url stringlengths 31 38 | title stringlengths 7 229 | abstract stringlengths 44 2.87k | text stringlengths 319 2.51M | meta dict |
|---|---|---|---|---|
https://arxiv.org/abs/0807.4369 | Some combinatorial properties of flag simplicial pseudomanifolds and spheres | A simplicial complex $\Delta$ is called flag if all minimal nonfaces of $\Delta$ have at most two elements. The following are proved: First, if $\Delta$ is a flag simplicial pseudomanifold of dimension $d-1$, then the graph of $\Delta$ (i) is $(2d-2)$-vertex-connected and (ii) has a subgraph which is a subdivision of t... | \section{Introduction}
\label{intro}
We will be interested in finite simplicial complexes. Such a complex
$\Delta$ is called \emph{flag} if every set of vertices which are
pairwise joined by edges in $\Delta$ is a face of $\Delta$. For instance,
every order complex (meaning the simplicial complex of all chains in a ... | {
"timestamp": "2009-05-28T09:37:25",
"yymm": "0807",
"arxiv_id": "0807.4369",
"language": "en",
"url": "https://arxiv.org/abs/0807.4369",
"abstract": "A simplicial complex $\\Delta$ is called flag if all minimal nonfaces of $\\Delta$ have at most two elements. The following are proved: First, if $\\Delta$ ... |
https://arxiv.org/abs/1712.09677 | Momentum and Stochastic Momentum for Stochastic Gradient, Newton, Proximal Point and Subspace Descent Methods | In this paper we study several classes of stochastic optimization algorithms enriched with heavy ball momentum. Among the methods studied are: stochastic gradient descent, stochastic Newton, stochastic proximal point and stochastic dual subspace ascent. This is the first time momentum variants of several of these metho... | \section{Introduction} \label{sec:intro}
Two of the most popular algorithmic ideas for solving optimization problems involving big volumes of data are {\em stochastic approximation} and {\em momentum}. By stochastic approximation we refer to the practice pioneered by Robins and Monro \cite{robbins1951stochastic} of ... | {
"timestamp": "2018-03-30T02:00:39",
"yymm": "1712",
"arxiv_id": "1712.09677",
"language": "en",
"url": "https://arxiv.org/abs/1712.09677",
"abstract": "In this paper we study several classes of stochastic optimization algorithms enriched with heavy ball momentum. Among the methods studied are: stochastic ... |
https://arxiv.org/abs/1702.01299 | On a question of Erdos and Faudree on the size Ramsey numbers | For given simple graphs $G_1$ and $G_2$, the size Ramsey number $\hat{R}(G_1,G_2)$ is the smallest positive integer $m$, where there exists a graph $G$ with $m$ edges such that in any edge coloring of $G$ with two colors red and blue, there is either a red copy of $G_1$ or a blue copy of $G_2$. In 1981, Erdős and Faudr... | \section{Introduction}\label{sec:intro}
In this paper, all graphs are finite, undirected and
simple. We also follow \cite{Boundy} for the terminology
and notation not defined here. Let $G,G_1,\ldots,G_n$ be given graphs. We write $G\rightarrow (G_1,\ldots,G_n)$, if in every edge coloring of $G$ with $n$ colors, the... | {
"timestamp": "2017-02-07T02:04:09",
"yymm": "1702",
"arxiv_id": "1702.01299",
"language": "en",
"url": "https://arxiv.org/abs/1702.01299",
"abstract": "For given simple graphs $G_1$ and $G_2$, the size Ramsey number $\\hat{R}(G_1,G_2)$ is the smallest positive integer $m$, where there exists a graph $G$ w... |
https://arxiv.org/abs/2208.12563 | Ramsey theory constructions from hypergraph matchings | We give asymptotically optimal constructions in generalized Ramsey theory using results about conflict-free hypergraph matchings. For example, we present an edge-coloring of $K_{n,n}$ with $2n/3 + o(n)$ colors such that each $4$-cycle receives at least three colors on its edges. This answers a question of Axenovich, Fü... | \section{Introduction}
Given graphs $G$ and $H$, an $(H,q)$-coloring of~$G$ is an edge-coloring of~$G$ such that every copy of~$H$ in~$G$ receives at least distinct~$q$ colors.
Let $r(G, H, q)$ be the
minimum number of colors in an $(H,q)$-coloring of $G$.
Classical Ramsey numbers
for multicolorings are the sp... | {
"timestamp": "2022-08-29T02:11:26",
"yymm": "2208",
"arxiv_id": "2208.12563",
"language": "en",
"url": "https://arxiv.org/abs/2208.12563",
"abstract": "We give asymptotically optimal constructions in generalized Ramsey theory using results about conflict-free hypergraph matchings. For example, we present ... |
https://arxiv.org/abs/2203.14129 | Nash, Conley, and Computation: Impossibility and Incompleteness in Game Dynamics | Under what conditions do the behaviors of players, who play a game repeatedly, converge to a Nash equilibrium? If one assumes that the players' behavior is a discrete-time or continuous-time rule whereby the current mixed strategy profile is mapped to the next, this becomes a problem in the theory of dynamical systems.... | \section{Background in Conley Theory}
In this section we review fundamental notions of dynamical systems, especially from the point of view of Conley theory. The standard reference for chain recurrence, Conley's decomposition theorem and the Conley index is the monograph \cite{conley1978isolated}. An approachable in... | {
"timestamp": "2022-03-29T02:18:41",
"yymm": "2203",
"arxiv_id": "2203.14129",
"language": "en",
"url": "https://arxiv.org/abs/2203.14129",
"abstract": "Under what conditions do the behaviors of players, who play a game repeatedly, converge to a Nash equilibrium? If one assumes that the players' behavior i... |
https://arxiv.org/abs/1702.06112 | A general framework for path convexities | In this work we deal with the so-called path convexities, defined over special collections of paths. For example, the collection of the shortest paths in a graph is associated with the well-known geodesic convexity, while the collection of the induced paths is associated with the monophonic convexity; and there are man... | \section{Introduction}
A {\em finite convexity space} is a pair $(V,\mathcal{C})$ consisting of a finite set $V$ and a family $\mathcal{C}$ of subsets of $V$ such that $\emptyset\in\mathcal{C}$, $V\in\mathcal{C}$, and $\mathcal{C}$ is closed under intersection. Members of $\mathcal{C}$ are called {\em convex sets}.
L... | {
"timestamp": "2017-02-21T02:13:14",
"yymm": "1702",
"arxiv_id": "1702.06112",
"language": "en",
"url": "https://arxiv.org/abs/1702.06112",
"abstract": "In this work we deal with the so-called path convexities, defined over special collections of paths. For example, the collection of the shortest paths in ... |
https://arxiv.org/abs/0808.1202 | Equidistribution of the Fekete points on the sphere | The Fekete points are the points that maximize a Vandermonde-type determinant that appears in the polynomial Lagrange interpolation formula. They are well suited points for interpolation formulas and numerical integration. We prove the asymptotic equidistribution of the Fekete points in the sphere. The way we proceed i... | \section{Introduction}
For any integer $\ell\ge 0,$ let $\mathcal{H}_{\ell}$ be the space of spherical
harmonics of degree $\ell$ in $\S^{d}$. For any integer $L\ge 0$ we denote the
space of spherical harmonics of degree not exceeding $L$ by $\Pi_{L}$ These
vector spaces have dimensions
\[
\dim \Pi_{L}=\frac{d+2L}... | {
"timestamp": "2008-08-08T14:34:38",
"yymm": "0808",
"arxiv_id": "0808.1202",
"language": "en",
"url": "https://arxiv.org/abs/0808.1202",
"abstract": "The Fekete points are the points that maximize a Vandermonde-type determinant that appears in the polynomial Lagrange interpolation formula. They are well s... |
https://arxiv.org/abs/2301.08701 | Smallest posets with given cyclic automorphism group | For each $n\ge 1$ we determine the minimum number of points in a poset with cyclic automorphism group of order $n$. | \section{Introduction}
In 1938 R. Frucht \cite{Fru} proved that any finite group can be realized as the automorphism group of a graph. Moreover, the graph can be taken with $3d|G|$ vertices, where $d$ is the cardinality of any generator set of $G$ (\cite[Theorems 3.2, 4.2]{Fru49}). In 1959 G. Sabidussi \cite{Sab} show... | {
"timestamp": "2023-01-23T02:15:02",
"yymm": "2301",
"arxiv_id": "2301.08701",
"language": "en",
"url": "https://arxiv.org/abs/2301.08701",
"abstract": "For each $n\\ge 1$ we determine the minimum number of points in a poset with cyclic automorphism group of order $n$.",
"subjects": "Combinatorics (math.... |
https://arxiv.org/abs/1009.2823 | Theory and applications of lattice point methods for binomial ideals | This survey of methods surrounding lattice point methods for binomial ideals begins with a leisurely treatment of the geometric combinatorics of binomial primary decomposition. It then proceeds to three independent applications whose motivations come from outside of commutative algebra: hypergeometric systems, combinat... | \section*{Introduction}\label{s:intro
Binomial ideals in polynomial rings over algebraically closed fields
admit \emph{binomial primary decompositions}: expressions as
intersections of primary binomial ideals. The algebra of these
decompositions is governed by the geometry of lattice points in
polyhedra and related ... | {
"timestamp": "2010-09-16T02:00:56",
"yymm": "1009",
"arxiv_id": "1009.2823",
"language": "en",
"url": "https://arxiv.org/abs/1009.2823",
"abstract": "This survey of methods surrounding lattice point methods for binomial ideals begins with a leisurely treatment of the geometric combinatorics of binomial pr... |
https://arxiv.org/abs/math/0603245 | Normal forms for orthogonal similarity classes of skew-symmetric matrices | Let F be an algebraically closed field of characteristic different from 2. We show that every nonsingular skew-symmetric n by n matrix X over F is orthogonally similar to a bidiagonal skew-symmetric matrix. In the singular case one has to allow some 4-diagonal blocks as well. If further the characteristic is 0, we cons... | \section{Introduction}
In this note $F$ denotes an algebraically closed field of
characteristic not $2$. By $M_n(F)$ we denote the algebra of $n$
by $n$ matrices over $F$, by ${\mbox{\rm GL}}_n(F)$ the group of invertible
elements of $M_n(F)$, and by $I_n$ the identity matrix of size
$n$. For any matrix $X$, let $X'$... | {
"timestamp": "2006-12-04T12:12:20",
"yymm": "0603",
"arxiv_id": "math/0603245",
"language": "en",
"url": "https://arxiv.org/abs/math/0603245",
"abstract": "Let F be an algebraically closed field of characteristic different from 2. We show that every nonsingular skew-symmetric n by n matrix X over F is ort... |
https://arxiv.org/abs/1911.03424 | Approximation Bounds for Interpolation and Normals on Triangulated Surfaces and Manifolds | How good is a triangulation as an approximation of a smooth curved surface or manifold? We provide bounds on the {\em interpolation error}, the error in the position of the surface, and the {\em normal error}, the error in the normal vectors of the surface, as approximated by a piecewise linearly triangulated surface w... | \section{Introduction}
Triangulations of surfaces are used heavily in computer graphics,
visualization, and geometric modeling;
they also find applications in scientific computing.
Also useful are triangulations of manifolds in
spaces of dimension higher than three---for example, as
a tool for studying the topology o... | {
"timestamp": "2019-11-11T02:28:26",
"yymm": "1911",
"arxiv_id": "1911.03424",
"language": "en",
"url": "https://arxiv.org/abs/1911.03424",
"abstract": "How good is a triangulation as an approximation of a smooth curved surface or manifold? We provide bounds on the {\\em interpolation error}, the error in ... |
https://arxiv.org/abs/1909.11441 | Sharp stability for the Riesz potential | In this paper we show the stability of the ball as maximizer of the Riesz potential among sets of given volume. The stability is proved with sharp exponent $1/2$, and is valid for any dimension $N\geq 2$ and any power $1<\alpha<N$. | \section{Introduction}
The celebrated \emph{Riesz inequality} states that for any two positive functions $f,\,g:\mathbb R^N\to \mathbb R^+$ and any positive, decreasing function $h:\mathbb R^+\to \mathbb R^+$, one has
\begin{equation}\label{Riesz}
\int_{\mathbb R^N}\int_{\mathbb R^N} f(z)g(y) h(|y-z|)\,dy\,dz \leq \int... | {
"timestamp": "2019-09-26T02:11:36",
"yymm": "1909",
"arxiv_id": "1909.11441",
"language": "en",
"url": "https://arxiv.org/abs/1909.11441",
"abstract": "In this paper we show the stability of the ball as maximizer of the Riesz potential among sets of given volume. The stability is proved with sharp exponen... |
https://arxiv.org/abs/1103.1412 | 2-strand twisting and knots with identical quantum knot homologies | Given a knot, we ask how its Khovanov and Khovanov-Rozansky homologies change under the operation of introducing twists in a pair of strands. We obtain long exact sequences in homology and further algebraic structure which is then used to derive topological and computational results. Two of our applications include giv... | \section{Introduction and results}
In this paper we consider $sl(n)$ Khovanov-Rozansky homology (Khovanov homology appears as $n=2$) under the operation of adding twists in a pair of strands. We observe stabilization of the homology as we add more twists and, looking a little deeper, reveal some further algebraic stru... | {
"timestamp": "2011-05-04T02:05:31",
"yymm": "1103",
"arxiv_id": "1103.1412",
"language": "en",
"url": "https://arxiv.org/abs/1103.1412",
"abstract": "Given a knot, we ask how its Khovanov and Khovanov-Rozansky homologies change under the operation of introducing twists in a pair of strands. We obtain long... |
https://arxiv.org/abs/1306.0256 | Distributions of Angles in Random Packing on Spheres | This paper studies the asymptotic behaviors of the pairwise angles among n randomly and uniformly distributed unit vectors in R^p as the number of points n -> infinity, while the dimension p is either fixed or growing with n. For both settings, we derive the limiting empirical distribution of the random angles and the ... | \section{Introduction}
\label{intro}
\setcounter{equation}{0}
The distribution of the Euclidean and geodesic distances between two random points on a unit sphere or other geometric objects has a wide range of applications including transportation networks, pattern recognition, molecular biology, geometric probability,... | {
"timestamp": "2013-06-04T02:02:37",
"yymm": "1306",
"arxiv_id": "1306.0256",
"language": "en",
"url": "https://arxiv.org/abs/1306.0256",
"abstract": "This paper studies the asymptotic behaviors of the pairwise angles among n randomly and uniformly distributed unit vectors in R^p as the number of points n ... |
https://arxiv.org/abs/1706.04410 | A strong converse bound for multiple hypothesis testing, with applications to high-dimensional estimation | In statistical inference problems, we wish to obtain lower bounds on the minimax risk, that is to bound the performance of any possible estimator. A standard technique to obtain risk lower bounds involves the use of Fano's inequality. In an information-theoretic setting, it is known that Fano's inequality typically doe... | \section{Introduction} \label{sec:intro}
When solving an inference problem, we would like to know if the algorithm we use is close to optimal. In statistical language we seek to give a lower bound on the performance
of any estimator over a class of problems (often called the minimax risk over the class).
In the l... | {
"timestamp": "2018-04-06T02:01:22",
"yymm": "1706",
"arxiv_id": "1706.04410",
"language": "en",
"url": "https://arxiv.org/abs/1706.04410",
"abstract": "In statistical inference problems, we wish to obtain lower bounds on the minimax risk, that is to bound the performance of any possible estimator. A stand... |
https://arxiv.org/abs/1410.6943 | A monotonicity property for generalized Fibonacci sequences | Given k>1, let a_n be the sequence defined by the recurrence a_n=c_1a_{n-1}+c_2a_{n-2}+...+c_ka_{n-k} for n>=k, with initial values a_0=a_1=...=a_{k-2}=0 and a_{k-1}= 1. We show under a couple of assumptions concerning the constants c_i that the ratio of the n-th root of a_n to the (n-1)-st root of a_{n-1} is strictly ... | \section{Introduction}
In 1982, Firoozbakht conjectured that the sequence $\{\sqrt[n]{p_n}\}_{n\geq1}$ is strictly decreasing, where $p_n$ denotes the $n$-th prime. A stronger conjecture was later made by Sun \cite{Sun} that in fact
$$\frac{\sqrt[n+1]{p_{n+1}}}{\sqrt[n]{p_n}}<1-\frac{\log\log n}{2n^2}, \qquad n>4,... | {
"timestamp": "2014-10-28T01:08:29",
"yymm": "1410",
"arxiv_id": "1410.6943",
"language": "en",
"url": "https://arxiv.org/abs/1410.6943",
"abstract": "Given k>1, let a_n be the sequence defined by the recurrence a_n=c_1a_{n-1}+c_2a_{n-2}+...+c_ka_{n-k} for n>=k, with initial values a_0=a_1=...=a_{k-2}=0 an... |
https://arxiv.org/abs/1807.10231 | Polyominoes with maximally many holes | What is the maximum number of holes that a polyomino with $n$ tiles can enclose? Call this number $f(n)$. We show that if $n_k = \left( 2^{2k+1} + 3 \cdot 2^{k+1}+4 \right) / 3$ and $h_k = \left( 2^{2k}-1 \right) /3$, then $f(n_k) = h_k$ for $k \ge 1$. We also give nearly matching upper and lower bounds for large $n$, ... | \section{Polyominoes.}
\emph{Polyominoes}, first studied systematically by Golomb \cite{golomb1954checker}, are shapes that can be made by gluing together finitely many unit squares, edge to edge. A polyomino with $n$ squares is sometimes called an \emph{$n$-omino}.
Tiling problems involving polyominoes are well studi... | {
"timestamp": "2018-07-27T02:12:09",
"yymm": "1807",
"arxiv_id": "1807.10231",
"language": "en",
"url": "https://arxiv.org/abs/1807.10231",
"abstract": "What is the maximum number of holes that a polyomino with $n$ tiles can enclose? Call this number $f(n)$. We show that if $n_k = \\left( 2^{2k+1} + 3 \\cd... |
https://arxiv.org/abs/1706.02661 | The spectral determination of the multicone graphs Kw+P | The main aim of this study is to characterize new classes of multicone graphs which are determined by their signless Laplacian spectra, their Laplacian spectra and their adjacency spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let P and Kw denote the Petersen graph and a complete ... | \section{Introduction}
Let $G = (V,E)$ be a simple graph with the vertex set $ V = V (G) = \left\{ {v_1, \cdots , v_n} \right\}$ and the edge set $E = E(G) = \left\{ {e_1, \cdots , e_m} \right\}$. Denote by $d(v)$ the degree of vertex $ v $. All notions on graphs that are not defined here may be found in \cite{LP, Ba... | {
"timestamp": "2017-11-27T02:07:34",
"yymm": "1706",
"arxiv_id": "1706.02661",
"language": "en",
"url": "https://arxiv.org/abs/1706.02661",
"abstract": "The main aim of this study is to characterize new classes of multicone graphs which are determined by their signless Laplacian spectra, their Laplacian sp... |
https://arxiv.org/abs/2007.13241 | Beyond the Worst-Case Analysis of Algorithms (Introduction) | One of the primary goals of the mathematical analysis of algorithms is to provide guidance about which algorithm is the "best" for solving a given computational problem. Worst-case analysis summarizes the performance profile of an algorithm by its worst performance on any input of a given size, implicitly advocating fo... | \section{The Worst-Case Analysis of Algorithms}\label{s:intro}
\subsection{Comparing Incomparable Algorithms}
Comparing different algorithms is hard. For almost any pair of
algorithms and measure of algorithm performance,
each algorithm will perform better
than the other on some inputs.
For example, the MergeSort a... | {
"timestamp": "2020-07-28T02:27:48",
"yymm": "2007",
"arxiv_id": "2007.13241",
"language": "en",
"url": "https://arxiv.org/abs/2007.13241",
"abstract": "One of the primary goals of the mathematical analysis of algorithms is to provide guidance about which algorithm is the \"best\" for solving a given compu... |
https://arxiv.org/abs/0712.4358 | Limit Theorems for Internal Aggregation Models | We study the scaling limits of three different aggregation models on the integer lattice Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each sit... | \chapter{Introduction}
\section{Three Models with the Same Scaling Limit}
\label{scalinglimitintro}
Given finite sets $A, B \subset \Z^d$, Diaconis and Fulton \cite{DF} defined the \emph{smash sum} $A \oplus B$ as a certain random set whose cardinality is the sum of the cardinalities of $A$ and $B$. Write $A \cap B ... | {
"timestamp": "2007-12-28T15:04:22",
"yymm": "0712",
"arxiv_id": "0712.4358",
"language": "en",
"url": "https://arxiv.org/abs/0712.4358",
"abstract": "We study the scaling limits of three different aggregation models on the integer lattice Z^d: internal DLA, in which particles perform random walks until re... |
https://arxiv.org/abs/1611.03840 | The Length of the Longest Common Subsequence of Two Independent Mallows Permutations | The Mallows measure is a probability measure on $S_n$ where the probability of a permutation $\pi$ is proportional to $q^{l(\pi)}$ with $q > 0$ being a parameter and $l(\pi)$ the number of inversions in $\pi$. We prove a weak law of large numbers for the length of the longest common subsequences of two independent perm... | \section{Introduction}\label{S1}
\subsection{Background}
The longest common subsequence(LCS) problem is a classical problem which has application in fields such as molecular biology (see, e.g., \cite{pevzner}) , data comparison and software version control. Most previous works on the LCS problem are focused on the cas... | {
"timestamp": "2017-01-30T02:02:05",
"yymm": "1611",
"arxiv_id": "1611.03840",
"language": "en",
"url": "https://arxiv.org/abs/1611.03840",
"abstract": "The Mallows measure is a probability measure on $S_n$ where the probability of a permutation $\\pi$ is proportional to $q^{l(\\pi)}$ with $q > 0$ being a ... |
https://arxiv.org/abs/2107.00837 | Investment AUM Fee Costs: Evaluating a Simple Formula | How much do financial management fees cost investors? This article studies an approximate formula for the cumulative costs of annual Assets Under Management (AUM) fees. The formula states that an investment paying an annual fee of $\epsilon$% of AUM over $N$ years loses almost $N\epsilon$% of its value, compared to an ... | \section{Result}
The main result is a simple formula to approximate portfolio loss to investment fees, in the absence of offsetting benefit:
\begin{center}
\emph{An investment with annual fee expense $\epsilon$\% compounding for $N$ years loses almost $N\epsilon$\% of its value to fees.}
\end{center}
E.g., a 30 year in... | {
"timestamp": "2021-07-05T02:08:23",
"yymm": "2107",
"arxiv_id": "2107.00837",
"language": "en",
"url": "https://arxiv.org/abs/2107.00837",
"abstract": "How much do financial management fees cost investors? This article studies an approximate formula for the cumulative costs of annual Assets Under Manageme... |
https://arxiv.org/abs/2104.12262 | GCD of sums of $k$ consecutive Fibonacci, Lucas, and generalized Fibonacci numbers | We explore the sums of $k$ consecutive terms in the generalized Fibonacci sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence $G_n = G_{n-1} + G_{n-2}$ for all $n \geq 2$ with integral initial conditions $G_0$ and $G_1$. In particular, we give precise values for the greatest common divisor (GCD) of all sums ... | \section{Introduction}\label{sec:introduction}
In the inaugural issue of the \textit{Fibonacci Quarterly} in 1963, I.~D.~Ruggles proposed the following problem in the Elementary Problems section: ``Show that the sum of twenty consecutive Fibonacci numbers is divisible by the $10^{\mathrm{th}}$ Fibonacci number $F_{1... | {
"timestamp": "2021-11-09T02:25:08",
"yymm": "2104",
"arxiv_id": "2104.12262",
"language": "en",
"url": "https://arxiv.org/abs/2104.12262",
"abstract": "We explore the sums of $k$ consecutive terms in the generalized Fibonacci sequence $\\left(G_n\\right)_{n \\geq 0}$ given by the recurrence $G_n = G_{n-1}... |
https://arxiv.org/abs/2101.03824 | Choosing points on cubic plane curves: rigidity and flexibility | Every smooth cubic plane curve has 9 flex points and 27 sextatic points. We study the following question asked by Farb: Is it true that the known algebraic structures give all the possible ways to continuously choose $n$ distinct points on every smooth cubic plane curve, for each given positive integer $n$? We give an ... | \section{Introduction}
It is a classical topic to study certain structures of special points on complex smooth cubic plane curves, for example, the 9 flex points (first attributed to Maclaurin; see Introduction of \cite{AD} for a brief history) and the 27 sextatic points (studied by Cayley \cite{Cayley}).
Inspired by ... | {
"timestamp": "2021-03-25T01:22:21",
"yymm": "2101",
"arxiv_id": "2101.03824",
"language": "en",
"url": "https://arxiv.org/abs/2101.03824",
"abstract": "Every smooth cubic plane curve has 9 flex points and 27 sextatic points. We study the following question asked by Farb: Is it true that the known algebrai... |
https://arxiv.org/abs/2004.06671 | On the Stability of Fourier Phase Retrieval | Phase retrieval is concerned with recovering a function $f$ from the absolute value of its Fourier transform $|\widehat{f}|$. We study the stability properties of this problem in Lebesgue spaces. Our main results shows that $$ \| f-g\|_{L^2(\mathbb{R}^n)} \leq 2\cdot \| |\widehat{f}| - |\widehat{g}| \|_{L^2(\mathbb{R}^... | \section{Introduction and Results}
\subsection{Introduction} Phase retrieval refers to a broad class of problems where one is given incomplete information about an object (often the size of the coefficients with respect to some basis expansion but not their phase) and then tries to reconstruct the object. In the case o... | {
"timestamp": "2021-03-29T02:26:05",
"yymm": "2004",
"arxiv_id": "2004.06671",
"language": "en",
"url": "https://arxiv.org/abs/2004.06671",
"abstract": "Phase retrieval is concerned with recovering a function $f$ from the absolute value of its Fourier transform $|\\widehat{f}|$. We study the stability prop... |
https://arxiv.org/abs/math/0603520 | Alternating permutations and symmetric functions | We use the theory of symmetric functions to enumerate various classes of alternating permutations w of {1,2,...,n}. These classes include the following: (1) both w and w^{-1} are alternating, (2) w has certain special shapes, such as (m-1,m-2,...,1), under the RSK algorithm, (3) w has a specified cycle type, and (4) w ... | \section{Introduction.} \label{sec1}
\indent This paper can be regarded as a sequel to the classic paper
\cite{foulkes} of H. O. Foulkes in which he relates the enumeration of
alternating permutations to the representation theory of the symmetric
group and the theory of symmetric functions. We assume familiarity
with s... | {
"timestamp": "2006-07-05T22:32:17",
"yymm": "0603",
"arxiv_id": "math/0603520",
"language": "en",
"url": "https://arxiv.org/abs/math/0603520",
"abstract": "We use the theory of symmetric functions to enumerate various classes of alternating permutations w of {1,2,...,n}. These classes include the followin... |
https://arxiv.org/abs/2111.13551 | Optimal Estimation of Schatten Norms of a rectangular Matrix | We consider the twin problems of estimating the effective rank and the Schatten norms $\|{\bf A}\|_{s}$ of a rectangular $p\times q$ matrix ${\bf A}$ from noisy observations. When $s$ is an even integer, we introduce a polynomial-time estimator of $\|{\bf A}\|_s$ that achieves the minimax rate $(pq)^{1/4}$. Interesting... | \section{Introduction}
In many modern problems, scientists are faced with a large data matrix $\mathbf{Y}$, which is often assumed to be sum of a signal matrix $\mathbf{A}$ and a background noise. Under some structural assumptions on the signal such as small rank, it is possible to precisely recover the signal matrix ... | {
"timestamp": "2021-11-29T02:37:50",
"yymm": "2111",
"arxiv_id": "2111.13551",
"language": "en",
"url": "https://arxiv.org/abs/2111.13551",
"abstract": "We consider the twin problems of estimating the effective rank and the Schatten norms $\\|{\\bf A}\\|_{s}$ of a rectangular $p\\times q$ matrix ${\\bf A}$... |
https://arxiv.org/abs/1612.02719 | A short proof of Rudnev's point-plane incidence bound | In this note we give a shortened proof of a theorem of Rudnev, which bounds the number of incidences between points and planes over an arbitrary field. Rudnev's proof uses a map that goes via the four-dimensional Klein quadric to a three-dimensional space, where it applies a bound of Guth and Katz on intersection point... | \section{Introduction}
Let $\mathbb F$ be any field.
Given a point set $P\subset \mathbb F^3$ and a set $Q$ of planes,
we define the number of \emph{incidences} between $P$ and $Q$ to be $I(P,Q) = |\{(p,q)\in P\times Q: p\in q\}|$.
Rudnev \cite{R} proved the following incidence bound.
\begin{theorem}\label{thm:rudne... | {
"timestamp": "2016-12-09T02:06:35",
"yymm": "1612",
"arxiv_id": "1612.02719",
"language": "en",
"url": "https://arxiv.org/abs/1612.02719",
"abstract": "In this note we give a shortened proof of a theorem of Rudnev, which bounds the number of incidences between points and planes over an arbitrary field. Ru... |
https://arxiv.org/abs/1312.0101 | Locating the first nodal set in higher dimensions | This paper estimates the location and the width of the nodal set of the first Neumann eigenfunctions on a smooth convex domain $\Omega \subset \mathbb R^n$, whose length is normalized to be 1 and whose cross-section is contained in a ball of radius $\epsilon$. In \cite{CJK2009}, an $O(\epsilon)$ bound was obtained by c... | \section{Introduction and Statement of Results}
This paper concerns the nodal set of the eigenfunction of the Laplacian. The nodal set is the set of zeros of the eigenfunction. The geometry of the nodal set is greatly affected by the domain on which the Laplacian is defined. In particular, if the domain is long and nar... | {
"timestamp": "2013-12-12T02:11:23",
"yymm": "1312",
"arxiv_id": "1312.0101",
"language": "en",
"url": "https://arxiv.org/abs/1312.0101",
"abstract": "This paper estimates the location and the width of the nodal set of the first Neumann eigenfunctions on a smooth convex domain $\\Omega \\subset \\mathbb R^... |
https://arxiv.org/abs/0905.0121 | Tropical Scaling of Polynomial Matrices | The eigenvalues of a matrix polynomial can be determined classically by solving a generalized eigenproblem for a linearized matrix pencil, for instance by writing the matrix polynomial in companion form. We introduce a general scaling technique, based on tropical algebra, which applies in particular to this companion f... | \section{Introduction}
\label{sec:1}
A classical problem is to compute the eigenvalues of a matrix polynomial
\[
P(\lambda)=A_{0}+A_{1}\lambda+\cdots +A_{d}\lambda^d
\]
where $A_{l}\in \mathbb{C}^{n\times n},l=0 \ldots d$ are given. The eigenvalues are
defined as the solutions of $\det(P(\lambda))=0$. If $\lambda$ is a... | {
"timestamp": "2009-05-01T19:04:37",
"yymm": "0905",
"arxiv_id": "0905.0121",
"language": "en",
"url": "https://arxiv.org/abs/0905.0121",
"abstract": "The eigenvalues of a matrix polynomial can be determined classically by solving a generalized eigenproblem for a linearized matrix pencil, for instance by w... |
https://arxiv.org/abs/1807.00891 | Optimality and Sub-optimality of PCA I: Spiked Random Matrix Models | A central problem of random matrix theory is to understand the eigenvalues of spiked random matrix models, introduced by Johnstone, in which a prominent eigenvector (or "spike") is planted into a random matrix. These distributions form natural statistical models for principal component analysis (PCA) problems throughou... | \section{Behavior near criticality}\label{app:critical}
In Gaussian Wigner settings where we have established contiguity for all $\lambda<1$, it is natural to ask whether the spiked and unspiked models remain contiguous for a sequence $\lambda = 1 + \delta_n$ with $\delta_n \to 0$ (here $\delta_n$ may be positive or n... | {
"timestamp": "2018-07-16T02:03:34",
"yymm": "1807",
"arxiv_id": "1807.00891",
"language": "en",
"url": "https://arxiv.org/abs/1807.00891",
"abstract": "A central problem of random matrix theory is to understand the eigenvalues of spiked random matrix models, introduced by Johnstone, in which a prominent e... |
https://arxiv.org/abs/1601.04412 | On Second Solutions to Second-Order Difference Equations | We investigate and derive second solutions to linear homogeneous second-order difference equations using a variety of methods, in each case going beyond the purely formal solution and giving explicit expressions for the second solution. We present a new implementation of d'Alembert's reduction of order method, applying... | \section{Introduction}
\makeatletter
\def\tagform@#1{\maketag@@@{\normalsize(#1)\@@italiccorr}}
\makeatother
There are a number of distinct methods for generating a second independent solution to a
second-order linear differential equation when one solution is known. These include
1) The extended Cauchy... | {
"timestamp": "2016-01-19T02:11:32",
"yymm": "1601",
"arxiv_id": "1601.04412",
"language": "en",
"url": "https://arxiv.org/abs/1601.04412",
"abstract": "We investigate and derive second solutions to linear homogeneous second-order difference equations using a variety of methods, in each case going beyond t... |
https://arxiv.org/abs/1805.05387 | Almost every $n$-vertex graph is determined by its $3 \log_2{n}$-vertex subgraphs | The paper shows that almost every $n$-vertex graph is such that the multiset of its induced subgraphs on $3 \log_2{n}$ vertices is sufficient to determine it up to isomorphism. Therefore, for checking the isomorphism of a pair of $n$-vertex graphs, almost surely the multiset of their $3 \log_2{n}$-vertex subgraphs is s... | \section{The main result }
As the main result of this paper, we show that the multiset of all induced subgraphs with $3 \log{n}$ vertices are sufficient to determine the isomorphism class of almost every $n$-vertex graph. It means that for a randomly chosen graph $G$, this multiset is almost surely unique among all g... | {
"timestamp": "2018-10-15T02:02:15",
"yymm": "1805",
"arxiv_id": "1805.05387",
"language": "en",
"url": "https://arxiv.org/abs/1805.05387",
"abstract": "The paper shows that almost every $n$-vertex graph is such that the multiset of its induced subgraphs on $3 \\log_2{n}$ vertices is sufficient to determin... |
https://arxiv.org/abs/1506.01656 | Internally heated convection and Rayleigh-Bénard convection | This work reviews basic features of both Rayleigh-Bénard (RB) convection and internally heated (IH) convection, along with findings on IH convection from laboratory experiments and numerical simulations. In the first chapter, six canonical models of convection are described: three configurations of IH convection driven... | \subsection*{Extremum principles}
\addcontentsline{toc}{subsection}{Extremum principles}
In each configuration with $T=0$ on a boundary, there holds a minimum principle giving pointwise, instantaneous lower bounds on $T(\mathbf{x},t)$. For simplicity we assume that solutions to the Boussinesq equations exist and remai... | {
"timestamp": "2015-11-20T02:13:51",
"yymm": "1506",
"arxiv_id": "1506.01656",
"language": "en",
"url": "https://arxiv.org/abs/1506.01656",
"abstract": "This work reviews basic features of both Rayleigh-Bénard (RB) convection and internally heated (IH) convection, along with findings on IH convection from ... |
https://arxiv.org/abs/1605.01418 | A Sampling Kaczmarz-Motzkin Algorithm for Linear Feasibility | We combine two iterative algorithms for solving large-scale systems of linear inequalities, the relaxation method of Agmon, Motzkin et al. and the randomized Kaczmarz method. In doing so, we obtain a family of algorithms that generalize and extend both techniques. We prove several convergence results, and our computati... | \section{Introduction}
We are interested solving large-scale systems of linear inequalities $ Ax \leq b$. Here $b \in \mathbb{R}^m$ and $A$ an $m \times n$ matrix; \dn{the regime $m\gg n$ is our setting of interest, where iterative methods are typically employed}. We denote the rows of $A$ by the vectors $a_1,a_2,\do... | {
"timestamp": "2017-03-01T02:09:46",
"yymm": "1605",
"arxiv_id": "1605.01418",
"language": "en",
"url": "https://arxiv.org/abs/1605.01418",
"abstract": "We combine two iterative algorithms for solving large-scale systems of linear inequalities, the relaxation method of Agmon, Motzkin et al. and the randomi... |
https://arxiv.org/abs/0811.4769 | Further Improvements of Lower Bounds for the Least Common Multiples of Arithmetic Progressions | For relatively prime positive integers u_0 and r, we consider the arithmetic progression {u_k := u_0+k*r} (0 <= k <= n). Define L_n := lcm{u_0,u_1,...,u_n} and let a >= 2 be any integer. In this paper, we show that, for integers alpha,r >= a and n >= 2*alpha*r, we have L_n >= u_0*r^{alpha+a-2}*(r+1)^n. In particular, l... | \section{Introduction}\label{intro}
The search for effective estimates on the least common multiples of
finite arithmetic progressions began with the work of
Hanson~\cite{Ha} and Nair~\cite{Na}, who respectively found upper
and lower bounds for $\lcm\{1,\ldots, n\}$.
Inspired by this work, Bateman, Kalb, and Stenger~... | {
"timestamp": "2009-06-16T07:01:53",
"yymm": "0811",
"arxiv_id": "0811.4769",
"language": "en",
"url": "https://arxiv.org/abs/0811.4769",
"abstract": "For relatively prime positive integers u_0 and r, we consider the arithmetic progression {u_k := u_0+k*r} (0 <= k <= n). Define L_n := lcm{u_0,u_1,...,u_n} ... |
https://arxiv.org/abs/0908.2107 | Unitary equivalence of a matrix to its transpose | Motivated by a problem of Halmos, we obtain a canonical decomposition for complex matrices which are unitarily equivalent to their transpose (UET). Surprisingly, the naive assertion that a matrix is UET if and only if it is unitarily equivalent to a complex symmetric matrix (i.e., $T = T^t$) holds for matrices 7x7 and ... | \section{Introduction}
In his problem book \cite[Pr.~159]{Halmos}, Halmos asks whether every square complex matrix is unitarily equivalent to
its transpose (UET). For example, every finite Toeplitz matrix is unitarily equivalent to its transpose via the
permutation matrix which reverses the order of the standard ... | {
"timestamp": "2010-06-24T02:02:04",
"yymm": "0908",
"arxiv_id": "0908.2107",
"language": "en",
"url": "https://arxiv.org/abs/0908.2107",
"abstract": "Motivated by a problem of Halmos, we obtain a canonical decomposition for complex matrices which are unitarily equivalent to their transpose (UET). Surprisi... |
https://arxiv.org/abs/1604.07976 | Smaller Extended Formulations for the Spanning Tree Polytope of Bounded-genus Graphs | We give an $O(g^{1/2} n^{3/2} + g^{3/2} n^{1/2})$-size extended formulation for the spanning tree polytope of an $n$-vertex graph embedded on a surface of genus $g$, improving on the known $O(n^2 + g n)$-size extended formulations following from Wong and Martin. | \section{Introduction}
An \emph{extended formulation} of a (convex) polytope $P \subseteq \mathbb{R}^d$ is a linear system $Ax + By \leqslant b,\ Cx + Dy = c$ in variables $x \in \mathbb{R}^d$ and $y \in \mathbb{R}^k$ that provides a description of $P$ in the sense that
$$
P = \{x \in \mathbb{R}^d \mid \exists y \in \... | {
"timestamp": "2017-01-10T02:10:59",
"yymm": "1604",
"arxiv_id": "1604.07976",
"language": "en",
"url": "https://arxiv.org/abs/1604.07976",
"abstract": "We give an $O(g^{1/2} n^{3/2} + g^{3/2} n^{1/2})$-size extended formulation for the spanning tree polytope of an $n$-vertex graph embedded on a surface of... |
https://arxiv.org/abs/1210.7119 | Relating Edelman-Greene insertion to the Little map | The Little map and the Edelman-Greene insertion algorithm, a generalization of the Robinson-Schensted correspondence, are both used for enumerating the reduced decompositions of an element of the symmetric group. We show the Little map factors through Edelman-Greene insertion and establish new results about each map as... | \section{Introduction}
\label{sec:introduction}
\subsection{Preliminaries}
\label{ssec:Preliminaries}
In this paper, we clarify the relationship between two algorithmic bijections, due respectively to Edelman-Greene~\cite{edelman1987balanced} and to Little~\cite{little2003combinatorial}, both of which deal with reduce... | {
"timestamp": "2013-01-15T02:00:38",
"yymm": "1210",
"arxiv_id": "1210.7119",
"language": "en",
"url": "https://arxiv.org/abs/1210.7119",
"abstract": "The Little map and the Edelman-Greene insertion algorithm, a generalization of the Robinson-Schensted correspondence, are both used for enumerating the redu... |
https://arxiv.org/abs/2009.08113 | A path formula for the sock sorting problem | Suppose $n$ different pairs of socks are put in a tumble dryer. When the dryer is finished socks are taken out one by one, if a sock matches one of the socks on the sorting table both are removed, otherwise it is put on the table until its partner emerges from the dryer. We note the number of socks on the table after e... | \section{Background and statement of the result}
Suppose that $n$ different pairs of socks are put in a tumble dryer. After operating the dryer the socks are taken out one by one,
if a sock matches one of the socks on the sorting table both are removed, otherwise it is put on the table until its partner emerges from t... | {
"timestamp": "2020-09-18T02:10:17",
"yymm": "2009",
"arxiv_id": "2009.08113",
"language": "en",
"url": "https://arxiv.org/abs/2009.08113",
"abstract": "Suppose $n$ different pairs of socks are put in a tumble dryer. When the dryer is finished socks are taken out one by one, if a sock matches one of the so... |
https://arxiv.org/abs/1510.03500 | Spacing Distribution of a Bernoulli Sampled Sequence | We investigate the spacing distribution of sequence \[S_n=\left\{0,\frac{1}{n},\frac{2}{n},\dots,\frac{n-1}{n},1\right\}\] after Bernoulli sampling. We describe the closed form expression of the probability mass function of the spacings, and show that the spacings converge in distribution to a geometric random variable... | \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": "2015-10-14T02:04:35",
"yymm": "1510",
"arxiv_id": "1510.03500",
"language": "en",
"url": "https://arxiv.org/abs/1510.03500",
"abstract": "We investigate the spacing distribution of sequence \\[S_n=\\left\\{0,\\frac{1}{n},\\frac{2}{n},\\dots,\\frac{n-1}{n},1\\right\\}\\] after Bernoulli sampl... |
https://arxiv.org/abs/2211.08124 | Symmetric polynomials over finite fields | It is shown that two vectors with coordinates in the finite $q$-element field of characteristic $p$ belong to the same orbit under the natural action of the symmetric group if each of the elementary symmetric polynomials of degree $p^k,2p^k,\dots,(q-1)p^k$, $k=0,1,2,\dots$ has the same value on them. This separating se... | \section{Introduction}\label{sec:intro}
Throughout this paper $F$ stands for an arbitrary field, $q$ stands for a power of a prime $p$, and $\mathbb{F}_q$ stands for the field of $q$ elements.
The symmetric group $S_n$ acts on the vector space $F^n$ by permuting coordinates: for $\pi\in S_n$ and $v=(v_1,\dots,v_n)\i... | {
"timestamp": "2022-11-30T02:19:31",
"yymm": "2211",
"arxiv_id": "2211.08124",
"language": "en",
"url": "https://arxiv.org/abs/2211.08124",
"abstract": "It is shown that two vectors with coordinates in the finite $q$-element field of characteristic $p$ belong to the same orbit under the natural action of t... |
https://arxiv.org/abs/1010.2551 | Fractional Repetition Codes for Repair in Distributed Storage Systems | We introduce a new class of exact Minimum-Bandwidth Regenerating (MBR) codes for distributed storage systems, characterized by a low-complexity uncoded repair process that can tolerate multiple node failures. These codes consist of the concatenation of two components: an outer MDS code followed by an inner repetition c... |
\section{Capacity under Exact Uncoded Repair} \label{sec:FRCapacity}
The universally good FR codes constructed in the previous sections are guaranteed to have a rate greater or equal to the capacity $C_{MBR}$ of the system under random access and functional repair. However, there exist cases where FR codes can ach... | {
"timestamp": "2010-10-14T02:01:03",
"yymm": "1010",
"arxiv_id": "1010.2551",
"language": "en",
"url": "https://arxiv.org/abs/1010.2551",
"abstract": "We introduce a new class of exact Minimum-Bandwidth Regenerating (MBR) codes for distributed storage systems, characterized by a low-complexity uncoded repa... |
https://arxiv.org/abs/1508.07531 | A Generalization of Zeckendorf's Theorem via Circumscribed $m$-gons | Zeckendorf's theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers, where the Fibonacci numbers satisfy $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3$, $F_1=1$ and $F_2=2$. The distribution of the number of summands in such decomposition converges to a Gaussian, the gap... | \section{Introduction} \label{sec:intro}
The Fibonacci numbers are a heavily studied sequence which arise in many different ways and places. By defining them by $F_1 = 1$, $F_2 = 2$ and $F_{n+1} = F_n + F_{n-1}$, we have the remarkable property that every positive integer can be uniquely written as a sum of non-consec... | {
"timestamp": "2015-12-02T02:14:49",
"yymm": "1508",
"arxiv_id": "1508.07531",
"language": "en",
"url": "https://arxiv.org/abs/1508.07531",
"abstract": "Zeckendorf's theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers, where the Fibonacci numbe... |
https://arxiv.org/abs/1909.12025 | The Approximation Ratio of the 2-Opt Heuristic for the Metric Traveling Salesman Problem | The 2-Opt heuristic is one of the simplest algorithms for finding good solutions to the metric Traveling Salesman Problem. It is the key ingredient to the well-known Lin-Kernighan algorithm and often used in practice. So far, only upper and lower bounds on the approximation ratio of the 2-Opt heuristic for the metric T... | \section{Introduction}
In the Traveling Salesman Problem (TSP), we are given $n$ cities with their pairwise
distances. The task is to find a shortest tour that visits each city exactly once.
The Traveling Salesman Problem is one of the most intensely studied problems in combinatorial optimization.
It is well known t... | {
"timestamp": "2020-03-16T01:09:12",
"yymm": "1909",
"arxiv_id": "1909.12025",
"language": "en",
"url": "https://arxiv.org/abs/1909.12025",
"abstract": "The 2-Opt heuristic is one of the simplest algorithms for finding good solutions to the metric Traveling Salesman Problem. It is the key ingredient to the... |
https://arxiv.org/abs/2005.01695 | Cosine polynomials with few zeros | In a celebrated paper, Borwein, Erdélyi, Ferguson and Lockhart constructed cosine polynomials of the form \[ f_A(x) = \sum_{a \in A} \cos(ax), \] with $A\subseteq \mathbb{N}$, $|A|= n$ and as few as $n^{5/6+o(1)}$ zeros in $[0,2\pi]$, thereby disproving an old conjecture of J.E. Littlewood. Here we give a sharp analysi... | \section{Introduction}
In his 1968 monograph, J.E. Littlewood \cite{JELw4} collected many interesting problems on the behavior of polynomials and trigonometric sums with restricted coefficients,
a subject on which he worked extensively throughout his life \cite{HardyLittlewood,JELw1,JEL62,JELw2,JELw3,JEL66,LO38,LO45,L... | {
"timestamp": "2020-05-11T02:09:53",
"yymm": "2005",
"arxiv_id": "2005.01695",
"language": "en",
"url": "https://arxiv.org/abs/2005.01695",
"abstract": "In a celebrated paper, Borwein, Erdélyi, Ferguson and Lockhart constructed cosine polynomials of the form \\[ f_A(x) = \\sum_{a \\in A} \\cos(ax), \\] wit... |
https://arxiv.org/abs/2111.09196 | The least doubling constant of a path graph | We study the least doubling constant $C_G$ among all possible doubling measures defined on a path graph $G$. We consider both finite and infinite cases and show that, if $G=\mathbb Z$, $C_{\mathbb Z}=3$, while for $G=L_n$, the path graph with $n$ vertices, one has $1+2\cos(\frac{\pi}{n+1})\leq C_{L_n}<3$, with equality... | \section{Introduction}
This paper is motivated by the study of doubling measures on graphs, already started in \cite{ST}. In order to properly explain our approach, we will first recall some terminology from the abstract theory of measure metric spaces: Given a metric space $(X,d)$, a Borel measure $\mu$ on $X$ is sai... | {
"timestamp": "2021-11-18T02:23:04",
"yymm": "2111",
"arxiv_id": "2111.09196",
"language": "en",
"url": "https://arxiv.org/abs/2111.09196",
"abstract": "We study the least doubling constant $C_G$ among all possible doubling measures defined on a path graph $G$. We consider both finite and infinite cases an... |
https://arxiv.org/abs/2103.04122 | A Quantitative Helly-type Theorem: Containment in a Homothet | We introduce a new variant of quantitative Helly-type theorems: the minimal \emph{"homothetic distance"} of the intersection of a family of convex sets to the intersection of a subfamily of a fixed size. As an application, we establish the following quantitative Helly-type result for the \emph{diameter}. If $K$ is the ... | \section{Introduction}
In \cite{barany1982quantitative} (see also \cite{barany1984helly}), B\'ar\'any, Katchalski and Pach proved the following two statements. According to the \textbf{Quantitative Volume Theorem},
{\it
if the intersection of a family of convex sets in $\Red$ is of volume one, then the intersection o... | {
"timestamp": "2021-03-09T02:12:44",
"yymm": "2103",
"arxiv_id": "2103.04122",
"language": "en",
"url": "https://arxiv.org/abs/2103.04122",
"abstract": "We introduce a new variant of quantitative Helly-type theorems: the minimal \\emph{\"homothetic distance\"} of the intersection of a family of convex sets... |
https://arxiv.org/abs/1703.06949 | On Leighton's Comparison Theorem | We give a simple proof of a fairly flexible comparison theorem for equations of the type $-(p(u'+su))'+rp(u'+su)+qu=0$ on a finite interval where $1/p$, $r$, $s$, and $q$ are real and integrable. Flexibility is provided by two functions which may be chosen freely (within limits) according to the situation at hand. We i... | \section{Introduction}
In 1836 Sturm published his paper \cite{Sturm1836} containing the celebrated comparison theorem.
For it he studied two equations of the form $-(pu')'+qu=0$ to conclude something about the zeros of the solutions of one equation from the zeros of some solution of the other equation.
In fact, Sturm'... | {
"timestamp": "2017-03-22T01:01:13",
"yymm": "1703",
"arxiv_id": "1703.06949",
"language": "en",
"url": "https://arxiv.org/abs/1703.06949",
"abstract": "We give a simple proof of a fairly flexible comparison theorem for equations of the type $-(p(u'+su))'+rp(u'+su)+qu=0$ on a finite interval where $1/p$, $... |
https://arxiv.org/abs/1209.3049 | Lower bounds on the global minimum of a polynomial | We extend the method of Ghasemi and Marshall [SIAM. J. Opt. 22(2) (2012), pp 460-473], to obtain a lower bound $f_{{\rm gp},M}$ for a multivariate polynomial $f(x) \in \mathbb{R}[x]$ of degree $ \le 2d$ in $n$ variables $x = (x_1,...,x_n)$ on the closed ball ${x \in \mathbb{R}^n : \sum x_i^{2d} \le M}$, computable by g... | \section{Introduction}
Computing a lower bound on the global minimum on $\mathbb{R}^n$ of a multivariate polynomial is
a standard problem of optimization with many potential applications. In the last decade,
results in polynomial optimization combined with semidefinite programming (for sums of squares representation),... | {
"timestamp": "2012-09-17T02:00:27",
"yymm": "1209",
"arxiv_id": "1209.3049",
"language": "en",
"url": "https://arxiv.org/abs/1209.3049",
"abstract": "We extend the method of Ghasemi and Marshall [SIAM. J. Opt. 22(2) (2012), pp 460-473], to obtain a lower bound $f_{{\\rm gp},M}$ for a multivariate polynomi... |
https://arxiv.org/abs/1402.0957 | Conditioning of Leverage Scores and Computation by QR Decomposition | The leverage scores of a full-column rank matrix A are the squared row norms of any orthonormal basis for range(A). We show that corresponding leverage scores of two matrices A and A + \Delta A are close in the relative sense, if they have large magnitude and if all principal angles between the column spaces of A and A... | \section{Introduction}
Leverage scores are scalar quantities associated with the column space of a matrix, and
can be computed from the rows of \textit{any} orthonormal basis for this space.
\subsubsection*{Leverage scores}
We restrict our discussion here to leverage scores of full column rank matrices.
\begin{de... | {
"timestamp": "2015-05-26T02:04:54",
"yymm": "1402",
"arxiv_id": "1402.0957",
"language": "en",
"url": "https://arxiv.org/abs/1402.0957",
"abstract": "The leverage scores of a full-column rank matrix A are the squared row norms of any orthonormal basis for range(A). We show that corresponding leverage scor... |
https://arxiv.org/abs/1703.03800 | The 4-girth-thickness of the complete graph | In this paper, we define the $4$-girth-thickness $\theta(4,G)$ of a graph $G$ as the minimum number of planar subgraphs of girth at least $4$ whose union is $G$. We obtain the $4$-girth-thickness of the arbitrary complete graph $K_n$ getting that $\theta(4,K_n)=\left\lceil \frac{n+2}{4}\right\rceil$ for $n\not=6,10$ an... | \section{Introduction}
A finite graph $G$ is \emph{planar} if it can be embedded in the plane without any two of its edges crossing. A planar graph of order $n$ and girth $g$ has size at most $\frac{g}{g-2}(n-2)$ (see \cite{MR2368647}), and an acyclic graph of order $n$ has size at most $n-1$, in this case, we define i... | {
"timestamp": "2017-06-21T02:02:12",
"yymm": "1703",
"arxiv_id": "1703.03800",
"language": "en",
"url": "https://arxiv.org/abs/1703.03800",
"abstract": "In this paper, we define the $4$-girth-thickness $\\theta(4,G)$ of a graph $G$ as the minimum number of planar subgraphs of girth at least $4$ whose union... |
https://arxiv.org/abs/2111.10716 | Implications between Induction Principles for $\mathbb{N}$ in Peano Arithmetic | In introductory books about natural numbers, a common kind of assertion - often left as exercise to the reader - is that certain forms of induction on $\mathbb{N}$ (regular/ordinary, complete/strong) are equivalent one to each other and to the well-ordering principle. This means that if P1 and P2 are two of these princ... | \section{Introduction.}
An exercise in~\cite[Chapter~4, Problem 2]{RS14} asks:
\begin{quote}
``(a) Using the principle of mathematical induction, prove the principle of complete mathematical induction. (b) Conversely, show that the principle of mathematical induction logically follows from the principle of compl... | {
"timestamp": "2021-11-23T02:16:41",
"yymm": "2111",
"arxiv_id": "2111.10716",
"language": "en",
"url": "https://arxiv.org/abs/2111.10716",
"abstract": "In introductory books about natural numbers, a common kind of assertion - often left as exercise to the reader - is that certain forms of induction on $\\... |
https://arxiv.org/abs/0708.3947 | Optimality and uniqueness of the (4,10,1/6) spherical code | Linear programming bounds provide an elegant method to prove optimality and uniqueness of an (n,N,t) spherical code. However, this method does not apply to the parameters (4,10,1/6). We use semidefinite programming bounds instead to show that the Petersen code, which consists of the midpoints of the edges of the regula... | \section{Introduction}
Let $C$ be an $N$-element subset of the unit sphere $\Sn \subseteq
\R^n$. It is called an {\em $(n, N, t)$ spherical code} if every two
distinct points $(c,c')$ of $C$ have inner product $c \cdot c'$ at
most $t$. An $(n,N,t)$ spherical code is called {\em optimal} if there
is no $(n,N,t')$ spher... | {
"timestamp": "2008-05-14T15:40:49",
"yymm": "0708",
"arxiv_id": "0708.3947",
"language": "en",
"url": "https://arxiv.org/abs/0708.3947",
"abstract": "Linear programming bounds provide an elegant method to prove optimality and uniqueness of an (n,N,t) spherical code. However, this method does not apply to ... |
https://arxiv.org/abs/2104.02348 | Bernstein- and Markov-type inequalities | This survey discusses the classical Bernstein and Markov inequalities for the derivatives of polynomials, as well as some of their extensions to general sets. | \section{The original Bernstein and Markov inequalities}\label{sect1}
In 1912 S. N. Bernstein proved in \cite{Bernstein}
his famous inequality
that now takes the form
\begin{equation}
|T_n'(\theta)|\le n\sup_t|T_n(t)|,\qquad \theta\in\mathbf{R},
\label{b0}
\end{equation}
where
\[
T_n(t)=
a_0+(a_1\cos t+b_1\sin t)+\cdot... | {
"timestamp": "2021-05-24T02:23:47",
"yymm": "2104",
"arxiv_id": "2104.02348",
"language": "en",
"url": "https://arxiv.org/abs/2104.02348",
"abstract": "This survey discusses the classical Bernstein and Markov inequalities for the derivatives of polynomials, as well as some of their extensions to general s... |
https://arxiv.org/abs/1603.05639 | Sensitivity of mixing times in Eulerian digraphs | Let $X$ be a lazy random walk on a graph $G$. If $G$ is undirected, then the mixing time is upper bounded by the maximum hitting time of the graph. This fails for directed chains, as the biased random walk on the cycle $\mathbb{Z}_n$ shows. However, we establish that for Eulerian digraphs, the mixing time is $O(mn)$, w... | \section{Introduction}\label{sec:intro}
Random walks on graphs have been thoroughly studied,
but many results are only known in the undirected setting,
where spectral methods and the connection with electrical networks is available. As noted in Aldous and Fill~\cite{AF}, many properties of random walk on undirected ... | {
"timestamp": "2016-03-18T01:14:22",
"yymm": "1603",
"arxiv_id": "1603.05639",
"language": "en",
"url": "https://arxiv.org/abs/1603.05639",
"abstract": "Let $X$ be a lazy random walk on a graph $G$. If $G$ is undirected, then the mixing time is upper bounded by the maximum hitting time of the graph. This f... |
https://arxiv.org/abs/2207.12366 | The arithmetical combinatorics of $k,l$-regular partitions | For all positive integers $k,l,n$, the Little Glaisher theorem states that the number of partitions of $n$ into parts not divisible by $k$ and occurring less than $l$ times is equal to the number of partitions of $n$ into parts not divisible by $l$ and occurring less than $k$ times. While this refinement of Glaisher th... | \section{Introduction}
An integer partition is a finite non-increasing sequence of positive integers, called parts of the partition. The weight of an integer partition consists of the sum its parts. In this paper, we enumerate the partitions according to the number of occurrences of positive integers and write the pa... | {
"timestamp": "2022-07-26T02:39:16",
"yymm": "2207",
"arxiv_id": "2207.12366",
"language": "en",
"url": "https://arxiv.org/abs/2207.12366",
"abstract": "For all positive integers $k,l,n$, the Little Glaisher theorem states that the number of partitions of $n$ into parts not divisible by $k$ and occurring l... |
https://arxiv.org/abs/2104.05256 | A Coq Formalization of Lebesgue Integration of Nonnegative Functions | Integration, just as much as differentiation, is a fundamental calculus tool that is widely used in many scientific domains. Formalizing the mathematical concept of integration and the associated results in a formal proof assistant helps in providing the highest confidence on the correctness of numerical programs invo... |
\section*{Acknowledgments}
We are deeply grateful to Stéphane Aubry for some proofs and tactics on
\coqe{Rbar} properties, and to Assia Mahboubi, Cyril Cohen, Guillaume
Melquiond, and Vincent Semeria for fruitful discussions about \coqe{Rbar} and
\coqe{R} in {\soft{Coq}}.
This work was partly supported by the Paris ... | {
"timestamp": "2021-12-10T02:14:36",
"yymm": "2104",
"arxiv_id": "2104.05256",
"language": "en",
"url": "https://arxiv.org/abs/2104.05256",
"abstract": "Integration, just as much as differentiation, is a fundamental calculus tool that is widely used in many scientific domains. Formalizing the mathematical... |
https://arxiv.org/abs/0906.5484 | On the possible orders of a basis for a finite cyclic group | We prove a conjecture of Dukes and Herke concerning the possible orders of a basis for the cyclic group Z_n, namely : For each k \in N there exists a constant c_k > 0 such that, for all n \in N, if A \subseteq Z_n is a basis of order greater than n/k, then the order of A is within c_k of n/l for some integer l \in [1,k... | \section{Introduction}
Let $G$ be an abelian group, written additively, and $A$ a subset of $G$.
For a positive integer $h$ we denote by $hA$ the subset of $G$ consisting
of all possible sums of $h$ not necessarily distinct element of $A$, i.e.:
\be
hA = \{a_1 + \cdots + a_h : a_i \in A \}.
\ee
This set is called the... | {
"timestamp": "2009-07-04T19:16:46",
"yymm": "0906",
"arxiv_id": "0906.5484",
"language": "en",
"url": "https://arxiv.org/abs/0906.5484",
"abstract": "We prove a conjecture of Dukes and Herke concerning the possible orders of a basis for the cyclic group Z_n, namely : For each k \\in N there exists a const... |
https://arxiv.org/abs/1706.06126 | Analytic approximation of solutions of parabolic partial differential equations with variable coefficients | A complete family of solutions for the one-dimensional reaction-diffusion equation \[ u_{xx}(x,t)-q(x)u(x,t) = u_t(x,t) \] with a coefficient $q$ depending on $x$ is constructed. The solutions represent the images of the heat polynomials under the action of a transmutation operator. Their use allows one to obtain an ex... | \section{Introduction}
In the present work a complete system of solutions of a one-dimensional
reaction-diffusion equation with a variable coefficient%
\begin{equation}
u_{xx}(x,t)-q(x)u(x,t) = u_t(x,t) \label{eq rd}
\end{equation}%
considered on $\Omega :=(-b,b)\times (0,\tau )$ is obtained. We assume that the potent... | {
"timestamp": "2017-08-03T02:01:21",
"yymm": "1706",
"arxiv_id": "1706.06126",
"language": "en",
"url": "https://arxiv.org/abs/1706.06126",
"abstract": "A complete family of solutions for the one-dimensional reaction-diffusion equation \\[ u_{xx}(x,t)-q(x)u(x,t) = u_t(x,t) \\] with a coefficient $q$ depend... |
https://arxiv.org/abs/math/0512035 | Increasing and Decreasing Subsequences of Permutations and Their Variants | We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1,2,...,n was obtained by Vershik-Kero... | \section{Introduction}
Let $\mathfrak{S}_n$ denote the symmetric group of all permutations of
$[n]:=\{1,2,\dots,n\}$. We write permutations $w\in\mathfrak{S}_n$ as
\emph{words}, i.e., $w=a_1 a_2\cdots a_n$, where $w(i)=a_i$. An
\emph{increasing subsequence} of $w$ is a subsequence $a_{i_1}\cdots
a_{i_k}$ satisfying $a_... | {
"timestamp": "2005-12-01T19:49:06",
"yymm": "0512",
"arxiv_id": "math/0512035",
"language": "en",
"url": "https://arxiv.org/abs/math/0512035",
"abstract": "We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algor... |
https://arxiv.org/abs/1603.07435 | Optimal transport via a Monge-Ampère optimization problem | We rephrase Monge's optimal transportation (OT) problem with quadratic cost--via a Monge-Ampère equation--as an infinite-dimensional optimization problem, which is in fact a convex problem when the target is a log-concave measure with convex support. We define a natural finite-dimensional discretization to the problem ... | \section{Introduction}
\label{IntroSec}
In this article we develop a scheme for the numerical solution of the
the Monge--Amp\`ere equation
governing optimal transport.
Our main theorem provides natural and computationally feasible approximations of optimal transportation (OT) maps as well as of their Brenier convex ... | {
"timestamp": "2016-04-21T02:02:59",
"yymm": "1603",
"arxiv_id": "1603.07435",
"language": "en",
"url": "https://arxiv.org/abs/1603.07435",
"abstract": "We rephrase Monge's optimal transportation (OT) problem with quadratic cost--via a Monge-Ampère equation--as an infinite-dimensional optimization problem,... |
https://arxiv.org/abs/1608.03685 | Separable determination in Banach spaces | We study a relation between three different formulations of theorems on separable determination - one using the concept of rich families, second via the concept of suitable models and third, a new one, suggested in this paper, using the notion of $\omega$-monotone mappings. In particular, we show that in Banach spaces ... | \section{Comparision of the concept of rich families with the concept of suitable models}\label{section:richVsSuitable}
In this section we compare the concept of rich families with the concept of suitable models. Our main result is that in Banach spaces both concepts are equivalent, see Theorem \ref{t:main1}. Actually... | {
"timestamp": "2016-10-06T02:02:24",
"yymm": "1608",
"arxiv_id": "1608.03685",
"language": "en",
"url": "https://arxiv.org/abs/1608.03685",
"abstract": "We study a relation between three different formulations of theorems on separable determination - one using the concept of rich families, second via the c... |
https://arxiv.org/abs/1608.01929 | A conjectured bound on the spanning tree number of bipartite graphs | The Ferrers bound conjecture is a natural graph-theoretic extension of the enumeration of spanning trees for Ferrers graphs. We document the current status of the conjecture and provide a further conjecture which implies it. | \section{Introduction}
Ferrers graphs are a bipartite analogue of threshold graphs.
They were introduced by Hammer, Peled, and Srinivasan~\cite{HPS90},
where they were called \emph{difference graphs}.
Ferrers graphs and threshold graphs are both realizations of
\emph{Ferrers digraphs}, which were introduced
by Rigu... | {
"timestamp": "2016-08-18T02:10:09",
"yymm": "1608",
"arxiv_id": "1608.01929",
"language": "en",
"url": "https://arxiv.org/abs/1608.01929",
"abstract": "The Ferrers bound conjecture is a natural graph-theoretic extension of the enumeration of spanning trees for Ferrers graphs. We document the current statu... |
https://arxiv.org/abs/1907.10022 | Non-commutative groups as prescribed polytopal symmetries | We study properties of the realizations of groups as the combinatorial automorphism group of a convex polytope. We show that for any non-abelian group $G$ with a central involution there is a centrally symmetric polytope with $G$ as its combinatorial automorphisms. We show that for each integer $n$, there are groups th... | \section{#1}\setcounter{lemma}{0}}
\title{Non-commutative groups as prescribed polytopal symmetries}
\author{Alexandru Chirvasitu, Frieder Ladisch, and Pablo Sober\'on}
\begin{document}
\date{}
\newcommand{\Addresses}{
\bigskip
\footnotesize
\textsc{Department of Mathematics, University at Buffalo, Buff... | {
"timestamp": "2020-04-27T02:03:39",
"yymm": "1907",
"arxiv_id": "1907.10022",
"language": "en",
"url": "https://arxiv.org/abs/1907.10022",
"abstract": "We study properties of the realizations of groups as the combinatorial automorphism group of a convex polytope. We show that for any non-abelian group $G$... |
https://arxiv.org/abs/math/0611436 | Symmetric Products, Duality and Homological Dimension of Configuration Spaces | We discuss various aspects of "braid spaces'' or configuration spaces of unordered points on manifolds. First we describe how the homology of these spaces is affected by puncturing the underlying manifold, hence extending some results of Fred Cohen, Goryunov and Napolitano. Next we obtain a precise bound for the cohomo... | \section{Introduction}
Braid spaces or configuration spaces of \textit{unordered pairwise
distinct} points on manifolds have important applications to a
number of areas of mathematics and physics. They were of crucial use
in the seventies in the work of Arnold on singularities and then
later in the eighties in work of... | {
"timestamp": "2008-07-06T23:11:33",
"yymm": "0611",
"arxiv_id": "math/0611436",
"language": "en",
"url": "https://arxiv.org/abs/math/0611436",
"abstract": "We discuss various aspects of \"braid spaces'' or configuration spaces of unordered points on manifolds. First we describe how the homology of these s... |
https://arxiv.org/abs/0806.4473 | Simultaneous packing and covering in sequence spaces | We adapt a construction of Klee (1981) to find a packing of unit balls in $\ell_p$ ($1\leq p<\infty$) which is efficient in the sense that enlarging the radius of each ball to any $R>2^{1-1/p}$ covers the whole space. We show that the value $2^{1-1/p}$ is optimal. | \section{Introduction}
The so-called simultaneous packing and covering constant of a convex body $C$ in Euclidean space is a certain measure of the efficiency of a packing or a covering by translates of $C$.
This notion was introduced in various equivalent forms by Rogers \cite{Rogers1950}, Ry\v{s}kov \cite{Ryvskov1974... | {
"timestamp": "2008-06-27T11:49:37",
"yymm": "0806",
"arxiv_id": "0806.4473",
"language": "en",
"url": "https://arxiv.org/abs/0806.4473",
"abstract": "We adapt a construction of Klee (1981) to find a packing of unit balls in $\\ell_p$ ($1\\leq p<\\infty$) which is efficient in the sense that enlarging the ... |
https://arxiv.org/abs/2211.04527 | Bounds on the differential uniformity of the Wan-Lidl polynomials | We study the differential uniformity of the Wan-Lidl polynomials over finite fields. A general upper bound, independent of the order of the field, is established. Additional bounds are established in settings where one of the parameters is restricted. In particular, we establish a class of permutation polynomials which... | \section{Introduction and The Main Results}
Throughout this paper $\ff{q}$ denotes the finite field of order $q$, with
$q=p^e$ for some prime $p$ and $e\in\nat$, and $\ffs{q}$ denotes the nonzero elements of $\ff{q}$.
We use $\gen$ to denote a primitive element of $\ff{q}$.
It follows from Lagrange Interpolation and ... | {
"timestamp": "2022-11-10T02:00:59",
"yymm": "2211",
"arxiv_id": "2211.04527",
"language": "en",
"url": "https://arxiv.org/abs/2211.04527",
"abstract": "We study the differential uniformity of the Wan-Lidl polynomials over finite fields. A general upper bound, independent of the order of the field, is esta... |
https://arxiv.org/abs/1702.05542 | The multivariate bisection algorithm | The aim of this paper is the study of the bisection method in $\mathbb{R}^n$. In this work we propose a multivariate bisection method supported by the Poincaré-Miranda theorem in order to solve non-linear system of equations. Given an initial cube verifying the hypothesis of Poincaré-Miranda theorem the algorithm perfo... | \section{Introduction}
The problem of finding numerical approximations to the roots of a non-linear system of equations was subject of various studies, different methodologies have been proposed between optimization and Newton's procedures. In \cite{Lehmer} D. H. Lehmer proposed a method for solving polynomial equation... | {
"timestamp": "2017-11-28T02:09:35",
"yymm": "1702",
"arxiv_id": "1702.05542",
"language": "en",
"url": "https://arxiv.org/abs/1702.05542",
"abstract": "The aim of this paper is the study of the bisection method in $\\mathbb{R}^n$. In this work we propose a multivariate bisection method supported by the Po... |
https://arxiv.org/abs/1508.00435 | Approximating continuous maps by isometries | The Nash-Kuiper Theorem states that the collection of $C^1$-isometric embeddings from a Riemannian manifold $M^n$ into $\mathbb{E}^N$ is $C^0$-dense within the collection of all smooth 1-Lipschitz embeddings provided that $n < N$. This result is now known to be a consequence of Gromov's more general $h$-principle. Ther... | \section*{This is an unnumbered first-level section head}
\section{Introduction}\label{section 1}
Let $(M^m,g)$ denote an $m$-dimensional Riemannian manifold.
The famous Nash-Kuiper Theorem (\cite{Nash1}, \cite{Kuiper}) states that any smooth 1-Lipschitz embedding $f: (M^m, g) \rightarrow \mathbb{E}^n$
is $\epsil... | {
"timestamp": "2016-02-01T02:00:29",
"yymm": "1508",
"arxiv_id": "1508.00435",
"language": "en",
"url": "https://arxiv.org/abs/1508.00435",
"abstract": "The Nash-Kuiper Theorem states that the collection of $C^1$-isometric embeddings from a Riemannian manifold $M^n$ into $\\mathbb{E}^N$ is $C^0$-dense with... |
https://arxiv.org/abs/2209.12098 | Lower bounds for the directional discrepancy with respect to an interval of rotations | We show that the lower bound for the optimal directional discrepancy with respect to the class of rectangles in $\mathbb{R}^2$ rotated in a restricted interval of directions $[-\theta, \theta]$ with $\theta < \frac{\pi}{4}$ is of the order at least $N^{1/5}$ with a constant depending on $\theta$. | \section{introduction}
In the present paper we discuss the directional discrepancy in the plane. Consider a set $\Omega \subset \left[ -\frac{\pi}{4},\frac{\pi}{4}\right]$, which we shall call the \emph{allowed rotation set}.
Let the class of sets $\mathcal{R}_{\Omega}$ contain all rectangles in $\mathbb{R}^2$ w... | {
"timestamp": "2022-09-27T02:12:49",
"yymm": "2209",
"arxiv_id": "2209.12098",
"language": "en",
"url": "https://arxiv.org/abs/2209.12098",
"abstract": "We show that the lower bound for the optimal directional discrepancy with respect to the class of rectangles in $\\mathbb{R}^2$ rotated in a restricted in... |
https://arxiv.org/abs/2301.13718 | Naturally emerging maps for derangements and nonderangements | A derangement is a permutation with no fixed point, and a nonderangement is a permutation with at least one fixed point. There is a one-term recurrence for the number of derangements of $n$ elements, and we describe a bijective proof of this recurrence which can be found using a recursive map. We then show the combinat... | \section{Introduction}
A \textit{derangement} is a permutation $\sigma \in \mathfrak{S}_n$ such that for all $i \in [n]$, $$\sigma(i) \neq i,$$ i.e. a permutation which does not fix any element.
We denote by $D_n$ the set of derangements on $n$ elements, and let $d_n = |D_n|$. Let $E_n$ be the set of permutations of $... | {
"timestamp": "2023-02-01T02:19:00",
"yymm": "2301",
"arxiv_id": "2301.13718",
"language": "en",
"url": "https://arxiv.org/abs/2301.13718",
"abstract": "A derangement is a permutation with no fixed point, and a nonderangement is a permutation with at least one fixed point. There is a one-term recurrence fo... |
https://arxiv.org/abs/1807.03973 | ReLU Deep Neural Networks and Linear Finite Elements | In this paper, we investigate the relationship between deep neural networks (DNN) with rectified linear unit (ReLU) function as the activation function and continuous piecewise linear (CPWL) functions, especially CPWL functions from the simplicial linear finite element method (FEM). We first consider the special case o... | \section{Introduction}
\label{sec:into}
In recent years, deep learning models have achieved unprecedented
success in various tasks of machine learning or artificial
intelligence, such as computer vision, natural language processing and
reinforcement learning \cite{lecun2015deep}. One main technique in
deep learning i... | {
"timestamp": "2018-07-26T02:05:48",
"yymm": "1807",
"arxiv_id": "1807.03973",
"language": "en",
"url": "https://arxiv.org/abs/1807.03973",
"abstract": "In this paper, we investigate the relationship between deep neural networks (DNN) with rectified linear unit (ReLU) function as the activation function an... |
https://arxiv.org/abs/0708.2668 | Zone and double zone diagrams in abstract spaces | A zone diagram is a relatively new concept which was first defined and studied by T. Asano, J. Matousek and T. Tokuyama. It can be interpreted as a state of equilibrium between several mutually hostile kingdoms. Formally, it is a fixed point of a certain mapping. These authors considered the Euclidean plane and proved ... | \section{Introduction and notation}
A zone diagram of order $n$ is a relatively new concept
which can be interpreted as a state of equilibrium
between $n$ mutually hostile kingdoms. More formally, let $(X,d)$ be a metric space, and suppose $P=(P_k)_{k\in K}$ is a given tuple of nonempty sets in $X$. A zone diagram... | {
"timestamp": "2007-08-20T20:42:39",
"yymm": "0708",
"arxiv_id": "0708.2668",
"language": "en",
"url": "https://arxiv.org/abs/0708.2668",
"abstract": "A zone diagram is a relatively new concept which was first defined and studied by T. Asano, J. Matousek and T. Tokuyama. It can be interpreted as a state of... |
https://arxiv.org/abs/1406.2835 | Heat flow, heat content and the isoparametric property | Let $M$ be a Riemannian manifold and $\Omega$ a compact domain of $M$ with smooth boundary. We study the solution of the heat equation on $\Omega$ having constant unit initial conditions and Dirichlet boundary conditions. The purpose of this paper is to study the geometry of domains for which, at any fixed value of tim... | \section{Introduction}
In this paper, we prove a rigidity result for Riemannian manifolds with boundary satisfying a certain overdetermined problem for the heat equation; the aim is to understand the conditions on the heat content and the heat flow which insure the isoparametric property of the boundary. In this intr... | {
"timestamp": "2014-06-12T02:07:17",
"yymm": "1406",
"arxiv_id": "1406.2835",
"language": "en",
"url": "https://arxiv.org/abs/1406.2835",
"abstract": "Let $M$ be a Riemannian manifold and $\\Omega$ a compact domain of $M$ with smooth boundary. We study the solution of the heat equation on $\\Omega$ having ... |
https://arxiv.org/abs/2004.03833 | Multiplication operators between discrete Hardy spaces on rooted trees | Muthukumar and Ponnusamy \cite{MP-Tp-spaces} studied the multiplication operators on $\mathbb{T}_p$ spaces. In this article, we mainly consider multiplication operators between $\mathbb{T}_p$ and $\mathbb{T}_q$ ($p\neq q$). In particular, we characterize bounded and compact multiplication operators from $\mathbb{T}_{p}... | \section{Introduction}
Let $X$ be a normed space of functions defined on a set $\Omega$. For a complex-valued map $\psi$ on $\Omega$, the inducing multiplication operator is denoted by $M_\psi$ and defined by
$$M_\psi f = \psi f \mbox{~for every~}f \in X.$$
The study of multiplication operators acts as bridge between... | {
"timestamp": "2020-04-09T02:07:27",
"yymm": "2004",
"arxiv_id": "2004.03833",
"language": "en",
"url": "https://arxiv.org/abs/2004.03833",
"abstract": "Muthukumar and Ponnusamy \\cite{MP-Tp-spaces} studied the multiplication operators on $\\mathbb{T}_p$ spaces. In this article, we mainly consider multipli... |
https://arxiv.org/abs/1412.1805 | On $\varepsilon$ Approximations of Persistence Diagrams | Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently observed in nature. In this paper a theoretical framework for the algorithmic comput... | \section{Introduction}
\label{sec::Intro}
The formation of complex patterns can be encountered in almost all sciences. Many patterns observed in nature are extremely complicated and it is usually impossible to completely understand them using analytical methods. Often a coarser but computationally tractable ... | {
"timestamp": "2014-12-09T02:09:22",
"yymm": "1412",
"arxiv_id": "1412.1805",
"language": "en",
"url": "https://arxiv.org/abs/1412.1805",
"abstract": "Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provid... |
https://arxiv.org/abs/2211.00804 | Analysis and object oriented implementation of the Kovacic algorithm | This paper gives a detailed overview and a number of worked out examples illustrating the Kovacic \cite{Kovacic86} algorithm for solving second order linear differential equation ${A(x) y"+ B(x) y' + C(x) y=0}$ where $A,B,C$ are rational functions with complex coefficients in the independent variable $x$. All three cas... |
\section{Introduction}
Kovacic~\cite{Kovacic86} gave an algorithm for finding a closed form Liouvillian\footnote{Wikipedia defines
Liouvillian function as function of one variable which is the composition of a finite number of
arithmetic operations $(+,-, \times, \div )$, exponentials, constants, solutions of algebra... | {
"timestamp": "2022-11-03T01:06:05",
"yymm": "2211",
"arxiv_id": "2211.00804",
"language": "en",
"url": "https://arxiv.org/abs/2211.00804",
"abstract": "This paper gives a detailed overview and a number of worked out examples illustrating the Kovacic \\cite{Kovacic86} algorithm for solving second order lin... |
https://arxiv.org/abs/1310.5201 | Homomesy in products of two chains | Many invertible actions $\tau$ on a set ${\mathcal{S}}$ of combinatorial objects, along with a natural statistic $f$ on ${\mathcal{S}}$, exhibit the following property which we dub \textbf{homomesy}: the average of $f$ over each $\tau$-orbit in ${\mathcal{S}}$ is the same as the average of $f$ over the whole set ${\mat... | \section{Introduction}
\label{sec:int}
We begin with the definition of our main unifying concept,
and supporting nomenclature.
\begin{definition}\label{def:ce}
Given a set ${\mathcal{S}}$,
an invertible map $\tau$ from ${\mathcal{S}}$ to itself such that each
$\tau$-orbit is finite,
and a function (or ``statistic'') ... | {
"timestamp": "2015-06-22T02:11:21",
"yymm": "1310",
"arxiv_id": "1310.5201",
"language": "en",
"url": "https://arxiv.org/abs/1310.5201",
"abstract": "Many invertible actions $\\tau$ on a set ${\\mathcal{S}}$ of combinatorial objects, along with a natural statistic $f$ on ${\\mathcal{S}}$, exhibit the foll... |
https://arxiv.org/abs/2111.07425 | The general position achievement game played on graphs | A general position set of a graph $G$ is a set of vertices $S$ in $G$ such that no three vertices from $S$ lie on a common shortest path. In this paper we introduce and study the general position achievement game. The game is played on a graph $G$ by players A and B who alternatively pick vertices of $G$. A selection o... | \section{Introduction}
\label{sec:intro}
The general position problem for graphs was independently introduced and researched in~\cite{manuel-2018a, ullas-2016}, but should be noted that in the case of hypercubes, it has been studied much earlier by K\"orner~\cite{korner-1995}. Among motives for introducing the proble... | {
"timestamp": "2021-11-16T02:22:29",
"yymm": "2111",
"arxiv_id": "2111.07425",
"language": "en",
"url": "https://arxiv.org/abs/2111.07425",
"abstract": "A general position set of a graph $G$ is a set of vertices $S$ in $G$ such that no three vertices from $S$ lie on a common shortest path. In this paper we... |
https://arxiv.org/abs/1207.5857 | Distance Distributions in Regular Polygons | This paper derives the exact cumulative density function of the distance between a randomly located node and any arbitrary reference point inside a regular $\el$-sided polygon. Using this result, we obtain the closed-form probability density function (PDF) of the Euclidean distance between any arbitrary reference point... | \section{Introduction}
Recently, the distance distributions in wireless networks have
received a lot of attention in the
literature~\cite{Srinivasa-2010,Moltchanov-2012,Fan-2007}. The
distance distributions can be applied to study important wireless
network characteristics such as interference, outage probability,
conn... | {
"timestamp": "2013-06-27T02:00:31",
"yymm": "1207",
"arxiv_id": "1207.5857",
"language": "en",
"url": "https://arxiv.org/abs/1207.5857",
"abstract": "This paper derives the exact cumulative density function of the distance between a randomly located node and any arbitrary reference point inside a regular ... |
https://arxiv.org/abs/1312.6840 | The k-metric dimension of a graph | As a generalization of the concept of a metric basis, this article introduces the notion of $k$-metric basis in graphs. Given a connected graph $G=(V,E)$, a set $S\subseteq V$ is said to be a $k$-metric generator for $G$ if the elements of any pair of different vertices of $G$ are distinguished by at least $k$ elements... | \section{Introduction}
The problem of uniquely determining the location of an intruder in a network was the principal motivation of introducing the concept of metric dimension in graphs by Slater in \cite{Slater1975,Slater1988}, where the metric generators were called locating sets. The concept of metric dimension of... | {
"timestamp": "2015-02-05T02:08:39",
"yymm": "1312",
"arxiv_id": "1312.6840",
"language": "en",
"url": "https://arxiv.org/abs/1312.6840",
"abstract": "As a generalization of the concept of a metric basis, this article introduces the notion of $k$-metric basis in graphs. Given a connected graph $G=(V,E)$, a... |
https://arxiv.org/abs/1207.6659 | The Supremum Norm of the Discrepancy Function: Recent Results and Connections | A great challenge in the analysis of the discrepancy function D_N is to obtain universal lower bounds on the L-infty norm of D_N in dimensions d \geq 3. It follows from the average case bound of Klaus Roth that the L-infty norm of D_N is at least (log N) ^{(d-1)/2}. It is conjectured that the L-infty bound is significa... | \section{Introduction}
We survey recent results on the sup-norm of the discrepancy function.
For integers $ d\ge 2$, and $ N\ge 1$, let $ \mathcal P_N \subset [0,1] ^{d} $ be a finite point set with cardinality $ \sharp \mathcal P_N=N $. Define the associated discrepancy function by
\begin{equation} \label{e.... | {
"timestamp": "2012-09-24T02:01:05",
"yymm": "1207",
"arxiv_id": "1207.6659",
"language": "en",
"url": "https://arxiv.org/abs/1207.6659",
"abstract": "A great challenge in the analysis of the discrepancy function D_N is to obtain universal lower bounds on the L-infty norm of D_N in dimensions d \\geq 3. It... |
https://arxiv.org/abs/1906.11326 | Approximating the pth Root by Composite Rational Functions | A landmark result from rational approximation theory states that $x^{1/p}$ on $[0,1]$ can be approximated by a type-$(n,n)$ rational function with root-exponential accuracy. Motivated by the recursive optimality property of Zolotarev functions (for the square root and sign functions), we investigate approximating $x^{1... | \section{Introduction}
Composing rational functions is an efficient way of generating a rational function $r(x)=r_k(\cdots r_2(r_1(x)))$ of high degree: if each $r_i$ is of type $(m,m)$, then $r$ is of type $(m^k,m^k)$. By choosing each $r_i$ appropriately, one can often obtain a function $r$ that approximates a desir... | {
"timestamp": "2019-06-28T02:02:05",
"yymm": "1906",
"arxiv_id": "1906.11326",
"language": "en",
"url": "https://arxiv.org/abs/1906.11326",
"abstract": "A landmark result from rational approximation theory states that $x^{1/p}$ on $[0,1]$ can be approximated by a type-$(n,n)$ rational function with root-ex... |
https://arxiv.org/abs/2002.12236 | Optimization of Graph Total Variation via Active-Set-based Combinatorial Reconditioning | Structured convex optimization on weighted graphs finds numerous applications in machine learning and computer vision. In this work, we propose a novel adaptive preconditioning strategy for proximal algorithms on this problem class. Our preconditioner is driven by a sharp analysis of the local linear convergence rate d... |
\section{Discussion and conclusion}
We presented an efficient reconditioning strategy for proximal
algorithms on graphs. By relying on a sharp analysis of the local linear convergence
rate we proposed an edge partitioning of the graph into forests which provably boosts
the linear convergence rate. The scaled dual u... | {
"timestamp": "2020-02-28T02:16:00",
"yymm": "2002",
"arxiv_id": "2002.12236",
"language": "en",
"url": "https://arxiv.org/abs/2002.12236",
"abstract": "Structured convex optimization on weighted graphs finds numerous applications in machine learning and computer vision. In this work, we propose a novel ad... |
https://arxiv.org/abs/1906.09006 | Endomorphism operads of functors | We consider the endomorphism operad of a functor, which is roughly the object of natural transformations from (monoidal) powers of that functor to itself. There are many examples from geometry, topology, and algebra where this object has already been implicitly studied. We ask whether the endomorphism operad of the for... |
\section{Introduction}
It is commonplace in contemporary mathematics to discuss the natural
endomorphisms of a functor. Cohomology operations, the center of
a category, and reconstruction theorems in the style of Tannaka
are examples of this line of study.
The raison d'être of this article is to discuss the natur... | {
"timestamp": "2019-07-04T02:04:48",
"yymm": "1906",
"arxiv_id": "1906.09006",
"language": "en",
"url": "https://arxiv.org/abs/1906.09006",
"abstract": "We consider the endomorphism operad of a functor, which is roughly the object of natural transformations from (monoidal) powers of that functor to itself.... |
https://arxiv.org/abs/1406.3681 | Enumeration of MOLS of small order | We report the results of a computer investigation of sets of mutually orthogonal latin squares (MOLS) of small order. For $n\le9$ we1. Determine the number of orthogonal mates for each species of latin square of order $n$.2. Calculate the proportion of latin squares of order $n$ that have an orthogonal mate, and the ex... | \section{Introduction}\label{s:intro}
A latin square of order $n$ is an $n \times n$ matrix in which $n$
distinct symbols are arranged so that each symbol occurs once in each
row and column. Two latin squares $A = [a_{ij}]$ and $B = [b_{ij}]$
of order $n$ are said to be \emph{orthogonal} if the $n^2$ ordered
pairs $(... | {
"timestamp": "2014-10-14T02:12:52",
"yymm": "1406",
"arxiv_id": "1406.3681",
"language": "en",
"url": "https://arxiv.org/abs/1406.3681",
"abstract": "We report the results of a computer investigation of sets of mutually orthogonal latin squares (MOLS) of small order. For $n\\le9$ we1. Determine the number... |
https://arxiv.org/abs/1911.05794 | On the Mean Subtree Order of Graphs Under Edge Addition | For a graph $G$, the mean subtree order of $G$ is the average order of a subtree of $G$. In this note, we provide counterexamples to a recent conjecture of Chin, Gordon, MacPhee, and Vincent, that for every connected graph $G$ and every pair of distinct vertices $u$ and $v$ of $G$, the addition of the edge between $u$ ... | \section{Introduction}
Throughout, we assume that \emph{graphs} are finite, loopless, and contain no multiple edges, while \emph{multigraphs} are finite, loopless, and may contain multiple edges. In particular, every graph is a multigraph. A \emph{subtree} of a multigraph $G$ is a (not necessarily induced) subgraph ... | {
"timestamp": "2019-11-15T02:02:05",
"yymm": "1911",
"arxiv_id": "1911.05794",
"language": "en",
"url": "https://arxiv.org/abs/1911.05794",
"abstract": "For a graph $G$, the mean subtree order of $G$ is the average order of a subtree of $G$. In this note, we provide counterexamples to a recent conjecture o... |
https://arxiv.org/abs/1803.06964 | A modern maximum-likelihood theory for high-dimensional logistic regression | Every student in statistics or data science learns early on that when the sample size largely exceeds the number of variables, fitting a logistic model produces estimates that are approximately unbiased. Every student also learns that there are formulas to predict the variability of these estimates which are used for t... | \section{Broader Implications and Future Directions}
This paper shows that in high-dimensions, classical ML theory is
unacceptable. Among other things, classical theory predicts that the
MLE is approximately unbiased when in reality it seriously
overestimates effect magnitudes. Since the purpose of logistic
mode... | {
"timestamp": "2018-06-19T02:04:45",
"yymm": "1803",
"arxiv_id": "1803.06964",
"language": "en",
"url": "https://arxiv.org/abs/1803.06964",
"abstract": "Every student in statistics or data science learns early on that when the sample size largely exceeds the number of variables, fitting a logistic model pr... |
https://arxiv.org/abs/1904.07227 | First passage time for Slepian process with linear barrier | In this paper we extend results of L.A. Shepp by finding explicit formulas for the first passage probability $F_{a,b}(T\, |\, x)={\rm Pr}(S(t)<a+bt \text{ for all } t\in[0,T]\,\, | \,\,S(0)=x)$, for all $T>0$, where $S(t)$ is a Gaussian process with mean 0 and covariance $\mathbb{E} S(t)S(t')=\max\{0,1-|t-t'|\}\,.$ We ... | \section{Introduction}
Let $T>0$ be a fixed real number and let $S(t)$, $ t\in [0,T]$, be a Gaussian process with mean 0 and covariance
$$
\mathbb{E} S(t)S(t')=\max\{0,1-|t-t'|\}\, .
$$
This process is often called Slepian process and can be expressed in terms of the standard Brownian motion $W(t)$ by
\begin{eqnarray... | {
"timestamp": "2019-04-17T02:00:22",
"yymm": "1904",
"arxiv_id": "1904.07227",
"language": "en",
"url": "https://arxiv.org/abs/1904.07227",
"abstract": "In this paper we extend results of L.A. Shepp by finding explicit formulas for the first passage probability $F_{a,b}(T\\, |\\, x)={\\rm Pr}(S(t)<a+bt \\t... |
https://arxiv.org/abs/1812.04117 | Triangulations and a discrete Brunn-Minkowski inequality in the plane | For a set $A$ of points in the plane, not all collinear, we denote by ${\rm tr}(A)$ the number of triangles in any triangulation of $A$; that is, ${\rm tr}(A) = 2i+b-2$ where $b$ and $i$ are the numbers of points of $A$ in the boundary and the interior of $[A]$ (we use $[A]$ to denote "convex hull of $A$"). We conjectu... | \section{Introduction}
In this paper we write $A,B$ to denote finite
subsets of ${\mathbb R}^d$,
and $|\cdot|$ stands for their cardinality. We say that $A\subset {\mathbb R}^d$ is $d$--{\it dimensional} if it is not contained in any affine hyperplane of ${\mathbb R}^d$. Equivalently, the real affine span of $A$ is ... | {
"timestamp": "2018-12-12T02:03:55",
"yymm": "1812",
"arxiv_id": "1812.04117",
"language": "en",
"url": "https://arxiv.org/abs/1812.04117",
"abstract": "For a set $A$ of points in the plane, not all collinear, we denote by ${\\rm tr}(A)$ the number of triangles in any triangulation of $A$; that is, ${\\rm ... |
https://arxiv.org/abs/1504.06970 | Asymptotics of the modes of the ordered Stirling numbers | It is known that the $S(n,k)$ Stirling numbers as well as the ordered Stirling numbers $k!S(n,k)$ form log-concave sequences. Although in the first case there are many estimations about the mode, for the ordered Stirling numbers such estimations are not known. In this short note we study this problem and some of its ge... | \section{Introduction}
It is a classical result that for a fixed $n$ the $S(n,k)$ Stirling numbers form strictly log-concave sequences (SLC), that is, for any $n>2$ it holds true that
\[S(n,1)<S(n,2)<\cdots<S(n,K_n^*)\ge S(n,K_n^*+1)>S(n,K_n^*+2)>\cdots>S(n,n)\]
for some index $K_n^*$. It is conjectured by Wegner that... | {
"timestamp": "2015-04-28T02:13:10",
"yymm": "1504",
"arxiv_id": "1504.06970",
"language": "en",
"url": "https://arxiv.org/abs/1504.06970",
"abstract": "It is known that the $S(n,k)$ Stirling numbers as well as the ordered Stirling numbers $k!S(n,k)$ form log-concave sequences. Although in the first case t... |
https://arxiv.org/abs/1110.4334 | On the vertex index of convex bodies | We introduce the vertex index, vein(K), of a given centrally symmetric convex body K, which, in a sense, measures how well K can be inscribed into a convex polytope with small number of vertices. This index is closely connected to the illumination parameter of a body, introduced earlier by the first named author, and, ... | \section{Introduction}
\label{zero}
Let ${\bf K}$ be a convex body symmetric about the origin $0$ in ${\mathbb R}^d, d\ge 2$
(such bodies below we call $0$-symmetric convex bodies). Now, we place ${\bf K}$
in a convex polytope, say ${\bf P}$, with vertices $p_1, p_2, \dots , p_n$, where
$n\ge d+1$. Then it is natural... | {
"timestamp": "2011-10-20T02:05:00",
"yymm": "1110",
"arxiv_id": "1110.4334",
"language": "en",
"url": "https://arxiv.org/abs/1110.4334",
"abstract": "We introduce the vertex index, vein(K), of a given centrally symmetric convex body K, which, in a sense, measures how well K can be inscribed into a convex ... |
https://arxiv.org/abs/2011.00862 | On the Existence of Zero-Sum Perfect Matchings of Complete Graphs | In this paper, we prove that given a 2-edge-coloured complete graph $K_{4n}$ that has the same number of edges of each colour, we can always find a perfect matching with an equal number of edges of each colour. This solves a problem posed by Caro, Hansberg, Lauri, and Zarb. The problem is also independently solved by E... | \section{Introduction}
Note that in this paper, we will use the word `matching' when in fact we mean `perfect matching'.
For an edge-colouring function $f:E(G)\to S$ of a graph $G$ where $S\subseteq\mathbb{Z}$ and a subgraph $H$ of $G$, if $\sum_{e\in E(H)} f(e)=0$ then $H$ is called a \emph{zero-sum subgraph} of $G$... | {
"timestamp": "2020-11-03T02:40:18",
"yymm": "2011",
"arxiv_id": "2011.00862",
"language": "en",
"url": "https://arxiv.org/abs/2011.00862",
"abstract": "In this paper, we prove that given a 2-edge-coloured complete graph $K_{4n}$ that has the same number of edges of each colour, we can always find a perfec... |
https://arxiv.org/abs/1803.01925 | Improved bounds on Brun's constant | Brun's constant is $B=\sum_{p \in P_{2}} p^{-1} + (p+2)^{-1}$, where the summation is over all twin primes. We improve the unconditional bounds on Brun's constant to $1.840503 < B < 2.288513$, which is about a 13\% improvement on the previous best published result. | \section{Introduction}
Brun \cite{Brun} showed that the sum of the reciprocals of the twin primes converges. That is, if $P_{2}$ denotes the set of primes $p$ such that $p+2$ is also prime, the sum $B:= \sum_{p\in P_{2}} 1/p + 1/(p+2)$ is finite.
Various estimates for Brun's constant have been given based on calculat... | {
"timestamp": "2018-03-07T02:02:27",
"yymm": "1803",
"arxiv_id": "1803.01925",
"language": "en",
"url": "https://arxiv.org/abs/1803.01925",
"abstract": "Brun's constant is $B=\\sum_{p \\in P_{2}} p^{-1} + (p+2)^{-1}$, where the summation is over all twin primes. We improve the unconditional bounds on Brun'... |
https://arxiv.org/abs/2006.03167 | Inject Machine Learning into Significance Test for Misspecified Linear Models | Due to its strong interpretability, linear regression is widely used in social science, from which significance test provides the significance level of models or coefficients in the traditional statistical inference. However, linear regression methods rely on the linear assumptions of the ground truth function, which d... |
\section{Experiment Details}
\label{Sec: Experiment Details}
\subsection{Experiment Settings}
\label{Sec: Experiment Settings}
We will compare our proposed method with traditional Linear Regression.
In linear regression, we focus on its LSE estimator, and use gradient descent to get $w^{LSE}$.
In our new proposed m... | {
"timestamp": "2020-06-08T02:03:45",
"yymm": "2006",
"arxiv_id": "2006.03167",
"language": "en",
"url": "https://arxiv.org/abs/2006.03167",
"abstract": "Due to its strong interpretability, linear regression is widely used in social science, from which significance test provides the significance level of mo... |
https://arxiv.org/abs/1507.08893 | On the Number of Divisors of $n^2 -1$ | We prove an asymptotic formula for the sum $\sum_{n \leq N} d(n^2 - 1)$, where $d(n)$ denotes the number of divisors of $n$. During the course of our proof, we also furnish an asymptotic formula for the sum $\sum_{d \leq N} g(d)$, where $g(d)$ denotes the number of solutions $x$ in $\mathbb{Z}_d$ to the equation $x^2 \... | \section{Introduction}
It is the main purpose of this note to prove the following theorem.
\begin{thm} \label{main}
Let $d(n)$ denote the number of divisors of $n$. Then
$$\sum_{n \leq N} d(n^2-1) \sim \frac{6}{\pi^2} N \log^2 N$$
as $N \rightarrow \infty$.
\end{thm}
In consideration of the more general sum $\sum... | {
"timestamp": "2015-08-03T02:09:32",
"yymm": "1507",
"arxiv_id": "1507.08893",
"language": "en",
"url": "https://arxiv.org/abs/1507.08893",
"abstract": "We prove an asymptotic formula for the sum $\\sum_{n \\leq N} d(n^2 - 1)$, where $d(n)$ denotes the number of divisors of $n$. During the course of our pr... |
https://arxiv.org/abs/0712.2796 | Central cross-sections make surfaces of revolution quadric | We prove here that when all planes transverse and nearly perpendicular to the axis of a surface of revolution intersect it in loops having central symmetry, the surface must be quadric. It follows that the quadrics are the only surfaces of revolution without skewloops. Similar statements hold for hypersurfaces of revol... | \section{Introduction}\label{sec:intro}
Quadric surfaces of revolution---ellipsoids, cones, paraboloids, cylinders,
and hyperboloids---are the most basic
nontrivial surfaces in $\,\mathbf{R}^{3}\,$.
Euler noted \cite{boyer} that one can
describe them all, up to rigid motion,
by appropriately choosing constants $\,a,... | {
"timestamp": "2008-06-06T22:34:44",
"yymm": "0712",
"arxiv_id": "0712.2796",
"language": "en",
"url": "https://arxiv.org/abs/0712.2796",
"abstract": "We prove here that when all planes transverse and nearly perpendicular to the axis of a surface of revolution intersect it in loops having central symmetry,... |
https://arxiv.org/abs/1807.07446 | Generators of Bieberbach groups with 2-generated holonomy group | An n-dimensional Bieberbach group is the fundamental group of a closed flat $n$-dimensional manifold. K. Dekimpe and P. Penninckx conjectured that an n-dimensional Bieberbach group can be generated by n elements. In this paper, we show that the conjecture is true if the holonomy group is 2-generated (e.g. dihedral grou... | \section{Introduction}
We first introduce the geometric definition of a crystallographic groups. A group $\Gamma$ is said to be an {\it $n$-dimensional crystallographic group} if it is a discrete subgroup of $\mathbb{R}^n\rtimes O(n)$, which is the group of isomotries of $\mathbb{R}^n$ and it acts cocompactly on $\mat... | {
"timestamp": "2018-07-20T02:09:35",
"yymm": "1807",
"arxiv_id": "1807.07446",
"language": "en",
"url": "https://arxiv.org/abs/1807.07446",
"abstract": "An n-dimensional Bieberbach group is the fundamental group of a closed flat $n$-dimensional manifold. K. Dekimpe and P. Penninckx conjectured that an n-di... |
https://arxiv.org/abs/0801.0798 | On the monochromatic Schur Triples type problem | We discuss a problem posed by Ronald Graham about the minimum number, over all 2-colorings of $[1,n]$, of monochromatic $\{x,y,x+ay\}$ triples for $a \geq 1$. We give a new proof of the original case of $a=1$. We show that the minimum number of such triples is at most $\frac{n^2}{2a(a^2+2a+3)} + O(n)$ when $a \geq 2$. ... | \section{Introduction} Recall that
the {\it Schur numbers}, $s(r)$ , denote the maximal integer $n$
such that there exists an $r$-coloring of $[1,n-1]$ that avoids a
monochromatic solution to $x+y=z$. For example $s(2) = 5$ and $s(3) = 14$.
$s(5)$ is unknown but is conjectured to be 161.\\
The original question
about ... | {
"timestamp": "2008-01-05T11:10:59",
"yymm": "0801",
"arxiv_id": "0801.0798",
"language": "en",
"url": "https://arxiv.org/abs/0801.0798",
"abstract": "We discuss a problem posed by Ronald Graham about the minimum number, over all 2-colorings of $[1,n]$, of monochromatic $\\{x,y,x+ay\\}$ triples for $a \\ge... |
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