url stringlengths 31 38 | title stringlengths 7 229 | abstract stringlengths 44 2.87k | text stringlengths 319 2.51M | meta dict |
|---|---|---|---|---|
https://arxiv.org/abs/2010.02918 | Some stable non-elementary classes of modules | Fisher [Fis75] and Baur [Bau75] showed independently in the seventies that if $T$ is a complete first-order theory extending the theory of modules, then the class of models of $T$ with pure embeddings is stable. In [Maz4, 2.12], it is asked if the same is true for any abstract elementary class $(K, \leq_p)$ such that $... | \section{Introduction}
An abstract elementary class $\mathbf{K}$ (AEC for short) is a pair $\mathbf{K}=(K \leap{\K})$ where $K$ is a class of structures and $\leap{\K}$ is a partial order on $K$ extending the substructure relation such that $\mathbf{K}$ is closed under direct limits and satisfies the coherence propert... | {
"timestamp": "2021-07-12T02:21:16",
"yymm": "2010",
"arxiv_id": "2010.02918",
"language": "en",
"url": "https://arxiv.org/abs/2010.02918",
"abstract": "Fisher [Fis75] and Baur [Bau75] showed independently in the seventies that if $T$ is a complete first-order theory extending the theory of modules, then t... |
https://arxiv.org/abs/2210.10148 | Bidiagonal Decompositions of Vandermonde-Type Matrices of Arbitrary Rank | We present a method to derive new explicit expressions for bidiagonal decompositions of Vandermonde and related matrices such as the (q-, h-) Bernstein-Vandermonde ones, among others. These results generalize the existing expressions for nonsingular matrices to matrices of arbitrary rank. For totally nonnegative matric... | \section{Introduction}
A matrix is totally nonnegative (TN) if all of its minors are nonnegative \cite{ando,fallatjohnsontp,karlin}. The bidiagonal decompositions of the TN matrices have become an important tool in the study of these matrices \cite{fallat01,gascapena92} and for performing matrix computations with them ... | {
"timestamp": "2022-10-20T02:02:49",
"yymm": "2210",
"arxiv_id": "2210.10148",
"language": "en",
"url": "https://arxiv.org/abs/2210.10148",
"abstract": "We present a method to derive new explicit expressions for bidiagonal decompositions of Vandermonde and related matrices such as the (q-, h-) Bernstein-Va... |
https://arxiv.org/abs/1810.03119 | Regularity of binomial edge ideals of chordal graphs | In this paper we prove the conjectured upper bound for Castelnuovo-Mumford regularity of binomial edge ideals posed in [23], in the case of chordal graphs. Indeed, we show that the regularity of any chordal graph G is bounded above by the number of maximal cliques of G, denoted by c(G). Moreover, we classify all chorda... | \section{Introduction}\label{introduction}
The binomial edge ideal of a graph was introduced in $2010$ by Herzog, Hibi, Hreinsd{\'o}ttir, Kahle and Rauh in \cite{HHHKR}, and at the same time by Ohtani in \cite{O}.
\par Let $G$ be a graph with the vertex set $V(G)=\{1,\ldots, n\}$ and the edge set $E(G)$. Let $S={\mat... | {
"timestamp": "2018-10-09T02:11:25",
"yymm": "1810",
"arxiv_id": "1810.03119",
"language": "en",
"url": "https://arxiv.org/abs/1810.03119",
"abstract": "In this paper we prove the conjectured upper bound for Castelnuovo-Mumford regularity of binomial edge ideals posed in [23], in the case of chordal graphs... |
https://arxiv.org/abs/2006.04701 | Determinants of binary matrices achieve every integral value up to $Ω(2^n/n)$ | This work shows that the smallest natural number $d_n$ that is not the determinant of some $n\times n$ binary matrix is at least $c\,2^n/n$ for $c=1/201$. That same quantity naturally lower bounds the number of distinct integers $D_n$ which can be written as the determinant of some $n\times n$ binary matrix. This asymp... | \section{Introduction}
We take a binary matrix $M$ to mean a matrix with all entries in $\{0,1\}$.
We investigate the range of the determinant, i.e. $\mathcal{D}_n=\{\det(M):M\in \{0,1\}^{n\times n}\}$
\footnote{Many papers take entries of binary matrices to be $\pm 1$.
There exists a one-to-one correspondence between ... | {
"timestamp": "2022-03-30T02:18:29",
"yymm": "2006",
"arxiv_id": "2006.04701",
"language": "en",
"url": "https://arxiv.org/abs/2006.04701",
"abstract": "This work shows that the smallest natural number $d_n$ that is not the determinant of some $n\\times n$ binary matrix is at least $c\\,2^n/n$ for $c=1/201... |
https://arxiv.org/abs/2302.13186 | Construction numbers: How to build a graph? | Counting the number of linear extensions of a partial order was considered by Stanley about 50 years ago. For the partial order on the vertices and edges of a graph determined by inclusion, we call such linear extensions {\it construction sequences} for the graph as each edge follows both of its endpoints. The number o... | \section{}
\begin{abstract}
\noindent We count the number of ways to build paths, stars, cycles, and complete graphs as a sequence of vertices and edges, where each edge follows both of its endpoints. The problem was considered 50 years ago by Stanley but the explicit sequences corresponding to graph families seem to ... | {
"timestamp": "2023-02-28T02:12:04",
"yymm": "2302",
"arxiv_id": "2302.13186",
"language": "en",
"url": "https://arxiv.org/abs/2302.13186",
"abstract": "Counting the number of linear extensions of a partial order was considered by Stanley about 50 years ago. For the partial order on the vertices and edges ... |
https://arxiv.org/abs/2211.13678 | On sumsets of nonbases of maximum size | Let $G$ be a finite abelian group. A nonempty subset $A$ in $G$ is called a basis of order $h$ if $hA=G$; when $hA \neq G$, it is called a nonbasis of order $h$. Our interest is in all possible sizes of $hA$ when $A$ is a nonbasis of order $h$ in $G$ of maximum size; we provide the complete answer when $h=2$ or $h=3$. | \section{Introduction}
Let $G$ be a finite abelian group of order $n \geq 2$, written in additive notation. For a positive integer $h$, the {\em Minkowski sum} of nonempty subsets $A_1, \ldots, A_h$ of $G$ is defined as
$$A_1+ \cdots + A_h = \{ a_1+ \cdots + a_h \; : \; a_1 \in A_1, \ldots, a_h \in A_h\}.$$ W... | {
"timestamp": "2022-11-28T02:14:22",
"yymm": "2211",
"arxiv_id": "2211.13678",
"language": "en",
"url": "https://arxiv.org/abs/2211.13678",
"abstract": "Let $G$ be a finite abelian group. A nonempty subset $A$ in $G$ is called a basis of order $h$ if $hA=G$; when $hA \\neq G$, it is called a nonbasis of or... |
https://arxiv.org/abs/2001.06557 | Magic Cayley-Sudoku Tables | A Cayley-sudoku table of a finite group G is a Cayley table for G, the body of which is partitioned into uniformly sized rectangular blocks in such a way that each group element appears exactly once in each block. A Cayley-sudoku table is pandiagonal magic provided the blocks are square and the "sum" (using the group o... | \section{Introduction}
Inspired by the popularity of sudoku puzzles along with the well-known fact that the body of the Cayley table\footnote{That is, the operation table of the group without the borders.} of any finite group already has $2/3$ of the properties of a sudoku table in that each element appears exactly on... | {
"timestamp": "2020-01-22T02:03:23",
"yymm": "2001",
"arxiv_id": "2001.06557",
"language": "en",
"url": "https://arxiv.org/abs/2001.06557",
"abstract": "A Cayley-sudoku table of a finite group G is a Cayley table for G, the body of which is partitioned into uniformly sized rectangular blocks in such a way ... |
https://arxiv.org/abs/2208.09050 | Large totally symmetric sets | A totally symmetric set is a subset of a group such that every permutation of the subset can be realized by conjugation in the group. The (non-)existence of large totally symmetric sets obstruct homomorphisms, so bounds on the sizes of totally symmetric sets are of particular use. In this paper, we prove that if a grou... |
\section{Introduction}
Kordek---Margalit \cite{Kordek-Margalit} introduced the notion of a totally symmetric set in a group as a means to study homomorphisms. Briefly, a subset $X \subset G$ of a group is totally symmetric if any permutation of $X$ can be realized by conjugation in $G$---for instance, the set of tran... | {
"timestamp": "2022-08-22T02:02:23",
"yymm": "2208",
"arxiv_id": "2208.09050",
"language": "en",
"url": "https://arxiv.org/abs/2208.09050",
"abstract": "A totally symmetric set is a subset of a group such that every permutation of the subset can be realized by conjugation in the group. The (non-)existence ... |
https://arxiv.org/abs/1908.11533 | A Newton algorithm for semi-discrete optimal transport with storage fees | We introduce and prove convergence of a damped Newton algorithm to approximate solutions of the semi-discrete optimal transport problem with storage fees, corresponding to a problem with hard capacity constraints. This is a variant of the optimal transport problem arising in queue penalization problems, and has applica... | \section{Introduction}
\subsection{Semi-discrete optimal transport with storage fees}
In this paper we deal with the following problem. Let $X\subset \R^n$, $n\geq 2$ be compact and $Y:=\{y_i\}_{i=1}^N \subset \R^n$ a fixed collection of finite points, along with a \emph{cost function} $c: X\times Y\to \R$ and a \em... | {
"timestamp": "2020-08-17T02:02:15",
"yymm": "1908",
"arxiv_id": "1908.11533",
"language": "en",
"url": "https://arxiv.org/abs/1908.11533",
"abstract": "We introduce and prove convergence of a damped Newton algorithm to approximate solutions of the semi-discrete optimal transport problem with storage fees,... |
https://arxiv.org/abs/2104.10841 | Uniqueness and stability for the solution of a nonlinear least squares problem | In this paper, we focus on the nonlinear least squares: $\mbox{min}_{\mathbf{x} \in \mathbb{H}^d}\| |A\mathbf{x}|-\mathbf{b}\|$ where $A\in \mathbb{H}^{m\times d}$, $\mathbf{b} \in \mathbb{R}^m$ with $\mathbb{H} \in \{\mathbb{R},\mathbb{C} \}$ and consider the uniqueness and stability of solutions. Such problem arises,... | \section{introduction}
\subsection{Problem setup}
Assume that $A:=[{\bm a}_1,\ldots,{\bm a}_m]^* \in {\mathbb H}^{m\times d}$ and
${\bm b}:=[b_1,\ldots,b_m]^*\in {\mathbb R}^m$ where ${\mathbb H}\in \{{\mathbb R},{\mathbb C}\}$. We are interested in the
following program
\begin{equation}\label{eq:least}
\Phi_A({\bm b}... | {
"timestamp": "2021-04-23T02:10:49",
"yymm": "2104",
"arxiv_id": "2104.10841",
"language": "en",
"url": "https://arxiv.org/abs/2104.10841",
"abstract": "In this paper, we focus on the nonlinear least squares: $\\mbox{min}_{\\mathbf{x} \\in \\mathbb{H}^d}\\| |A\\mathbf{x}|-\\mathbf{b}\\|$ where $A\\in \\mat... |
https://arxiv.org/abs/1505.06054 | Each normalized state is a member of an orthonormal basis: A simple proof | In a finite dimensional Hilbert space, each normalized vector (state) can be chosen as a member of an orthonormal basis of the space. We give a proof of this statement in a manner that seems to be more comprehensible for physics students than the formal abstract one. |
\section{Introduction}
Finite dimensional Hilbert spaces, e.g. spin of a particle or polarization of a photon, is of great interest in quantum mechanics, specially in quantum information and computation theory \cite{1}. So, studying the mathematical properties of such spaces is necessary for physics students, special... | {
"timestamp": "2015-05-25T02:08:44",
"yymm": "1505",
"arxiv_id": "1505.06054",
"language": "en",
"url": "https://arxiv.org/abs/1505.06054",
"abstract": "In a finite dimensional Hilbert space, each normalized vector (state) can be chosen as a member of an orthonormal basis of the space. We give a proof of t... |
https://arxiv.org/abs/1610.00780 | A subcopula based dependence measure | A dependence measure for arbitrary type pairs of random variables is proposed and analyzed, which in the particular case where both random variables are continuous turns out to be a concordance measure. Also, a sample version of the proposed dependence measure based on the empirical subcopula is provided, along with an... | \section{Introduction}
If $(X,Y)$ is a bivariate random vector with joint probability distribution $F_{X,Y}(x,y)=P(X\leq x, Y\leq y),$ the outstanding theorem by Sklar (1959) ensures that there exists a unique functional relationship $S$ between $F_{X,Y}$ and its marginal univariate probability distribution functions ... | {
"timestamp": "2017-02-07T02:12:10",
"yymm": "1610",
"arxiv_id": "1610.00780",
"language": "en",
"url": "https://arxiv.org/abs/1610.00780",
"abstract": "A dependence measure for arbitrary type pairs of random variables is proposed and analyzed, which in the particular case where both random variables are c... |
https://arxiv.org/abs/0903.4567 | Pancyclicity of Hamiltonian and highly connected graphs | A graph G on n vertices is Hamiltonian if it contains a cycle of length n and pancyclic if it contains cycles of length $\ell$ for all $3 \le \ell \le n$. Write $\alpha(G)$ for the independence number of $G$, i.e. the size of the largest subset of the vertex set that does not contain an edge, and $\kappa(G)$ for the (v... | \section{Introduction}
A {\em Hamilton cycle} is a spanning cycle in a graph, i.e.\ a cycle passing through all vertices.
A graph is called {\em Hamiltonian} if it contains such a cycle. Hamiltonicity is one of the most
fundamental notions in graph theory, tracing its origins to Sir William Rowan Hamilton in the 1850'... | {
"timestamp": "2009-03-26T14:16:51",
"yymm": "0903",
"arxiv_id": "0903.4567",
"language": "en",
"url": "https://arxiv.org/abs/0903.4567",
"abstract": "A graph G on n vertices is Hamiltonian if it contains a cycle of length n and pancyclic if it contains cycles of length $\\ell$ for all $3 \\le \\ell \\le n... |
https://arxiv.org/abs/2107.06328 | Searching for Rigidity in Algebraic Starscapes | We create plots of algebraic integers in the complex plane, exploring the effect of sizing the integers according to various arithmetic invariants. We focus on Galois theoretic invariants, in particular creating plots which emphasize algebraic integers whose Galois group is not the full symmetric group--these integers ... | \section{Introduction}
In what follows, we visually explore the set of complex numbers that are roots of monic polynomials with integer coefficients, the so-called \emph{algebraic integers}. The creation and study of plots of algebraic integers has a rich and collaborative history, bringing together pure and computati... | {
"timestamp": "2021-07-15T02:01:34",
"yymm": "2107",
"arxiv_id": "2107.06328",
"language": "en",
"url": "https://arxiv.org/abs/2107.06328",
"abstract": "We create plots of algebraic integers in the complex plane, exploring the effect of sizing the integers according to various arithmetic invariants. We foc... |
https://arxiv.org/abs/2110.06357 | Tangent Space and Dimension Estimation with the Wasserstein Distance | Consider a set of points sampled independently near a smooth compact submanifold of Euclidean space. We provide mathematically rigorous bounds on the number of sample points required to estimate both the dimension and the tangent spaces of that manifold with high confidence. The algorithm for this estimation is Local P... | \section{Introduction}
In this paper, we study the problem of estimating tangent spaces and the intrinsic dimension of a data manifold with high confidence. Our goal is to provide mathematically rigorous, explicit and practical bounds on the number of sample points required for such estimations. In data science terms,... | {
"timestamp": "2022-09-16T02:17:27",
"yymm": "2110",
"arxiv_id": "2110.06357",
"language": "en",
"url": "https://arxiv.org/abs/2110.06357",
"abstract": "Consider a set of points sampled independently near a smooth compact submanifold of Euclidean space. We provide mathematically rigorous bounds on the numb... |
https://arxiv.org/abs/1501.06320 | Conditions on Ramsey non-equivalence | Given a graph H, a graph G is called a Ramsey graph of H if there is a monochromatic copy of H in every coloring of the edges of G with two colors. Two graphs G, H are called Ramsey equivalent if they have the same set of Ramsey graphs. Fox et al. [J. Combin. Theory Ser. B 109 (2014), 120--133] asked whether there are ... | \section{Introduction}
Given a graph $H$, a graph $G$ is called a \emph{Ramsey graph of $H$} if there is a monochromatic copy of $H$ in every coloring of the edges of $G$ with two colors.
If $G$ is a Ramsey graph of $H$, we write $G\to H$ and say that $G$ arrows $H$.
We denote by $\mathcal{R}(H)$ the set of all gr... | {
"timestamp": "2015-03-25T01:10:22",
"yymm": "1501",
"arxiv_id": "1501.06320",
"language": "en",
"url": "https://arxiv.org/abs/1501.06320",
"abstract": "Given a graph H, a graph G is called a Ramsey graph of H if there is a monochromatic copy of H in every coloring of the edges of G with two colors. Two gr... |
https://arxiv.org/abs/2206.08054 | Generalized Leverage Scores: Geometric Interpretation and Applications | In problems involving matrix computations, the concept of leverage has found a large number of applications. In particular, leverage scores, which relate the columns of a matrix to the subspaces spanned by its leading singular vectors, are helpful in revealing column subsets to approximately factorize a matrix with qua... | \section*{Appendix}
\subsection*{Proofs missing from the main text}
\input{proof_cssp}
\subsection*{Quality-efficiency tradeoff results for the rest of the datasets}
In Figures~\ref{fig:objective ratio} and~\ref{fig:running times} we
plot the objective ratio and the running time for a larger number of
datasets... | {
"timestamp": "2022-06-17T02:15:45",
"yymm": "2206",
"arxiv_id": "2206.08054",
"language": "en",
"url": "https://arxiv.org/abs/2206.08054",
"abstract": "In problems involving matrix computations, the concept of leverage has found a large number of applications. In particular, leverage scores, which relate ... |
https://arxiv.org/abs/1711.06818 | Convex Set of Doubly Substochastic Matrices | Denote $\mathcal{A}$ as the set of all doubly substochastic $m \times n$ matrices and let $k$ be a positive integer. Let $\mathcal{A}_k$ be the set of all $1/k$-bounded doubly substochastic $m \times n$ matrices, i.e., $\mathcal{A}_k \triangleq \{E \in \mathcal{A}: e_{i,j} \in [0, 1/k], \forall i=1,2,\cdots,m, j = 1,2,... | \section{Introduction}
An $n \times n$ matrix $E=(e_{i,j})$ is \emph{doubly stochastic} if it satisfies the following conditions:
\begin{equation}
\left\{
\begin{array}{ll}
e_{i,j} \ge 0, & \hbox{$\forall i,j=1,2,\cdots, n$;} \\
\sum_{j=1}^{n} e_{i,j} = 1, & \hbox{$\forall i = 1,2,\cdots,n$;} \\
\sum_{i=1... | {
"timestamp": "2017-11-21T02:05:24",
"yymm": "1711",
"arxiv_id": "1711.06818",
"language": "en",
"url": "https://arxiv.org/abs/1711.06818",
"abstract": "Denote $\\mathcal{A}$ as the set of all doubly substochastic $m \\times n$ matrices and let $k$ be a positive integer. Let $\\mathcal{A}_k$ be the set of ... |
https://arxiv.org/abs/2007.08639 | Nested formulas for cosine and inverse cosine functions based on Viète's formula for $π$ | In this article, we develop nested representations for cosine and inverse cosine functions, which is a generalization of Viète's formula for $\pi$. We explore a natural inverse relationship between these representations and develop numerical algorithms to compute them. Throughout this article, we perform numerical comp... | \section{Introduction}
In 1593s, Vi\`{e}te developed an infinite nested radical product formula for $\pi$ given as
\begin{equation}\label{eq:1}
\frac{2}{\pi}= \frac{\sqrt{2}}{2}\frac{\sqrt{2+\sqrt{2}}}{2}\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdots,
\end{equation}
which, historically, may have been the first infinite se... | {
"timestamp": "2020-07-20T02:03:48",
"yymm": "2007",
"arxiv_id": "2007.08639",
"language": "en",
"url": "https://arxiv.org/abs/2007.08639",
"abstract": "In this article, we develop nested representations for cosine and inverse cosine functions, which is a generalization of Viète's formula for $\\pi$. We ex... |
https://arxiv.org/abs/1710.10989 | An improved algorithm to compute the exponential of a matrix | In this work, we present a new way to compute the Taylor polynomial of the matrix exponential which reduces the number of matrix multiplications in comparison with the de-facto standard Patterson-Stockmeyer method. This reduction is sufficient to make the method superior in performance to Padé approximants by 10-30% ov... | \section{Introduction}
The search for efficient algorithms to compute the exponential of an square matrix has a tremendous interest, given the wide range of its applications in many branches of science. Its
importance is clearly revealed by the great impact achieved by various reference reviews devoted to the subject,... | {
"timestamp": "2017-10-31T01:18:36",
"yymm": "1710",
"arxiv_id": "1710.10989",
"language": "en",
"url": "https://arxiv.org/abs/1710.10989",
"abstract": "In this work, we present a new way to compute the Taylor polynomial of the matrix exponential which reduces the number of matrix multiplications in compar... |
https://arxiv.org/abs/2201.13417 | In praise (and search) of J. V. Uspensky | The two of us have shared a fascination with James Victor Uspensky's 1937 textbook $Introduction \, to \, Mathematical \, Probability$ ever since our graduate student days: it contains many interesting results not found in other books on the same subject in the English language, together with many non-trivial examples,... | \section{Introduction}
In 1927, when Harald Cram\'{e}r visited England and mentioned to G. H. Hardy (his friend and former teacher) he had become interested in probability theory, Hardy replied ``there was no mathematically satisfactory book in English on this subject, and encouraged me to write one'' (Cram\'{e}r, 197... | {
"timestamp": "2022-02-01T02:54:19",
"yymm": "2201",
"arxiv_id": "2201.13417",
"language": "en",
"url": "https://arxiv.org/abs/2201.13417",
"abstract": "The two of us have shared a fascination with James Victor Uspensky's 1937 textbook $Introduction \\, to \\, Mathematical \\, Probability$ ever since our g... |
https://arxiv.org/abs/1308.2698 | Positroids and non-crossing partitions | We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground set, and then freely placing the structure of a connected positroid on each of t... | \section{Introduction}
\label{sec:intro}
A \emph{positroid} is a matroid on an ordered set which can be represented
by the columns of a full rank $d \times n$ real matrix such that all its maximal
minors are nonnegative. Such matroids were first considered by
Postnikov \cite{postnikov} in his study of the \emph{tot... | {
"timestamp": "2013-09-17T02:08:06",
"yymm": "1308",
"arxiv_id": "1308.2698",
"language": "en",
"url": "https://arxiv.org/abs/1308.2698",
"abstract": "We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every posit... |
https://arxiv.org/abs/1610.07992 | Numerical analysis of strongly nonlinear PDEs | We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and nonconvex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental result in this area which states that stable, consistent, and monotone schemes con... |
\section{Discretizations of convex second--order elliptic equations}
\label{sec:convex}
Up to this point we have discussed general issues regarding stability and convergence of numerical methods
for general fully nonlinear equations, as well as the construction of suitable schemes for linear problems in nondivergenc... | {
"timestamp": "2016-10-26T02:08:27",
"yymm": "1610",
"arxiv_id": "1610.07992",
"language": "en",
"url": "https://arxiv.org/abs/1610.07992",
"abstract": "We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and nonconvex fully nonlinear equatio... |
https://arxiv.org/abs/1910.06386 | All of Linear Regression | Least squares linear regression is one of the oldest and widely used data analysis tools. Although the theoretical analysis of the ordinary least squares (OLS) estimator is as old, several fundamental questions are yet to be answered. Suppose regression observations $(X_1,Y_1),\ldots,(X_n,Y_n)\in\mathbb{R}^d\times\math... | \section{Introduction}
Linear regression is one of the oldest and most widely practiced data analysis method. In many real data settings least squares linear regression leads to performance in par with state-of-the-art (and often far more complicated) methods while remaining amenable to interpretation. These advantages... | {
"timestamp": "2019-10-16T02:02:10",
"yymm": "1910",
"arxiv_id": "1910.06386",
"language": "en",
"url": "https://arxiv.org/abs/1910.06386",
"abstract": "Least squares linear regression is one of the oldest and widely used data analysis tools. Although the theoretical analysis of the ordinary least squares ... |
https://arxiv.org/abs/1402.3945 | Approximating gradients with continuous piecewise polynomial functions | Motivated by conforming finite element methods for elliptic problems of second order, we analyze the approximation of the gradient of a target function by continuous piecewise polynomial functions over a simplicial mesh. The main result is that the global best approximation error is equivalent to an appropriate sum in ... | \section{Introduction}
\label{S:Intro}
Finite element methods are one of the most successful tools for the
numerical solution of partial differential equations. In their
simplest form they are Galerkin methods where the discrete space is
given by elements that are appropriately coupled. This piecewise
structure allo... | {
"timestamp": "2014-02-18T02:11:23",
"yymm": "1402",
"arxiv_id": "1402.3945",
"language": "en",
"url": "https://arxiv.org/abs/1402.3945",
"abstract": "Motivated by conforming finite element methods for elliptic problems of second order, we analyze the approximation of the gradient of a target function by c... |
https://arxiv.org/abs/2203.16462 | Convergence of gradient descent for deep neural networks | This article presents a criterion for convergence of gradient descent to a global minimum, which is then used to show that gradient descent with proper initialization converges to a global minimum when training any feedforward neural network with smooth and strictly increasing activation functions, provided that the in... | \section{A convergence criterion for gradient descent}\label{introsec}
The goal of gradient descent is to find a minimum of a differentiable function $f:\mathbb{R}^p\to \mathbb{R}$ by starting at some $x_0\in \mathbb{R}^p$, and then iteratively moving in the direction of steepest descent, as
\[
x_{k+1} = x_k -\eta_k \n... | {
"timestamp": "2022-12-20T02:10:32",
"yymm": "2203",
"arxiv_id": "2203.16462",
"language": "en",
"url": "https://arxiv.org/abs/2203.16462",
"abstract": "This article presents a criterion for convergence of gradient descent to a global minimum, which is then used to show that gradient descent with proper in... |
https://arxiv.org/abs/2012.14979 | Contour Integral Methods for Nonlinear Eigenvalue Problems: A Systems Theoretic Approach | Contour integral methods for nonlinear eigenvalue problems seek to compute a subset of the spectrum in a bounded region of the complex plane. We briefly survey this class of algorithms, establishing a relationship to system realization techniques in control theory. This connection motivates a new general framework for ... |
\section{Conclusions}
Contour integral methods provide an effective tool for computing
eigenvalues in a bounded region of the complex plane.
By casting these algorithms in the framework of systems theory,
we have proposed several new Loewner matrix methods inspired by
rational interpolation for system realization an... | {
"timestamp": "2021-01-01T02:17:35",
"yymm": "2012",
"arxiv_id": "2012.14979",
"language": "en",
"url": "https://arxiv.org/abs/2012.14979",
"abstract": "Contour integral methods for nonlinear eigenvalue problems seek to compute a subset of the spectrum in a bounded region of the complex plane. We briefly s... |
https://arxiv.org/abs/1909.03770 | Correlation for permutations | In this note we investigate correlation inequalities for `up-sets' of permutations, in the spirit of the Harris--Kleitman inequality. We focus on two well-studied partial orders on $S_n$, giving rise to differing notions of up-sets. Our first result shows that, under the strong Bruhat order on $S_n$, up-sets are positi... | \section{Introduction}
Let $X=\{1,2,\dots,n\}=[n]$. A family $\mathcal{F} \subset {\cal P}(X) = \{A: A \subset X\}$ is an \emph{up-set} if given $F \in \mathcal{F}$ and $F \subset G \subset X$ then $G \in \mathcal{F}$. The well-known and very useful Harris--Kleitman inequality \cite{harris,kleitman} guarantees that an... | {
"timestamp": "2020-04-22T02:09:31",
"yymm": "1909",
"arxiv_id": "1909.03770",
"language": "en",
"url": "https://arxiv.org/abs/1909.03770",
"abstract": "In this note we investigate correlation inequalities for `up-sets' of permutations, in the spirit of the Harris--Kleitman inequality. We focus on two well... |
https://arxiv.org/abs/2111.06877 | Continuity, Uniqueness and Long-Term Behavior of Nash Flows Over Time | We consider a dynamic model of traffic that has received a lot of attention in the past few years. Users control infinitesimal flow particles aiming to travel from a source to destination as quickly as possible. Flow patterns vary over time, and congestion effects are modeled via queues, which form whenever the inflow ... |
\section{Conclusion}\label{sec:conclusion}
\modified{
As already remarked, our continuity results can be seen as necessary requirements for the fluid queueing model to have any bearing on understanding real traffic networks.
Were it the case that continuity did not hold, arbitrarily small deviations from the (simpli... | {
"timestamp": "2021-11-15T02:25:11",
"yymm": "2111",
"arxiv_id": "2111.06877",
"language": "en",
"url": "https://arxiv.org/abs/2111.06877",
"abstract": "We consider a dynamic model of traffic that has received a lot of attention in the past few years. Users control infinitesimal flow particles aiming to tr... |
https://arxiv.org/abs/1803.00724 | Transformation representations of sandwich semigroups | Let $a$ be an element of a semigroup $S$. The local subsemigroup of $S$ with respect to $a$ is the subsemigroup $aSa$ of $S$. The variant of $S$ with respect to $a$ is the semigroup with underlying set $S$ and operation $\star_a$ defined by $x\star_ay=xay$ for $x,y\in S$. We show that the following classes contain prec... | \section{Introduction and statement of main result}\label{sect:intro}
The purpose of this note is to establish a link between two well-known semigroup constructions, namely \emph{semigroup variants} and \emph{local subsemigroups}, both to be defined shortly. Our main result (Theorem \ref{thm:main} below) shows tha... | {
"timestamp": "2018-03-05T02:07:00",
"yymm": "1803",
"arxiv_id": "1803.00724",
"language": "en",
"url": "https://arxiv.org/abs/1803.00724",
"abstract": "Let $a$ be an element of a semigroup $S$. The local subsemigroup of $S$ with respect to $a$ is the subsemigroup $aSa$ of $S$. The variant of $S$ with resp... |
https://arxiv.org/abs/1811.05200 | Polynomial Schur's theorem | We resolve the Ramsey problem for $\{x,y,z:x+y=p(z)\}$ for all polynomials $p$ over $\mathbb{Z}$. In particular, we characterise all polynomials that are $2$-Ramsey, that is, those $p(z)$ such that any $2$-colouring of $\mathbb{N}$ contains infinitely many monochromatic solutions for $x+y=p(z)$. For polynomials that ar... | \section{Introduction}
The study of Ramsey theory searches for monochromatic patterns in finite colourings of ${\mathbb N}$. A pattern is \emph{$k$-Ramsey}, $k\in{\mathbb N}$, if it appears \emph{infinitely} often in any $k$-colouring of ${\mathbb N}$; and \emph{Ramsey} if this holds for every $k\in{\mathbb N}$. Ramsey... | {
"timestamp": "2018-11-14T02:09:57",
"yymm": "1811",
"arxiv_id": "1811.05200",
"language": "en",
"url": "https://arxiv.org/abs/1811.05200",
"abstract": "We resolve the Ramsey problem for $\\{x,y,z:x+y=p(z)\\}$ for all polynomials $p$ over $\\mathbb{Z}$. In particular, we characterise all polynomials that a... |
https://arxiv.org/abs/2009.11413 | An elementary approach for minimax estimation of Bernoulli proportion in the restricted parameter space | We present an elementary mathematical method to find the minimax estimator of the Bernoulli proportion $\theta$ under the squared error loss when $\theta$ belongs to the restricted parameter space of the form $\Omega = [0, \eta]$ for some pre-specified constant $0 \leq \eta \leq 1$. This problem is inspired from the pr... | \section{Introduction}
\label{sec:intro}
Point estimation of model parameters have been an important topic in statistics, machine learning, and data science.
Besides the maximum likelihood estimator (MLE), the method of moment (MOM), and Bayesian method, another important approach
is the minimax estimator that minim... | {
"timestamp": "2020-09-25T02:04:51",
"yymm": "2009",
"arxiv_id": "2009.11413",
"language": "en",
"url": "https://arxiv.org/abs/2009.11413",
"abstract": "We present an elementary mathematical method to find the minimax estimator of the Bernoulli proportion $\\theta$ under the squared error loss when $\\thet... |
https://arxiv.org/abs/2103.02112 | Should multilevel methods for discontinuous Galerkin discretizations use discontinuous interpolation operators? | Multi-level preconditioners for Discontinuous Galerkin (DG) discretizations are widely used to solve elliptic equations, and a main ingredient of such solvers is the interpolation operator to transfer information from the coarse to the fine grid. Classical interpolation operators give continuous interpolated values, bu... | \section{Discontinuous Interpolation for a Model Problem}
Interpolation operators are very important for the construction of a
multigrid method. Since multigrid's inception by Fedorenko
\cite{Fedorenko1964}, interpolation was identified as key, deserving
an entire appendix in Brandt's seminal work \cite{Brandt1977}:
... | {
"timestamp": "2021-03-04T02:09:19",
"yymm": "2103",
"arxiv_id": "2103.02112",
"language": "en",
"url": "https://arxiv.org/abs/2103.02112",
"abstract": "Multi-level preconditioners for Discontinuous Galerkin (DG) discretizations are widely used to solve elliptic equations, and a main ingredient of such sol... |
https://arxiv.org/abs/1903.04532 | When can a link be obtained from another using crossing exchanges and smoothings? | Let $L$ be a fixed link. Given a link diagram $D$, is there a sequence of crossing exchanges and smoothings on $D$ that yields a diagram of $L$? We approach this problem from the computational complexity point of view. It follows from work by Endo, Itoh, and Taniyama that if $L$ is a prime link with crossing number at ... | \section{Introduction}\label{sec:intro}
We work in the piecewise linear category. All links under consideration are nonsplit, unordered, unoriented and contained in the 3-sphere ${S}^3$. We remark that when we speak of a link $L$ we include the possibility that $L$ is a link with only one component, that is, a knot. A... | {
"timestamp": "2019-03-14T01:07:44",
"yymm": "1903",
"arxiv_id": "1903.04532",
"language": "en",
"url": "https://arxiv.org/abs/1903.04532",
"abstract": "Let $L$ be a fixed link. Given a link diagram $D$, is there a sequence of crossing exchanges and smoothings on $D$ that yields a diagram of $L$? We approa... |
https://arxiv.org/abs/2106.09606 | Cardinality Minimization, Constraints, and Regularization: A Survey | We survey optimization problems that involve the cardinality of variable vectors in constraints or the objective function. We provide a unified viewpoint on the general problem classes and models, and give concrete examples from diverse application fields such as signal and image processing, portfolio selection, or mac... |
\section{Introduction}\label{sec:intro}
The cardinality of variable vectors occurs in a plethora of optimization
problems, in either constraints or the objective function. In the
following, we attempt to describe the broad landscape of such problems with
a general emphasis on \emph{continuous} variables. This restrict... | {
"timestamp": "2021-06-18T02:26:57",
"yymm": "2106",
"arxiv_id": "2106.09606",
"language": "en",
"url": "https://arxiv.org/abs/2106.09606",
"abstract": "We survey optimization problems that involve the cardinality of variable vectors in constraints or the objective function. We provide a unified viewpoint ... |
https://arxiv.org/abs/1601.07458 | Computing partial traces and reduced density matrices | Taking partial traces for computing reduced density matrices, or related functions, is a ubiquitous procedure in the quantum mechanics of composite systems. In this article, we present a thorough description of this function and analyze the number of elementary operations (ops) needed, under some possible alternative i... | \section{Introduction}
When calculating certain functions of quantum systems, in many instances
the running time of classical computers increases exponentially with
the number of elementary parts that compose those systems. This issue
is a hurdle to current research in many areas of science. But it is
also a motive fo... | {
"timestamp": "2016-08-25T02:00:35",
"yymm": "1601",
"arxiv_id": "1601.07458",
"language": "en",
"url": "https://arxiv.org/abs/1601.07458",
"abstract": "Taking partial traces for computing reduced density matrices, or related functions, is a ubiquitous procedure in the quantum mechanics of composite system... |
https://arxiv.org/abs/1805.06954 | A better bound for ordinary triangles | Let $P$ be a finite set of points in the plane. A c-ordinary triangle is a set of three non-collinear points of $P$ such that each line spanned by the points contains at most $c$ points of $P$. We show that if $P$ is not contained in the union of two lines and $|P|$ is sufficiently large, then it contains an 11-ordinar... | \section{Introduction}
Let $P$ be a set of $n$ points in $\mathbb R^2$. A line which contains two or more points of $P$ is called a \emph{spanned line}, and we call a spanned line \emph{c-ordinary} if it contains at most $c$ points of $P$. The Sylvester-Gallai Theorem states that there exists a 2-ordinary line in any ... | {
"timestamp": "2018-06-28T02:01:47",
"yymm": "1805",
"arxiv_id": "1805.06954",
"language": "en",
"url": "https://arxiv.org/abs/1805.06954",
"abstract": "Let $P$ be a finite set of points in the plane. A c-ordinary triangle is a set of three non-collinear points of $P$ such that each line spanned by the poi... |
https://arxiv.org/abs/1911.04522 | Relating Notions of Convergence in Geometric Analysis | We relate $L^p$ convergence of metric tensors or volume convergence to a given smooth metric to Intrinsic Flat and Gromov-Hausdorff convergence for sequences of Riemannian manifolds. We present many examples of sequences of conformal metrics which demonstrate that these notions of convergence do not agree in general ev... | \section{Introduction}
There are many settings in Riemannian geometry where one must examine a sequence of Riemannian manifolds and prove that they converge in some sense to a limit space. These situations arise when one is seeking a canonical metric in a given class, or examining how manifolds evolve under flows, o... | {
"timestamp": "2020-06-01T02:13:35",
"yymm": "1911",
"arxiv_id": "1911.04522",
"language": "en",
"url": "https://arxiv.org/abs/1911.04522",
"abstract": "We relate $L^p$ convergence of metric tensors or volume convergence to a given smooth metric to Intrinsic Flat and Gromov-Hausdorff convergence for sequen... |
https://arxiv.org/abs/2204.04436 | Error Guarantees for Least Squares Approximation with Noisy Samples in Domain Adaptation | Given $n$ samples of a function $f\colon D\to\mathbb C$ in random points drawn with respect to a measure $\varrho_S$ we develop theoretical analysis of the $L_2(D, \varrho_T)$-approximation error. For a parituclar choice of $\varrho_S$ depending on $\varrho_T$, it is known that the weighted least squares method from fi... | \section{Introduction}
In this paper we study the reconstruction of complex-valued functions on a $d$-dimensional domain $D\subset \mathds R^d$ from possibly noisy function values
\begin{align*}
\bm y = \bm f + \bm \varepsilon = (f(\bm x^1) + \varepsilon_1, \dots, f(\bm x^n)+\varepsilon_n)^{\mathsf T}\,,
\end{align... | {
"timestamp": "2022-04-12T02:10:17",
"yymm": "2204",
"arxiv_id": "2204.04436",
"language": "en",
"url": "https://arxiv.org/abs/2204.04436",
"abstract": "Given $n$ samples of a function $f\\colon D\\to\\mathbb C$ in random points drawn with respect to a measure $\\varrho_S$ we develop theoretical analysis o... |
https://arxiv.org/abs/1901.01075 | Activity measures of dynamical systems over non-archimedean fields | Toward the understanding of bifurcation phenomena of dynamics on the Berkovich projective line $\mathbb{P}^{1,an}$ over non-archimedean fields, we study the stability (or passivity) of critical points of families of polynomials parametrized by analytic curves. We construct the activity measure of a critical point of a ... | \section{Introduction}
Let $K$ be an algebraically closed field with complete, non-trivial and non-archimedean valuation. Let us consider an analytic family of rational functions $f : V\times \mathbb{P}^{1, an}\to\mathbb{P}^{1, an}$ of degree $d\geq2$ parametrized by a strictly $K$-analytic curve $V$ and a marked criti... | {
"timestamp": "2019-09-10T02:10:38",
"yymm": "1901",
"arxiv_id": "1901.01075",
"language": "en",
"url": "https://arxiv.org/abs/1901.01075",
"abstract": "Toward the understanding of bifurcation phenomena of dynamics on the Berkovich projective line $\\mathbb{P}^{1,an}$ over non-archimedean fields, we study ... |
https://arxiv.org/abs/1907.10196 | A reverse Aldous/Broder algorithm | The Aldous--Broder algorithm provides a way of sampling a uniformly random spanning tree for finite connected graphs using simple random walk. Namely, start a simple random walk on a connected graph and stop at the cover time. The tree formed by all the first-entrance edges has the law of a uniform spanning tree. Here ... | \section{Introduction}\label{sec:intro}
Let $G=(V,E)$ be a finite connected graph and let $\mathcal{T}$ be the set of spanning trees of $G$. Obviously $\mathcal{T}$ is a finite set. The \notion{uniform spanning tree} is the uniform measure on $\mathcal{T}$, which is denoted by $\mathsf{UST}(G)$.
In the late... | {
"timestamp": "2021-03-29T02:23:36",
"yymm": "1907",
"arxiv_id": "1907.10196",
"language": "en",
"url": "https://arxiv.org/abs/1907.10196",
"abstract": "The Aldous--Broder algorithm provides a way of sampling a uniformly random spanning tree for finite connected graphs using simple random walk. Namely, sta... |
https://arxiv.org/abs/1606.02328 | Drawing the Almost Convex Set in an Integer Grid of Minimum Size | In 2001, Károlyi, Pach and Tóth introduced a family of point sets to solve an Erdős-Szekeres type problem; which have been used to solve several other Edős-Szekeres type problems. In this paper we refer to these sets as nested almost convex sets. A nested almost convex set $\mathcal{X}$ has the property that the interi... | \section{Introduction.}
We say that a set of points in the plane is in \emph{general position}
if no three of them are collinear.
Throughout this paper all points sets are in general position.
In \cite{Erdos_EmptyConvex_1978}, Erd\H{o}s asked for the minimum integer $E(s,l)$ that satisfies the following.
Every set o... | {
"timestamp": "2016-06-09T02:01:44",
"yymm": "1606",
"arxiv_id": "1606.02328",
"language": "en",
"url": "https://arxiv.org/abs/1606.02328",
"abstract": "In 2001, Károlyi, Pach and Tóth introduced a family of point sets to solve an Erdős-Szekeres type problem; which have been used to solve several other Edő... |
https://arxiv.org/abs/1804.05952 | Erdős-Szekeres On-Line | In 1935, Erdős and Szekeres proved that $(m-1)(k-1)+1$ is the minimum number of points in the plane which definitely contain an increasing subset of $m$ points or a decreasing subset of $k$ points (as ordered by their $x$-coordinates). We consider their result from an on-line game perspective: Let points be determined ... | \section{Introduction}
In \cite{ErdosSzekeres1935}, Erd\H{o}s and Szekeres proved that $(m-1)(k-1)+1$ is the minimum number of points in the plane (ordered by their $x$-coordinates) that guarantees an increasing (in terms of $y$-coordinates) subset of $m$ points or a decreasing subset of $k$ points.
We refer to this n... | {
"timestamp": "2018-04-19T02:05:03",
"yymm": "1804",
"arxiv_id": "1804.05952",
"language": "en",
"url": "https://arxiv.org/abs/1804.05952",
"abstract": "In 1935, Erdős and Szekeres proved that $(m-1)(k-1)+1$ is the minimum number of points in the plane which definitely contain an increasing subset of $m$ p... |
https://arxiv.org/abs/2212.00446 | Nontrivial lower bounds for the $p$-adic valuations of some type of rational numbers | In this paper, we will show that the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers is unusually large. This generalizes the very recent results by the author and by A. Dubickas, which are both related to the special case $p = 2$. The crucial point for obtaining our main result ... | \section{Introduction and Notation}\label{sec1}
Throughout this paper, we let ${\mathbb N}$ denote the set of positive integers and ${\mathbb N}_0 := {\mathbb N} \cup \{0\}$ denote the set of non-negative integers. For $x \in {\mathbb R}$, we let $\lfloor x\rfloor$ denote the integer part of $x$. For a given prime num... | {
"timestamp": "2022-12-02T02:12:04",
"yymm": "2212",
"arxiv_id": "2212.00446",
"language": "en",
"url": "https://arxiv.org/abs/2212.00446",
"abstract": "In this paper, we will show that the $p$-adic valuation (where $p$ is a given prime number) of some type of rational numbers is unusually large. This gene... |
https://arxiv.org/abs/2105.07388 | Fast randomized numerical rank estimation for numerically low-rank matrices | Matrices with low-rank structure are ubiquitous in scientific computing. Choosing an appropriate rank is a key step in many computational algorithms that exploit low-rank structure. However, estimating the rank has been done largely in an ad-hoc fashion in large-scale settings. In this work we develop a randomized algo... | \section{Introduction}\label{sec:intro}
Low-rank matrices are ubiquitous in scientific computing and data science.
Sometimes a matrix of interest can be shown to be of numerically low rank~\cite{beckermann2017singular,udell2019big}, for example by showing that the singular values decay rapidly.
More often, matrices t... | {
"timestamp": "2021-05-18T02:17:45",
"yymm": "2105",
"arxiv_id": "2105.07388",
"language": "en",
"url": "https://arxiv.org/abs/2105.07388",
"abstract": "Matrices with low-rank structure are ubiquitous in scientific computing. Choosing an appropriate rank is a key step in many computational algorithms that ... |
https://arxiv.org/abs/1809.03049 | Comparing the power of cops to zombies in pursuit-evasion games | We compare two kinds of pursuit-evasion games played on graphs. In Cops and Robbers, the cops can move strategically to adjacent vertices as they please, while in a new variant, called deterministic Zombies and Survivors, the zombies (the counterpart of the cops) are required to always move towards the survivor (the co... | \section{Introduction}
The game of Cops and Robbers is a perfect-information two-player pursuit-evasion game played on a graph. One player controls a group of cops and the other player controls a single robber. To begin the game, the cops and robber each choose vertices to occupy, with the cops choosing first. Pl... | {
"timestamp": "2018-09-11T02:12:35",
"yymm": "1809",
"arxiv_id": "1809.03049",
"language": "en",
"url": "https://arxiv.org/abs/1809.03049",
"abstract": "We compare two kinds of pursuit-evasion games played on graphs. In Cops and Robbers, the cops can move strategically to adjacent vertices as they please, ... |
https://arxiv.org/abs/1712.00709 | Simulated Annealing Algorithm for Graph Coloring | The goal of this Random Walks project is to code and experiment the Markov Chain Monte Carlo (MCMC) method for the problem of graph coloring. In this report, we present the plots of cost function \(\mathbf{H}\) by varying the parameters like \(\mathbf{q}\) (Number of colors that can be used in coloring) and \(\mathbf{c... | \section{Introduction}
\subsection{Graph Coloring}
Graph coloring is one of the most important concepts in graph theory and is used in many real time applications in computer science. It can be defined as a problem of how to assign colors to certain elements of a graph given some constraints. It is especially used in... | {
"timestamp": "2017-12-05T02:08:57",
"yymm": "1712",
"arxiv_id": "1712.00709",
"language": "en",
"url": "https://arxiv.org/abs/1712.00709",
"abstract": "The goal of this Random Walks project is to code and experiment the Markov Chain Monte Carlo (MCMC) method for the problem of graph coloring. In this repo... |
https://arxiv.org/abs/2107.07643 | The principal Erdős--Gallai differences of a degree sequence | The Erdős--Gallai criteria for recognizing degree sequences of simple graphs involve a system of inequalities. Given a fixed degree sequence, we consider the list of differences of the two sides of these inequalities. These differences have appeared in varying contexts, including characterizations of the split and thre... | \section{Erd\H{o}s--Gallai differences} \label{sec: intro}
Let $d=(d_1,\dots,d_n)$ be the degree sequence of an arbitrary finite, simple graph, and suppose that the terms of $d$ are listed in nonincreasing order. Let $m(d) = \max\{i: d_i \geq i-1\}$; this parameter is called the \emph{modified Durfee number} of $d$. ... | {
"timestamp": "2021-07-19T02:05:28",
"yymm": "2107",
"arxiv_id": "2107.07643",
"language": "en",
"url": "https://arxiv.org/abs/2107.07643",
"abstract": "The Erdős--Gallai criteria for recognizing degree sequences of simple graphs involve a system of inequalities. Given a fixed degree sequence, we consider ... |
https://arxiv.org/abs/1507.01364 | Proof of a conjecture on the zero forcing number of a graph | Amos et al. (Discrete Appl. Math. 181 (2015) 1-10) introduced the notion of the $k$-forcing number of graph for a positive integer $k$ as the generalization of the zero forcing number of a graph. The $k$-forcing number of a simple graph $G$, denoted by $F_k(G)$, is the minimum number of vertices that need to be initial... | \section{\large Introduction}
We consider undirected finite simple connected graphs only. For
notation and terminology not defined here, we refer to \cite{Bo}.
For a graph $G = (V(G), E(G))$, $|V(G)|$ and $|E(G)|$ are its order
and size, respectively. For a vertex $v\in V(G)$, the neighborhood
$N(v)$ of $v$ is def... | {
"timestamp": "2015-07-07T02:14:12",
"yymm": "1507",
"arxiv_id": "1507.01364",
"language": "en",
"url": "https://arxiv.org/abs/1507.01364",
"abstract": "Amos et al. (Discrete Appl. Math. 181 (2015) 1-10) introduced the notion of the $k$-forcing number of graph for a positive integer $k$ as the generalizati... |
https://arxiv.org/abs/1803.03240 | The maximum number of $P_\ell$ copies in $P_k$-free graphs | Generalizing Turán's classical extremal problem, Alon and Shikhelman investigated the problem of maximizing the number of $T$ copies in an $H$-free graph, for a pair of graphs $T$ and $H$. Whereas Alon and Shikhelman were primarily interested in determining the order of magnitude for large classes of graphs $H$, we foc... | \section{Introduction}
For a graph $G$, we let $e(G)$ denote the number of edges in $G$, and for a given graph $H$, we let $\mathcal{N}(H,G)$ denote the number of (not necessarily induced) copies of $H$ in $G$. If there is no copy of $H$ in $G$, we say that $G$ is $H$-free. We denote the path with $k$ edges by $P_k$ a... | {
"timestamp": "2019-07-12T02:10:29",
"yymm": "1803",
"arxiv_id": "1803.03240",
"language": "en",
"url": "https://arxiv.org/abs/1803.03240",
"abstract": "Generalizing Turán's classical extremal problem, Alon and Shikhelman investigated the problem of maximizing the number of $T$ copies in an $H$-free graph,... |
https://arxiv.org/abs/1507.08021 | A Counterexample and Fix to a Minimum Distance Duality Theorem | We consider dual optimization problems to the fundamental problem of finding the minimum distance from a point to a subspace. We provide a counterexample to a theorem which has appeared in the literature, relating the minimum distance problem to a maximization problem in the predual space. The theorem was stated in a s... | \section{Introduction}
We consider minimum distance problems in real normed vector spaces.
A well-known duality result for these problems is a generalization of
the projection theorem in the dual space.
We consider the validity of similar duality results in the predual space.
This could be of use for problems arising ... | {
"timestamp": "2015-08-27T02:02:42",
"yymm": "1507",
"arxiv_id": "1507.08021",
"language": "en",
"url": "https://arxiv.org/abs/1507.08021",
"abstract": "We consider dual optimization problems to the fundamental problem of finding the minimum distance from a point to a subspace. We provide a counterexample ... |
https://arxiv.org/abs/2008.10922 | Positivity Conditions for Cubic, Quartic and Quintic Polynomials | We present a necessary and sufficient condition for a cubic polynomial to be positive for all positive reals. We identify the set where the cubic polynomial is nonnegative but not all positive for all positive reals, and explicitly give the points where the cubic polynomial attains zero. We then reformulate a necessary... | \section{Introduction}
In 1988, Schmidt and He\ss \cite{SH88} presented a necessary and sufficient condition for a cubic polynomial to be nonnegative for all positive reals. In 1994, Ulrich and Watson \cite{UW94} presented a necessary and sufficient condition for a quartic polynomial to be nonnegative for all positi... | {
"timestamp": "2020-09-21T02:13:17",
"yymm": "2008",
"arxiv_id": "2008.10922",
"language": "en",
"url": "https://arxiv.org/abs/2008.10922",
"abstract": "We present a necessary and sufficient condition for a cubic polynomial to be positive for all positive reals. We identify the set where the cubic polynomi... |
https://arxiv.org/abs/2012.05297 | Persistence of Morse decompositions over grid resolution for maps and time series | We can approximate a continuous self-map $f$ of a compact metric space by discretizing the space into a grid. Through either the map itself or a time series, $f$ induces a multivalued grid map $\mathcal F$. The dynamical properties of $\mathcal F$ depend on the resolution of the grid, and we study the persistence of th... | \section{Introduction}
In order to understand the dynamics of a continuous map $f:X\to X$ of a compact metric space, we can create a finite discretization of the space, then use a computer to create a multivalued map ${\mathcal F}$ on the discretization to approximate $f$.
The relationship between the dynamics of $f$ ... | {
"timestamp": "2020-12-11T02:01:51",
"yymm": "2012",
"arxiv_id": "2012.05297",
"language": "en",
"url": "https://arxiv.org/abs/2012.05297",
"abstract": "We can approximate a continuous self-map $f$ of a compact metric space by discretizing the space into a grid. Through either the map itself or a time seri... |
https://arxiv.org/abs/2001.02991 | A note on the minimization of a Tikhonov functional with $\ell^1$-penalty | In this paper, we consider the minimization of a Tikhonov functional with an $\ell_1$ penalty for solving linear inverse problems with sparsity constraints. One of the many approaches used to solve this problem uses the Nemskii operator to transform the Tikhonov functional into one with an $\ell_2$ penalty term but a n... |
\section{Introduction}
In this paper, we consider linear operator equations of the form
\begin{equation}\label{Ax=y}
Ax = y \,,
\end{equation}
where $A \, : \, {\ell^2} \to {\ell^2}$ is a bounded linear operator on the (infinite-dimensional) sequence space ${\ell^2}$. Note that by using a suitable basis or frame,... | {
"timestamp": "2020-06-25T02:15:54",
"yymm": "2001",
"arxiv_id": "2001.02991",
"language": "en",
"url": "https://arxiv.org/abs/2001.02991",
"abstract": "In this paper, we consider the minimization of a Tikhonov functional with an $\\ell_1$ penalty for solving linear inverse problems with sparsity constrain... |
https://arxiv.org/abs/2109.08392 | Arbitrary-precision computation of the gamma function | We discuss the best methods available for computing the gamma function $\Gamma(z)$ in arbitrary-precision arithmetic with rigorous error bounds. We address different cases: rational, algebraic, real or complex arguments; large or small arguments; low or high precision; with or without precomputation. The methods also c... | \section{Introduction}
The gamma function
\begin{equation}
\label{eq:gammadef}
\Gamma(z) = \int_0^{\infty} t^{z-1} e^{-t} dt \quad (\operatorname{Re}(z) > 0), \quad \Gamma(z) = \frac{\Gamma(z+1)}{z},
\end{equation}
is arguably the most important higher transcendental function,
having a tendency to crop up in any setti... | {
"timestamp": "2021-09-20T02:11:29",
"yymm": "2109",
"arxiv_id": "2109.08392",
"language": "en",
"url": "https://arxiv.org/abs/2109.08392",
"abstract": "We discuss the best methods available for computing the gamma function $\\Gamma(z)$ in arbitrary-precision arithmetic with rigorous error bounds. We addre... |
https://arxiv.org/abs/2210.11116 | A new approach for computing the distance and the diameter in circulant graphs | The diameter of a graph is the maximum distance among all pairs of vertices. Thus a graph $G$ has diameter $d$ if any two vertices are at distance at most $d$ and there are two vertices at distance $d$. We are interested in studying the diameter of circulant graphs $C_n(1,s)$, i.e., graphs with the set $\{0,1,\ldots, n... | \section{Introduction}
\label{Intro}
Circulant graphs form an important and very well-studied class of graphs \cite{monakhova2012survey}. They find applications to the computer network design, telecommunication networking, distributed computation, parallel processing architectures, and VLSI design.
For $n\in \mathbb... | {
"timestamp": "2022-10-21T02:11:01",
"yymm": "2210",
"arxiv_id": "2210.11116",
"language": "en",
"url": "https://arxiv.org/abs/2210.11116",
"abstract": "The diameter of a graph is the maximum distance among all pairs of vertices. Thus a graph $G$ has diameter $d$ if any two vertices are at distance at most... |
https://arxiv.org/abs/1703.01620 | Direction sets, Lipschitz graphs and density | We consider the direction set determined by various subsets $E$ of Euclidean space and show that there is a trichotomy: Either (i) The subset is the graph of a Lipschitz function and the direction set is not dense in the sphere, (ii) The subset is the graph of a non-Lipschitz function and the direction set is dense but... | \section{Introduction}
The purpose of this paper is study directions sets determined by subsets of the Euclidean space. Informally, direction sets consists of direction vectors determined by pairs of vectors from a given set. More precisely, we have the following definitions.
\begin{definition} Fix a subset $E \sub... | {
"timestamp": "2017-03-07T02:08:05",
"yymm": "1703",
"arxiv_id": "1703.01620",
"language": "en",
"url": "https://arxiv.org/abs/1703.01620",
"abstract": "We consider the direction set determined by various subsets $E$ of Euclidean space and show that there is a trichotomy: Either (i) The subset is the graph... |
https://arxiv.org/abs/2002.03075 | Some arithmetical problems that are obtained by analyzing proofs and infinite graphs | Applying Baaz's Generalization Method and a new technique to, respectively, proofs and denumerable simple graphs, diverse arithmetical patterns are observed. In particular, sufficient conditions for a number to be a divisor of a Fermat number are provided. The accuracy of such observations is asked in several subsequen... | \section{Introduction}
Roughly speaking, Baaz's Generalization Method (see \cite{Baaz}) is a procedure that, given a proof of a certain theorem, allows to obtain a more general proof (and, in particular, a more general theorem). In this paper the idea is applied to six proofs that fifth Fermat number is divisible by 6... | {
"timestamp": "2020-02-11T02:05:02",
"yymm": "2002",
"arxiv_id": "2002.03075",
"language": "en",
"url": "https://arxiv.org/abs/2002.03075",
"abstract": "Applying Baaz's Generalization Method and a new technique to, respectively, proofs and denumerable simple graphs, diverse arithmetical patterns are observ... |
https://arxiv.org/abs/math/0603493 | Extremal metrics and stabilities on polarized manifolds | The Hitchin-Kobayashi correspondence for vector bundles, established by Donaldson, Kobayashi, Luebke, Uhlenbeck and Yau, states that an indecomposable holomorphic vector bundle over a compact Kaehler manifold is stable in the sense of Takemoto-Mumford if and only if the vector bundle admits a Hermitian-Einstein metric.... | \section{Introduction}
Let $M$ be a compact complex connected manifold.
As an introduction to our subjects,
we recall the following well-known conjecture of Calabi \cite{Cal1}:
\medskip\noindent
{\bf Conjecture}.
{\rm (i)}
{\em If $\,c_1 (M)_{\Bbb R} < 0$, then $M$ admits a unique K\"ahler-Einstein metric $\omega$ ... | {
"timestamp": "2006-04-07T14:24:15",
"yymm": "0603",
"arxiv_id": "math/0603493",
"language": "en",
"url": "https://arxiv.org/abs/math/0603493",
"abstract": "The Hitchin-Kobayashi correspondence for vector bundles, established by Donaldson, Kobayashi, Luebke, Uhlenbeck and Yau, states that an indecomposable... |
https://arxiv.org/abs/2110.06008 | A variational principle for Gaussian lattice sums | We consider a two-dimensional analogue of Jacobi theta functions and prove that, among all lattices $\Lambda \subset \mathbb{R}^2$ with fixed density, the minimal value is maximized by the hexagonal lattice. This result can be interpreted as the dual of a 1988 result of Montgomery who proved that the hexagonal lattice ... | \section{Introduction}\label{sec_intro}
\subsection{Main Result}
The purpose of this paper is to characterize optimizers for a variational problem with applications in various fields. Let $\Lambda$ be a lattice in $\mathbb{R}^2$ and consider the function
\begin{equation}\label{eq:translatedtheta}
E_\L (z;\alpha) = \... | {
"timestamp": "2021-10-13T02:21:51",
"yymm": "2110",
"arxiv_id": "2110.06008",
"language": "en",
"url": "https://arxiv.org/abs/2110.06008",
"abstract": "We consider a two-dimensional analogue of Jacobi theta functions and prove that, among all lattices $\\Lambda \\subset \\mathbb{R}^2$ with fixed density, ... |
https://arxiv.org/abs/2001.04870 | Notes on the Neighborhood Polynomials | The neighborhood polynomial of graph $G$, denoted by $N(G,x)$, is the generating function for the number of vertex subsets of $G$ which are subsets of open neighborhoods of vertices in $G$. For any graph polynomial, it can be useful to generate a new family of polynomials by introducing some restrictions and characteri... | \section{Introduction}
\noindent
Let $G=(V,E)$ be a simple, finite and undirected graph. For a vertex $v\in V$, the \emph{open neighborhood} of $v$, denoted by $N(v)$, is defined by $N(v)=\{u\mid \{u,v\}\in E\}$.
\vspace{2mm}
\noindent
The \emph{neighborhood complex} of $G$, denoted by $\mathcal{N}(G)$ and introduced ... | {
"timestamp": "2020-01-17T02:07:36",
"yymm": "2001",
"arxiv_id": "2001.04870",
"language": "en",
"url": "https://arxiv.org/abs/2001.04870",
"abstract": "The neighborhood polynomial of graph $G$, denoted by $N(G,x)$, is the generating function for the number of vertex subsets of $G$ which are subsets of ope... |
https://arxiv.org/abs/0809.2511 | Integral and isocapacitary inequalities | It is shown by a counterexample that isocapacitary and isoperimetric constants of a multi-dimensional Euclidean domain starshaped with respect to a ball are not equivalent. Sharp integral inequalities involving the harmonic capacity which imply Faber-Krahn property of the fundamental Dirichlet-Laplace eigenvalue are ob... | \section{Introduction}
Isocapacitary inequalities are intimately connected with various properties of Sobolev spaces, especially with norms of embedding operators \cite{Gr}, \cite{M1}--\cite{M4}, \cite{M6}, \cite{M7}, \cite{M9}, \cite{M14}. For instance, the best constants in some of these inequalities give two-sided ... | {
"timestamp": "2008-09-15T14:46:38",
"yymm": "0809",
"arxiv_id": "0809.2511",
"language": "en",
"url": "https://arxiv.org/abs/0809.2511",
"abstract": "It is shown by a counterexample that isocapacitary and isoperimetric constants of a multi-dimensional Euclidean domain starshaped with respect to a ball are... |
https://arxiv.org/abs/2105.05467 | Strong $BV$-extension and $W^{1,1}$-extension domains | We show that a bounded domain in a Euclidean space is a $W^{1,1}$-extension domain if and only if it is a strong $BV$-extension domain. In the planar case, bounded and strong $BV$-extension domains are shown to be exactly those $BV$-extension domains for which the set $\partial\Omega \setminus \bigcup_{i} \overline{\Om... | \section{Introduction}
Let $\Omega\subset\mathbb{R}^n$ be a domain for some $n\geq 2$. For every $1\leq p\leq \infty$, we define the Sobolev space $W^{1,p}(\Omega)$ to be
$$W^{1,p}(\Omega)=\{u\in L^p(\Omega):\, \nabla u\in L^p(\Omega;\mathbb{R}^n)\}, $$
where $\nabla u$ denotes the distributional gradient of $u$. We ... | {
"timestamp": "2021-10-07T02:19:26",
"yymm": "2105",
"arxiv_id": "2105.05467",
"language": "en",
"url": "https://arxiv.org/abs/2105.05467",
"abstract": "We show that a bounded domain in a Euclidean space is a $W^{1,1}$-extension domain if and only if it is a strong $BV$-extension domain. In the planar case... |
https://arxiv.org/abs/1202.5108 | Upper bounds for Steklov eigenvalues on surfaces | We give explicit isoperimetric upper bounds for all Steklov eigenvalues of a compact orientable surface with boundary, in terms of the genus, the length of the boundary, and the number of boundary components. Our estimates generalize a recent result of Fraser-Schoen, as well as the classical inequalites obtained by Her... | \section{Introduction}
\subsection{Steklov spectrum}
Let $\Sigma$ be a compact orientable surface with
boundary, and let
$\Delta$
be the Laplace--Beltrami operator associated with a Riemannian
metric on $\Sigma$. The Steklov eigenvalue problem on $\Sigma$ is given by:
\begin{gather*}
\Delta u =0 \,\,\, {\rm in}\,\,... | {
"timestamp": "2012-02-24T02:01:19",
"yymm": "1202",
"arxiv_id": "1202.5108",
"language": "en",
"url": "https://arxiv.org/abs/1202.5108",
"abstract": "We give explicit isoperimetric upper bounds for all Steklov eigenvalues of a compact orientable surface with boundary, in terms of the genus, the length of ... |
https://arxiv.org/abs/1804.06863 | Combinatorics of orbit configuration spaces | From a group action on a space, define a variant of the configuration space by insisting that no two points inhabit the same orbit. When the action is almost free, this "orbit configuration space" is the complement of an arrangement of subvarieties inside the cartesian product, and we use this structure to study its to... | \section{Introduction}\label{sec:intro}
\subsection{Orbit configuration spaces}
A fundamental topological object attached to a topological space $X$ is its
ordered configuration space $\Conf_n(X)$ of $n$ distinct points in $X$.
Analogously, given a group $G$ acting freely on $X$ one defines the
\emph{orbit configura... | {
"timestamp": "2020-01-31T02:12:27",
"yymm": "1804",
"arxiv_id": "1804.06863",
"language": "en",
"url": "https://arxiv.org/abs/1804.06863",
"abstract": "From a group action on a space, define a variant of the configuration space by insisting that no two points inhabit the same orbit. When the action is alm... |
https://arxiv.org/abs/1809.10925 | Data depth and floating body | Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the depth stands as a generalization of a measure of symmetry for convex sets, well ... | \section{Introduction}
Halfspace depth and floating body are the same concept. The first is extensively studied in nonparametric statistics, the second is of great importance in convex geometry. Until recently, work on data depth has not been recognized by the convex geometry community, and that in convex geometry no... | {
"timestamp": "2018-10-01T02:09:35",
"yymm": "1809",
"arxiv_id": "1809.10925",
"language": "en",
"url": "https://arxiv.org/abs/1809.10925",
"abstract": "Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Half... |
https://arxiv.org/abs/1701.08394 | Analysis of the gift exchange problem | In the gift exchange game there are n players and n wrapped gifts. When a player's number is called, that person can either choose one of the remaining wrapped gifts, or can "steal" a gift from someone who has already unwrapped it, subject to the restriction that no gift can be stolen more than a total of sigma times. ... | \section{The problem}\label{Sec1}
The following game is sometimes played at parties.
A number $\sigma$ (typically $1$ or $2$) is fixed in advance.
Each of the $n$ guests brings a wrapped gift, the gifts are
placed on a table (this is the ``pool'' of gifts),
and slips of paper containing the numbers $1$ to $n$
are dist... | {
"timestamp": "2017-01-31T02:05:46",
"yymm": "1701",
"arxiv_id": "1701.08394",
"language": "en",
"url": "https://arxiv.org/abs/1701.08394",
"abstract": "In the gift exchange game there are n players and n wrapped gifts. When a player's number is called, that person can either choose one of the remaining wr... |
https://arxiv.org/abs/2107.07885 | Variations on the Erdős distinct-sums problem | Let $\{a_1, . . . , a_n\}$ be a set of positive integers with $a_1 < \dots < a_n$ such that all $2^n$ subset sums are distinct. A famous conjecture by Erdős states that $a_n>c\cdot 2^n$ for some constant $c$, while the best result known to date is of the form $a_n>c\cdot 2^n/\sqrt{n}$. In this paper, we weaken the cond... | \section{Introduction}
For any $n\geq 1$, consider sets $\{a_1, . . . , a_n\}$ of positive integers with $a_1 < \dots < a_n$ whose subset sums are all distinct. A famous conjecture, due to Paul Erd\H{o}s, is that $a_n \geq c \cdot 2^n$ for some constant $c > 0$. Using the variance method, Erd\H{o}s and Moser \cite{Erdo... | {
"timestamp": "2021-07-20T02:30:26",
"yymm": "2107",
"arxiv_id": "2107.07885",
"language": "en",
"url": "https://arxiv.org/abs/2107.07885",
"abstract": "Let $\\{a_1, . . . , a_n\\}$ be a set of positive integers with $a_1 < \\dots < a_n$ such that all $2^n$ subset sums are distinct. A famous conjecture by ... |
https://arxiv.org/abs/2008.10398 | Recursively abundant and recursively perfect numbers | The divisor function $\sigma(n)$ sums the divisors of $n$. We call $n$ abundant when $\sigma(n) - n > n$ and perfect when $\sigma(n) - n = n$. I recently introduced the recursive divisor function $a(n)$, the recursive analog of the divisor function. It measures the extent to which a number is highly divisible into part... | \subsection{Recursive divisor function}
\section{Introduction}
\subsection{Parts into parts into parts}
A man wants to leave a pot of gold coins to his future children.
The coins are to be equally split among his children, even though he doesn't yet know how many children he will have.
The coins cannot be subdivided.... | {
"timestamp": "2020-08-25T02:28:41",
"yymm": "2008",
"arxiv_id": "2008.10398",
"language": "en",
"url": "https://arxiv.org/abs/2008.10398",
"abstract": "The divisor function $\\sigma(n)$ sums the divisors of $n$. We call $n$ abundant when $\\sigma(n) - n > n$ and perfect when $\\sigma(n) - n = n$. I recent... |
https://arxiv.org/abs/1312.5180 | Maximal induced matchings in triangle-free graphs | An induced matching in a graph is a set of edges whose endpoints induce a $1$-regular subgraph. It is known that any $n$-vertex graph has at most $10^{n/5} \approx 1.5849^n$ maximal induced matchings, and this bound is best possible. We prove that any $n$-vertex triangle-free graph has at most $3^{n/3} \approx 1.4423^n... | \section{Introduction}
A celebrated result due Moon and Moser~\cite{MM65} states that any graph on $n$ vertices has at most $3^{n/3}\approx 1.4423^n$ maximal independent sets. Moon and Moser also proved that this bound is best possible by characterizing the extremal graphs as follows: a graph on $n$ vertices has exactl... | {
"timestamp": "2013-12-19T02:10:00",
"yymm": "1312",
"arxiv_id": "1312.5180",
"language": "en",
"url": "https://arxiv.org/abs/1312.5180",
"abstract": "An induced matching in a graph is a set of edges whose endpoints induce a $1$-regular subgraph. It is known that any $n$-vertex graph has at most $10^{n/5} ... |
https://arxiv.org/abs/1301.6862 | A new cubic nonconforming finite element on rectangles | A new nonconforming rectangle element with cubic convergence for the energy norm is introduced. The degrees of freedom (DOFs) are defined by the twelve values at the three Gauss points on each of the four edges. Due to the existence of one linear relation among the above DOFs, it turns out the DOFs are eleven. The nonc... | \section{Introduction}
It has been well known that the standard lowest order conforming elements can
produce numerical locking and checker-board solutions in the approximation of
solid and fluid mechanics problems: see for instance \cite{bs92b, bs92a, brezzi-fortin,
ciar, gira-ravi} and the references therein. An eff... | {
"timestamp": "2013-01-30T02:01:43",
"yymm": "1301",
"arxiv_id": "1301.6862",
"language": "en",
"url": "https://arxiv.org/abs/1301.6862",
"abstract": "A new nonconforming rectangle element with cubic convergence for the energy norm is introduced. The degrees of freedom (DOFs) are defined by the twelve valu... |
https://arxiv.org/abs/1704.05373 | Strengthened Euler's Inequality in Spherical and Hyperbolic Geometries | Euler's inequality is a well known inequality relating the inradius and circumradius of a triangle. In Euclidean geometry, this inequality takes the form $R \geq 2r$ where $R$ is the circumradius and $r$ is the inradius. In spherical geometry, the inequality takes the form $\tan(R) \geq 2\tan(r)$ as proved in \cite{MPV... | \section{Introduction}
In Euclidean geometry, as discussed in \cite{SV}, given a triangle with side lengths $a,b,c$, circumradius $R$, and inradius $r$, we may strengthen Euler's inequality to give:
\begin{theorem}[\cite{SU},\cite{VW}]\label{original}
\begin{subequations}
\begin{align}
\frac{R}{r} &\geq \frac{abc + ... | {
"timestamp": "2017-04-19T02:06:30",
"yymm": "1704",
"arxiv_id": "1704.05373",
"language": "en",
"url": "https://arxiv.org/abs/1704.05373",
"abstract": "Euler's inequality is a well known inequality relating the inradius and circumradius of a triangle. In Euclidean geometry, this inequality takes the form ... |
https://arxiv.org/abs/2206.14280 | The numerical solution of fractional integral equations via orthogonal polynomials in fractional powers | We present a spectral method for one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including ones with irrational order, multiple fractional orders, non-trivial variable coefficients, and initial-boundary conditions. The method u... | \section*{Acknowledgements}
We thank Sheehan Olver for inspiring our interest in spectral methods for FIEs and FDEs and for many useful suggestions.
\section*{Code Availability}
\texttt{Julia} code for all figures presented in this paper are available at \url{https://github.com/putianyi889/JFP-demo}
\section{C... | {
"timestamp": "2022-07-04T02:03:44",
"yymm": "2206",
"arxiv_id": "2206.14280",
"language": "en",
"url": "https://arxiv.org/abs/2206.14280",
"abstract": "We present a spectral method for one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a va... |
https://arxiv.org/abs/math/0607238 | Explicit solutions to certain inf max problems from Turan power sum theory | Let s_v denote the pure power sum \sum_{k=1}^n z_k^v. In a previous paper we proved that \sqrt n <= \inf_{|z_k| => 1} \max_{v=1,...,n^2} |s_v| <= \sqrt{n+1} when n+1 is prime. In this paper we prove that \inf_{|z_k| = 1} \max_{v=1,...,n^2-n} |s_v| = \sqrt{n-1} when n-1 is a prime power, and if 2 <= i <= n-1 and n => 3 ... | \section{Introduction}
In his paper \cite{Turan2} Tur\'an shows that
\begin{gather*}
\inf_{\abs{z_k} \geq 1} \max_{\nu=1,\ldots,n} \abs{\sum_{k=1}^n z_k^\nu} =1.
\end{gather*}
Furthermore he gives an explicit construction
\begin{gather*}
z_k=e \p{\frac {k} {n+1}} \qquad \qquad \p{e(x)=e^{2 \pi i x}}
\end{gather... | {
"timestamp": "2007-04-05T01:23:18",
"yymm": "0607",
"arxiv_id": "math/0607238",
"language": "en",
"url": "https://arxiv.org/abs/math/0607238",
"abstract": "Let s_v denote the pure power sum \\sum_{k=1}^n z_k^v. In a previous paper we proved that \\sqrt n <= \\inf_{|z_k| => 1} \\max_{v=1,...,n^2} |s_v| <= ... |
https://arxiv.org/abs/2105.09285 | Proof of Cayley-Hamilton theorem using polynomials over the algebra of module endomorphisms | If $R$ is a commutative unital ring and $M$ is a unital $R$-module, then each element of $\operatorname{End}_R(M)$ determines a left $\operatorname{End}_{R}(M)[X]$-module structure on $\operatorname{End}_{R}(M)$, where $\operatorname{End}_{R}(M)$ is the $R$-algebra of endomorphisms of $M$ and $\operatorname{End}_{R}(M)... | \section{Introduction}
\begin{theorem}[Cayley-Hamilton theorem]
Let\/ $R$ be a commutative unital ring\/
\textup(i.e.{}\textup, with $1_R$\/\textup)
and\/ $M$ be a finite-rank free unital\/ $R$-module
\textup(i.e.{}\textup, which respects $1_R$\/\textup)\textup.
Let\/ $a\colon M\to M$ be an endomorphism of\/... | {
"timestamp": "2022-02-03T02:04:37",
"yymm": "2105",
"arxiv_id": "2105.09285",
"language": "en",
"url": "https://arxiv.org/abs/2105.09285",
"abstract": "If $R$ is a commutative unital ring and $M$ is a unital $R$-module, then each element of $\\operatorname{End}_R(M)$ determines a left $\\operatorname{End}... |
https://arxiv.org/abs/2010.13010 | On the ordering of the Markov numbers | The Markov numbers are the positive integers that appear in the solutions of the equation $x^2+y^2+z^2=3xyz$. These numbers are a classical subject in number theory and have important ramifications in hyperbolic geometry, algebraic geometry and combinatorics.It is known that the Markov numbers can be labeled by the lat... | \section{Introduction}
In 1879, Andrey Markov studied the equation
\begin{equation}
\label{eq M}
x^2+y^2+z^2=3xyz,
\end{equation}
which is now known as the \emph{Markov equation}. A positive integer solution $(m_1,m_2,m_3)$ of (\ref{eq M}) is called a \emph{Markov triple} and the integers that appear in the Markov ... | {
"timestamp": "2020-10-27T01:21:01",
"yymm": "2010",
"arxiv_id": "2010.13010",
"language": "en",
"url": "https://arxiv.org/abs/2010.13010",
"abstract": "The Markov numbers are the positive integers that appear in the solutions of the equation $x^2+y^2+z^2=3xyz$. These numbers are a classical subject in num... |
https://arxiv.org/abs/2110.10000 | The interval posets of permutations seen from the decomposition tree perspective | The interval poset of a permutation is the set of intervals of a permutation, ordered with respect to inclusion. It has been introduced and studied recently in [B. Tenner,arXiv:2007.06142]. We study this poset from the perspective of the decomposition trees of permutations, describing a procedure to obtain the former f... | \section{Introduction}
Recently, Bridget Tenner defined the \emph{interval posets} associated with permutations,
and described some properties of these posets in~\cite{Bridget}.
We noted that the so-called \emph{decomposition trees} of permutations, which encode their \emph{substitution decomposition}, provide an alt... | {
"timestamp": "2021-11-08T02:22:31",
"yymm": "2110",
"arxiv_id": "2110.10000",
"language": "en",
"url": "https://arxiv.org/abs/2110.10000",
"abstract": "The interval poset of a permutation is the set of intervals of a permutation, ordered with respect to inclusion. It has been introduced and studied recent... |
https://arxiv.org/abs/2111.11817 | More on co-even domination number | Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is the domination number of $G$. A dominating set $D$ is called co-even dominating... | \section{Introduction}
Let $G = (V,E)$ be a graph with vertex set $V$ and edge set $E$. Throughout this paper, we consider only simple graphs.
For each vertex $v\in V$, the set $N_G(v)=\{u\in V | uv \in E\}$ refers to the open neighbourhood of $v$ and the set $N_G[v]=N_G(v)\cup \{v\}$ refers to the closed neighbourho... | {
"timestamp": "2021-11-24T02:18:14",
"yymm": "2111",
"arxiv_id": "2111.11817",
"language": "en",
"url": "https://arxiv.org/abs/2111.11817",
"abstract": "Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex ... |
https://arxiv.org/abs/2002.07520 | Gradient $\ell_1$ Regularization for Quantization Robustness | We analyze the effect of quantizing weights and activations of neural networks on their loss and derive a simple regularization scheme that improves robustness against post-training quantization. By training quantization-ready networks, our approach enables storing a single set of weights that can be quantized on-deman... |
\section{First-Order Quantization-Robust Models}
\label{sec:method}
In this section, we propose a regularization technique for robustness to quantization noise. We first propose an appropriate model for quantization noise. Then, we show how we can effectively control the first-order, i.e., the linear part of the outp... | {
"timestamp": "2020-02-19T02:12:32",
"yymm": "2002",
"arxiv_id": "2002.07520",
"language": "en",
"url": "https://arxiv.org/abs/2002.07520",
"abstract": "We analyze the effect of quantizing weights and activations of neural networks on their loss and derive a simple regularization scheme that improves robus... |
https://arxiv.org/abs/1510.08132 | On mapping theorems for numerical range | Let $T$ be an operator on a Hilbert space $H$ with numerical radius $w(T)\le1$. According to a theorem of Berger and Stampfli, if $f$ is a function in the disk algebra such that $f(0)=0$, then $w(f(T))\le\|f\|_\infty$. We give a new and elementary proof of this result using finite Blaschke products.A well-known result ... | \section{Introduction}
Let $H$ be a complex Hilbert space
and $T$ be a bounded linear operator on~$H$.
The \emph{numerical range} of $T$ is defined by
\[
W(T):=\{\langle Tx,x\rangle: x\in H,~\|x\|=1\}.
\]
It is a convex set
whose closure contains the spectrum of $T$.
If $\dim H<\infty$, then $W(T)$ is compact.
The \... | {
"timestamp": "2015-10-29T01:03:30",
"yymm": "1510",
"arxiv_id": "1510.08132",
"language": "en",
"url": "https://arxiv.org/abs/1510.08132",
"abstract": "Let $T$ be an operator on a Hilbert space $H$ with numerical radius $w(T)\\le1$. According to a theorem of Berger and Stampfli, if $f$ is a function in th... |
https://arxiv.org/abs/0706.4134 | Bounds on the number of real solutions to polynomial equations | We use Gale duality for polynomial complete intersections and adapt the proof of the fewnomial bound for positive solutions to obtain the bound (e^4+3) 2^(k choose 2) n^k/4 for the number of non-zero real solutions to a system of n polynomials in n variables having n+k+1 monomials whose exponent vectors generate a subg... | \section*{Introduction}
In~\cite{BBS}, the sharp bound of $2n{+}1$ was obtained for the number of
non-zero real solutions to a system of $n$ polynomial equations in $n$
variables having $n{+}2$ monomials whose exponents affinely span the lattice $\mathbb{Z}^n$.
In~\cite{Bi07}, the sharp bound of $n{+}1$ was given for... | {
"timestamp": "2007-10-03T23:34:27",
"yymm": "0706",
"arxiv_id": "0706.4134",
"language": "en",
"url": "https://arxiv.org/abs/0706.4134",
"abstract": "We use Gale duality for polynomial complete intersections and adapt the proof of the fewnomial bound for positive solutions to obtain the bound (e^4+3) 2^(k... |
https://arxiv.org/abs/1509.03845 | Preventing blow up by convective terms in dissipative PDEs | We study the impact of the convective terms on the global solvability or finite time blow up of solutions of dissipative PDEs. We consider the model examples of 1D Burger's type equations, convective Cahn-Hilliard equation, generalized Kuramoto-Sivashinsky equation and KdV type equations, we establish the following com... | \section{Introduction}
It is well-known that the solutions of nonlinear evolutionary PDEs may blow up in a finite time. The most studied is the case of a semilinear heat equation, for instance, all positive solutions of the problem
\begin{equation}\label{0.heat}
\partial_t u=\partial^2_x u+u^2,\ \ u\big|_{x=\pm1}=0,\ \... | {
"timestamp": "2015-09-15T02:09:06",
"yymm": "1509",
"arxiv_id": "1509.03845",
"language": "en",
"url": "https://arxiv.org/abs/1509.03845",
"abstract": "We study the impact of the convective terms on the global solvability or finite time blow up of solutions of dissipative PDEs. We consider the model examp... |
https://arxiv.org/abs/2206.08606 | The span of singular tuples of a tensor beyond the boundary format | A singular $k$-tuple of a tensor $T$ of format $(n_1,\ldots,n_k)$ is essentially a complex critical point of the distance function from $T$ constrained to the cone of tensors of format $(n_1,\ldots,n_k)$ of rank at most one. A generic tensor has finitely many complex singular $k$-tuples, and their number depends only o... | \section{Introduction}\label{sec: intro}
A {\em singular $k$-tuple} of an order-$k$ tensor is the generalization of the notion of singular pair of a rectangular matrix. Singular $k$-tuples preserve important properties of singular pairs. For instance, in the problem of minimizing the distance between a given tensor $T... | {
"timestamp": "2022-06-20T02:11:14",
"yymm": "2206",
"arxiv_id": "2206.08606",
"language": "en",
"url": "https://arxiv.org/abs/2206.08606",
"abstract": "A singular $k$-tuple of a tensor $T$ of format $(n_1,\\ldots,n_k)$ is essentially a complex critical point of the distance function from $T$ constrained t... |
https://arxiv.org/abs/2005.08698 | Flip and Neimark-Sacker Bifurcations in a Coupled Logistic Map System | In this paper we consider a system of strongly coupled logistic maps involving two parameters. We classify and investigate the stability of its fixed points. A local bifurcation analysis of the system using Center Manifold is undertaken and then supported by numerical computations.This reveals the existence of reverse ... | \section{Introduction}
Coupled logistic maps originally gained attention in the mathematical biology literature via their utility in models of, for instance, populations of migrating species and environmental heterogeneity \cite{gyllenberg,kendall}. Recent years, however, have seen a renewed interest in the dynamics of... | {
"timestamp": "2020-05-19T02:32:26",
"yymm": "2005",
"arxiv_id": "2005.08698",
"language": "en",
"url": "https://arxiv.org/abs/2005.08698",
"abstract": "In this paper we consider a system of strongly coupled logistic maps involving two parameters. We classify and investigate the stability of its fixed poin... |
https://arxiv.org/abs/1902.06627 | A Combinatorial Identity for Rooted Labeled Forests | In this brief note a straightforward combinatorial proof for an identity directly connecting rooted forests and unordered set partitions is provided. Furthermore, references that put this type of identity in the context of forest volumes and multinomial identities are given. | \section{The Identity}
The aim of this note is to provide an elementary and purely combinatorial proof
for an identity stated and proved (via induction) by
Dorlas, Rebenko and Savoie in~\cite{Dorlas-Rebenko-Savoie:2019:forest-identity}.
Let $m$ and $p$ be positive integers with $p\leq m$ and define $\Omega = \{x_1, ... | {
"timestamp": "2019-07-11T02:15:35",
"yymm": "1902",
"arxiv_id": "1902.06627",
"language": "en",
"url": "https://arxiv.org/abs/1902.06627",
"abstract": "In this brief note a straightforward combinatorial proof for an identity directly connecting rooted forests and unordered set partitions is provided. Furt... |
https://arxiv.org/abs/1902.05055 | Covering graphs by monochromatic trees and Helly-type results for hypergraphs | How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given $r$-edge-coloured graph $G$? These problems were introduced in the 1960s and were intensively studied by various researchers over the last 50 years. In this paper, we establish a connection between this problem and the ... | \section{Introduction}
\vspace{-0.1cm}
Given an $r$-edge-coloured graph $G$, how many monochromatic paths, cycles or general trees does one need to cover all vertices of $G$? The study of such problems has a very rich history going back to the 1960's when Gerencs\'er and Gy\'arf\'as \cite{gerencser67} showed that for a... | {
"timestamp": "2020-08-05T02:20:23",
"yymm": "1902",
"arxiv_id": "1902.05055",
"language": "en",
"url": "https://arxiv.org/abs/1902.05055",
"abstract": "How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given $r$-edge-coloured graph $G$? These problems were intr... |
https://arxiv.org/abs/1907.06601 | On circles enclosing many points | We prove that every set of $n$ red and $n$ blue points in the plane contains a red and a blue point such that every circle through them encloses at least $n(1-\frac{1}{\sqrt{2}}) -o(n)$ points of the set. This is a two-colored version of a problem posed by Neumann-Lara and Urrutia. We also show that every set $S$ of $n... | \section{Introduction}
\label{intro}
Neumann-Lara and Urrutia~\cite{NU88} posed the following problem: Prove that every set $S$ of $n$ points in the plane contains two points $p$ and $q$ such that any circle which passes through $p$ and $q$ encloses ``many" points of $S$. The question is to quantify this number of enc... | {
"timestamp": "2019-07-31T02:10:26",
"yymm": "1907",
"arxiv_id": "1907.06601",
"language": "en",
"url": "https://arxiv.org/abs/1907.06601",
"abstract": "We prove that every set of $n$ red and $n$ blue points in the plane contains a red and a blue point such that every circle through them encloses at least ... |
https://arxiv.org/abs/1411.5894 | On the norm of products of polynomials on ultraproduct of Banach spaces | The purpose of this article is to study the problem of finding sharp lower bounds for the norm of the product of polynomials in the ultraproducts of Banach spaces $(X_i)_{\mathfrak U}$. We show that, under certain hypotheses, there is a strong relation between this problem and the same problem for the spaces $X_i$. | \section{Introduction}
In this article we study the factor problem in the context of ultraproducts of Banach spaces. This problem can be stated as follows: for a Banach space $X$ over a field $\mathbb K$ (with $\mathbb K=\mathbb R$ or $\mathbb K=\mathbb C$) and natural numbers $k_1,\cdots, k_n$ find the optimal consta... | {
"timestamp": "2014-11-24T02:11:56",
"yymm": "1411",
"arxiv_id": "1411.5894",
"language": "en",
"url": "https://arxiv.org/abs/1411.5894",
"abstract": "The purpose of this article is to study the problem of finding sharp lower bounds for the norm of the product of polynomials in the ultraproducts of Banach ... |
https://arxiv.org/abs/1608.08964 | An elementary inductive proof that $AB=I$ implies $BA=I$ for matrices | In this note we give an elementary demonstration of the fact that AB=I implies BA=I for square matrices A,B with coefficients in a field K. By elementary we mean that our proof follows from the very definitions of matrix and product of a matrix, with no extra help of more sophisticated results, as the use of dimensions... | \section{Introduction}
Let $\mathbb{K}$ be a division ring and $M_n(\mathbb{K})$ be the ring of square matrices of order $n$ with coefficients in $\mathbb{K}$. Let us denote by $I_n$ de identity matrix of orden $n$, which is the unit element of $M_n(\mathbb{K})$. A very basic important fact about matrices is that they ... | {
"timestamp": "2016-09-01T02:07:03",
"yymm": "1608",
"arxiv_id": "1608.08964",
"language": "en",
"url": "https://arxiv.org/abs/1608.08964",
"abstract": "In this note we give an elementary demonstration of the fact that AB=I implies BA=I for square matrices A,B with coefficients in a field K. By elementary ... |
https://arxiv.org/abs/2010.02722 | Bounded and finite factorization domains | An integral domain is atomic if every nonzero nonunit factors into irreducibles. Let $R$ be an integral domain. We say that $R$ is a bounded factorization domain if it is atomic and for every nonzero nonunit $x \in R$, there is a positive integer $N$ such that for any factorization $x = a_1 \cdots a_n$ of $x$ into irre... | \section{Introduction}
\label{sec:intro}
During the last three decades, the study of factorizations based on Diagram~\eqref{diag:AAZ's atomic chain} has earned significant attention among researchers in commutative algebra and semigroup theory. This diagram of classes of integral domains satisfying conditions weaker t... | {
"timestamp": "2020-10-07T02:22:21",
"yymm": "2010",
"arxiv_id": "2010.02722",
"language": "en",
"url": "https://arxiv.org/abs/2010.02722",
"abstract": "An integral domain is atomic if every nonzero nonunit factors into irreducibles. Let $R$ be an integral domain. We say that $R$ is a bounded factorization... |
https://arxiv.org/abs/0904.4507 | Rotor Walks and Markov Chains | The rotor walk is a derandomized version of the random walk on a graph. On successive visits to any given vertex, the walker is routed to each of the neighboring vertices in some fixed cyclic order, rather than to a random sequence of neighbors. The concept generalizes naturally to Markov chains on a countable state sp... | \section{Introduction} \label{results}
Let $X_0,X_1,\ldots$ be a Markov chain on a countable set
$V$ with transition probabilities $p:V\times V\to[0,1]$
(see e.g.~\cite{norris} for background).
We call the elements of $V$ {\bf\boldmath vertices}.
We write $\mathbb{P}_u$ for the law of the Markov chain
started at verte... | {
"timestamp": "2010-04-08T02:00:24",
"yymm": "0904",
"arxiv_id": "0904.4507",
"language": "en",
"url": "https://arxiv.org/abs/0904.4507",
"abstract": "The rotor walk is a derandomized version of the random walk on a graph. On successive visits to any given vertex, the walker is routed to each of the neighb... |
https://arxiv.org/abs/1608.03818 | Super-convergence and post-processing for mixed finite element approximations of the wave equation | We consider the numerical approximation of acoustic wave propagation problems by mixed BDM(k+1)-P(k) finite elements on unstructured meshes. Optimal convergence of the discrete velocity and super-convergence of the pressure by one order are established. Based on these results, we propose a post-processing strategy that... | \section{Introduction} \label{sec:intro}
The propagation of pressure waves of small amplitude in a Newtonian fluid or an elastic solid can be modeled by hyperbolic systems of the form
\begin{align}
a \partial_t p + \mathrm{div}\, u &= 0, \label{eq:sys1}\\
b \partial_t u + \nabla p &= 0. \label{eq:sys2}
\end{align}
Her... | {
"timestamp": "2016-08-15T02:07:04",
"yymm": "1608",
"arxiv_id": "1608.03818",
"language": "en",
"url": "https://arxiv.org/abs/1608.03818",
"abstract": "We consider the numerical approximation of acoustic wave propagation problems by mixed BDM(k+1)-P(k) finite elements on unstructured meshes. Optimal conve... |
https://arxiv.org/abs/1603.00673 | Eventually stable rational functions | For a field K, rational function phi in K(z) of degree at least two, and alpha in P^1(K), we study the polynomials in K[z] whose roots are given by the solutions to phi^n(z) = alpha, where phi^n denotes the nth iterate of phi. When the number of irreducible factors of these polynomials stabilizes as n grows, the pair (... | \section{Introduction}
Given a field $K$ and $f \in K[z]$, many authors have studied the question of whether $f$ is \textit{stable over $K$}, that is, if all iterates $f^n(z)$ for $n \geq 1$ are irreducible over $K$. See for example \cite{shparostafe, danielson, ostafe2, quaddiv, itconst}, and also \cite[Sections 1 an... | {
"timestamp": "2017-05-03T02:05:47",
"yymm": "1603",
"arxiv_id": "1603.00673",
"language": "en",
"url": "https://arxiv.org/abs/1603.00673",
"abstract": "For a field K, rational function phi in K(z) of degree at least two, and alpha in P^1(K), we study the polynomials in K[z] whose roots are given by the so... |
https://arxiv.org/abs/1204.6207 | Spectra of edge-independent random graphs | Let $G$ be a random graph on the vertex set $\{1,2,..., n\}$ such that edges in $G$ are determined by independent random indicator variables, while the probability $p_{ij}$ for $\{i,j\}$ being an edge in $G$ is not assumed to be equal. Spectra of the adjacency matrix and the normalized Laplacian matrix of $G$ are recen... | \section{Introduction}
Given an $n \times n$ symmetric matrix $M$, let $\lambda_1(M)$,
$\lambda_2(M),\ldots, \lambda_n(M)$ be the list of eigenvalues of $M$
in the non-decreasing order. What can we say about these eigenvalues
if $M$ is a matrix associated with a random graph $G$? Here $M$ could be
the adjacency ma... | {
"timestamp": "2012-04-30T02:02:01",
"yymm": "1204",
"arxiv_id": "1204.6207",
"language": "en",
"url": "https://arxiv.org/abs/1204.6207",
"abstract": "Let $G$ be a random graph on the vertex set $\\{1,2,..., n\\}$ such that edges in $G$ are determined by independent random indicator variables, while the pr... |
https://arxiv.org/abs/0706.2250 | Sign lemma for dimension shifting | There is a surprising occurrence of some minus signs in the isomorphisms produced in the well-known technique of dimension shifting in calculating derived functors in homological algebra. We explicitly determine these signs. Getting these signs right is important in order to avoid basic contradictions. We illustrate th... | \subsection*{\hbox{}\hfill{\normalsize\sl #1}\hfill\hbox{}}}
\textheight 23truecm
\textwidth 15truecm
\addtolength{\oddsidemargin}{-1.05truecm}
\addtolength{\topmargin}{-2truecm}
\makeatletter \def\l@section{\@dottedtocline{1}{0em}{1.2em}} \makeatother
\begin{document}
\centerline{\Large\bf Sign lemma for dimens... | {
"timestamp": "2007-06-15T11:38:24",
"yymm": "0706",
"arxiv_id": "0706.2250",
"language": "en",
"url": "https://arxiv.org/abs/0706.2250",
"abstract": "There is a surprising occurrence of some minus signs in the isomorphisms produced in the well-known technique of dimension shifting in calculating derived f... |
https://arxiv.org/abs/1705.01349 | An improvement of the Kolmogorov-Riesz compactness theorem | The purpose of this short note is to provide a new and very short proof of a result by Sudakov, offering an important improvement of the classical result by Kolmogorov-Riesz on compact subsets of Lebesgue spaces. | \section*{Introduction} \label{sec:intro}
The classical compactness theorem of Kolmogorov--Riesz reads as follows \cite{HOH}:
A subset $\mathcal{F}$ of $L^p(\mathbb{R}^n)$, with $1\le p<\infty$, is totally bounded if,
and only if,
\begin{enumerate}
\renewcommand{\theenumi}{\alph{enumi}}
\item $\mathcal{F}$ i... | {
"timestamp": "2019-03-22T01:18:06",
"yymm": "1705",
"arxiv_id": "1705.01349",
"language": "en",
"url": "https://arxiv.org/abs/1705.01349",
"abstract": "The purpose of this short note is to provide a new and very short proof of a result by Sudakov, offering an important improvement of the classical result ... |
https://arxiv.org/abs/1001.3356 | Equivalence of polynomial conjectures in additive combinatorics | We study two conjectures in additive combinatorics. The first is the polynomial Freiman-Ruzsa conjecture, which relates to the structure of sets with small doubling. The second is the inverse Gowers conjecture for $U^3$, which relates to functions which locally look like quadratics. In both cases a weak form, with expo... | \section{Introduction}
Additive combinatorics studies subsets of abelian groups, with the
main examples are subsets of the integers and of vector spaces
over finite fields. The main problems entail connecting various
properties related to the additive structure of the space, to
structural properties of the subsets. In ... | {
"timestamp": "2010-01-19T17:28:13",
"yymm": "1001",
"arxiv_id": "1001.3356",
"language": "en",
"url": "https://arxiv.org/abs/1001.3356",
"abstract": "We study two conjectures in additive combinatorics. The first is the polynomial Freiman-Ruzsa conjecture, which relates to the structure of sets with small ... |
https://arxiv.org/abs/1401.6755 | On power graphs of finite groups with forbidden induced subgraphs | The power graph $\mathcal{P}(G)$ of a finite group $G$ is a graph whose vertex set is the group $G$ and distinct elements $x,y\in G$ are adjacent if one is a power of the other, that is, $x$ and $y$ are adjacent if $x\in\langle y\rangle$ or $y\in\langle x\rangle$. We characterize all finite groups $G$ whose power graph... | \section{Introduction}
The \textit{power graph} $\mathcal{P}(G)$ of a group $G$ is a graph with elements of $G$ as its vertices such that two distinct elements $x$ and $y$ are adjacent if $y=x^m$ or $x=y^m$ for some positive integer $m$. Clearly, for finite (torsion) groups two distinct elements $x$ and $y$ are adjacen... | {
"timestamp": "2014-01-28T02:12:26",
"yymm": "1401",
"arxiv_id": "1401.6755",
"language": "en",
"url": "https://arxiv.org/abs/1401.6755",
"abstract": "The power graph $\\mathcal{P}(G)$ of a finite group $G$ is a graph whose vertex set is the group $G$ and distinct elements $x,y\\in G$ are adjacent if one i... |
https://arxiv.org/abs/1507.06356 | Korenblum-Type Extremal Problems in Bergman Spaces | We shall study non-linear extremal problems in Bergman space $\mathcal{A}^2(\mathbb{D})$. We show the existence of the solution and that the extremal functions are bounded. Further, we shall discuss special cases for polynomials, investigate the properties of the solution and provide a bound for the solution. This prob... | \section{Korenblum's Maximum Principle: History and recent results}
Let $\mathbb{D}=\{z\in\mathbb{C}:\ |z|<1\}$ be the open unit disk and $A(c,1)=\{z\in\mathbb{C}:\, c<|z|<1\}$ be the annulus defined in the complex plane $\mathbb{C}$. Then, the Bergman space $\mathcal{A}^2(\mathbb{D})$ is the class of functions $f$ an... | {
"timestamp": "2015-07-24T02:03:21",
"yymm": "1507",
"arxiv_id": "1507.06356",
"language": "en",
"url": "https://arxiv.org/abs/1507.06356",
"abstract": "We shall study non-linear extremal problems in Bergman space $\\mathcal{A}^2(\\mathbb{D})$. We show the existence of the solution and that the extremal fu... |
https://arxiv.org/abs/0711.3727 | The iterated Aluthge transforms of a matrix converge | Given an $r\times r$ complex matrix $T$, if $T=U|T|$ is the polar decomposition of $T$, then, the Aluthge transform is defined by $$ \Delta(T)= |T|^{1/2} U |T |^{1/2}. $$ Let $\Delta^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, i.e. $\Delta^{0}(T)=T$ and $\Delta^{n}(T)=\Delta(\Delta^{n-1}(T))$, $n\in\m... | \section{Introduction}
Let $\mathcal{H}$ be a Hilbert space and $T$ a bounded operator defined on $\mathcal{H}$ whose polar
decomposition is $T=U|T|$. The \textit{Aluthge transform} of $T$ is the operator
$\alu{T}=|T|^{1/2}U\ |T|^{1/2}$. This transform was introduced in \cite{[Aluthge]}
to study p-hyponormal and lo... | {
"timestamp": "2007-11-23T14:49:21",
"yymm": "0711",
"arxiv_id": "0711.3727",
"language": "en",
"url": "https://arxiv.org/abs/0711.3727",
"abstract": "Given an $r\\times r$ complex matrix $T$, if $T=U|T|$ is the polar decomposition of $T$, then, the Aluthge transform is defined by $$ \\Delta(T)= |T|^{1/2} ... |
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