url stringlengths 31 38 | title stringlengths 7 229 | abstract stringlengths 44 2.87k | text stringlengths 319 2.51M | meta dict |
|---|---|---|---|---|
https://arxiv.org/abs/2105.10615 | Convergence directions of the randomized Gauss--Seidel method and its extension | The randomized Gauss--Seidel method and its extension have attracted much attention recently and their convergence rates have been considered extensively. However, the convergence rates are usually determined by upper bounds, which cannot fully reflect the actual convergence. In this paper, we make a detailed analysis ... | \section{Introduction}
Linear least squares problem is a ubiquitous problem arising frequently in data analysis and scientific computing. Specifically, given a data matrix $A\in R^{m\times n}$ and a data vector $b\in R^{m}$, a linear least squares problem can be written as follows
\begin{equation}
\label{ls}
\min \limi... | {
"timestamp": "2021-05-25T02:05:27",
"yymm": "2105",
"arxiv_id": "2105.10615",
"language": "en",
"url": "https://arxiv.org/abs/2105.10615",
"abstract": "The randomized Gauss--Seidel method and its extension have attracted much attention recently and their convergence rates have been considered extensively.... |
https://arxiv.org/abs/1912.01763 | A note on semi-infinite program bounding methods | Semi-infinite programs are a class of mathematical optimization problems with a finite number of decision variables and infinite constraints. As shown by Blankenship and Falk (Blankenship and Falk. "Infinitely constrained optimization problems." Journal of Optimization Theory and Applications 19.2 (1976): 261-281.), a ... |
\section{Introduction}
This note discusses methods for the global solution of semi-infinite programs (SIP).
Specifically, the method from \cite{mitsos11} is considered, and it is shown with a counterexample that the lower bounds do not always converge.
Throughout we use notation as close as possible to that used in \c... | {
"timestamp": "2019-12-05T02:06:35",
"yymm": "1912",
"arxiv_id": "1912.01763",
"language": "en",
"url": "https://arxiv.org/abs/1912.01763",
"abstract": "Semi-infinite programs are a class of mathematical optimization problems with a finite number of decision variables and infinite constraints. As shown by ... |
https://arxiv.org/abs/math/0610707 | A fixed point theorem for the infinite-dimensional simplex | We define the infinite dimensional simplex to be the closure of the convex hull of the standard basis vectors in R^infinity, and prove that this space has the 'fixed point property': any continuous function from the space into itself has a fixed point. Our proof is constructive, in the sense that it can be used to find... | \section{Introduction}
In finite dimensions, one of the simplest methods for proving the
Brouwer fixed point theorem is via a combinatorial result known as
Sperner's lemma \cite{Sper28}, which is a statement about labelled
triangulations of a simplex in $\ensuremath{\mathbb{R}} ^n$. In this paper, we use
Sperner's le... | {
"timestamp": "2006-10-24T03:10:17",
"yymm": "0610",
"arxiv_id": "math/0610707",
"language": "en",
"url": "https://arxiv.org/abs/math/0610707",
"abstract": "We define the infinite dimensional simplex to be the closure of the convex hull of the standard basis vectors in R^infinity, and prove that this space... |
https://arxiv.org/abs/1911.12009 | Involution pipe dreams | Involution Schubert polynomials represent cohomology classes of $K$-orbit closures in the complete flag variety, where $K$ is the orthogonal or symplectic group. We show they also represent $T$-equivariant cohomology classes of subvarieties defined by upper-left rank conditions in the spaces of symmetric or skew-symmet... | \section{Introduction}
One can identify the equivariant cohomology rings for the spaces of symmetric and skew-symmetric complex matrices with multivariate polynomial rings.
Under this identification, we show
that the classes of certain natural subvarieties of (skew-)symmetric matrices are given by the \emph{involutio... | {
"timestamp": "2020-08-17T02:05:50",
"yymm": "1911",
"arxiv_id": "1911.12009",
"language": "en",
"url": "https://arxiv.org/abs/1911.12009",
"abstract": "Involution Schubert polynomials represent cohomology classes of $K$-orbit closures in the complete flag variety, where $K$ is the orthogonal or symplectic... |
https://arxiv.org/abs/1410.6535 | A New Fractional Derivative with Classical Properties | We introduce a new fractional derivative which obeys classical properties including: linearity, product rule, quotient rule, power rule, chain rule, vanishing derivatives for constant functions, the Rolle's Theorem and the Mean Value Theorem. The definition, \[ D^\alpha (f)(t) = \lim_{\epsilon \rightarrow 0} \frac{f(te... | \section{Introduction}
The derivative of non-integer order has been an interesting research topic for several centuries. The idea was motivated by the question, ``What does it mean by $\frac{d^n f}{dx^n}$, if $n=\frac{1}{2}$ ?'', asked by L'Hospital in 1695 in his letters to Leibniz \cite{letter1,letter2, letter3}. Sin... | {
"timestamp": "2014-11-11T02:08:51",
"yymm": "1410",
"arxiv_id": "1410.6535",
"language": "en",
"url": "https://arxiv.org/abs/1410.6535",
"abstract": "We introduce a new fractional derivative which obeys classical properties including: linearity, product rule, quotient rule, power rule, chain rule, vanishi... |
https://arxiv.org/abs/2204.00383 | A visualisation for conveying the dynamics of iterative eigenvalue algorithms over PSD matrices | We propose a new way of visualising the dynamics of iterative eigenvalue algorithms such as the QR algorithm, over the important special case of PSD (positive semi-definite) matrices. Many subtle and important properties of such algorithms are easily found this way. We believe that this may have pedagogical value to bo... | \section{Simple iterative eigenvalue algorithms}
The (naive) QR algorithm {\cite{francis1961qr,wilkinson1968global,GoluVanl96}}
evaluated on matrix $M$ begins with setting $M_0 = M$ and then repeating the
following two steps until convergence:
\begin{enumeratenumeric}
\item Find the QR decomposition $M_n = Q_n R_n$.... | {
"timestamp": "2022-04-04T02:21:37",
"yymm": "2204",
"arxiv_id": "2204.00383",
"language": "en",
"url": "https://arxiv.org/abs/2204.00383",
"abstract": "We propose a new way of visualising the dynamics of iterative eigenvalue algorithms such as the QR algorithm, over the important special case of PSD (posi... |
https://arxiv.org/abs/2010.15204 | Shortest closed curve to inspect a sphere | We show that in Euclidean 3-space any closed curve which lies outside the unit sphere and contains the sphere within its convex hull has length at least $4\pi$. Equality holds only when the curve is composed of $4$ semicircles of length $\pi$, arranged in the shape of a baseball seam, as conjectured by V. A. Zalgaller ... | \section{Introduction}
What is the shortest closed orbit a satellite may take to inspect the entire surface of a round asteroid?
This is a well-known optimization problem \cite{zalgaller:1996,orourkeMO,hiriart:2008,ghomi:lwr,cfg,finch&wetzel} in classical differential geometry and convexity theory, which may be prec... | {
"timestamp": "2021-07-23T02:04:13",
"yymm": "2010",
"arxiv_id": "2010.15204",
"language": "en",
"url": "https://arxiv.org/abs/2010.15204",
"abstract": "We show that in Euclidean 3-space any closed curve which lies outside the unit sphere and contains the sphere within its convex hull has length at least $... |
https://arxiv.org/abs/2107.02428 | Browder's Theorem through Brouwer's Fixed Point Theorem | One of the conclusions of Browder (1960) is a parametric version of Brouwer's Fixed Point Theorem, stating that for every continuous function $f : ([0,1] \times X) \to X$, where $X$ is a simplex in a Euclidean space, the set of fixed points of $f$, namely, the set $\{(t,x) \in [0,1] \times X \colon f(t,x) = x\}$, has a... | \section{Introduction}
Brouwer's Fixed Point Theorem (Hadamard, 1910, Brouwer, 1911) states that every continuous function from a finite dimensional simplex into itself
has a fixed point.
This result was later generalized to nonempty, convex, and compact subsets of more general topological vector spaces, see, e.g.... | {
"timestamp": "2021-07-07T02:12:17",
"yymm": "2107",
"arxiv_id": "2107.02428",
"language": "en",
"url": "https://arxiv.org/abs/2107.02428",
"abstract": "One of the conclusions of Browder (1960) is a parametric version of Brouwer's Fixed Point Theorem, stating that for every continuous function $f : ([0,1] ... |
https://arxiv.org/abs/2107.14079 | Density of binary disc packings: lower and upper bounds | We provide, for any $r\in (0,1)$, lower and upper bounds on the maximal density of a packing in the Euclidean plane of discs of radius $1$ and $r$. The lower bounds are mostly folk, but the upper bounds improve the best previously known ones for any $r\in[0.11,0.74]$. For many values of $r$, this gives a fairly good id... | \section{Introduction}
A {\em disc packing} (or {\em circle packing}) is a set of interior-disjoint discs in the Euclidean plane.
Its {\em density} $\delta$ is the proportion of the plane covered by the discs:
$$
\delta:=\limsup_{k\to \infty}\frac{\textrm{area of the square $[-k,k]^2$ covered by discs}}{\textrm{area o... | {
"timestamp": "2022-06-07T02:28:12",
"yymm": "2107",
"arxiv_id": "2107.14079",
"language": "en",
"url": "https://arxiv.org/abs/2107.14079",
"abstract": "We provide, for any $r\\in (0,1)$, lower and upper bounds on the maximal density of a packing in the Euclidean plane of discs of radius $1$ and $r$. The l... |
https://arxiv.org/abs/math/0411020 | Resolutions of small sets of fat points | We investigate the minimal graded free resolutions of ideals of at most n+1 fat points in general position in P^n. Our main theorem is that these ideals are componentwise linear. This result yields a number of corollaries, including the multiplicity conjecture of Herzog, Huneke, and Srinivasan in this case. On the comp... | \section[1]{Introduction}\label{s:intro}
The study of sets of fat points in projective space is a classical topic that has continued to receive significant attention recently. Many researchers have investigated the Hilbert function, minimal graded free resolution, and other invariants of ideals of fat points, usually ... | {
"timestamp": "2005-06-01T19:10:15",
"yymm": "0411",
"arxiv_id": "math/0411020",
"language": "en",
"url": "https://arxiv.org/abs/math/0411020",
"abstract": "We investigate the minimal graded free resolutions of ideals of at most n+1 fat points in general position in P^n. Our main theorem is that these idea... |
https://arxiv.org/abs/1902.05034 | Generalized ergodic problems: existence and uniqueness structures of solutions | We study a generalized ergodic problem (E), which is a Hamilton-Jacobi equation of contact type, in the flat $n$-dimensional torus. We first obtain existence of solutions to this problem under quite general assumptions. Various examples are presented and analyzed to show that (E) does not have unique solutions in gener... | \section{Introduction}
In this paper, we focus on the following equation
\[
{\rm (E)} \qquad
H(x,u,Du) = c \qquad \text{ in } \mathbb{T}^n.
\]
Here, $\mathbb{T}^n =\mathbb{R}^n/\mathbb{Z}^n$ is the flat $n$-dimensional torus,
and the Hamiltonian $H=H(x,r,p):\mathbb{T}^n \times \mathbb{R} \times \mathbb{R}^n \to \mathb... | {
"timestamp": "2019-02-14T02:20:49",
"yymm": "1902",
"arxiv_id": "1902.05034",
"language": "en",
"url": "https://arxiv.org/abs/1902.05034",
"abstract": "We study a generalized ergodic problem (E), which is a Hamilton-Jacobi equation of contact type, in the flat $n$-dimensional torus. We first obtain existe... |
https://arxiv.org/abs/1406.0774 | Set Theory or Higher Order Logic to Represent Auction Concepts in Isabelle? | When faced with the question of how to represent properties in a formal proof system any user has to make design decisions. We have proved three of the theorems from Maskin's 2004 survey article on Auction Theory using the Isabelle/HOL system, and we have produced verified code for combinatorial Vickrey auctions. A fun... | \section{Introduction} \label{sec:introduction}
\label{RefIntro}
\lnote{MC: Rephrased after R3 noticed there is also type theory as a third strong competitor.}
When representing mathematics in formal proof systems, alternative foundations can be used, with two important examples being set theory (e.g., Mizar{} takes th... | {
"timestamp": "2014-06-04T02:10:21",
"yymm": "1406",
"arxiv_id": "1406.0774",
"language": "en",
"url": "https://arxiv.org/abs/1406.0774",
"abstract": "When faced with the question of how to represent properties in a formal proof system any user has to make design decisions. We have proved three of the theo... |
https://arxiv.org/abs/1811.06389 | Semi-perfect 1-Factorizations of the Hypercube | A 1-factorization $\mathcal{M} = \{M_1,M_2,\ldots,M_n\}$ of a graph $G$ is called perfect if the union of any pair of 1-factors $M_i, M_j$ with $i \ne j$ is a Hamilton cycle. It is called $k$-semi-perfect if the union of any pair of 1-factors $M_i, M_j$ with $1 \le i \le k$ and $k+1 \le j \le n$ is a Hamilton cycle.We ... | \section{Introduction}
A 1-factorization of a graph $H$ is a partition of the edges of $H$ into disjoint perfect matchings $\{M_1,M_2,\ldots,M_n\}$, also known as 1-factors.
Let $\mathcal{M} = \{M_1,M_2,\ldots,M_n\}$ be such a 1-factorization.
We say that $\mathcal{M}$ is a perfect factorization if every pair $M_i \c... | {
"timestamp": "2018-11-16T02:13:36",
"yymm": "1811",
"arxiv_id": "1811.06389",
"language": "en",
"url": "https://arxiv.org/abs/1811.06389",
"abstract": "A 1-factorization $\\mathcal{M} = \\{M_1,M_2,\\ldots,M_n\\}$ of a graph $G$ is called perfect if the union of any pair of 1-factors $M_i, M_j$ with $i \\n... |
https://arxiv.org/abs/1407.4339 | Extension from Precoloured Sets of Edges | We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and Shannon's bounds on the chromatic index of (multi)graphs in terms of their maximum degree $\Delta$. We are especially interested in the following question: when is i... | \section{Introduction}
Let $G = (V,E)$ be a (multi)graph and let $\mathcal{K}=[K]=\{1,\dots,K\}$ be a
palette of available colours. (In this paper, a \emph{multigraph} can have
multiple edges, but no loops; while a \emph{graph} is always simple.) We
consider the following question: given a subset $S\subseteq E$ of edg... | {
"timestamp": "2014-07-17T02:09:22",
"yymm": "1407",
"arxiv_id": "1407.4339",
"language": "en",
"url": "https://arxiv.org/abs/1407.4339",
"abstract": "We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and S... |
https://arxiv.org/abs/2207.09323 | Thin polytopes: Lattice polytopes with vanishing local $h^*$-polynomial | In this paper we study the novel notion of thin polytopes: lattice polytopes whose local $h^*$-polynomials vanish. The local $h^*$-polynomial is an important invariant in modern Ehrhart theory. Its definition goes back to Stanley with fundamental results achieved by Karu, Borisov & Mavlyutov, Schepers, and Katz & Stapl... | \section{Introduction}
In this paper we propose to investigate {\em thin polytopes}: lattice polytopes with vanishing local $h^*$-polynomials. Local $h^*$-polynomials are also called $l^*$-polynomials or $\tilde{S}$-polynomials. In the case of lattice simplices, they equal the so-called box polynomial, see Example~\re... | {
"timestamp": "2022-07-20T02:20:00",
"yymm": "2207",
"arxiv_id": "2207.09323",
"language": "en",
"url": "https://arxiv.org/abs/2207.09323",
"abstract": "In this paper we study the novel notion of thin polytopes: lattice polytopes whose local $h^*$-polynomials vanish. The local $h^*$-polynomial is an import... |
https://arxiv.org/abs/math/0602191 | On the maximum number of cliques in a graph | A \emph{clique} is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for the following graph classes: (1) graphs with $n$ vertices and $m$ edges; (2) graphs with $n$ vertices, $m$ edges, and maximum degree $\Delta$; (3) $d$-degenerate graphs with $n$ vertices and $m$ ... | \section{Introduction}
\seclabel{Intro}
The typical question of extremal graph theory asks for the maximum number of edges in a graph in a certain family; see the surveys \citep{SimonSos-DM01, Simonovits97, Bollobas95, Simonovits83}. For example, a celebrated theorem of \citet{Turan41} states that the maximum number o... | {
"timestamp": "2007-03-02T11:56:57",
"yymm": "0602",
"arxiv_id": "math/0602191",
"language": "en",
"url": "https://arxiv.org/abs/math/0602191",
"abstract": "A \\emph{clique} is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for the following graph clas... |
https://arxiv.org/abs/1911.06619 | Monotone Sobolev functions in planar domains: level sets and smooth approximation | We prove that almost every level set of a Sobolev function in a planar domain consists of points, Jordan curves, or homeomorphic copies of an interval. For monotone Sobolev functions in the plane we have the stronger conclusion that almost every level set is an embedded $1$-dimensional topological submanifold of the pl... | \section{Introduction}\label{Section:Introduction}
The classical theorem of Sard \cite{Sard:Theorem} asserts that for a $C^2$-smooth function $f$ in a planar domain $\Omega$ almost every value is a regular value. That is, for almost all $t\in \mathbb R$ the set $f^{-1}(t)$ does not intersect the critical set of $f$, a... | {
"timestamp": "2019-11-18T02:10:07",
"yymm": "1911",
"arxiv_id": "1911.06619",
"language": "en",
"url": "https://arxiv.org/abs/1911.06619",
"abstract": "We prove that almost every level set of a Sobolev function in a planar domain consists of points, Jordan curves, or homeomorphic copies of an interval. Fo... |
https://arxiv.org/abs/1004.2446 | Spanning and independence properties of frame partitions | We answer a number of open problems in frame theory concerning the decomposition of frames into linearly independent and/or spanning sets. We prove that in finite dimensional Hilbert spaces, Parseval frames with norms bounded away from 1 can be decomposed into a number of sets whose complements are spanning, where the ... | \section{}
\section{Introduction}
A family of vectors $\{f_i\}_{i\in I}$ is a \textit{frame} for a Hilbert space
$\mathbb{H}$ if there are constants $0<A\le B<\infty$ satisfying
\[ A\|x\|^2 \le \sum_{i\in I}|\langle x,f_i\rangle|^2 \le B \|x\|^2,\ \
\mbox{for all $x\in \mathbb{H}$}.\]
The theory of frames in Hilbe... | {
"timestamp": "2010-04-15T02:01:34",
"yymm": "1004",
"arxiv_id": "1004.2446",
"language": "en",
"url": "https://arxiv.org/abs/1004.2446",
"abstract": "We answer a number of open problems in frame theory concerning the decomposition of frames into linearly independent and/or spanning sets. We prove that in ... |
https://arxiv.org/abs/math/0505080 | Conway's napkin problem | The napkin problem was first posed by John H. Conway, and written up as a `toughie' in "Mathematical Puzzles: A Connoisseur's Collection," by Peter Winkler. To paraphrase Winkler's book, there is a banquet dinner to be served at a mathematics conference. At a particular table, $n$ men are to be seated around a circular... | \section{Introduction}
The problem studied in this article first appeared in the book
``Mathematical Puzzles: A Connoisseur's Collection," by Peter Winkler
\cite{Winkler}, and was inspired by a true story. Rather than
recounting the problem and the story ourselves, we prefer to quote
directly from ``Mathematical Puzzle... | {
"timestamp": "2006-01-26T23:23:46",
"yymm": "0505",
"arxiv_id": "math/0505080",
"language": "en",
"url": "https://arxiv.org/abs/math/0505080",
"abstract": "The napkin problem was first posed by John H. Conway, and written up as a `toughie' in \"Mathematical Puzzles: A Connoisseur's Collection,\" by Peter ... |
https://arxiv.org/abs/2208.07459 | Nesterov smoothing for sampling without smoothness | We study the problem of sampling from a target distribution in $\mathbb{R}^d$ whose potential is not smooth. Compared with the sampling problem with smooth potentials, this problem is much less well-understood due to the lack of smoothness. In this paper, we propose a novel sampling algorithm for a class of non-smooth ... | \section{Introduction}
\looseness=-1 Sampling from a target distribution $\pi (x ) \propto \exp(-s(x)) $ known up to a normalization constant is an important problem in many areas such as machine learning, due to its pivotal role in Bayesian statistics and inference~\citep{gelman1995bayesian,durmus2018efficient,krauth2... | {
"timestamp": "2022-08-17T02:03:41",
"yymm": "2208",
"arxiv_id": "2208.07459",
"language": "en",
"url": "https://arxiv.org/abs/2208.07459",
"abstract": "We study the problem of sampling from a target distribution in $\\mathbb{R}^d$ whose potential is not smooth. Compared with the sampling problem with smoo... |
https://arxiv.org/abs/0811.1531 | Homotopy groups and twisted homology of arrangements | Recent work of M. Yoshinaga shows that in some instances certain higher homotopy groups of arrangements map onto non-resonant homology. This is in contrast to the usual Hurewicz map to untwisted homology, which is always the zero homomorphism in degree greater than one. In this work we examine this dichotomy, generaliz... | \section{Introduction}
Let $\mathcal{A}$ be an arrangement of hyperplanes in $\mathbf{C}
^{\ell }$. Thus $\mathcal{A}$ is a finite non-empty collection $
\left\{ H_{1},\ldots ,H_{n}\right\} $ where $H_{i}=\alpha _{i}^{-1}(b_i)$ with $b_i \in \mathbf{C}$ and
each $\alpha _{i}$ is a linear homogeneous form in the variab... | {
"timestamp": "2009-06-02T23:07:39",
"yymm": "0811",
"arxiv_id": "0811.1531",
"language": "en",
"url": "https://arxiv.org/abs/0811.1531",
"abstract": "Recent work of M. Yoshinaga shows that in some instances certain higher homotopy groups of arrangements map onto non-resonant homology. This is in contrast ... |
https://arxiv.org/abs/1507.04571 | GPU-based visualization of domain-coloured algebraic Riemann surfaces | We examine an algorithm for the visualization of domain-coloured Riemann surfaces of plane algebraic curves. The approach faithfully reproduces the topology and the holomorphic structure of the Riemann surface. We discuss how the algorithm can be implemented efficiently in OpenGL with geometry shaders, and (less effici... | \section{Introduction}
\subsection{Mathematical background}
\label{sec:mathematical-background}
The following basic example illustrates what we would like to visualize.
\begin{example}
Let $y$ be the square root of $x$, \[y = \sqrt{x}.\]
If $x$ is a non-negative real number, we typically define $y$ as the
non-negati... | {
"timestamp": "2015-11-11T02:08:14",
"yymm": "1507",
"arxiv_id": "1507.04571",
"language": "en",
"url": "https://arxiv.org/abs/1507.04571",
"abstract": "We examine an algorithm for the visualization of domain-coloured Riemann surfaces of plane algebraic curves. The approach faithfully reproduces the topolo... |
https://arxiv.org/abs/2003.03197 | Bounding probability of small deviation on sum of independent random variables: Combination of moment approach and Berry-Esseen theorem | In the context of bounding probability of small deviation, there are limited general tools. However, such bounds have been widely applied in graph theory and inventory management. We introduce a common approach to substantially sharpen such inequality bounds by combining the semidefinite optimization approach of moment... | \section{Introduction}
\label{}
The problem of upper bounding
\begin{equation} \label{sum version}
\text{Prob}\left[ \sum_{i=1}^n X_i \geq \sum_{i=1}^n \mathop{\mathbb{E}}(X_i) + \delta \right].
\end{equation}
for independent random variables $X_i$ and a given constant $\delta>0$, has been studied for years.
Many cla... | {
"timestamp": "2020-03-09T01:11:15",
"yymm": "2003",
"arxiv_id": "2003.03197",
"language": "en",
"url": "https://arxiv.org/abs/2003.03197",
"abstract": "In the context of bounding probability of small deviation, there are limited general tools. However, such bounds have been widely applied in graph theory ... |
https://arxiv.org/abs/2202.02092 | Couplings and Matchings: Combinatorial notes on Strassen's theorem | Some mathematical theorems represent ideas that are discovered again and again in different forms. One such theorem is Hall's marriage theorem. This theorem is equivalent to several other theorems in combinatorics and optimization theory, in the sense that these results can easily be derived from each other. In this pa... | \section{Introduction}
In the original paper from \citeyear{hall1935representatives} \citeauthor{hall1935representatives} already mentions a similarity between his \emph{marriage theorem} and a result by \citeauthor{konig1916graphen} from \citeyear{konig1916graphen}.
Since then numerous other results have been found th... | {
"timestamp": "2022-02-07T02:14:59",
"yymm": "2202",
"arxiv_id": "2202.02092",
"language": "en",
"url": "https://arxiv.org/abs/2202.02092",
"abstract": "Some mathematical theorems represent ideas that are discovered again and again in different forms. One such theorem is Hall's marriage theorem. This theor... |
https://arxiv.org/abs/1807.04670 | The Hausdorff-Young inequality on Lie groups | We prove several results about the best constants in the Hausdorff-Young inequality for noncommutative groups. In particular, we establish a sharp local central version for compact Lie groups, and extend known results for the Heisenberg group. In addition, we prove a universal lower bound to the best constant for gener... | \section{Introduction}
For $f \in L^1(\mathbb{R}^n)$, define the Fourier transform $\hat f$ of $f$ by
\[
\hat f(\xi) = \int_{\mathbb{R}^n} f(x) \, e^{2\pi i \xi \cdot x} \, dx \qquad\forall \xi \in \mathbb{R}^n.
\]
Then the Riemann--Lebesgue lemma states that $\hat f \in C_0(\mathbb{R}^n)$ and
\[
\| \hat f \|_\infty ... | {
"timestamp": "2018-12-21T02:22:07",
"yymm": "1807",
"arxiv_id": "1807.04670",
"language": "en",
"url": "https://arxiv.org/abs/1807.04670",
"abstract": "We prove several results about the best constants in the Hausdorff-Young inequality for noncommutative groups. In particular, we establish a sharp local c... |
https://arxiv.org/abs/1412.0840 | Natural operations on differential forms | We prove that the only natural operations between differential forms are those obtained using linear combinations, the exterior product and the exterior differential. Our result generalises work by Palais and Freed-Hopkins.As an application, we also deduce a theorem, originally due to Kolar, that determines those natur... | \section*{Introduction}
Let $\, \omega_1\dots,\omega_k\,$ be differential forms on a smooth manifold, of positive degree. In this paper we determine those differential forms that can be associated in a {\it natural} way to the given forms $\, \omega_1, \ldots , \omega_k$. Loosely speaking, our result says that the onl... | {
"timestamp": "2014-12-03T02:11:59",
"yymm": "1412",
"arxiv_id": "1412.0840",
"language": "en",
"url": "https://arxiv.org/abs/1412.0840",
"abstract": "We prove that the only natural operations between differential forms are those obtained using linear combinations, the exterior product and the exterior dif... |
https://arxiv.org/abs/1607.04813 | Infinite families of 2-designs and 3-designs from linear codes | The interplay between coding theory and $t$-designs started many years ago. While every $t$-design yields a linear code over every finite field, the largest $t$ for which an infinite family of $t$-designs is derived directly from a linear or nonlinear code is $t=3$. Sporadic $4$-designs and $5$-designs were derived fro... | \section{Introduction}
Let ${\mathcal{P}}$ be a set of $v \ge 1$ elements, and let ${\mathcal{B}}$ be a set of $k$-subsets of ${\mathcal{P}}$, where $k$ is
a positive integer with $1 \leq k \leq v$. Let $t$ be a positive integer with $t \leq k$. The pair
${\mathbb{D}} = ({\mathcal{P}}, {\mathcal{B}})$ is called a $t$-... | {
"timestamp": "2016-07-19T02:05:49",
"yymm": "1607",
"arxiv_id": "1607.04813",
"language": "en",
"url": "https://arxiv.org/abs/1607.04813",
"abstract": "The interplay between coding theory and $t$-designs started many years ago. While every $t$-design yields a linear code over every finite field, the large... |
https://arxiv.org/abs/2006.04355 | $K_{r+1}$-saturated graphs with small spectral radius | For a graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain $H$ as a subgraph but for any $e \in E(\overline{G})$, $G+e$ contains $H$. In this note, we prove a sharp lower bound for the number of paths and walks on length $2$ in $n$-vertex $K_{r+1}$-saturated graphs. We then use this bound to give a lower bou... | \section{Introduction}
\subsection{Notation and preliminaries}
In this note we deal with finite undirected graphs with no loops or multiple edges.
For a graph $H$, a graph $G$ is $H$-{\em saturated} if $H$ is not a subgraph of
$G$ but after adding to $G$ any edge results in a graph containing $H$.
For a positive ... | {
"timestamp": "2020-06-09T02:24:49",
"yymm": "2006",
"arxiv_id": "2006.04355",
"language": "en",
"url": "https://arxiv.org/abs/2006.04355",
"abstract": "For a graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain $H$ as a subgraph but for any $e \\in E(\\overline{G})$, $G+e$ contains $H$. In this... |
https://arxiv.org/abs/2210.05520 | On the convexity of the quaternionic essential numerical range | The numerical range in the quaternionic setting is, in general, a non convex subset of the quaternions. The essential numerical range is a refinement of the numerical range that only keeps the elements that have, in a certain sense, infinite multiplicity. We prove that the essential numerical range of a bounded linear ... | \section*{Introduction}
Let $\mathbb{F}$ be the field of complex numbers or the skew field $\mathbb{H}$ of Hamilton quaternions. Let $\mathcal{H}$ be a Hilbert space over $\mathbb{F}$ and let $T$ be a bounded linear operator on $\mathcal{H}$. The numerical range of $T$ is the set
\[
W(T)=W_{\mathbb{F}}(T)=\{\langle Tx... | {
"timestamp": "2022-10-12T02:18:17",
"yymm": "2210",
"arxiv_id": "2210.05520",
"language": "en",
"url": "https://arxiv.org/abs/2210.05520",
"abstract": "The numerical range in the quaternionic setting is, in general, a non convex subset of the quaternions. The essential numerical range is a refinement of t... |
https://arxiv.org/abs/1712.03568 | On a Detail in Hales's "Dense Sphere Packings: A Blueprint for Formal Proofs" | In "Dense Sphere Packings: A Blueprint for Formal Proofs" Hales proves that for every packing of unit spheres, the density in a ball of radius $r$ is at most $\pi/\sqrt{18}+c/r$ for some constant $c$. When $r$ tends to infinity, this gives a proof to the famous Kepler conjecture. As formulated by Hales, $c$ depends on ... | \section{Introduction}
\label{sec:Introduction}
In \cite{Blueprint} Hales proves that for every packing of infinitely many unit spheres into three dimensional space, there exists a constant $c$ such that the \emph{density} inside a ball of radius $r$ is upper bounded by $\pi/\sqrt{18}+c/r$. Here, the density is define... | {
"timestamp": "2017-12-12T02:11:01",
"yymm": "1712",
"arxiv_id": "1712.03568",
"language": "en",
"url": "https://arxiv.org/abs/1712.03568",
"abstract": "In \"Dense Sphere Packings: A Blueprint for Formal Proofs\" Hales proves that for every packing of unit spheres, the density in a ball of radius $r$ is at... |
https://arxiv.org/abs/2003.07234 | On the fixed volume discrepancy of the Korobov point sets | This paper is devoted to the study of a discrepancy-type characteristic -- the fixed volume discrepancy -- of the Korobov point sets in the unit cube. It was observed recently that this new characteristic allows us to obtain optimal rate of dispersion from numerical integration results. This observation motivates us to... | \section{Introduction}
\label{I}
This paper is a follow up to the recent paper \cite{VT172}. It is devoted to the study of a discrepancy-type characteristic -- the fixed volume discrepancy --
of a point set in the unit cube $\Omega_d:=[0,1)^d$.
We refer the reader to the following books and survey papers on discre... | {
"timestamp": "2020-03-17T01:24:13",
"yymm": "2003",
"arxiv_id": "2003.07234",
"language": "en",
"url": "https://arxiv.org/abs/2003.07234",
"abstract": "This paper is devoted to the study of a discrepancy-type characteristic -- the fixed volume discrepancy -- of the Korobov point sets in the unit cube. It ... |
https://arxiv.org/abs/1503.02352 | Infinite-dimensional $\ell^1$ minimization and function approximation from pointwise data | We consider the problem of approximating a smooth function from finitely-many pointwise samples using $\ell^1$ minimization techniques. In the first part of this paper, we introduce an infinite-dimensional approach to this problem. Three advantages of this approach are as follows. First, it provides interpolatory appro... | \section{Introduction}\label{s:introduction}
Many problems in science and engineering require the approximation of a smooth function from a finite set of pointwise samples. Although a classical problem in approximation theory, in the last several years there has been an increasing focus on the use of convex optimizat... | {
"timestamp": "2016-12-16T02:07:56",
"yymm": "1503",
"arxiv_id": "1503.02352",
"language": "en",
"url": "https://arxiv.org/abs/1503.02352",
"abstract": "We consider the problem of approximating a smooth function from finitely-many pointwise samples using $\\ell^1$ minimization techniques. In the first part... |
https://arxiv.org/abs/1405.4320 | Parameter selection and numerical approximation properties of Fourier extensions from fixed data | Fourier extensions have been shown to be an effective means for the approximation of smooth, nonperiodic functions on bounded intervals given their values on an equispaced, or in general, scattered grid. Related to this method are two parameters. These are the extension parameter $T$ (the ratio of the size of the exten... | \section{Introduction}\label{s:introduction}
In many problems, one is faced with the task of recovering a smooth function $f: [-1,1] \rightarrow \bbC$ to high accuracy from its pointwise samples on an equispaced, or in general, scattered grid. This problem is challenging, unless the grid points have a specific distrib... | {
"timestamp": "2014-05-20T02:01:30",
"yymm": "1405",
"arxiv_id": "1405.4320",
"language": "en",
"url": "https://arxiv.org/abs/1405.4320",
"abstract": "Fourier extensions have been shown to be an effective means for the approximation of smooth, nonperiodic functions on bounded intervals given their values o... |
https://arxiv.org/abs/1402.3413 | The colourful simplicial depth conjecture | Given $d+1$ sets of points, or colours, $S_1,\ldots,S_{d+1}$ in $\mathbb R^d$, a colourful simplex is a set $T\subseteq\bigcup_{i=1}^{d+1}S_i$ such that $|T\cap S_i|\leq 1$, for all $i\in\{1,\ldots,d+1\}$. The colourful Carathéodory theorem states that, if $\mathbf 0$ is in the convex hull of each $S_i$, then there exi... | \section{Introduction}
A {\em colourful point configuration} is a collection of $d+1$ sets of points $\mathbf{S}_1,\ldots,\mathbf{S}_{d+1}$ in $\mathbb R^d$. A {\em colourful simplex} is a subset $T$ of $\bigcup_{i=1}^{d+1}\mathbf{S}_i$ such that $|T\cap \mathbf{S}_i|\leq 1$.
The colourful Carath\'eodory theorem, pr... | {
"timestamp": "2014-04-16T02:08:13",
"yymm": "1402",
"arxiv_id": "1402.3413",
"language": "en",
"url": "https://arxiv.org/abs/1402.3413",
"abstract": "Given $d+1$ sets of points, or colours, $S_1,\\ldots,S_{d+1}$ in $\\mathbb R^d$, a colourful simplex is a set $T\\subseteq\\bigcup_{i=1}^{d+1}S_i$ such that... |
https://arxiv.org/abs/0809.0022 | Lagrangians for dissipative nonlinear oscillators: the method of Jacobi Last Multiplier | We present a method devised by Jacobi to derive Lagrangians of any second-order differential equation: it consists in finding a Jacobi Last Multiplier. We illustrate the easiness and the power of Jacobi's method by applying it to several equations and also a class of equations studied by Musielak with his own method [M... | \section{Introduction}
It should be well-known that the knowledge of a Jacobi Last
Multiplier always yields a Lagrangian of any second-order ordinary
differential equation \cite{JacobiVD}, \cite{Whittaker}. Yet many
distinguished scientists seem to be unaware of this classical
result. In this paper we present again the... | {
"timestamp": "2008-08-30T00:15:36",
"yymm": "0809",
"arxiv_id": "0809.0022",
"language": "en",
"url": "https://arxiv.org/abs/0809.0022",
"abstract": "We present a method devised by Jacobi to derive Lagrangians of any second-order differential equation: it consists in finding a Jacobi Last Multiplier. We i... |
https://arxiv.org/abs/1710.10875 | Shifts of the prime divisor function of Alladi and Erdős | We introduce a variation on the prime divisor function $B(n)$ of Alladi and Erdős, a close relative of the sum of proper divisors function $s(n)$. After proving some basic properties regarding these functions, we study the dynamics of its iterates and discover behaviour that is reminiscent of aliquot sequences. We prov... | \section{Introduction}
Let $n$ be a positive integer with prime factorization $n=p_1^{r_1}\dots p_k^{r_k}$. Consider the sum of prime divisors function $B(n) = \sum_{i=1}^kr_ip_i$ and the sum of distinct prime divisors function $\beta(n) = \sum_{i=1}^k p_i$. They can be viewed as variants of the sum of proper divisor... | {
"timestamp": "2017-10-31T01:16:57",
"yymm": "1710",
"arxiv_id": "1710.10875",
"language": "en",
"url": "https://arxiv.org/abs/1710.10875",
"abstract": "We introduce a variation on the prime divisor function $B(n)$ of Alladi and Erdős, a close relative of the sum of proper divisors function $s(n)$. After p... |
https://arxiv.org/abs/0706.4100 | Embedding nearly-spanning bounded degree trees | We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1-\epsilon)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d\geq 2 and 0<\epsilon<1, there exists a constant c=c(d,\epsilon) such that a random grap... | \section{Introduction}
In this paper we obtain a sufficient condition for a sparse graph $G$
to contain a copy of every nearly-spanning tree $T$ of bounded
maximum degree, in terms of the expansion properties of $G$.
The restriction on the degree of $T$ comes naturally from the fact that we consider graphs
of consta... | {
"timestamp": "2007-06-27T23:13:21",
"yymm": "0706",
"arxiv_id": "0706.4100",
"language": "en",
"url": "https://arxiv.org/abs/0706.4100",
"abstract": "We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1-\\epsilon)n vertices, in... |
https://arxiv.org/abs/1803.09633 | Direct Proofs of the Fundamental Theorem of Calculus for the Omega Integral | When introduced in a 2018 article in the American Mathematical Monthly, the omega integral was shown to be an extension of the Riemann integral. Although results for continuous functions such as the Fundamental Theorem of Calculus follow immediately, a much more satisfying approach would be to provide direct proofs not... | \section{Introduction}
The omega integral, which makes use of the hyperreals to integrate real functions, was introduced in \cite{monthly} and proven to be an extension of the Riemann integral. Theorems relating to continuous functions, such as integrability, additivity, and the fundamental theorem follow immediately... | {
"timestamp": "2018-03-28T02:20:57",
"yymm": "1803",
"arxiv_id": "1803.09633",
"language": "en",
"url": "https://arxiv.org/abs/1803.09633",
"abstract": "When introduced in a 2018 article in the American Mathematical Monthly, the omega integral was shown to be an extension of the Riemann integral. Although ... |
https://arxiv.org/abs/0802.3523 | On linear versions of some addition theorems | Let K \subset L be a field extension. Given K-subspaces A,B of L, we study the subspace spanned by the product set AB = {ab | a \in A, b \in B}. We obtain some lower bounds on the dimension of this subspace and on dim B^n in terms of dim A, dim B and n. This is achieved by establishing linear versions of constructions ... | \section{Introduction}
Let $G$ be a group, written multiplicatively. Given subsets $A,B \subset G$,
we denote by
\begin{equation*}
AB = \{ab \mid a \in A, b\in B \}
\end{equation*}
the \textit{product set} of $A,B$. For $A,B$ finite, several results in
additive number theory give estimates on the cardinality of $AB$ ... | {
"timestamp": "2008-02-24T18:27:17",
"yymm": "0802",
"arxiv_id": "0802.3523",
"language": "en",
"url": "https://arxiv.org/abs/0802.3523",
"abstract": "Let K \\subset L be a field extension. Given K-subspaces A,B of L, we study the subspace spanned by the product set AB = {ab | a \\in A, b \\in B}. We obtai... |
https://arxiv.org/abs/1708.08085 | Valuations, arithmetic progressions, and prime numbers | In this short note, we give two proofs of the infinitude of primes via valuation theory and give a new proof of the divergence of the sum of prime reciprocals by Roth's theorem and Euler-Legendre's theorem for arithmetic progressions. | \section{The infinitude of primes via valuation theory}
We cite Neukirch's book \cite{N} for the facts in valuation theory.
\begin{Ostrowski}[{\cite[p.~119, (3.7)]{N}}]
Every non-trivial valuation on the field of rational numbers is equivalent to either the usual absolute value or the $p$-adic valuation for some p... | {
"timestamp": "2018-02-13T02:18:33",
"yymm": "1708",
"arxiv_id": "1708.08085",
"language": "en",
"url": "https://arxiv.org/abs/1708.08085",
"abstract": "In this short note, we give two proofs of the infinitude of primes via valuation theory and give a new proof of the divergence of the sum of prime recipro... |
https://arxiv.org/abs/1707.04926 | Theoretical insights into the optimization landscape of over-parameterized shallow neural networks | In this paper we study the problem of learning a shallow artificial neural network that best fits a training data set. We study this problem in the over-parameterized regime where the number of observations are fewer than the number of parameters in the model. We show that with quadratic activations the optimization la... | \section{Numerical experiments}
\begin{figure}[!t]
\centering
\begin{subfigure}[b]{.43\textwidth}
\includegraphics[width=\linewidth]{SoftPlus-varyK.pdf}
\hspace{1cm}
\caption{$d =10$}\label{SoftPlus-varyK}
\end{subfigure}
\hspace{1cm}
\begin{subfigure... | {
"timestamp": "2018-11-09T02:04:30",
"yymm": "1707",
"arxiv_id": "1707.04926",
"language": "en",
"url": "https://arxiv.org/abs/1707.04926",
"abstract": "In this paper we study the problem of learning a shallow artificial neural network that best fits a training data set. We study this problem in the over-p... |
https://arxiv.org/abs/2105.07472 | Lexicographic Enumeration of Set Partitions | In this report, we summarize the set partition enumeration problems and thoroughly explain the algorithms used to solve them. These algorithms iterate through the partitions in lexicographic order and are easy to understand and implement in modern high-level programming languages, without recursive structures and jump ... | \section{Introduction}
\label{sec:introduction}
\citet[Section 7.2.1.5]{knuth2014art} discusses the problem of partition enumeration, which consists in enumerating the number of ways that a set can be partitioned into non-empty subsets (or blocks). Following Knuth's notation, each partition of a set $U=\{u_1,u_2,\dot... | {
"timestamp": "2021-05-18T02:20:53",
"yymm": "2105",
"arxiv_id": "2105.07472",
"language": "en",
"url": "https://arxiv.org/abs/2105.07472",
"abstract": "In this report, we summarize the set partition enumeration problems and thoroughly explain the algorithms used to solve them. These algorithms iterate thr... |
https://arxiv.org/abs/0905.2563 | Stationary map coloring | We consider a planar Poisson process and its associated Voronoi map. We show that there is a proper coloring with 6 colors of the map which is a deterministic isometry-equivariant function of the Poisson process. As part of the proof we show that the 6-core of the corresponding Delaunay triangulation is empty.Generaliz... | \section{Introduction}
The Poisson-Voronoi map is a natural random planar map. Being planar, a
specific instance can always be colored with 4 colors with adjacent cells
having distinct colors. The question we consider here is whether such a
coloring can be realized in a way that would be isometry-equivariant, that
is,... | {
"timestamp": "2009-05-15T17:31:28",
"yymm": "0905",
"arxiv_id": "0905.2563",
"language": "en",
"url": "https://arxiv.org/abs/0905.2563",
"abstract": "We consider a planar Poisson process and its associated Voronoi map. We show that there is a proper coloring with 6 colors of the map which is a determinist... |
https://arxiv.org/abs/1903.08214 | A tighter bound on the number of relevant variables in a bounded degree Boolean function | A classical theorem of Nisan and Szegedy says that a boolean function with degree $d$ as a real polynomial depends on at most $d2^{d-1}$ of its variables. In recent work by Chiarelli, Hatami and Saks, this upper bound was improved to $C \cdot 2^d$, where $C = 6.614$. Here we refine their argument to show that one may t... | \section{Introduction}
Given a Boolean function $f: \{0,1\}^n \to \{0,1\}$, there is a unique multilinear polynomial in $\R[x_1, \dots, x_n]$ which agrees with $f$ on every input in $\{0,1\}^n$. One important feature of this polynomial is its degree, denoted $\deg(f)$, which is known to be polynomially related to many... | {
"timestamp": "2019-03-22T01:09:00",
"yymm": "1903",
"arxiv_id": "1903.08214",
"language": "en",
"url": "https://arxiv.org/abs/1903.08214",
"abstract": "A classical theorem of Nisan and Szegedy says that a boolean function with degree $d$ as a real polynomial depends on at most $d2^{d-1}$ of its variables.... |
https://arxiv.org/abs/2008.08172 | Curves on the torus intersecting at most k times | We show that any set of distinct homotopy classes of simple closed curves on the torus that pairwise intersect at most $k$ times has size $k + O(\sqrt{k} \log k)$. Prior to this work, a lemma of Agol, together with the state of the art bounds for the size of prime gaps, implied the error term $O(k^{21/40})$, and in fac... | \section{Introduction}
Let $T \approx \mathbb{R}^2/\mathbb{Z}^2$ be the closed oriented surface of genus one.
We indicate the homotopy class of an embedding of $S^1$ briefly by `curve'.
By pulling a curve tight and lifting it to the universal cover, the collection of curves on $T$ is in one-to-one correspondence with... | {
"timestamp": "2020-08-20T02:03:26",
"yymm": "2008",
"arxiv_id": "2008.08172",
"language": "en",
"url": "https://arxiv.org/abs/2008.08172",
"abstract": "We show that any set of distinct homotopy classes of simple closed curves on the torus that pairwise intersect at most $k$ times has size $k + O(\\sqrt{k}... |
https://arxiv.org/abs/1710.06002 | Covering compact metric spaces greedily | A general greedy approach to construct coverings of compact metric spaces by metric balls is given and analyzed. The analysis is a continuous version of Chvatal's analysis of the greedy algorithm for the weighted set cover problem. The approach is demonstrated in an exemplary manner to construct efficient coverings of ... | \section{Introduction}
Let $X$ be a compact metric space having metric $d$. Given a scalar
$r\in\mathbb{R}_{\geq 0}$ we define the \emph{closed ball} of radius $r$
around center $x \in X$ by
\[
B(x,r) = \{ y \in X : d(x,y) \leq r\}.
\]
The \emph{covering number} of the space $X$ and a positive number $r$
is
\[
\mathc... | {
"timestamp": "2018-02-05T02:07:28",
"yymm": "1710",
"arxiv_id": "1710.06002",
"language": "en",
"url": "https://arxiv.org/abs/1710.06002",
"abstract": "A general greedy approach to construct coverings of compact metric spaces by metric balls is given and analyzed. The analysis is a continuous version of C... |
https://arxiv.org/abs/1201.0425 | Spectral gaps of random graphs and applications | We study the spectral gap of the Erdős--Rényi random graph through the connectivity threshold. In particular, we show that for any fixed $\delta > 0$ if $$p \ge \frac{(1/2 + \delta) \log n}{n},$$ then the normalized graph Laplacian of an Erdős--Rényi graph has all of its nonzero eigenvalues tightly concentrated around ... | \section{Introduction}
Studying the spectral properties of random matrices has played a central role in probability theory ever since Wigner's paper establishing the semi-circular law for symmetric matrices with independent centered entries above the diagonal~\cite{wigner}. The theory of these matrices is rich and we... | {
"timestamp": "2013-12-24T02:10:18",
"yymm": "1201",
"arxiv_id": "1201.0425",
"language": "en",
"url": "https://arxiv.org/abs/1201.0425",
"abstract": "We study the spectral gap of the Erdős--Rényi random graph through the connectivity threshold. In particular, we show that for any fixed $\\delta > 0$ if $$... |
https://arxiv.org/abs/1605.07832 | Antistrong digraphs | An antidirected trail in a digraph is a trail (a walk with no arc repeated) in which the arcs alternate between forward and backward arcs. An antidirected path is an antidirected trail where no vertex is repeated. We show that it is NP-complete to decide whether two vertices $x,y$ in a digraph are connected by an antid... | \section{Introduction}
We refer the reader to~\cite{bang2009} for notation and terminology
not explicitly defined in this paper. An {\bf antidirected path} in
a digraph $D$ is a path in which the arcs alternate between forward
and backward arcs. The digraph $D$ is said to be {\bf anticonnected}
if it contains an antid... | {
"timestamp": "2016-05-26T02:09:23",
"yymm": "1605",
"arxiv_id": "1605.07832",
"language": "en",
"url": "https://arxiv.org/abs/1605.07832",
"abstract": "An antidirected trail in a digraph is a trail (a walk with no arc repeated) in which the arcs alternate between forward and backward arcs. An antidirected... |
https://arxiv.org/abs/1301.4590 | Computing Hypermatrix Spectra with the Poisson Product Formula | We compute the spectrum of the "all ones" hypermatrix using the Poisson product formula. This computation includes a complete description of the eigenvalues' multiplicities, a seemingly elusive aspect of the spectral theory of tensors. We also give a general distributional picture of the spectrum as a point-set in the ... | \section{Introduction}
There are few extant techniques that allow one to determine the characteristic polynomial of a symmetric hypermatrix in the sense of Qi (\cite{Qi_2005}). Indeed, the lack of a simple formula for the symmetric hyperdeterminant makes many questions about hypermatrices much more difficult than the... | {
"timestamp": "2013-01-22T02:01:11",
"yymm": "1301",
"arxiv_id": "1301.4590",
"language": "en",
"url": "https://arxiv.org/abs/1301.4590",
"abstract": "We compute the spectrum of the \"all ones\" hypermatrix using the Poisson product formula. This computation includes a complete description of the eigenvalu... |
https://arxiv.org/abs/2111.04339 | Sharp Sobolev regularity of restricted X-ray transforms | We study $L^p$-Sobolev regularity estimate for the restricted X-ray transforms generated by nondegenerate curves. Making use of the inductive strategy in the recent work by the authors, we establish the sharp $L^p$-regularity estimates for the restricted X-ray transforms in $\mathbb R^{d+1}$, $d\ge 3$. This extends the... | \section{Introduction}
Let $\gamma$ be a smooth curve from $I=[-1,1]$ to $\mathbb R^{d}$.
We consider
\[
\mathfrak R f(x,s)=\psi(s)\int f(x+t\gamma(s),t)\chi(t)dt, \quad f\in \mathcal S(\mathbb{R}^{d+1}),
\]
where $\psi$ and $\chi$ are smooth functions supported in the interior of the intervals $I$ and $[1,2]$, res... | {
"timestamp": "2021-11-10T02:07:12",
"yymm": "2111",
"arxiv_id": "2111.04339",
"language": "en",
"url": "https://arxiv.org/abs/2111.04339",
"abstract": "We study $L^p$-Sobolev regularity estimate for the restricted X-ray transforms generated by nondegenerate curves. Making use of the inductive strategy in ... |
https://arxiv.org/abs/1904.01714 | A LeVeque-Type Inequality on the ring of $p$-adic integers | We derive an inequality on the discrepancy of sequences on the ring of $p$-adic integers $\ZZ_p$ using techniques from Fourier analysis. The inequality is used to obtain an upper bound on the discrepancy of the sequence $\alpha_n = na +b$, where $a$ and $b$ are elements of $\ZZ_p$. This is a $p$-adic analogue of the cl... | \section{Introduction}
\label{section: introduction}
The theory of equidistribution of sequences modulo one was initiated by Hermann Weyl in $1916$. Since then, it has spurred a lot of interest in many areas of mathematics, including number theory, harmonic analysis, and ergodic theory. The standard reference in this ... | {
"timestamp": "2019-04-04T02:04:39",
"yymm": "1904",
"arxiv_id": "1904.01714",
"language": "en",
"url": "https://arxiv.org/abs/1904.01714",
"abstract": "We derive an inequality on the discrepancy of sequences on the ring of $p$-adic integers $\\ZZ_p$ using techniques from Fourier analysis. The inequality i... |
https://arxiv.org/abs/2103.11330 | Expected Extinction Times of Epidemics with State-Dependent Infectiousness | We model an epidemic where the per-person infectiousness in a network of geographic localities changes with the total number of active cases. This would happen as people adopt more stringent non-pharmaceutical precautions when the population has a larger number of active cases. We show that there exists a sharp thresho... | \section{Conclusion}
\label{sec:conclusion}
We have developed a model for epidemic spread within and across
population centers with state-dependent infectiousness.
In this model, we directly prove (without mean-field assumptions) that
there exists a sharp threshold for the curing rate \(\delta\)
such that when \(\del... | {
"timestamp": "2021-12-07T02:20:29",
"yymm": "2103",
"arxiv_id": "2103.11330",
"language": "en",
"url": "https://arxiv.org/abs/2103.11330",
"abstract": "We model an epidemic where the per-person infectiousness in a network of geographic localities changes with the total number of active cases. This would h... |
https://arxiv.org/abs/1310.1570 | Local Maxima of Quadratic Boolean Functions | How many strict local maxima can a real quadratic function on $\{0,1\}^n$ have? Holzman conjectured a maximum of $n \choose \lfloor n/2 \rfloor$. The aim of this paper is to prove this conjecture. Our approach is via a generalization of Sperner's theorem that may be of independent interest. | \section{Introduction}
Let $\Theta$ be a real quadratic function (polynomial of total degree $\le 2$) in $n$ variables $x_1,\ldots x_n$. A \textit{strict local maximum} (or just \textit{local maximum}) of $\Theta$ on the discrete cube $Q_n=\{0,1\}^n$ is a point whose value is strictly larger than all of its neighbors.
... | {
"timestamp": "2013-10-08T02:06:17",
"yymm": "1310",
"arxiv_id": "1310.1570",
"language": "en",
"url": "https://arxiv.org/abs/1310.1570",
"abstract": "How many strict local maxima can a real quadratic function on $\\{0,1\\}^n$ have? Holzman conjectured a maximum of $n \\choose \\lfloor n/2 \\rfloor$. The a... |
https://arxiv.org/abs/2204.01947 | Tournaments and Even Graphs are Equinumerous | A graph is called odd if there is an orientation of its edges and an automorphism that reverses the sense of an odd number of its edges, and even otherwise. Pontus von Brömssen (né Andersson) showed that the existence of such an automorphism is independent of the orientation, and considered the question of counting pai... | \section{Introduction}
In a paper on the asymptotics of random tournaments, Pontus Andersson \cite[page 252]{MR1662785} introduced the concept of an \emph{even graph} in the following fashion: Given a graph $X$, assign an arbitrary orientation to its edges. Then an automorphism $g \in \mathrm{Aut}(X)$ \emph{reverses... | {
"timestamp": "2022-04-06T02:13:00",
"yymm": "2204",
"arxiv_id": "2204.01947",
"language": "en",
"url": "https://arxiv.org/abs/2204.01947",
"abstract": "A graph is called odd if there is an orientation of its edges and an automorphism that reverses the sense of an odd number of its edges, and even otherwis... |
https://arxiv.org/abs/0709.4046 | Equivalence Theorems in Numerical Analysis : Integration, Differentiation and Interpolation | We show that if a numerical method is posed as a sequence of operators acting on data and depending on a parameter, typically a measure of the size of discretization, then consistency, convergence and stability can be related by a Lax-Richtmyer type equivalence theorem -- a consistent method is convergent if and only i... | \section{Introduction}
For a numerical method the three most important aspects are its
consistency, convergence and stability. These three were related in
the well known equivalence theorem of Lax and Richtmyer for finite
difference methods for certain partial differential equations
\cite{LaRi1956}. We show that in a v... | {
"timestamp": "2007-09-26T16:11:07",
"yymm": "0709",
"arxiv_id": "0709.4046",
"language": "en",
"url": "https://arxiv.org/abs/0709.4046",
"abstract": "We show that if a numerical method is posed as a sequence of operators acting on data and depending on a parameter, typically a measure of the size of discr... |
https://arxiv.org/abs/1902.07137 | Recovery of a mixture of Gaussians by sum-of-norms clustering | Sum-of-norms clustering is a method for assigning $n$ points in $\mathbb{R}^d$ to $K$ clusters, $1\le K\le n$, using convex optimization. Recently, Panahi et al.\ proved that sum-of-norms clustering is guaranteed to recover a mixture of Gaussians under the restriction that the number of samples is not too large. The pu... | \section{Introduction}
Clustering is perhaps the most central problem in unsupervised machine learning and has been studied for over 60 years \cite{shais}. The problem may be stated informally as follows. One is given $n$ points, $\a_1,\ldots,\a_n$ lying in $\R^d$. One seeks to partition $\{1,\ldots,n\}$ into $K$ se... | {
"timestamp": "2019-02-20T02:17:48",
"yymm": "1902",
"arxiv_id": "1902.07137",
"language": "en",
"url": "https://arxiv.org/abs/1902.07137",
"abstract": "Sum-of-norms clustering is a method for assigning $n$ points in $\\mathbb{R}^d$ to $K$ clusters, $1\\le K\\le n$, using convex optimization. Recently, Pan... |
https://arxiv.org/abs/1804.07575 | Optimal Sorting with Persistent Comparison Errors | We consider the problem of sorting $n$ elements in the case of \emph{persistent} comparison errors. In this model (Braverman and Mossel, SODA'08), each comparison between two elements can be wrong with some fixed (small) probability $p$, and \emph{comparisons cannot be repeated}. Sorting perfectly in this model is impo... | \section{Introduction}\label{sec-introduction}
\newcommand{w.h.p.}{w.h.p.}
\newcommand{exp.}{exp.}
We study the problem of \emph{sorting} $n$ distinct elements under \emph{persistent} random comparison \emph{errors}. This problem arises naturally when sorting is applied to real life scenarios. For example, one could... | {
"timestamp": "2018-04-23T02:10:55",
"yymm": "1804",
"arxiv_id": "1804.07575",
"language": "en",
"url": "https://arxiv.org/abs/1804.07575",
"abstract": "We consider the problem of sorting $n$ elements in the case of \\emph{persistent} comparison errors. In this model (Braverman and Mossel, SODA'08), each c... |
https://arxiv.org/abs/1412.6639 | A geometric Hall-type theorem | We introduce a geometric generalization of Hall's marriage theorem. For any family $F = \{X_1, \dots, X_m\}$ of finite sets in $\mathbb{R}^d$, we give conditions under which it is possible to choose a point $x_i\in X_i$ for every $1\leq i \leq m$ in such a way that the points $\{x_1,...,x_m\}\subset \mathbb{R}^d$ are i... | \section{Introduction}
\subsection{Background} Let $F = \{S_1, \dots, S_m\}$ be a family of finite subsets of a common ground set $E$. A {\em system of distinct representatives} is an $m$-element subset $\{x_1, \dots, x_m\}\subset E$ such that $x_i \in S_i$ for all $1\leq i \leq m$. A classical result in combinatorics... | {
"timestamp": "2015-01-15T02:07:11",
"yymm": "1412",
"arxiv_id": "1412.6639",
"language": "en",
"url": "https://arxiv.org/abs/1412.6639",
"abstract": "We introduce a geometric generalization of Hall's marriage theorem. For any family $F = \\{X_1, \\dots, X_m\\}$ of finite sets in $\\mathbb{R}^d$, we give c... |
https://arxiv.org/abs/1105.5572 | Lagrange's Theorem for Hopf Monoids in Species | Following Radford's proof of Lagrange's theorem for pointed Hopf algebras, we prove Lagrange's theorem for Hopf monoids in the category of connected species. As a corollary, we obtain necessary conditions for a given subspecies K of a Hopf monoid H to be a Hopf submonoid: the quotient of any one of the generating serie... | \section*{Introduction}
Lagrange's theorem states that for any subgroup $K$ of a group $H$,
$H\cong K\times Q$ as (left) $K$-sets, where $Q=H/K$. In particular,
if $H$ is finite, then $|K|$ divides $|H|$.
Passing to group algebras over a field $\Bbbk$, we have that
$\Bbbk H \cong \Bbbk K \otimes \Bbbk Q$ as (left) ... | {
"timestamp": "2012-08-02T02:01:00",
"yymm": "1105",
"arxiv_id": "1105.5572",
"language": "en",
"url": "https://arxiv.org/abs/1105.5572",
"abstract": "Following Radford's proof of Lagrange's theorem for pointed Hopf algebras, we prove Lagrange's theorem for Hopf monoids in the category of connected species... |
https://arxiv.org/abs/math/0502249 | A characterization of spaces with discrete topological fundamental group | We offer a counterexample to a theorem in the literature and then repair the theorem as follows: The fundamental group of a locally path connected metric space inherits the discrete topology in a natural way if and only if the underlying space is semilocally simply connected. | \section{Introduction}
In general $\pi _{1}(X,p),$ the based fundamental group of a space $X,$
admits a canonical topology and becomes the \textit{topological fundamental
group.} To date significant progress has been made in the investigation of
the topological fundamental group of certain non semilocally simply
conne... | {
"timestamp": "2005-02-11T23:53:46",
"yymm": "0502",
"arxiv_id": "math/0502249",
"language": "en",
"url": "https://arxiv.org/abs/math/0502249",
"abstract": "We offer a counterexample to a theorem in the literature and then repair the theorem as follows: The fundamental group of a locally path connected met... |
https://arxiv.org/abs/1411.0867 | On the equality of Hausdorff measure and Hausdorff content | We are interested in situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result shows that this equality holds for any subset of a self-similar set corresponding to a nontrivial cylinder of an irreducible subshift of finite type, and thus also for any self... | \section{Introduction}
Hausdorff measure and dimension are among the most important notions in fractal geometry and geometric measure theory used to quantify the size of a set. The Hausdorff content is a concept closely related to the Hausdorff measure, but perhaps less popular in the context of classical measure... | {
"timestamp": "2015-07-30T02:07:59",
"yymm": "1411",
"arxiv_id": "1411.0867",
"language": "en",
"url": "https://arxiv.org/abs/1411.0867",
"abstract": "We are interested in situations where the Hausdorff measure and Hausdorff content of a set are equal in the critical dimension. Our main result shows that t... |
https://arxiv.org/abs/2001.01792 | Proof of the simplicity conjecture | In the 1970s, Fathi, having proven that the group of compactly supported volume-preserving homeomorphisms of the $n$-ball is simple for $n \ge 3$, asked if the same statement holds in dimension $2$. We show that the group of compactly supported area-preserving homeomorphisms of the two-disc is not simple. This settles ... | \section{Introduction}
\subsection{The main theorem} \label{sec:intro_main_theo}
Let $(S,\omega)$ be a surface equipped with an area form. An {\bf area-preserving homeomorphism} is a homeomorphism which preserves the measure induced by $\omega$.
Let $\Homeo_c(\D, \omega)$ denote the group of area-preserving
home... | {
"timestamp": "2020-06-18T02:05:03",
"yymm": "2001",
"arxiv_id": "2001.01792",
"language": "en",
"url": "https://arxiv.org/abs/2001.01792",
"abstract": "In the 1970s, Fathi, having proven that the group of compactly supported volume-preserving homeomorphisms of the $n$-ball is simple for $n \\ge 3$, asked ... |
https://arxiv.org/abs/1303.4966 | Automorphisms of groups and converse of Schur's theorem | An automorphism of a group G is called an IA-automorphism if it induces the identity automorphism on the abelianized group G/G'. Let IA(G) denote the group of all IA-automorphisms of G. We classify all finitely generated nilpotent groups G of class 2 for which IA(G) is isomorphic to Inn(G). In particular, we classify a... | \section{\normalsize Introduction}
Let $G$ be any group and let $G^{\prime}$ and $Z(G)$ respectively denote the derived group and the center of $G$. By $\mathrm{Inn}(G)$ we denote the inner automorphism group of $G$. Following Bachmuth [2], we call an automorphism of $G$ an IA-automorphism if it induces the identity ... | {
"timestamp": "2013-03-21T01:02:33",
"yymm": "1303",
"arxiv_id": "1303.4966",
"language": "en",
"url": "https://arxiv.org/abs/1303.4966",
"abstract": "An automorphism of a group G is called an IA-automorphism if it induces the identity automorphism on the abelianized group G/G'. Let IA(G) denote the group ... |
https://arxiv.org/abs/1611.01655 | Twenty (simple) questions | A basic combinatorial interpretation of Shannon's entropy function is via the "20 questions" game. This cooperative game is played by two players, Alice and Bob: Alice picks a distribution $\pi$ over the numbers $\{1,\ldots,n\}$, and announces it to Bob. She then chooses a number $x$ according to $\pi$, and Bob attempt... | \section{Introduction} \label{sec:introduction}
A basic combinatorial and operational interpretation of Shannon's entropy function, which is often taught in introductory courses on information theory, is via the ``20 questions'' game (see for example the well-known textbook~\cite{DBLP:books/daglib/0016881}).
This game... | {
"timestamp": "2017-04-26T02:05:09",
"yymm": "1611",
"arxiv_id": "1611.01655",
"language": "en",
"url": "https://arxiv.org/abs/1611.01655",
"abstract": "A basic combinatorial interpretation of Shannon's entropy function is via the \"20 questions\" game. This cooperative game is played by two players, Alice... |
https://arxiv.org/abs/2105.12806 | A Universal Law of Robustness via Isoperimetry | Classically, data interpolation with a parametrized model class is possible as long as the number of parameters is larger than the number of equations to be satisfied. A puzzling phenomenon in deep learning is that models are trained with many more parameters than what this classical theory would suggest. We propose a ... | \section{Introduction}
Solving $n$ equations generically requires only $n$ unknowns\footnote{As in, for instance, the inverse function theorem in analysis or B\'ezout's theorem in algebraic geometry. See also \cite{yun2019small, BELM20} for versions of this claim with neural networks.}. However, the revolutionary deep ... | {
"timestamp": "2021-10-25T02:07:40",
"yymm": "2105",
"arxiv_id": "2105.12806",
"language": "en",
"url": "https://arxiv.org/abs/2105.12806",
"abstract": "Classically, data interpolation with a parametrized model class is possible as long as the number of parameters is larger than the number of equations to ... |
https://arxiv.org/abs/1311.2984 | Edge-maximality of power graphs of finite cyclic groups | We show that among all finite groups of any given order, the cyclic group of that order has the maximum number of edges in its power graph. Contains corrections to published version. | \section{Introduction}
In this paper, we resolve in the affirmative a conjecture of
Mirzargar et al. \cite[Conjecture 2]{m} concerning the number of
edges in the power graph of a finite group. Motivated by the
work of Kelarev and Quinn \cite{KQ1,KQ2,k,KQS}, Chakrabarty,
Ghosh, and Sen \cite{ch} introduced undire... | {
"timestamp": "2014-11-11T02:14:39",
"yymm": "1311",
"arxiv_id": "1311.2984",
"language": "en",
"url": "https://arxiv.org/abs/1311.2984",
"abstract": "We show that among all finite groups of any given order, the cyclic group of that order has the maximum number of edges in its power graph. Contains correct... |
https://arxiv.org/abs/2102.13193 | Minimum Spanning Tree Cycle Intersection Problem | Consider a connected graph $G$ and let $T$ be a spanning tree of $G$. Every edge $e \in G-T$ induces a cycle in $T \cup \{e\}$. The intersection of two distinct such cycles is the set of edges of $T$ that belong to both cycles. We consider the problem of finding a spanning tree that has the least number of such non-emp... | \section{Introduction}
In this article we present what we believe is a novel problem in
graph theory, namely the Minimum Spanning Tree Cycle Intersection
(\emph{MSTCI}) problem.
\bigskip
The problem can be expressed as follows. Let $G$ be a graph
and $T$ a spanning tree of $G$. Every edge $e \in G-T$ induces a
c... | {
"timestamp": "2021-03-01T02:03:32",
"yymm": "2102",
"arxiv_id": "2102.13193",
"language": "en",
"url": "https://arxiv.org/abs/2102.13193",
"abstract": "Consider a connected graph $G$ and let $T$ be a spanning tree of $G$. Every edge $e \\in G-T$ induces a cycle in $T \\cup \\{e\\}$. The intersection of tw... |
https://arxiv.org/abs/1507.00829 | Anti-concentration for polynomials of independent random variables | We prove anti-concentration results for polynomials of independent random variables with arbitrary degree. Our results extend the classical Littlewood-Offord result for linear polynomials, and improve several earlier estimates.We discuss applications in two different areas. In complexity theory, we prove near optimal l... | \section{Introduction}
Let $\xi$ be a Rademacher random variable (taking value $\pm 1$ with probability $1/2$)
and $A=\{a_1,\dots,a_n\}$ be a multi-set in $\R$ (here $n \rightarrow \infty$). Consider the random sum
$$S := a_1 \xi_1 + \dots + a_n \xi_n $$ where $\xi_i$ are iid copies of $\xi$.
In 1943, Little... | {
"timestamp": "2015-08-11T02:01:52",
"yymm": "1507",
"arxiv_id": "1507.00829",
"language": "en",
"url": "https://arxiv.org/abs/1507.00829",
"abstract": "We prove anti-concentration results for polynomials of independent random variables with arbitrary degree. Our results extend the classical Littlewood-Off... |
https://arxiv.org/abs/1503.07933 | On Limits of Dense Packing of Equal Spheres in a Cube | We examine packing of $n$ congruent spheres in a cube when $n$ is close but less than the number of spheres in a regular cubic close-packed (ccp) arrangement of $\lceil p^{3}/2\rceil$ spheres. For this family of packings, the previous best-known arrangements were usually derived from a ccp by omission of a certain numb... | \section{Introduction}
We consider the problem of finding the densest packings of congruent, non-overlapping, spheres in a cube.
Equivalently, we can search for an arrangement of points inside a unit cube so that the minimum
distance between any two points is as large as possible.
The maximum separation distance of $n... | {
"timestamp": "2015-03-30T02:03:47",
"yymm": "1503",
"arxiv_id": "1503.07933",
"language": "en",
"url": "https://arxiv.org/abs/1503.07933",
"abstract": "We examine packing of $n$ congruent spheres in a cube when $n$ is close but less than the number of spheres in a regular cubic close-packed (ccp) arrangem... |
https://arxiv.org/abs/1406.5295 | Rows vs Columns for Linear Systems of Equations - Randomized Kaczmarz or Coordinate Descent? | This paper is about randomized iterative algorithms for solving a linear system of equations $X \beta = y$ in different settings. Recent interest in the topic was reignited when Strohmer and Vershynin (2009) proved the linear convergence rate of a Randomized Kaczmarz (RK) algorithm that works on the rows of $X$ (data p... | \section{Introduction}
Solving linear systems of equations is a classical topic and our interest in this problem is fairly limited. While we do compare two algorithms - Randomized Kaczmarz (RK) (see \citet{StrVer09}) and Randomized Coordinate Descent (RCD) (see \citet{LevLew10}) - to each other, we will not be presentl... | {
"timestamp": "2014-06-23T02:05:38",
"yymm": "1406",
"arxiv_id": "1406.5295",
"language": "en",
"url": "https://arxiv.org/abs/1406.5295",
"abstract": "This paper is about randomized iterative algorithms for solving a linear system of equations $X \\beta = y$ in different settings. Recent interest in the to... |
https://arxiv.org/abs/1609.06473 | Explicit formulas for enumeration of lattice paths: basketball and the kernel method | This article deals with the enumeration of directed lattice walks on the integers with any finite set of steps, starting at a given altitude $j$ and ending at a given altitude $k$, with additional constraints such as, for example, to never attain altitude $0$ in-between. We first discuss the case of walks on the intege... | \section{Introduction}\label{Section1}
While analysing permutations sortable by a stack,
Knuth~\cite[Ex.~1--4 in Sec.~2.2.1]{knuth}
showed they were counted by Catalan numbers,
and were therefore in bijection with Dyck paths (lattice paths with
steps $(1,1)$ and $(1,-1)$ in the plane integer lattice, from the
origin... | {
"timestamp": "2017-02-06T02:06:47",
"yymm": "1609",
"arxiv_id": "1609.06473",
"language": "en",
"url": "https://arxiv.org/abs/1609.06473",
"abstract": "This article deals with the enumeration of directed lattice walks on the integers with any finite set of steps, starting at a given altitude $j$ and endin... |
https://arxiv.org/abs/2108.02275 | Admissibility and Frame Homotopy for Quaternionic Frames | We consider the following questions: when do there exist quaternionic frames with given frame spectrum and given frame vector norms? When such frames exist, is it always possible to interpolate between any two while fixing their spectra and norms? In other words, the first question is the admissibility question for qua... | \section{Introduction}\label{sec:intro}
In a finite-dimensional real or complex Hilbert space $\mathfrak{H}$, an (ordered) \emph{frame} is simply a collection $f_1, \dots , f_N \in \mathfrak{H}$ that spans $\mathfrak{H}$. When $N=d:=\dim \mathfrak{H}$, this is just a basis for $\mathfrak{H}$, but when $N > d$ a frame ... | {
"timestamp": "2022-01-12T02:23:34",
"yymm": "2108",
"arxiv_id": "2108.02275",
"language": "en",
"url": "https://arxiv.org/abs/2108.02275",
"abstract": "We consider the following questions: when do there exist quaternionic frames with given frame spectrum and given frame vector norms? When such frames exis... |
https://arxiv.org/abs/1610.00084 | Spectral asymptotics for Kac-Murdock-Szegő matrices | Szegő's First Limit Theorem provides the limiting statistical distribution (LSD) of the eigenvalues of large Toeplitz matrices. Szegő's Second (or Strong) Limit Theorem for Toeplitz matrices gives a second order correction to the First Limit Theorem, and allows one to calculate asymptotics for the determinants of large... | \section{Introduction}
\subsection{The problem and its history}
For any function $a$ of two variables with Fourier series
\begin{equation} \label{FS1}
a(x,t) =\sum_{k \in {\mathbb Z}} \hat{a}_k(x) e^{ikt},
\end{equation}
Kac, Murdock and Szeg\H{o} \cite{kamusz53, grsz58} introduced in 1953 what they called \textit{g... | {
"timestamp": "2016-10-04T02:01:52",
"yymm": "1610",
"arxiv_id": "1610.00084",
"language": "en",
"url": "https://arxiv.org/abs/1610.00084",
"abstract": "Szegő's First Limit Theorem provides the limiting statistical distribution (LSD) of the eigenvalues of large Toeplitz matrices. Szegő's Second (or Strong)... |
https://arxiv.org/abs/1807.02194 | An algorithm for enumerating difference sets | The DifSets package for GAP implements an algorithm for enumerating all difference sets in a group up to equivalence and provides access to a library of results. The algorithm functions by finding difference sums, which are potential images of difference sets in quotient groups of the original group, and searching thei... | \section{Introduction}\label{sec:introduction}
Let $G$ be a finite group of order $v$ and $D$ a subset of $G$ with $k$ elements.
Then $D$ is a \emph{$(v, k, \lambda)$-difference set} if each nonidentity element of $G$ can be written as $d_id_j^{-1}$ for $d_i, d_j \in D$ in exactly $\lambda$ different ways.
Difference ... | {
"timestamp": "2019-03-14T01:07:24",
"yymm": "1807",
"arxiv_id": "1807.02194",
"language": "en",
"url": "https://arxiv.org/abs/1807.02194",
"abstract": "The DifSets package for GAP implements an algorithm for enumerating all difference sets in a group up to equivalence and provides access to a library of r... |
https://arxiv.org/abs/1401.2597 | Multivariate Density Estimation via Adaptive Partitioning (I): Sieve MLE | We study a non-parametric approach to multivariate density estimation. The estimators are piecewise constant density functions supported by binary partitions. The partition of the sample space is learned by maximizing the likelihood of the corresponding histogram on that partition. We analyze the convergence rate of th... | \section{Estimation of functions of bounded variation}
In image analysis, the denoised image is usually assumed to be a function of
bounded variation. Obtaining an approximation is a crucial procedure
before any downstream analysis. Currently, nonlinear approximation,
such as wavelet compression \cite{DeVore} and wavel... | {
"timestamp": "2015-08-21T02:01:14",
"yymm": "1401",
"arxiv_id": "1401.2597",
"language": "en",
"url": "https://arxiv.org/abs/1401.2597",
"abstract": "We study a non-parametric approach to multivariate density estimation. The estimators are piecewise constant density functions supported by binary partition... |
https://arxiv.org/abs/1102.1984 | Deformation Retracts of Neighborhood Complexes of Stable Kneser Graphs | In 2003, A. Bjorner and M. de Longueville proved that the neighborhood complex of the stable Kneser graph SG_{n,k} is homotopy equivalent to a k-sphere. Further, for n=2 they showed that the neighborhood complex deformation retracts to a subcomplex isomorphic to the associahedron. They went on to ask whether or not, fo... | \section{Introduction and Main Result}
In 1978, L. Lov\'{a}sz proved in \cite{LovaszChromaticNumberHomotopy} M. Kneser's conjecture that if one partitions all the subsets of size $n$ of a $(2n + k)$-element set into $(k+1)$ classes, then one of the classes must contain two disjoint subsets.
Lov\'{a}sz proved this con... | {
"timestamp": "2011-02-11T02:00:14",
"yymm": "1102",
"arxiv_id": "1102.1984",
"language": "en",
"url": "https://arxiv.org/abs/1102.1984",
"abstract": "In 2003, A. Bjorner and M. de Longueville proved that the neighborhood complex of the stable Kneser graph SG_{n,k} is homotopy equivalent to a k-sphere. Fur... |
https://arxiv.org/abs/1804.11003 | Gradient Sampling Methods for Nonsmooth Optimization | This paper reviews the gradient sampling methodology for solving nonsmooth, nonconvex optimization problems. An intuitively straightforward gradient sampling algorithm is stated and its convergence properties are summarized. Throughout this discussion, we emphasize the simplicity of gradient sampling as an extension of... | \section{Introduction}\label{sec.introduction}
The Gradient Sampling (GS) algorithm is a conceptually simple descent method for solving nonsmooth, nonconvex optimization problems, yet it is one that possesses a solid theoretical foundation and has been employed to substantial success in a wide variety of applications.... | {
"timestamp": "2018-05-01T02:10:54",
"yymm": "1804",
"arxiv_id": "1804.11003",
"language": "en",
"url": "https://arxiv.org/abs/1804.11003",
"abstract": "This paper reviews the gradient sampling methodology for solving nonsmooth, nonconvex optimization problems. An intuitively straightforward gradient sampl... |
https://arxiv.org/abs/2205.04734 | Brezzi--Douglas--Marini interpolation on anisotropic simplices and prisms | The Brezzi--Douglas--Marini interpolation error on anisotropic elements has been analyzed in two recent publications, the first focusing on simplices with estimates in $L^2$, the other considering parallelotopes with estimates in terms of $L^p$-norms. This contribution provides generalized estimates for anisotropic sim... | \section{Introduction}
The Brezzi--Douglas--Marini (BDM) finite element \cite{Nedelec1986} was introduced to approximate $\vec{H}_{\mathrm{div}}$ by polynomials.
This proves useful for problems of incompressible fluid flow, where recent approaches employ $\vec{H}_{\mathrm{div}}$-conforming discretizations to approximat... | {
"timestamp": "2022-05-11T02:11:46",
"yymm": "2205",
"arxiv_id": "2205.04734",
"language": "en",
"url": "https://arxiv.org/abs/2205.04734",
"abstract": "The Brezzi--Douglas--Marini interpolation error on anisotropic elements has been analyzed in two recent publications, the first focusing on simplices with... |
https://arxiv.org/abs/1903.00465 | A note on congruence properties of the generalized bi-periodic Horadam sequence | In this paper, we consider a generalization of Horadam sequence {w_n} which is defined by the recurrence w_n = aw_n-1 + cw_n-2; if n is even, w_n = bw_n-1 + cw_n-2; if n is odd with arbitrary initial conditions w_0, w_1 and nonzero real numbers a, b, and c. We investigate some congruence properties of the generalized H... | \section{Introduction}
The generalized bi-periodic Horadam sequence $\left\{ w_{n}\right\} $ is
defined by the recurrence relatio
\begin{equation*}
w_{n}=\left\{
\begin{array}{ll}
aw_{n-1}+cw_{n-2}, & \mbox{ if }n\mbox{ is even} \\
bw_{n-1}+cw_{n-2}, & \mbox{ if }n\mbox{ is odd
\end{array
\right. ,\text{ }n\g... | {
"timestamp": "2019-03-04T02:21:55",
"yymm": "1903",
"arxiv_id": "1903.00465",
"language": "en",
"url": "https://arxiv.org/abs/1903.00465",
"abstract": "In this paper, we consider a generalization of Horadam sequence {w_n} which is defined by the recurrence w_n = aw_n-1 + cw_n-2; if n is even, w_n = bw_n-1... |
https://arxiv.org/abs/1310.2691 | An incomplete variant of Wilson's congruence | This article examines the nontrivial solutions of the congruence \[ (p-1)\cdots(p-r) \equiv -1 \pmod p. \] We discuss heuristics for the proportion of primes $p$ that have exactly $N$ solutions to this congruence. We supply numerical evidence in favour of these conjectures, and discuss the algorithms used in our calcul... | \section{Heuristics and conjectures}
Wilson's Theorem \cite[Theorems 80 and 81]{HW} states that
\[ (p-1)! \equiv -1 \pmod p \]
if and only if $p$ is a prime. Now truncate the factorial after $r$ terms. For which primes $p$ is there an $r$ for which
\begin{equation}\label{wit}
(p-1)\cdots(p-r) \equiv -1 \pmod p ?... | {
"timestamp": "2013-10-11T02:04:08",
"yymm": "1310",
"arxiv_id": "1310.2691",
"language": "en",
"url": "https://arxiv.org/abs/1310.2691",
"abstract": "This article examines the nontrivial solutions of the congruence \\[ (p-1)\\cdots(p-r) \\equiv -1 \\pmod p. \\] We discuss heuristics for the proportion of ... |
https://arxiv.org/abs/1310.6013 | Matchings and Hamilton Cycles with Constraints on Sets of Edges | The aim of this paper is to extend and generalise some work of Katona on the existence of perfect matchings or Hamilton cycles in graphs subject to certain constraints. The most general form of these constraints is that we are given a family of sets of edges of our graph and are not allowed to use all the edges of any ... | \section{Introduction}
Many results in graph theory concern establishing conditions on a graph $G$ which guarantee that $G$ must contain some particular spanning structure. A classical example of this is Dirac's theorem \cite{Dirac}:
\begin{theorem}[Dirac]\label{dirac:thm}
Every graph on $n$ vertices with minimum deg... | {
"timestamp": "2013-10-23T02:09:42",
"yymm": "1310",
"arxiv_id": "1310.6013",
"language": "en",
"url": "https://arxiv.org/abs/1310.6013",
"abstract": "The aim of this paper is to extend and generalise some work of Katona on the existence of perfect matchings or Hamilton cycles in graphs subject to certain ... |
https://arxiv.org/abs/1111.5749 | Accuracy analysis of the box-counting algorithm | Accuracy of the box-counting algorithm for numerical computation of the fractal exponents is investigated. To this end several sample mathematical fractal sets are analyzed. It is shown that the standard deviation obtained for the fit of the fractal scaling in the log-log plot strongly underestimates the actual error. ... | \section{Introduction}
In last decades computations of fractal dimensions (exponents)
have become very popular in various areas of physics, as well
as in interdisciplinary research. Fractal structures have been
found in wide spectrum of problems, ranging from high energy physics
\cite{Bialas1988} to cosmology \cit... | {
"timestamp": "2011-11-28T02:01:37",
"yymm": "1111",
"arxiv_id": "1111.5749",
"language": "en",
"url": "https://arxiv.org/abs/1111.5749",
"abstract": "Accuracy of the box-counting algorithm for numerical computation of the fractal exponents is investigated. To this end several sample mathematical fractal s... |
https://arxiv.org/abs/1912.09763 | Optimizing Sparsity over Lattices and Semigroups | Motivated by problems in optimization we study the sparsity of the solutions to systems of linear Diophantine equations and linear integer programs, i.e., the number of non-zero entries of a solution, which is often referred to as the $\ell_0$-norm. Our main results are improved bounds on the $\ell_0$-norm of sparse so... |
\section{Introduction}
This paper discusses the problem of finding sparse solutions to systems of linear Diophantine equations
and integer linear programs.
We investigate the $\ell_0$-norm $\|\boldsymbol x\|_0 := |\setcond{ i }{x_i \ne 0}|$, a function widely used in the theory of {\em compressed sensing} \cite{CS_... | {
"timestamp": "2020-08-06T02:20:35",
"yymm": "1912",
"arxiv_id": "1912.09763",
"language": "en",
"url": "https://arxiv.org/abs/1912.09763",
"abstract": "Motivated by problems in optimization we study the sparsity of the solutions to systems of linear Diophantine equations and linear integer programs, i.e.,... |
https://arxiv.org/abs/1508.06170 | Recent advances on Dirac-type problems for hypergraphs | A fundamental question in graph theory is to establish conditions that ensure a graph contains certain spanning subgraphs. Two well-known examples are Tutte's theorem on perfect matchings and Dirac's theorem on Hamilton cycles. Generalizations of Dirac's theorem, and related matching and packing problems for hypergraph... | \section{Introduction}
Given two (hyper)graphs $F$ and $H$, which conditions guarantee $H$ contains $F$ as a subgraph?
When $|V(F)|= |V(H)|$, the decision problem of whether $H$ contains $F$ is often NP-complete, {\em e.g.}, deciding if a graph $H$ contains a Hamilton cycle is a well-known NP-complete problem. Therefor... | {
"timestamp": "2015-08-26T02:10:04",
"yymm": "1508",
"arxiv_id": "1508.06170",
"language": "en",
"url": "https://arxiv.org/abs/1508.06170",
"abstract": "A fundamental question in graph theory is to establish conditions that ensure a graph contains certain spanning subgraphs. Two well-known examples are Tut... |
https://arxiv.org/abs/2008.03385 | Solving two-parameter eigenvalue problems using an alternating method | We present a new approach to compute selected eigenvalues and eigenvectors of the two-parameter eigenvalue problem. Our method requires computing generalized eigenvalue problems of the same size as the matrices of the initial two-parameter eigenvalue problem. The method is applicable for right definite problems, possib... | \section{Introduction}\label{setting}
In this work we consider the \emph{two-parameter eigenvalue problem}
\begin{equation}\label{2paraev}
\begin{aligned}
(A_1+\lambda B_1+\mu C_1)u&=0,\\
(A_2+\lambda B_2+\mu C_2)v&=0
\end{aligned}
\end{equation}
with matrices $A_1,B_1,C_1\in\mathbb{R}^{n\times n}$ and $A_2,B_2,C_2\... | {
"timestamp": "2021-05-12T02:22:48",
"yymm": "2008",
"arxiv_id": "2008.03385",
"language": "en",
"url": "https://arxiv.org/abs/2008.03385",
"abstract": "We present a new approach to compute selected eigenvalues and eigenvectors of the two-parameter eigenvalue problem. Our method requires computing generali... |
https://arxiv.org/abs/1311.3644 | Inversion and subspaces of a finite field | Let $A$ and $B$ two $F_q$-subspaces of a finite field, of the same size, and let $A^{-1}$ denote the set of inverses of the nonzero elements of $A$. Mattarei proved that $A^{-1}$ can only be contained in $A$ if either $A$ is a subfield, or $A$ is the set of trace zero elements in a quadratic extension of a field. Csajb... | \section{Introduction}
In response to a question of Andrea Caranti, for use in~\cite{CDVS:AES}, the author determined in~\cite{Mat:inverse-closed}
the additive subgroups of a field which are closed with respect to inverting nonzero elements.
The more general question with a division ring instead of a field was ind... | {
"timestamp": "2013-12-10T02:07:37",
"yymm": "1311",
"arxiv_id": "1311.3644",
"language": "en",
"url": "https://arxiv.org/abs/1311.3644",
"abstract": "Let $A$ and $B$ two $F_q$-subspaces of a finite field, of the same size, and let $A^{-1}$ denote the set of inverses of the nonzero elements of $A$. Mattare... |
https://arxiv.org/abs/1007.1760 | Factorization of banded permutations | We consider the factorization of permutations into bandwidth 1 permutations, which are products of mutually nonadjacent simple transpositions. We exhibit an upper bound on the minimal number of such factors and thus prove a conjecture of Gilbert Strang: a banded permutation of bandwidth $w$ can be represented as the pr... | \section{Introduction}\label{section:intro}
Computational efficiency very often requires us to represent matrices as products of certain special, easily computable, matrices using as few factors as possible. Matrices of bounded bandwidth are often seen in practical applications. In \cite{Strang1} and \cite{Strang2} G... | {
"timestamp": "2012-01-17T02:01:57",
"yymm": "1007",
"arxiv_id": "1007.1760",
"language": "en",
"url": "https://arxiv.org/abs/1007.1760",
"abstract": "We consider the factorization of permutations into bandwidth 1 permutations, which are products of mutually nonadjacent simple transpositions. We exhibit an... |
https://arxiv.org/abs/2001.05992 | Provable Benefit of Orthogonal Initialization in Optimizing Deep Linear Networks | The selection of initial parameter values for gradient-based optimization of deep neural networks is one of the most impactful hyperparameter choices in deep learning systems, affecting both convergence times and model performance. Yet despite significant empirical and theoretical analysis, relatively little has been p... | \section{Conclusion} \label{sec:conclu}
In this work, we studied the effect of the initialization parameter values of deep linear neural networks on the convergence time of gradient descent. We found that when the initial weights are iid Gaussian, the convergence time grows exponentially in the depth unless the width i... | {
"timestamp": "2020-01-17T02:17:02",
"yymm": "2001",
"arxiv_id": "2001.05992",
"language": "en",
"url": "https://arxiv.org/abs/2001.05992",
"abstract": "The selection of initial parameter values for gradient-based optimization of deep neural networks is one of the most impactful hyperparameter choices in d... |
https://arxiv.org/abs/1405.6894 | On the number of monotone sequences | One of the most classical results in Ramsey theory is the theorem of Erdős and Szekeres from 1935, which says that every sequence of more than $k^2$ numbers contains a monotone subsequence of length $k+1$. We address the following natural question motivated by this result: Given integers $k$ and $n$ with $n \geq k^2+1$... | \section{Introduction}
\label{sec:introduction}
A typical problem in extremal combinatorics has the following form: What is the largest size of a structure which does not contain any forbidden configurations? Once this extremal value is known, it is very natural to ask how many forbidden configurations one is guarant... | {
"timestamp": "2014-05-28T02:08:36",
"yymm": "1405",
"arxiv_id": "1405.6894",
"language": "en",
"url": "https://arxiv.org/abs/1405.6894",
"abstract": "One of the most classical results in Ramsey theory is the theorem of Erdős and Szekeres from 1935, which says that every sequence of more than $k^2$ numbers... |
https://arxiv.org/abs/0907.1019 | The algebraic crossing number and the braid index of knots and links | It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links.The Morton-Franks-Williams inequality gives a lower bound for braid index. And sharpness of the inequality on a knot type implies the t... | \section{Introduction}
The {\em braid index} is one of the classical invariants of knots and links.
Any knot and link type is presented as a braid closure.
The braid index of a link type is the least number of braid strands
needed for that.
The {\em algebraic crossing number} (or writhe) is an integer
associated to ... | {
"timestamp": "2009-07-06T17:41:39",
"yymm": "0907",
"arxiv_id": "0907.1019",
"language": "en",
"url": "https://arxiv.org/abs/0907.1019",
"abstract": "It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for man... |
https://arxiv.org/abs/2204.07885 | On the principal minors of the powers of a matrix | We show that if $A$ is an $n\times n$-matrix, then the diagonal entries of each power $A^{m}$ are uniquely determined by the principal minors of $A$, and can be written as universal (integral) polynomials in the latter. Furthermore, if the latter all equal $1$, then so do the former. These results are inspired by Probl... | \section{Introduction}
Let $R$ be a commutative ring. Let $A$ be an $n\times n$-matrix over $R$,
where $n$ is a nonnegative integer.
A \emph{principal submatrix} of $A$ means a matrix obtained from $A$ by
removing some rows and the corresponding columns (i.e., removing the $i_{1
$-th, $i_{2}$-th, $\ldots$, $i_{... | {
"timestamp": "2022-04-19T02:23:00",
"yymm": "2204",
"arxiv_id": "2204.07885",
"language": "en",
"url": "https://arxiv.org/abs/2204.07885",
"abstract": "We show that if $A$ is an $n\\times n$-matrix, then the diagonal entries of each power $A^{m}$ are uniquely determined by the principal minors of $A$, and... |
https://arxiv.org/abs/2006.12545 | A variant of Cauchy's argument principle for analytic functions which applies to curves containing zeroes | It is known that the Cauchy's argument principle, applied to an holomorphic function $f$, requires that $f$ has no zeros on the curve of integration. In this short note, we give a generalization of such a principle which covers the case when $f$ has zeros on the curve, as well as an application. | \section{Introduction, statement of results, and an application}
The argument principle, one of the fundamental results in complex analysis, can be formulated as follows (see \cite{Rudin2001} or \cite{remmert2012theory}).
\begin{thm} \label{classarg}
Suppose that $\gamma$ is a smooth Jordan curve in a domain $U$, and ... | {
"timestamp": "2020-06-24T02:01:18",
"yymm": "2006",
"arxiv_id": "2006.12545",
"language": "en",
"url": "https://arxiv.org/abs/2006.12545",
"abstract": "It is known that the Cauchy's argument principle, applied to an holomorphic function $f$, requires that $f$ has no zeros on the curve of integration. In t... |
https://arxiv.org/abs/1905.02600 | The Pseudoforest analogue for the Strong Nine Dragon Tree Conjecture is True | We prove that for any positive integers $k$ and $d$, if a graph $G$ has maximum average degree at most $2k + \frac{2d}{d+k+1}$, then $G$ decomposes into $k+1$ pseudoforests $C_{1},\ldots,C_{k+1}$ such that there is an $i$ such that for every connected component $C$ of $C_{i}$, we have that $e(C) \leq d$. | \section{Introduction}
For any graph $G$, a \textit{decomposition} of $G$ is a set of edge disjoint subgraphs of $G$ such that the union of their edges sets is the edge set of the graph. Graph decompositions are a particularly heavily studied area of graph theory, and one of the most beautiful results about graph dec... | {
"timestamp": "2020-03-02T02:00:25",
"yymm": "1905",
"arxiv_id": "1905.02600",
"language": "en",
"url": "https://arxiv.org/abs/1905.02600",
"abstract": "We prove that for any positive integers $k$ and $d$, if a graph $G$ has maximum average degree at most $2k + \\frac{2d}{d+k+1}$, then $G$ decomposes into ... |
https://arxiv.org/abs/2007.08079 | Unique determination of ellipsoids by their dual volumes and the moment problem | Gusakova and Zaporozhets conjectured that ellipsoids in $\mathbb R^n$ are uniquely determined (up to an isometry) by their Steiner polynomials. Petrov and Tarasov confirmed this conjecture in $\mathbb R^3$. In this paper we solve the dual problem. We show that any ellipsoid in $\mathbb{R}^n$ centered at the origin is u... | \section{Introduction}
The study of behavior of volume under the Minkowski (vector) addition is the main focus of the Brunn-Minkowski theory.
Let $K$ be a convex body in $\mathbb R^n$. Denote by $B_2^n$ the unit Euclidean ball in $\mathbb R^n$. The classical Steiner formula asserts that for every $\epsilon > 0$,
... | {
"timestamp": "2020-07-17T02:06:59",
"yymm": "2007",
"arxiv_id": "2007.08079",
"language": "en",
"url": "https://arxiv.org/abs/2007.08079",
"abstract": "Gusakova and Zaporozhets conjectured that ellipsoids in $\\mathbb R^n$ are uniquely determined (up to an isometry) by their Steiner polynomials. Petrov an... |
https://arxiv.org/abs/2004.06093 | Topology of deep neural networks | We study how the topology of a data set $M = M_a \cup M_b \subseteq \mathbb{R}^d$, representing two classes $a$ and $b$ in a binary classification problem, changes as it passes through the layers of a well-trained neural network, i.e., with perfect accuracy on training set and near-zero generalization error ($\approx 0... | \section{Overview}
A key insight of topological data analysis is that ``\emph{data has shape}'' \cite{focm,acta}. That data sets often have nontrivial topologies, which may be exploited in their analysis, is now a widely accepted principle with abundant examples across multiple disciplines: dynamical systems \cite{kh... | {
"timestamp": "2020-04-14T02:25:47",
"yymm": "2004",
"arxiv_id": "2004.06093",
"language": "en",
"url": "https://arxiv.org/abs/2004.06093",
"abstract": "We study how the topology of a data set $M = M_a \\cup M_b \\subseteq \\mathbb{R}^d$, representing two classes $a$ and $b$ in a binary classification prob... |
https://arxiv.org/abs/cs/0509013 | On the variational distance of independently repeated experiments | Let P and Q be two probability distributions which differ only for values with non-zero probability. We show that the variational distance between the n-fold product distributions P^n and Q^n cannot grow faster than the square root of n. | \section{Preliminaries}
Let $P$ be a probability distribution with range ${\mathcal{Z}}$ and let $n \in
{\mathbb{N}}$. We denote by $P^n$ the $n$-fold \emph{product
distribution}, that is,
\[
P^n(z_1, \ldots, z_n) = \prod_{i=1}^n P(z_i)
\]
for any $z_1, \ldots, z_n \in {\mathcal{Z}}$. Note that $P^n$ describes $n... | {
"timestamp": "2005-09-05T13:38:42",
"yymm": "0509",
"arxiv_id": "cs/0509013",
"language": "en",
"url": "https://arxiv.org/abs/cs/0509013",
"abstract": "Let P and Q be two probability distributions which differ only for values with non-zero probability. We show that the variational distance between the n-f... |
https://arxiv.org/abs/2206.13334 | Butler's Method applied to $\mathbb{Z}_p[C_p\times C_p]$-permutation modules | Let $G$ be a finite $p$-group with normal subgroup $N$ of order $p$. The first author and Zalesskii have previously given a characterization of permutation modules for $\mathbb{Z}_pG$ in terms of modules for $G/N$, but the necessity of their conditions was not known. We apply a correspondence due to Butler to demonstra... | \section{Introduction}
Let $R$ be a complete discrete valuation ring whose residue field has characteristic $p$ ($R$ will almost always be $\Z_p$ or $\mathbb{F}_p$) and let $G$ be a finite $p$-group. An $RG$-module is a \emph{lattice} if it is free of finite $R$-rank. An $RG$-lattice is a \emph{permutation module} i... | {
"timestamp": "2022-06-30T02:18:08",
"yymm": "2206",
"arxiv_id": "2206.13334",
"language": "en",
"url": "https://arxiv.org/abs/2206.13334",
"abstract": "Let $G$ be a finite $p$-group with normal subgroup $N$ of order $p$. The first author and Zalesskii have previously given a characterization of permutatio... |
https://arxiv.org/abs/1708.03069 | Two-vertex generators of Jacobians of graphs | We give necessary and sufficient conditions under which the Jacobian of a graph is generated by a divisor that is the difference of two vertices. This answers a question posed by Becker and Glass and allows us to prove various other propositions about the order of divisors that are the difference of two vertices. We co... | \section{Introduction}
Given a finite, undirected, connected multigraph $ G$ without loops, a \textit{divisor} is an assignment of integer values to the vertices. The \textit{degree} of a divisor is the sum of these values. The \textit{Jacobian} of $ G$, denoted $ \text{Jac}(G)$, is a finite abelian group defined as t... | {
"timestamp": "2017-09-12T02:05:13",
"yymm": "1708",
"arxiv_id": "1708.03069",
"language": "en",
"url": "https://arxiv.org/abs/1708.03069",
"abstract": "We give necessary and sufficient conditions under which the Jacobian of a graph is generated by a divisor that is the difference of two vertices. This ans... |
https://arxiv.org/abs/2005.02529 | Regarding two conjectures on clique and biclique partitions | For a graph $G$, let $cp(G)$ denote the minimum number of cliques of $G$ needed to cover the edges of $G$ exactly once. Similarly, let $bp_k(G)$ denote the minimum number of bicliques (i.e. complete bipartite subgraphs of $G$) needed to cover each edge of $G$ exactly $k$ times. We consider two conjectures -- one regard... | \section{Introduction}
For a fixed family of graphs $\mathcal{F}$, an $\mathcal{F}$-partition of a graph $G$ is a collection $\mathcal{C} = \{H_1, \dots, H_\ell\}$ of subgraphs $H_i \subset G$ such that each edge of $G$ belongs to exactly one $H_i \in \mathcal{C}$, and each $H_i$ is isomorphic to some graph in $\math... | {
"timestamp": "2020-05-07T02:06:48",
"yymm": "2005",
"arxiv_id": "2005.02529",
"language": "en",
"url": "https://arxiv.org/abs/2005.02529",
"abstract": "For a graph $G$, let $cp(G)$ denote the minimum number of cliques of $G$ needed to cover the edges of $G$ exactly once. Similarly, let $bp_k(G)$ denote th... |
https://arxiv.org/abs/1304.0143 | Which alternating and symmetric groups are unit groups? | We prove there is no ring with unit group isomorphic to S_n for n \geq 5 and that there is no ring with unit group isomorphic to A_n for n \geq 5, n \neq 8. We give examples of rings with unit groups isomorphic to S_1, S_2, S_3, S_4, A_1, A_2, A_3, A_4, and A_8. We expect our methods to work similarly for other groups ... | \section{Introduction}
In this paper we will consider a special case of the general question: For what finite groups $G$ is there a ring with unit group isomorphic to $G$?
We shall see in the following example that this is a nontrivial condition.
\begin{Eg}
There does not exist a ring whose unit group is cyclic of ord... | {
"timestamp": "2013-04-02T02:02:55",
"yymm": "1304",
"arxiv_id": "1304.0143",
"language": "en",
"url": "https://arxiv.org/abs/1304.0143",
"abstract": "We prove there is no ring with unit group isomorphic to S_n for n \\geq 5 and that there is no ring with unit group isomorphic to A_n for n \\geq 5, n \\neq... |
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