url stringlengths 31 38 | title stringlengths 7 229 | abstract stringlengths 44 2.87k | text stringlengths 319 2.51M | meta dict |
|---|---|---|---|---|
https://arxiv.org/abs/1710.03745 | Erdos-Hajnal conjecture for graphs with bounded VC-dimension | The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every $n$-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least $e^{(\log n)^{1 - o(1)}}$. The depen... | \section{Introduction}
During the relatively short history of computational geometry, there were many breakthroughs that originated from results in extremal combinatorics \cite{GRT17}. Range searching turned out to be closely related to discrepancy theory \cite{Ch00}, linear programming to McMullen's Upper Bound the... | {
"timestamp": "2017-10-11T02:11:47",
"yymm": "1710",
"arxiv_id": "1710.03745",
"language": "en",
"url": "https://arxiv.org/abs/1710.03745",
"abstract": "The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its... |
https://arxiv.org/abs/2108.11536 | Factorizations in evaluation monoids of Laurent semirings | For a positive real number $\alpha$, let $\mathbb{N}_0[\alpha,\alpha^{-1}]$ be the semiring of all real numbers $f(\alpha)$ for $f(x)$ lying in $\mathbb{N}_0[x,x^{-1}]$, which is the semiring of all Laurent polynomials over the set of nonnegative integers $\mathbb{N}_0$. In this paper, we study various factorization pr... | \section{Introduction}
The purpose of this paper is to understand the (additive) factorization properties of the commutative semirings $\mathbb{N}_0[\alpha, \alpha^{-1}]$ for any $\alpha \in \mathbb{R}_{> 0}$. To be more precise, let $\mathbb{N}_0[x,x^{-1}]$ denote the set of Laurent polynomials with coefficients in... | {
"timestamp": "2021-08-27T02:06:50",
"yymm": "2108",
"arxiv_id": "2108.11536",
"language": "en",
"url": "https://arxiv.org/abs/2108.11536",
"abstract": "For a positive real number $\\alpha$, let $\\mathbb{N}_0[\\alpha,\\alpha^{-1}]$ be the semiring of all real numbers $f(\\alpha)$ for $f(x)$ lying in $\\ma... |
https://arxiv.org/abs/2108.08299 | Restricted Dyck Paths on Valleys Sequence | In this paper we study a subfamily of a classic lattice path, the \emph{Dyck paths}, called \emph{restricted $d$-Dyck} paths, in short $d$-Dyck. A valley of a Dyck path $P$ is a local minimum of $P$; if the difference between the heights of two consecutive valleys (from left to right) is at least $d$, we say that $P$ i... | \section{Introduction}
A classic concept, the \emph{Dyck paths}, has been widely studied. Recently, a subfamily of these paths, non-decreasing Dyck paths, has received certain
level of interest due to the good behavior of its recursive relations and generating functions. In this paper we keep
studying a generalizatio... | {
"timestamp": "2021-08-20T02:00:13",
"yymm": "2108",
"arxiv_id": "2108.08299",
"language": "en",
"url": "https://arxiv.org/abs/2108.08299",
"abstract": "In this paper we study a subfamily of a classic lattice path, the \\emph{Dyck paths}, called \\emph{restricted $d$-Dyck} paths, in short $d$-Dyck. A valle... |
https://arxiv.org/abs/1907.01634 | On some P-Q mixed modular equations of degree 5 | In his second notebook, Ramanujan recorded total of 23 P-Q modular equations involving theta-functions $f(-q)$, $\varphi(q)$ and $\psi(q)$. In this paper, modular equations analogous to those recorded by Ramanujan are obtained involving $f(-q)$. As a consequence, values of certain quotients of theta-function are evalua... | \section{Introduction}
For $|q|<1,$ let $(a;q)_\infty$
denote the infinite product $\displaystyle \prod_{n=0}^\infty(1-aq^{n})$, where $a$, $q $ are complex numbers and $f(a,b)$ be the Ramanujan theta-function:
\begin{equation*}
f(a,b):=\sum_{n=-\infty}^{\infty}a^{n(n+1)/2} b^{n(n-1)/2},\,\,\,|ab|<1,
\end{equation... | {
"timestamp": "2019-08-27T02:22:55",
"yymm": "1907",
"arxiv_id": "1907.01634",
"language": "en",
"url": "https://arxiv.org/abs/1907.01634",
"abstract": "In his second notebook, Ramanujan recorded total of 23 P-Q modular equations involving theta-functions $f(-q)$, $\\varphi(q)$ and $\\psi(q)$. In this pape... |
https://arxiv.org/abs/1406.7229 | Dimension-Free $L^p$-Maximal Inequalities in $\mathbb{Z}_{m+1}^N$ | For $m \geq 2$, let $(\mathbb{Z}_{m+1}^N, |\cdot|)$ denote the group equipped with the so-called $l^0$ metric,\[ |y| = \left| \big( y(1), \dots, y(N) \big) \right| := | \{1 \leq i \leq N : y(i) \neq 0 \} |,\] and define the $L^1$-normalized indicator of the $r$-sphere, \[ \sigma_r := \frac{1}{|\{|x| = r\}|} 1_{\{|x| =r... | \section{Introduction}\label{intro}
In $\RR^N$, let
\[ M_{B}^{\RR^N}f(x) := \sup_{r > 0} \frac{c_N}{r^N} \int_{|y| \leq r} |f(x-y)| \ dy,\]
denote the standard Hardy-Littlewood maximal function, where $c_N^{-1}$ is the volume of the $N$-dimensional Euclidean unit ball.
A celebrated result of Stein and Str\"{o}mbe... | {
"timestamp": "2014-12-02T02:07:00",
"yymm": "1406",
"arxiv_id": "1406.7229",
"language": "en",
"url": "https://arxiv.org/abs/1406.7229",
"abstract": "For $m \\geq 2$, let $(\\mathbb{Z}_{m+1}^N, |\\cdot|)$ denote the group equipped with the so-called $l^0$ metric,\\[ |y| = \\left| \\big( y(1), \\dots, y(N)... |
https://arxiv.org/abs/1908.10348 | Characterisation of the weak-star symmetric strong diameter 2 property in Lipschitz spaces | We give a characterisation of the weak* symmetric strong diameter 2 property for Lipschitz function spaces in terms of a property of the corresponding metric space. Using this characterisation we show that the weak* symmetric strong diameter 2 property is different from the weak* strong diameter 2 property in Lipschitz... | \section{Introduction}
We consider only real Banach spaces. We start by fixing some notation. Given a metric space $M$ and a point $x$ in $M$, we denote by $B(x,r)$ the open ball in $M$ centered at $x$ of radius $r$. Let $X$ be a Banach space. We denote the closed unit ball, the unit sphere, and the dual space of $X$ b... | {
"timestamp": "2019-08-28T02:19:43",
"yymm": "1908",
"arxiv_id": "1908.10348",
"language": "en",
"url": "https://arxiv.org/abs/1908.10348",
"abstract": "We give a characterisation of the weak* symmetric strong diameter 2 property for Lipschitz function spaces in terms of a property of the corresponding met... |
https://arxiv.org/abs/2109.01735 | Enumerating $k$-Naples Parking Functions Through Catalan Objects | This paper studies a generalization of parking functions named $k$-Naples parking functions, where backward movement is allowed. One consequence of backward movement is that the number of ascending $k$-Naples is not the same as the number of descending $k$-Naples. This paper focuses on generalizing the bijections of as... | \section{Introduction}\label{sec:Introduction}
Parking functions are special types of integer sequences that were proposed independently by Ronald Pyke \cite{Pyke} as well as by Alan Konheim and Benjamin Weiss \cite{KonheimAndWeiss} in order to study hashing problems in computer science.
If we have a sequence of $n$ i... | {
"timestamp": "2021-09-07T02:03:44",
"yymm": "2109",
"arxiv_id": "2109.01735",
"language": "en",
"url": "https://arxiv.org/abs/2109.01735",
"abstract": "This paper studies a generalization of parking functions named $k$-Naples parking functions, where backward movement is allowed. One consequence of backwa... |
https://arxiv.org/abs/2109.01930 | Geometric bijections between spanning subgraphs and orientations of a graph | Let $G$ be a connected finite graph. Backman, Baker, and Yuen have constructed a family of explicit and easy-to-describe bijections $g_{\sigma,\sigma^*}$ between spanning trees of $G$ and $(\sigma,\sigma^*)$-compatible orientations, where the $(\sigma,\sigma^*)$-compatible orientations are the representatives of equiva... | \section{Introduction}\label{intro}
\subsection{Introduction to the main combinatorial results}
Let $G$ be a connected finite graph and let $E$ be its edge set.
This paper examines correspondence between spanning subgraphs and orientations of $G$. Obviously, the number of spanning subgraphs of $G$ equals the number of ... | {
"timestamp": "2021-10-18T02:06:20",
"yymm": "2109",
"arxiv_id": "2109.01930",
"language": "en",
"url": "https://arxiv.org/abs/2109.01930",
"abstract": "Let $G$ be a connected finite graph. Backman, Baker, and Yuen have constructed a family of explicit and easy-to-describe bijections $g_{\\sigma,\\sigma^*}... |
https://arxiv.org/abs/1506.02797 | Abelian Powers and Repetitions in Sturmian Words | Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79-95, 2011) proved that at every position of a Sturmian word starts an abelian power of exponent $k$ for every $k > 0$. We improve on this result by studying the maximum exponents of abelian powers and abelian repetitions (an abelian repetition is an analogue of a f... | \section{Introduction}
Sturmian words are infinite words having exactly $n+1$ distinct factors of each length $n\geq 0$. By the celebrated theorem of Morse and Hedlund \cite{MoHe38}, they are the aperiodic binary words with minimal factor complexity. Every Sturmian word is characterized by an irrational number $\alpha... | {
"timestamp": "2016-04-19T02:16:47",
"yymm": "1506",
"arxiv_id": "1506.02797",
"language": "en",
"url": "https://arxiv.org/abs/1506.02797",
"abstract": "Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79-95, 2011) proved that at every position of a Sturmian word starts an abelian power of exponent $k$... |
https://arxiv.org/abs/1502.06635 | Small random instances of the stable roommates problem | Let $p_n$ denote the probability that a random instance of the stable roommates problem of size $n$ admits a solution. We derive an explicit formula for $p_n$ and compute exact values of $p_n$ for $n\leq 12$. | \section{Introduction}
\label{sec:intro}
Matching under preferences is a topic of great practical importance,
deep mathematical structure, and elegant algorithmics
\cite{manlove:book,gusfield:irving:book}. A paradigmatic example is
the stable roommates problem \cite{gale:shapley:62}. Consider an even
number $n$ of par... | {
"timestamp": "2015-02-25T02:02:39",
"yymm": "1502",
"arxiv_id": "1502.06635",
"language": "en",
"url": "https://arxiv.org/abs/1502.06635",
"abstract": "Let $p_n$ denote the probability that a random instance of the stable roommates problem of size $n$ admits a solution. We derive an explicit formula for $... |
https://arxiv.org/abs/1210.3772 | An Extension Theorem for Real Kahler Submanifolds in Codimension Four | In this article, we prove a Kahler extension theorem for real Kahler submanifolds of codimension 4 and rank at least 5. Our main theorem states that such a manifold is a holomorphic hypersurface in another real Kahler submanifold of codimension 2. This generalizes a result of Dajczer and Gromoll in 1997 which states th... | \section{Introduction}
\vspace{0.4cm}
Submanifold theory, and especially the study of Riemannian submanifolds in Euclidean spaces, have been a classic subarea in differential geometry. The Nash embedding theorem \cite{Nash} guarantees that any complete Riemannian manifold can be isometrically embedded into an Euclid... | {
"timestamp": "2012-10-16T02:03:04",
"yymm": "1210",
"arxiv_id": "1210.3772",
"language": "en",
"url": "https://arxiv.org/abs/1210.3772",
"abstract": "In this article, we prove a Kahler extension theorem for real Kahler submanifolds of codimension 4 and rank at least 5. Our main theorem states that such a ... |
https://arxiv.org/abs/1611.07111 | The Total Acquisition Number of Random Geometric Graphs | Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, provided that the weight on $v$ is at least as large as the weight on $u$. The total acquisition number of $G$, denoted by $a_t(G)$, is the minimum cardinality of the set... | \section{Introduction}
Gossiping and broadcasting are two well studied problems involving information dissemination in a group of individuals connected by a communication network~\cite{HHL}. In the gossip problem, each member has a unique piece of information which she would like to pass to everyone else. In the broad... | {
"timestamp": "2016-11-23T02:01:48",
"yymm": "1611",
"arxiv_id": "1611.07111",
"language": "en",
"url": "https://arxiv.org/abs/1611.07111",
"abstract": "Let $G$ be a graph in which each vertex initially has weight 1. In each step, the weight from a vertex $u$ to a neighbouring vertex $v$ can be moved, prov... |
https://arxiv.org/abs/2207.00768 | Sum-of-Max Partition under a Knapsack Constraint | Sequence partition problems arise in many fields, such as sequential data analysis, information transmission, and parallel computing. In this paper, we study the following partition problem variant: given a sequence of $n$ items $1,\ldots,n$, where each item $i$ is associated with weight $w_i$ and another parameter $s_... | \section{Introduction}
Sequence and tree partition problems have been studied extensively since 1970s,
due to their importance in parallel processing \cite{seq-2,app-parallel-1,app-parallel-2},
task scheduling \cite{app-task-scheduling-1,app-task-scheduling-2},
sequential data analysis \cite{app-data-analy... | {
"timestamp": "2022-07-05T02:07:32",
"yymm": "2207",
"arxiv_id": "2207.00768",
"language": "en",
"url": "https://arxiv.org/abs/2207.00768",
"abstract": "Sequence partition problems arise in many fields, such as sequential data analysis, information transmission, and parallel computing. In this paper, we st... |
https://arxiv.org/abs/1103.4346 | Quantized algebras of functions on homogeneous spaces with Poisson stabilizers | Let G be a simply connected semisimple compact Lie group with standard Poisson structure, K a closed Poisson-Lie subgroup, 0<q<1. We study a quantization C(G_q/K_q) of the algebra of continuous functions on G/K. Using results of Soibelman and Dijkhuizen-Stokman we classify the irreducible representations of C(G_q/K_q) ... | \section*{Introduction}
Following the foundational works of Woronowicz \cite{W} and Soibelman and Vaksman \cite{SV1}, the algebras of functions on $q$-deformations of compact groups and their homogeneous spaces were extensively studied in the 90s. Later the interest moved more towards noncommutative geometry of these ... | {
"timestamp": "2011-04-19T02:04:10",
"yymm": "1103",
"arxiv_id": "1103.4346",
"language": "en",
"url": "https://arxiv.org/abs/1103.4346",
"abstract": "Let G be a simply connected semisimple compact Lie group with standard Poisson structure, K a closed Poisson-Lie subgroup, 0<q<1. We study a quantization C(... |
https://arxiv.org/abs/1910.06201 | Graphs in which the Maxine heuristic produces a maximum independent set | The residue of a graph is the number of zeros left after iteratively applying the Havel-Hakimi algorithm to its degree sequence. Favaron, Mahéo, and Saclé showed that the residue is a lower bound on the independence number. The Maxine heuristic reduces a graph to an independent set of size $M$. It has been shown that g... | \section{Introduction} \label{sec: intro}
We will be considering simple graphs and we will let $N(v)$ represent the neighborhood of a vertex $v$ in a graph, and let $u\sim v$ mean that $u$ and $v$ are adjacent in the graph. For such a graph $G$ and subset of vertices $U$ in the graph, let $G[U]$ be the induced subgraph... | {
"timestamp": "2019-10-15T02:29:01",
"yymm": "1910",
"arxiv_id": "1910.06201",
"language": "en",
"url": "https://arxiv.org/abs/1910.06201",
"abstract": "The residue of a graph is the number of zeros left after iteratively applying the Havel-Hakimi algorithm to its degree sequence. Favaron, Mahéo, and Saclé... |
https://arxiv.org/abs/0812.1064 | Graph Minors and Minimum Degree | Let $\mathcal{D}_k$ be the class of graphs for which every minor has minimum degree at most $k$.Then $\mathcal{D}_k$ is closed under taking minors.By the Robertson-Seymour graph minor theorem, $\mathcal{D}_k$ is characterised by a finite family of minor-minimal forbidden graphs, which we denote by $\widehat{\mathcal{D}... | \section{Introduction}
The theory of graph minors developed by \citet{RS-GraphMinors} is one of the most important in graph theory influencing many branches of mathematics. Let \ensuremath{\mathcal{X}}\ be a minor-closed class of graphs\footnote{All graphs considered in this paper are undirected, simple, and finite.... | {
"timestamp": "2008-12-05T03:14:21",
"yymm": "0812",
"arxiv_id": "0812.1064",
"language": "en",
"url": "https://arxiv.org/abs/0812.1064",
"abstract": "Let $\\mathcal{D}_k$ be the class of graphs for which every minor has minimum degree at most $k$.Then $\\mathcal{D}_k$ is closed under taking minors.By the ... |
https://arxiv.org/abs/2006.11815 | Quantum trees which maximize higher eigenvalues are unbalanced | The isoperimetric problem of maximizing all eigenvalues of the Laplacian on a metric tree graph within the class of trees of a given average edge length is studied. It turns out that, up to rescaling, the unique maximizer of the $k$-th positive eigenvalue is the star graph with three edges of lengths $2 k - 1$, $1$ and... | \section{Introduction}
Within spectral geometry, isoperimetric problems for eigenvalues have a long history that reaches back at least as far as to Lord Rayleigh's famous book {\it The Theory of Sound} \cite[§210]{Rayleigh}. This class of problems deals with finding a shape which maximizes or minimizes (functions of... | {
"timestamp": "2020-09-03T02:02:40",
"yymm": "2006",
"arxiv_id": "2006.11815",
"language": "en",
"url": "https://arxiv.org/abs/2006.11815",
"abstract": "The isoperimetric problem of maximizing all eigenvalues of the Laplacian on a metric tree graph within the class of trees of a given average edge length i... |
https://arxiv.org/abs/1211.0765 | Holomorphic flexibility properties of the space of cubic rational maps | For each natural number d, the space R_d of rational maps of degree d on the Riemann sphere has the structure of a complex manifold. The topology of these manifolds has been extensively studied. The recent development of Oka theory raises some new and interesting questions about their complex structure. We apply geomet... | \section{Introduction and statement of results} \label{section:intro}
The space of rational maps on the Riemann sphere
can be given the structure of a complex manifold.
The topology of this manifold
(the compact-open topology)
has been studied extensively,
beginning with the work of Segal~\cite{Segal-1979}.
In this pa... | {
"timestamp": "2012-11-13T02:05:15",
"yymm": "1211",
"arxiv_id": "1211.0765",
"language": "en",
"url": "https://arxiv.org/abs/1211.0765",
"abstract": "For each natural number d, the space R_d of rational maps of degree d on the Riemann sphere has the structure of a complex manifold. The topology of these m... |
https://arxiv.org/abs/2009.10458 | Lower bounds for multicolor Ramsey numbers | We give an exponential improvement to the lower bound on diagonal Ramsey numbers for any fixed number of colors greater than two. | \section{Introduction}
The Ramsey number $r(t; \ell)$ is the smallest natural number $n$ such that every $\ell$-coloring of the edges of the complete graph $K_n$ contains a monochromatic $K_t$. For $\ell = 2$, the problem of determining $r(t) := r(t;2)$ is arguably one of the most famous in combinatorics. The bounds
\... | {
"timestamp": "2020-11-30T02:02:44",
"yymm": "2009",
"arxiv_id": "2009.10458",
"language": "en",
"url": "https://arxiv.org/abs/2009.10458",
"abstract": "We give an exponential improvement to the lower bound on diagonal Ramsey numbers for any fixed number of colors greater than two.",
"subjects": "Combina... |
https://arxiv.org/abs/2212.12492 | An ODE characterisation of multi-marginal optimal transport with pairwise cost functions | The purpose of this paper is to introduce a new numerical method to solve multi-marginal optimal transport problems with pairwise interaction costs. The complexity of multi-marginal optimal transport generally scales exponentially in the number of marginals $m$. We introduce a one parameter family of cost functions tha... | \section{Introduction}
The theory of optimal transport plays an important role in many applications (see \cite{Villani-OptimalTransport-09,Villani-TOT2003,santambook,peyre2017computational}). Its generalization to the multi-marginal case consists in minimizing the functional
\[\gamma\mapsto\int_{X^1 \times ...\times X^... | {
"timestamp": "2022-12-26T02:13:55",
"yymm": "2212",
"arxiv_id": "2212.12492",
"language": "en",
"url": "https://arxiv.org/abs/2212.12492",
"abstract": "The purpose of this paper is to introduce a new numerical method to solve multi-marginal optimal transport problems with pairwise interaction costs. The c... |
https://arxiv.org/abs/2008.13200 | Arithmetic of Some Sequences Via $2$-determinants | We extend our investigation of $2$-determinants, which we defined in a previous paper. For a linear homogenous recurrence of the second order, we consider relations between different sequences satisfying the same linear homogeneous recurrence of the second order. After we prove a generalized identity of d'Ocagne, we de... | \section{Introduction and preliminaries}
We continue our investigation of $2$-determinants,
defined in \cite{ja1}. For a linear homogenous recurrence of the second order, we consider
relations between different sequences satisfying the same recurrence. Linear homogenous recurrences of the second order are much studied... | {
"timestamp": "2021-05-12T02:29:33",
"yymm": "2008",
"arxiv_id": "2008.13200",
"language": "en",
"url": "https://arxiv.org/abs/2008.13200",
"abstract": "We extend our investigation of $2$-determinants, which we defined in a previous paper. For a linear homogenous recurrence of the second order, we consider... |
https://arxiv.org/abs/1204.2180 | A regularity lemma and twins in words | For a word $S$, let $f(S)$ be the largest integer $m$ such that there are two disjoints identical (scattered) subwords of length $m$. Let $f(n, \Sigma) = \min \{f(S): S \text{is of length} n, \text{over alphabet} \Sigma \}$. Here, it is shown that \[2f(n, \{0,1\}) = n-o(n)\] using the regularity lemma for words.I.e., a... | \section{Introduction}
Let $S=s_1 \ldots s_n $ be a word of length $n$, i.e., a sequence $s_1, s_2, \ldots, s_n$.
A (scattered) {\it subword} of
$S$ is a word $S'= s_{i_1} s_{i_2} \ldots s_{i_s}$, where
$i_1<i_2<\cdots < i_s$. This notion was largely investigated in
combinatorics on words and formal languages theor... | {
"timestamp": "2012-04-11T02:02:57",
"yymm": "1204",
"arxiv_id": "1204.2180",
"language": "en",
"url": "https://arxiv.org/abs/1204.2180",
"abstract": "For a word $S$, let $f(S)$ be the largest integer $m$ such that there are two disjoints identical (scattered) subwords of length $m$. Let $f(n, \\Sigma) = \... |
https://arxiv.org/abs/1810.07462 | Halfway to Rota's basis conjecture | In 1989, Rota made the following conjecture. Given $n$ bases $B_{1},\dots,B_{n}$ in an $n$-dimensional vector space $V$, one can always find $n$ disjoint bases of $V$, each containing exactly one element from each $B_{i}$ (we call such bases transversal bases). Rota's basis conjecture remains wide open despite its appa... | \section{Introduction}
Given bases $B_{1},\dots,B_{n}$ in an $n$-dimensional vector space
$V$, a \emph{transversal basis }is a basis of $V$ containing a single
distinguished vector from each of $B_{1},\dots,B_{n}$. Two transversal
bases are said to be \emph{disjoint} if their distinguished vectors
from $B_{i}$ are dis... | {
"timestamp": "2020-04-06T02:07:43",
"yymm": "1810",
"arxiv_id": "1810.07462",
"language": "en",
"url": "https://arxiv.org/abs/1810.07462",
"abstract": "In 1989, Rota made the following conjecture. Given $n$ bases $B_{1},\\dots,B_{n}$ in an $n$-dimensional vector space $V$, one can always find $n$ disjoint... |
https://arxiv.org/abs/1609.06083 | A classification of anisotropic Besov spaces | We study (homogeneous and inhomogeneous) anisotropic Besov spaces associated to expansive dilation matrices $A \in {\rm GL}(d,\mathbb{R})$, with the goal of clarifying when two such matrices induce the same scale of Besov spaces. For this purpose, we first establish that anisotropic Besov spaces have an alternative des... | \section{Introduction}\label{introduction}
Let $A \in \mathbb{R}^{d \times d}$ denote a matrix whose eigenvalues all have modulus $>1$. Matrices of this kind, often with additional properties (such as integer entries) were the basis of the study of discrete wavelet systems (frames or bases) obtained from dilations by ... | {
"timestamp": "2016-09-21T02:04:06",
"yymm": "1609",
"arxiv_id": "1609.06083",
"language": "en",
"url": "https://arxiv.org/abs/1609.06083",
"abstract": "We study (homogeneous and inhomogeneous) anisotropic Besov spaces associated to expansive dilation matrices $A \\in {\\rm GL}(d,\\mathbb{R})$, with the go... |
https://arxiv.org/abs/1011.3506 | Dynamics of a rational multi-parameter second order difference equation with cubic numerator and quadratic monomial denominator | The asymptotic behavior (such as convergence to an equilibrium, convergence to a 2-cycle, and divergence to infinity) of solutions of the following multi-parameter, rational, second order difference equation x_{n+1} =(ax_{n}^3+ bx_{n}^2x_{n-1}+cx_{n}x_{n-1}^2+dx_{n-1}^3)/x_{n}^2, x_{-1},x_{0}\in R, is studied in this p... | \section{Introduction}
Most of the work about rational difference equations treat the case where both numerator and denominator are linear polynomials. For second order rational difference equations with linear numerator and denominator we refer the reader to the monograph of Kulenovic and Ladass (\cite{KL}). In 2008,... | {
"timestamp": "2010-11-17T02:02:01",
"yymm": "1011",
"arxiv_id": "1011.3506",
"language": "en",
"url": "https://arxiv.org/abs/1011.3506",
"abstract": "The asymptotic behavior (such as convergence to an equilibrium, convergence to a 2-cycle, and divergence to infinity) of solutions of the following multi-pa... |
https://arxiv.org/abs/1703.00827 | Sandpiles on the square lattice | We give a non-trivial upper bound for the critical density when stabilizing i.i.d. distributed sandpiles on the lattice $\mathbb{Z}^2$. We also determine the asymptotic spectral gap, asymptotic mixing time and prove a cutoff phenomenon for the recurrent state abelian sandpile model on the torus $\left( \mathbb{Z} / m\m... | \section{Introduction}
\subsection{Stabilization of i.i.d.~sandpiles}
A \emph{sandpile} on the integer lattice $\mathbb{Z}^2$ is a function $\sigma : \mathbb{Z}^2 \to \mathbb{Z}_{\geq 0}$, where $\sigma(x)$ represents the number of grains of sand at the site $x$. The sandpile $\sigma$ is \emph{stable} if each $\sigma(... | {
"timestamp": "2018-07-12T02:11:17",
"yymm": "1703",
"arxiv_id": "1703.00827",
"language": "en",
"url": "https://arxiv.org/abs/1703.00827",
"abstract": "We give a non-trivial upper bound for the critical density when stabilizing i.i.d. distributed sandpiles on the lattice $\\mathbb{Z}^2$. We also determine... |
https://arxiv.org/abs/0801.2987 | The minimum rank problem over finite fields | The structure of all graphs having minimum rank at most k over a finite field with q elements is characterized for any possible k and q. A strong connection between this characterization and polarities of projective geometries is explained. Using this connection, a few results in the minimum rank problem are derived by... |
\section{Introduction}
Given a field $F$ and a simple undirected graph $G$ on $n$ vertices (i.e., an
undirected graph without loops or multiple edges), let
$S(F,G)$ be the set of symmetric $n\times n$ matrices $A$ with entries
in $F$ satisfying $a_{ij} \neq 0$, $i \neq j$, if and only if $ij$ is
an edge in $G... | {
"timestamp": "2008-01-18T23:56:30",
"yymm": "0801",
"arxiv_id": "0801.2987",
"language": "en",
"url": "https://arxiv.org/abs/0801.2987",
"abstract": "The structure of all graphs having minimum rank at most k over a finite field with q elements is characterized for any possible k and q. A strong connection... |
https://arxiv.org/abs/2212.02365 | Error reduction using machine learning on Ising worm simulation | We develop a method to improve on the statistical errors for higher moments using machine learning techniques. We present here results for the dual representation of the Ising model with an external field, derived via the high temperature expansion and simulated by the worm algorithm. We compare two ways of measuring t... | \section{Ising dual representation}
The dual representation of the Ising model is derived by introducing bond variables and integrating out the spin degrees of freedom \cite{Prokof:2001,Wolff2008,Gabriel:2002}:
\begin{align}
Z_{\rm Ising}&=\sum_{\{s\}} e^{-\beta H(s)},\qquad H=-J \sum_{\langle i,j \rangle}s_i s_j+h\s... | {
"timestamp": "2022-12-06T02:28:31",
"yymm": "2212",
"arxiv_id": "2212.02365",
"language": "en",
"url": "https://arxiv.org/abs/2212.02365",
"abstract": "We develop a method to improve on the statistical errors for higher moments using machine learning techniques. We present here results for the dual repres... |
https://arxiv.org/abs/2107.00364 | Implicit Acceleration and Feature Learning in Infinitely Wide Neural Networks with Bottlenecks | We analyze the learning dynamics of infinitely wide neural networks with a finite sized bottle-neck. Unlike the neural tangent kernel limit, a bottleneck in an otherwise infinite width network al-lows data dependent feature learning in its bottle-neck representation. We empirically show that a single bottleneck in infi... | \section{Introduction}
The study of infinitely wide neural networks is one of the most actively researched topics in deep learning theory \cite{NTK,gp1,gp2,gp3,gp4,gp5,gp6,gp7,yang,yang2,yang3,TP2b,Littwin2020OnRK,Littwin2020OnTO}. Previous work identified distinct training regimes that are determined by the networks h... | {
"timestamp": "2021-07-05T02:11:25",
"yymm": "2107",
"arxiv_id": "2107.00364",
"language": "en",
"url": "https://arxiv.org/abs/2107.00364",
"abstract": "We analyze the learning dynamics of infinitely wide neural networks with a finite sized bottle-neck. Unlike the neural tangent kernel limit, a bottleneck ... |
https://arxiv.org/abs/2005.09690 | Invariance of the tame fundamental group under base change between algebraically closed fields | We show that the tame étale fundamental group of a connected normal finite type separated scheme remains invariant upon base change between algebraically closed fields of characteristic $p \geq 0$. | \section{Statement of theorem}
For $X$ a scheme, we let $\pi_1(X)$ denote the \'etale fundamental group of $X$,
where we leave the base point implicit,
and we let $\pi_1^{(p)}(X)$ denote the maximal prime to $p$ quotient of $\pi_1(X)$,
again with an implicit choice of base point.
For convenience of notation, we let $\... | {
"timestamp": "2020-05-21T02:01:02",
"yymm": "2005",
"arxiv_id": "2005.09690",
"language": "en",
"url": "https://arxiv.org/abs/2005.09690",
"abstract": "We show that the tame étale fundamental group of a connected normal finite type separated scheme remains invariant upon base change between algebraically ... |
https://arxiv.org/abs/1901.03794 | Analyzing a Maximum Principle for Finite Horizon State Constrained Problems via Parametric Examples. Part 1: Problems with Unilateral State Constraints | In the present paper, the maximum principle for finite horizon state constrained problems from the book by R. Vinter [\textit{Optimal Control}, Birkhäuser, Boston, 2000; Theorem~9.3.1] is analyzed via parametric examples. The latter has origin in a recent paper by V.~Basco, P.~Cannarsa, and H.~Frankowska, and resembles... | \section{Introduction}
It is well known that optimal control problems with state constraints are models of importance, but one usually faces with a lot of difficulties in analyzing them. These models have been considered since the early days of the optimal control theory. For instance, the whole Chapter VI of the class... | {
"timestamp": "2019-01-15T02:04:49",
"yymm": "1901",
"arxiv_id": "1901.03794",
"language": "en",
"url": "https://arxiv.org/abs/1901.03794",
"abstract": "In the present paper, the maximum principle for finite horizon state constrained problems from the book by R. Vinter [\\textit{Optimal Control}, Birkhäuse... |
https://arxiv.org/abs/0711.1979 | Galilean Classification of Curves | In this paper, we classify space-time curves up to Galilean group of transformations with Cartan's method of equivalence. As an aim, we elicit invariats from action of special Galilean group on space-time curves, that are, in fact, conservation laws in physics. We also state a necessary and sufficient condition for equ... | \section{Introduction}
Galilean transformation group has an important place in classic
and modern physics for instance: in quantum theory, gauge
transformations in electromagnetism, in mechanics \cite{AM}, and
conductivity tensors in fluid dynamics \cite{Ga}, also, in
mathematical fields such as Lagrangian mechanics, d... | {
"timestamp": "2007-11-13T14:26:22",
"yymm": "0711",
"arxiv_id": "0711.1979",
"language": "en",
"url": "https://arxiv.org/abs/0711.1979",
"abstract": "In this paper, we classify space-time curves up to Galilean group of transformations with Cartan's method of equivalence. As an aim, we elicit invariats fro... |
https://arxiv.org/abs/1903.08571 | Identifying Maximal Non-Redundant Integer Cone Generators | A non-redundant integer cone generator (NICG) of dimension $d$ is a set $S$ of vectors from $\{0,1\}^d$ whose vector sum cannot be generated as a positive integer linear combination of a proper subset of $S$. The largest possible cardinality of NICG of a dimension $d$, denoted by $N(d)$, provides an upper bound on the ... |
\section{Conclusions}
{\m{\small QFBAPA}} has the small model property. We are interested in deriving a number $N(d)$,
where $N(d)$ is the smallest number such that the following property holds: if formula has a solution, then
it also has a solution of the size $N(d)$.
In this paper we computed the values $N(4)$, $N... | {
"timestamp": "2019-03-21T01:20:45",
"yymm": "1903",
"arxiv_id": "1903.08571",
"language": "en",
"url": "https://arxiv.org/abs/1903.08571",
"abstract": "A non-redundant integer cone generator (NICG) of dimension $d$ is a set $S$ of vectors from $\\{0,1\\}^d$ whose vector sum cannot be generated as a positi... |
https://arxiv.org/abs/hep-th/9312023 | On free differentials on associative algebras | A free differential for an arbitrary associative algebra is defined as a differential with a uniqueness property. The existence problem for such a differential is posed. The notion of optimal calculi for given commutation rules is introduced and an explicit construction of it for a homogenous case is provided. Some exa... | \section{Introduction}
A differential $d:R \rightarrow\, _{R}\!M\!_{R}$ is called free if the differential
of any element $v$ has a unique presentation of the form $dv=dx^{i}\cdot v_{i}$,
where $x^1,\ldots ,x^n$ are generators of the algebra and $dx^1,\ldots ,dx^n$ their
differentials. Any free differential defines a c... | {
"timestamp": "1997-11-14T00:50:47",
"yymm": "9312",
"arxiv_id": "hep-th/9312023",
"language": "en",
"url": "https://arxiv.org/abs/hep-th/9312023",
"abstract": "A free differential for an arbitrary associative algebra is defined as a differential with a uniqueness property. The existence problem for such a... |
https://arxiv.org/abs/1804.11342 | Hyperreal Numbers for Infinite Divergent Series | Treating divergent series properly has been an ongoing issue in mathematics. However, many of the problems in divergent series stem from the fact that divergent series were discovered prior to having a number system which could handle them. The infinities that resulted from divergent series led to contradictions within... | \section{The Problem of Infinite Series}
Historically, infinities have led to many problems in mathematics.
Infinities, when not handled carefully, easily lead to contradictions and indeterminacies.
Therefore, caution has always been urged when dealing with infinite series.
This is especially true with divergent in... | {
"timestamp": "2018-07-30T02:03:07",
"yymm": "1804",
"arxiv_id": "1804.11342",
"language": "en",
"url": "https://arxiv.org/abs/1804.11342",
"abstract": "Treating divergent series properly has been an ongoing issue in mathematics. However, many of the problems in divergent series stem from the fact that div... |
https://arxiv.org/abs/0711.0540 | Umkehr Maps | In this note we study umkehr maps in generalized (co)homology theories arising from the Pontrjagin-Thom construction, from integrating along fibers, pushforward homomorphisms, and other similar constructions. We consider the basic properties of these constructions and develop axioms which any umkehr homomorphism must s... | \section{Introduction}
The classical umkehr homomorphism of Hopf \cite{Hopf},
assigns to a map $f\: M \to N$
of closed manifolds of the same dimension a ``wrong way'' homomorphism
$f_!\: H_*(N)\to H_*(M)$ on singular homology. Hopf showed that
this map is compatible with intersection pairings.
Freudenthal \cite{Freud}... | {
"timestamp": "2007-11-04T20:48:14",
"yymm": "0711",
"arxiv_id": "0711.0540",
"language": "en",
"url": "https://arxiv.org/abs/0711.0540",
"abstract": "In this note we study umkehr maps in generalized (co)homology theories arising from the Pontrjagin-Thom construction, from integrating along fibers, pushfor... |
https://arxiv.org/abs/1308.1700 | A simple combinatorial interpretation of certain generalized Bell and Stirling numbers | In a series of papers, P. Blasiak et al. developed a wide-ranging generalization of Bell numbers (and of Stirling numbers of the second kind) that appears to be relevant to the so-called Boson normal ordering problem. They provided a recurrence and, more recently, also offered a (fairly complex) combinatorial interpret... | \section{Introduction}
\label{sec:intro}
In \cite{blasiak1,blasiak2,blasiak3,blasiak4} P. Blasiak et al.
introduced coefficients $B_{r,s}(n)$, and $S_{r,s}(n,k)$ that
provide a wide-ranging generalization of Bell numbers, and of
Stirling numbers of the second kind, respectively. In particular
they defined the generali... | {
"timestamp": "2013-08-09T02:00:46",
"yymm": "1308",
"arxiv_id": "1308.1700",
"language": "en",
"url": "https://arxiv.org/abs/1308.1700",
"abstract": "In a series of papers, P. Blasiak et al. developed a wide-ranging generalization of Bell numbers (and of Stirling numbers of the second kind) that appears t... |
https://arxiv.org/abs/2202.00754 | On Wilson's theorem about domains of attraction and tubular neighborhoods | In this paper, we show that the domain of attraction of a compact asymptotically stable submanifold of a finite-dimensional smooth manifold of an autonomous system is homeomorphic to its tubular neighborhood. The compactness of the attractor is crucial, without which this result is false; two counterexamples are provid... | \section{Introduction}
The domain of attraction of an attractor of a continuous dynamical system has been widely studied. An \emph{attractor} is a closed invariant set of which there exists an open neighborhood such that every trajectory of the dynamical system starting within the neighborhood eventually converges to t... | {
"timestamp": "2022-02-03T02:03:57",
"yymm": "2202",
"arxiv_id": "2202.00754",
"language": "en",
"url": "https://arxiv.org/abs/2202.00754",
"abstract": "In this paper, we show that the domain of attraction of a compact asymptotically stable submanifold of a finite-dimensional smooth manifold of an autonomo... |
https://arxiv.org/abs/2211.06291 | Do Bayesian Neural Networks Need To Be Fully Stochastic? | We investigate the benefit of treating all the parameters in a Bayesian neural network stochastically and find compelling theoretical and empirical evidence that this standard construction may be unnecessary. To this end, we prove that expressive predictive distributions require only small amounts of stochasticity. In ... | \section{Proofs}\label{sec:supplement:proofs}
We provide a proof of Theorem~\ref{theorem:one_bias_is_all_you_need}, which states that a number of architectures are universal conditional distribution approximators (UCDAs). First, we restate the architectures that we consider and our theorem statement for convenience. Th... | {
"timestamp": "2022-11-14T02:13:43",
"yymm": "2211",
"arxiv_id": "2211.06291",
"language": "en",
"url": "https://arxiv.org/abs/2211.06291",
"abstract": "We investigate the benefit of treating all the parameters in a Bayesian neural network stochastically and find compelling theoretical and empirical eviden... |
https://arxiv.org/abs/1902.10901 | Robust and Local Optimal A Priori Error Estimates for Interface Problems with Low Regularity: Mixed Finite Element Approximations | For elliptic interface problems in two- and three-dimensions with a possible very low regularity, this paper establishes a priori error estimates for the Raviart-Thomas and Brezzi-Douglas-Marini mixed finite element approximations. These estimates are robust with respect to the diffusion coefficient and optimal with re... |
\section{Introduction}\label{intro}
\setcounter{equation}{0}
As a prototype of problems with interface singularities, this paper studies {\em a priori} error estimates of mixed finite element methods for the following interface problem (i.e., the diffusion problem with discontinuous coefficients):
\begin{equation}\la... | {
"timestamp": "2019-03-01T02:09:36",
"yymm": "1902",
"arxiv_id": "1902.10901",
"language": "en",
"url": "https://arxiv.org/abs/1902.10901",
"abstract": "For elliptic interface problems in two- and three-dimensions with a possible very low regularity, this paper establishes a priori error estimates for the ... |
https://arxiv.org/abs/2012.09055 | Degree Counting Theorems for 2x2 non-symmetric singular Liouville Systems | Let $(M,g)$ be a compact Riemann surface with no boundary and $u=(u_1,u_2)$ be a solution of the following singular Liouville system: $$\Delta_g u_i+\sum_{j=1}^2 a_{ij}\rho_j(\frac{h_je^{u_j}}{\int_M h_je^{u_j}dV_g}-1)=\sum_{l=1}^{N}4\pi\gamma_l(\delta_{p_l}-1), $$ where $h_1,h_2$ are positive smooth functions, $p_1,\c... | \section{Introduction}
In this article we study the following Liouville system defined on a compact Riemann surface $(M,g)$ with no boundary:
\begin{equation}\label{1.1}
\begin{aligned}
\Delta_g u_1^*+a_{11}\rho_1(\frac{h_1^*e^{u_1^*}}{\int_M h_1^*e^{u_1^*}}-1)+a_{12}\rho_2(\frac{h_2^*e^{u_2^*}}{\int_M h_2^*e^{u_2^*... | {
"timestamp": "2020-12-17T02:22:38",
"yymm": "2012",
"arxiv_id": "2012.09055",
"language": "en",
"url": "https://arxiv.org/abs/2012.09055",
"abstract": "Let $(M,g)$ be a compact Riemann surface with no boundary and $u=(u_1,u_2)$ be a solution of the following singular Liouville system: $$\\Delta_g u_i+\\su... |
https://arxiv.org/abs/math/9906085 | Comments on Lagrange Partial Differential Equation | The relations between solutions of the three types of totally linear partial differential equations of first order are presented. The approach is based on factorization of a non-homogeneous first order differential operator to products consisting of a scalar function, a homogeneous first order differential operator and... | \section{Introduction}
The method for solution of Lagrange partial differential equation is well
known, and is found almost in every text book on partial differential
equations\cite{Piaggio,Ayres,Carrier}. Our goal in this article is to show
how the factorization of a non-homogeneous first order differential operator
... | {
"timestamp": "1999-06-12T23:57:24",
"yymm": "9906",
"arxiv_id": "math/9906085",
"language": "en",
"url": "https://arxiv.org/abs/math/9906085",
"abstract": "The relations between solutions of the three types of totally linear partial differential equations of first order are presented. The approach is base... |
https://arxiv.org/abs/2108.07863 | Computations of relative topological coHochschild homology | Hess and Shipley defined an invariant of coalgebra spectra called topological coHochschild homology, and Bohmann-Gerhardt-Høgenhaven-Shipley-Ziegenhagen developed a coBökstedt spectral sequence to compute the homology of coTHH for coalgebras over the sphere spectrum. We construct a relative coBökstedt spectral sequence... | \section{Introduction}
We develop computational tools for studying topological coHochschild homology ($\mathrm{coTHH}$). Recent work of Hess and Shipley \cite{hess2018invariance} defines this invariant to study coalgebra spectra. Work of Malkiewich \cite{malkiewich2017cyclotomic} and Hess-Shipley \cite{hess2018invarian... | {
"timestamp": "2021-08-19T02:03:36",
"yymm": "2108",
"arxiv_id": "2108.07863",
"language": "en",
"url": "https://arxiv.org/abs/2108.07863",
"abstract": "Hess and Shipley defined an invariant of coalgebra spectra called topological coHochschild homology, and Bohmann-Gerhardt-Høgenhaven-Shipley-Ziegenhagen d... |
https://arxiv.org/abs/1908.10989 | Revisiting a Cutting Plane Method for Perfect Matchings | In 2016, Chandrasekaran, Végh, and Vempala published a method to solve the minimum-cost perfect matching problem on an arbitrary graph by solving a strictly polynomial number of linear programs. However, their method requires a strong uniqueness condition, which they imposed by using perturbations of the form $c(i)=c_0... | \section{Introduction}
Given a graph $G=(V, E)$ with edge cost function $c$, the minimum-cost
(or minimum-weight) perfect matching problem is to find a perfect matching
$E' \subseteq E$ (a subset such that every vertex $v \in V$ is covered
by exactly one $uv \in E'$) so that the sum of the costs of $E'$ is minimized.
... | {
"timestamp": "2019-08-30T02:04:46",
"yymm": "1908",
"arxiv_id": "1908.10989",
"language": "en",
"url": "https://arxiv.org/abs/1908.10989",
"abstract": "In 2016, Chandrasekaran, Végh, and Vempala published a method to solve the minimum-cost perfect matching problem on an arbitrary graph by solving a strict... |
https://arxiv.org/abs/2102.02737 | Insight of the Green's function as a defect state in a boundary value problem | A new perspective of the Green's function in a boundary value problem as the only eigenstate in an auxiliary formulation is introduced. In this treatment, the Green's function can be perceived as a defect state in the presence of a $\delta$-function potential, the height of which depends on the Green's function itself.... | \section{Introduction}
Initially distributed to just 51 private members of a subscription library in 1828 \cite{green_book}, the essay of George Green on the application of mathematical analysis to the theories of electricity and magnetism has inspired generations of physicists and mathematicians. By studying the thre... | {
"timestamp": "2021-02-05T02:19:53",
"yymm": "2102",
"arxiv_id": "2102.02737",
"language": "en",
"url": "https://arxiv.org/abs/2102.02737",
"abstract": "A new perspective of the Green's function in a boundary value problem as the only eigenstate in an auxiliary formulation is introduced. In this treatment,... |
https://arxiv.org/abs/1604.06127 | The HOMFLY Polynomial of Links in Closed Braid Form | It is well known that any link can be represented by the closure of a braid. The minimum number of strings needed in a braid whose closure represents a given link is called the braid index of the link and the well known Morton-Frank-Williams inequality reveals a close relationship between the HOMFLY polynomial of a lin... | \section{Introduction}
\medskip
It is well known that any link can be represented by the closure of a braid. The minimum
number of strands needed in a braid whose closure represents a given link is called the
{\em braid index} of the link. Defined as the extreme value of a quantity over an infinite family of links t... | {
"timestamp": "2016-12-08T02:02:05",
"yymm": "1604",
"arxiv_id": "1604.06127",
"language": "en",
"url": "https://arxiv.org/abs/1604.06127",
"abstract": "It is well known that any link can be represented by the closure of a braid. The minimum number of strings needed in a braid whose closure represents a gi... |
https://arxiv.org/abs/2009.13730 | A Fully Parallel Primal-Dual Algorithm for Centralized and Distributed Optimization | In this paper, a centralized two-block separable optimization is considered for which a fully parallel primal-dual discrete-time algorithm with fixed step size is derived based on monotone operator splitting method. In this algorithm, the primal variables are updated in an alternating fashion like Alternating Direction... | \section{Introduction}
In this paper, we consider the following two-block decomposable optimization (and its extension to multi-block optimization) with affine constraint\footnote{For better comparison between ADMM and our proposed algorithm, we use the same notations for (\ref{1}) as in \cite{boydADMM}.}:
\begin{equ... | {
"timestamp": "2020-09-30T02:08:42",
"yymm": "2009",
"arxiv_id": "2009.13730",
"language": "en",
"url": "https://arxiv.org/abs/2009.13730",
"abstract": "In this paper, a centralized two-block separable optimization is considered for which a fully parallel primal-dual discrete-time algorithm with fixed step... |
https://arxiv.org/abs/1706.00439 | Tensor Contraction Layers for Parsimonious Deep Nets | Tensors offer a natural representation for many kinds of data frequently encountered in machine learning. Images, for example, are naturally represented as third order tensors, where the modes correspond to height, width, and channels. Tensor methods are noted for their ability to discover multi-dimensional dependencie... |
\section{Introduction}
Following their successful application to computer vision,
speech recognition, and natural language processing,
deep neural networks have become ubiquitous
in the machine learning community.
And yet many questions remain unanswered:
Why do deep neural networks work?
How many parameters are r... | {
"timestamp": "2017-06-05T02:00:36",
"yymm": "1706",
"arxiv_id": "1706.00439",
"language": "en",
"url": "https://arxiv.org/abs/1706.00439",
"abstract": "Tensors offer a natural representation for many kinds of data frequently encountered in machine learning. Images, for example, are naturally represented a... |
https://arxiv.org/abs/2208.03941 | A high-resolution dynamical view on momentum methods for over-parameterized neural networks | Due to the simplicity and efficiency of the first-order gradient method, it has been widely used in training neural networks. Although the optimization problem of the neural network is non-convex, recent research has proved that the first-order method is capable of attaining a global minimum for training over-parameter... | \section{Introduction}
Momentum methods utilize the history of gradients, which exhibit accelerated convergence rates compared to the gradient descent method~(GD).
Momentum methods are frequently used in training neural networks due to the high computational cost of deep learning.
The convergence properties of two wel... | {
"timestamp": "2022-09-07T02:15:36",
"yymm": "2208",
"arxiv_id": "2208.03941",
"language": "en",
"url": "https://arxiv.org/abs/2208.03941",
"abstract": "Due to the simplicity and efficiency of the first-order gradient method, it has been widely used in training neural networks. Although the optimization pr... |
https://arxiv.org/abs/1711.04954 | Impartial Triangular Chocolate Bar Games | Chocolate bar games are variants of the game of Nim in which the goal is to leave your opponent with the single bitter part of the chocolate bar. The rectangular chocolate bar game is a thinly disguised form of classical multi-heap Nim. In this work, we investigate the mathematical structure of triangular chocolate bar... | \section{Introduction}\label{intro}
The original chocolate bar game \cite{robin} consists of square boxes in which one square is blue and other squares are brown.
Brown squares are sweet, and the blue square is considered too bitter to eat. For example, see Figure \ref{robinchoco1}.
Each player takes turns breaking the... | {
"timestamp": "2017-11-15T02:40:11",
"yymm": "1711",
"arxiv_id": "1711.04954",
"language": "en",
"url": "https://arxiv.org/abs/1711.04954",
"abstract": "Chocolate bar games are variants of the game of Nim in which the goal is to leave your opponent with the single bitter part of the chocolate bar. The rect... |
https://arxiv.org/abs/1503.00725 | Intrinsic random walks and sub-Laplacians in sub-Riemannian geometry | On a sub-Riemannian manifold we define two type of Laplacians. The \emph{macroscopic Laplacian} $\Delta_\omega$, as the divergence of the horizontal gradient, once a volume $\omega$ is fixed, and the \emph{microscopic Laplacian}, as the operator associated with a sequence of geodesic random walks. We consider a general... | \section{Convergence of random walks}\label{a:randomwalk}
We let $M$ be a smooth, geodesically complete (sub)-Riemannian manifold. Our goal here is to give fairly general conditions for a sequence of random walks on $M$ to converge. In particular, we will work with a larger class of random walks than those treated els... | {
"timestamp": "2016-01-11T02:09:01",
"yymm": "1503",
"arxiv_id": "1503.00725",
"language": "en",
"url": "https://arxiv.org/abs/1503.00725",
"abstract": "On a sub-Riemannian manifold we define two type of Laplacians. The \\emph{macroscopic Laplacian} $\\Delta_\\omega$, as the divergence of the horizontal gr... |
https://arxiv.org/abs/2008.02489 | On a minimax principle in spectral gaps | The minimax principle for eigenvalues in gaps of the essential spectrum in the form presented by Griesemer, Lewis, and Siedentop in [Doc. Math. 4 (1999), 275--283] is adapted to cover certain abstract perturbative settings with bounded or unbounded perturbations, in particular ones that are off-diagonal with respect to... | \section{Introduction and main result}\label{sec:intro}
The standard Courant minimax values $\lambda_k(A)$ of a lower semibounded operator $A$ on a Hilbert space ${\mathcal H}$ are given by
\begin{equation*}
\lambda_k(A)
=
\inf_{\substack{{\mathfrak M}\subset\Dom(A)\\ \dim{\mathfrak M}=k}} \sup_{\substack{x\in{\... | {
"timestamp": "2020-08-07T02:09:53",
"yymm": "2008",
"arxiv_id": "2008.02489",
"language": "en",
"url": "https://arxiv.org/abs/2008.02489",
"abstract": "The minimax principle for eigenvalues in gaps of the essential spectrum in the form presented by Griesemer, Lewis, and Siedentop in [Doc. Math. 4 (1999), ... |
https://arxiv.org/abs/1803.10918 | A simplified presentation of Specht modules | Fulton and Kraskiewicz gave a presentation of Specht modules as a quotient of the space of column tabloids by dual Garnir relations. We simplify this presentation by showing that it can be generated by a single relation for each pair of columns of a tableau with ordered columns, thereby significantly reducing the numbe... | \section{Introduction}
Representations of the symmetric group $S_m$ have a long and beautiful history in mathematics. Partitions of $m$ biject with the irreducible representations of $S_{m}$ given by Specht modules; these representations have a basis corresponding to standard Young tableaux. The relations that allow us... | {
"timestamp": "2018-03-30T02:04:42",
"yymm": "1803",
"arxiv_id": "1803.10918",
"language": "en",
"url": "https://arxiv.org/abs/1803.10918",
"abstract": "Fulton and Kraskiewicz gave a presentation of Specht modules as a quotient of the space of column tabloids by dual Garnir relations. We simplify this pres... |
https://arxiv.org/abs/2111.11860 | A Discrete-Time Compartmental Epidemiological Model for COVID-19 with a Case Study for Portugal | In [Ecological Complexity 44 (2020) Art. 100885, DOI:https://doi.org/10.1016/j.ecocom.2020.100885] a continuous-time compartmental mathematical model for the spread of the Coronavirus disease 2019 (COVID-19) is presented with Portugal as case study, from 2 March to 4 May 2020, and the local stability of the Disease Fre... | \section{Introduction}
The coronavirus belong to the family of \emph{Coronaviridae}. It is a virus that causes infection
to humans, other mammals and birds. The infection usually affects the respiratory system,
and the symptoms may vary from simple colds to pneumonia. To date, eight coronaviruses are known.
The new... | {
"timestamp": "2021-11-24T02:19:43",
"yymm": "2111",
"arxiv_id": "2111.11860",
"language": "en",
"url": "https://arxiv.org/abs/2111.11860",
"abstract": "In [Ecological Complexity 44 (2020) Art. 100885, DOI:https://doi.org/10.1016/j.ecocom.2020.100885] a continuous-time compartmental mathematical model for ... |
https://arxiv.org/abs/1909.01690 | Characterization of $k-$smooth operators between Banach spaces | We study $k-$smoothness of bounded linear operators defined between arbitrary Banach spaces. As an application, we characterize $k-$smooth operators defined from $\ell_1^n$ to an arbitrary Banach space. We also completely characterize $k-$smooth operators defined between arbitrary two-dimensional Banach spaces. | \section{Introduction}
The characterization of smoothness of operator between Banach spaces is a rich, intricate problem to study. It helps to understand the geometry of operator space. Over the years several mathematicians have been studying the smoothness of operators defined between Banach spaces. The readers may g... | {
"timestamp": "2019-09-05T02:13:08",
"yymm": "1909",
"arxiv_id": "1909.01690",
"language": "en",
"url": "https://arxiv.org/abs/1909.01690",
"abstract": "We study $k-$smoothness of bounded linear operators defined between arbitrary Banach spaces. As an application, we characterize $k-$smooth operators defin... |
https://arxiv.org/abs/2204.04970 | Non-Convex Optimization with Certificates and Fast Rates Through Kernel Sums of Squares | We consider potentially non-convex optimization problems, for which optimal rates of approximation depend on the dimension of the parameter space and the smoothness of the function to be optimized. In this paper, we propose an algorithm that achieves close to optimal a priori computational guarantees, while also provid... | \section{Introduction}
A well-designed optimization algorithm provides two important types of guarantees. First, it guarantees {\em a priori} that its output will achieve a certain degree of accuracy, with computational complexity that is hopefully adaptive to the specific properties of function to be optimized and pos... | {
"timestamp": "2022-04-12T02:36:46",
"yymm": "2204",
"arxiv_id": "2204.04970",
"language": "en",
"url": "https://arxiv.org/abs/2204.04970",
"abstract": "We consider potentially non-convex optimization problems, for which optimal rates of approximation depend on the dimension of the parameter space and the ... |
https://arxiv.org/abs/1209.3617 | Strategy complexity of finite-horizon Markov decision processes and simple stochastic games | Markov decision processes (MDPs) and simple stochastic games (SSGs) provide a rich mathematical framework to study many important problems related to probabilistic systems. MDPs and SSGs with finite-horizon objectives, where the goal is to maximize the probability to reach a target state in a given finite time, is a cl... | \section{Introduction}
\smallskip\noindent{\bf Markov decision process and simple stochastic games.}
The class of \emph{Markov decision processes (MDPs)} is a classical model for probabilistic
systems that exhibit both stochastic and and deterministic
behavior~\cite{Howard}.
MDPs have been widely used to model and ... | {
"timestamp": "2012-09-18T02:04:38",
"yymm": "1209",
"arxiv_id": "1209.3617",
"language": "en",
"url": "https://arxiv.org/abs/1209.3617",
"abstract": "Markov decision processes (MDPs) and simple stochastic games (SSGs) provide a rich mathematical framework to study many important problems related to probab... |
https://arxiv.org/abs/2006.14078 | Machine learning the real discriminant locus | Parameterized systems of polynomial equations arise in many applications in science and engineering with the real solutions describing, for example, equilibria of a dynamical system, linkages satisfying design constraints, and scene reconstruction in computer vision. Since different parameter values can have a differen... | \section{Introduction}
Systems of polynomial equations are a collection of multivariate nonlinear equations in which each equation is a multivariate polynomial. Such systems arise naturally in many areas of science and engineering ranging from
chemistry, particle physics, string theory, mathematical biology, phylogen... | {
"timestamp": "2020-06-26T02:04:45",
"yymm": "2006",
"arxiv_id": "2006.14078",
"language": "en",
"url": "https://arxiv.org/abs/2006.14078",
"abstract": "Parameterized systems of polynomial equations arise in many applications in science and engineering with the real solutions describing, for example, equil... |
https://arxiv.org/abs/1201.2853 | Renormalized energy concentration in random matrices | We define a "renormalized energy" as an explicit functional on arbitrary point configurations of constant average density in the plane and on the real line. The definition is inspired by ideas of [SS1,SS3]. Roughly speaking, it is obtained by subtracting two leading terms from the Coulomb potential on a growing number ... | \section{Introduction}
The aim of this paper is to introduce and compute a function, called the ``renormalized energy", for some specific random point processes that arise in random matrix models, and in this way to associate to each of these processes a unique number, which is expected to measure its ``disorder".
... | {
"timestamp": "2012-10-24T02:04:38",
"yymm": "1201",
"arxiv_id": "1201.2853",
"language": "en",
"url": "https://arxiv.org/abs/1201.2853",
"abstract": "We define a \"renormalized energy\" as an explicit functional on arbitrary point configurations of constant average density in the plane and on the real lin... |
https://arxiv.org/abs/1902.08249 | A new stability test for linear neutral differential equations | We obtain new explicit exponential stability conditions for the linear scalar neutral equation with two bounded delays $
\dot{x}(t)-a(t)\dot{x}(g(t))+b(t)x(h(t))=0, $
where $
0\leq a(t)\leq A_0<1$, $0<b_0\leq b(t)\leq B$, using the Bohl-Perron theorem and a transformation of the neutral equation into a differential equ... | \section{Introduction}
Many applied problems lead to neutral differential equations as their mathematical models, for example,
a model of a controlled motion of a rigid body, a distributed network (a long line with tunnel diodes), models of infection diseases, a price model in economic dynamics, see, for example, \cit... | {
"timestamp": "2019-02-25T02:02:23",
"yymm": "1902",
"arxiv_id": "1902.08249",
"language": "en",
"url": "https://arxiv.org/abs/1902.08249",
"abstract": "We obtain new explicit exponential stability conditions for the linear scalar neutral equation with two bounded delays $\n\\dot{x}(t)-a(t)\\dot{x}(g(t))+b... |
https://arxiv.org/abs/1906.08944 | Explicit Artin maps into ${\rm PGL}_2$ | Let $G$ be a subgroup of ${\rm PGL}_2({\mathbb F}_q)$, where $q$ is any prime power, and let $Q \in {\mathbb F}_q[x]$ such that ${\mathbb F}_q(x)/{\mathbb F}_q(Q(x))$ is a Galois extension with group $G$. By explicitly computing the Artin map on unramified degree-1 primes in ${\mathbb F}_q(Q)$ for various groups $G$, i... | \section{Introduction} \label{sec:intro}
Let $K$ be a field and $G$ a finite subgroup of $\PGL_2(K)$. It is
well known, and will be proved in Section~\ref{sec:Q}, that there is
$Q \in K(x)$
such that $K(x)/K(Q(x))$ has Galois group~$G$. Normalize $Q$ so that
$Q(\infty) = \infty$; $Q$ will be called a {\it quoti... | {
"timestamp": "2021-04-05T02:02:04",
"yymm": "1906",
"arxiv_id": "1906.08944",
"language": "en",
"url": "https://arxiv.org/abs/1906.08944",
"abstract": "Let $G$ be a subgroup of ${\\rm PGL}_2({\\mathbb F}_q)$, where $q$ is any prime power, and let $Q \\in {\\mathbb F}_q[x]$ such that ${\\mathbb F}_q(x)/{\\... |
https://arxiv.org/abs/2006.01400 | Approximation Guarantees of Local Search Algorithms via Localizability of Set Functions | This paper proposes a new framework for providing approximation guarantees of local search algorithms. Local search is a basic algorithm design technique and is widely used for various combinatorial optimization problems. To analyze local search algorithms for set function maximization, we propose a new notion called l... | \section{Introduction}\label{sec:local-background}
Local search is a widely used technique to design efficient algorithms for optimization problems.
Roughly speaking, local search algorithms start with an initial solution and gradually increase the objective value by repeatedly moving the solution to a nearby point.
Wh... | {
"timestamp": "2020-06-03T02:09:45",
"yymm": "2006",
"arxiv_id": "2006.01400",
"language": "en",
"url": "https://arxiv.org/abs/2006.01400",
"abstract": "This paper proposes a new framework for providing approximation guarantees of local search algorithms. Local search is a basic algorithm design technique ... |
https://arxiv.org/abs/2212.03501 | Eight times four bialgebras of hypergraphs, cointeractions, and chromatic polynomials | We consider the bialgebra of hypergraphs, a generalization of Schmitt's Hopf algebra of graphs, and show it has a cointeracting bialgebra. So one has a double bialgebra in the sense of L. Foissy, who recently proved there is then a unique double bialgebra morphism to the double bialgebra structure on the polynomial rin... | \section{Introduction}
\label{sec:intro}
We introduce twelve bialgebras of hypergraphs, or three quartets of
bialgebras. Two of these quartets come as four double bialgebras.
This latter notion was recently introduced and their general theory developed
by L.~Foissy \cite{Fo22}. A double bialgebra is the same as two
co... | {
"timestamp": "2023-01-02T02:05:24",
"yymm": "2212",
"arxiv_id": "2212.03501",
"language": "en",
"url": "https://arxiv.org/abs/2212.03501",
"abstract": "We consider the bialgebra of hypergraphs, a generalization of Schmitt's Hopf algebra of graphs, and show it has a cointeracting bialgebra. So one has a do... |
https://arxiv.org/abs/1709.09738 | Formulations of the PFR Conjecture over $\mathbb{Z}$ | The polynomial Fre\uıman--Ruzsa conjecture is a fundamental open question in additive combinatorics. However, over the integers (or more generally $\mathbb{R}^d$ or $\mathbb{Z}^d$) the optimal formulation has not been fully pinned down.The conjecture states that a set of small doubling is controlled by a very structure... | \section{Introduction}
A celebrated theorem of Fre\u{\i}man \cite{freiman} states that if $A \subseteq \ZZ$ is a finite set satisfying the small doubling hypothesis $|A+A| \le K |A|$ for some small $K$, where $A +A$ is the sumset $\{x+y \colon x,y \in A\}$, then $A$ must be contained in a generalized arithmetic progre... | {
"timestamp": "2017-09-29T02:02:44",
"yymm": "1709",
"arxiv_id": "1709.09738",
"language": "en",
"url": "https://arxiv.org/abs/1709.09738",
"abstract": "The polynomial Fre\\uıman--Ruzsa conjecture is a fundamental open question in additive combinatorics. However, over the integers (or more generally $\\mat... |
https://arxiv.org/abs/math/9506224 | Topological conjugacy of circle diffeomorphisms | The classical criterion for a circle diffeomorphism to be topologically conjugate to an irrational rigid rotation was given by A. Denjoy. In 1985, one of us (Sullivan) gave a new criterion. There is an example satisfying Denjoy's bounded variation condition rather than Sullivan's Zygmund condition and vice versa. This ... | \section{Introduction}
Given a circle orientation preserving homeomorphism $f:
S^{1}\rightarrow S^{1}$,
the rotation
number \[\rho(f)=\lim_{n\rightarrow \infty}\frac{F^{n}(x)-x}{n} \;\;mod
\;\; 1\]
is independent of $x$ and the lift $F$ of $f$, where
$F:R^{1}\rightarrow R^{1}$ is a lift of $f$ and $x\in
R^{1}$. And it ... | {
"timestamp": "1999-11-25T20:34:38",
"yymm": "9506",
"arxiv_id": "math/9506224",
"language": "en",
"url": "https://arxiv.org/abs/math/9506224",
"abstract": "The classical criterion for a circle diffeomorphism to be topologically conjugate to an irrational rigid rotation was given by A. Denjoy. In 1985, one... |
https://arxiv.org/abs/2104.02635 | Quantitative ergodic theorems for actions of groups of polynomial growth | We strengthen the maximal ergodic theorem for actions of groups of polynomial growth to a form involving jump quantity, which is the sharpest result among the family of variational or maximal ergodic theorems. As a consequence, we deduce in this setting the quantitative ergodic theorem, in particular, the upcrossing in... | \section{Introduction}\label{ST1}
\subsection{Background and main results}
In the past few decades, a great deal of significant results related to the pointwise ergodic theorems for group actions have been established. The earliest pointwise ergodic theorems, to our knowledge, was obtained by Birkhoff~\cite{Birkhoff31}... | {
"timestamp": "2021-04-07T02:23:38",
"yymm": "2104",
"arxiv_id": "2104.02635",
"language": "en",
"url": "https://arxiv.org/abs/2104.02635",
"abstract": "We strengthen the maximal ergodic theorem for actions of groups of polynomial growth to a form involving jump quantity, which is the sharpest result among... |
https://arxiv.org/abs/1311.4450 | Counting subgraphs in hyperbolic graphs with symmetry | This note addresses some questions that arise in the series of works by Kyoji Saito on the growth functions of graphs. We study "hyperbolike" graphs, which include Cayley graphs of hyperbolic groups. We generalize some well-known results on hyperbolic groups to the hyperbolike setting, including rationality of generati... | \section{Introduction}
This note addresses some questions that arise in the series of works
by Kyoji Saito on the growth functions of graphs \cite{Sa}, \cite{Sa2}.
We study ``hyperbolike'' graphs, which include Cayley graphs of
hyperbolic groups. We generalize some well-known results on hyperbolic
groups to the hyperbo... | {
"timestamp": "2013-11-19T02:15:13",
"yymm": "1311",
"arxiv_id": "1311.4450",
"language": "en",
"url": "https://arxiv.org/abs/1311.4450",
"abstract": "This note addresses some questions that arise in the series of works by Kyoji Saito on the growth functions of graphs. We study \"hyperbolike\" graphs, whic... |
https://arxiv.org/abs/1709.00313 | A Simple Proof Characterizing Interval Orders with Interval Lengths between 1 and $k$ | A poset $P= (X, \prec)$ has an interval representation if each $x \in X$ can be assigned a real interval $I_x$ so that $x \prec y$ in $P$ if and only if $I_x$ lies completely to the left of $I_y$. Such orders are called \emph{interval orders}. Fishburn proved that for any positive integer $k$, an interval order has a r... | \section{Introduction}
\subsection{Posets and Interval Orders}
A poset $P$ consists of a set $X$ of \emph{points} and a relation $\prec$ that is irreflexive and transitive, and therefore antisymmetric. It is sometimes convenient to write $y \succ x$ instead of $x \prec y$. If $x \prec y$ or $y \prec x$, we say tha... | {
"timestamp": "2017-09-04T02:08:42",
"yymm": "1709",
"arxiv_id": "1709.00313",
"language": "en",
"url": "https://arxiv.org/abs/1709.00313",
"abstract": "A poset $P= (X, \\prec)$ has an interval representation if each $x \\in X$ can be assigned a real interval $I_x$ so that $x \\prec y$ in $P$ if and only i... |
https://arxiv.org/abs/1909.07270 | A Weighted $\ell_1$-Minimization Approach For Wavelet Reconstruction of Signals and Images | In this effort, we propose a convex optimization approach based on weighted $\ell_1$-regularization for reconstructing objects of interest, such as signals or images, that are sparse or compressible in a wavelet basis. We recover the wavelet coefficients associated to the functional representation of the object of inte... | \section{Introduction}
We investigate recovering an object of interest (OoI) from either a small number of samples
or a noisy version using a weighted $\ell_1$-norm regularized convex optimization scheme
with a specific choice of weights.
Throughout this effort, the functional representation of an OoI is given by
\b... | {
"timestamp": "2019-09-17T02:30:38",
"yymm": "1909",
"arxiv_id": "1909.07270",
"language": "en",
"url": "https://arxiv.org/abs/1909.07270",
"abstract": "In this effort, we propose a convex optimization approach based on weighted $\\ell_1$-regularization for reconstructing objects of interest, such as signa... |
https://arxiv.org/abs/2006.10430 | The turnpike property and the long-time behavior of the Hamilton-Jacobi-Bellman equation for finite-dimensional LQ control problems | We analyze the consequences that the so-called turnpike property has on the long-time behavior of the value function corresponding to a finite-dimensional linear-quadratic optimal control problem with general terminal cost and constrained controls.We prove that, when the time horizon $T$ tends to infinity, the value fu... | \section{Introduction}
\subsection{Motivation and setting}
We are interested in the asymptotic behavior of the value function associated to an optimal control problem, when the time-horizon tends to infinity.
In particular, we want to deduce it as a consequence of a property that is satisfied by a large class of op... | {
"timestamp": "2021-11-23T02:22:06",
"yymm": "2006",
"arxiv_id": "2006.10430",
"language": "en",
"url": "https://arxiv.org/abs/2006.10430",
"abstract": "We analyze the consequences that the so-called turnpike property has on the long-time behavior of the value function corresponding to a finite-dimensional... |
https://arxiv.org/abs/1511.04580 | On Erasure Combinatorial Batch Codes | Combinatorial batch codes were defined by Paterson, Stinson, and Wei as purely combinatorial versions of the batch codes introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai. There are $n$ items and $m$ servers, each of which stores a subset of the items. A batch code is an arrangement for storing items on servers so... | \section{Introduction}
We study a class of combinatorial objects that we call \emph{combinatorial batch codes with redundancy}. These are motivated by a data retrieval problem in which a collection of items (such as files) are stored, with possible duplication, on a collection of servers. After the items are stored, ... | {
"timestamp": "2015-11-17T02:07:24",
"yymm": "1511",
"arxiv_id": "1511.04580",
"language": "en",
"url": "https://arxiv.org/abs/1511.04580",
"abstract": "Combinatorial batch codes were defined by Paterson, Stinson, and Wei as purely combinatorial versions of the batch codes introduced by Ishai, Kushilevitz,... |
https://arxiv.org/abs/1103.4258 | $k$-Sum Decomposition of Strongly Unimodular Matrices | Networks are frequently studied algebraically through matrices. In this work, we show that networks may be studied in a more abstract level using results from the theory of matroids by establishing connections to networks by decomposition results of matroids. First, we present the implications of the decomposition of r... | \section{Introduction}
Totally unimodular (TU) matrices form an important class of matrices for integer and linear programming due to the integrality properties of
the associated polyhedron. A matrix $A$ is \emph{totally unimodular} if each square submatrix of $A$ has determinant $0,+1,$ or $-1$.
The class of TU matr... | {
"timestamp": "2011-03-23T01:01:37",
"yymm": "1103",
"arxiv_id": "1103.4258",
"language": "en",
"url": "https://arxiv.org/abs/1103.4258",
"abstract": "Networks are frequently studied algebraically through matrices. In this work, we show that networks may be studied in a more abstract level using results fr... |
https://arxiv.org/abs/1708.08024 | Analyticity of Bounded Solutions of Analytic State-Dependent Delay Differential Equations | We study the analyticity of bounded solutions of systems of analytic state-dependent delay differential equations. We obtain the analyticity of solutions by transforming the system of state-dependent delay equations into an abstract ordinary differential equation in a subspace of the sequence space $l^{\infty}(\mathbb{... | \section{Introduction}\label{SOPS-4-1}
The analyticity of bounded solutions of delay differential equations with constant delay such as the well-known Wright's equation was established in work of Nussbaum \cite{Nussbaum-analyticity}. It is natural to conjecture that this analyticity result holds true for many diffe... | {
"timestamp": "2017-08-29T02:06:00",
"yymm": "1708",
"arxiv_id": "1708.08024",
"language": "en",
"url": "https://arxiv.org/abs/1708.08024",
"abstract": "We study the analyticity of bounded solutions of systems of analytic state-dependent delay differential equations. We obtain the analyticity of solutions ... |
https://arxiv.org/abs/1811.06380 | Embeddings of Free Magmas with Applications to the Study of Free Non-Associative Algebras | We introduce an embedding of the free magma on a set A into the direct product of the free magma on a singleton set and the free semigroup on A. This embedding is then used to prove several theorems related to algebraic independence of subsets of the free non-associative algebra on A. Among these theorems is a generali... | \section{Preliminaries}
Throughout our entire discussion, we will let $A$ denote a fixed, nonempty set and we will let $\mathscr{M}[A]$ denote the free magma on $A$. Associated with each element $m$ of $\mathscr{M}[A]$ is a positive integer $\text{deg}(m)$ called the \emph{degree} of $m$. If $\circ_M$ denotes the pr... | {
"timestamp": "2018-11-16T02:13:30",
"yymm": "1811",
"arxiv_id": "1811.06380",
"language": "en",
"url": "https://arxiv.org/abs/1811.06380",
"abstract": "We introduce an embedding of the free magma on a set A into the direct product of the free magma on a singleton set and the free semigroup on A. This embe... |
https://arxiv.org/abs/1802.08246 | Characterizing Implicit Bias in Terms of Optimization Geometry | We study the implicit bias of generic optimization methods, such as mirror descent, natural gradient descent, and steepest descent with respect to different potentials and norms, when optimizing underdetermined linear regression or separable linear classification problems. We explore the question of whether the specifi... | \section{Introduction}\label{sec:intro}
Implicit bias from the optimization algorithm plays a crucial
role in learning deep neural networks as it introduces effective capacity control not directly specified in the objective \citep{neyshabur2015search,neyshabur2015path,zhang2017understanding, keskar2016large,wilson201... | {
"timestamp": "2018-10-23T02:20:38",
"yymm": "1802",
"arxiv_id": "1802.08246",
"language": "en",
"url": "https://arxiv.org/abs/1802.08246",
"abstract": "We study the implicit bias of generic optimization methods, such as mirror descent, natural gradient descent, and steepest descent with respect to differe... |
https://arxiv.org/abs/0811.1701 | Sufficient enlargements of minimal volume for finite dimensional normed linear spaces | Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is called a {\it sufficient enlargement} for $X$ if, for an arbitrary isometric embedding of $X$ into a Banach space $Y$, there exists a linear projection $P:Y\to X$ ... | \section{Introduction}
This paper is devoted to a generalization of the main results of
\cite{Ost04}, where similar results were proved in the dimension
two. We refer to \cite{Ost04,Ost07+} for more background and
motivation.
\subsection{Notation and definitions}
All linear spaces considered in this paper will be ov... | {
"timestamp": "2008-11-11T20:37:11",
"yymm": "0811",
"arxiv_id": "0811.1701",
"language": "en",
"url": "https://arxiv.org/abs/0811.1701",
"abstract": "Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric, bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$ is... |
https://arxiv.org/abs/1703.07951 | Sums of quadratic functions with two discriminants | Zagier in [4] discusses a construction of a function $F_{k,D}(x)$ defined for an even integer $k \geq 2$, and a positive discriminant $D$. This construction is intimately related to half-integral weight modular forms. In particular, the average value of this function is a constant multiple of the $D$-th Fourier coeffic... | \section{Introduction} \label{intro}
Let $\mathfrak{Q}_D$ be the set of all quadratic functions $Q=ax^2+bx+c=[a,b,c]$ with integer coefficients and of discriminant $D=b^2-4ac>0$. For an even positive integer $k \geq 2$, Zagier \cite{Zagier} defines the function $F_{k,D}: \mathbb{R} \to \mathbb{R}$ by
\begin{equation... | {
"timestamp": "2017-03-24T01:03:22",
"yymm": "1703",
"arxiv_id": "1703.07951",
"language": "en",
"url": "https://arxiv.org/abs/1703.07951",
"abstract": "Zagier in [4] discusses a construction of a function $F_{k,D}(x)$ defined for an even integer $k \\geq 2$, and a positive discriminant $D$. This construct... |
https://arxiv.org/abs/0810.0092 | On the pre-image of a point under an isogeny | Given a rational point on a curve in a rational isogeny class, a natural question concerns the field of definition of its pre-images. The multiplication by m endomorphism is a powerful and much-used tool. The pre-images for this map are found by factorizing a monic polynomial of degree m^2. For m = 2, Everest and King ... | \section{Introduction}
Given an elliptic curve $E/ \mathbb{Q}$, the set of all curves $E'$ isogenous to $E$ over $\mathbb{Q}$ is finite (up to
isomorphism) and is known as an isogeny class. V\'{e}lu's formulae \cite{1} and the Weierstrass parameterization of
the elliptic curve can be used to find an isogeny class. Thi... | {
"timestamp": "2008-10-01T09:38:04",
"yymm": "0810",
"arxiv_id": "0810.0092",
"language": "en",
"url": "https://arxiv.org/abs/0810.0092",
"abstract": "Given a rational point on a curve in a rational isogeny class, a natural question concerns the field of definition of its pre-images. The multiplication by ... |
https://arxiv.org/abs/2104.02853 | An SEIR epidemic model of fractional order to analyze the evolution of the COVID-19 epidemic in Argentina | A pandemic caused by a new coronavirus (COVID-19) has spread worldwide, inducing an epidemic still active in Argentina. In this chapter, we present a case study using an SEIR (Susceptible-Exposed-Infected-Recovered) diffusion model of fractional order in time to analyze the evolution of the epidemic in Buenos Aires and... | \section{Introduction}\label{intro}
We present an SEIR subdiffusion model
of fractional order $\nu$, with $0 <\nu \le 1$ to analyze the time evolution of the COVID-19 epidemic in Buenos Aires and neighboring areas (Region Metropolitana de Buenos Aires, (RMBA)) with a population of about 15 million inhabitants.
... | {
"timestamp": "2021-04-08T02:07:44",
"yymm": "2104",
"arxiv_id": "2104.02853",
"language": "en",
"url": "https://arxiv.org/abs/2104.02853",
"abstract": "A pandemic caused by a new coronavirus (COVID-19) has spread worldwide, inducing an epidemic still active in Argentina. In this chapter, we present a case... |
https://arxiv.org/abs/math/0605754 | String cohomology groups of complex projective spaces | Let X be a space and write LX for its free loop space equipped with the action of the circle group T given by dilation. We compute the equivariant cohomology H^*(LX_hT; Z/p) as a module over H^*(BT; Z/p) when X=CP^r for any positive integer r and any prime number p. The computation implies that the associated mod p Ser... | \section{Introduction}
Let $LX$ be the space of maps from the circle to a space
$X$. The circle acts on itself by rotation, and this action
induces an action of the circle group ${{\mathbb T}}=S^1=SO(2)$ on $LX$.
(The action extends to an $O(2)$-action, but we will not consider
the extended action in this paper).
Th... | {
"timestamp": "2006-05-30T14:48:01",
"yymm": "0605",
"arxiv_id": "math/0605754",
"language": "en",
"url": "https://arxiv.org/abs/math/0605754",
"abstract": "Let X be a space and write LX for its free loop space equipped with the action of the circle group T given by dilation. We compute the equivariant coh... |
https://arxiv.org/abs/1004.1596 | Strict inequalities of critical probabilities on Gilbert's continuum percolation graph | Any infinite graph has site and bond percolation critical probabilities satisfying $p_c^{site}\geq p_c^{bond}$. The strict version of this inequality holds for many, but not all, infinite graphs.In this paper, the class of graphs for which the strict inequality holds is extended to a continuum percolation model. In Gi... | \section{Introduction}
\label{secintro}
Consider an infinite connected graph $G$ and perform bond
percolation by independently marking each edge open with probability
$p$ and closed otherwise. The critical probability $p_c^{\rm bond}$
refers to the value of $p$ above which there exists almost surely
(a.s.) an infinite ... | {
"timestamp": "2010-04-30T02:01:50",
"yymm": "1004",
"arxiv_id": "1004.1596",
"language": "en",
"url": "https://arxiv.org/abs/1004.1596",
"abstract": "Any infinite graph has site and bond percolation critical probabilities satisfying $p_c^{site}\\geq p_c^{bond}$. The strict version of this inequality hold... |
https://arxiv.org/abs/1705.03851 | Rotational subsets of the circle | A rotational subset, relative to a continuous transformation $T: \mathbb{T} \to \mathbb{T}$ on the unit circle, is a closed, invariant subset of $\mathbb{T}$ that is minimal and on which $T$ respects the standard orientation of the unit circle. In the case where $T$ is the standard angle doubling map, such subsets were... | \section*{Introduction}
In what follows, $\mathbb{T}$ denotes the unit circle with the standard orientation.
\begin{defn}
Let $X \subset \mathbb{T}$ and $f: X \to X$ be a continuous transformation. The
map $f$ \emph{preserves cyclic order} if, for any $P, Q, R \in X$ with
distinct images, the arcs $P\,Q\,R$ and $f... | {
"timestamp": "2017-12-19T02:02:33",
"yymm": "1705",
"arxiv_id": "1705.03851",
"language": "en",
"url": "https://arxiv.org/abs/1705.03851",
"abstract": "A rotational subset, relative to a continuous transformation $T: \\mathbb{T} \\to \\mathbb{T}$ on the unit circle, is a closed, invariant subset of $\\mat... |
https://arxiv.org/abs/0812.4977 | Decay of mass for nonlinear equation with fractional Laplacian | The large time behavior of nonnegative solutions to the reaction-diffusion equation $\partial_t u=-(-\Delta)^{\alpha/2}u - u^p,$ $(\alpha\in(0,2], p>1)$ posed on $\mathbb{R}^N$ and supplemented with an integrable initial condition is studied. We show that the anomalous diffusion term determines the large time asymptoti... | \section{Introduction}
We study the behavior, as $t\to\infty$, of solutions to the
following initial value problem for the reaction-diffusion equation with
the anomalous diffusion
\begin{eqnarray}\label{eq}
\partial_t u &=& -\Lambda^\alpha u + \lambda u^p,\qquad x\in\mathbb{R}^N,t>0,\\
u(x,0) &=& u_0(x),\label{ini... | {
"timestamp": "2008-12-29T22:45:11",
"yymm": "0812",
"arxiv_id": "0812.4977",
"language": "en",
"url": "https://arxiv.org/abs/0812.4977",
"abstract": "The large time behavior of nonnegative solutions to the reaction-diffusion equation $\\partial_t u=-(-\\Delta)^{\\alpha/2}u - u^p,$ $(\\alpha\\in(0,2], p>1)... |
https://arxiv.org/abs/1605.04829 | The degree of commutativity and lamplighter groups | The degree of commutativity of a group $G$ measures the probability of choosing two elements in $G$ which commute. There are many results studying this for finite groups. In [AMV17], this was generalised to infinite groups. In this note, we compute the degree of commutativity for wreath products of the form $\mathbb{Z}... | \section{Introduction}
Let $F$ be a finite group. Then degree of commutativity of $F$, denoted $\dc(F)$, is the probability of choosing two elements in $F$ which commute i.e.\
\[\dc(F):=\frac{|\{(a, b) \in F^2 : ab=ba\}|}{|F|^2}.\]
This definition was generalised to infinite groups in \cite{dcA} in the following way.... | {
"timestamp": "2016-05-17T02:16:22",
"yymm": "1605",
"arxiv_id": "1605.04829",
"language": "en",
"url": "https://arxiv.org/abs/1605.04829",
"abstract": "The degree of commutativity of a group $G$ measures the probability of choosing two elements in $G$ which commute. There are many results studying this fo... |
https://arxiv.org/abs/1910.10828 | Superconvergent flux recovery of the Rannacher-Turek nonconforming element | This work presents superconvergence estimates of the nonconforming Rannacher--Turek element for second order elliptic equations on any cubical meshes in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$. In particular, a corrected numerical flux is shown to be superclose to the Raviart--Thomas interpolant of the exact flux. We the... | \section{Introduction and preliminaries}\label{sec1}
Finite element superconvergent recovery is quite popular in practice for its simplicity and ability to develop asymptotically exact a posteriori error estimators. The theory of superconvergent recovery for conforming Lagrange elements is well-established, see, e.g., ... | {
"timestamp": "2021-03-10T02:26:37",
"yymm": "1910",
"arxiv_id": "1910.10828",
"language": "en",
"url": "https://arxiv.org/abs/1910.10828",
"abstract": "This work presents superconvergence estimates of the nonconforming Rannacher--Turek element for second order elliptic equations on any cubical meshes in $... |
https://arxiv.org/abs/2009.04266 | The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation | Comparing metric measure spaces (i.e. a metric space endowed with aprobability distribution) is at the heart of many machine learning problems. The most popular distance between such metric measure spaces is theGromov-Wasserstein (GW) distance, which is the solution of a quadratic assignment problem. The GW distance is... |
\section{Conclusion}
This paper defines two Unbalanced Gromov-Wasserstein formulations. We prove that they are both positive and definite. We provide a scalable, GPU-friendly algorithm to compute one of them, and show that the other is a distance between mm-spaces up to isometry.
These divergences and distances allow... | {
"timestamp": "2021-06-09T02:13:09",
"yymm": "2009",
"arxiv_id": "2009.04266",
"language": "en",
"url": "https://arxiv.org/abs/2009.04266",
"abstract": "Comparing metric measure spaces (i.e. a metric space endowed with aprobability distribution) is at the heart of many machine learning problems. The most p... |
https://arxiv.org/abs/2005.02185 | A note on total co-independent domination in trees | A set $D$ of vertices of a graph $G$ is a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $D$. The total domination number of $G$ is the minimum cardinality of any total dominating set of $G$ and is denoted by $\gamma_t(G)$. The total dominating set $D$ is called a total co-independent... | \section{Introduction} \label{Intro}
Throughout this work we consider $G=(V(G),E(G))$ as a simple graph of order $n=|V(G)|$. That is, graphs that are finite, undirected, and without loops or multiple edges. Given a vertex $v$ of $G$, $N_G(v)$ represents the \emph{open neighborhood} of $v$, \emph{i.e.}, the set of all ... | {
"timestamp": "2020-05-06T02:15:11",
"yymm": "2005",
"arxiv_id": "2005.02185",
"language": "en",
"url": "https://arxiv.org/abs/2005.02185",
"abstract": "A set $D$ of vertices of a graph $G$ is a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $D$. The total domination numb... |
https://arxiv.org/abs/2211.05611 | Modular forms via invariant theory | We discuss two simple but useful observations that allow the construction of modular forms from given ones using invariant theory. The first one deals with elliptic modular forms and their derivatives, and generalizes the Rankin-Cohen bracket, while the second one deals with vector-valued modular forms of genus greater... | \section*{Acknowledgements}
The first author was supported by Simons Foundation Award
546235 at the Institute for Computational and
Experimental Research in Mathematics at Brown University.
The second author thanks ICERM for hospitality enjoyed during a visit
when this paper was written. We thank Carel Faber for useful... | {
"timestamp": "2022-11-11T02:14:28",
"yymm": "2211",
"arxiv_id": "2211.05611",
"language": "en",
"url": "https://arxiv.org/abs/2211.05611",
"abstract": "We discuss two simple but useful observations that allow the construction of modular forms from given ones using invariant theory. The first one deals wit... |
https://arxiv.org/abs/2103.08463 | How to distribute data across tasks for meta-learning? | Meta-learning models transfer the knowledge acquired from previous tasks to quickly learn new ones. They are trained on benchmarks with a fixed number of data points per task. This number is usually arbitrary and it is unknown how it affects performance at testing. Since labelling of data is expensive, finding the opti... | \section{Introduction}
Deep learning (DL) models require a large amount of data in order to perform well, when trained from scratch, but labeling data is expensive and time consuming.
An effective approach to avoid the costs of collecting and labeling a large amount of data is transfer learning: train a model on one b... | {
"timestamp": "2022-04-11T02:11:34",
"yymm": "2103",
"arxiv_id": "2103.08463",
"language": "en",
"url": "https://arxiv.org/abs/2103.08463",
"abstract": "Meta-learning models transfer the knowledge acquired from previous tasks to quickly learn new ones. They are trained on benchmarks with a fixed number of ... |
https://arxiv.org/abs/0811.4576 | Concentration of the integral norm of idempotents | This is a companion paper of a recent one, entitled {\sl Integral concentration of idempotent trigonometric polynomials with gaps}. New results of the present work concern $L^1$ concentration, while the above mentioned paper deals with $L^p$-concentration.Our aim here is two-fold. At the first place we try to explain m... | \section{Introduction and statement of results}\label{sec:intro}
The problem of $p$-concentration
on the torus for idempotent polynomials has been considered first
in \cite{first}, \cite{CRMany}, \cite{CRSome}, \cite{DPQ}. We use
the notation $\mathbb T:=\mathbb R/\mathbb Z$ for the torus. Then $e(t):=e^{2\pi i
t}$ ... | {
"timestamp": "2008-11-27T17:07:01",
"yymm": "0811",
"arxiv_id": "0811.4576",
"language": "en",
"url": "https://arxiv.org/abs/0811.4576",
"abstract": "This is a companion paper of a recent one, entitled {\\sl Integral concentration of idempotent trigonometric polynomials with gaps}. New results of the pres... |
https://arxiv.org/abs/1701.07569 | Data-Driven Sparse Sensor Placement for Reconstruction | Optimal sensor placement is a central challenge in the design, prediction, estimation, and control of high-dimensional systems. High-dimensional states can often leverage a latent low-dimensional representation, and this inherent compressibility enables sparse sensing. This article explores optimized sensor placement f... | \subsection{Extensions to dynamics, control, and multiscale physics}
Data-driven sensor selection is generally used for instantaneous full-state reconstruction, despite the fact that many signals are generated by a dynamical system~\cite{guckenheimer_holmes,HLBR_turb}.
Even in reduced-order models, sensors are typica... | {
"timestamp": "2017-08-21T02:06:02",
"yymm": "1701",
"arxiv_id": "1701.07569",
"language": "en",
"url": "https://arxiv.org/abs/1701.07569",
"abstract": "Optimal sensor placement is a central challenge in the design, prediction, estimation, and control of high-dimensional systems. High-dimensional states ca... |
https://arxiv.org/abs/0704.0656 | Necessary optimality conditions for the calculus of variations on time scales | We study more general variational problems on time scales. Previous results are generalized by proving necessary optimality conditions for (i) variational problems involving delta derivatives of more than the first order, and (ii) problems of the calculus of variations with delta-differential side conditions (Lagrange ... | \section{Introduction}
The theory of time scales is a relatively new area, that unify and
generalize difference and differential equations \cite{livro}. It
was initiated by Stefan Hilger in the nineties of the XX century
\cite{Hilger90,Hilger97}, and is now subject of strong current
research in many different fields i... | {
"timestamp": "2007-04-04T22:31:16",
"yymm": "0704",
"arxiv_id": "0704.0656",
"language": "en",
"url": "https://arxiv.org/abs/0704.0656",
"abstract": "We study more general variational problems on time scales. Previous results are generalized by proving necessary optimality conditions for (i) variational p... |
https://arxiv.org/abs/1002.4432 | Adjoint action of automorphism groups on radical endomorphisms, generic equivalence and Dynkin quivers | Let $Q$ be a connected quiver with no oriented cycles, $k$ the field of complex numbers and $P$ a projective representation of $Q$. We study the adjoint action of the automorphism group $\Aut_{kQ} P$ on the space of radical endomorphisms $\radE_{kQ}P$. Using generic equivalence, we show that the quiver $Q$ has the prop... | \section*{Introduction}
Let $\Delta$ be a quiver and let $P$ be a projective representation.
We study generic orbits for the adjoint action of $Aut P$ on the
radical endomorphisms $radEndP$. If $\Delta$ is of type $\mathbb{A}$
with linear orientation, then $End P$ is a parabolic subalgebra in
$\mathfrak{gl}_n$ and a d... | {
"timestamp": "2010-03-10T02:00:33",
"yymm": "1002",
"arxiv_id": "1002.4432",
"language": "en",
"url": "https://arxiv.org/abs/1002.4432",
"abstract": "Let $Q$ be a connected quiver with no oriented cycles, $k$ the field of complex numbers and $P$ a projective representation of $Q$. We study the adjoint act... |
https://arxiv.org/abs/2012.14480 | On free subalgebras of varieties | We show that some results of L. Makar-Limanov, P. Malcolmson and Z. Reichstein on the existence of free associative algebras are valid in the more general context of varieties of algebras. | \section*{Introduction}
As stated in \cite[Conjecture~1.1]{Agata}, L. Makar-Limanov made the following conjecture:
\begin{conjecture}
Let $K$ be a field, $A$ be an associative $K$-algebra and $F$ be a field extension of $K$.
If $F\otimes_KA$ contains a free $K$-algebra on at least two free generators, then $A$ also ... | {
"timestamp": "2021-01-01T02:01:27",
"yymm": "2012",
"arxiv_id": "2012.14480",
"language": "en",
"url": "https://arxiv.org/abs/2012.14480",
"abstract": "We show that some results of L. Makar-Limanov, P. Malcolmson and Z. Reichstein on the existence of free associative algebras are valid in the more general... |
https://arxiv.org/abs/2302.08661 | Subsampling Suffices for Adaptive Data Analysis | Ensuring that analyses performed on a dataset are representative of the entire population is one of the central problems in statistics. Most classical techniques assume that the dataset is independent of the analyst's query and break down in the common setting where a dataset is reused for multiple, adaptively chosen, ... |
\subsubsection{A mechanism for statistical queries}
\label{subsec:SQ}
Our main application is an extremely simple and accurate mechanism for the broad class of \emph{statistical queries}. Statistical queries, introduced by Kearns \cite{Kea98}, are parameterized by a function $\phi: X \to [0,1]$. A valid answer to suc... | {
"timestamp": "2023-02-20T02:05:46",
"yymm": "2302",
"arxiv_id": "2302.08661",
"language": "en",
"url": "https://arxiv.org/abs/2302.08661",
"abstract": "Ensuring that analyses performed on a dataset are representative of the entire population is one of the central problems in statistics. Most classical tec... |
https://arxiv.org/abs/2106.01608 | A Discussion On the Validity of Manifold Learning | Dimensionality reduction (DR) and manifold learning (ManL) have been applied extensively in many machine learning tasks, including signal processing, speech recognition, and neuroinformatics. However, the understanding of whether DR and ManL models can generate valid learning results remains unclear. In this work, we i... | \section{Appendix one: Analysis of FPLM and Geometric guarantees} \label{Append1}
In this Appendix one, we first provide the algebraic solution of FPLM; then prove that the graph from the triangulation on 2-manifold is planar; thirdly, we show that the constraints that we assign to each round of FPLM can ensure that F... | {
"timestamp": "2021-06-04T02:11:14",
"yymm": "2106",
"arxiv_id": "2106.01608",
"language": "en",
"url": "https://arxiv.org/abs/2106.01608",
"abstract": "Dimensionality reduction (DR) and manifold learning (ManL) have been applied extensively in many machine learning tasks, including signal processing, spee... |
https://arxiv.org/abs/2112.05745 | A Simple and Efficient Sampling-based Algorithm for General Reachability Analysis | In this work, we analyze an efficient sampling-based algorithm for general-purpose reachability analysis, which remains a notoriously challenging problem with applications ranging from neural network verification to safety analysis of dynamical systems. By sampling inputs, evaluating their images in the true reachable ... | \section{Introduction}\label{sec:intro}
\begin{wrapfigure}{!R}{0.44\linewidth}
\begin{minipage}{0.95\linewidth}
\vspace{-12.5mm}
\includegraphics[width=1\linewidth,trim=0 0 0 120, clip]{figs/RandUP4.png}
\vspace{-5mm}
\caption{$\epsilon$-\textsc{RandUP}\xspace consists of three simple steps: 1) sampling $M$ in... | {
"timestamp": "2021-12-13T02:25:14",
"yymm": "2112",
"arxiv_id": "2112.05745",
"language": "en",
"url": "https://arxiv.org/abs/2112.05745",
"abstract": "In this work, we analyze an efficient sampling-based algorithm for general-purpose reachability analysis, which remains a notoriously challenging problem ... |
https://arxiv.org/abs/1608.01189 | On the power propagation time of a graph | In this paper, we give Nordhaus-Gaddum upper and lower bounds on the sum of the power propagation time of a graph and its complement, and we consider the effects of edge subdivisions and edge contractions on the power propagation time of a graph. We also study a generalization of power propagation time, known as $k-$po... | \subsection{General Bounds}
\abstract {In this paper, we characterize all graphs $G$ with extreme $k-$power propagation time $|G|-1$ or $|G|-2$ for $k\geq 1,$ and $|G|-3$ for $k\geq 2$. We determine all trees $T$ whose $1-$power propagation time (also called power propagation time or {\em standard} pow... | {
"timestamp": "2016-08-04T02:08:28",
"yymm": "1608",
"arxiv_id": "1608.01189",
"language": "en",
"url": "https://arxiv.org/abs/1608.01189",
"abstract": "In this paper, we give Nordhaus-Gaddum upper and lower bounds on the sum of the power propagation time of a graph and its complement, and we consider the ... |
https://arxiv.org/abs/2106.02452 | Proving Equivalence Between Complex Expressions Using Graph-to-Sequence Neural Models | We target the problem of provably computing the equivalence between two complex expression trees. To this end, we formalize the problem of equivalence between two such programs as finding a set of semantics-preserving rewrite rules from one into the other, such that after the rewrite the two programs are structurally i... |
\section{Introduction}
Text of paper \ldots
\begin{acks}
This material is based upon work supported by the
\grantsponsor{GS100000001}{National Science
Foundation}{http://dx.doi.org/10.13039/100000001} under Grant
No.~\grantnum{GS10... | {
"timestamp": "2021-06-10T02:09:15",
"yymm": "2106",
"arxiv_id": "2106.02452",
"language": "en",
"url": "https://arxiv.org/abs/2106.02452",
"abstract": "We target the problem of provably computing the equivalence between two complex expression trees. To this end, we formalize the problem of equivalence bet... |
https://arxiv.org/abs/1801.10607 | Hypercube Packings and Coverings with Higher Dimensional Rooks | We introduce a generalization of classical $q$-ary codes by allowing points to cover other points that are Hamming distance $1$ or $2$ in a freely chosen subset of all directions. More specifically, we generalize the notion of $1$-covering, $1$-packing, and $2$-packing in the case of $q$-ary codes. In the covering case... | \section{Introduction}
Consider a set of $n$ football matches which each end in either a win, draw, or loss. How many bets are necessary for an individual to guarantee that they predict at least $n-1$ of outcomes of the games correctly? What about having at least $n-k$ outcomes correct?
The above problem is the classi... | {
"timestamp": "2018-02-01T02:12:26",
"yymm": "1801",
"arxiv_id": "1801.10607",
"language": "en",
"url": "https://arxiv.org/abs/1801.10607",
"abstract": "We introduce a generalization of classical $q$-ary codes by allowing points to cover other points that are Hamming distance $1$ or $2$ in a freely chosen ... |
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