url stringlengths 31 38 | title stringlengths 7 229 | abstract stringlengths 44 2.87k | text stringlengths 319 2.51M | meta dict |
|---|---|---|---|---|
https://arxiv.org/abs/1610.01568 | The ratio of domination and independent domination numbers on trees | Let $\gamma(G)$ and $i(G)$ be the domination number and the independent domination number of $G$, respectively. In 1977, Hedetniemi and Mitchell began with the comparison of of $i(G)$ and $\gamma(G)$ and recently Rad and Volkmann posted a conjecture that $i(G)/ \gamma(G) \leq \Delta(G)/2$, where $\Delta(G)$ is the maxi... | \section{Introduction}
Throughout this paper $G = (V, E)$ is a simple undirected graph with vertex set $ V(G)$ and edge set $ E(G)$. For $v\in V(G)$, $N_G(v)=\{ w\in V(G): vw\in E(G)\}$ is
the open neighborhood of $v$ and $N_G[v]=N_G(v) \cup \{v\}$ is
the closed neighborhood of $v$ in $G$. If $N_G(v) = \p... | {
"timestamp": "2016-10-06T02:07:46",
"yymm": "1610",
"arxiv_id": "1610.01568",
"language": "en",
"url": "https://arxiv.org/abs/1610.01568",
"abstract": "Let $\\gamma(G)$ and $i(G)$ be the domination number and the independent domination number of $G$, respectively. In 1977, Hedetniemi and Mitchell began wi... |
https://arxiv.org/abs/2203.13870 | A simple mnemonic to compute sums of powers | We give a simple recursive formula to obtain the general sum of the first $N$ natural numbers to the $r$th power. Our method allows one to obtain the general formula for the $(r+1)$th power once one knows the general formula for the $r$th power. The method is very simple to remember owing to an analogy with differentia... | \section{Introduction}
Sums of powers have fascinated mathematicians for centuries. The sum of the first $N$ natural numbers is given by the simple well-known formula
\begin{equation}
\sum_{n=1}^N n = 1+2+\cdots + N = \frac{N(N+1)}{2} \ .
\label{sum r=1}
\end{equation}
On the other hand, while occasionally use... | {
"timestamp": "2022-03-29T02:02:42",
"yymm": "2203",
"arxiv_id": "2203.13870",
"language": "en",
"url": "https://arxiv.org/abs/2203.13870",
"abstract": "We give a simple recursive formula to obtain the general sum of the first $N$ natural numbers to the $r$th power. Our method allows one to obtain the gene... |
https://arxiv.org/abs/1903.05910 | Efficient evaluation of noncommutative polynomials using tensor and noncommutative Waring decompositions | This paper analyses a Waring type decomposition of a noncommuting (NC) polynomial $p$ with respect to the goal of evaluating $p$ efficiently on tuples of matrices. Such a decomposition can reduce the number of matrix multiplications needed to evaluate a noncommutative polynomial and is valuable when a single polynomial... | \section{Introduction}
\ \ \
The Waring problem has had a long history since it was first proposed in 1770 by Edward Waring. Initially it was a question about integers, asking whether a natural number could be written as the sum of powers of natural numbers. Later it was extended to polynomials. It concerns the questio... | {
"timestamp": "2019-03-15T01:15:45",
"yymm": "1903",
"arxiv_id": "1903.05910",
"language": "en",
"url": "https://arxiv.org/abs/1903.05910",
"abstract": "This paper analyses a Waring type decomposition of a noncommuting (NC) polynomial $p$ with respect to the goal of evaluating $p$ efficiently on tuples of ... |
https://arxiv.org/abs/1608.01666 | A central limit theorem for a new statistic on permutations | This paper does three things: It proves a central limit theorem for novel permutation statistics (for example, the number of descents plus the number of descents in the inverse). It provides a clear illustration of a new approach to proving central limit theorems more generally. It gives us an opportunity to acknowledg... | \section{Introduction}
Let $S_n$ be the group of all $n!$ permutations of $\{1,\ldots, n\}$. A variety of statistics $T(\pi)$ are used to enable tasks such as tests of randomness of a time series, comparison of voter profiles when candidates are ranked, non-parametric statistical tests and evaluation of search engine r... | {
"timestamp": "2016-10-28T02:01:23",
"yymm": "1608",
"arxiv_id": "1608.01666",
"language": "en",
"url": "https://arxiv.org/abs/1608.01666",
"abstract": "This paper does three things: It proves a central limit theorem for novel permutation statistics (for example, the number of descents plus the number of d... |
https://arxiv.org/abs/1805.01412 | Upper bounds for the regularity of powers of edge ideals of graphs | Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. In this paper, we obtain upper bounds for the Castelnuovo-Mumford regularity of $I(G)^q$ in terms of certain combinatorial invariants associated with $G$. We also prove a weaker version of a conjecture by Alilooee, Banerjee, Beyarslan and ... | \section{Introduction}
Let $I$ be a homogeneous ideal of a polynomial ring
$R = \K[x_1,\ldots,x_n]$ over a field $\K$ with usual grading.
In \cite{BEL91}, Bertram, Ein and Lazarsfeld
have initiated the study of the Castelnuovo-Mumford regularity of $I^q$
as a function of $q$ by proving that if $I$ is the defining... | {
"timestamp": "2018-05-04T02:12:24",
"yymm": "1805",
"arxiv_id": "1805.01412",
"language": "en",
"url": "https://arxiv.org/abs/1805.01412",
"abstract": "Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. In this paper, we obtain upper bounds for the Castelnuovo-Mumford regular... |
https://arxiv.org/abs/1805.10510 | A modification of the Chang-Wilson-Wolff Inequality via the Bellman Function | We describe the Bellman function technique for proving sharp inequalities in harmonic analysis. To provide an example along with historical context, we present how it was originally used by Donald Burkholder to prove $L^p$ boundedness of the $\pm 1$ martingale transform. Finally, with Burkholder's result as a blueprint... | \section{Introduction}
The Bellman function technique, named for applied mathematician Richard Bellman, is a tool that has been imported from the applied field of stochastic optimal control, and is now being used to tackle problems in probability and harmonic analysis. It was introduced to the world of analysis by Don... | {
"timestamp": "2018-05-29T02:06:56",
"yymm": "1805",
"arxiv_id": "1805.10510",
"language": "en",
"url": "https://arxiv.org/abs/1805.10510",
"abstract": "We describe the Bellman function technique for proving sharp inequalities in harmonic analysis. To provide an example along with historical context, we pr... |
https://arxiv.org/abs/2009.07893 | Largest small polygons: A sequential convex optimization approach | A small polygon is a polygon of unit diameter. The maximal area of a small polygon with $n=2m$ vertices is not known when $m\ge 7$. Finding the largest small $n$-gon for a given number $n\ge 3$ can be formulated as a nonconvex quadratically constrained quadratic optimization problem. We propose to solve this problem wi... | \section{Introduction}
The {\em diameter} of a polygon is the largest Euclidean distance between pairs of its vertices. A polygon is said to be {\em small} if its diameter equals one. For a given integer $n \ge 3$, the maximal area problem consists in finding the small $n$-gon with the largest area. The problem was fir... | {
"timestamp": "2021-06-02T02:11:56",
"yymm": "2009",
"arxiv_id": "2009.07893",
"language": "en",
"url": "https://arxiv.org/abs/2009.07893",
"abstract": "A small polygon is a polygon of unit diameter. The maximal area of a small polygon with $n=2m$ vertices is not known when $m\\ge 7$. Finding the largest s... |
https://arxiv.org/abs/2209.13199 | Normal Bundles of Rational Normal Curves on Hypersurfaces | Let $C$ be the rational normal curve of degree $e$ in $\mathbb{P}^n$, and let $X\subset \mathbb{P}^n$ be a degree $d\ge 2$ hypersurface containing $C$. In previous work, I. Coskun and E. Riedl showed that the normal bundle $N_{C/X}$ is balanced for a general $X$. H. Larson studied the case of lines ($e=1$) and computed... | \section{Introduction}
Rational curves play an important role in the study of birational and arithmetic geometry of projective varieties. The local structure of the space of rational curves on a variety is determined by its normal bundle. In this paper, we study the possible splitting types of the normal bundle of rat... | {
"timestamp": "2022-09-28T02:12:50",
"yymm": "2209",
"arxiv_id": "2209.13199",
"language": "en",
"url": "https://arxiv.org/abs/2209.13199",
"abstract": "Let $C$ be the rational normal curve of degree $e$ in $\\mathbb{P}^n$, and let $X\\subset \\mathbb{P}^n$ be a degree $d\\ge 2$ hypersurface containing $C$... |
https://arxiv.org/abs/1804.02614 | Randomized subspace iteration: Analysis of canonical angles and unitarily invariant norms | This paper is concerned with the analysis of the randomized subspace iteration for the computation of low-rank approximations. We present three different kinds of bounds. First, we derive both bounds for the canonical angles between the exact and the approximate singular subspaces. Second, we derive bounds for the low-... | \section{Introduction}
The computation of low-rank approximations of large-scale matrices is a vital step in many applications in data analysis and scientific computing. These applications include principal component analysis, facial recognition, spectral clustering, model reduction techniques such as proper orthogonal... | {
"timestamp": "2018-11-13T02:13:48",
"yymm": "1804",
"arxiv_id": "1804.02614",
"language": "en",
"url": "https://arxiv.org/abs/1804.02614",
"abstract": "This paper is concerned with the analysis of the randomized subspace iteration for the computation of low-rank approximations. We present three different ... |
https://arxiv.org/abs/2102.00996 | Compositions that are palindromic modulo $m$ | In recent work, G. E. Andrews and G. Simay prove a surprising relation involving parity palindromic compositions, and ask whether a combinatorial proof can be found. We extend their results to a more general class of compositions that are palindromic modulo $m$, that includes the parity palindromic case when $m=2$. We ... | \section{Introduction}
Let $\sigma=(\sigma_1,\sigma_2,\ldots,\sigma_k)$ be a sequence of positive integers such that $\sum \sigma_i = n$. The sequence $\sigma$ is called a \textit{composition} of $n$ of length $k$. The numbers $\sigma_i$ are called the \textit{parts} of the composition. If $\sigma_i=\sigma_{k-i+1}$ fo... | {
"timestamp": "2021-09-29T02:08:40",
"yymm": "2102",
"arxiv_id": "2102.00996",
"language": "en",
"url": "https://arxiv.org/abs/2102.00996",
"abstract": "In recent work, G. E. Andrews and G. Simay prove a surprising relation involving parity palindromic compositions, and ask whether a combinatorial proof ca... |
https://arxiv.org/abs/1505.04151 | Minkowski Symmetrizations of Star Shaped Sets | We provide sharp upper bounds for the number of symmetrizations required to transform a star shaped set in ${\mathbb R}^n$ arbitrarily close (in the Hausdorff metric) to the Euclidean ball. | \section{Introduction and results}\label{Sec_Intro}
A non empty compact set $K\subset {\cal R}^n$ is called {\em star shaped}
if $x\in K$ implies $[0,x]\subseteq K$. We denote the family of star
shaped sets in ${\cal R}^n$ by $\mathscr{S}^n$. Recall that given a set $K$ and a
direction $u \in S^{n-1}$, it's Minkowski s... | {
"timestamp": "2015-05-18T02:11:08",
"yymm": "1505",
"arxiv_id": "1505.04151",
"language": "en",
"url": "https://arxiv.org/abs/1505.04151",
"abstract": "We provide sharp upper bounds for the number of symmetrizations required to transform a star shaped set in ${\\mathbb R}^n$ arbitrarily close (in the Haus... |
https://arxiv.org/abs/1906.12085 | Tutorial: Complexity analysis of Singular Value Decomposition and its variants | We compared the regular Singular Value Decomposition (SVD), truncated SVD, Krylov method and Randomized PCA, in terms of time and space complexity. It is well-known that Krylov method and Randomized PCA only performs well when k << n, i.e. the number of eigenpair needed is far less than that of matrix size. We compared... | \section{Introduction}
Dimensionality reduction has always been a trendy topic in machine learning. Linear subspace method for reduction, e.g., Principal Component Analysis and its variation have been widely studied\cite{Xu1995Robust, Partridge2002Robust, Zou2006Sparse}, and some pieces of literature introduce probabil... | {
"timestamp": "2019-10-15T02:24:05",
"yymm": "1906",
"arxiv_id": "1906.12085",
"language": "en",
"url": "https://arxiv.org/abs/1906.12085",
"abstract": "We compared the regular Singular Value Decomposition (SVD), truncated SVD, Krylov method and Randomized PCA, in terms of time and space complexity. It is ... |
https://arxiv.org/abs/1210.6582 | Minimal Periods for Ordinary Differential Equations in Strictly Convex Banach Spaces and Explicit Bounds for some l^p-Spaces | Let x(t) be a non-constant T-periodic solution to the ordinary differential equation x'= f(x) in a Banach space X where f is assumed to be Lipschitz continuous with constant L. Then there exists a constant c such that T L >= c, with c only depending on X. It is known that c >= 6 in any Banach space and that c = 2{\pi} ... | \section{Introduction}
Consider the ordinary differential equation $\dot{x}=f(x)$ in a Banach space $X$, where $f$ is Lipschitz continuous with constant $L$, that is for any $x,y\in X$
$$
\|f(x)-f(y)\|_X\leq L\|x-y\|_X.
$$
In this case one can relate the period $T$ of any non-constant periodic orbit to the Lipschitz co... | {
"timestamp": "2013-07-24T02:08:42",
"yymm": "1210",
"arxiv_id": "1210.6582",
"language": "en",
"url": "https://arxiv.org/abs/1210.6582",
"abstract": "Let x(t) be a non-constant T-periodic solution to the ordinary differential equation x'= f(x) in a Banach space X where f is assumed to be Lipschitz continu... |
https://arxiv.org/abs/1101.0388 | Demystifying a divisibility property of the Kostant partition function | We study a family of identities regarding a divisibility property of the Kostant partition function which first appeared in a paper of Baldoni and Vergne. To prove the identities, Baldoni and Vergne used techniques of residues and called the resulting divisibility property "mysterious." We prove these identities entire... | \section{Introduction}
\label{sec:in}
The objective of this paper is to provide a natural combinatorial explanation of a divisibility property of the Kostant partition function. The question of evaluating Kostant partition functions has been subject of much interest, without a satisfactory combinatorial answer. To me... | {
"timestamp": "2011-01-04T02:03:52",
"yymm": "1101",
"arxiv_id": "1101.0388",
"language": "en",
"url": "https://arxiv.org/abs/1101.0388",
"abstract": "We study a family of identities regarding a divisibility property of the Kostant partition function which first appeared in a paper of Baldoni and Vergne. T... |
https://arxiv.org/abs/2002.07555 | Convergence analysis of multi-level spectral deferred corrections | The spectral deferred correction (SDC) method is class of iterative solvers for ordinary differential equations (ODEs). It can be interpreted as a preconditioned Picard iteration for the collocation problem. The convergence of this method is well-known, for suitable problems it gains one order per iteration up to the o... | \section{Numerical Results}
\label{sec:examples}
In this section, the convergence behavior of MLSDC, theoretically analyzed in the previous section, is verified by numerical examples. The method is applied to three different initial value problems and the results are compared to those from classical, single-level SDC... | {
"timestamp": "2020-08-17T02:00:52",
"yymm": "2002",
"arxiv_id": "2002.07555",
"language": "en",
"url": "https://arxiv.org/abs/2002.07555",
"abstract": "The spectral deferred correction (SDC) method is class of iterative solvers for ordinary differential equations (ODEs). It can be interpreted as a precond... |
https://arxiv.org/abs/1805.00343 | Teaching Differentiation: A Rare Case for the Problem of the Slope of the Tangent Line | In this article we discuss an important students' misconception about derivatives, that the expression of the derivative of the function contains the information as to whether the function is differentiable or not where the expression is undefined. As a working example we consider a typical Calculus problem of finding ... | \section{Introduction}
Derivate is one of the most important topics not only in mathematics, but also in physics, chemistry, economics and engineering. Every standard Calculus course provides a variety of exercises for the students to learn how to apply the concept of derivative. The types of problems range from findi... | {
"timestamp": "2018-05-02T02:09:26",
"yymm": "1805",
"arxiv_id": "1805.00343",
"language": "en",
"url": "https://arxiv.org/abs/1805.00343",
"abstract": "In this article we discuss an important students' misconception about derivatives, that the expression of the derivative of the function contains the info... |
https://arxiv.org/abs/1809.04072 | The 21 Card Trick and its Generalization | The 21 card trick is well known. It was recently shown in an episode of the popular YouTube channel Numberphile. In that trick, the audience is asked to remember a card, and through a series of steps, the magician is able to find the card. In this article, we look into the mathematics behind the trick, and look at a co... | \section{Introduction to the 21 card trick}
The \textbf{21 card trick} (21CT) is a very popular card trick. It was also recently shown in an episode of the popular YouTube channel \href{https://www.youtube.com/watch?v=d7dg7gVDWyg}{Numberphile}. We first explain how this trick is performed in a series of steps. For the... | {
"timestamp": "2018-10-22T02:11:28",
"yymm": "1809",
"arxiv_id": "1809.04072",
"language": "en",
"url": "https://arxiv.org/abs/1809.04072",
"abstract": "The 21 card trick is well known. It was recently shown in an episode of the popular YouTube channel Numberphile. In that trick, the audience is asked to r... |
https://arxiv.org/abs/2007.14389 | Asymptotic behaviour of minimal complements | The notion of minimal complements was introduced by Nathanson in 2011 as a natural group-theoretic analogue of the metric concept of nets. Given two non-empty subsets $W,W'$ in a group $G$, the set $W'$ is said to be a complement to $W$ if $W\cdot W'=G$ and it is minimal if no proper subset of $W'$ is a complement to $... | \section{Introduction}
\subsection{Motivation}
Let $A, B$ be non-empty subsets in a group $G$. The set $A$ is said to be a left (resp. right) complement to $B$ if $A \cdot B = G$ (resp. $B\cdot A = G$). The set $A$ is a minimal left complement to $B$ if $$A \cdot B = G \textnormal{ and } (A\setminus \lbrace a\rbrace )... | {
"timestamp": "2020-07-29T02:23:49",
"yymm": "2007",
"arxiv_id": "2007.14389",
"language": "en",
"url": "https://arxiv.org/abs/2007.14389",
"abstract": "The notion of minimal complements was introduced by Nathanson in 2011 as a natural group-theoretic analogue of the metric concept of nets. Given two non-e... |
https://arxiv.org/abs/1504.07156 | On consecutive sums in permutations | We study the number of values taken by the sums $\sum_{i=u}^{v-1} a_i$, where $a_1,a_2,\dots,a_n$ is a permutation of $1,2,\dots,n$ and $1 \leq u < v \leq n+1$. In particular, we show that for a random choice of a permutation, with high probability there are $(\frac{1+e^{-2}}{4} +o(1)) n^2$ such sums. This answers an o... | \section{Introduction}
For a sequence $a = (a_i)_{i=1}^n $ with $a_i \in \mathbb{Z}$, let $S(a)$ denote the set of all distinct sums $\sum_{i=u}^v a_i$ with $1 \leq u \leq v \leq n$. We shall mostly be interested in the size of $S(a)$, where $a_i$ is a permutation of the set $[n] := \{1,2,\dots,n\}$. A trivial upper... | {
"timestamp": "2015-05-21T02:00:46",
"yymm": "1504",
"arxiv_id": "1504.07156",
"language": "en",
"url": "https://arxiv.org/abs/1504.07156",
"abstract": "We study the number of values taken by the sums $\\sum_{i=u}^{v-1} a_i$, where $a_1,a_2,\\dots,a_n$ is a permutation of $1,2,\\dots,n$ and $1 \\leq u < v ... |
https://arxiv.org/abs/1608.07247 | Minimal number of points on a grid forming line segments of equal length | We consider the minimal number of points on a regular grid on the plane that generates $n$ line segments of points of exactly length $k$. We illustrate how this is related to the $n$-queens problem on the toroidal chessboard and show that this number is upper bounded by $kn/3$ and approaches $kn/4$ as $n\rightarrow\inf... | \section{Introduction}
We consider points on a regular grid on the plane which form horizontal, vertical or diagonal blocks of exactly $k$ points (which we will call {\em patterns})\footnote{We use the convention that an isolated point corresponds to $4$ patterns of length $1$; a horizontal, a vertical and 2 diagonal ... | {
"timestamp": "2017-07-20T02:01:08",
"yymm": "1608",
"arxiv_id": "1608.07247",
"language": "en",
"url": "https://arxiv.org/abs/1608.07247",
"abstract": "We consider the minimal number of points on a regular grid on the plane that generates $n$ line segments of points of exactly length $k$. We illustrate ho... |
https://arxiv.org/abs/1901.07685 | A Reider-type Result for Smooth Projective Toric Surfaces | Let $L$ be an ample line bundle over a smooth projective toric surface $X$. Then $L$ corresponds to a very ample lattice polytope $P$ that encodes many geometric properties of $L$. In this article, by studying $P$, we will give some necessary and sufficient numerical criteria for the adjoint series $|K_X+L|$ to be eith... | \section{Introduction}
The problem of determining whether a line bundle is nef or (very) ample is an important question in algebraic geometry. The Nakai-Moishezon criterion \cite{Nakai1963, Moishezon1964} states that a Cartier divisor $D$ on a proper scheme $X$ over an algebraically closed field is ample if and only ... | {
"timestamp": "2019-01-24T02:06:09",
"yymm": "1901",
"arxiv_id": "1901.07685",
"language": "en",
"url": "https://arxiv.org/abs/1901.07685",
"abstract": "Let $L$ be an ample line bundle over a smooth projective toric surface $X$. Then $L$ corresponds to a very ample lattice polytope $P$ that encodes many ge... |
https://arxiv.org/abs/2111.03191 | A note on using the mass matrix as a preconditioner for the Poisson equation | We show that the mass matrix derived from finite elements can be effectively used as a preconditioner for iteratively solving the linear system arising from finite-difference discretization of the Poisson equation, using the conjugate gradient method. We derive analytically the condition number of the preconditioned op... | \section{Introduction}
\label{sec:intro}
Consider a standard finite difference discretization of the Poisson equation in one, two and three dimensions:
\begin{equation}
-\Delta u = f
\label{eq:poisson}
\end{equation}
on a simple domain $\Omega$, e.g., the unit interval, square or cube respectively, and subject to ... | {
"timestamp": "2021-11-08T02:06:06",
"yymm": "2111",
"arxiv_id": "2111.03191",
"language": "en",
"url": "https://arxiv.org/abs/2111.03191",
"abstract": "We show that the mass matrix derived from finite elements can be effectively used as a preconditioner for iteratively solving the linear system arising fr... |
https://arxiv.org/abs/1403.3431 | Minimal TSP Tour is coNP-Complete | The problem of deciding if a Traveling Salesman Problem (TSP) tour is minimal was proved to be coNP-complete by Papadimitriou and Steiglitz. We give an alternative proof based on a polynomial time reduction from 3SAT. Like the original proof, our reduction also shows that given a graph $G$ and an Hamiltonian path of $G... | \section{Introduction}
The \emph{Traveling Salesman Problem} (TSP{}) is a well--known problem from
graph theory \cite{PapadimitriouComplexity},\cite{GJ}: we are given $n$
cities and a nonnegative integer distance $d_{ij}$ between
any two cities $i$ and $j$ (assume that the distances are symmetric, i.e.
for all $i,j, ... | {
"timestamp": "2014-03-24T01:06:10",
"yymm": "1403",
"arxiv_id": "1403.3431",
"language": "en",
"url": "https://arxiv.org/abs/1403.3431",
"abstract": "The problem of deciding if a Traveling Salesman Problem (TSP) tour is minimal was proved to be coNP-complete by Papadimitriou and Steiglitz. We give an alte... |
https://arxiv.org/abs/1905.05312 | Books versus triangles at the extremal density | A celebrated result of Mantel shows that every graph on $n$ vertices with $\lfloor n^2/4 \rfloor + 1$ edges must contain a triangle. A robust version of this result, due to Rademacher, says that there must in fact be at least $\lfloor n/2 \rfloor$ triangles in any such graph. Another strengthening, due to the combined ... | \section{Introduction}
Mantel's theorem \cite{Man} from 1907 is among the earliest results in extremal graph theory. It states that the maximum number of edges that a triangle-free graph on $n$ vertices can have is $\lfloor n^2/4 \rfloor$, with equality if and only if the graph is the balanced complete bipartite graph... | {
"timestamp": "2019-10-22T02:18:05",
"yymm": "1905",
"arxiv_id": "1905.05312",
"language": "en",
"url": "https://arxiv.org/abs/1905.05312",
"abstract": "A celebrated result of Mantel shows that every graph on $n$ vertices with $\\lfloor n^2/4 \\rfloor + 1$ edges must contain a triangle. A robust version of... |
https://arxiv.org/abs/math/0412443 | Minimum Perimeter Rectangles That Enclose Congruent Non-Overlapping Circles | We use computational experiments to find the rectangles of minimum perimeter into which a given number n of non-overlapping congruent circles can be packed. No assumption is made on the shape of the rectangles. In many of the packings found, the circles form the usual regular square-grid or hexagonal patterns or their ... | \section{\@startsection {section}{1}{\z@}{-3.5ex plus -1ex minus
-.2ex}{2.3ex plus .2ex}{\normalsize\bf}}
\begin{document}
\title{Minimum Perimeter Rectangles That Enclose\\
Congruent Non-Overlapping Circles}
\date{}
\maketitle
\begin{center}
\author{Boris D. Lubachevsky \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \... | {
"timestamp": "2008-05-30T11:34:11",
"yymm": "0412",
"arxiv_id": "math/0412443",
"language": "en",
"url": "https://arxiv.org/abs/math/0412443",
"abstract": "We use computational experiments to find the rectangles of minimum perimeter into which a given number n of non-overlapping congruent circles can be p... |
https://arxiv.org/abs/1412.7702 | Convergence from below suffices | An elementary application of Fatou's lemma gives a strengthened version of the monotone convergence theorem. We call this the convergence from below theorem. We make the case that this result should be better known, and deserves a place in any introductory course on measure and integration. | \section{The convergence from below theorem}
Three famous convergence-related results appear in most introductory courses on measure and integration:
the monotone convergence theorem, Fatou's lemma and the dominated convergence theorem.
In teaching this material it is common to follow the approach taken in, for example... | {
"timestamp": "2014-12-25T02:09:06",
"yymm": "1412",
"arxiv_id": "1412.7702",
"language": "en",
"url": "https://arxiv.org/abs/1412.7702",
"abstract": "An elementary application of Fatou's lemma gives a strengthened version of the monotone convergence theorem. We call this the convergence from below theorem... |
https://arxiv.org/abs/1911.09792 | Minority Voter Distributions and Partisan Gerrymandering | Many people believe that it is disadvantageous for members aligning with a minority party to cluster in cities, as this makes it easier for the majority party to gerrymander district boundaries to diminish the representation of the minority. We examine this effect by exhaustively computing the average representation fo... |
\section{Introduction and Motivation}
\input{sections/intro.tex}
\section{Basic Definitions and Terminology}
\input{sections/definitions.tex}
\section{Grid}
\input{sections/grid.tex}
\newpage
\section{Metaheuristics for Optimization}
\input{sections/metaheuristics.tex}
\section{Conclusion and Further ... | {
"timestamp": "2019-11-25T02:05:50",
"yymm": "1911",
"arxiv_id": "1911.09792",
"language": "en",
"url": "https://arxiv.org/abs/1911.09792",
"abstract": "Many people believe that it is disadvantageous for members aligning with a minority party to cluster in cities, as this makes it easier for the majority p... |
https://arxiv.org/abs/1906.04970 | Unified Viscous-to-inertial Scaling in Liquid Droplet Coalescence | This letter presents a theory on the coalescence of two spherical liquid droplets that are initially stationary. The evolution of the radius of a liquid neck formed upon coalescence was formulated as an initial value problem and then solved to yield an exact solution without free parameters, with its two asymptotic app... | \section{\label{sec:level1}First-level heading}
Droplet Coalescence \cite{EggersJ:99a,AartsDGAL:05a,Thoroddsen:05a,PaulsenJD:11a,Xia:17a} is a ubiquitous phenomenon involving impact or contact of dispersed two-phase flows \cite{YarinAL:06a,ZhangP:11a,ThoravalMJ:12a,TranT:13a,KavehpourHP:15a,ZhangP:16a,Xia:19a}. Among ... | {
"timestamp": "2019-06-13T02:07:41",
"yymm": "1906",
"arxiv_id": "1906.04970",
"language": "en",
"url": "https://arxiv.org/abs/1906.04970",
"abstract": "This letter presents a theory on the coalescence of two spherical liquid droplets that are initially stationary. The evolution of the radius of a liquid n... |
https://arxiv.org/abs/1901.05369 | Improving linear quantile regression for replicated data | This paper deals with improvement of linear quantile regression, when there are a few distinct values of the covariates but many replicates. On can improve asymptotic efficiency of the estimated regression coefficients by using suitable weights in quantile regression, or simply by using weighted least squares regressio... | \section{Introduction}
Consider a quantile regression problem with a handful of distinct values of covariates, where each covariate profile is replicated many times. A linear regression model for the quantiles are often preferred for such data. If one ignores the fact of replications, the linear quantile regression es... | {
"timestamp": "2019-01-17T02:16:59",
"yymm": "1901",
"arxiv_id": "1901.05369",
"language": "en",
"url": "https://arxiv.org/abs/1901.05369",
"abstract": "This paper deals with improvement of linear quantile regression, when there are a few distinct values of the covariates but many replicates. On can improv... |
https://arxiv.org/abs/1806.07518 | Eakin-Sathaye type theorems for joint reductions and good filtrations of ideals | Analogues of Eakin-Sathaye theorem for reductions of ideals are proved for ${\mathbb N}^s$-graded good filtrations. These analogues yield bounds on joint reduction vectors for a family of ideals and reduction numbers for $\mathbb N$-graded filtrations. Several examples related to lex-segment ideals, contracted ideals i... | \section{Introduction}
The objective of this paper is to prove Eakin-Sathaye type theorems \cite{ES1976} for joint reductions and good filtrations of ideals. Recall that an ideal $J$ contained in an ideal $I$ in a commutative ring $R$ is called a reduction of $I$ if there is a non-negative integer $n$ such that $JI^n=... | {
"timestamp": "2019-10-10T02:07:23",
"yymm": "1806",
"arxiv_id": "1806.07518",
"language": "en",
"url": "https://arxiv.org/abs/1806.07518",
"abstract": "Analogues of Eakin-Sathaye theorem for reductions of ideals are proved for ${\\mathbb N}^s$-graded good filtrations. These analogues yield bounds on joint... |
https://arxiv.org/abs/2105.10116 | Approximate analytical solution for transient heat and mass transfer across an irregular interface | Motivated by practical applications in heat conduction and contaminant transport, we consider heat and mass diffusion across a perturbed interface separating two finite regions of distinct diffusivity. Under the assumption of continuity of the solution and diffusive flux at the interface, we use perturbation theory to ... | \section{Introduction}
\noindent The diffusion equation is fundamental to applied mathematics with numerous applications in engineering and the physical and life sciences \cite{liu_1998,pontrelli_2020,simpson_2017,zhao_2016,mantzavinos_2016}. Most practical applications of the diffusion equation involve a complex heter... | {
"timestamp": "2022-01-12T02:05:55",
"yymm": "2105",
"arxiv_id": "2105.10116",
"language": "en",
"url": "https://arxiv.org/abs/2105.10116",
"abstract": "Motivated by practical applications in heat conduction and contaminant transport, we consider heat and mass diffusion across a perturbed interface separat... |
https://arxiv.org/abs/1808.06569 | Splitter Theorems for Graph Immersions | We establish splitter theorems for graph immersions for two families of graphs, $k$-edge-connected graphs, with $k$ even, and 3-edge-connected, internally 4-edge-connected graphs. As a corollary, we prove that every $3$-edge-connected, internally $4$-edge-connected graph on at least seven vertices that immerses $K_5$ a... | \section{Introduction}
\label{intro}
Throughout the paper, we use standard definitions and notation for graphs as in \cite{MR1367739}. Let $G$ be a graph with a certain connectivity. One natural question is that whether there is a way to ``reduce" $G$ while preserving the same connectivity, and possibly also the presen... | {
"timestamp": "2018-08-21T02:19:21",
"yymm": "1808",
"arxiv_id": "1808.06569",
"language": "en",
"url": "https://arxiv.org/abs/1808.06569",
"abstract": "We establish splitter theorems for graph immersions for two families of graphs, $k$-edge-connected graphs, with $k$ even, and 3-edge-connected, internally... |
https://arxiv.org/abs/1909.06882 | Lagrange interpolation over division rings | For a division ring $\mathbb F$, the polynomials $f\in\mathbb F$ can be evaluated "on the left" and "on the right" giving rise to left and right Lagrange interpolation problems. The problems containig interpolation conditions of the same type were considered in \cite{lam1} where the solvability criterion was given in t... | \section{Introduction}
\setcounter{equation}{0}
Given distinct nodes $\alpha_1,\ldots,\alpha_n$ and target values $c_1,\ldots,c_n$ in a field $\mathbb F$,
the classical Lagrange interpolation problem consists of finding a polynomial $f\in\mathbb F[z]$ such that
\begin{equation}
f(\alpha_i)=c_i\quad\mbox{for}\quad i=1... | {
"timestamp": "2019-09-17T02:18:49",
"yymm": "1909",
"arxiv_id": "1909.06882",
"language": "en",
"url": "https://arxiv.org/abs/1909.06882",
"abstract": "For a division ring $\\mathbb F$, the polynomials $f\\in\\mathbb F$ can be evaluated \"on the left\" and \"on the right\" giving rise to left and right La... |
https://arxiv.org/abs/0810.1378 | Long-time behaviour of discretizations of the simple pendulum equation | We compare the performance of several discretizations of the simple pendulum equation in a series of numerical experiments. The stress is put on the long-time behaviour. We choose for the comparison numerical schemes which preserve the qualitative features of solutions (like periodicity). All these schemes are either s... | \section{Introduction}
New but more and more important direction in the numerical analysis is
geometric numerical integration \cite{HLW,Is,IZ,MQ}.
Numerical methods within this approach are
tailored for specific equations rather than for large general classes of equations.
The aim is to preserve qualitative features, ... | {
"timestamp": "2008-10-08T10:20:53",
"yymm": "0810",
"arxiv_id": "0810.1378",
"language": "en",
"url": "https://arxiv.org/abs/0810.1378",
"abstract": "We compare the performance of several discretizations of the simple pendulum equation in a series of numerical experiments. The stress is put on the long-ti... |
https://arxiv.org/abs/1311.7441 | Almost involutive Hopf algebras | We define the concept of \emph{companion automorphism} of a Hopf algebra $H$ as an automorphism $\sigma:H \rightarrow H$: $\sigma^2=S^2$ --where $S$ denotes the antipode--. A Hopf algebra is said to be \emph{almost involutive} (AI) if it admits a companion automorphism that can be viewed as a special additional symmetr... | \section{Introduction.}
We start by summarizing the contents of this paper.
\bigskip
\noindent
In Section \ref{section:defejemplos} we introduce the definition of an \emph{almost involutive Hopf algebra}, show that
Sweedler's Hopf algebra is an example and observe that compact quantum groups are also examples --in ... | {
"timestamp": "2013-12-02T02:12:27",
"yymm": "1311",
"arxiv_id": "1311.7441",
"language": "en",
"url": "https://arxiv.org/abs/1311.7441",
"abstract": "We define the concept of \\emph{companion automorphism} of a Hopf algebra $H$ as an automorphism $\\sigma:H \\rightarrow H$: $\\sigma^2=S^2$ --where $S$ den... |
https://arxiv.org/abs/2008.09405 | A Tipping Point for the Planarity of Small and Medium Sized Graphs | This paper presents an empirical study of the relationship between the density of small-medium sized random graphs and their planarity. It is well known that, when the number of vertices tends to infinite, there is a sharp transition between planarity and non-planarity for edge density d=0.5. However, this asymptotic p... | \section{Introduction}\label{se:intro}
Several popular Graph Drawing algorithms devised to draw graphs of small-medium size assume that the graph to be drawn is planar both in the static setting~\cite{t-hdg-63,fpp-hdpgg-90,s-epgg-90} and in the dynamic one~\cite{10.1007/s00454-018-0018-9,ddfpr-upm-tcs-20,bdfp-gssa-j-2... | {
"timestamp": "2020-08-27T02:01:24",
"yymm": "2008",
"arxiv_id": "2008.09405",
"language": "en",
"url": "https://arxiv.org/abs/2008.09405",
"abstract": "This paper presents an empirical study of the relationship between the density of small-medium sized random graphs and their planarity. It is well known t... |
https://arxiv.org/abs/1401.4969 | Local and Parallel Finite Element Algorithm Based On Multilevel Discretization for Eigenvalue Problem | A local and parallel algorithm based on the multilevel discretization is proposed in this paper to solve the eigenvalue problem by the finite element method. With this new scheme, solving the eigenvalue problem in the finest grid is transferred to solutions of the eigenvalue problems on the coarsest mesh and a series o... | \section{Introduction}
Solving large scale eigenvalue problems becomes a fundamental problem in modern science and engineering society.
However, it is always a very difficult task to solve high-dimensional eigenvalue problems which come from
physical and chemistry sciences. Xu and Zhou \cite{XuZhou_Eigen} give a type... | {
"timestamp": "2014-01-21T02:16:15",
"yymm": "1401",
"arxiv_id": "1401.4969",
"language": "en",
"url": "https://arxiv.org/abs/1401.4969",
"abstract": "A local and parallel algorithm based on the multilevel discretization is proposed in this paper to solve the eigenvalue problem by the finite element method... |
https://arxiv.org/abs/1301.4157 | On the Product Rule for Classification Problems | We discuss theoretical aspects of the product rule for classification problems in supervised machine learning for the case of combining classifiers. We show that (1) the product rule arises from the MAP classifier supposing equivalent priors and conditional independence given a class; (2) under some conditions, the pro... | \section{Introduction}\label{sec:introduction}
\vspace{-0.25cm}
With the advance of the Machine Learning field, and the discovery
of many different techniques,
the subject of \emph{combining multiple learners} \cite{Alpaydin2004} eventually
drove attention, in particular the problem of \emph{combining classifiers}.
Man... | {
"timestamp": "2013-01-18T02:02:51",
"yymm": "1301",
"arxiv_id": "1301.4157",
"language": "en",
"url": "https://arxiv.org/abs/1301.4157",
"abstract": "We discuss theoretical aspects of the product rule for classification problems in supervised machine learning for the case of combining classifiers. We show... |
https://arxiv.org/abs/0712.0864 | A New Error Bound for Shifted Surface Spline Interpolation | A New Error Bound for shifted surface spline interpolation is presented. This error bound probably is the most powerful one up to now. | \section{Introduction}
In the theory of radial basis functions, it's well known that any conditionally positive definite radial function can form an interpolant for any set of scattered data. We make a simple sketch of this process as follows.
Suppose $h$ is a continuous function on $R^{n}$ which is strictly ... | {
"timestamp": "2007-12-06T03:22:54",
"yymm": "0712",
"arxiv_id": "0712.0864",
"language": "en",
"url": "https://arxiv.org/abs/0712.0864",
"abstract": "A New Error Bound for shifted surface spline interpolation is presented. This error bound probably is the most powerful one up to now.",
"subjects": "Nume... |
https://arxiv.org/abs/2301.10803 | Evaluating Probabilistic Classifiers: The Triptych | Probability forecasts for binary outcomes, often referred to as probabilistic classifiers or confidence scores, are ubiquitous in science and society, and methods for evaluating and comparing them are in great demand. We propose and study a triptych of diagnostic graphics that focus on distinct and complementary aspect... | \section{Introduction} \label{sec:introduction}
Across science and society, probability forecasts for the occurrence of a binary outcome, also referred to as probabilistic classifiers or confidence scores, are widely used. Prominent examples include a patient's recovery or survival, weather events, solar flares, the... | {
"timestamp": "2023-01-27T02:00:48",
"yymm": "2301",
"arxiv_id": "2301.10803",
"language": "en",
"url": "https://arxiv.org/abs/2301.10803",
"abstract": "Probability forecasts for binary outcomes, often referred to as probabilistic classifiers or confidence scores, are ubiquitous in science and society, and... |
https://arxiv.org/abs/2103.11893 | Thresholding Greedy Pursuit for Sparse Recovery Problems | We study here sparse recovery problems in the presence of additive noise. We analyze a thresholding version of the CoSaMP algorithm, named Thresholding Greedy Pursuit (TGP). We demonstrate that an appropriate choice of thresholding parameter, even without the knowledge of sparsity level of the signal and strength of th... | \section{Introduction} \label{section1}
In this section, we introduce our algorithm and associated theorems for theoretical guarantees. We also give an overview of sparse recovery algorithms and related literatures.
\subsection{Sparsity Promoting Optimization}
We are interested in finding sparse \textit{signals} $\... | {
"timestamp": "2021-03-23T01:38:14",
"yymm": "2103",
"arxiv_id": "2103.11893",
"language": "en",
"url": "https://arxiv.org/abs/2103.11893",
"abstract": "We study here sparse recovery problems in the presence of additive noise. We analyze a thresholding version of the CoSaMP algorithm, named Thresholding Gr... |
https://arxiv.org/abs/2004.04593 | The Importance of Good Starting Solutions in the Minimum Sum of Squares Clustering Problem | The clustering problem has many applications in Machine Learning, Operations Research, and Statistics. We propose three algorithms to create starting solutions for improvement algorithms for this problem. We test the algorithms on 72 instances that were investigated in the literature. Forty eight of them are relatively... | \section{Introduction}
{\color{black}We present three new approaches
that may be used to generate good starting solutions to clustering problems and then we} show that these
starting solutions vastly improve the performance of popular local searches (improvement algorithms)
used in clustering such as k-means \ci... | {
"timestamp": "2020-04-10T02:14:48",
"yymm": "2004",
"arxiv_id": "2004.04593",
"language": "en",
"url": "https://arxiv.org/abs/2004.04593",
"abstract": "The clustering problem has many applications in Machine Learning, Operations Research, and Statistics. We propose three algorithms to create starting solu... |
https://arxiv.org/abs/1908.10962 | Optimal transport mapping via input convex neural networks | In this paper, we present a novel and principled approach to learn the optimal transport between two distributions, from samples. Guided by the optimal transport theory, we learn the optimal Kantorovich potential which induces the optimal transport map. This involves learning two convex functions, by solving a novel mi... | \subsection{High dimensional experiments}
\label{sec:high-dim}
\input{figure_high_dim.tex}
We consider the challenging task of learning optimal transport maps on high dimensional distributions. In particular, we consider both synthetic and real world high dimensional datasets and provide quantitative and qualitative... | {
"timestamp": "2020-06-19T02:02:30",
"yymm": "1908",
"arxiv_id": "1908.10962",
"language": "en",
"url": "https://arxiv.org/abs/1908.10962",
"abstract": "In this paper, we present a novel and principled approach to learn the optimal transport between two distributions, from samples. Guided by the optimal tr... |
https://arxiv.org/abs/1709.09764 | Rigidity of tilting modules in category O | In this note we give an overview of rigidity properties and Loewy lengths of tilting modules in the BGG category O associated to a reductive Lie algebra. These results are well-known by several specialists, but seem difficult to find in the existing literature. | \section*{Introduction}
In highest weight categories, tilting modules are the modules which have both a standard and costandard filtration, see \cite{Ringel}. In category $\mathcal O$ specifically, they are the self-dual modules with Verma filtration, see \cite{CI, ES}.
In each of the three `classical' Lie-theoretic s... | {
"timestamp": "2017-10-11T02:02:05",
"yymm": "1709",
"arxiv_id": "1709.09764",
"language": "en",
"url": "https://arxiv.org/abs/1709.09764",
"abstract": "In this note we give an overview of rigidity properties and Loewy lengths of tilting modules in the BGG category O associated to a reductive Lie algebra. ... |
https://arxiv.org/abs/1910.11442 | The thresholding scheme for mean curvature flow and de Giorgi's ideas for minimizing movements | We consider the thresholding scheme and explore its connection to De Giorgi's ideas on gradient flows in metric spaces; here applied to mean curvature flow as the steepest descent of the interfacial area.The basis of our analysis is the observation by Esedoglu and the second author that thresholding can be interpreted ... | \section{Introduction and context}
The purpose of these notes is to draw a connection between De Giorgi's tools for
minimizing movements, that is, gradient flows in metric spaces on the one
hand, and the very popular thresholding scheme for flow of a hyper-surface by its mean curvature
on the other hand.
While we have... | {
"timestamp": "2019-10-28T01:04:29",
"yymm": "1910",
"arxiv_id": "1910.11442",
"language": "en",
"url": "https://arxiv.org/abs/1910.11442",
"abstract": "We consider the thresholding scheme and explore its connection to De Giorgi's ideas on gradient flows in metric spaces; here applied to mean curvature flo... |
https://arxiv.org/abs/1108.2452 | Sequential Auctions and Externalities | In many settings agents participate in multiple different auctions that are not necessarily implemented simultaneously. Future opportunities affect strategic considerations of the players in each auction, introducing externalities. Motivated by this consideration, we study a setting of a market of buyers and sellers, w... | \section{Iterated elimination of dominated
strategies}\label{appendix:refinements}
First we define precisely the concept of a strategy profile that survives
iterated elimination of weakly dominated strategies. Then we characterize such
profiles for the first-price auction with externalities using a
graph-theoretical a... | {
"timestamp": "2011-12-30T02:04:53",
"yymm": "1108",
"arxiv_id": "1108.2452",
"language": "en",
"url": "https://arxiv.org/abs/1108.2452",
"abstract": "In many settings agents participate in multiple different auctions that are not necessarily implemented simultaneously. Future opportunities affect strategi... |
https://arxiv.org/abs/q-alg/9605025 | Quantum Lie algebras; their existence, uniqueness and $q$-antisymmetry | Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules of the quantized enveloping algebras. On them the quantum Lie bracket is given by the quantum adjoint action.Here we define for any finite-dimen... | \section{Introduction}
Lie algebras play an important role in the description of many classical
physical theories. This is particularly pronounced in integrable
models which are described entirely in terms of Lie algebraic data.
However, when quantizing a classical theory the Lie algebraic
description seems to be dest... | {
"timestamp": "1996-10-10T21:36:38",
"yymm": "9605",
"arxiv_id": "q-alg/9605025",
"language": "en",
"url": "https://arxiv.org/abs/q-alg/9605025",
"abstract": "Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concr... |
https://arxiv.org/abs/1608.01305 | Elephant Random Walks and their connection to Pólya-type urns | In this paper, we explain the connection between the Elephant Random Walk (ERW) and an urn model à la Pólya and derive functional limit theorems for the former. The ERW model was introduced by Schütz and Trimper [2004] to study memory effects in a one-dimensional discrete-time random walk with a complete memory of its ... | \section{Introduction}
Random walks and, more generally, diffusion processes are widely used in
theoretical physics to describe phenomena of traveling motion and mass
transport. Due to the fractal structure of nature and space and temporal
long-range correlations in particle movements (see,
e.g.,~\cite{Ma,MaNe,MeKl,ScM... | {
"timestamp": "2016-09-07T02:00:43",
"yymm": "1608",
"arxiv_id": "1608.01305",
"language": "en",
"url": "https://arxiv.org/abs/1608.01305",
"abstract": "In this paper, we explain the connection between the Elephant Random Walk (ERW) and an urn model à la Pólya and derive functional limit theorems for the f... |
https://arxiv.org/abs/1703.04520 | Lipschitz Normal Embeddings in the Space of Matrices | The germ of an algebraic variety is naturally equipped with two different metrics up to bilipschitz equivalence. The inner metric and the outer metric. One calls a germ of a variety Lipschitz normally embedded if the two metrics are bilipschitz equivalent. In this article we prove Lipschitz normal embeddedness of some ... | \section{Introduction}
If $(X,0)$ is the germ of an algebraic (analytic) variety over $\mathbbm{K} =
\mathbb{R}$ or $\mathbb{C}$, then one
can define two natural metrics on it. Both are defined by choosing an
embedding of $(X,0)$ into $(\mathbbm{K}^N,0)$. The first is the \emph{outer
metric}, where the distance betw... | {
"timestamp": "2017-03-14T01:15:20",
"yymm": "1703",
"arxiv_id": "1703.04520",
"language": "en",
"url": "https://arxiv.org/abs/1703.04520",
"abstract": "The germ of an algebraic variety is naturally equipped with two different metrics up to bilipschitz equivalence. The inner metric and the outer metric. On... |
https://arxiv.org/abs/1110.4896 | Strengthened Brooks Theorem for digraphs of girth three | Brooks' Theorem states that a connected graph $G$ of maximum degree $\Delta$ has chromatic number at most $\Delta$, unless $G$ is an odd cycle or a complete graph. A result of Johansson (1996) shows that if $G$ is triangle-free, then the chromatic number drops to $O(\Delta / \log \Delta)$. In this paper, we derive a we... | \section{Introduction}
Brooks' Theorem states that if $G$ is a connected graph with
maximum degree $\Delta$, then $\chi(G) \leq \Delta + 1$, where
equality is attained only for odd cycles and complete graphs. The
presence of triangles has significant influence on the chromatic
number of a graph. A result of Johansson ... | {
"timestamp": "2011-10-25T02:00:14",
"yymm": "1110",
"arxiv_id": "1110.4896",
"language": "en",
"url": "https://arxiv.org/abs/1110.4896",
"abstract": "Brooks' Theorem states that a connected graph $G$ of maximum degree $\\Delta$ has chromatic number at most $\\Delta$, unless $G$ is an odd cycle or a comple... |
https://arxiv.org/abs/2201.09691 | Multidimensional Manhattan Preferences | A preference profile with $m$ alternatives and $n$ voters is $d$-Manhattan (resp. $d$-Euclidean) if both the alternatives and the voters can be placed into the $d$-dimensional space such that between each pair of alternatives, every voter prefers the one which has a shorter Manhattan (resp. Euclidean) distance to the v... | \section{Introduction}\label{sec:intro}
Modelling voters' linear preferences as geometric distances is an approach popular in many research fields such as economics~\cite{Hotelling1929,Downs1957,Eguia2011},
political and social sciences~\cite{Sto1963,Poo1989,Ene1990,BoLa2006},
and psychology~\cite{Coombs1964,BorGroMai... | {
"timestamp": "2022-01-25T02:41:04",
"yymm": "2201",
"arxiv_id": "2201.09691",
"language": "en",
"url": "https://arxiv.org/abs/2201.09691",
"abstract": "A preference profile with $m$ alternatives and $n$ voters is $d$-Manhattan (resp. $d$-Euclidean) if both the alternatives and the voters can be placed int... |
https://arxiv.org/abs/1011.2991 | Parity conjectures for elliptic curves over global fields of positive characteristic | We prove the $p$-parity conjecture for elliptic curves over global fields of characteristic $p > 3$. We also present partial results on the $\ell$-parity conjecture for primes $\ell \neq p$. | \section{Introduction}
Let $K$ be a global field and let $E$ be an elliptic curve defined over $K$. The conjecture of Birch and Swinnerton-Dyer asserts that the rank of the Mordell-Weil group $E(K)$ is equal to the order of vanishing of the Hasse-Weil $L$-function $L(E/K,s)$ as $s=1$. A weaker question is to know whet... | {
"timestamp": "2010-12-15T02:02:30",
"yymm": "1011",
"arxiv_id": "1011.2991",
"language": "en",
"url": "https://arxiv.org/abs/1011.2991",
"abstract": "We prove the $p$-parity conjecture for elliptic curves over global fields of characteristic $p > 3$. We also present partial results on the $\\ell$-parity c... |
https://arxiv.org/abs/0804.0224 | Characterization of the critical values of branching random walks on weighted graphs through infinite-type branching processes | We study the branching random walk on weighted graphs; site-breeding and edge-breeding branching random walks on graphs are seen as particular cases. We describe the strong critical value in terms of a geometrical parameter of the graph. We characterize the weak critical value and relate it to another geometrical param... | \section{Introduction}\label{sec:intro}
\setcounter{equation}{0}
We consider
the branching random walk (briefly BRW) as a continuous-time process
where particles live on an at most countable set $X$ (the set of sites).
Each particle lives on a site and, independently of the others, has a random lifespan; during its l... | {
"timestamp": "2008-09-07T14:38:00",
"yymm": "0804",
"arxiv_id": "0804.0224",
"language": "en",
"url": "https://arxiv.org/abs/0804.0224",
"abstract": "We study the branching random walk on weighted graphs; site-breeding and edge-breeding branching random walks on graphs are seen as particular cases. We des... |
https://arxiv.org/abs/2112.14943 | Non-jumping Turán densities of hypergraphs | A real number $\alpha\in [0, 1)$ is a jump for an integer $r\ge 2$ if there exists $c>0$ such that no number in $(\alpha , \alpha + c)$ can be the Turán density of a family of $r$-uniform graphs. A classical result of Erd\H os and Stone \cite{ES} implies that that every number in $[0, 1)$ is a jump for $r=2$. Erd\H os ... | \section{Introduction}
For a finite set $V$ and a positive integer $r$ we denote by ${{V \choose r}}$ the family of all $r$-subsets of $V$. An {\em $r$-uniform graph} ({\em $r$-graph}) $G$ is a set $V(G)$ of vertices together with a set $E(G) \subseteq {V(G) \choose r}$ of edges. The {\em density} of $G$ is defined by ... | {
"timestamp": "2022-01-03T02:07:02",
"yymm": "2112",
"arxiv_id": "2112.14943",
"language": "en",
"url": "https://arxiv.org/abs/2112.14943",
"abstract": "A real number $\\alpha\\in [0, 1)$ is a jump for an integer $r\\ge 2$ if there exists $c>0$ such that no number in $(\\alpha , \\alpha + c)$ can be the Tu... |
https://arxiv.org/abs/1806.00079 | Open and closed random walks with fixed edgelengths in $\mathbb{R}^d$ | In this paper, we consider fixed edgelength $n$-step random walks in $\mathbb{R}^d$. We give an explicit construction for the closest closed equilateral random walk to almost any open equilateral random walk based on the geometric median, providing a natural map from open polygons to closed polygons of the same edgelen... | \section{Introduction}
Random walks in space with fixed edgelengths have been of interest to statistical physicists and chemists since Lord Rayleigh's day. These walks model polymers in solution (at least under $\theta$-solvent conditions)~\cite{Rayleigh:1919do,hughes1995random,FloryPaulJ1969Smoc} and are similarly int... | {
"timestamp": "2018-06-04T02:02:35",
"yymm": "1806",
"arxiv_id": "1806.00079",
"language": "en",
"url": "https://arxiv.org/abs/1806.00079",
"abstract": "In this paper, we consider fixed edgelength $n$-step random walks in $\\mathbb{R}^d$. We give an explicit construction for the closest closed equilateral ... |
https://arxiv.org/abs/2209.11517 | An order-theoretic perspective on modes and maximum a posteriori estimation in Bayesian inverse problems | It is often desirable to summarise a probability measure on a space $X$ in terms of a mode, or MAP estimator, i.e.\ a point of maximum probability. Such points can be rigorously defined using masses of metric balls in the small-radius limit. However, the theory is not entirely straightforward: the literature contains m... |
\section{Introduction}
\label{sec:introduction}
In diverse applications such as statistical inference and the analysis of transition paths of random dynamical systems it is desirable to summarise a complicated probability measure $\mu$ on a space $X$ by a single distinguished point $x^{\star} \in X$ that is, in s... | {
"timestamp": "2022-09-26T02:09:46",
"yymm": "2209",
"arxiv_id": "2209.11517",
"language": "en",
"url": "https://arxiv.org/abs/2209.11517",
"abstract": "It is often desirable to summarise a probability measure on a space $X$ in terms of a mode, or MAP estimator, i.e.\\ a point of maximum probability. Such ... |
https://arxiv.org/abs/2009.06486 | Oblique Derivative Problems for Elliptic Equations on Conical Domains | We study the oblique derivative problem for uniformly elliptic equations on cone domains. Under the assumption of axi-symmetry of the solution, we find sufficient conditions on the angle of the oblique vector for Hölder regularity of the gradient to hold up to the vertex of the cone. The proof of regularity is based on... | \section{Introduction}
The aim of this paper is to study the regularity up to the boundary for solutions of oblique derivative problems for uniformly elliptic equations on domains with conical singularities. As well as being of interest from the point of view of elliptic PDE theory, oblique derivative problems on co... | {
"timestamp": "2020-09-15T02:32:56",
"yymm": "2009",
"arxiv_id": "2009.06486",
"language": "en",
"url": "https://arxiv.org/abs/2009.06486",
"abstract": "We study the oblique derivative problem for uniformly elliptic equations on cone domains. Under the assumption of axi-symmetry of the solution, we find su... |
https://arxiv.org/abs/1611.09806 | Squarefree values of polynomial discriminants I | We determine the density of monic integer polynomials of given degree $n>1$ that have squarefree discriminant; in particular, we prove for the first time that the lower density of such polynomials is positive. Similarly, we prove that the density of monic integer polynomials $f(x)$, such that $f(x)$ is irreducible and ... | \section{Introduction}
The pupose of this paper is to determine the density of monic integer
polynomials of given degree whose discriminant is squarefree. For
polynomials $f(x)=x^n+a_1x^{n-1}+\cdots+a_n$, the term $(-1)^ia_i$
represents the sum of the $i$-fold products of the roots of $f$. It is
thus natural to order... | {
"timestamp": "2016-12-01T02:10:41",
"yymm": "1611",
"arxiv_id": "1611.09806",
"language": "en",
"url": "https://arxiv.org/abs/1611.09806",
"abstract": "We determine the density of monic integer polynomials of given degree $n>1$ that have squarefree discriminant; in particular, we prove for the first time ... |
https://arxiv.org/abs/2207.14098 | Convergence of iterates in nonlinear Perron-Frobenius theory | Let $C$ be a closed cone with nonempty interior $C^\circ$ in a Banach space. Let $f:C^\circ \rightarrow C^\circ$ be an order-preserving subhomogeneous function with a fixed point in $C^\circ$. We introduce a condition which guarantees that the iterates $f^k(x)$ converge to a fixed point for all $x \in C^\circ$. This co... | \section{Introduction}
Let $X$ be a real Banach space with norm $\| \cdot \|$. A \emph{closed cone} is a closed convex set $C \subset X$ such that (i) $\lambda C \subset C$ for all $\lambda \ge 0$ and (ii) $C \cap (-C) = \{0\}$. A closed cone $C$ induces the following partial order on $X$. We say that $x \le_C y$ wh... | {
"timestamp": "2022-07-29T02:21:24",
"yymm": "2207",
"arxiv_id": "2207.14098",
"language": "en",
"url": "https://arxiv.org/abs/2207.14098",
"abstract": "Let $C$ be a closed cone with nonempty interior $C^\\circ$ in a Banach space. Let $f:C^\\circ \\rightarrow C^\\circ$ be an order-preserving subhomogeneous... |
https://arxiv.org/abs/1111.0094 | Generalization of a few results in Integer Partitions | In this paper, we generalize a few important results in Integer Partitions; namely the results known as Stanley's theorem and Elder's theorem, and the congruence results proposed by Ramanujan for the partition function. We generalize the results of Stanley and Elder from a fixed integer to an array of subsequent intege... | \section{Introduction}
Partitioning a positive integer $n$ as sum of certain positive integers is a well known problem in the domain of number theory and combinatorics. A {\em partition} of a positive integer $n$ is any non-increasing sequence of positive integers that add up to $n$. The partition function $P(n)$ is de... | {
"timestamp": "2011-11-02T01:01:01",
"yymm": "1111",
"arxiv_id": "1111.0094",
"language": "en",
"url": "https://arxiv.org/abs/1111.0094",
"abstract": "In this paper, we generalize a few important results in Integer Partitions; namely the results known as Stanley's theorem and Elder's theorem, and the congr... |
https://arxiv.org/abs/1405.2844 | Covering Array Bounds Using Analytical Techniques | A $t$-covering array with entries from the alphabet ${\cal Q}=\{0,1,\ldots,q-1\}$ is a $k\times n$ stack, so that for any choice of $t$ (typically non-consecutive) columns, each of the $q^{t}$ possible $t$-letter words over ${\cal Q}$ appear at least once among the rows of the selected columns. We will show how a combi... | \section{Introduction}
A $t$-covering array with entries from the alphabet ${\cal Q}=\{0,1,\ldots,q-1\}$ is a $k\times n$ stack, so that for any choice of $t$ typically non-consecutive columns, each of the $q^{t}$ possible $t$-letter words over ${\cal Q}$ appear at least once among the rows of the selected columns... | {
"timestamp": "2014-05-13T02:18:20",
"yymm": "1405",
"arxiv_id": "1405.2844",
"language": "en",
"url": "https://arxiv.org/abs/1405.2844",
"abstract": "A $t$-covering array with entries from the alphabet ${\\cal Q}=\\{0,1,\\ldots,q-1\\}$ is a $k\\times n$ stack, so that for any choice of $t$ (typically non-... |
https://arxiv.org/abs/1804.09411 | Stable-Matching Voronoi Diagrams: Combinatorial Complexity and Algorithms | We study algorithms and combinatorial complexity bounds for \emph{stable-matching Voronoi diagrams}, where a set, $S$, of $n$ point sites in the plane determines a stable matching between the points in $\mathbb{R}^2$ and the sites in $S$ such that (i) the points prefer sites closer to them and sites prefer points close... | \section{Introduction}\label{smvd:sec:intro}
The \emph{Voronoi diagram} is a well-known geometric structure with a
broad spectrum of applications in computational geometry
and other areas of Computer Science, e.g.,
see~\cite{Aurenhammer:1991,bookAurenhammer,Brandt1992, Kise1998, Meguerdichian2001, Petrek2007, St... | {
"timestamp": "2020-02-11T02:29:09",
"yymm": "1804",
"arxiv_id": "1804.09411",
"language": "en",
"url": "https://arxiv.org/abs/1804.09411",
"abstract": "We study algorithms and combinatorial complexity bounds for \\emph{stable-matching Voronoi diagrams}, where a set, $S$, of $n$ point sites in the plane de... |
https://arxiv.org/abs/2205.10010 | Some identities on degenerate hyperharmonic numbers | The aim of this paper is to investigate some properties, recurrence relations and identities involving degenerate hyperharmonic numbers, hyperharmonic numbers and degenerate harmonic numbers. In particular, we derive an explicit expression of the degenerate hyperharmonic numbers in terms of the degenerate harmonic numb... | \section{Introduction}
Recent explorations for various degenerate versions of quite a few special numbers and polynomials have been fascinating and fruitful, which began with the pioneering work of Carlitz in [2,3].
These quests for degenerat versons are not only limited to special polynomials and numbers but also ex... | {
"timestamp": "2022-05-23T02:12:35",
"yymm": "2205",
"arxiv_id": "2205.10010",
"language": "en",
"url": "https://arxiv.org/abs/2205.10010",
"abstract": "The aim of this paper is to investigate some properties, recurrence relations and identities involving degenerate hyperharmonic numbers, hyperharmonic num... |
https://arxiv.org/abs/1607.02564 | Base-$b$ analogues of classic combinatorial objects | We study the properties of the base-$b$ binomial coefficient defined by Jiu and the second author, introduced in the context of a digital binomial theorem. After introducing a general summation formula, we derive base-$b$ analogues of the Stirling numbers of the second kind, the Fibonacci numbers and the classical expo... | \section{Introduction}
\label{Introduction}
Recently, Jiu and the second author have conducted work \cite{Vignat1} concerning the base-$b$ binomial coefficient. We let $n=\sum_{i=0}^{N_n-1}n_i b^i$ and $k=\sum_{i=0}^{N_k-1}k_i b^i$ be the base-$b$ expansions of $n$ and $k$ respectively. Then we define $N:=\max\{N_n, N... | {
"timestamp": "2016-11-18T02:01:55",
"yymm": "1607",
"arxiv_id": "1607.02564",
"language": "en",
"url": "https://arxiv.org/abs/1607.02564",
"abstract": "We study the properties of the base-$b$ binomial coefficient defined by Jiu and the second author, introduced in the context of a digital binomial theorem... |
https://arxiv.org/abs/1809.01405 | Noncrossing Arc Diagrams, Tamari Lattices, and Parabolic Quotients of the Symmetric Group | Ordering permutations by containment of inversion sets yields a fascinating partial order on the symmetric group: the weak order. This partial order is, among other things, a semidistributive lattice. As a consequence, every permutation has a canonical representation as a join of other permutations. Combinatorially, th... | \section{Introduction}
\label{sec:introduction}
\subsection{Noncrossing partition lattices}
\label{sec:intro_noncrossing}
The \defn{noncrossing partition lattice} $\mathcal{N\!C}_{n}$ was introduced in the 1970s by G.~Kreweras as a subposet of the lattice of all set partitions of $[n]\stackrel{\mathrm{def}}{=}\{1,2,\... | {
"timestamp": "2020-01-20T02:10:52",
"yymm": "1809",
"arxiv_id": "1809.01405",
"language": "en",
"url": "https://arxiv.org/abs/1809.01405",
"abstract": "Ordering permutations by containment of inversion sets yields a fascinating partial order on the symmetric group: the weak order. This partial order is, a... |
https://arxiv.org/abs/2103.03138 | Numerical reconstruction of curves from their Jacobians | We approach the Torelli problem of recostructing a curve from its Jacobian from a computational point of view. Following Dubrovin, we design a machinery to solve this problem effectively, which builds on methods in numerical algebraic geometry. We verify this methods via numerical experiments with curves up to genus 7. | \section{Introduction}
\label{introduction}
The Torelli theorem is a classical and foundational result in algebraic geometry, stating that a Riemann surface, or smooth algebraic curve, $C$ is uniquely determined by its Jacobian variety $J(C)$. More concretely, the theorem says that a Riemann surface of genus $g$ can b... | {
"timestamp": "2021-03-05T02:42:47",
"yymm": "2103",
"arxiv_id": "2103.03138",
"language": "en",
"url": "https://arxiv.org/abs/2103.03138",
"abstract": "We approach the Torelli problem of recostructing a curve from its Jacobian from a computational point of view. Following Dubrovin, we design a machinery t... |
https://arxiv.org/abs/2005.05461 | Geometry and Algebra of the Deltoid Map | The geometry of the deltoid curve gives rise to a self-map of $\mathbb{C}^2$ that is expressed in coordinates by $f(x,y) = (y^2 - 2x, x^2 - 2y)$. This is one in a family of maps that generalize Chebyshev polynomials to several variables. We use this example to illustrate two important objects in complex dynamics: the J... | \section{Introduction.}
Complex dynamics is perhaps best known for the fractal
images it produces. For instance, given a polynomial
function $\C \to \C$, an important set to consider is
the \emph{Julia set}, whose points behave ``chaotically''
under iteration of the function; for most polynomials,
the Julia set i... | {
"timestamp": "2020-05-13T02:05:19",
"yymm": "2005",
"arxiv_id": "2005.05461",
"language": "en",
"url": "https://arxiv.org/abs/2005.05461",
"abstract": "The geometry of the deltoid curve gives rise to a self-map of $\\mathbb{C}^2$ that is expressed in coordinates by $f(x,y) = (y^2 - 2x, x^2 - 2y)$. This is... |
https://arxiv.org/abs/2110.10299 | Archimedean Zeta Functions and Oscillatory Integrals | This note is a short survey of two topics: Archimedean zeta functions and Archimedean oscillatory integrals. We have tried to portray some of the history of the subject and some of its connections with similar devices in mathematics. We present some of the main results of the theory and at the end we discuss some gener... | \section{Introduction}\label{Sec:Intro}
Probably the first Archimedean zeta function was the Gamma function, introduced by Euler in a letter to Golbach in 1729 with the aim to interpolate the factorial, see \cite{Da}.
If the real part of the complex number $s$ is positive, then the Gamma function is defined via the ... | {
"timestamp": "2021-10-22T02:06:49",
"yymm": "2110",
"arxiv_id": "2110.10299",
"language": "en",
"url": "https://arxiv.org/abs/2110.10299",
"abstract": "This note is a short survey of two topics: Archimedean zeta functions and Archimedean oscillatory integrals. We have tried to portray some of the history ... |
https://arxiv.org/abs/1410.3997 | On reversible maps and symmetric periodic points | In reversible dynamical systems, it is frequently of importance to understand symmetric features. The aim of this paper is to explore symmetric periodic points of reversible maps on planar domains invariant under a reflection. We extend Franks' theorem on a dichotomy of the number of periodic points of area preserving ... | \section{Main results and applications}
The study on symmetric features of reversible dynamical systems is of great importance. This paper is majorly concerned with symmetric periodic points of reversible maps on planar domains, which can be applied to find symmetric periodic orbits of reversible dynamical systems with... | {
"timestamp": "2014-10-16T02:09:07",
"yymm": "1410",
"arxiv_id": "1410.3997",
"language": "en",
"url": "https://arxiv.org/abs/1410.3997",
"abstract": "In reversible dynamical systems, it is frequently of importance to understand symmetric features. The aim of this paper is to explore symmetric periodic poi... |
https://arxiv.org/abs/1910.02316 | Superiority of Bayes estimators over the MLE in high dimensional multinomial models and its implication for nonparametric Bayes theory | This article focuses on the performance of Bayes estimators, in comparison with the MLE, in multinomial models with a relatively large number of cells. The prior for the Bayes estimator is taken to be the conjugate Dirichlet, i.e., the multivariate Beta, with exchangeable distributions over the coordinates, including t... | \section{Introduction}
The present article shows by analytical computations and simulations that Bayes estimators even in moderately high dimensional multinomial models outperform the MLE on most of the parameter space. High dimensional multinomial models are useful for industrial planning. For example, for planning... | {
"timestamp": "2019-10-08T02:10:54",
"yymm": "1910",
"arxiv_id": "1910.02316",
"language": "en",
"url": "https://arxiv.org/abs/1910.02316",
"abstract": "This article focuses on the performance of Bayes estimators, in comparison with the MLE, in multinomial models with a relatively large number of cells. Th... |
https://arxiv.org/abs/1503.00374 | A Randomized Algorithm for Approximating the Log Determinant of a Symmetric Positive Definite Matrix | We introduce a novel algorithm for approximating the logarithm of the determinant of a symmetric positive definite (SPD) matrix. The algorithm is randomized and approximates the traces of a small number of matrix powers of a specially constructed matrix, using the method of Avron and Toledo~\cite{AT11}. From a theoreti... |
\section{Conclusions}
\label{sec:conclusions}
There are few approximation algorithms for the logarithm of the determinant of
a symmetric positive definite matrix.
Unfortunately, those algorithms either do not work for all SPD matrices, or do
not admit a worst-case theoretical analysis, or both.
In this work, we pre... | {
"timestamp": "2015-03-03T02:14:50",
"yymm": "1503",
"arxiv_id": "1503.00374",
"language": "en",
"url": "https://arxiv.org/abs/1503.00374",
"abstract": "We introduce a novel algorithm for approximating the logarithm of the determinant of a symmetric positive definite (SPD) matrix. The algorithm is randomiz... |
https://arxiv.org/abs/1404.2321 | Distinct distance estimates and low degree polynomial partitioning | We give a shorter proof of a slightly weaker version of a theorem of Nets Katz and the author. We prove that if a set of $L$ lines in $\mathbb{R}^3$ contains at most $L^{1/2}$ lines in any low degree algebraic surface, then the number of $r$-rich points is at most $C_\epsilon L^{(3/2) + \epsilon} r^{-2}$. Nets and I us... | \section{Background and notation}
Our proof is based on polynomial partitioning. Here we restate the partitioning theorem with an extra condition bounding the number of cells $O_i$.
\begin{theorem} \label{polyham} For each dimension $n$ and each degree $D \ge 1$, the following holds. For any finite set $S \subset \... | {
"timestamp": "2014-11-12T02:06:19",
"yymm": "1404",
"arxiv_id": "1404.2321",
"language": "en",
"url": "https://arxiv.org/abs/1404.2321",
"abstract": "We give a shorter proof of a slightly weaker version of a theorem of Nets Katz and the author. We prove that if a set of $L$ lines in $\\mathbb{R}^3$ contai... |
https://arxiv.org/abs/2202.02384 | A proof of the Erdős primitive set conjecture | A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in 1935 that the series $f(A) = \sum_{a\in A}1/(a \log a)$ is uniformly bounded over all choices of primitive sets $A$. In 1986 he asked if this bound is attained for the set of prime numbers. In this article we answer i... | \section{Introduction}
A set of integers $A\subset \mathbb{Z}_{>1}$ is {\it primitive} if no member in $A$ divides another. For example, the integers in a dyadic interval $(x,2x]$ form a primitive set. Similarly the set of primes is primitive, along with the set $\mathbb{N}_k$ of numbers with exactly $k$ prime factor... | {
"timestamp": "2022-02-08T02:02:48",
"yymm": "2202",
"arxiv_id": "2202.02384",
"language": "en",
"url": "https://arxiv.org/abs/2202.02384",
"abstract": "A set of integers greater than 1 is primitive if no member in the set divides another. Erdős proved in 1935 that the series $f(A) = \\sum_{a\\in A}1/(a \\... |
https://arxiv.org/abs/2301.02043 | Orbifold braid groups | The orbifold braid groups of two dimensional orbifolds were defined in [1] (arXiv:math/9907194) to understand certain Artin groups as subgroups of some suitable orbifold braid groups.We studied orbifold braid groups in some more detail in [17] (arXiv:2006.07106) and [18] (arXiv:2106.08110), to prove the Farrell-Jones I... | \section{Introduction}
Let $M$ be a connected two dimensional orbifold (see \cite{Sco83}). Consider the following
hyperplane complement in $M^n$.
$$PB_n(M)=\{(x_1,x_2,\ldots, x_n)\in M^n\ |\ x_i\neq x_j\ \text{for}\ i\neq j\}.$$
\noindent
$PB_n(M)$ is an orbifold and called the {\it configuration space} of ordered $... | {
"timestamp": "2023-01-06T02:09:40",
"yymm": "2301",
"arxiv_id": "2301.02043",
"language": "en",
"url": "https://arxiv.org/abs/2301.02043",
"abstract": "The orbifold braid groups of two dimensional orbifolds were defined in [1] (arXiv:math/9907194) to understand certain Artin groups as subgroups of some su... |
https://arxiv.org/abs/math/0510604 | Monte Carlo comparisons of the self-avoiding walk and SLE as parameterized curves | The scaling limit of the two-dimensional self-avoiding walk (SAW) is believed to be given by the Schramm-Loewner evolution (SLE) with the parameter kappa equal to 8/3. The scaling limit of the SAW has a natural parameterization and SLE has a standard parameterization using the half-plane capacity. These two parameteriz... | \section{Introduction}
\label{intro}
The scaling limit of the two dimensional self-avoiding walk (SAW)
is believed to be the Schramm-Loewner evolution (SLE)
with $\kappa=8/3$ \cite{lsw_saw}.
Previous Monte Carlo studies have compared the scaling limit of the SAW and
SLE$_{8/3}$ as unparameterized curves \cite{tk_saw... | {
"timestamp": "2005-10-27T16:36:23",
"yymm": "0510",
"arxiv_id": "math/0510604",
"language": "en",
"url": "https://arxiv.org/abs/math/0510604",
"abstract": "The scaling limit of the two-dimensional self-avoiding walk (SAW) is believed to be given by the Schramm-Loewner evolution (SLE) with the parameter ka... |
https://arxiv.org/abs/2212.13251 | Robust computation of optimal transport by $β$-potential regularization | Optimal transport (OT) has become a widely used tool in the machine learning field to measure the discrepancy between probability distributions. For instance, OT is a popular loss function that quantifies the discrepancy between an empirical distribution and a parametric model. Recently, an entropic penalty term and th... | \section{Introduction}
Many machine learning problems such as density estimation and generative modeling are often formulated by a discrepancy between probability distributions \citep{leastsquare,GAN}. As a common choice, the Kullback-Leibler (KL) divergence \citep{KLdivergence} has been widely used since minimizing th... | {
"timestamp": "2022-12-27T02:19:18",
"yymm": "2212",
"arxiv_id": "2212.13251",
"language": "en",
"url": "https://arxiv.org/abs/2212.13251",
"abstract": "Optimal transport (OT) has become a widely used tool in the machine learning field to measure the discrepancy between probability distributions. For insta... |
https://arxiv.org/abs/1606.08393 | Location of the Adsorption Transition for Lattice Polymers | We consider various lattice models of polymers: lattice trees, lattice animals, and self-avoiding walks. The polymer interacts with a surface (hyperplane), receiving a unit energy reward for each site in the surface. There is an adsorption transition of the polymer at a critical value of $\beta$, the inverse temperatur... | \section{Introduction}
\label{sec.intro}
We shall work in the $d$-dimensional hypercubic lattice $\mathbb{L}^d$ ($d\geq 2$),
with sites $x=(x_1,\ldots,x_d)\in \mathbb{Z}^d$ and edges connecting nearest neighbours.
Let $\mathbb{L}^d_+$ be the part of $\mathbb{L}^d$ in the half-space $x_1\geq 0$.
Here is our ``big... | {
"timestamp": "2016-06-28T02:20:12",
"yymm": "1606",
"arxiv_id": "1606.08393",
"language": "en",
"url": "https://arxiv.org/abs/1606.08393",
"abstract": "We consider various lattice models of polymers: lattice trees, lattice animals, and self-avoiding walks. The polymer interacts with a surface (hyperplane)... |
https://arxiv.org/abs/1511.09141 | Consecutive Integers and the Collatz Conjecture | Pairs of consecutive integers have the same height in the Collatz problem with surprising frequency. Garner gave a conjectural family of conditions for exactly when this occurs. Our main result is an infinite family of counterexamples to Garner's conjecture. | \section{\@startsection {section}{1}{\z@}
{-30pt \@plus -1ex \@minus -.2ex}
{2.3ex \@plus.2ex}
{\normalfont\normalsize\bfseries}}
\renewcommand\subsection{\@startsection{subsection}{2}{\z@}
{-3.25ex\@plus -1ex \@minus -.2ex}
{1.5ex \@plus .2ex}
{\normalfont\normalsize\bfseries}}
\renewcommand{\@seccntformat}[1]... | {
"timestamp": "2015-12-01T02:15:04",
"yymm": "1511",
"arxiv_id": "1511.09141",
"language": "en",
"url": "https://arxiv.org/abs/1511.09141",
"abstract": "Pairs of consecutive integers have the same height in the Collatz problem with surprising frequency. Garner gave a conjectural family of conditions for ex... |
https://arxiv.org/abs/1108.1938 | The classification of weighted projective spaces | We obtain two classifications of weighted projective spaces; up to homeomorphism and up to homotopy equivalence. We show that the former coincides with Al Amrani's classification up to isomorphism of algebraic varieties, and deduce the latter by proving that the Mislin genus of any weighted projective space is rigid. | \section{Introduction}
Weighted projective spaces are the simplest projective toric varieties
that exhibit orbifold singularities. They have been extensively
investigated by algebraic geometers, but have attracted only fleeting
attention from algebraic topologists
since Kawasaki's pioneering work~\cite{Kawasaki:1973}... | {
"timestamp": "2012-12-04T02:04:32",
"yymm": "1108",
"arxiv_id": "1108.1938",
"language": "en",
"url": "https://arxiv.org/abs/1108.1938",
"abstract": "We obtain two classifications of weighted projective spaces; up to homeomorphism and up to homotopy equivalence. We show that the former coincides with Al A... |
https://arxiv.org/abs/1204.1718 | Computational complexity and memory usage for multi-frontal direct solvers in structured mesh finite elements | The multi-frontal direct solver is the state-of-the-art algorithm for the direct solution of sparse linear systems. This paper provides computational complexity and memory usage estimates for the application of the multi-frontal direct solver algorithm on linear systems resulting from B-spline-based isogeometric finite... | \section{Introduction}\input{intro.tex}
\section{Multi-frontal direct solver algorithm}\input{alg.tex}
\section{Computational complexity and memory usage}\input{complexity.tex}
\section{Numerical Results}\input{results.tex}
\section{Conclusions}\input{conc.tex}
\section{Acknowledgements}
DP has been partially supported... | {
"timestamp": "2012-04-10T02:01:49",
"yymm": "1204",
"arxiv_id": "1204.1718",
"language": "en",
"url": "https://arxiv.org/abs/1204.1718",
"abstract": "The multi-frontal direct solver is the state-of-the-art algorithm for the direct solution of sparse linear systems. This paper provides computational comple... |
https://arxiv.org/abs/1304.1312 | On nonlinear potential theory, and regular boundary points, for the p-Laplacian in N space variables | We turn back to some pioneering results concerning, in particular, nonlinear potential theory and non-homogeneous boundary value problems for the so called p-Laplacian operator. Unfortunately these results, obtained at the very beginning of the seventies, were kept in the shade. We believe that our proofs are still of ... | \part{Foreword}\label{foreword}
\vspace{0.2cm}
\section{Introduction}\label{introduction}
At the very beginning of the seventies we proved a set of results
concerning nonlinear potential theory related to the so-called
$\,p-$Laplace operator. Following \cite{ricmat}, here we use the
symbol $"\,t\,"$ instead of the no... | {
"timestamp": "2013-04-05T02:01:54",
"yymm": "1304",
"arxiv_id": "1304.1312",
"language": "en",
"url": "https://arxiv.org/abs/1304.1312",
"abstract": "We turn back to some pioneering results concerning, in particular, nonlinear potential theory and non-homogeneous boundary value problems for the so called ... |
https://arxiv.org/abs/2008.06267 | On the homology of independence complexes | The independence complex $\mathrm{Ind}(G)$ of a graph $G$ is the simplicial complex formed by its independent sets. This article introduces a deformation of the simplicial boundary map of $\mathrm{Ind}(G)$ that gives rise to a double complex with trivial homology. Filtering this double complex in the right direction in... | \section{Introduction}
An \textit{independent set} in a graph $G=(V,E)$ is a subset of its vertices $I \subset V$ such that no two elements in $I$ are adjacent. More generally, a subset $I\subset V$ is \textit{$r$-independent} if every connected component of the \textit{induced subgraph} $G[I]:=(I,E')$ with $E'=\{e \i... | {
"timestamp": "2020-10-01T02:13:40",
"yymm": "2008",
"arxiv_id": "2008.06267",
"language": "en",
"url": "https://arxiv.org/abs/2008.06267",
"abstract": "The independence complex $\\mathrm{Ind}(G)$ of a graph $G$ is the simplicial complex formed by its independent sets. This article introduces a deformation... |
https://arxiv.org/abs/1806.02408 | On the existence of symmetric minimizers | In this note we revisit a less known symmetrization method for functions with respect to a topological group $G$, which we call $G$-averaging. We note that, although quite non-technical in nature, this method yields $G$-invariant minimizers of functionals satisfying some relaxed convexity properties. We give an abstrac... |
\section{Introduction}
Identifying symmetries of problems has always been of importance in the deeper understanding of mathematical and physical problems. As Kawohl wrote in \cite{Kawohl1998}: ``Many problems in analysis appear symmetric, yet in fact their solutions are sometimes nonsymmetric.'' It has been a subject ... | {
"timestamp": "2018-07-03T02:16:58",
"yymm": "1806",
"arxiv_id": "1806.02408",
"language": "en",
"url": "https://arxiv.org/abs/1806.02408",
"abstract": "In this note we revisit a less known symmetrization method for functions with respect to a topological group $G$, which we call $G$-averaging. We note tha... |
https://arxiv.org/abs/0910.0281 | Hypergraphic LP Relaxations for Steiner Trees | We investigate hypergraphic LP relaxations for the Steiner tree problem, primarily the partition LP relaxation introduced by Koenemann et al. [Math. Programming, 2009]. Specifically, we are interested in proving upper bounds on the integrality gap of this LP, and studying its relation to other linear relaxations. Our r... | \section{Introduction}\label{sec:intro}
In the {\em Steiner tree} problem, we are given an undirected graph
$G=(V,E)$, non-negative costs $c_e$ for all edges $e \in E$, and a set
of {\em terminal} vertices $R \subseteq V$. The goal is to find a
minimum-cost tree $T$ spanning $R$, and possibly some {\em Steiner
verti... | {
"timestamp": "2010-03-09T02:00:31",
"yymm": "0910",
"arxiv_id": "0910.0281",
"language": "en",
"url": "https://arxiv.org/abs/0910.0281",
"abstract": "We investigate hypergraphic LP relaxations for the Steiner tree problem, primarily the partition LP relaxation introduced by Koenemann et al. [Math. Program... |
https://arxiv.org/abs/math/0612104 | Representations of finite groups | This book is an introduction to a fast developing branch of mathematics - the theory of representations of groups. It presents classical results of this theory concerning finite groups. | \chapter{1}
REPRESENTATIONS OF GROUPS
\endtitle
\endtopmatter
\leftheadtext{CHAPTER \uppercase\expandafter{\romannumeral 1}.
REPRESENTATIONS OF GROUPS.}
\document
\headline=\truehead
\head
\SectionNum{1}{5} Representations of groups and their homomorphisms.
\endhead
\rightheadtext{\S\,1. Representations and their homom... | {
"timestamp": "2006-12-04T20:49:57",
"yymm": "0612",
"arxiv_id": "math/0612104",
"language": "en",
"url": "https://arxiv.org/abs/math/0612104",
"abstract": "This book is an introduction to a fast developing branch of mathematics - the theory of representations of groups. It presents classical results of th... |
https://arxiv.org/abs/1305.6031 | Congruences for Generalized Frobenius Partitions with an Arbitrarily Large Number of Colors | In his 1984 AMS Memoir, George Andrews defined the family of $k$--colored generalized Frobenius partition functions. These are denoted by $c\phi_k(n)$ where $k\geq 1$ is the number of colors in question. In that Memoir, Andrews proved (among many other things) that, for all $n\geq 0,$ $c\phi_2(5n+3) \equiv 0\pmod{5}.$ ... | \section{Introduction}
In his 1984 AMS Memoir, George Andrews \cite{AndMem} defined the family of $k$--colored generalized Frobenius partition functions which are denoted by $c\phi_k(n)$ where $k\geq 1$ is the number of colors in question. Among many things,
Andrews \cite[Corollary 10.1]{AndMem} proved that, for a... | {
"timestamp": "2013-05-28T02:02:17",
"yymm": "1305",
"arxiv_id": "1305.6031",
"language": "en",
"url": "https://arxiv.org/abs/1305.6031",
"abstract": "In his 1984 AMS Memoir, George Andrews defined the family of $k$--colored generalized Frobenius partition functions. These are denoted by $c\\phi_k(n)$ wher... |
https://arxiv.org/abs/1710.08480 | The generalized Sierpiński Arrowhead Curve | We define special Hamiltonian-paths and special permutations of the up-facing dark tiles on a checked triangular grid related to the generalized Sierpiński Gasket. Our definitions and observations make possible the generalization of the Sierpiński Arrowhead Curve for all orders. We produce these symmetric recursive cur... | \section{The bijective relation between paths and tilings}
In this section we define two kinds of special paths related to the same generator pattern on the triangular grid. We observe their transformability into each other and prove their bijective relation.
\subsection{Checked generator patterns}
Let us consider a... | {
"timestamp": "2017-10-25T02:01:40",
"yymm": "1710",
"arxiv_id": "1710.08480",
"language": "en",
"url": "https://arxiv.org/abs/1710.08480",
"abstract": "We define special Hamiltonian-paths and special permutations of the up-facing dark tiles on a checked triangular grid related to the generalized Sierpińsk... |
https://arxiv.org/abs/1811.09988 | Spectral form factor and semi-circle law in the time direction | We study the time derivative of the connected part of spectral form factor, which we call the slope of ramp, in Gaussian matrix model. We find a closed formula of the slope of ramp at finite $N$ with non-zero inverse temperature. Using this exact result, we confirm numerically that the slope of ramp exhibits a semi-cir... | \section{Introduction \label{sec:intro}}
After the seminal work \cite{Garcia-Garcia:2016mno,Cotler:2016fpe},
the spectral form factor is intensively studied as a diagnostic
of the quantum chaotic behavior of the Sachdev-Ye-Kitaev (SYK) model \cite{KitaevTalks,Sachdev,Maldacena:2016hyu}, which is a solvable example of t... | {
"timestamp": "2019-02-15T02:03:10",
"yymm": "1811",
"arxiv_id": "1811.09988",
"language": "en",
"url": "https://arxiv.org/abs/1811.09988",
"abstract": "We study the time derivative of the connected part of spectral form factor, which we call the slope of ramp, in Gaussian matrix model. We find a closed fo... |
https://arxiv.org/abs/2104.07028 | On Missing Mass Variance | The missing mass refers to the probability of elements not observed in a sample, and since the work of Good and Turing during WWII, has been studied extensively in many areas including ecology, linguistic, networks and information theory.This work determines what is the \emph{maximal variance of the missing mass}, for ... | \section{Introduction}
\subsection{Background}
Let $X_1,\ldots,X_n\sim^{IID} p$ be a sample from a distribution $(p_s)_{s
\in S}$ on a countable alphabet $S$, and $f_s = \#\{i: X_i = s\}$ be the empirical (observed) frequencies. The missing mass
\begin{align}
M_0 = \sum_{s}p_s\mathbb{I}(f_s=0),
\end{align}
which ... | {
"timestamp": "2021-04-16T02:00:07",
"yymm": "2104",
"arxiv_id": "2104.07028",
"language": "en",
"url": "https://arxiv.org/abs/2104.07028",
"abstract": "The missing mass refers to the probability of elements not observed in a sample, and since the work of Good and Turing during WWII, has been studied exten... |
https://arxiv.org/abs/2205.07346 | Optimal Error-Detecting Codes for General Asymmetric Channels via Sperner Theory | Several communication models that are of relevance in practice are asymmetric in the way they act on the transmitted "objects". Examples include channels in which the amplitudes of the transmitted pulses can only be decreased, channels in which the symbols can only be deleted, channels in which non-zero symbols can onl... | \section{Introduction}
\label{sec:intro}
Several important channel models possess an intrinsic asymmetry in the
way they act on the transmitted ``objects''.
A classical example is the binary $ \mathsf{Z} $-channel in which the
transmitted $ 1 $'s may be received as $ 0 $'s, but not vice versa.
In the present article w... | {
"timestamp": "2022-05-17T02:24:49",
"yymm": "2205",
"arxiv_id": "2205.07346",
"language": "en",
"url": "https://arxiv.org/abs/2205.07346",
"abstract": "Several communication models that are of relevance in practice are asymmetric in the way they act on the transmitted \"objects\". Examples include channel... |
https://arxiv.org/abs/math/9907060 | Polynomial Homotopies for Dense, Sparse and Determinantal Systems | Numerical homotopy continuation methods for three classes of polynomial systems are presented. For a generic instance of the class, every path leads to a solution and the homotopy is optimal. The counting of the roots mirrors the resolution of a generic system that is used to start up the deformations. Software and app... | \section{Introduction}
Solving polynomial systems numerically means computing approximations
to all isolated solutions.
Homotopy continuation methods provide paths to approximate solutions.
The idea is to break up the original system into simpler problems.
To solve the original system, the solutions of the simpler sys... | {
"timestamp": "1999-07-13T00:47:25",
"yymm": "9907",
"arxiv_id": "math/9907060",
"language": "en",
"url": "https://arxiv.org/abs/math/9907060",
"abstract": "Numerical homotopy continuation methods for three classes of polynomial systems are presented. For a generic instance of the class, every path leads t... |
https://arxiv.org/abs/1910.01164 | Insight in the Rumin Cohomology and Orientability Properties of the Heisenberg Group | The purpose of this study is to analyse two related topics: the Rumin cohomology and the $\mathbb{H}$-orientability in the Heisenberg group $\mathbb{H}^n$. In the first three chapters we carefully describe the Rumin cohomology with particular emphasis at the second order differential operator $D$, giving examples in th... | \chapter{Preliminaries}
In this chapter we will first provide definitions and notions about Lie and Carnot groups (Section \ref{liegroups}), as well as introduce the Heisenberg group and some basic properties and automorphisms associated to it. Second, we will present the standard basis of
vector fields in the Heisen... | {
"timestamp": "2019-10-04T02:01:27",
"yymm": "1910",
"arxiv_id": "1910.01164",
"language": "en",
"url": "https://arxiv.org/abs/1910.01164",
"abstract": "The purpose of this study is to analyse two related topics: the Rumin cohomology and the $\\mathbb{H}$-orientability in the Heisenberg group $\\mathbb{H}^... |
https://arxiv.org/abs/math/0608118 | Finiteness of Hilbert functions and bounds for Castelnuovo-Mumford regularity of initial ideals | Bounds for the Castelnuovo-Mumford regularity and Hilbert coefficients are given in terms of the arithmetic degree (if the ring is reduced) or in terms of the defining degrees. From this it follows that there exists only a finite number of Hilbert functions associated to reduced algebras over an algebraically closed fi... | \section*{Introduction}\smallskip
In the famous book SGA6, Kleiman proved that given two positive
integers $e$ and $d$, there exists only a finite number of Hilbert
functions associated to reduced and equidimensional $K$-algebras
$S$ over an algebraically closed field such that $\operatorname{deg} S \le e$
and $\di... | {
"timestamp": "2006-08-04T07:30:14",
"yymm": "0608",
"arxiv_id": "math/0608118",
"language": "en",
"url": "https://arxiv.org/abs/math/0608118",
"abstract": "Bounds for the Castelnuovo-Mumford regularity and Hilbert coefficients are given in terms of the arithmetic degree (if the ring is reduced) or in term... |
https://arxiv.org/abs/2012.10350 | Scaling Laws for Gaussian Random Many-Access Channels | This paper considers a Gaussian multiple-access channel with random user activity where the total number of users $\ell_n$ and the average number of active users $k_n$ may grow with the blocklength $n$. For this channel, it studies the maximum number of bits that can be transmitted reliably per unit-energy as a functio... | \section{Capacity per Unit-Energy of Non-Random Many-Access Channels}
\label{Sec_nonrandom}
\input{nonrandom.tex}
\input{random.tex}
\input{discussion.tex}
\section{Conclusion}
\label{Sec_conclusion}
In this work, we analyzed scaling laws
of a Gaussian random \edit{MnAC} where the total number of users as w... | {
"timestamp": "2021-09-03T02:23:12",
"yymm": "2012",
"arxiv_id": "2012.10350",
"language": "en",
"url": "https://arxiv.org/abs/2012.10350",
"abstract": "This paper considers a Gaussian multiple-access channel with random user activity where the total number of users $\\ell_n$ and the average number of acti... |
https://arxiv.org/abs/2005.11751 | On 3-strand singular pure braid group | In the present paper we study the singular pure braid group $SP_{n}$ for $n=2, 3$. We find generators, defining relations and the algebraical structure of these groups. In particular, we prove that $SP_{3}$ is a semi-direct product $SP_{3} = \widetilde{V}_3 \leftthreetimes \mathbb{Z}$, where $\widetilde{V}_3$ is an HNN... | \section{Introduction}
E. Artin introduced the braid group in 1925 and gave a presentation for this group in 1947 \cite{A}. The braid groups have applications in a wide variety of areas of mathematics, physics and biology. The notion of singular braids was introduced independently by John Baez \cite{Baez} and Joan B... | {
"timestamp": "2020-05-26T02:17:01",
"yymm": "2005",
"arxiv_id": "2005.11751",
"language": "en",
"url": "https://arxiv.org/abs/2005.11751",
"abstract": "In the present paper we study the singular pure braid group $SP_{n}$ for $n=2, 3$. We find generators, defining relations and the algebraical structure of... |
https://arxiv.org/abs/1712.07801 | Density Estimation with Contaminated Data: Minimax Rates and Theory of Adaptation | This paper studies density estimation under pointwise loss in the setting of contamination model. The goal is to estimate $f(x_0)$ at some $x_0\in\mathbb{R}$ with i.i.d. observations, $$ X_1,\dots,X_n\sim (1-\epsilon)f+\epsilon g, $$ where $g$ stands for a contamination distribution. In the context of multiple testing,... | \section{Results for Arbitrary Contamination}\label{sec:ma}
\subsection{Minimax Rates}
In this section, we study the contamination model without any structural assumption on the contamination distribution:
\begin{align*}
X_1,\dots,X_n \sim (1-\epsilon)P_f+\epsilon G
\end{align*}
where $P_f$ is a distribution on $\ma... | {
"timestamp": "2018-07-30T02:04:39",
"yymm": "1712",
"arxiv_id": "1712.07801",
"language": "en",
"url": "https://arxiv.org/abs/1712.07801",
"abstract": "This paper studies density estimation under pointwise loss in the setting of contamination model. The goal is to estimate $f(x_0)$ at some $x_0\\in\\mathb... |
https://arxiv.org/abs/1509.09272 | On stationary solutions of KdV and mKdV equations | Stationary solutions on a bounded interval for an initial-boundary value problem to Korteweg--de~Vries and modified Korteweg--de~Vries equation (for the last one both in focusing and defocusing cases) are constructed. The method of the study is based on the theory of conservative systems with one degree of freedom. The... | \section*{}
Both Korteweg--de~Vries equation (KdV)
$$
u_t+au_x+u_{xxx}+uu_x=0
$$
and modified Korteweg--de~Vries equation (mKdV)
$$
u_t+au_x+u_{xxx}\pm u^2u_x=0
$$
(the sign "+" stands for the focusing case and the sign "-" -- for the defocusing one) describe propagation of long nonlinear waves in dispersive media. We ... | {
"timestamp": "2015-10-01T02:12:48",
"yymm": "1509",
"arxiv_id": "1509.09272",
"language": "en",
"url": "https://arxiv.org/abs/1509.09272",
"abstract": "Stationary solutions on a bounded interval for an initial-boundary value problem to Korteweg--de~Vries and modified Korteweg--de~Vries equation (for the l... |
https://arxiv.org/abs/1407.5280 | Density and spectrum of minimal submanifolds in space forms | Let $M^m$ be a minimal properly immersed submanifold in an ambient space close, in a suitable sense, to the space form $\mathbb{N}^n_k$ of curvature $-k\le 0$. In this paper, we are interested in the relation between the density function $\Theta(r)$ of $M^m$ and the spectrum of the Laplace-Beltrami operator. In particu... | \section{Introduction}
\label{intro}
Let $M^m$ be a minimal, properly immersed submanifold in a complete ambient space $N^n$. In the present paper, we are interested in the case when $N$ is close, in a sense made precise below, to a space form $\mathbb{N}_k^n$ of curvature $-k\le 0$. In particular, our focus is the st... | {
"timestamp": "2016-02-29T02:12:47",
"yymm": "1407",
"arxiv_id": "1407.5280",
"language": "en",
"url": "https://arxiv.org/abs/1407.5280",
"abstract": "Let $M^m$ be a minimal properly immersed submanifold in an ambient space close, in a suitable sense, to the space form $\\mathbb{N}^n_k$ of curvature $-k\\l... |
https://arxiv.org/abs/2005.13210 | Census of bounded curvature paths | A bounded curvature path is a continuously differentiable piece-wise $C^2$ path with bounded absolute curvature connecting two points in the tangent bundle of a surface. These paths have been widely considered in computer science and engineering since the bound on curvature models the trajectory of the motion of robots... | \section{Prelude}
It is well known that any two plane curves both closed or with different endpoints are homotopic. Graustein, and Whitney in 1937, independently proved that not any two planar closed curves are regularly homotopic (homotopic through immersions) \cite{whitney}. Markov in 1857 considered several optimi... | {
"timestamp": "2020-05-28T02:11:02",
"yymm": "2005",
"arxiv_id": "2005.13210",
"language": "en",
"url": "https://arxiv.org/abs/2005.13210",
"abstract": "A bounded curvature path is a continuously differentiable piece-wise $C^2$ path with bounded absolute curvature connecting two points in the tangent bundl... |
https://arxiv.org/abs/0912.1825 | Alexandrov curvature of convex hypersurfaces in Hilbert space | It is shown that convex hypersurfaces in Hilbert spaces have nonnegative Alexandrov curvature. This extends an earlier result of Buyalo for convex hypersurfaces in Riemannian manifolds of finite dimension. | \section{Introduction}
In this paper, the following result is established:
\begin{thm}\label{maintheorem} If $C$ is an open set in a Hilbert space $H$
and $\overline C$ is locally convex, then
$\partial C$ is a nonnegatively curved Alexandrov space under the induced length metric.
\end{thm}
Questions of this sort go... | {
"timestamp": "2009-12-15T19:43:54",
"yymm": "0912",
"arxiv_id": "0912.1825",
"language": "en",
"url": "https://arxiv.org/abs/0912.1825",
"abstract": "It is shown that convex hypersurfaces in Hilbert spaces have nonnegative Alexandrov curvature. This extends an earlier result of Buyalo for convex hypersurf... |
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