url stringlengths 31 38 | title stringlengths 7 229 | abstract stringlengths 44 2.87k | text stringlengths 319 2.51M | meta dict |
|---|---|---|---|---|
https://arxiv.org/abs/2003.01423 | A uniform result for the dimension of fractional Brownian motion level sets | Let $B =\{ B_t \, : \, t \geq 0 \}$ be a real-valued fractional Brownian motion of index $H \in (0,1)$. We prove that the macroscopic Hausdorff dimension of the level sets $\mathcal{L}_x = \left\{ t \in \mathbb{R}_+ \, : \, B_t=x \right\}$ is, with probability one, equal to $1-H$ for all $x\in\mathbb{R}$. | \section{Introduction}
Let $B =\{ B_t \, : \, t \geq 0 \}$ be a fractional Brownian motion of index $H \in (0,1)$, that is, a centered, real-valued Gaussian process with covariance function
\begin{align}
R(s,t)= \mathbb{E} \left( B_s B_t\right)= \dfrac{1}{2}\left( \lvert s \rvert ^{2H} + \lvert t \rvert ^{2H} - \lvert... | {
"timestamp": "2020-03-04T02:11:54",
"yymm": "2003",
"arxiv_id": "2003.01423",
"language": "en",
"url": "https://arxiv.org/abs/2003.01423",
"abstract": "Let $B =\\{ B_t \\, : \\, t \\geq 0 \\}$ be a real-valued fractional Brownian motion of index $H \\in (0,1)$. We prove that the macroscopic Hausdorff dime... |
https://arxiv.org/abs/0812.1817 | Least-Squares Approximation by Elements from Matrix Orbits Achieved by Gradient Flows on Compact Lie Groups | Let $S(A)$ denote the orbit of a complex or real matrix $A$ under a certain equivalence relation such as unitary similarity, unitary equivalence, unitary congruences etc. Efficient gradient-flow algorithms are constructed to determine the best approximation of a given matrix $A_0$ by the sum of matrices in $S(A_1), ...... | \section{Introduction}
\setcounter{equation}{0}
Motivated by problems in pure and applied areas, there has been a
great deal of interest in studying equivalence classes on matrices, say,
under compact Lie group actions. For instance,
(a) the unitary (orthogonal)
similarity orbit of a complex (real) square matrix $A$ ... | {
"timestamp": "2008-12-10T00:48:37",
"yymm": "0812",
"arxiv_id": "0812.1817",
"language": "en",
"url": "https://arxiv.org/abs/0812.1817",
"abstract": "Let $S(A)$ denote the orbit of a complex or real matrix $A$ under a certain equivalence relation such as unitary similarity, unitary equivalence, unitary co... |
https://arxiv.org/abs/2006.04162 | The q-voter model on the torus | In the $q$-voter model, the voter at $x$ changes its opinion at rate $f_x^q$, where $f_x$ is the fraction of neighbors with the opposite opinion. Mean-field calculations suggest that there should be coexistence between opinions if $q<1$ and clustering if $q>1$. This model has been extensively studied by physicists, but... | \section{Introduction}
In the linear voter model, the state at time $t$ is $\xi_t :\mathbb{Z}^d \to \{0,1\}$, where 0 and 1 are two opinions. The individual at $x$ changes opinion at a rate equal to the fraction $f_x$ of its neighbors with the opposite opinion. For the last decade physicists have studied the $q$-voter... | {
"timestamp": "2020-06-09T02:18:51",
"yymm": "2006",
"arxiv_id": "2006.04162",
"language": "en",
"url": "https://arxiv.org/abs/2006.04162",
"abstract": "In the $q$-voter model, the voter at $x$ changes its opinion at rate $f_x^q$, where $f_x$ is the fraction of neighbors with the opposite opinion. Mean-fie... |
https://arxiv.org/abs/1406.0716 | A strict undirected model for the $k$-nearest neighbour graph | Let $G=G_{n,k}$ denote the graph formed by placing points in a square of area $n$ according to a Poisson process of density 1 and joining each pair of points which are both $k$ nearest neighbours of each other. Then $G_{n,k}$ can be used as a model for wireless networks, and has some advantages in terms of applications... | \section{Introduction}
Let $G_{n,k}$ be the graph formed by placing points in $S_{n}$, a $\sqrt{n}\times\sqrt{n}$ square, according to a Poisson process of density $1$ and connecting two points if they are both $k$-nearest neighbours of each other (i.e. one of the $k$-nearest points in $S_{n}$). We will refer to this a... | {
"timestamp": "2014-06-04T02:09:17",
"yymm": "1406",
"arxiv_id": "1406.0716",
"language": "en",
"url": "https://arxiv.org/abs/1406.0716",
"abstract": "Let $G=G_{n,k}$ denote the graph formed by placing points in a square of area $n$ according to a Poisson process of density 1 and joining each pair of point... |
https://arxiv.org/abs/1412.4595 | Maximal-clique partitions and the Roller Coaster Conjecture | A graph $G$ is {\em well-covered} if every maximal independent set has the same cardinality $q$. Let $i_k(G)$ denote the number of independent sets of cardinality $k$ in $G$. Brown, Dilcher, and Nowakowski conjectured that the independence sequence $(i_0(G), i_1(G), \ldots, i_q(G))$ was unimodal for any well-ordered gr... | \section{Introduction}\label{S:introduction}
The behavior of the coefficients of the independence polynomial of graphs in various classes has produced many interesting problems. For a graph $G$, we let $\mathcal{I}(G)$ be the set of independent sets in $G$, i.e., $\mathcal{I}(G)=\setof{I\subseteq V(G)}{E(G[I])=\empty... | {
"timestamp": "2014-12-16T02:19:22",
"yymm": "1412",
"arxiv_id": "1412.4595",
"language": "en",
"url": "https://arxiv.org/abs/1412.4595",
"abstract": "A graph $G$ is {\\em well-covered} if every maximal independent set has the same cardinality $q$. Let $i_k(G)$ denote the number of independent sets of card... |
https://arxiv.org/abs/1903.01845 | Maximal orthogonal sets of unimodular vectors over finite local rings of odd characteristic | Let $R$ be a finite local ring of odd characteristic and $\beta$ a non-degenerate symmetric bilinear form on $R^2$.In this short note, we determine the largest possible cardinality of pairwise orthogonal sets of unimodular vectors in $R^2$. | \section{Introduction}
Two famous distinct distances and unit distances problems for the plane $\mathbb{R}^2$ were posed by Erd{\H o}s \cite{Erdos1946}. The first problem asks for the minimum number of distinct distances among $n$ points in the plane while the latter problem asks for the maximum number of the unit d... | {
"timestamp": "2019-03-06T02:20:44",
"yymm": "1903",
"arxiv_id": "1903.01845",
"language": "en",
"url": "https://arxiv.org/abs/1903.01845",
"abstract": "Let $R$ be a finite local ring of odd characteristic and $\\beta$ a non-degenerate symmetric bilinear form on $R^2$.In this short note, we determine the l... |
https://arxiv.org/abs/1510.08417 | Monotone Projection Lower Bounds from Extended Formulation Lower Bounds | In this short note, we reduce lower bounds on monotone projections of polynomials to lower bounds on extended formulations of polytopes. Applying our reduction to the seminal extended formulation lower bounds of Fiorini, Massar, Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014; J. ACM, 2017)... | \section{Introduction} \label{sec:intro}
The permanent
\[
\perm_n(X) = \sum_{\pi \in S_n} x_{1,\pi(1)} x_{2,\pi(2)} \dotsb x_{n,\pi(n)}
\]
(where $S_n$ denotes the symmetric group of all permutations of $\{1,\dotsc,n\}$) has long fascinated combinatorialists \cite{minc,vanLintWilson,muirMetzler}, more recently physicis... | {
"timestamp": "2016-12-26T02:04:55",
"yymm": "1510",
"arxiv_id": "1510.08417",
"language": "en",
"url": "https://arxiv.org/abs/1510.08417",
"abstract": "In this short note, we reduce lower bounds on monotone projections of polynomials to lower bounds on extended formulations of polytopes. Applying our redu... |
https://arxiv.org/abs/1805.01368 | Homological stability for spaces of commuting elements in Lie groups | In this paper we study homological stability for spaces ${\rm Hom}(\mathbb{Z}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that for each $n\geqslant 1$, these spaces satisfy rational homological stability as $G$ ranges through any of the classical sequences of compact, connected Lie groups, or th... | \section{Introduction}
Let $G$ be a compact, connected Lie group. The focus of this article is the space ${\rm Hom}({\mathbb Z}^n,G)$
consisting of all group homomorphisms $\rho: {\mathbb Z}^n \to G$.
The standard basis for ${\mathbb Z}^n$ defines an embedding of ${\rm Hom}({\mathbb Z}^n, G)\hookrightarrow G^n$, an... | {
"timestamp": "2020-04-30T02:02:37",
"yymm": "1805",
"arxiv_id": "1805.01368",
"language": "en",
"url": "https://arxiv.org/abs/1805.01368",
"abstract": "In this paper we study homological stability for spaces ${\\rm Hom}(\\mathbb{Z}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that f... |
https://arxiv.org/abs/2012.01503 | A density bound for triangle-free $4$-critical graphs | We prove that every triangle-free $4$-critical graph $G$ satisfies $e(G) \geq \frac{5v(G)+2}{3}$. This result gives a unified proof that triangle-free planar graphs are $3$-colourable, and that graphs of girth at least five which embed in either the projective plane, torus, or Klein Bottle are $3$-colourable, which are... | \section{Introduction}
Given two graphs $G$ and $H$, a \textit{homomorphism} from $G$ to $H$ is a map $f:V(G) \to V(H)$ such that for any edge $xy \in E(G)$, we have that $f(x)f(y) \in E(H)$. A \textit{$k$-colouring} of a graph $G$ is a homomorphism from $G$ to $K_{k}$. A graph $G$ is $k$-critical if $G$ is $k$-colou... | {
"timestamp": "2020-12-04T02:03:11",
"yymm": "2012",
"arxiv_id": "2012.01503",
"language": "en",
"url": "https://arxiv.org/abs/2012.01503",
"abstract": "We prove that every triangle-free $4$-critical graph $G$ satisfies $e(G) \\geq \\frac{5v(G)+2}{3}$. This result gives a unified proof that triangle-free p... |
https://arxiv.org/abs/1311.2965 | Derived subdivisions make every PL sphere polytopal | We give a simple proof that some iterated derived subdivision of every PL sphere is combinatorially equivalent to the boundary of a simplicial polytope, thereby resolving a problem of Billera (personal communication). | \subsection{Making any PL sphere polytopal}
A \Defn{subdivision} of a simplicial complex $\Delta$ is a simplicial complex
$\Delta'$ with the same underlying space as $\Delta$, such that for every face
$D'$ of $\Delta'$ there is some face $D$ of $\Delta$ for which $D' \subset D$.
One also says that $\Delta'$ is a \De... | {
"timestamp": "2014-03-21T01:10:34",
"yymm": "1311",
"arxiv_id": "1311.2965",
"language": "en",
"url": "https://arxiv.org/abs/1311.2965",
"abstract": "We give a simple proof that some iterated derived subdivision of every PL sphere is combinatorially equivalent to the boundary of a simplicial polytope, the... |
https://arxiv.org/abs/2109.05556 | Family of $\mathscr{D}$-modules and representations with a boundedness property | In the representation theory of real reductive Lie groups, many objects have finiteness properties. For example, the lengths of Verma modules and principal series representations are finite, and more precisely, they are bounded. In this paper, we introduce a notion of uniformly bounded families of holonomic $\mathscr{D... | \section{Application to representation theory}\label{sect:applications}
In this section, we define a notion of uniformly bounded family of $\lie{g}$-modules.
A typical example is a family of Harish-Chandra modules with bounded lengths.
As an application of results about uniformly bounded families of $\mathscr{D}$-modu... | {
"timestamp": "2021-09-22T02:15:53",
"yymm": "2109",
"arxiv_id": "2109.05556",
"language": "en",
"url": "https://arxiv.org/abs/2109.05556",
"abstract": "In the representation theory of real reductive Lie groups, many objects have finiteness properties. For example, the lengths of Verma modules and principa... |
https://arxiv.org/abs/0809.2124 | Iterated function systems, moments, and transformations of infinite matrices | We study the moments of equilibrium measures for iterated function systems (IFSs) and draw connections to operator theory. Our main object of study is the infinite matrix which encodes all the moment data of a Borel measure on R^d or C. To encode the salient features of a given IFS into precise moment data, we establis... | \chapter{A transformation of moment matrices: the affine case}\label{Sec:Exist}
In this chapter we use the maps from an iterated function system to describe a corresponding transformation on moment matrices. In the case of an affine IFS, we show that the matrix transformation is a sum of triple products of
infinite... | {
"timestamp": "2008-09-12T04:51:26",
"yymm": "0809",
"arxiv_id": "0809.2124",
"language": "en",
"url": "https://arxiv.org/abs/0809.2124",
"abstract": "We study the moments of equilibrium measures for iterated function systems (IFSs) and draw connections to operator theory. Our main object of study is the i... |
https://arxiv.org/abs/2010.11450 | Optimal Approximation -- Smoothness Tradeoffs for Soft-Max Functions | A soft-max function has two main efficiency measures: (1) approximation - which corresponds to how well it approximates the maximum function, (2) smoothness - which shows how sensitive it is to changes of its input. Our goal is to identify the optimal approximation-smoothness tradeoffs for different measures of approxi... | \section{Introduction} \label{sec:intro}
A soft-max function is a mechanism for choosing one out of a number of options, given the
value of each option. Such functions have applications in many areas of computer science and
machine learning, such as deep learning
(as the final layer of a neural network classifier... | {
"timestamp": "2020-10-23T02:10:42",
"yymm": "2010",
"arxiv_id": "2010.11450",
"language": "en",
"url": "https://arxiv.org/abs/2010.11450",
"abstract": "A soft-max function has two main efficiency measures: (1) approximation - which corresponds to how well it approximates the maximum function, (2) smoothne... |
https://arxiv.org/abs/1809.03070 | Rectangle Coincidences and Sweepouts | We prove an integral formula for continuous paths of rectangles inscribed in a piecewise smooth loop. We then use this integral formula to show that (with a very mild genericity hypothesis) the number of rectangle coincidences, informally described as the number of inscribed rectangles minus the number of isometry clas... |
\section{Introduction}
A {\it Jordan loop\/} is the image of a circle under a
continuous injective map into the plane.
Toeplitz conjectured in 1911 that every Jordan loop contains $4$
points which are the vertices of a square. This is sometimes called
the {\it Square Peg Problem\/}.
For historical details and a lo... | {
"timestamp": "2018-11-28T02:06:09",
"yymm": "1809",
"arxiv_id": "1809.03070",
"language": "en",
"url": "https://arxiv.org/abs/1809.03070",
"abstract": "We prove an integral formula for continuous paths of rectangles inscribed in a piecewise smooth loop. We then use this integral formula to show that (with... |
https://arxiv.org/abs/1609.07352 | Some results on the Signature and Cubature of the Fractional Brownian motion for $H>\frac{1}{2}$ | In this work we present different results concerning the signature and the cubature of fractional Brownian motion (fBm). The first result regards the rate of convergence of the expected signature of the linear piecewise approximation of the fBm to its exact value, for a value of the Hurst parameter $H\in(\frac{1}{2},1)... | \section*{Some results on the Signature and Cubature of the Fractional Brownian motion for $H>\frac{1}{2}$}
\subsection*{Riccardo Passeggeri\footnote[1]{Imperial College London, UK. Email: riccardo.passeggeri14@imperial.ac.uk}}
\end{center}
\begin{abstract}
In this work we present different results concerning the signa... | {
"timestamp": "2017-11-20T02:10:29",
"yymm": "1609",
"arxiv_id": "1609.07352",
"language": "en",
"url": "https://arxiv.org/abs/1609.07352",
"abstract": "In this work we present different results concerning the signature and the cubature of fractional Brownian motion (fBm). The first result regards the rate... |
https://arxiv.org/abs/1712.07368 | Notes on noncommutative Fitting invariants | To each finitely presented module $M$ over a commutative ring $R$ one can associate an $R$-ideal $\mathrm{Fitt}_{R}(M)$, which is called the (zeroth) Fitting ideal of $M$ over $R$. This is of interest because it is always contained in the $R$-annihilator $\mathrm{Ann}_{R}(M)$ of $M$, but is often much easier to compute... | \section*{Introduction}
Let $R$ be a commutative unitary ring and let $M$ be a finitely presented
$R$-module. This means that there is an exact sequence
\begin{equation} \label{eqn:finite-presentation-comm}
R^a \stackrel{h}{\longrightarrow} R^b \longrightarrow
M \longrightarrow 0,
\end{equation}
where $a$ and $b$ are ... | {
"timestamp": "2018-09-11T02:19:41",
"yymm": "1712",
"arxiv_id": "1712.07368",
"language": "en",
"url": "https://arxiv.org/abs/1712.07368",
"abstract": "To each finitely presented module $M$ over a commutative ring $R$ one can associate an $R$-ideal $\\mathrm{Fitt}_{R}(M)$, which is called the (zeroth) Fit... |
https://arxiv.org/abs/1204.1687 | Recursively determined representing measures for bivariate truncated moment sequences | A theorem of Bayer and Teichmann implies that if a finite real multisequence \beta = \beta^(2d) has a representing measure, then the associated moment matrix M_d admits positive, recursively generated moment matrix extensions M_(d+1), M_(d+2),... For a bivariate recursively determinate M_d, we show that the existence o... | \section{Introduction}\label{Intro}
\setcounter{equation}{0}
Let $\beta\equiv \beta^{(2d)}:= \{\beta_{ij}\}_{i,j\ge 0, i+j\le 2d}$
denote a real bivariate moment sequence of degree $2d$. \ The Truncated
Moment Problem seeks conditions on $\beta$ for the existence of a
positive Borel measure $\mu$ on $\mathbb{R}^{2... | {
"timestamp": "2012-04-10T02:01:34",
"yymm": "1204",
"arxiv_id": "1204.1687",
"language": "en",
"url": "https://arxiv.org/abs/1204.1687",
"abstract": "A theorem of Bayer and Teichmann implies that if a finite real multisequence \\beta = \\beta^(2d) has a representing measure, then the associated moment mat... |
https://arxiv.org/abs/2206.09189 | List Arboricity of Finitary Matroids: A Generalization of Seymour's Result | Seymour proved that the chromatic numbers and the list chromatic numbers of loop-free finite matroids are the same. In this paper we prove the same statement for infinite, loop-free finitary matroids. | \section{Introduction}
Matroids are important objects of finite combinatorics, that can represent the basic properties of independency and rank. The theorems about matroids can be applied in a wide range of fields of mathematics, such as linear algebra or graph theory. One of the most interesting fact about matroids i... | {
"timestamp": "2022-06-22T02:09:43",
"yymm": "2206",
"arxiv_id": "2206.09189",
"language": "en",
"url": "https://arxiv.org/abs/2206.09189",
"abstract": "Seymour proved that the chromatic numbers and the list chromatic numbers of loop-free finite matroids are the same. In this paper we prove the same statem... |
https://arxiv.org/abs/1701.08237 | An Efficient Algebraic Solution to the Perspective-Three-Point Problem | In this work, we present an algebraic solution to the classical perspective-3-point (P3P) problem for determining the position and attitude of a camera from observations of three known reference points. In contrast to previous approaches, we first directly determine the camera's attitude by employing the corresponding ... |
\section{Introduction}
\label{sec:intro}
The Perspective-n-Point (PnP) is the problem of determining the 3D position and orientation (pose) of a camera from
observations of known point features.
The PnP is typically formulated and solved linearly by employing lifting (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot,~\c... | {
"timestamp": "2017-01-31T02:01:59",
"yymm": "1701",
"arxiv_id": "1701.08237",
"language": "en",
"url": "https://arxiv.org/abs/1701.08237",
"abstract": "In this work, we present an algebraic solution to the classical perspective-3-point (P3P) problem for determining the position and attitude of a camera fr... |
https://arxiv.org/abs/2302.00453 | Width and Depth Limits Commute in Residual Networks | We show that taking the width and depth to infinity in a deep neural network with skip connections, when branches are scaled by $1/\sqrt{depth}$ (the only nontrivial scaling), result in the same covariance structure no matter how that limit is taken. This explains why the standard infinite-width-then-depth approach pro... | \section{Introduction}
In recent years, deep neural networks have achieved remarkable success in a variety of tasks, such as image classification and natural language processing. However, the behavior of these networks in the limit of large depth and large width is still not fully understood.
The success of large la... | {
"timestamp": "2023-02-02T02:14:45",
"yymm": "2302",
"arxiv_id": "2302.00453",
"language": "en",
"url": "https://arxiv.org/abs/2302.00453",
"abstract": "We show that taking the width and depth to infinity in a deep neural network with skip connections, when branches are scaled by $1/\\sqrt{depth}$ (the onl... |
https://arxiv.org/abs/1911.03858 | Arıkan meets Shannon: Polar codes with near-optimal convergence to channel capacity | Let $W$ be a binary-input memoryless symmetric (BMS) channel with Shannon capacity $I(W)$ and fix any $\alpha > 0$. We construct, for any sufficiently small $\delta > 0$, binary linear codes of block length $O(1/\delta^{2+\alpha})$ and rate $I(W)-\delta$ that enable reliable communication on $W$ with quasi-linear time ... | \section{Inverse sub-exponential decoding error probability}
\label{sec:exponential-decoding}
In this section we finish proving our main result (Theorem~\ref{thm:intro-main}), by showing how to obtain inverse sub-exponential $\exp(-N^{\alpha})$ probability of error decoding within our construction of polar codes, while... | {
"timestamp": "2020-07-30T02:03:36",
"yymm": "1911",
"arxiv_id": "1911.03858",
"language": "en",
"url": "https://arxiv.org/abs/1911.03858",
"abstract": "Let $W$ be a binary-input memoryless symmetric (BMS) channel with Shannon capacity $I(W)$ and fix any $\\alpha > 0$. We construct, for any sufficiently sm... |
https://arxiv.org/abs/1704.02030 | Using stacking to average Bayesian predictive distributions | The widely recommended procedure of Bayesian model averaging is flawed in the M-open setting in which the true data-generating process is not one of the candidate models being fit. We take the idea of stacking from the point estimation literature and generalize to the combination of predictive distributions, extending ... | \section{Introduction}
A general challenge in statistics is prediction in the presence of multiple candidate models or learning algorithms $\mathcal{M}=(M_1,\dots, M_K)$. Model selection---picking one model that can give optimal performance for future data---can be unstable and wasteful of information \citep[see, e.g.... | {
"timestamp": "2017-09-19T02:03:09",
"yymm": "1704",
"arxiv_id": "1704.02030",
"language": "en",
"url": "https://arxiv.org/abs/1704.02030",
"abstract": "The widely recommended procedure of Bayesian model averaging is flawed in the M-open setting in which the true data-generating process is not one of the c... |
https://arxiv.org/abs/2107.01424 | On the semitotal dominating sets of graphs | A set $D$ of vertices in an isolate-free graph $G$ is a semitotal dominating set of $G$ if $D$ is a dominating set of $G$ and every vertex in $D$ is within distance $2$ from another vertex of $D$.The semitotal domination number of $G$ is the minimum cardinality of a semitotal dominating set of $G$ and is denoted by $\g... | \section{Introduction}
A dominating set of a graph $G=(V,E)$ is any subset $S$ of $V$ such that every vertex not in $S$ is adjacent to at least one member of $S$.
The minimum cardinality of all dominating sets of $G$ is called the domination number of $G$ and is denoted by $\gamma(G)$. This parameter has been ex... | {
"timestamp": "2021-07-06T02:10:24",
"yymm": "2107",
"arxiv_id": "2107.01424",
"language": "en",
"url": "https://arxiv.org/abs/2107.01424",
"abstract": "A set $D$ of vertices in an isolate-free graph $G$ is a semitotal dominating set of $G$ if $D$ is a dominating set of $G$ and every vertex in $D$ is withi... |
https://arxiv.org/abs/math/0412102 | The arithmetic and the geometry of Kobayashi hyperbolicity | We survey the properties of Brody and Kobayashi hyperbolic manifolds. | \section{Introductory remarks about hyperbolicity}
In this section we define Brody and Kobayashi hyperbolicity and state
their basic properties. We also give examples of hyperbolic and
non-hyperbolic manifolds. The reader can refer to
\cite{lang:hyperbolic} and \cite{demailly:hyperbolic} for more
details.\footnote{Aft... | {
"timestamp": "2004-12-05T23:49:50",
"yymm": "0412",
"arxiv_id": "math/0412102",
"language": "en",
"url": "https://arxiv.org/abs/math/0412102",
"abstract": "We survey the properties of Brody and Kobayashi hyperbolic manifolds.",
"subjects": "Algebraic Geometry (math.AG); Complex Variables (math.CV)",
"... |
https://arxiv.org/abs/math/0509667 | The homotopy invariance of the string topology loop product and string bracket | Let M be a closed, oriented, n -manifold, and LM its free loop space.Chas and Sullivan defined a commutative algebra structure in the homology of LM, and a Lie algebra structure in its equivariant homology. These structures are known as the string topology loop product and string bracket, respectively.In this paper we ... | \section*{Introduction}
The term ``string topology" refers to multiplicative structures on the (generalized) homology of spaces of paths and loops in a manifold. Let $M^n$ be a closed, oriented, smooth $n$-manifold. The basic ``loop homology algebra" is defined by a product
$$
\mu : H_*(LM) \otimes H_*(LM) \lon... | {
"timestamp": "2008-10-18T01:50:12",
"yymm": "0509",
"arxiv_id": "math/0509667",
"language": "en",
"url": "https://arxiv.org/abs/math/0509667",
"abstract": "Let M be a closed, oriented, n -manifold, and LM its free loop space.Chas and Sullivan defined a commutative algebra structure in the homology of LM, ... |
https://arxiv.org/abs/2211.08973 | Smooth integers and the Dickman $ρ$ function | We establish an asymptotic formula for $\Psi(x,y)$ whose shape is $x \rho(\log x/\log y)$ times correction factors. These factors take into account the contributions of zeta zeros and prime powers and the formula can be regarded as an (approximate) explicit formula for $\Psi(x,y)$. With this formula at hand we prove os... | \section{Introduction}
A positive integer is called $y$-friable (or $y$-smooth) if all its prime factors do not exceed $y$. We denote the number of $y$-friable integers up to $x$ by $\Psi(x,y)$. We assume throughout $x \ge y \ge 2$.
We denote by $\rho\colon [0,\infty) \to (0,\infty)$ the Dickman function, defined as $... | {
"timestamp": "2022-11-30T02:19:40",
"yymm": "2211",
"arxiv_id": "2211.08973",
"language": "en",
"url": "https://arxiv.org/abs/2211.08973",
"abstract": "We establish an asymptotic formula for $\\Psi(x,y)$ whose shape is $x \\rho(\\log x/\\log y)$ times correction factors. These factors take into account th... |
https://arxiv.org/abs/1508.02878 | Fullerenes with distant pentagons | For each $d>0$, we find all the smallest fullerenes for which the least distance between two pentagons is $d$. We also show that for each $d$ there is an $h_d$ such that fullerenes with pentagons at least distance $d$ apart and any number of hexagons greater than or equal to $h_d$ exist.We also determine the number of ... | \section{Introduction}
A \textit{fullerene}~\cite{kroto_85} is a cubic plane graph where all faces are pentagons or hexagons. Euler's formula implies that a
fullerene with $n$ vertices contains exactly 12 pentagons and $n/2 - 10$ hexagons.
The
\textit{dual} of a fullerene is the plane graph
obtained by exchanging the ... | {
"timestamp": "2015-08-13T02:08:18",
"yymm": "1508",
"arxiv_id": "1508.02878",
"language": "en",
"url": "https://arxiv.org/abs/1508.02878",
"abstract": "For each $d>0$, we find all the smallest fullerenes for which the least distance between two pentagons is $d$. We also show that for each $d$ there is an ... |
https://arxiv.org/abs/0812.2978 | Generalizations of Chung-Feller Theorem | The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with flaws $m$ is the $n$-th Catalan number and independent on $m$. L. Shapiro [7] found the Chung-Feller properties for the Motzkin paths. In this paper, we find the connections between these two Chung-Feller theorems. We focus... | \section{Introduction}
Let $\mathcal{S}$ be a subset of the set $\mathbb{Z}\times
\mathbb{Z}\setminus\{(0,0)\}$, where $\mathbb{Z}$ is the set of the
integers. We call $\mathcal{S}$ the {\it step set}. Let $k$ be an
integer.
\begin{defn} An $(\mathcal{S},k)$-lattice path is a path in
$\mathbb{Z}\times \mathbb{Z}$ which... | {
"timestamp": "2008-12-16T06:46:18",
"yymm": "0812",
"arxiv_id": "0812.2978",
"language": "en",
"url": "https://arxiv.org/abs/0812.2978",
"abstract": "The classical Chung-Feller theorem [2] tells us that the number of Dyck paths of length $n$ with flaws $m$ is the $n$-th Catalan number and independent on $... |
https://arxiv.org/abs/1803.03705 | Geodesic Obstacle Representation of Graphs | An obstacle representation of a graph is a mapping of the vertices onto points in the plane and a set of connected regions of the plane (called obstacles) such that the straight-line segment connecting the points corresponding to two vertices does not intersect any obstacles if and only if the vertices are adjacent in ... |
\section{Introduction}
\label{sec:introduction}
\input{introduction.tex}
\section{Notation and Preliminaries}
\label{sec:prelimins}
\input{prelimins.tex}
\section{General Representations}
\label{sec:general}
\input{general.tex}
\section{Non-Crossing Representations}
\label{sec:nonCrossing}
\input{nonCrossing.t... | {
"timestamp": "2018-03-13T01:02:40",
"yymm": "1803",
"arxiv_id": "1803.03705",
"language": "en",
"url": "https://arxiv.org/abs/1803.03705",
"abstract": "An obstacle representation of a graph is a mapping of the vertices onto points in the plane and a set of connected regions of the plane (called obstacles)... |
https://arxiv.org/abs/1705.01652 | Polluted Bootstrap Percolation with Threshold Two in All Dimensions | In the polluted bootstrap percolation model, the vertices of a graph are independently declared initially occupied with probability p or closed with probability q. At subsequent steps, a vertex becomes occupied if it is not closed and it has at least r occupied neighbors. On the cubic lattice Z^d of dimension d>=3 with... | \section{Introduction}\label{sec-intro}
Bootstrap percolation is a fundamental cellular
automaton model for nucleation and growth from sparse
random initial seeds. In this article we address how the
model is affected by the presence of pollution in the form
of sparse random permanent
obstacles.
Let $\mathbb Z^d$ be ... | {
"timestamp": "2017-05-05T02:02:08",
"yymm": "1705",
"arxiv_id": "1705.01652",
"language": "en",
"url": "https://arxiv.org/abs/1705.01652",
"abstract": "In the polluted bootstrap percolation model, the vertices of a graph are independently declared initially occupied with probability p or closed with proba... |
https://arxiv.org/abs/1911.00574 | Local regularity result for an optimal transportation problem with rough measures in the plane | We investigate the properties of convex functions in the plane that satisfy a local inequality which generalizes the notion of sub-solution of Monge-Ampere equation for a Monge-Kantorovich problem with quadratic cost between non-absolutely continuous measures. For each measure, we introduce a discrete scale so that the... | \section{Introduction}
\subsection{Generalized Monge-Amp\`ere equation for rough measures}
In this paper, we investigate the properties of convex functions in $\mathbb R^2$ that can be seen as {\em local} one-sided Kantorovich potentials. More specifically, we consider a (continuous) convex function $\psi:\mathbb R^2\... | {
"timestamp": "2021-04-08T02:25:38",
"yymm": "1911",
"arxiv_id": "1911.00574",
"language": "en",
"url": "https://arxiv.org/abs/1911.00574",
"abstract": "We investigate the properties of convex functions in the plane that satisfy a local inequality which generalizes the notion of sub-solution of Monge-Amper... |
https://arxiv.org/abs/1909.10027 | Invariant solutions of a nonlinear wave equation with a small dissipation obtained via approximate symmetries | In this paper, it is shown how a combination of approximate symmetries of a nonlinear wave equation with small dissipations and singularity analysis provides exact analytic solutions. We perform the analysis using the Lie symmetry algebra of this equation and identify the conjugacy classes of the one-dimensional subalg... | \section{Introduction}
A systematic computational method for constructing an approximate symmetry group for a given system of partial differential equations (PDEs) has been extensively developed by many authors, see e.g. \cite{Ames,Bluman1,Fushchich}. A broad review of recent developments in this subject can be found ... | {
"timestamp": "2019-09-24T02:15:13",
"yymm": "1909",
"arxiv_id": "1909.10027",
"language": "en",
"url": "https://arxiv.org/abs/1909.10027",
"abstract": "In this paper, it is shown how a combination of approximate symmetries of a nonlinear wave equation with small dissipations and singularity analysis provi... |
https://arxiv.org/abs/1103.4125 | The geometric stability of Voronoi diagrams with respect to small changes of the sites | Voronoi diagrams appear in many areas in science and technology and have numerous applications. They have been the subject of extensive investigation during the last decades. Roughly speaking, they are a certain decomposition of a given space into cells, induced by a distance function and by a tuple of subsets called t... | \section{Introduction}\label{sec:Intro}
\subsection{Background} The Voronoi diagram (the Voronoi tessellation, the Voronoi decomposition, the Dirichlet tessellation) is one of the basic structures
in computational geometry. Roughly speaking, it is a certain decomposition of a given space $X$ into cells, induced by ... | {
"timestamp": "2011-04-08T02:00:21",
"yymm": "1103",
"arxiv_id": "1103.4125",
"language": "en",
"url": "https://arxiv.org/abs/1103.4125",
"abstract": "Voronoi diagrams appear in many areas in science and technology and have numerous applications. They have been the subject of extensive investigation during... |
https://arxiv.org/abs/2212.13440 | Contraction and $k$-contraction in Lurie systems with applications to networked systems | A Lurie system is the interconnection of a linear time-invariant system and a nonlinear feedback function. We derive a new sufficient condition for $k$-contraction of a Lurie system. For $k=1$, our sufficient condition reduces to the standard stability condition based on the bounded real lemma and a small gain conditio... | \section{Introduction}
Consider a nonlinear system obtained by connecting
a linear time-invariant~(LTI) system with state vector~$x\in \mathbb R^n$, input~$u\in\mathbb R^m$ and output~$y\in \mathbb R^q$:
\begin{equation}\label{initial}
\begin{array}{l}
\dot x(t)=Ax(t)+ Bu(t) ,\\%[1em]
y(t)=Cx(t) ,
\end{array}
\end{e... | {
"timestamp": "2022-12-29T02:05:49",
"yymm": "2212",
"arxiv_id": "2212.13440",
"language": "en",
"url": "https://arxiv.org/abs/2212.13440",
"abstract": "A Lurie system is the interconnection of a linear time-invariant system and a nonlinear feedback function. We derive a new sufficient condition for $k$-co... |
https://arxiv.org/abs/2106.06012 | Learning distinct features helps, provably | We study the diversity of the features learned by a two-layer neural network trained with the least squares loss. We measure the diversity by the average $L_2$-distance between the hidden-layer features and theoretically investigate how learning non-redundant distinct features affects the performance of the network. To... | \section{Introduction}
Neural networks are a powerful class of non-linear function approximators that have been successfully used to tackle a wide range of problems. They have enabled breakthroughs in many tasks, such as image classification \citep{krizhevsky2012imagenet}, speech recognition \citep{hinton2012deep}, and... | {
"timestamp": "2022-02-16T02:01:47",
"yymm": "2106",
"arxiv_id": "2106.06012",
"language": "en",
"url": "https://arxiv.org/abs/2106.06012",
"abstract": "We study the diversity of the features learned by a two-layer neural network trained with the least squares loss. We measure the diversity by the average ... |
https://arxiv.org/abs/1909.09043 | A note on minimal art galleries | We will consider some extensions of the polygonal art gallery problem. In a recent paper Morrison has shown the smallest (9 sides) example of an art gallery that cannot be observed by guards placed in every third corner. Author also mentioned two related problems, for which the minimal examples are not known. We will s... | \section{Introduction}
Original art gallery problem is posed as following: given a polygon with $n$ sides choose $x$ points called guards inside it such that any point of polygon can be observed by at least one guard (precisely, for any $p$ in the polygon there exists guard $q$ such that the line segment $\overline{pq}... | {
"timestamp": "2019-09-20T02:22:18",
"yymm": "1909",
"arxiv_id": "1909.09043",
"language": "en",
"url": "https://arxiv.org/abs/1909.09043",
"abstract": "We will consider some extensions of the polygonal art gallery problem. In a recent paper Morrison has shown the smallest (9 sides) example of an art galle... |
https://arxiv.org/abs/1906.03058 | Robust subgaussian estimation of a mean vector in nearly linear time | We construct an algorithm, running in time $\tilde{\mathcal O}(N d + uK d)$, which is robust to outliers and heavy-tailed data and which achieves the subgaussian rate from [Lugosi, Mendelson] \begin{equation}\label{eq:intro_subgaus_rate} \sqrt{\frac{{\rm Tr}(\Sigma)}{N}}+\sqrt{\frac{||\Sigma||_{op}K}{N}} \end{equation}... |
\section{Introduction on the robust mean vector estimation problem}
\label{sec:introduction_on_the_mean_vector_problem}
Estimating the mean of a random variable in a $d$-dimensional space when given some of its realizations is arguably the oldest and most fundamental problem of statistics. In the past few years, it h... | {
"timestamp": "2019-06-28T02:09:35",
"yymm": "1906",
"arxiv_id": "1906.03058",
"language": "en",
"url": "https://arxiv.org/abs/1906.03058",
"abstract": "We construct an algorithm, running in time $\\tilde{\\mathcal O}(N d + uK d)$, which is robust to outliers and heavy-tailed data and which achieves the su... |
https://arxiv.org/abs/1809.09442 | Local biquandles and Niebrzydowski's tribracket theory | We introduce a new algebraic structure called \textit{local biquandles} and show how colorings of oriented classical link diagrams and of broken surface diagrams are related to tribracket colorings. We define a (co)homology theory for local biquandles and show that it is isomorphic to Niebrzydowski's tribracket (co)hom... | \section*{Introduction}
Invariants of knots and knotted surfaces defined in terms of colorings by
algebraic structures have a long history, including colorings by algebraic
structures such as groups, quandles, biquandles and more \cite{ElhamdadiNelson}.
Enhancements, called cocycle invariants, of these invariants usi... | {
"timestamp": "2019-02-19T02:19:50",
"yymm": "1809",
"arxiv_id": "1809.09442",
"language": "en",
"url": "https://arxiv.org/abs/1809.09442",
"abstract": "We introduce a new algebraic structure called \\textit{local biquandles} and show how colorings of oriented classical link diagrams and of broken surface ... |
https://arxiv.org/abs/1806.09062 | A simplified and unified generalization of some majorization results | We consider positive, integral-preserving linear operators acting on $L^1$ space, known as stochastic operators or Markov operators. We show that, on finite-dimensional spaces, any stochastic operator can be approximated by a sequence of stochastic integral operators (such operators arise naturally when considering mat... | \section{Introduction}\label{sec:Intro}
In this work, we connect several generalizations of majorization in reference to vector-valued measurable functions; notably, matrix majorization, multivariate majorization, mixing distance, $f$-divergence, and coarse graining.
While some results are known, they appear rath... | {
"timestamp": "2019-06-13T02:17:52",
"yymm": "1806",
"arxiv_id": "1806.09062",
"language": "en",
"url": "https://arxiv.org/abs/1806.09062",
"abstract": "We consider positive, integral-preserving linear operators acting on $L^1$ space, known as stochastic operators or Markov operators. We show that, on fini... |
https://arxiv.org/abs/1212.3893 | A sufficient condition for congruency of orbits of Lie groups and some applications | We give a sufficient condition for isometric actions to have the congruency of orbits, that is, all orbits are isometrically congruent to each other. As applications, we give simple and unified proofs for some known congruence results, and also provide new examples of isometric actions on symmetric spaces of noncompact... | \section{Introduction}
Isometric actions of Lie groups on Riemannian manifolds $M$
and submanifold geometry of their orbits are fundamental topics in geometry.
In this paper, we consider isometric actions which have the congruency of orbits,
that is, all of whose orbits are isometrically congruent to each other... | {
"timestamp": "2012-12-18T02:03:52",
"yymm": "1212",
"arxiv_id": "1212.3893",
"language": "en",
"url": "https://arxiv.org/abs/1212.3893",
"abstract": "We give a sufficient condition for isometric actions to have the congruency of orbits, that is, all orbits are isometrically congruent to each other. As app... |
https://arxiv.org/abs/1102.3515 | On Gromov's Method of Selecting Heavily Covered Points | A result of Boros and Füredi ($d=2$) and of Bárány (arbitrary $d$) asserts that for every $d$ there exists $c_d>0$ such that for every $n$-point set $P\subset \R^d$, some point of $\R^d$ is covered by at least $c_d{n\choose d+1}$ of the $d$-simplices spanned by the points of $P$. The largest possible value of $c_d$ has... | \section{Introduction}
Let $P\subset {\mathbb{R}}^2$ be a set of $n$ points in general position (i.e., no three points collinear). Boros and F\"uredi~\cite{BorosFuredi:PlanarSelectionLemma-84} showed that there always exists a point $a\in{\mathbb{R}}^2$ contained in a positive fraction of all the $\binom{n}{3}$ triangl... | {
"timestamp": "2011-02-18T02:01:12",
"yymm": "1102",
"arxiv_id": "1102.3515",
"language": "en",
"url": "https://arxiv.org/abs/1102.3515",
"abstract": "A result of Boros and Füredi ($d=2$) and of Bárány (arbitrary $d$) asserts that for every $d$ there exists $c_d>0$ such that for every $n$-point set $P\\sub... |
https://arxiv.org/abs/2009.05100 | The Complete Positivity of Symmetric Tridiagonal and Pentadiagonal Matrices | We provide a decomposition that is sufficient in showing when a symmetric tridiagonal matrix $A$ is completely positive. Our decomposition can be applied to a wide range of matrices. We give alternate proofs for a number of related results found in the literature in a simple, straightforward manner. We show that the cp... | \section{Preliminaries}
All matrices herein will be real-valued. Let $A$ be an $n\times n$ symmetric tridiagonal matrix:
$$A=\begin{pmatrix}a_1&b_1&&&& \\ b_1 & a_2 & b_2 &&&\\ &\ddots&\ddots&\ddots&&& \\& &\ddots&\ddots&\ddots&& \\ &&&b_{n-3}&a_{n-2}&b_{n-2}& \\&&&&b_{n-2}&a_{n-1}&b_{n-1} \\&&&&&b_{n-1}&a_n \end{pma... | {
"timestamp": "2021-03-11T02:24:35",
"yymm": "2009",
"arxiv_id": "2009.05100",
"language": "en",
"url": "https://arxiv.org/abs/2009.05100",
"abstract": "We provide a decomposition that is sufficient in showing when a symmetric tridiagonal matrix $A$ is completely positive. Our decomposition can be applied ... |
https://arxiv.org/abs/1911.08213 | Cohomology of contact loci | We construct a spectral sequence converging to the cohomology with compact support of the m-th contact locus of a complex polynomial. The first page is explicitly described in terms of a log resolution and coincides with the first page of McLean's spectral sequence converging to the Floer cohomology of the m-th iterate... | \section{Introduction and main results}
The motivation for this note comes from a question of Seidel, and a subsequent question by McLean. In \cite{DL} Denef and Loeser proved that the Euler characteristic of the contact loci of a complex polynomial $f$ coincides with the Lefschetz number of the $m$-th iterate of the... | {
"timestamp": "2020-09-08T02:07:22",
"yymm": "1911",
"arxiv_id": "1911.08213",
"language": "en",
"url": "https://arxiv.org/abs/1911.08213",
"abstract": "We construct a spectral sequence converging to the cohomology with compact support of the m-th contact locus of a complex polynomial. The first page is ex... |
https://arxiv.org/abs/1301.1503 | Application of semidefinite programming to maximize the spectral gap produced by node removal | The smallest positive eigenvalue of the Laplacian of a network is called the spectral gap and characterizes various dynamics on networks. We propose mathematical programming methods to maximize the spectral gap of a given network by removing a fixed number of nodes. We formulate relaxed versions of the original problem... | \section{Introduction}
An undirected and unweighted network (i.e., graph) on $N$ nodes is equivalent to an $N \times N$
symmetric
adjacency matrix $A=(A_{ij})$,
where $A_{ij}=1$ when nodes (also called vertices) $i$ and $j$ form
a link (also called edge), and
$A_{ij}=0$ otherwise. We define the Laplacian matri... | {
"timestamp": "2013-01-09T02:01:45",
"yymm": "1301",
"arxiv_id": "1301.1503",
"language": "en",
"url": "https://arxiv.org/abs/1301.1503",
"abstract": "The smallest positive eigenvalue of the Laplacian of a network is called the spectral gap and characterizes various dynamics on networks. We propose mathema... |
https://arxiv.org/abs/1304.2809 | On partial sparse recovery | We consider the problem of recovering a partially sparse solution of an underdetermined system of linear equations by minimizing the $\ell_1$-norm of the part of the solution vector which is known to be sparse. Such a problem is closely related to a classical problem in Compressed Sensing where the $\ell_1$-norm of the... | \section{Introduction}\label{sec:introduction}
\IEEEPARstart{I}{n} Compressed Sensing one is interested in recovering a sparse
solution~$\bar x\in\RR^N$ of an underdetermined system of the form $y=A \bar x$, given
a vector $y\in\RR^k$ and a matrix $A \in \RR^{k\times N}$
with far fewer rows than columns $(k\ll N... | {
"timestamp": "2013-04-11T02:01:02",
"yymm": "1304",
"arxiv_id": "1304.2809",
"language": "en",
"url": "https://arxiv.org/abs/1304.2809",
"abstract": "We consider the problem of recovering a partially sparse solution of an underdetermined system of linear equations by minimizing the $\\ell_1$-norm of the p... |
https://arxiv.org/abs/2007.02008 | A spectral extremal problem on graphs with given size and matching number | Brualdi and Hoffman (1985) proposed the problem of determining the maximal spectral radius of graphs with given size. In this paper, we consider the Brualdi-Hoffman type problem of graphs with given matching number. The maximal $Q$-spectral radius of graphs with given size and matching number is obtained, and the corre... | \section{Introduction}\label{s-1}
Unless stated otherwise, we follow \cite{Boundy2008,Cvetkovic2010} for terminology and notations. All graphs considered here are simple and undirected. Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The order of $G$ is the number of vertices in $V(G)$. The number of ed... | {
"timestamp": "2020-07-07T02:05:25",
"yymm": "2007",
"arxiv_id": "2007.02008",
"language": "en",
"url": "https://arxiv.org/abs/2007.02008",
"abstract": "Brualdi and Hoffman (1985) proposed the problem of determining the maximal spectral radius of graphs with given size. In this paper, we consider the Brual... |
https://arxiv.org/abs/1905.13292 | Spanning Trees and Domination in Hypercubes | Let $L(G)$ denote the maximum number of leaves in any spanning tree of a connected graph $G$. We show the (known) result that for the $n$-cube $Q_n$, $L(Q_n) \sim 2^n = |V(Q_n)|$ as $n\rightarrow \infty$. Examining this more carefully, consider the minimum size of a connected dominating set of vertices $\gamma_c(Q_n)$,... | \section{Introduction}\label{sec:Over}
The $n$-cube graph $Q_n$ has $2^n$ vertices, the strings
$a_1\ldots a_n$ on $n$ bits, where two vertices
are adjacent if and only if their strings differ in exactly
one coordinate (where one vertex has 0 and the other has 1).
The $n$-cube is frequently used as a structure f... | {
"timestamp": "2019-06-03T02:03:14",
"yymm": "1905",
"arxiv_id": "1905.13292",
"language": "en",
"url": "https://arxiv.org/abs/1905.13292",
"abstract": "Let $L(G)$ denote the maximum number of leaves in any spanning tree of a connected graph $G$. We show the (known) result that for the $n$-cube $Q_n$, $L(Q... |
https://arxiv.org/abs/1204.5551 | A lower bound on seller revenue in single buyer monopoly auctions | We consider a monopoly seller who optimally auctions a single object to a single potential buyer, with a known distribution of valuations. We show that a tight lower bound on the seller's expected revenue is $1/e$ times the geometric expectation of the buyer's valuation, and that this bound is uniquely achieved for the... | \section{Introduction}
Consider a monopoly seller, selling a single object to a single
potential buyer. We assume that the buyer has a valuation for the
object which is unknown to the seller, and that the seller's
uncertainty is quantified by a probability distribution, from which it
believes the buyer picks its valuat... | {
"timestamp": "2013-10-08T02:11:30",
"yymm": "1204",
"arxiv_id": "1204.5551",
"language": "en",
"url": "https://arxiv.org/abs/1204.5551",
"abstract": "We consider a monopoly seller who optimally auctions a single object to a single potential buyer, with a known distribution of valuations. We show that a ti... |
https://arxiv.org/abs/1109.3115 | Log-concavity of complexity one Hamiltonian torus actions | Let $(M,\omega)$ be a closed $2n$-dimensional symplectic manifold equipped with a Hamiltonian $T^{n-1}$-action. Then Atiyah-Guillemin-Sternberg convexity theorem implies that the image of the moment map is an $(n-1)$-dimensional convex polytope. In this paper, we show that the density function of the Duistermaat-Heckma... | \section{Introduction}
In statistic physics, the relation $S(E) = k \log{W(E)}$ is
called Boltzmann's principle where $W$ is
the number of states with given values of macroscopic parameters $E$
(like energy, temperature, ..), $k$ is the Boltzmann's constant,
and $S$ is the entropy of the system
which measures t... | {
"timestamp": "2011-09-15T02:04:00",
"yymm": "1109",
"arxiv_id": "1109.3115",
"language": "en",
"url": "https://arxiv.org/abs/1109.3115",
"abstract": "Let $(M,\\omega)$ be a closed $2n$-dimensional symplectic manifold equipped with a Hamiltonian $T^{n-1}$-action. Then Atiyah-Guillemin-Sternberg convexity t... |
https://arxiv.org/abs/1111.4587 | A Complete Characterization of the Gap between Convexity and SOS-Convexity | Our first contribution in this paper is to prove that three natural sum of squares (sos) based sufficient conditions for convexity of polynomials, via the definition of convexity, its first order characterization, and its second order characterization, are equivalent. These three equivalent algebraic conditions, hencef... | \section{Introduction}
\subsection{Nonnegativity and sum of squares}
One of the cornerstones of real algebraic geometry is Hilbert's
seminal paper in 1888~\cite{Hilbert_1888}, where he gives a
complete characterization of the degrees and dimensions \aaan{for}
which nonnegative polynomials can be written as sums of squa... | {
"timestamp": "2012-07-17T02:02:18",
"yymm": "1111",
"arxiv_id": "1111.4587",
"language": "en",
"url": "https://arxiv.org/abs/1111.4587",
"abstract": "Our first contribution in this paper is to prove that three natural sum of squares (sos) based sufficient conditions for convexity of polynomials, via the d... |
https://arxiv.org/abs/2006.00742 | Error bounds for overdetermined and underdetermined generalized centred simplex gradients | Using the Moore--Penrose pseudoinverse, this work generalizes the gradient approximation technique called centred simplex gradient to allow sample sets containing any number of points. This approximation technique is called the \emph{generalized centred simplex gradient}. We develop error bounds and, under a full-rank ... | \section{Introduction}\label{sec:intro}
Derivative-free optimization (DFO) focuses on the study of optimization algorithms that do not use first-order information within the algorithm. Recent advances in their applications, convergence analysis and practical implementations have fuelled a surge in DFO research (see \ci... | {
"timestamp": "2020-06-02T02:27:14",
"yymm": "2006",
"arxiv_id": "2006.00742",
"language": "en",
"url": "https://arxiv.org/abs/2006.00742",
"abstract": "Using the Moore--Penrose pseudoinverse, this work generalizes the gradient approximation technique called centred simplex gradient to allow sample sets co... |
https://arxiv.org/abs/1702.07077 | Min-oo conjecture for fully nonlinear conformally invariant equations | In this paper we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard $n$-sphere $\mathbb S^n$ under suitable conditions along the boundary. We emphasize that our results do not assume concavity assumption on the fully nonlinear equations we... | \section{Introduction}
In 1995, Min-Oo \cite{O}, inspired by the work of Schoen and Yau \cite{SY1,SY2} on the Positive Mass Theorem, conjectured that if $(M^n,g)$
is a compact Riemannian manifold with boundary such that the scalar curvature of $M$ is at least $n(n-1)$ and whose boundary $\partial M$ is totally geodesi... | {
"timestamp": "2018-11-26T02:08:10",
"yymm": "1702",
"arxiv_id": "1702.07077",
"language": "en",
"url": "https://arxiv.org/abs/1702.07077",
"abstract": "In this paper we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard $n$-s... |
https://arxiv.org/abs/1406.0762 | Sobolev orthogonal polynomials on product domains | Orthogonal polynomials on the product domain $[a_1,b_1] \times [a_2,b_2]$ with respect to the inner product $$\langle f,g \rangle_S = \int_{a_1}^{b_1} \int_{a_2}^{b_2} \nabla f(x,y)\cdot \nabla g(x,y)\, w_1(x)w_2(y)\,dx\, dy + \lambda f(c_1,c_2)g(c_1,c_2) $$ are constructed, where $w_i$ is a weight function on $[a_i,b_... | \section{Introduction}
\setcounter{equation}{0}
Let $w_i(x)$ be a nonnegative weight function defined on an interval $[a_i,b_i]$, where $i =1,2$. Let $W$ be
the product weight function
\begin{equation} \label{eq:W}
W(x,y):= w_1(x) w_2(y), \qquad (x,y) \in \Omega : = [a_1,b_1] \times [a_2,b_2].
\end{equation}
The p... | {
"timestamp": "2014-06-04T02:10:04",
"yymm": "1406",
"arxiv_id": "1406.0762",
"language": "en",
"url": "https://arxiv.org/abs/1406.0762",
"abstract": "Orthogonal polynomials on the product domain $[a_1,b_1] \\times [a_2,b_2]$ with respect to the inner product $$\\langle f,g \\rangle_S = \\int_{a_1}^{b_1} \... |
https://arxiv.org/abs/1908.07092 | Linear stability analysis for large dynamical systems on directed random graphs | We present a linear stability analysis of stationary states (or fixed points) in large dynamical systems defined on random directed graphs with a prescribed distribution of indegrees and outdegrees. We obtain two remarkable results for such dynamical systems: First, infinitely large systems on directed graphs can be st... | \section{Introduction}
Scientists use networks to characterize the causal interactions between the constituents of large dynamical systems \cite{barrat2008dynamical, newman2010networks, barthelemy2018spatial, dorogovtsev2013evolution, barabasi2016network}. An important question is how network archit... | {
"timestamp": "2019-08-21T02:03:51",
"yymm": "1908",
"arxiv_id": "1908.07092",
"language": "en",
"url": "https://arxiv.org/abs/1908.07092",
"abstract": "We present a linear stability analysis of stationary states (or fixed points) in large dynamical systems defined on random directed graphs with a prescrib... |
https://arxiv.org/abs/2007.08956 | Vertex distinction with subgraph centrality: a proof of Estrada's conjecture and some generalizations | Centrality measures are used in network science to identify the most important vertices for transmission of information and dynamics on a graph. One of these measures, introduced by Estrada and collaborators, is the $\beta$-subgraph centrality, which is based on the exponential of the matrix $\beta A$, where $A$ is the... | \section{Introduction}
Centrality measures have been used to determine the importance of a vertex in a graph, with many applications in biology, finance, sociology, epidemiology, and more generally in network science. Among many such measures, we focus here on subgraph centrality, which is based on counting the numbe... | {
"timestamp": "2021-06-08T02:27:37",
"yymm": "2007",
"arxiv_id": "2007.08956",
"language": "en",
"url": "https://arxiv.org/abs/2007.08956",
"abstract": "Centrality measures are used in network science to identify the most important vertices for transmission of information and dynamics on a graph. One of th... |
https://arxiv.org/abs/1703.06214 | Free 2-step nilpotent Lie algebras and indecomposable modules | Given an algebraically closed field $F$ of characteristic 0 and an $F$-vector space $V$, let $L(V)=V\oplus\Lambda^2(V)$ denote the free 2-step nilpotent Lie algebra associated to $V$. In this paper, we classify all uniserial representations of the solvable Lie algebra $\mathfrak g=\langle x\rangle\ltimes L(V)$, where $... | \section{Introduction}
\label{intro}
We fix throughout an algebraically closed field $F$ of characteristic zero. All Lie
algebras and representations considered in this paper are assumed
to be finite dimensional over $F$, unless explicitly stated
otherwise.
According to \cite{M} (see also \cite{GP}), the task of cla... | {
"timestamp": "2017-03-21T01:02:02",
"yymm": "1703",
"arxiv_id": "1703.06214",
"language": "en",
"url": "https://arxiv.org/abs/1703.06214",
"abstract": "Given an algebraically closed field $F$ of characteristic 0 and an $F$-vector space $V$, let $L(V)=V\\oplus\\Lambda^2(V)$ denote the free 2-step nilpotent... |
https://arxiv.org/abs/1411.6606 | A Baxter class of a different kind, and other bijective results using tableau sequences ending with a row shape | Tableau sequences of bounded height have been central to the analysis of k-noncrossing set partitions and matchings. We show here that familes of sequences that end with a row shape are particularly compelling and lead to some interesting connections. First, we prove that hesitating tableaux of height at most two endin... | \section{Introduction}
\label{sec:introduction}
The counting sequence for Baxter permutations, whose elements
\[
B_n=\sum_{k=1}^n\frac{\binom{n+1}{k-1}\binom{n+1}{k}\binom{n+1}{k+1}}{\binom{n+1}{1}\binom{n+1}{2}}
\] are known as Baxter
numbers, is a fascinating combinatorial entity, enumerating a
diverse selection of c... | {
"timestamp": "2015-05-13T02:12:12",
"yymm": "1411",
"arxiv_id": "1411.6606",
"language": "en",
"url": "https://arxiv.org/abs/1411.6606",
"abstract": "Tableau sequences of bounded height have been central to the analysis of k-noncrossing set partitions and matchings. We show here that familes of sequences ... |
https://arxiv.org/abs/1203.3868 | Optimal covers with Hamilton cycles in random graphs | A packing of a graph G with Hamilton cycles is a set of edge-disjoint Hamilton cycles in G. Such packings have been studied intensively and recent results imply that a largest packing of Hamilton cycles in G_n,p a.a.s. has size \lfloor delta(G_n,p) /2 \rfloor. Glebov, Krivelevich and Szabó recently initiated research o... | \section{Introduction}
Given graphs $H$ and $G$, an $H$-decomposition of $G$ is a set of edge-disjoint copies of $H$ in $G$ which cover all edges of $G$.
The study of such decompositions forms an important area of Combinatorics but it is notoriously difficult.
Often an $H$-decomposition does not exist (or it may be out... | {
"timestamp": "2013-07-25T02:04:46",
"yymm": "1203",
"arxiv_id": "1203.3868",
"language": "en",
"url": "https://arxiv.org/abs/1203.3868",
"abstract": "A packing of a graph G with Hamilton cycles is a set of edge-disjoint Hamilton cycles in G. Such packings have been studied intensively and recent results i... |
https://arxiv.org/abs/2010.15519 | Spanning trees at the connectivity threshold | We present an explicit connected spanning structure that appears in a random graph just above the connectivity threshold with high probability. | \section{Introduction}
\label{sec:intro}
The \defn{binomial random graph} $G(n,p)$ is a graph on $n$ vertices, in which every pair of vertices is connected independently with probability $p$.
It is a well known and thoroughly studied model (see, e.g.,~\cites{Bol,FK,JLR}).
A fundamental result, due to Erd\H{o}s and R\'e... | {
"timestamp": "2021-11-29T02:39:25",
"yymm": "2010",
"arxiv_id": "2010.15519",
"language": "en",
"url": "https://arxiv.org/abs/2010.15519",
"abstract": "We present an explicit connected spanning structure that appears in a random graph just above the connectivity threshold with high probability.",
"subje... |
https://arxiv.org/abs/1509.03990 | Raising The Bar For Vertex Cover: Fixed-parameter Tractability Above A Higher Guarantee | We investigate the following above-guarantee parameterization of the classical Vertex Cover problem: Given a graph $G$ and $k\in\mathbb{N}$ as input, does $G$ have a vertex cover of size at most $(2LP-MM)+k$? Here $MM$ is the size of a maximum matching of $G$, $LP$ is the value of an optimum solution to the relaxed (st... | \section{The Algorithm}\label{sec:algorithm}
In this section we describe our algorithm which solves \name{Vertex Cover Above Lov\'{a}sz-Plummer}\xspace in
\(\ensuremath{\mathcal{O}^{\star}}\xspace(3^{\hat{k}})\) time. We start with an overview of the
algorithm. We then state the reduction and branching rules which
we u... | {
"timestamp": "2015-09-15T02:12:56",
"yymm": "1509",
"arxiv_id": "1509.03990",
"language": "en",
"url": "https://arxiv.org/abs/1509.03990",
"abstract": "We investigate the following above-guarantee parameterization of the classical Vertex Cover problem: Given a graph $G$ and $k\\in\\mathbb{N}$ as input, do... |
https://arxiv.org/abs/1712.02492 | Rates of convergence in $W^2_p$-norm for the Monge-Ampère equation | We develop discrete $W^2_p$-norm error estimates for the Oliker-Prussner method applied to the Monge-Ampère equation. This is obtained by extending discrete Alexandroff estimates and showing that the contact set of a nodal function contains information on its second order difference. In addition, we show that the size ... | \section{Introduction}
\thispagestyle{empty}
In this paper we develop
discrete $W^2_p$ error estimates
for numerical approximations of
the Monge-Amp\`ere equation
with Dirichlet boundary conditions:
\begin{subequations}\label{MA}
\begin{alignat}{2}\label{MA1}
\det(D^2 u) & = f\qquad \text{in }\Omega,\\
\label{MA2}... | {
"timestamp": "2017-12-08T02:05:14",
"yymm": "1712",
"arxiv_id": "1712.02492",
"language": "en",
"url": "https://arxiv.org/abs/1712.02492",
"abstract": "We develop discrete $W^2_p$-norm error estimates for the Oliker-Prussner method applied to the Monge-Ampère equation. This is obtained by extending discre... |
https://arxiv.org/abs/2004.02837 | Near-linear convergence of the Random Osborne algorithm for Matrix Balancing | We revisit Matrix Balancing, a pre-conditioning task used ubiquitously for computing eigenvalues and matrix exponentials. Since 1960, Osborne's algorithm has been the practitioners' algorithm of choice and is now implemented in most numerical software packages. However, its theoretical properties are not well understoo... | \section*{Acknowledgements.}
JA thanks Enric Boix-Adsera and Jonathan Niles-Weed for helpful conversations.
\section{Introduction}\label{sec:intro}
Let $\mathbf{1}$ denote the all-ones vector in $\R^n$. A nonnegative square matrix $A \in \R_{\geq 0}^{n \times n}$ is said to be \emph{balanced} if its row sums $r(A) :=... | {
"timestamp": "2020-04-07T02:32:43",
"yymm": "2004",
"arxiv_id": "2004.02837",
"language": "en",
"url": "https://arxiv.org/abs/2004.02837",
"abstract": "We revisit Matrix Balancing, a pre-conditioning task used ubiquitously for computing eigenvalues and matrix exponentials. Since 1960, Osborne's algorithm ... |
https://arxiv.org/abs/1011.0180 | Independent sets in random graphs from the weighted second moment method | We prove new lower bounds on the likely size of a maximum independent set in a random graph with a given average degree. Our method is a weighted version of the second moment method, where we give each independent set a weight based on the total degree of its vertices. | \section{Introduction}
We are interested in the likely size of the largest independent set $S$ in a random graph with a given average degree. Azuma's inequality implies that $|S|$ is tightly concentrated around its expectation. Moreover, it is easy to see that $|S|=\Theta(n)$ whenever the average degree is constant.... | {
"timestamp": "2011-06-20T02:00:37",
"yymm": "1011",
"arxiv_id": "1011.0180",
"language": "en",
"url": "https://arxiv.org/abs/1011.0180",
"abstract": "We prove new lower bounds on the likely size of a maximum independent set in a random graph with a given average degree. Our method is a weighted version of... |
https://arxiv.org/abs/2001.09383 | Orientable Hamiltonian Embeddings of the Hypercube Graph | A Hamiltonian embedding is an embedding of a graph $G$ such that the boundary of each face is a Hamiltonian cycle of $G$. It is shown that the hypercube graph $Q_n$ admits such an embedding on an orientable surface when $n$ is a power of 2. Basic necessary conditions on Hamiltonian embeddings for $Q_n$ and conjectures ... | \section{Introduction}
The hypercube graph $Q_n$ is an example of a graph with a large amount of symmetry. We shall define $Q_n$ in two different ways.
\de{1.1} The $n$-dimensional hypercube $Q_n$, for $n \geq 1$ is given by
\[ V(Q_n) = \{0,1\}^n \qquad E(Q_n) = \{xy: x \mbox{ and $y$ differ in exactly one coordinate} ... | {
"timestamp": "2020-01-28T02:10:14",
"yymm": "2001",
"arxiv_id": "2001.09383",
"language": "en",
"url": "https://arxiv.org/abs/2001.09383",
"abstract": "A Hamiltonian embedding is an embedding of a graph $G$ such that the boundary of each face is a Hamiltonian cycle of $G$. It is shown that the hypercube g... |
https://arxiv.org/abs/2101.12708 | Between steps: Intermediate relaxations between big-M and convex hull formulations | This work develops a class of relaxations in between the big-M and convex hull formulations of disjunctions, drawing advantages from both. The proposed "P-split" formulations split convex additively separable constraints into P partitions and form the convex hull of the partitioned disjuncts. Parameter P represents the... | \section{Introduction}
There are well-known trade-offs between the big-M and convex hull relaxations of disjunctions in terms of problem size and relaxation tightness. Convex hull formulations \cite{balas1998disjunctive,ben2001lectures,ceria1999convex,helton2009sufficient,jeroslow1984modelling,stubbs1999branch} provide... | {
"timestamp": "2021-02-01T02:18:27",
"yymm": "2101",
"arxiv_id": "2101.12708",
"language": "en",
"url": "https://arxiv.org/abs/2101.12708",
"abstract": "This work develops a class of relaxations in between the big-M and convex hull formulations of disjunctions, drawing advantages from both. The proposed \"... |
https://arxiv.org/abs/1704.04752 | Further and stronger analogy between sampling and optimization: Langevin Monte Carlo and gradient descent | In this paper, we revisit the recently established theoretical guarantees for the convergence of the Langevin Monte Carlo algorithm of sampling from a smooth and (strongly) log-concave density. We improve the existing results when the convergence is measured in the Wasserstein distance and provide further insights on t... | \section{Introduction}
Let $p$ be a positive integer and $f:\mathbb{R}^p\to\mathbb{R}$ be a measurable function such that the
integral $\int_{\mathbb{R}^p} \exp\{-f(\boldsymbol{\theta})\}\,d\boldsymbol{\theta}$ is finite. In various applications,
one is faced with the problems of finding the minimum point of $f$ or ... | {
"timestamp": "2017-07-31T02:08:55",
"yymm": "1704",
"arxiv_id": "1704.04752",
"language": "en",
"url": "https://arxiv.org/abs/1704.04752",
"abstract": "In this paper, we revisit the recently established theoretical guarantees for the convergence of the Langevin Monte Carlo algorithm of sampling from a smo... |
https://arxiv.org/abs/math/0606646 | A quasisymmetric function for matroids | A new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant (1) defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric functions, which is surjective if one uses rational coefficients, (2) is a multivariate generating function for integer weig... | \section{Definition as generating function}
\label{definition-section}
We begin by defining the new matroid invariant. For matroid terminology
undefined here, we refer the reader to some of the standard references, such
as \cite{CrapoRota, Oxley, Welsh, White1, White2, White3}.
Let $M=(E,{\mathcal B})$ be a matroid ... | {
"timestamp": "2009-01-12T21:15:06",
"yymm": "0606",
"arxiv_id": "math/0606646",
"language": "en",
"url": "https://arxiv.org/abs/math/0606646",
"abstract": "A new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant (1) defines a Hopf morphism from the H... |
https://arxiv.org/abs/2005.01260 | A conformal characterization of manifolds of constant sectional curvature | A special case of the main result states that a complete $1$-connected Riemannian manifold $(M^n,g)$ is isometric to one of the models $\mathbb R^n$, $S^n(c)$, $\mathbb H^n(-c)$ of constant curvature if and only if every $p\in M^n$ is a non-degenerate maximum of a germ of smooth functions whose Riemannian gradient is a... | \section{Introduction}
The goal of the present paper is to offer a characterization of connected Riemannian manifolds of constant sectional curvature in terms of the existence of certain special germs of functions.
For conceptual clarity, it is convenient to single out the following notion:
\begin{defn} \rm
A ... | {
"timestamp": "2020-05-05T02:23:49",
"yymm": "2005",
"arxiv_id": "2005.01260",
"language": "en",
"url": "https://arxiv.org/abs/2005.01260",
"abstract": "A special case of the main result states that a complete $1$-connected Riemannian manifold $(M^n,g)$ is isometric to one of the models $\\mathbb R^n$, $S^... |
https://arxiv.org/abs/1408.4082 | Higher Affine Connections | For a smooth manifold $M$, it was shown in \cite{BPH} that every affine connection on the tangent bundle $TM$ naturally gives rise to covariant differentiation of multivector fields (MVFs) and differential forms along MVFs. In this paper, we generalize the covariant derivative of \cite{BPH} and construct covariant deri... | \section{Introduction}
Let $M$ be a manfiold. It was shown in \cite{BPH} that every affine connection $\nabla$ on the tangent bundle $TM$ naturally gives rise to covariant differentiation of multivector fields (MVFs) and differential forms along MVFs. For covariant differentiation of MVFs along MVFs, the covariant de... | {
"timestamp": "2014-12-30T02:17:18",
"yymm": "1408",
"arxiv_id": "1408.4082",
"language": "en",
"url": "https://arxiv.org/abs/1408.4082",
"abstract": "For a smooth manifold $M$, it was shown in \\cite{BPH} that every affine connection on the tangent bundle $TM$ naturally gives rise to covariant differentia... |
https://arxiv.org/abs/1602.03773 | A sequence of triangle-free pseudorandom graphs | A construction of Alon yields a sequence of highly pseudorandom triangle-free graphs with edge density significantly higher than one might expect from comparison with random graphs. We give an alternative construction for such graphs. | \section{Introduction}
A graph $G$ is said to be $(p, \beta)$-jumbled if
\[\left|e(X) - p \binom{|X|}{2}\right| \leq \beta |X|\]
for all $X \subseteq V(G)$. For example, the binomial random graph $G_{n,p}$ is $(p, \beta)$-jumbled with $\beta = O(\sqrt{pn})$. It is not hard to show~\cite{EGPS88, ES71} that this is esse... | {
"timestamp": "2016-08-05T02:03:15",
"yymm": "1602",
"arxiv_id": "1602.03773",
"language": "en",
"url": "https://arxiv.org/abs/1602.03773",
"abstract": "A construction of Alon yields a sequence of highly pseudorandom triangle-free graphs with edge density significantly higher than one might expect from com... |
https://arxiv.org/abs/1203.2449 | Tropical matrix groups | We study the subgroup structure of the semigroup of finitary tropical matrices under multiplication. We show that every maximal subgroup is isomorphic to the full linear automorphism group of a related tropical polytope, and that each of these groups is the direct product of the real numbers with a finite group. We als... | \section{Introduction}
Tropical algebra is the algebra of the real numbers (sometimes augmented
with an extra element denoted by $-\infty$) under the operations of
addition and maximum. It has applications in areas such as combinatorial
optimisation and scheduling, control theory, and algebraic geometry to
name bu... | {
"timestamp": "2012-03-13T01:04:03",
"yymm": "1203",
"arxiv_id": "1203.2449",
"language": "en",
"url": "https://arxiv.org/abs/1203.2449",
"abstract": "We study the subgroup structure of the semigroup of finitary tropical matrices under multiplication. We show that every maximal subgroup is isomorphic to th... |
https://arxiv.org/abs/2110.05194 | A note on some properties of the $λ$-Polynomial | The expression $a^n + b^n$ can be factored as $(a+b)(a^{n-1} - a^{n-2} b + a^{n-3} b^2 - ... + b^{n-1})$ when $n$ is an odd integer greater than one. This paper focuses on proving a few properties of the longer factor above, which we call $\lambda_n(a,b)$. One such property is that the primes which divide $\lambda_n(a,... | \section{Preliminary lemmas}
\textbf{Lemma 1.1} Let $a,b \in \mathbb{Z}$ and let $d \in \mathbb{Z}$ divide $a$ but not $b$. Then $d$ does not divide $a+b$.\\
\textit{Proof.} Suppose otherwise. We can then write $a = kd$ and $a+b = hd$ for some $k,h \in \mathbb{Z}$. As a consequence $b = hd - kd = d(h-k)$ which implie... | {
"timestamp": "2021-10-28T02:28:05",
"yymm": "2110",
"arxiv_id": "2110.05194",
"language": "en",
"url": "https://arxiv.org/abs/2110.05194",
"abstract": "The expression $a^n + b^n$ can be factored as $(a+b)(a^{n-1} - a^{n-2} b + a^{n-3} b^2 - ... + b^{n-1})$ when $n$ is an odd integer greater than one. This... |
https://arxiv.org/abs/1708.00833 | tt-geometry of filtered modules | We compute the tensor triangular spectrum of perfect complexes of filtered modules over a commutative ring, and deduce a classification of the thick tensor ideals. We give two proofs: one by reducing to perfect complexes of graded modules which have already been studied in the literature, and one more direct for which ... |
\section{Introduction}
\label{sec:intro}
One of the age-old problems mathematicians engage in is to classify their objects of study, up to an appropriate equivalence relation. In contexts in which the domain is organized in a category with compatible tensor and triangulated structure (we call this a \emph{tt-category... | {
"timestamp": "2017-08-08T02:09:20",
"yymm": "1708",
"arxiv_id": "1708.00833",
"language": "en",
"url": "https://arxiv.org/abs/1708.00833",
"abstract": "We compute the tensor triangular spectrum of perfect complexes of filtered modules over a commutative ring, and deduce a classification of the thick tenso... |
https://arxiv.org/abs/2010.02906 | The Index Theorem for Toeplitz Operators as a Corollary of Bott Periodicity | This is an expository paper about the index of Toeplitz operators, and in particular Boutet de Monvel's theorem. We prove Boutet de Monvel's theorem as a corollary of Bott periodicity, and independently of the Atiyah-Singer index theorem. | \section{Introduction}
This is an expository paper about the index of Toeplitz operators,
and in particular Boutet de Monvel's theorem \cite{Bo79}.
We prove Boutet de Monvel's theorem as a corollary of Bott periodicity,
and independently of the Atiyah-Singer index theorem.
Let $M$ be an odd dimensional closed Spin$... | {
"timestamp": "2021-01-01T02:13:53",
"yymm": "2010",
"arxiv_id": "2010.02906",
"language": "en",
"url": "https://arxiv.org/abs/2010.02906",
"abstract": "This is an expository paper about the index of Toeplitz operators, and in particular Boutet de Monvel's theorem. We prove Boutet de Monvel's theorem as a ... |
https://arxiv.org/abs/2109.11769 | Non-Euclidean Self-Organizing Maps | Self-Organizing Maps (SOMs, Kohonen networks) belong to neural network models of the unsupervised class. In this paper, we present the generalized setup for non-Euclidean SOMs. Most data analysts take it for granted to use some subregions of a flat space as their data model; however, by the assumption that the underlyi... | \section{Introduction}
Self-Organizing Maps (SOMs, also known as Kohonen networks) belong to neural network models of the unsupervised class allowing for dimension reduction in data without a significant loss of information. SOMs preserve the underlying topology of high-dimensional input and transform the information ... | {
"timestamp": "2022-05-03T02:37:41",
"yymm": "2109",
"arxiv_id": "2109.11769",
"language": "en",
"url": "https://arxiv.org/abs/2109.11769",
"abstract": "Self-Organizing Maps (SOMs, Kohonen networks) belong to neural network models of the unsupervised class. In this paper, we present the generalized setup f... |
https://arxiv.org/abs/1909.03203 | Geometry of planar curves intersecting many lines in a few points | The local Lipschitz property is shown for the graph avoiding multiple point intersection with lines directed in a given cone. The assumption is much stronger than those of Marstrand's well-known theorem, but the conclusion is much stronger too. Additionally, a continuous curve with a similar property is $\sigma$-finite... | \section{The statement of the problem}
The problem we consider in this note grew from a question in perturbation theory of self-adjoint operators, see \cite{LTV}. The question was to better understand the structure of Borel sets in $\mathbb{R}^n$ which have a small intersection with a whole cone of lines. Marstrand's... | {
"timestamp": "2019-09-10T02:05:59",
"yymm": "1909",
"arxiv_id": "1909.03203",
"language": "en",
"url": "https://arxiv.org/abs/1909.03203",
"abstract": "The local Lipschitz property is shown for the graph avoiding multiple point intersection with lines directed in a given cone. The assumption is much stron... |
https://arxiv.org/abs/2003.03311 | Covering cycles in sparse graphs | Let $k \geq 2$ be an integer. Kouider and Lonc proved that the vertex set of every graph $G$ with $n \geq n_0(k)$ vertices and minimum degree at least $n/k$ can be covered by $k - 1$ cycles. Our main result states that for every $\alpha > 0$ and $p = p(n) \in (0, 1]$, the same conclusion holds for graphs $G$ with minim... | \section{The Absorber Lemma}\label{sec:absorbers}
This section is dedicated to the construction of the absorbers and the proof of
the Absorbing Lemma (Lemma~\ref{lem:absorbing-lemma}). We recall the definition
of an absorber first.
\absorber*
The most `natural' way to construct an absorber would be to first find
str... | {
"timestamp": "2021-11-18T02:23:07",
"yymm": "2003",
"arxiv_id": "2003.03311",
"language": "en",
"url": "https://arxiv.org/abs/2003.03311",
"abstract": "Let $k \\geq 2$ be an integer. Kouider and Lonc proved that the vertex set of every graph $G$ with $n \\geq n_0(k)$ vertices and minimum degree at least $... |
https://arxiv.org/abs/2111.02967 | An Empirical Comparison of the Quadratic Sieve Factoring Algorithm and the Pollard Rho Factoring Algorithm | One of the most significant challenges on cryptography today is the problem of factoring large integers since there are no algorithms that can factor in polynomial time, and factoring large numbers more than some limits(200 digits) remain difficult. The security of the current cryptosystems depends on the hardness of f... | \section{Introduction}
The idea of public-key cryptography was first introduced in 1975 by Martin Hellman, Ralph Merkle, and Whitfield Diffie at Stanford University~\cite{ten_years}. Before the public key cryptosystem era, if two people want to exchange secret information without anybody else knowing, they have to agre... | {
"timestamp": "2021-11-05T01:20:09",
"yymm": "2111",
"arxiv_id": "2111.02967",
"language": "en",
"url": "https://arxiv.org/abs/2111.02967",
"abstract": "One of the most significant challenges on cryptography today is the problem of factoring large integers since there are no algorithms that can factor in p... |
https://arxiv.org/abs/0711.2541 | Schubert calculus and cohomology of Lie groups. Part I. 1-connected Lie groups | Let $G$ be a compact and $1$--connected Lie group with a maximal torus $T$. Based on Schubert calculus on the flag manifold $G/T$ [15] we construct the integral cohomology ring $H^{\ast}(G)$ uniformly for all $G$. | \section{Introduction}
Let $G$ be a compact, $1$--connected and simple Lie group, namely, $G$ is one
of the classical groups $SU(n),Spin(n),Sp(n)$, or one of the exceptional
groups $G_{2},F_{4},E_{6},E_{7},E_{8}$. These groups constitute the
cornerstones in all compact Lie groups in views of Cartan's classificati... | {
"timestamp": "2010-08-31T02:01:14",
"yymm": "0711",
"arxiv_id": "0711.2541",
"language": "en",
"url": "https://arxiv.org/abs/0711.2541",
"abstract": "Let $G$ be a compact and $1$--connected Lie group with a maximal torus $T$. Based on Schubert calculus on the flag manifold $G/T$ [15] we construct the inte... |
https://arxiv.org/abs/2110.08453 | Voting Theory in the Lean Theorem Prover | There is a long tradition of fruitful interaction between logic and social choice theory. In recent years, much of this interaction has focused on computer-aided methods such as SAT solving and interactive theorem proving. In this paper, we report on the development of a framework for formalizing voting theory in the L... | \section{Introduction}
There is a long tradition of fruitful interaction between logic and social choice theory. Both Kenneth Arrow \cite[p.~154]{Arrow2014} and Amartya Sen \cite[p.~108]{Sen2017} have noted the influence of mathematical logic on their thinking about the foundations of social choice theory. Early wo... | {
"timestamp": "2021-10-19T02:07:18",
"yymm": "2110",
"arxiv_id": "2110.08453",
"language": "en",
"url": "https://arxiv.org/abs/2110.08453",
"abstract": "There is a long tradition of fruitful interaction between logic and social choice theory. In recent years, much of this interaction has focused on compute... |
https://arxiv.org/abs/1704.06255 | On the gonality, treewidth, and orientable genus of a graph | We examine connections between the gonality, treewidth, and orientable genus of a graph. Especially, we find that hyperelliptic graphs in the sense of Baker and Norine are planar. We give a notion of a bielliptic graph and show that each of these must embed into a closed orientable surface of genus one. We also find, f... | \section{Preliminaries on the involutions of graphs and hyperelliptic graphs}
Although the notion of a hyperelliptic graph is well-established,
we prefer to use the
equivalent definition furnished by the hyperelliptic involution \cite{BNHypell}.
\begin{dfn} A \newword{mixing involution} on a graph $G$ is an order-two... | {
"timestamp": "2017-04-21T02:08:42",
"yymm": "1704",
"arxiv_id": "1704.06255",
"language": "en",
"url": "https://arxiv.org/abs/1704.06255",
"abstract": "We examine connections between the gonality, treewidth, and orientable genus of a graph. Especially, we find that hyperelliptic graphs in the sense of Bak... |
https://arxiv.org/abs/1008.4768 | The absence of phase transition for the classical XY-model on Sierpinski pyramid with fractal dimension D=2 | For the spin models with continuous symmetry on regular lattices and finite range of interactions the lower critical dimension is d=2. In two dimensions the classical XY-model displays Berezinskii-Kosterlitz-Thouless transition associated with unbinding of topological defects (vortices and antivortices). We perform a M... | \section{Introduction}
One of the powerful predictions of the renormalization group theory of critical phenomena is universality
according to which the critical behavior of a system is determined by: (1) symmetry group of the Hamiltonian,
(2) spatial dimensionality $d$ and (3) whether or not the interactions are sho... | {
"timestamp": "2010-08-30T02:02:05",
"yymm": "1008",
"arxiv_id": "1008.4768",
"language": "en",
"url": "https://arxiv.org/abs/1008.4768",
"abstract": "For the spin models with continuous symmetry on regular lattices and finite range of interactions the lower critical dimension is d=2. In two dimensions the... |
https://arxiv.org/abs/math/0411171 | New refinements of the McKay conjecture for arbitrary finite groups | Let $G$ be an arbitrary finite group and fix a prime number $p$. The McKay conjecture asserts that $G$ and the normalizer in $G$ of a Sylow $p$-subgroup have equal numbers of irreducible characters with degrees not divisible by $p$. The Alperin-McKay conjecture is a version of this as applied to individual Brauer $p$-b... | \section{Introduction and Conjecture A}
Let $G$ be an arbitrary finite group and fix a prime number $p$. As is well
known, there seem to be some mysterious and unexplained connections
between the representation theory of $G$ and that of certain of its
$p$-local subgroups. For example, it appears to be true that if $P$... | {
"timestamp": "2004-11-08T19:27:11",
"yymm": "0411",
"arxiv_id": "math/0411171",
"language": "en",
"url": "https://arxiv.org/abs/math/0411171",
"abstract": "Let $G$ be an arbitrary finite group and fix a prime number $p$. The McKay conjecture asserts that $G$ and the normalizer in $G$ of a Sylow $p$-subgro... |
https://arxiv.org/abs/1108.2684 | Gabor frames with rational density | We consider the frame property of the Gabor system G(g, {\alpha}, {\beta}) = {e2{\pi}i{\beta}nt g(t - {\alpha}m) : m, n \in Z} for the case of rational oversampling, i.e. {\alpha}, {\beta} \in Q. A 'rational' analogue of the Ron-Shen Gramian is constructed, and prove that for any odd window function g the system G(g, {... | \section{Introduction}\label{S:I}
One of the fundamental problems of Gabor analysis can be stated as
follows: given a window function $g\in L^2({\mathbb R})$, determine the set of
lattice parameters $\alpha,\beta {>}0$ such that the Gabor system
$\mathcal{G}(g{,}\alpha{,}\beta) {=} \{e^{2i\pi \beta n t}
g(t{-}\alpha m)... | {
"timestamp": "2011-08-15T02:03:07",
"yymm": "1108",
"arxiv_id": "1108.2684",
"language": "en",
"url": "https://arxiv.org/abs/1108.2684",
"abstract": "We consider the frame property of the Gabor system G(g, {\\alpha}, {\\beta}) = {e2{\\pi}i{\\beta}nt g(t - {\\alpha}m) : m, n \\in Z} for the case of rationa... |
https://arxiv.org/abs/2210.00279 | Failure-informed adaptive sampling for PINNs | Physics-informed neural networks (PINNs) have emerged as an effective technique for solving PDEs in a wide range of domains. It is noticed, however, the performance of PINNs can vary dramatically with different sampling procedures. For instance, a fixed set of (prior chosen) training points may fail to capture the effe... | \section{Introduction}
Partial differential equations (PDEs) are important tools for modeling many real-world phenomena. Traditional numeric solvers such as finite difference method and finite element method have been widely used to solve PDEs for many decades.
However, for high-dimensional problems, the computationa... | {
"timestamp": "2022-10-11T02:15:32",
"yymm": "2210",
"arxiv_id": "2210.00279",
"language": "en",
"url": "https://arxiv.org/abs/2210.00279",
"abstract": "Physics-informed neural networks (PINNs) have emerged as an effective technique for solving PDEs in a wide range of domains. It is noticed, however, the p... |
https://arxiv.org/abs/cond-mat/0508040 | Width of percolation transition in complex networks | It is known that the critical probability for the percolation transition is not a sharp threshold, actually it is a region of non-zero width $\Delta p_c$ for systems of finite size. Here we present evidence that for complex networks $\Delta p_c \sim \frac{p_c}{l}$, where $l \sim N^{\nu_{opt}}$ is the average length of ... | \section{\label{sec:Introduction}Introduction:}
Recently the subject of networks has received much attention. It was
realized that many systems in the real world, such as the Internet,
can be successfully modeled as networks. Other examples include social
networks such as the web of social contacts, and biological ne... | {
"timestamp": "2005-08-01T12:25:46",
"yymm": "0508",
"arxiv_id": "cond-mat/0508040",
"language": "en",
"url": "https://arxiv.org/abs/cond-mat/0508040",
"abstract": "It is known that the critical probability for the percolation transition is not a sharp threshold, actually it is a region of non-zero width $... |
https://arxiv.org/abs/2104.05200 | A Note on the Performance of Algorithms for Solving Linear Diophantine Equations in the Naturals | We implement four algorithms for solving linear Diophantine equations in the naturals: a lexicographic enumeration algorithm, a completion procedure, a graph-based algorithm, and the Slopes algorithm. As already known, the lexicographic enumeration algorithm and the completion procedure are slower than the other two al... |
\section{Introduction}\label{sec:intro}
Solving linear Diophantine equations in the naturals is at the core of
AC-unification algorithms. AC-unification reduces to top-most
unification problems of the form
\[ f^*(u_1, \ldots, u_l) = f^*(v_1, \ldots, v_k), \] where
\( u_1, \ldots, u_l, v_1, \ldots, v_k \) are variab... | {
"timestamp": "2021-04-13T02:25:43",
"yymm": "2104",
"arxiv_id": "2104.05200",
"language": "en",
"url": "https://arxiv.org/abs/2104.05200",
"abstract": "We implement four algorithms for solving linear Diophantine equations in the naturals: a lexicographic enumeration algorithm, a completion procedure, a gr... |
https://arxiv.org/abs/1904.05717 | The MOMMS Family of Matrix Multiplication Algorithms | As the ratio between the rate of computation and rate with which data can be retrieved from various layers of memory continues to deteriorate, a question arises: Will the current best algorithms for computing matrix-matrix multiplication on future CPUs continue to be (near) optimal? This paper provides compelling analy... |
\subsection{An I/O lower bound for MMM}
Smith et al.~\cite{smith2019tight} starts with a simple model of memory with two layers of memory: a small, fast memory with capacity of $ M $ elements and a large, slow memory with unlimited capacity. It shows that any algorithm for ordinary MMM%
\footnote{We only consider a... | {
"timestamp": "2019-04-12T02:14:44",
"yymm": "1904",
"arxiv_id": "1904.05717",
"language": "en",
"url": "https://arxiv.org/abs/1904.05717",
"abstract": "As the ratio between the rate of computation and rate with which data can be retrieved from various layers of memory continues to deteriorate, a question ... |
https://arxiv.org/abs/1502.02120 | Testing uniformity on high-dimensional spheres against monotone rotationally symmetric alternatives | We consider the problem of testing uniformity on high-dimensional unit spheres. We are primarily interested in non-null issues. We show that rotationally symmetric alternatives lead to two Local Asymptotic Normality (LAN) structures. The first one is for fixed modal location $\theta$ and allows to derive locally asympt... | \section{Introduction}
In directional statistics, inference is based on $p$-variate observations lying on the unit sphere~$\mathcal{S}^{p-1}:=\{\ensuremath{\mathbf{x}}\in\mathbb R^{p} : \|\ensuremath{\mathbf{x}}\|=\sqrt{\ensuremath{\mathbf{x}}'\ensuremath{\mathbf{x}}}=1\}$. This is relevant in various situations. (i) ... | {
"timestamp": "2016-04-28T02:12:39",
"yymm": "1502",
"arxiv_id": "1502.02120",
"language": "en",
"url": "https://arxiv.org/abs/1502.02120",
"abstract": "We consider the problem of testing uniformity on high-dimensional unit spheres. We are primarily interested in non-null issues. We show that rotationally ... |
https://arxiv.org/abs/1403.5186 | The spectral Phase-Amplitude representation of a wave function revisited | The phase and amplitude (Ph-A) of a wave function vary slowly and monotonically with distance, in contrast to the wave function that can be highly oscillatory. Hence an attractive feature of the Ph-A representation is that it requires far fewer meshpoints than for the wave function itself. In 1930 Milne developed an eq... | \section{Introduction}
When the Phase-Amplitude (Ph-A) method was first introduced by Milne in 1930
\cite{MILNE} , and then taken up by many authors, see Ref. \cite{KORSCH}, the
main motivation was the paucity of numerical mesh points required, compared to
the calculation of the wave function itself. This is beca... | {
"timestamp": "2014-05-29T02:11:06",
"yymm": "1403",
"arxiv_id": "1403.5186",
"language": "en",
"url": "https://arxiv.org/abs/1403.5186",
"abstract": "The phase and amplitude (Ph-A) of a wave function vary slowly and monotonically with distance, in contrast to the wave function that can be highly oscillato... |
https://arxiv.org/abs/1302.1699 | Sharp deviation bounds for quadratic forms | This note presents sharp inequalities for deviation probability of a general quadratic form of a random vector \(\xiv\) with finite exponential moments. The obtained deviation bounds are similar to the case of a Gaussian random vector. The results are stated under general conditions and do not suppose any special struc... |
\section{#1}}
\newcommand{\Section}[1]{\subsection{#1}}
\newcommand{\Subsection}[1]{\subsubsection{#1}}
\def\Chname{Section }
\def\chname{section }
\def\chapsect{Section}
}
{
\newcommand{\Chapter}[1]{\chapter{#1}}
\newcommand{\Section}[1]{\section{#1}}
\newcommand{\Subsec... | {
"timestamp": "2013-02-08T02:01:52",
"yymm": "1302",
"arxiv_id": "1302.1699",
"language": "en",
"url": "https://arxiv.org/abs/1302.1699",
"abstract": "This note presents sharp inequalities for deviation probability of a general quadratic form of a random vector \\(\\xiv\\) with finite exponential moments. ... |
https://arxiv.org/abs/1901.04405 | Quadratization in discrete optimization and quantum mechanics | A book about turning high-degree optimization problems into quadratic optimization problems that maintain the same global minimum (ground state). This book explores quadratizations for pseudo-Boolean optimization, perturbative gadgets used in QMA completeness theorems, and also non-perturbative k-local to 2-local trans... | \section{Introduction}
\noindent\rule{\textwidth}{0.4pt}
\vspace{-1.25mm}
\indent When optimizing discrete functions, it is often easier when the function is quadratic than if it is of higher degree. But notice that the cubic and quadratic functions:
\vspace{-4mm}
\begin{align}
b_1b_2+b_2b_3+b_3b_4 -4b_1b_2b_3 & \hs... | {
"timestamp": "2019-09-24T02:20:31",
"yymm": "1901",
"arxiv_id": "1901.04405",
"language": "en",
"url": "https://arxiv.org/abs/1901.04405",
"abstract": "A book about turning high-degree optimization problems into quadratic optimization problems that maintain the same global minimum (ground state). This boo... |
https://arxiv.org/abs/1104.4992 | Boundedness of trajectories for weakly reversible, single linkage class reaction systems | This paper is concerned with the dynamical properties of deterministically modeled chemical reaction systems with mass-action kinetics. Such models are ubiquitously found in chemistry, population biology, and the burgeoning field of systems biology. A basic question, whose answer remains largely unknown, is the followi... | \section{Introduction}
Building off the work of Fritz Horn, Roy Jackson, and Martin Feinberg
\cite{FeinbergLec79, Feinberg87, FeinHorn1972, Horn72, Horn74, HornJack72} the
mathematical theory termed ``Chemical Reaction Network Theory'' has, over the past 40 years, determined many of the basic qualitative properties of... | {
"timestamp": "2011-06-20T02:00:30",
"yymm": "1104",
"arxiv_id": "1104.4992",
"language": "en",
"url": "https://arxiv.org/abs/1104.4992",
"abstract": "This paper is concerned with the dynamical properties of deterministically modeled chemical reaction systems with mass-action kinetics. Such models are ubiq... |
https://arxiv.org/abs/1503.01587 | On debiasing restoration algorithms: applications to total-variation and nonlocal-means | Bias in image restoration algorithms can hamper further analysis, typically when the intensities have a physical meaning of interest, e.g., in medical imaging. We propose to suppress a part of the bias -- the method bias -- while leaving unchanged the other unavoidable part -- the model bias. Our debiasing technique ca... | \section{Bias of reconstruction algorithms}
Due to the ill-posedness of our observation model and without any assumptions on $u_0$,
one cannot ensure the noise variance to be reduced while
keeping the solution $\uf^\star$ unbiased.
Recall that the statistical bias is defined as the difference
\begin{equation}
\text{... | {
"timestamp": "2015-03-06T02:07:32",
"yymm": "1503",
"arxiv_id": "1503.01587",
"language": "en",
"url": "https://arxiv.org/abs/1503.01587",
"abstract": "Bias in image restoration algorithms can hamper further analysis, typically when the intensities have a physical meaning of interest, e.g., in medical ima... |
https://arxiv.org/abs/1911.04552 | New approaches to finite generation of cohomology rings | In support variety theory, representations of a finite dimensional (Hopf) algebra $A$ can be studied geometrically by associating any representation of $A$ to an algebraic variety using the cohomology ring of $A$. An essential assumption in this theory is the finite generation condition for the cohomology ring of $A$ a... | \section{Introduction}
Hochschild cohomology was introduced by Hochschild in 1945 \cite{GHoch59} for any associative algebra. Gerstenhaber \cite{Ger63} showed that this cohomology has a graded algebra structure (via cup product) and a graded Lie algebra structure (via a Lie bracket or Gerstenhaber bracket). These two ... | {
"timestamp": "2021-08-17T02:16:26",
"yymm": "1911",
"arxiv_id": "1911.04552",
"language": "en",
"url": "https://arxiv.org/abs/1911.04552",
"abstract": "In support variety theory, representations of a finite dimensional (Hopf) algebra $A$ can be studied geometrically by associating any representation of $A... |
https://arxiv.org/abs/2011.11289 | Sparse Inpainting with Smoothed Particle Hydrodynamics | Digital image inpainting refers to techniques used to reconstruct a damaged or incomplete image by exploiting available image information. The main goal of this work is to perform the image inpainting process from a set of sparsely distributed image samples with the Smoothed Particle Hydrodynamics (SPH) technique. As, ... | \section{Introduction}
Image inpainting aims at restoring partially damaged image or missing
parts of an image in a
visually appealing manner \cite{BS00}. It has a wide
number of practical applications such as
art restoration \cite{KM13,BF08,CA18,RC11},
object removal \cite{CP04},
medical imaging \cite{TB19},
inpa... | {
"timestamp": "2021-08-25T02:04:24",
"yymm": "2011",
"arxiv_id": "2011.11289",
"language": "en",
"url": "https://arxiv.org/abs/2011.11289",
"abstract": "Digital image inpainting refers to techniques used to reconstruct a damaged or incomplete image by exploiting available image information. The main goal o... |
https://arxiv.org/abs/2103.11917 | The digrundy number of digraphs | We extend the Grundy number and the ochromatic number, parameters on graph colorings, to digraph colorings, we call them {\emph{digrundy number}} and {\emph{diochromatic number}}, respectively. First, we prove that for every digraph the diochromatic number equals the digrundy number (as it happen for graphs). Then, we ... | \section{Introduction}
It is common that classical results or problems on graph theory provide us interesting questions on digraph theory. An interesting question is what is the natural generalization of the chromatic number in the class of digraphs. In 1982 Neumann-Lara introduced the concept of dichromatic number as... | {
"timestamp": "2021-03-23T01:39:20",
"yymm": "2103",
"arxiv_id": "2103.11917",
"language": "en",
"url": "https://arxiv.org/abs/2103.11917",
"abstract": "We extend the Grundy number and the ochromatic number, parameters on graph colorings, to digraph colorings, we call them {\\emph{digrundy number}} and {\\... |
https://arxiv.org/abs/2109.02241 | Supervised DKRC with Images for Offline System Identification | Koopman spectral theory has provided a new perspective in the field of dynamical systems in recent years. Modern dynamical systems are becoming increasingly non-linear and complex, and there is a need for a framework to model these systems in a compact and comprehensive representation for prediction and control. The ce... | \section{Introduction}
\label{sec:introduction}
This document is a template for \LaTeX. If you are
reading a paper or PDF version of this document, please download the
electronic file, trans\_jour.tex, from the IEEE Web site at \underline
{http://www.ieee.org/authortools/trans\_jour.tex} so you can use it to prepare ... | {
"timestamp": "2021-09-07T02:25:42",
"yymm": "2109",
"arxiv_id": "2109.02241",
"language": "en",
"url": "https://arxiv.org/abs/2109.02241",
"abstract": "Koopman spectral theory has provided a new perspective in the field of dynamical systems in recent years. Modern dynamical systems are becoming increasing... |
https://arxiv.org/abs/1308.5173 | Independence ratio and random eigenvectors in transitive graphs | A theorem of Hoffman gives an upper bound on the independence ratio of regular graphs in terms of the minimum $\lambda_{\min}$ of the spectrum of the adjacency matrix. To complement this result we use random eigenvectors to gain lower bounds in the vertex-transitive case. For example, we prove that the independence rat... |
\section{Introduction}
\subsection{The independence ratio and the minimum eigenvalue}
An \emph{independent set} is a set of vertices in a graph, no two of which are adjacent.
The \emph{independence ratio} of a graph $G$ is the size of
its largest independent set divided by the total number of vertices.
If $G... | {
"timestamp": "2013-08-26T02:06:26",
"yymm": "1308",
"arxiv_id": "1308.5173",
"language": "en",
"url": "https://arxiv.org/abs/1308.5173",
"abstract": "A theorem of Hoffman gives an upper bound on the independence ratio of regular graphs in terms of the minimum $\\lambda_{\\min}$ of the spectrum of the adja... |
https://arxiv.org/abs/1911.01724 | Connector-Breaker games on random boards | By now, the Maker-Breaker connectivity game on a complete graph $K_n$ or on a random graph $G\sim G_{n,p}$ is well studied. Recently, London and Pluhár suggested a variant in which Maker always needs to choose her edges in such a way that her graph stays connected. By their results it follows that for this connected ve... | \section{Introduction}
A positional game is a perfect information game played by two players on a {\em board} $X$ equipped with a family of subsets $\mathcal F \subset 2^X$, which represent {\em winning sets}. During each round of such a game both players claim previously unclaimed elements of the board. For instance,... | {
"timestamp": "2019-11-06T02:14:17",
"yymm": "1911",
"arxiv_id": "1911.01724",
"language": "en",
"url": "https://arxiv.org/abs/1911.01724",
"abstract": "By now, the Maker-Breaker connectivity game on a complete graph $K_n$ or on a random graph $G\\sim G_{n,p}$ is well studied. Recently, London and Pluhár s... |
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