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url stringlengths 31 38	title stringlengths 7 229	abstract stringlengths 44 2.87k	text stringlengths 319 2.51M	meta dict
https://arxiv.org/abs/1912.08327	Extreme Values of the Fiedler Vector on Trees	Let $G$ be a connected tree on $n$ vertices and let $L = D-A$ denote the Laplacian matrix on $G$. The second-smallest eigenvalue $\lambda_{2}(G) > 0$, also known as the algebraic connectivity, as well as the associated eigenvector $\phi_2$ have been of substantial interest. We investigate the question of when the maxim...	\section{Introduction} \subsection{Introduction.} Let $G=(V,E)$ be a simple, undirected, connected tree on $n$ vertices $\left\{v_1, \dots, v_n\right\}$. The degree matrix $D$ is the diagonal matrix $d_{ii} = \deg(v_i)$, the adjacency matrix $A$ encodes the connections between the vertices. The matrix $$ L = D - A$$ i...	{ "timestamp": "2019-12-19T02:04:28", "yymm": "1912", "arxiv_id": "1912.08327", "language": "en", "url": "https://arxiv.org/abs/1912.08327", "abstract": "Let $G$ be a connected tree on $n$ vertices and let $L = D-A$ denote the Laplacian matrix on $G$. The second-smallest eigenvalue $\\lambda_{2}(G) > 0$, al...
https://arxiv.org/abs/1906.12272	Weighing the Sun with five photographs	With only five photographs of the Sun at different dates we show that the mass of Sun can be calculated by using a telescope, a camera, and the Kepler's third law. With these photographs we are able to calculate the distance between Sun and Earth at different dates in a period of time of about three months. These dista...	\section{Introduction} Our solar system is completly dominated by the Sun. Its huge mass bounds the eight planets in elliptical orbits around it. The proper explanation of the motion of each planet can be found from the Newton's law of universal gravitation. One of the greatest triumph of Newton's law is to provide t...	{ "timestamp": "2019-07-01T02:18:31", "yymm": "1906", "arxiv_id": "1906.12272", "language": "en", "url": "https://arxiv.org/abs/1906.12272", "abstract": "With only five photographs of the Sun at different dates we show that the mass of Sun can be calculated by using a telescope, a camera, and the Kepler's t...
https://arxiv.org/abs/1607.00420	Coloring the power graph of a semigroup	Let $G$ be a semigroup. The vertices of the power graph $\mathcal{P}(G)$ are the elements of $G$, and two elements are adjacent if and only if one of them is a power of the other. We show that the chromatic number of $\mathcal{P}(G)$ is at most countable, answering a recent question of Aalipour et al.	\section{Introduction} This note is devoted to the graph constructed in a special way from a given semigroup $G$. This graph is called the \textit{power graph} of $G$, denoted by $\mathcal{P}(G)$, and its vertices are the elements of $G$. Elements $g,h\in G$ are adjacent if and only if one of them is a power of the ot...	{ "timestamp": "2016-07-05T02:01:01", "yymm": "1607", "arxiv_id": "1607.00420", "language": "en", "url": "https://arxiv.org/abs/1607.00420", "abstract": "Let $G$ be a semigroup. The vertices of the power graph $\\mathcal{P}(G)$ are the elements of $G$, and two elements are adjacent if and only if one of the...
https://arxiv.org/abs/2008.01693	Stable and accurate numerical methods for generalized Kirchhoff-Love plates	Efficient and accurate numerical algorithms are developed to solve a generalized Kirchhoff-Love plate model subject to three common physical boundary conditions: (i) clamped; (ii) simply supported; and (iii) free. We solve the model equation by discretizing the spatial derivatives using second-order finite-difference s...	\section{Stability analysis and time step determination}\label{sec:analysis} We study the stability of the schemes and use the analytical results to determine stable time steps in practical computations. As is already pointed out in \cite{Newmark59} that the implicit NB2 time-stepping scheme is unconditionally s...	{ "timestamp": "2020-08-05T02:21:56", "yymm": "2008", "arxiv_id": "2008.01693", "language": "en", "url": "https://arxiv.org/abs/2008.01693", "abstract": "Efficient and accurate numerical algorithms are developed to solve a generalized Kirchhoff-Love plate model subject to three common physical boundary cond...
https://arxiv.org/abs/2212.10586	A combinatorial proof of a tantalizing symmetry on Catalan objects	We investigate a tantalizing symmetry on Catalan objects. In terms of Dyck paths, this symmetry is interpreted in the following way: if $w_{n,k,m}$ is the number of Dyck paths of semilength $n$ with $k$ occurrences of $UD$ and $m$ occurrences of $UUD$, then $w_{2k+1,k,m}=w_{2k+1,k,k+1-m}$. We give two proofs of this sy...	\section{Introduction} Lattice path enumeration is an active area of investigation in combinatorics. A \emph{lattice path} is a path in the discrete integer lattice $\mathbb{Z}^n$ consisting of a sequence of steps from a prescribed \emph{step set} and satisfying prescribed restrictions. Among classical lattice paths,...	{ "timestamp": "2022-12-22T02:00:20", "yymm": "2212", "arxiv_id": "2212.10586", "language": "en", "url": "https://arxiv.org/abs/2212.10586", "abstract": "We investigate a tantalizing symmetry on Catalan objects. In terms of Dyck paths, this symmetry is interpreted in the following way: if $w_{n,k,m}$ is the...
https://arxiv.org/abs/1401.5766	On matrix balancing and eigenvector computation	Balancing a matrix is a preprocessing step while solving the nonsymmetric eigenvalue problem. Balancing a matrix reduces the norm of the matrix and hopefully this will improve the accuracy of the computation. Experiments have shown that balancing can improve the accuracy of the computed eigenval- ues. However, there ex...	\section{Introduction} For a given vector norm $\\|\cdot\\|$, an $n$-by-$n$ (square) matrix $A$ is said to be {\it balanced} if and only if, for all $i$ from 1 to $n$, the norm of its $i$-th column and the norm of its $i$-th row are equal. $A$ and $\widetilde{A}$ are {\em diagonally similar} means that there exists a d...	{ "timestamp": "2014-01-23T02:11:44", "yymm": "1401", "arxiv_id": "1401.5766", "language": "en", "url": "https://arxiv.org/abs/1401.5766", "abstract": "Balancing a matrix is a preprocessing step while solving the nonsymmetric eigenvalue problem. Balancing a matrix reduces the norm of the matrix and hopefull...
https://arxiv.org/abs/0710.5611	Universal cycles for permutations	A universal cycle for permutations is a word of length n! such that each of the n! possible relative orders of n distinct integers occurs as a cyclic interval of the word. We show how to construct such a universal cycle in which only n+1 distinct integers are used. This is best possible and proves a conjecture of Chung...	\section{Introduction} A \emph{de Bruijn cycle} of order $n$ is a word in $\{0,1\}^{2^n}$ in which each $n$-tuple in $\{0,1\}^n$ appears exactly once as a cyclic interval (see \cite{dB}). The idea of a universal cycle generalizes the notion of a de Bruijn cycle. Suppose that $\mathcal{F}$ is a family of combinatoria...	{ "timestamp": "2007-10-30T12:22:53", "yymm": "0710", "arxiv_id": "0710.5611", "language": "en", "url": "https://arxiv.org/abs/0710.5611", "abstract": "A universal cycle for permutations is a word of length n! such that each of the n! possible relative orders of n distinct integers occurs as a cyclic interv...
https://arxiv.org/abs/2101.08038	Infinitely many twin prime polynomials of odd degree	While the twin prime conjecture is still famously open, it holds true in the setting of finite fields: There are infinitely many pairs of monic irreducible polynomials over $\mathbb{F}_q$ that differ by a fixed constant, for each $q \geq 3$. Elementary, constructive proofs were given for different cases by Hall and Pol...	\section{Introduction.} Let $\mathbb{F}_q$ be a finite field of $q\geq3$ elements, where $q$ is a prime power. The ring of integers $\mathbb{Z}$ and the polynomial ring $\mathbb{F}_q[X]$ exhibit a number of common features, including both being unique factorization domains. A prime (polynomial) in the latter setting i...	{ "timestamp": "2021-01-21T02:15:18", "yymm": "2101", "arxiv_id": "2101.08038", "language": "en", "url": "https://arxiv.org/abs/2101.08038", "abstract": "While the twin prime conjecture is still famously open, it holds true in the setting of finite fields: There are infinitely many pairs of monic irreducibl...
https://arxiv.org/abs/1101.0612	Optimal Meshes for Finite Elements of Arbitrary Order	Given a function f defined on a bidimensional bounded domain and a positive integer N, we study the properties of the triangulation that minimizes the distance between f and its interpolation on the associated finite element space, over all triangulations of at most N elements. The error is studied in the Lp norm and w...	\section{Introduction.} \subsection{Optimal mesh adaptation} In finite element approximation, a usual distinction is between {\it uniform} and {\it adaptive} methods. In the latter, the elements defining the mesh may vary strongly in size and shape for a better adaptation to the local features of the approximated fu...	{ "timestamp": "2011-01-05T02:00:08", "yymm": "1101", "arxiv_id": "1101.0612", "language": "en", "url": "https://arxiv.org/abs/1101.0612", "abstract": "Given a function f defined on a bidimensional bounded domain and a positive integer N, we study the properties of the triangulation that minimizes the dista...
https://arxiv.org/abs/2207.10512	The shape of $x^2\bmod n$	We examine the graphs generated by the map $x\mapsto x^2\bmod n$ for various $n$, present some results on the structure of these graphs, and compute some very cool examples.	\section{Overview} For any $n$, we consider the map \begin{equation} f_n(x) := x^2\bmod n. \end{equation} From this map, we can generate a (directed) graph $\gr n$ whose vertices are the set $\{0,1,\dots, n-1\}$ and edges from $x$ to $f_n(x)$ for each $x$. We show a few examples of such graphs later in the pape...	{ "timestamp": "2022-07-22T02:22:33", "yymm": "2207", "arxiv_id": "2207.10512", "language": "en", "url": "https://arxiv.org/abs/2207.10512", "abstract": "We examine the graphs generated by the map $x\\mapsto x^2\\bmod n$ for various $n$, present some results on the structure of these graphs, and compute som...
https://arxiv.org/abs/2007.09719	Rainbow odd cycles	We prove that every family of (not necessarily distinct) odd cycles $O_1, \dots, O_{2\lceil n/2 \rceil-1}$ in the complete graph $K_n$ on $n$ vertices has a rainbow odd cycle (that is, a set of edges from distinct $O_i$'s, forming an odd cycle). As part of the proof, we characterize those families of $n$ odd cycles in ...	\section{Introduction}\label{sec:introduction} Given a family $\mathcal{E}$ of sets, an $\mathcal{E}$-\emph{rainbow set} is a set $R \subseteq \union\mathcal{E}$ with an injection $\sigma\colon R \to \mathcal{E}$ such that $e \in \sigma(e)$ for all $e \in R$. The term rainbow set originates in viewing every member of ...	{ "timestamp": "2021-02-19T02:03:52", "yymm": "2007", "arxiv_id": "2007.09719", "language": "en", "url": "https://arxiv.org/abs/2007.09719", "abstract": "We prove that every family of (not necessarily distinct) odd cycles $O_1, \\dots, O_{2\\lceil n/2 \\rceil-1}$ in the complete graph $K_n$ on $n$ vertices ...
https://arxiv.org/abs/2010.02211	Likelihood-based solution to the Monty Hall puzzle and a related 3-prisoner paradox	The Monty Hall puzzle has been solved and dissected in many ways, but always using probabilistic arguments, so it is considered a probability puzzle. In this paper the puzzle is set up as an orthodox statistical problem involving an unknown parameter, a probability model and an observation. This means we can compute a ...	\section{The puzzle and the paradox} First, here is the Monty Hall puzzle: \begin{quote} You are a contestant in a game show and presented with 3 closed doors. Behind one is a car, and behind the others only goats. You pick one door (let's call that Door 1), then the host \textit{will} open another door that reveal...	{ "timestamp": "2020-10-07T02:00:10", "yymm": "2010", "arxiv_id": "2010.02211", "language": "en", "url": "https://arxiv.org/abs/2010.02211", "abstract": "The Monty Hall puzzle has been solved and dissected in many ways, but always using probabilistic arguments, so it is considered a probability puzzle. In t...
https://arxiv.org/abs/1610.07227	Sums of squares in Quaternion rings	Lagrange's Four Squares Theorem states that any positive integer can be expressed as the sum of four integer squares. We investigate the analogous question over Quaternion rings, focusing on squares of elements of Quaternion rings with integer coefficients. We determine the minimum necessary number of squares for infin...	\section{Introduction and Definitions} \subsection*{Waring's Problem} \begin{theorem}[Waring's Problem/Hilbert-Waring Theorem] For every integer $k \geq 2$ there exists a positive integer $g(k)$ such that every positive integer is the sum of at most $g(k)$ $k$-th powers of integers. \end{theorem} Generalizations of ...	{ "timestamp": "2016-10-25T02:06:55", "yymm": "1610", "arxiv_id": "1610.07227", "language": "en", "url": "https://arxiv.org/abs/1610.07227", "abstract": "Lagrange's Four Squares Theorem states that any positive integer can be expressed as the sum of four integer squares. We investigate the analogous questio...
https://arxiv.org/abs/1507.01565	Torsion and ground state maxima: close but not the same	Could the location of the maximum point for a positive solution of a semilinear Poisson equation on a convex domain be independent of the form of the nonlinearity? Cima and Derrick found certain evidence for this surprising conjecture.We construct counterexamples on the half-disk, by working with the torsion function a...	\section{\bf Introduction} \label{sec:intro} Suppose the Poisson equation \[ \begin{cases} -\Delta u = f(u) & \text{in $\Omega$,} \\ \quad \ \ u = 0 & \text{on $\partial \Omega$,} \end{cases} \] has a positive solution on the bounded convex plane domain $\Omega$. Here the nonlinearity $f$ is assumed to be Lipschitz...	{ "timestamp": "2015-07-07T02:20:23", "yymm": "1507", "arxiv_id": "1507.01565", "language": "en", "url": "https://arxiv.org/abs/1507.01565", "abstract": "Could the location of the maximum point for a positive solution of a semilinear Poisson equation on a convex domain be independent of the form of the nonl...
https://arxiv.org/abs/0707.3450	Stability and intersection properties of solutions to the nonlinear biharmonic equation	We study the positive, regular, radially symmetric solutions to the nonlinear biharmonic equation $\Delta^2 \phi = \phi^p$. First, we show that there exists a critical value $p_c$, depending on the space dimension, such that the solutions are linearly unstable if $p<p_c$ and linearly stable if $p\geq p_c$. Then, we foc...	\section{Introduction} Consider the positive, regular, radially symmetric solutions of the equation \begin{equation}\label{ee} \Delta^2 \phi (x) = \phi(x)^p, \quad\quad x\in {\mathbb R}^n. \end{equation} Such solutions are known to exist when $n>4$ and $p\geq \frac{n+4}{n-4}$, but they fail to exist, otherwise. Our mai...	{ "timestamp": "2007-07-23T22:43:24", "yymm": "0707", "arxiv_id": "0707.3450", "language": "en", "url": "https://arxiv.org/abs/0707.3450", "abstract": "We study the positive, regular, radially symmetric solutions to the nonlinear biharmonic equation $\\Delta^2 \\phi = \\phi^p$. First, we show that there exi...
https://arxiv.org/abs/1510.06492	Generalized Shortest Path Kernel on Graphs	We consider the problem of classifying graphs using graph kernels. We define a new graph kernel, called the generalized shortest path kernel, based on the number and length of shortest paths between nodes. For our example classification problem, we consider the task of classifying random graphs from two well-known fami...	\section{Analysis} \label{sec:Analysis} In this section we give some approximated analysis of random feature vectors in order to give theoretical support for our experimental observations. We first show that one-cluster and two-cluster graphs have quite similar SPI feature vectors (as their expectations). Then we nex...	{ "timestamp": "2015-10-23T02:06:02", "yymm": "1510", "arxiv_id": "1510.06492", "language": "en", "url": "https://arxiv.org/abs/1510.06492", "abstract": "We consider the problem of classifying graphs using graph kernels. We define a new graph kernel, called the generalized shortest path kernel, based on the...
https://arxiv.org/abs/1406.3838	A fast 25/6-approximation for the minimum unit disk cover problem	Given a point set P in 2D, the problem of finding the smallest set of unit disks that cover all of P is NP-hard. We present a simple algorithm for this problem with an approximation factor of 25/6 in the Euclidean norm and 2 in the max norm, by restricting the disk centers to lie on parallel lines. The run time and spa...	\section{Introduction} Given a point set $P$ in $\mathbb{R}^2$, the \textit{unit disk cover problem} (UDC) seeks to find the smallest set of unit disks that cover all of $P$. This problem arises in applications to facility location, motion planning, and image processing \cite{fowler, hochbaummaass}. In both the $L_2$ ...	{ "timestamp": "2014-06-17T02:08:55", "yymm": "1406", "arxiv_id": "1406.3838", "language": "en", "url": "https://arxiv.org/abs/1406.3838", "abstract": "Given a point set P in 2D, the problem of finding the smallest set of unit disks that cover all of P is NP-hard. We present a simple algorithm for this prob...
https://arxiv.org/abs/2106.09262	Componentwise linear ideals in Veronese rings	In this article, we study the componentwise linear ideals in the Veronese subrings of $R=K[x_1,\ldots,x_n]$. If char$(K)=0$, then we give a characterization for graded ideals in the $c^{th}$ Veronese ring $R^{(c)}$ to be componentwise linear. This characterization is an analogue of that over $R$ due to Aramova, Herzog ...	\section{Introduction} Let $R=K[x_1,\ldots,x_n]$, where $K$ is a field. For $c\in \mathbb N$, the $c^{th}$ {\it Veronese ring} of $R$ is the ring $$R^{(c)}:=K[\mbox{all monomials of degree equal to $c$ in } R].$$ The componentwise linear ideals in $R$ are well studied by various authors. Their graded Betti numbers ar...	{ "timestamp": "2021-06-18T02:11:05", "yymm": "2106", "arxiv_id": "2106.09262", "language": "en", "url": "https://arxiv.org/abs/2106.09262", "abstract": "In this article, we study the componentwise linear ideals in the Veronese subrings of $R=K[x_1,\\ldots,x_n]$. If char$(K)=0$, then we give a characterizat...
https://arxiv.org/abs/2203.03832	The splitting algorithms by Ryu, by Malitsky-Tam, and by Campoy applied to normal cones of linear subspaces converge strongly to the projection onto the intersection	Finding a zero of a sum of maximally monotone operators is a fundamental problem in modern optimization and nonsmooth analysis. Assuming that the resolvents of the operators are available, this problem can be tackled with the Douglas-Rachford algorithm. However, when dealing with three or more operators, one must work ...	\section{Introduction} Throughout the paper, we assume that \begin{equation} \text{$X$ is a real Hilbert space} \end{equation} with inner product $\scal{\cdot}{\cdot}$ and induced norm $\\|\cdot\\|$. Let $A_1,\ldots,A_n$ be maximally monotone operators on $X$. (See, e.g., \cite{BC2017} for background on maximally mono...	{ "timestamp": "2022-03-09T02:09:46", "yymm": "2203", "arxiv_id": "2203.03832", "language": "en", "url": "https://arxiv.org/abs/2203.03832", "abstract": "Finding a zero of a sum of maximally monotone operators is a fundamental problem in modern optimization and nonsmooth analysis. Assuming that the resolven...
https://arxiv.org/abs/2207.11741	Induced subgraphs of zero-divisor graphs	The zero-divisor graph of a finite commutative ring with unity is the graph whose vertex set is the set of zero-divisors in the ring, with $a$ and $b$ adjacent if $ab=0$. We show that the class of zero-divisor graphs is universal, in the sense that every finite graph is isomorphic to an induced subgraph of a zero-divis...	\section{Introduction} In this paper, ``ring'' means ``finite commutative ring with unity'', while ``graph'' means ``finite simple undirected graph'' (except in the penultimate section, where finiteness will be relaxed). The \emph{zero-divisor graph} $\Gamma(R)$ of a ring $R$ has vertices the zero-divisors in $R$ (the ...	{ "timestamp": "2022-07-26T02:17:51", "yymm": "2207", "arxiv_id": "2207.11741", "language": "en", "url": "https://arxiv.org/abs/2207.11741", "abstract": "The zero-divisor graph of a finite commutative ring with unity is the graph whose vertex set is the set of zero-divisors in the ring, with $a$ and $b$ adj...
https://arxiv.org/abs/0711.0906	Multivariate Fuss-Catalan numbers	Catalan numbers $C(n)=\frac{1}{n+1}{2n\choose n}$ enumerate binary trees and Dyck paths. The distribution of paths with respect to their number $k$ of factors is given by ballot numbers $B(n,k)=\frac{n-k}{n+k}{n+k\choose n}$. These integers are known to satisfy simple recurrence, which may be visualised in a ``Catalan ...	\section{Catalan triangle, binary trees, and Dyck paths} We recall in this section well-known results about Catalan numbers and ballot numbers. The {\em Catalan numbers} $$C(n)=\frac{1}{n+1}{2n\choose n}$$ are integers that appear in many combinatorial problems. These numbers first appeared in Euler's work as the nu...	{ "timestamp": "2007-11-06T16:40:29", "yymm": "0711", "arxiv_id": "0711.0906", "language": "en", "url": "https://arxiv.org/abs/0711.0906", "abstract": "Catalan numbers $C(n)=\\frac{1}{n+1}{2n\\choose n}$ enumerate binary trees and Dyck paths. The distribution of paths with respect to their number $k$ of fac...
https://arxiv.org/abs/1107.3852	Sums of Ceiling Functions Solve Nested Recursions	It is known that, for given integers s \geq 0 and j > 0, the nested recursion R(n) = R(n - s - R(n - j)) + R(n - 2j - s - R(n - 3j)) has a closed form solution for which a combinatorial interpretation exists in terms of an infinite, labeled tree. For s = 0, we show that this solution sequence has a closed form as the s...	\section{Introduction} \label{sec:Intro} This paper investigates the occurrence of sums of ceiling functions as solutions to nested recursions of the form \begin{align} R(n) = R(n - s_1 -R(n-a_1)) + R(n-s_2-R(n-a_2)) \label{Hn} \end{align} with $s_i,a_i$ integers, $a_i > 0$, and specified initial conditions. We adopt t...	{ "timestamp": "2011-07-21T02:00:18", "yymm": "1107", "arxiv_id": "1107.3852", "language": "en", "url": "https://arxiv.org/abs/1107.3852", "abstract": "It is known that, for given integers s \\geq 0 and j > 0, the nested recursion R(n) = R(n - s - R(n - j)) + R(n - 2j - s - R(n - 3j)) has a closed form solu...
https://arxiv.org/abs/2106.07882	Do the Hodge spectra distinguish orbifolds from manifolds? Part 1	We examine the relationship between the singular set of a compact Riemannian orbifold and the spectrum of the Hodge Laplacian on $p$-forms by computing the heat invariants associated to the $p$-spectrum. We show that the heat invariants of the $0$-spectrum together with those of the $1$-spectrum for the corresponding H...	\section*{Introduction} A (Riemannian) orbifold is a versatile generalization of a (Riemannian) manifold that permits the presence of well-structured singular points. Orbifolds appear in a variety of mathematical areas and have applications in physics, in particular to string theory. Orbifolds are loc...	{ "timestamp": "2021-06-16T02:11:09", "yymm": "2106", "arxiv_id": "2106.07882", "language": "en", "url": "https://arxiv.org/abs/2106.07882", "abstract": "We examine the relationship between the singular set of a compact Riemannian orbifold and the spectrum of the Hodge Laplacian on $p$-forms by computing th...
https://arxiv.org/abs/1607.03014	The middle hedgehog of a planar convex body	A convexity point of a convex body is a point with the property that the union of the body and its reflection in the point is convex. It is proved that in the plane a typical convex body (in the sense of Baire category) has infinitely many convexity points. The proof makes use of the `middle hedgehog' of a planar conve...	\section{Introduction}\label{sec1} The following question was posed to me by Shiri Artstein--Avidan: `Does every convex body $K$ in the plane have a point $z$ such that the union of $K$ and its reflection in $z$ is convex?' After some surprise about never having come across this simple question, and after some fruitle...	{ "timestamp": "2016-07-12T02:15:54", "yymm": "1607", "arxiv_id": "1607.03014", "language": "en", "url": "https://arxiv.org/abs/1607.03014", "abstract": "A convexity point of a convex body is a point with the property that the union of the body and its reflection in the point is convex. It is proved that in...
https://arxiv.org/abs/math/0501230	Crossings and Nestings of Matchings and Partitions	We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric...	\section{Introduction} A (complete) matching on $[2n]=\{1,2,\dots,2n\}$ is a partition of $[2n]$ of type $(2,2, \dots, 2)$. It can be represented by listing its $n$ blocks, as $\{(i_1, j_1), (i_2, j_2), \dots, (i_n, j_n)\}$ where $i_r < j_r$ for $ 1 \leq r \leq n$. Two blocks (also called arcs) $(i_r, j_r)$ and $(i_s...	{ "timestamp": "2005-11-14T21:18:50", "yymm": "0501", "arxiv_id": "math/0501230", "language": "en", "url": "https://arxiv.org/abs/math/0501230", "abstract": "We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating...
https://arxiv.org/abs/1609.01321	Backward Error Analysis for Perturbation Methods	We demonstrate via several examples how the backward error viewpoint can be used in the analysis of solutions obtained by perturbation methods. We show that this viewpoint is quite general and offers several important advantages. Perhaps the most important is that backward error analysis can be used to demonstrate the ...	\section{Introduction} As the title suggests, the main idea of this paper is to use backward error analysis (BEA) to assess and interpret solutions obtained by perturbation methods. The idea will seem natural, perhaps even obvious, to those who are familiar with the way in which backward error analysis has s...	{ "timestamp": "2016-09-07T02:00:44", "yymm": "1609", "arxiv_id": "1609.01321", "language": "en", "url": "https://arxiv.org/abs/1609.01321", "abstract": "We demonstrate via several examples how the backward error viewpoint can be used in the analysis of solutions obtained by perturbation methods. We show th...
https://arxiv.org/abs/1801.06747	Connectivity of cubical polytopes	A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. We deal with the connectivity of the graphs of cubical polytopes. We first establish that, for any $d\ge 3$, the graph of a cubical $d$-polytope with minimum degree $\delta$ is $\min\{\delta,2d-2\}$-connected. Second, we sho...	\section{Introduction} The $k$-dimensional {\it skeleton} of a polytope $P$ is the set of all its faces of dimension of at most $k$. The 1-skeleton of $P$ is the {\it graph} $G(P)$ of $P$. We denote by $V(P)$ the vertex set of $P$. This paper studies the (vertex) connectivity of a cubical polytope, the (vertex) conne...	{ "timestamp": "2019-07-16T02:20:39", "yymm": "1801", "arxiv_id": "1801.06747", "language": "en", "url": "https://arxiv.org/abs/1801.06747", "abstract": "A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. We deal with the connectivity of the graphs of cubical pol...
https://arxiv.org/abs/2209.14474	A slight generalization of Steffensen Method for Solving Non Linear Equations	In this article, we present an iterative method to find simple roots of nonlinear equations, that is, to solving an equation of the form $f(x) = 0$. Different from Newton's method, the method we purpose do not require evaluation of derivatives. The method is based on the classical Steffensen's method and it is a slight...	\section{Introduction} Iterative methods for solving nonlinear real equations of the form $f(x) = 0$ have been widely studied by many researchers around the world. Newton's (or Newton-Raphson's) method certainly is the best known iterative method and studied in any numerical calculus course. From a initial guess $x_0...	{ "timestamp": "2022-09-30T02:05:12", "yymm": "2209", "arxiv_id": "2209.14474", "language": "en", "url": "https://arxiv.org/abs/2209.14474", "abstract": "In this article, we present an iterative method to find simple roots of nonlinear equations, that is, to solving an equation of the form $f(x) = 0$. Diffe...
https://arxiv.org/abs/1910.08189	Digital Fundamental Groups and Edge Groups of Clique Complexes	In previous work, we have defined---intrinsically, entirely within the digital setting---a fundamental group for digital images. Here, we show that this group is isomorphic to the edge group of the clique complex of the digital image considered as a graph. The clique complex is a simplicial complex and its edge group i...	\section{Introduction} A \emph{digital image} $X$ is a finite subset $X \subseteq \mathbb{Z}^n$ of the integral lattice in some $n$-dimensional Euclidean space, together with a particular adjacency relation on the set of points. This is an abstraction of an actual digital image which consists of pixels (in the plane...	{ "timestamp": "2019-10-21T02:03:47", "yymm": "1910", "arxiv_id": "1910.08189", "language": "en", "url": "https://arxiv.org/abs/1910.08189", "abstract": "In previous work, we have defined---intrinsically, entirely within the digital setting---a fundamental group for digital images. Here, we show that this g...
https://arxiv.org/abs/2103.14002	Ramanujan's Beautiful Integrals	Throughout his entire mathematical life, Ramanujan loved to evaluate definite integrals. One can find them in his problems submitted to the \emph{Journal of the Indian Mathematical Society}, notebooks, Quarterly Reports to the University of Madras, letters to Hardy, published papers and the Lost Notebook. His evaluatio...	\section{Introduction} Ramanujan loved infinite series and integrals. They permeate almost all of his work from the years he recorded his findings in notebooks \cite{nb} until the end of his life in 1920 at the age of 32. In this paper we provide a survey of some of his most beautiful theorems on integrals. Of course...	{ "timestamp": "2021-03-26T01:30:16", "yymm": "2103", "arxiv_id": "2103.14002", "language": "en", "url": "https://arxiv.org/abs/2103.14002", "abstract": "Throughout his entire mathematical life, Ramanujan loved to evaluate definite integrals. One can find them in his problems submitted to the \\emph{Journal...
https://arxiv.org/abs/1904.10765	Equipartitions and Mahler volumes of symmetric convex bodies	Following ideas of Iriyeh and Shibata we give a short proof of the three-dimensional Mahler conjecture {\mf for symmetric convex bodies}. Our contributions include, in particular, simple self-contained proofs of their two key statements. The first of these is an equipartition (ham sandwich type) theorem which refines a...	\section{Introduction} A {\it convex body} is a compact convex subset of $\mathbb R^n$ with non empty interior. We say that $K$ is {\it symmetric} if it is centrally symmetric with its center at the origin, i.e. $K=-K$. We write $\|L\|$ for the $k$-dimensional Lebesgue measure (volume) of a measurable set $L\subs...	{ "timestamp": "2021-01-21T02:18:53", "yymm": "1904", "arxiv_id": "1904.10765", "language": "en", "url": "https://arxiv.org/abs/1904.10765", "abstract": "Following ideas of Iriyeh and Shibata we give a short proof of the three-dimensional Mahler conjecture {\\mf for symmetric convex bodies}. Our contributio...
https://arxiv.org/abs/math/0510568	Winning rate in the full-information best choice problem	Following a long-standing suggestion by Gilbert and Mosteller, we derive an explicit formula for the asymptotic winning rate in the full-information problem of the best choice.	\section{Introduction} Let $X_1,X_2\ldots$ be a sequence of independent uniform $[0,1]$ random variables. The full-information best choice problem, as introduced by Gilbert and Mosteller \cite{GM}, asks one to find a stopping rule $\tau_n$ to maximise the probability \begin{equation}\label{stop} P_n(\tau):=\math...	{ "timestamp": "2005-11-19T22:44:31", "yymm": "0510", "arxiv_id": "math/0510568", "language": "en", "url": "https://arxiv.org/abs/math/0510568", "abstract": "Following a long-standing suggestion by Gilbert and Mosteller, we derive an explicit formula for the asymptotic winning rate in the full-information p...
https://arxiv.org/abs/2205.00879	An invitation to formal power series	This is a lecture on the theory of formal power series developed entirely without any analytic machinery. Combining ideas from various authors we are able to prove Newton's binomial theorem, Jacobi's triple product, the Rogers--Ramanujan identities and many other prominent results. We apply these methods to derive seve...	\section{Introduction} In a first course on abstract algebra students learn the difference between polynomial (real-valued) functions familiar from high school and formal polynomials defined over arbitrary fields. In courses on analysis they learn further that certain “well-behaved” functions possess a Taylor series e...	{ "timestamp": "2022-05-03T02:41:53", "yymm": "2205", "arxiv_id": "2205.00879", "language": "en", "url": "https://arxiv.org/abs/2205.00879", "abstract": "This is a lecture on the theory of formal power series developed entirely without any analytic machinery. Combining ideas from various authors we are able...
https://arxiv.org/abs/1703.02775	Cubical Covers of Sets in $\mathbb{R}^n$	Wild sets in $\mathbb{R}^n$ can be tamed through the use of various representations though sometimes this taming removes features considered important. Finding the wildest sets for which it is still true that the representations faithfully inform us about the original set is the focus of this rather playful, expository...	\section{Introduction} \label{sec:intro} In this paper we explain and illuminate a few ideas for (1) representing sets and (2) learning from those representations. Though some of the ideas and results we explain are likely written down elsewhere (though we are not aware of those references), our purpose is not to clai...	{ "timestamp": "2017-11-15T02:01:30", "yymm": "1703", "arxiv_id": "1703.02775", "language": "en", "url": "https://arxiv.org/abs/1703.02775", "abstract": "Wild sets in $\\mathbb{R}^n$ can be tamed through the use of various representations though sometimes this taming removes features considered important. F...
https://arxiv.org/abs/1303.5466	An O(N) Direct Solver for Integral Equations on the Plane	An efficient direct solver for volume integral equations with O(N) complexity for a broad range of problems is presented. The solver relies on hierarchical compression of the discretized integral operator, and exploits that off-diagonal blocks of certain dense matrices have numerically low rank. Technically, the solver...	\subsection{Notation and Preliminaries} We view an $N \times N$ matrix $A$ as a kernel function $K = K(p,q)$ evaluated at pairs of sample points. As it typically comes from an integral formulation of an elliptic PDE, aside from a low-rank block structure we also expect Green's identities to hold for $K$; this however...	{ "timestamp": "2013-05-16T02:00:21", "yymm": "1303", "arxiv_id": "1303.5466", "language": "en", "url": "https://arxiv.org/abs/1303.5466", "abstract": "An efficient direct solver for volume integral equations with O(N) complexity for a broad range of problems is presented. The solver relies on hierarchical ...
https://arxiv.org/abs/1507.00810	A Refinement of Vietoris Inequality for Cosine Polynomials	The classical Vietoris cosine inequality is refined by establishing a positive polynomial lower bound.	\section{Introduction} In 1958, Vietoris \cite{V} published the following ``surprising and quite deep result" \cite[p. 1]{K} on inequalities for a class of sine and cosine polynomials. \vspace{0.3cm} \noindent {\bf{Proposition 1.}} \emph{If the real numbers $a_k$ $(k=0,1,...,n)$ satisfy $$ a_0\geq a_1 \geq \cdot...	{ "timestamp": "2015-07-06T02:04:02", "yymm": "1507", "arxiv_id": "1507.00810", "language": "en", "url": "https://arxiv.org/abs/1507.00810", "abstract": "The classical Vietoris cosine inequality is refined by establishing a positive polynomial lower bound.", "subjects": "Classical Analysis and ODEs (math....
https://arxiv.org/abs/math/0701149	Sums of Consecutive Integers	A decomposition of a natural number n is a sequence of consecutive natural numbers that sums to n. We construct a one-to-one correspondence between the odd factors of a natural number and its decompositions. We study the decompositions by their lengths and introduce the concept of length spectrum. Also, we use diagrams...	\section*{The Proof} We start by defining a {\em decomposition of a natural number $ n$} to be a sequence of consecutive natural numbers whose sum is $n$. The number of terms is called the {\em length of the decomposition}, and a decomposition of length $1$ is called {\em trivial}. Further, a decomposition is called ...	{ "timestamp": "2007-01-05T00:19:01", "yymm": "0701", "arxiv_id": "math/0701149", "language": "en", "url": "https://arxiv.org/abs/math/0701149", "abstract": "A decomposition of a natural number n is a sequence of consecutive natural numbers that sums to n. We construct a one-to-one correspondence between th...
https://arxiv.org/abs/1101.5652	Completeness of Ordered Fields	The main goal of this project is to prove the equivalency of several characterizations of completeness of Archimedean ordered fields; some of which appear in most modern literature as theorems following from the Dedekind completeness of the real numbers, while a couple are not as well known and have to do with other ar...	\section*{Introduction} In most textbooks, the set of real numbers $\mathbb{R}$ is commonly taken to be a totally ordered Dedekind complete field. Following from this definition, one can then establish the basic properties of $\mathbb{R}$ such as the Bolzano-Weierstrass property, the Monotone Convergence property, the ...	{ "timestamp": "2011-02-01T02:00:30", "yymm": "1101", "arxiv_id": "1101.5652", "language": "en", "url": "https://arxiv.org/abs/1101.5652", "abstract": "The main goal of this project is to prove the equivalency of several characterizations of completeness of Archimedean ordered fields; some of which appear i...
https://arxiv.org/abs/0808.2160	Local energy estimates for the finite element method on sharply varying grids	Local energy error estimates for the finite element method for elliptic problems were originally proved in 1974 by Nitsche and Schatz. These estimates show that the local energy error may be bounded by a local approximation term, plus a global "pollution" term that measures the influence of solution quality from outsid...	\section{Introduction} In this note we prove local energy error estimates for the finite element method for second-order linear elliptic problems on highly refined triangulations. Most a priori error analyses for the finite element method in norms other than the global energy norm place severe restrictions on the mesh...	{ "timestamp": "2008-08-15T18:04:14", "yymm": "0808", "arxiv_id": "0808.2160", "language": "en", "url": "https://arxiv.org/abs/0808.2160", "abstract": "Local energy error estimates for the finite element method for elliptic problems were originally proved in 1974 by Nitsche and Schatz. These estimates show ...
https://arxiv.org/abs/2012.05138	On the minimum value of the condition number of polynomials	In 1993, Shub and Smale posed the problem of finding a sequence of univariate polynomials of degree $N$ with condition number bounded above by $N$. In a previous paper by C. Beltán, U. Etayo, J. Marzo and J. Ortega-Cerdà, it was proved that the optimal value of the condition number is of the form $O(\sqrt{N})$, and the...	\section{Introduction} \subsection{Statement of the main problem} The condition number of a polynomial at a root is a measure for the first order variation of the root under small perturbations of the polynomial. It has different formulas and properties depending on how these changes are measured, see for example \cite...	{ "timestamp": "2020-12-10T02:25:18", "yymm": "2012", "arxiv_id": "2012.05138", "language": "en", "url": "https://arxiv.org/abs/2012.05138", "abstract": "In 1993, Shub and Smale posed the problem of finding a sequence of univariate polynomials of degree $N$ with condition number bounded above by $N$. In a p...