and for two...	{ "timestamp": "2020-04-09T02:03:31", "yymm": "1912", "arxiv_id": "1912.11063", "language": "en", "url": "https://arxiv.org/abs/1912.11063", "abstract": "Focusing on semiclassical systems, we show that the parametrically long exponential growth of out-of-time order correlators (OTOCs), also known as scrambl...
https://arxiv.org/abs/2004.05177	Dissipation-assisted operator evolution method for capturing hydrodynamic transport	We introduce the dissipation-assisted operator evolution (DAOE) method for calculating transport properties of strongly interacting lattice systems in the high temperature regime. DAOE is based on evolving observables in the Heisenberg picture, and applying an artificial dissipation that reduces the weight on non-local...	\section{Additional data for the Ising chain and XX ladder models} \subsection{Convergence with bond dimension} In the main text, we showed that the dissipation leads to a decay of the operator entanglement at long times. Crucially, this makes the maximal operator entanglement encountered during the evolution indepe...	{ "timestamp": "2020-04-14T02:00:16", "yymm": "2004", "arxiv_id": "2004.05177", "language": "en", "url": "https://arxiv.org/abs/2004.05177", "abstract": "We introduce the dissipation-assisted operator evolution (DAOE) method for calculating transport properties of strongly interacting lattice systems in the...
https://arxiv.org/abs/1703.04973	Real interpolation with variable exponent	We present the real interpolation with variable exponent and we prove the basic properties in analogy to the classical real interpolation. More precisely, we prove that under some additional conditions, this method can be reduced to the case of fixed exponent. An application, we give the real interpolation of variable ...	\section*{{\normalsize \bf #2}}\list {[\arabic{enumi}]}{\settowidth\labelwidth{[#1]}\leftmargin\labelwidth \advance\leftmargin\labelsep \usecounter{enumi}} \def\newblock{\hskip .11em plus .33em minus -.07em} \sloppy \sfcode`\.=1000\relax} \let\endthebibliograph=\endlist \newcommand{\...	{ "timestamp": "2017-03-16T01:07:59", "yymm": "1703", "arxiv_id": "1703.04973", "language": "en", "url": "https://arxiv.org/abs/1703.04973", "abstract": "We present the real interpolation with variable exponent and we prove the basic properties in analogy to the classical real interpolation. More precisely,...
https://arxiv.org/abs/2004.11160	Calculating permutation entropy without permutations	A method for analyzing sequential data sets, similar to the permutation entropy one, is discussed. The characteristic features of this method are as follows: it preserves information about equal values, if any, in the embedding vectors; it is exempt of combinatorics; it delivers the same entropy value as does the permu...	\section{Introduction} Due to technical progress in the areas of sensors and storage devices a huge amount of raw data about time course of different processes, such as ECG, EEG, climate data recordings, stock market data have become available. These data are redundant. The data processing and classification, aimed a...	{ "timestamp": "2020-08-14T02:11:58", "yymm": "2004", "arxiv_id": "2004.11160", "language": "en", "url": "https://arxiv.org/abs/2004.11160", "abstract": "A method for analyzing sequential data sets, similar to the permutation entropy one, is discussed. The characteristic features of this method are as follo...
https://arxiv.org/abs/1002.3934	Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics	We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces, prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces, and prove t...	\section{Introduction} \subsection{Definitions and the statement of the problem} Consider a pseudo-Riemmanian metric $g=(g_{ij})$ on a surface $M^2$. A function $F:{T^}M\to \mathbb{R}$ is called {\it an integral} of the geodesic flow of $g$, if $\{ H, F\}=0$, where $H:= \tfrac{1}{2} \sum_{i,j} g^{ij} p_ip_j:T^...	{ "timestamp": "2010-02-20T20:34:24", "yymm": "1002", "arxiv_id": "1002.3934", "language": "en", "url": "https://arxiv.org/abs/1002.3934", "abstract": "We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we sol...