url stringlengths 31 38 | title stringlengths 7 229 | abstract stringlengths 44 2.87k | text stringlengths 319 2.51M | meta dict |
|---|---|---|---|---|
https://arxiv.org/abs/2206.14911 | Minimum Weight Euclidean $(1+\varepsilon)$-Spanners | Given a set $S$ of $n$ points in the plane and a parameter $\varepsilon>0$, a Euclidean $(1+\varepsilon)$-spanner is a geometric graph $G=(S,E)$ that contains, for all $p,q\in S$, a $pq$-path of weight at most $(1+\varepsilon)\|pq\|$. We show that the minimum weight of a Euclidean $(1+\varepsilon)$-spanner for $n$ poin... | \section{Introduction} \label{sec:intro}
For a set $S$ of $n$ points in a metric space, a graph $G=(S,E)$ is a \emph{$t$-spanner} if $G$ contains, between any two points $p,q\in S$, a $pq$-path of weight at most $t\cdot \|pq\|$, where $t\geq 1$ is the \emph{stretch factor} of the spanner. In other words, a $t$-span... | {
"timestamp": "2022-07-01T02:03:43",
"yymm": "2206",
"arxiv_id": "2206.14911",
"language": "en",
"url": "https://arxiv.org/abs/2206.14911",
"abstract": "Given a set $S$ of $n$ points in the plane and a parameter $\\varepsilon>0$, a Euclidean $(1+\\varepsilon)$-spanner is a geometric graph $G=(S,E)$ that co... |
https://arxiv.org/abs/1905.12560 | On the equivalence between graph isomorphism testing and function approximation with GNNs | Graph Neural Networks (GNNs) have achieved much success on graph-structured data. In light of this, there have been increasing interests in studying their expressive power. One line of work studies the capability of GNNs to approximate permutation-invariant functions on graphs, and another focuses on the their power as... | \section{Conclusions}
In this work we address the important question of organizing the fast-growing zoo
of GNN architectures in terms of what functions they can and cannot represent.
We follow the approach via the graph isomorphism test, and show that is equivalent
to the other perspective via function approximatio... | {
"timestamp": "2019-05-30T02:21:13",
"yymm": "1905",
"arxiv_id": "1905.12560",
"language": "en",
"url": "https://arxiv.org/abs/1905.12560",
"abstract": "Graph Neural Networks (GNNs) have achieved much success on graph-structured data. In light of this, there have been increasing interests in studying their... |
https://arxiv.org/abs/1711.08791 | Cantor set arithmetic | Every element $u$ of $[0,1]$ can be written in the form $u=x^2y$, where $x,y$ are elements of the Cantor set $C$. In particular, every real number between zero and one is the product of three elements of the Cantor set. On the other hand the set of real numbers $v$ that can be written in the form $v=xy$ with $x$ and $y... | \section{Introduction.}
One of the first exotic mathematical objects encountered by the
post-calculus student is the Cantor set
\begin{equation}\label{Steinhaus}
C = \left\{\sum_{k= 1}^\infty \alpha_k3^{-k}, \alpha_k \in \{0,2\}\right\}.
\end{equation}
(See {Section 2} for
several equivalent definitions of $C$.) One o... | {
"timestamp": "2017-11-27T02:09:00",
"yymm": "1711",
"arxiv_id": "1711.08791",
"language": "en",
"url": "https://arxiv.org/abs/1711.08791",
"abstract": "Every element $u$ of $[0,1]$ can be written in the form $u=x^2y$, where $x,y$ are elements of the Cantor set $C$. In particular, every real number between... |
https://arxiv.org/abs/2112.04654 | Unimodular triangulations of sufficiently large dilations | An integral polytope is a polytope whose vertices have integer coordinates. A unimodular triangulation of an integral polytope in $\mathbb{R}^d$ is a triangulation in which all simplices are integral with volume $1/d!$. A classic result of Knudsen, Mumford, and Waterman states that for every integral polytope $P$, ther... | \section{Introduction}
Unimodular triangulations are elementary objects which arise naturally in algebra and combinatorics. We refer to the paper by Haase et al.\ \cite{HPPS} for an extensive survey on the subject. In this paper we answer a longstanding question on the existence of certain unimodular triangulations.
... | {
"timestamp": "2021-12-10T02:06:53",
"yymm": "2112",
"arxiv_id": "2112.04654",
"language": "en",
"url": "https://arxiv.org/abs/2112.04654",
"abstract": "An integral polytope is a polytope whose vertices have integer coordinates. A unimodular triangulation of an integral polytope in $\\mathbb{R}^d$ is a tri... |
https://arxiv.org/abs/1603.00677 | Karhunen-Loeve expansions of Levy processes | Karhunen-Loeve expansions (KLE) of stochastic processes are important tools in mathematics, the sciences, economics, and engineering. However, the KLE is primarily useful for those processes for which we can identify the necessary components, i.e., a set of basis functions, and the distribution of an associated set of ... | \section{Introduction}
Fourier series are powerful tools in mathematics and many other fields. The Karhunen-Lo{\`e}ve theorem (KLT) allows us to create generalized Fourier series from stochastic processes in an, in some sense, optimal way. Arguably the most famous application of the KLT is to derive the classic sine se... | {
"timestamp": "2016-03-03T02:09:04",
"yymm": "1603",
"arxiv_id": "1603.00677",
"language": "en",
"url": "https://arxiv.org/abs/1603.00677",
"abstract": "Karhunen-Loeve expansions (KLE) of stochastic processes are important tools in mathematics, the sciences, economics, and engineering. However, the KLE is ... |
https://arxiv.org/abs/2008.06577 | Cycles of a given length in tournaments | We study the asymptotic behavior of the maximum number of directed cycles of a given length in a tournament: let $c(\ell)$ be the limit of the ratio of the maximum number of cycles of length $\ell$ in an $n$-vertex tournament and the expected number of cycles of length $\ell$ in the random $n$-vertex tournament, when $... | \section{Introduction}
\label{sec:intro}
In this paper, we address one of the most natural extremal problems concerning tournaments:
\emph{What is the maximum number of cycles of a given length that can be contained in an $n$-vertex tournament?}
The cases of cycles of length three and four are well-understood.
An $n$-... | {
"timestamp": "2022-07-25T02:10:38",
"yymm": "2008",
"arxiv_id": "2008.06577",
"language": "en",
"url": "https://arxiv.org/abs/2008.06577",
"abstract": "We study the asymptotic behavior of the maximum number of directed cycles of a given length in a tournament: let $c(\\ell)$ be the limit of the ratio of t... |
https://arxiv.org/abs/1201.6649 | Discriminant coamoebas through homology | Understanding the complement of the coamoeba of a (reduced) A-discriminant is one approach to studying the monodromy of solutions to the corresponding system of A-hypergeometric differential equations. Nilsson and Passare described the structure of the coamoeba and its complement (a zonotope) when the reduced A-discrim... | \section*{Introduction}
$A$-hypergeometric functions, which are solutions to $A$-hypergeometric systems of
differential equations~\cite{GKZ89,GKZ94,SST}, enjoy two complimentary analytical formulae
which together give an approach to studying the monodromy of the
solutions~\cite{Beukers} at non-resonant parameters.... | {
"timestamp": "2012-08-03T02:04:43",
"yymm": "1201",
"arxiv_id": "1201.6649",
"language": "en",
"url": "https://arxiv.org/abs/1201.6649",
"abstract": "Understanding the complement of the coamoeba of a (reduced) A-discriminant is one approach to studying the monodromy of solutions to the corresponding syste... |
https://arxiv.org/abs/1404.2710 | On Collatz' Words, Sequences and Trees | Motivated by a recent work of Trümper we consider the general Collatz word (up-down pattern) and the sequences following this pattern. The recurrences for the first and last sequence entries are given, obtained from repeated application of the general solution of a binary linear inhomogeneous Diophantine equation. Thes... | \section{Introduction}
The Collatz map $C$ for natural numbers maps an odd number $m$ to $3\,m\,+\, 1$ and an even number to \dstyle{\frac{m}{2}}. The {\sl Collatz} conjecture \cite{Lagarias}, \cite{WikiC}, \cite{WeissteinC} is the claim that every natural number $n$ ends up, after sufficient iterations of the map $C... | {
"timestamp": "2014-04-11T02:05:00",
"yymm": "1404",
"arxiv_id": "1404.2710",
"language": "en",
"url": "https://arxiv.org/abs/1404.2710",
"abstract": "Motivated by a recent work of Trümper we consider the general Collatz word (up-down pattern) and the sequences following this pattern. The recurrences for t... |
https://arxiv.org/abs/2301.02645 | The Generalized Kauffman-Harary Conjecture is True | For a reduced alternating diagram of a knot with a prime determinant $p,$ the Kauffman-Harary conjecture states that every non-trivial Fox $p$-coloring of the knot assigns different colors to its arcs. In this paper, we prove a generalization of the conjecture stated nineteen years ago by Asaeda, Przytycki, and Sikora:... | \section{History of the alternation conjecture}
In 1998, Louis H. Kauffman and Frank Harary formulated the following conjecture \cite{HK}:
\begin{conjecture*}
Let $D$ be a reduced, alternating diagram of a knot $K$ having determinant $p$, where $p$ is prime. Then every non-trivial $p$-coloring of $D$ assigns dif... | {
"timestamp": "2023-01-09T02:13:55",
"yymm": "2301",
"arxiv_id": "2301.02645",
"language": "en",
"url": "https://arxiv.org/abs/2301.02645",
"abstract": "For a reduced alternating diagram of a knot with a prime determinant $p,$ the Kauffman-Harary conjecture states that every non-trivial Fox $p$-coloring of... |
https://arxiv.org/abs/2106.12190 | Closed-Form, Provable, and Robust PCA via Leverage Statistics and Innovation Search | The idea of Innovation Search, which was initially proposed for data clustering, was recently used for outlier detection. In the application of Innovation Search for outlier detection, the directions of innovation were utilized to measure the innovation of the data points. We study the Innovation Values computed by the... | \section{Introduction}
Principal Component Analysis (PCA) has been extensively
used for linear dimensionality reduction.
While PCA is useful when the data has
low intrinsic dimension, its output is sensitive to outliers
in the sense that the subspace found by PCA can arbitrarily deviate from
the true underlying subsp... | {
"timestamp": "2021-06-24T02:12:16",
"yymm": "2106",
"arxiv_id": "2106.12190",
"language": "en",
"url": "https://arxiv.org/abs/2106.12190",
"abstract": "The idea of Innovation Search, which was initially proposed for data clustering, was recently used for outlier detection. In the application of Innovation... |
https://arxiv.org/abs/2106.03070 | Linear Rescaling to Accurately Interpret Logarithms | The standard approximation of a natural logarithm in statistical analysis interprets a linear change of \(p\) in \(\ln(X)\) as a \((1+p)\) proportional change in \(X\), which is only accurate for small values of \(p\). I suggest base-\((1+p)\) logarithms, where \(p\) is chosen ahead of time. A one-unit change in \(\log... | \section{The Traditional Interpretation of
Logarithms}\label{the-traditional-interpretation-of-logarithms}}
It is common practice in many statistical applications, especially in
regression analysis, to transform variables using the natural logarithm
\(\ln(X)\). This can be done for statistical reasons, for example to ... | {
"timestamp": "2021-10-07T02:12:48",
"yymm": "2106",
"arxiv_id": "2106.03070",
"language": "en",
"url": "https://arxiv.org/abs/2106.03070",
"abstract": "The standard approximation of a natural logarithm in statistical analysis interprets a linear change of \\(p\\) in \\(\\ln(X)\\) as a \\((1+p)\\) proporti... |
https://arxiv.org/abs/2208.02559 | Equivalence between Time Series Predictability and Bayes Error Rate | Predictability is an emerging metric that quantifies the highest possible prediction accuracy for a given time series, being widely utilized in assessing known prediction algorithms and characterizing intrinsic regularities in human behaviors. Lately, increasing criticisms aim at the inaccuracy of the estimated predict... | \section*{Theorem}
\begin{theorem}
\label{theorem:1}
Given a M-state time series, its predictability $\Pi$ is equivalent to the Bayes error rate $R$ in a M-classification problem as
\begin{equation}
\Pi = 1 - R, \label{eqn:1}
\end{equation}
if we treat each state as a class, and the series before the state as ... | {
"timestamp": "2022-08-05T02:10:13",
"yymm": "2208",
"arxiv_id": "2208.02559",
"language": "en",
"url": "https://arxiv.org/abs/2208.02559",
"abstract": "Predictability is an emerging metric that quantifies the highest possible prediction accuracy for a given time series, being widely utilized in assessing ... |
https://arxiv.org/abs/1212.1412 | Antiderivatives Exist without Integration | We present a proof that any continuous function with domain including a closed interval yields an antiderivative of that function on that interval. This is done without the need of any integration comparable to that of Riemann, Cauchy, or Darboux. The proof is based on one given by Lebesgue in 1905. | \section*{Lebesgue's Construction}
Throughout, let $f$ be a function that is continuous on an interval $[a, b]$.
We use Lebesgue's notation and give a modern rendition of his construction. Suppose a set of $n$ points $\{(a_{0},d_{0}), (a_{1},d_{1}), \dots, (a_{n},d_{n})\}$, $a=a_{0} < a_{1} < \dots < a_{n}=b$ are... | {
"timestamp": "2012-12-07T02:04:39",
"yymm": "1212",
"arxiv_id": "1212.1412",
"language": "en",
"url": "https://arxiv.org/abs/1212.1412",
"abstract": "We present a proof that any continuous function with domain including a closed interval yields an antiderivative of that function on that interval. This is ... |
https://arxiv.org/abs/2002.00080 | Convergence rate analysis and improved iterations for numerical radius computation | The main two algorithms for computing the numerical radius are the level-set method of Mengi and Overton and the cutting-plane method of Uhlig. Via new analyses, we explain why the cutting-plane approach is sometimes much faster or much slower than the level-set one and then propose a new hybrid algorithm that remains ... | \section{Introduction}
Consider the discrete-time dynamical system
\begin{equation}
\label{eq:ode_disc}
x_{k+1} = Ax_k,
\end{equation}
where $A \in \mathbb{C}^{n \times n}$ and $x_k \in \mathbb{C}^n$.
The asymptotic behavior of \eqref{eq:ode_disc} is of course characterized by the moduli of $\Lambda(A)$.
Given the \e... | {
"timestamp": "2021-10-28T02:26:16",
"yymm": "2002",
"arxiv_id": "2002.00080",
"language": "en",
"url": "https://arxiv.org/abs/2002.00080",
"abstract": "The main two algorithms for computing the numerical radius are the level-set method of Mengi and Overton and the cutting-plane method of Uhlig. Via new an... |
https://arxiv.org/abs/1707.01883 | Hermann Hankel's "On the general theory of motion of fluids", an essay including an English translation of the complete Preisschrift from 1861 | The present is a companion paper to "A contemporary look at Hermann Hankel's 1861 pioneering work on Lagrangian fluid dynamics" by Frisch, Grimberg and Villone (2017). Here we present the English translation of the 1861 prize manuscript from Göttingen University "Zur allgemeinen Theorie der Bewegung der Flüssigkeiten" ... | \section{Introductory notes}
\label{s:intro}
\deffootnotemark{\textsuperscript{\thefootnotemark}}\deffootnote{2em}{1.6em}{${}^\thefootnotemark$\hspace{0.0319cm}\enskip}
\setcounter{footnote}{0}
Here we present, with some supplementary documents, a full translation from German of
the winning essay, originally writ... | {
"timestamp": "2017-09-19T02:14:27",
"yymm": "1707",
"arxiv_id": "1707.01883",
"language": "en",
"url": "https://arxiv.org/abs/1707.01883",
"abstract": "The present is a companion paper to \"A contemporary look at Hermann Hankel's 1861 pioneering work on Lagrangian fluid dynamics\" by Frisch, Grimberg and ... |
https://arxiv.org/abs/0805.4174 | On a generalization of Christoffel words: epichristoffel words | Sturmian sequences are well-known as the ones having minimal complexity over a 2-letter alphabet. They are also the balanced sequences over a 2-letter alphabet and the sequences describing discrete lines. They are famous and have been extensively studied since the 18th century. One of the {extensions} of these sequence... | \section{Introduction}
As far as we know, Sturmian sequences first appeared in the literature at the 18th century in the precursory works of the astronomer Bernoulli \cite{jb1772}. They later appeared in the 19th century in Christoffel \cite{ebc1875} and Markov \cite{am1882} works. The first deep study of these sequ... | {
"timestamp": "2009-04-24T10:50:25",
"yymm": "0805",
"arxiv_id": "0805.4174",
"language": "en",
"url": "https://arxiv.org/abs/0805.4174",
"abstract": "Sturmian sequences are well-known as the ones having minimal complexity over a 2-letter alphabet. They are also the balanced sequences over a 2-letter alpha... |
https://arxiv.org/abs/2302.02872 | Limiting distributions of conjugate algebraic integers | Let $\Sigma \subset \mathbb{C}$ be a compact subset of the complex plane, and $\mu$ be a probability distribution on $\Sigma$. We give necessary and sufficient conditions for $\mu$ to be the weak* limit of a sequence of uniform probability measures on a complete set of conjugate algebraic integers lying eventually in a... | \section{Introduction}
\subsection{Background}
This paper is motivated by the following question. What compact sets $\Sigma$ in the complex plane $\mathbb{C}$ contain infinitely many sets of conjugate algebraic integers, and how are they distributed in $\Sigma$? The ideas of this project originated in the works of many... | {
"timestamp": "2023-02-07T02:31:16",
"yymm": "2302",
"arxiv_id": "2302.02872",
"language": "en",
"url": "https://arxiv.org/abs/2302.02872",
"abstract": "Let $\\Sigma \\subset \\mathbb{C}$ be a compact subset of the complex plane, and $\\mu$ be a probability distribution on $\\Sigma$. We give necessary and ... |
https://arxiv.org/abs/1907.02004 | On Hamiltonian cycles in balanced $k$-partite graphs | For all integers $k$ with $k\geq 2$, if $G$ is a balanced $k$-partite graph on $n\geq 3$ vertices with minimum degree at least \[ \left\lceil\frac{n}{2}\right\rceil+\left\lfloor\frac{n+2}{2\lceil\frac{k+1}{2}\rceil}\right\rfloor-\frac{n}{k}=\begin{cases} \lceil\frac{n}{2}\rceil+\lfloor\frac{n+2}{k+1}\rfloor-\frac{n}{k}... | \section{Introduction}
The study of Hamiltonian cycles in balanced $k$-partite graphs begins with the following classic results of Dirac, and Moon and Moser. Dirac \cite{D} proved that for all graphs $G$ on $n\geq 3$ vertices, if $\delta(G)\geq \ceiling{\frac{n}{2}}$, then $G$ has a Hamiltonian cycle. Moon and Moser ... | {
"timestamp": "2020-05-28T02:07:59",
"yymm": "1907",
"arxiv_id": "1907.02004",
"language": "en",
"url": "https://arxiv.org/abs/1907.02004",
"abstract": "For all integers $k$ with $k\\geq 2$, if $G$ is a balanced $k$-partite graph on $n\\geq 3$ vertices with minimum degree at least \\[ \\left\\lceil\\frac{n... |
https://arxiv.org/abs/1710.03249 | Optimal Graphs for Independence and $k$-Independence Polynomials | The independence polynomial $I(G,x)$ of a finite graph $G$ is the generating function for the sequence of the number of independent sets of each cardinality. We investigate whether, given a fixed number of vertices and edges, there exists optimally-least (optimally-greatest) graphs, that are least (respectively, greate... | \section{Introduction}
Given a property of subsets of the vertex or edge set -- such as independent, complete, dominating for vertices and matching for edges -- one is often interested in maximizing or minimizing the size of the set in a given graph $G$. However, one can get a much more nuanced study of the subsets by... | {
"timestamp": "2017-10-11T02:00:41",
"yymm": "1710",
"arxiv_id": "1710.03249",
"language": "en",
"url": "https://arxiv.org/abs/1710.03249",
"abstract": "The independence polynomial $I(G,x)$ of a finite graph $G$ is the generating function for the sequence of the number of independent sets of each cardinali... |
https://arxiv.org/abs/1811.06493 | Intermediate dimensions | We introduce a continuum of dimensions which are `intermediate' between the familiar Hausdorff and box dimensions. This is done by restricting the families of allowable covers in the definition of Hausdorff dimension by insisting that $|U| \leq |V|^\theta$ for all sets $U, V$ used in a particular cover, where $\theta \... | \section{Intermediate dimensions: definitions and background}
\setcounter{equation}{0}
\setcounter{theo}{0}
We work with subsets of $\mathbb{R}^n$ throughout, although much of what we establish also holds in more general metric spaces. We denote the {\em diameter} of a set $F$ by $|F|$, and when we refer to a {\em cov... | {
"timestamp": "2019-11-07T02:15:12",
"yymm": "1811",
"arxiv_id": "1811.06493",
"language": "en",
"url": "https://arxiv.org/abs/1811.06493",
"abstract": "We introduce a continuum of dimensions which are `intermediate' between the familiar Hausdorff and box dimensions. This is done by restricting the familie... |
https://arxiv.org/abs/math/9806076 | On the volume of the polytope of doubly stochastic matrices | We study the calculation of the volume of the polytope B_n of n by n doubly stochastic matrices; that is, the set of real non-negative matrices with all row and column sums equal to one.We describe two methods. The first involves a decomposition of the polytope into simplices. The second involves the enumeration of ``m... | \section{Introduction}
\label{sec:intro}
We study the calculation of the volume of the polytope $B_n$ of
$n \times n$ doubly stochastic matrices; that is, the set of real nonnegative
matrices with all row and column sums equal to one. This polytope is
sometimes known as the Birkhoff polytope or the assignment polytop... | {
"timestamp": "1998-06-13T04:05:13",
"yymm": "9806",
"arxiv_id": "math/9806076",
"language": "en",
"url": "https://arxiv.org/abs/math/9806076",
"abstract": "We study the calculation of the volume of the polytope B_n of n by n doubly stochastic matrices; that is, the set of real non-negative matrices with a... |
https://arxiv.org/abs/2109.01697 | The double-bubble problem on the square lattice | We investigate minimal-perimeter configurations of two finite sets of points on the square lattice. This corresponds to a lattice version of the classical double-bubble problem. We give a detailed description of the fine geometry of minimisers and, in some parameter regime, we compute the optimal perimeter as a functio... | \section{Introduction}
The classical double-bubble problem is concerned with the shape of
two sets of given volume under minimisation of their surface area. In the
Euclidean space, minimisers are enclosed by three spherical
caps, intersecting at an angle of $2\pi/3$. The proof of this fact in
${\mathbb R}^2$ dates... | {
"timestamp": "2022-03-29T02:44:44",
"yymm": "2109",
"arxiv_id": "2109.01697",
"language": "en",
"url": "https://arxiv.org/abs/2109.01697",
"abstract": "We investigate minimal-perimeter configurations of two finite sets of points on the square lattice. This corresponds to a lattice version of the classical... |
https://arxiv.org/abs/1612.00214 | A remark on local fractional calculus and ordinary derivatives | In this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental prop... | \section{Introduction}
Fractional calculus is a generalization of ordinary calculus, where derivatives and integrals of arbitrary real or complex order are defined. These fractional operators may model more efficiently certain real world phenomena, specially when the dynamics is affected by constraints inherent to the... | {
"timestamp": "2016-12-02T02:04:31",
"yymm": "1612",
"arxiv_id": "1612.00214",
"language": "en",
"url": "https://arxiv.org/abs/1612.00214",
"abstract": "In this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of ... |
https://arxiv.org/abs/2207.02060 | A sharp Korn's inequality for piecewise $H^1$ space and its application | In this paper, we revisit Korn's inequality for the piecewise $H^1$ space based on general polygonal or polyhedral decompositions of the domain. Our Korn's inequality is expressed with minimal jump terms. These minimal jump terms are identified by characterizing the restriction of rigid body mode to edge/face of the pa... | \section{Introduction}\Label{Intro}
The Korn's inequality played a fundamental role in the development of linear elasticity. There is a work reviewing Korn's inequality and its applications in continuum mechanics \cite{horgan1995korn}. In fact, there are a lot of works on proving classical Korn's inequality \cite{nitsc... | {
"timestamp": "2022-07-06T02:17:20",
"yymm": "2207",
"arxiv_id": "2207.02060",
"language": "en",
"url": "https://arxiv.org/abs/2207.02060",
"abstract": "In this paper, we revisit Korn's inequality for the piecewise $H^1$ space based on general polygonal or polyhedral decompositions of the domain. Our Korn'... |
https://arxiv.org/abs/2011.12449 | Iterations for the Unitary Sign Decomposition and the Unitary Eigendecomposition | We construct fast, structure-preserving iterations for computing the sign decomposition of a unitary matrix $A$ with no eigenvalues equal to $\pm i$. This decomposition factorizes $A$ as the product of an involutory matrix $S = \operatorname{sign}(A) = A(A^2)^{-1/2}$ times a matrix $N = (A^2)^{1/2}$ with spectrum conta... | \section{Introduction} \label{sec:intro}
Every matrix $A \in \mathbb{C}^{n \times n}$ with no purely imaginary eigenvalues can be written uniquely as a product
\[
A = SN,
\]
where $S \in \mathbb{C}^{n \times n}$ is involutory ($S^2=I$), $N \in \mathbb{C}^{n \times n}$ has spectrum in the open right half of the complex... | {
"timestamp": "2020-11-26T02:07:00",
"yymm": "2011",
"arxiv_id": "2011.12449",
"language": "en",
"url": "https://arxiv.org/abs/2011.12449",
"abstract": "We construct fast, structure-preserving iterations for computing the sign decomposition of a unitary matrix $A$ with no eigenvalues equal to $\\pm i$. Thi... |
https://arxiv.org/abs/1303.6012 | Are the Snapshot Difference Quotients Needed in the Proper Orthogonal Decomposition? | This paper presents a theoretical and numerical investigation of the following practical question: Should the time difference quotients of the snapshots be used to generate the proper orthogonal decomposition basis functions? The answer to this question is important, since some published numerical studies use the time ... | \section{Conclusions}
\label{sec:conclusions}
The effect of using or not the snapshot DQs in the generation of the POD basis (the $DQ$ and the $no\_DQ$ cases, respectively) was investigated theoretically and numerically.
The criterion used in this theoretical and numerical investigation was the rate of convergence wi... | {
"timestamp": "2013-06-18T02:00:31",
"yymm": "1303",
"arxiv_id": "1303.6012",
"language": "en",
"url": "https://arxiv.org/abs/1303.6012",
"abstract": "This paper presents a theoretical and numerical investigation of the following practical question: Should the time difference quotients of the snapshots be ... |
https://arxiv.org/abs/2012.00525 | The algebraic classification of nilpotent algebras | We give the complete algebraic classification of all complex 4-dimensional nilpotent algebras. The final list has 234 (parametric families of) isomorphism classes of algebras, 66 of which are new in the literature. | \section*{Introduction}
The description, up to isomorphism, of all the algebras of some fixed dimension satisfying certain properties (the so-called algebraic classification) is a classical problem in algebra.
There are many papers devoted to algebraic classification of small-dimensional algebras in several varieties... | {
"timestamp": "2020-12-02T02:23:37",
"yymm": "2012",
"arxiv_id": "2012.00525",
"language": "en",
"url": "https://arxiv.org/abs/2012.00525",
"abstract": "We give the complete algebraic classification of all complex 4-dimensional nilpotent algebras. The final list has 234 (parametric families of) isomorphism... |
https://arxiv.org/abs/1711.07288 | When Fourth Moments Are Enough | This note concerns a somewhat innocent question motivated by an observation concerning the use of Chebyshev bounds on sample estimates of $p$ in the binomial distribution with parameters $n,p$. Namely, what moment order produces the best Chebyshev estimate of $p$? If $S_n(p)$ has a binomial distribution with parameters... | \section{Introduction}
This note concerns a somewhat innocent question motivated by an observation
concerning the use of Chebyshev bounds on sample estimates of $p$ in the
binomial distribution with parameters $n,p$. Namely, what moment order
produces the best Chebyshev estimate of $p$? Chebyshev is arguably the m... | {
"timestamp": "2017-11-21T02:17:22",
"yymm": "1711",
"arxiv_id": "1711.07288",
"language": "en",
"url": "https://arxiv.org/abs/1711.07288",
"abstract": "This note concerns a somewhat innocent question motivated by an observation concerning the use of Chebyshev bounds on sample estimates of $p$ in the binom... |
https://arxiv.org/abs/2205.10968 | Eigenvalue bounds of the Kirchhoff Laplacian | We prove that each eigenvalue l(k) of the Kirchhoff Laplacian K of a graph or quiver is bounded above by d(k)+d(k-1) for all k in {1,...,n}. Here l(1),...,l(n) is a non-decreasing list of the eigenvalues of K and d(1),..,d(n) is a non-decreasing list of vertex degrees with the additional assumption d(0)=0. We also prov... | \section{The theorem}
\paragraph{}
Let $G=(V,E)$ be a {\bf finite simple graph} with $n$ vertices.
Denote by $\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n$ the ordered
list of eigenvalues of the {\bf Kirchhoff matrix} $K=B-A$, where $B$ is the
diagonal vertex degree matrix with ordered vertex degrees
$d_1 \leq ... | {
"timestamp": "2022-05-27T02:04:52",
"yymm": "2205",
"arxiv_id": "2205.10968",
"language": "en",
"url": "https://arxiv.org/abs/2205.10968",
"abstract": "We prove that each eigenvalue l(k) of the Kirchhoff Laplacian K of a graph or quiver is bounded above by d(k)+d(k-1) for all k in {1,...,n}. Here l(1),...... |
https://arxiv.org/abs/1610.07836 | Classification of crescent configurations | Let $n$ points be in crescent configurations in $\mathbb{R}^d$ if they lie in general position in $\mathbb{R}^d$ and determine $n-1$ distinct distances, such that for every $1 \leq i \leq n-1$ there is a distance that occurs exactly $i$ times. Since Erdős' conjecture in 1989 on the existence of $N$ sufficiently large s... | \section{Introduction}
Erd\H{o}s once wrote, ``my most striking contribution to geometry is, no doubt, my problem on the number of distinct distances,'' \cite{Erd96}. The referred question, which asks what is the minimum number of distinct distances determined by $n$ points, was first asked in 1946 \cite{Erd46} and mar... | {
"timestamp": "2016-10-26T02:05:01",
"yymm": "1610",
"arxiv_id": "1610.07836",
"language": "en",
"url": "https://arxiv.org/abs/1610.07836",
"abstract": "Let $n$ points be in crescent configurations in $\\mathbb{R}^d$ if they lie in general position in $\\mathbb{R}^d$ and determine $n-1$ distinct distances,... |
https://arxiv.org/abs/1806.09810 | On Representer Theorems and Convex Regularization | We establish a general principle which states that regularizing an inverse problem with a convex function yields solutions which are convex combinations of a small number of atoms. These atoms are identified with the extreme points and elements of the extreme rays of the regularizer level sets. An extension to a broade... | \section{Application to some popular regularizers}
\label{sec:app}
We now show that the extreme points and extreme rays of numerous convex regularizers can be described analytically, allowing to describe important analytical properties of the solutions of some popular problems.
The list given below is far from being ... | {
"timestamp": "2018-11-27T02:29:11",
"yymm": "1806",
"arxiv_id": "1806.09810",
"language": "en",
"url": "https://arxiv.org/abs/1806.09810",
"abstract": "We establish a general principle which states that regularizing an inverse problem with a convex function yields solutions which are convex combinations o... |
https://arxiv.org/abs/1909.08687 | Unusual elementary axiomatizations for abelian groups | One of the most studied algebraic structures with one operation is the Abelian group, which is defined as a structure whose operation satisfies the associative and commutative properties, has identical element and every element has an inverse element. In this article, we characterize the Abelian groups with other prope... | \section*{Introduction}
\noindent The axiomatic presentation of mathematical theories allows the selection of different sets of axioms for its development. This choice depends on criteria of economy, elegance, simplicity or pedagogy.
The definition of an abelian group was initially formulated for finite groups (with ... | {
"timestamp": "2019-09-20T02:02:16",
"yymm": "1909",
"arxiv_id": "1909.08687",
"language": "en",
"url": "https://arxiv.org/abs/1909.08687",
"abstract": "One of the most studied algebraic structures with one operation is the Abelian group, which is defined as a structure whose operation satisfies the associ... |
https://arxiv.org/abs/2202.11875 | Characterizing Spectral Properties of Bridge | The Bridge graph is a special type of graph which are constructed by connecting identical connected graphs with path graphs. We discuss different types of bridge graphs $B_{n\times l}^{m\times k}$ in this paper. In particular, we discuss the following: complete-type bridge graphs, star-type bridge graphs, and full bina... | \section{Introduction}
Spectral graph theory is the process of characterizing graphs by means of the eigenvalues and eigenvectors of the graph Laplacian. It connects graphs to matrices, and allows us to understand properties of graph using more analytic means. Recently, spectral graph theory has found application in ma... | {
"timestamp": "2023-01-03T02:06:27",
"yymm": "2202",
"arxiv_id": "2202.11875",
"language": "en",
"url": "https://arxiv.org/abs/2202.11875",
"abstract": "The Bridge graph is a special type of graph which are constructed by connecting identical connected graphs with path graphs. We discuss different types of... |
https://arxiv.org/abs/1308.0861 | Bounds of incidences between points and algebraic curves | We prove new bounds on the number of incidences between points and higher degree algebraic curves. The key ingredient is an improved initial bound, which is valid for all fields. Then we apply the polynomial method to obtain global bounds on $\mathbb{R}$ and $\mathbb{C}$. | \section{introduction}
The Szemer\'{e}di--Trotter theorem \cite{szemeredi1983extremal} says that for a finite set, $L$, of lines and a finite set, $P$, of points in $\mathbb{R}^2$, the number of incidences is less than a constant times $|P|^{\frac{2}{3}} |L|^{\frac{2}{3}} + |P| + |L|$. There have been several generali... | {
"timestamp": "2015-03-31T02:19:36",
"yymm": "1308",
"arxiv_id": "1308.0861",
"language": "en",
"url": "https://arxiv.org/abs/1308.0861",
"abstract": "We prove new bounds on the number of incidences between points and higher degree algebraic curves. The key ingredient is an improved initial bound, which is... |
https://arxiv.org/abs/1504.03029 | The covering radius of randomly distributed points on a manifold | We derive fundamental asymptotic results for the expected covering radius $\rho(X_N)$ for $N$ points that are randomly and independently distributed with respect to surface measure on a sphere as well as on a class of smooth manifolds. For the unit sphere $\mathbb{S}^d \subset \mathbb{R}^{d+1}$, we obtain the precise a... | \section{Introduction and Notation}
The purpose of this paper is to obtain asymptotic results for the expected value of the covering
radius of $N$ points $X_N=\{x_1, x_2,\ldots, x_N\}$ that are randomly and independently distributed with respect to a given measure $\mu$
over a metric space $(\mathcal{X}, m)$. By the \e... | {
"timestamp": "2015-04-14T02:09:57",
"yymm": "1504",
"arxiv_id": "1504.03029",
"language": "en",
"url": "https://arxiv.org/abs/1504.03029",
"abstract": "We derive fundamental asymptotic results for the expected covering radius $\\rho(X_N)$ for $N$ points that are randomly and independently distributed with... |
https://arxiv.org/abs/1602.05298 | On critical points of random polynomials and spectrum of certain products of random matrices | In the first part we study critical points of random polynomials. We choose two deterministic sequences of complex numbers,whose empirical measures converge to the same probability measure in complex plane. We make a sequence of polynomials whose zeros are chosen from either of sequences at random. We show that the lim... | \chapter{Critical points of random polynomials}
\label{ch:criticalpoints4}
\section{Introduction}In this chapter we will investigate the distribution of the critical points in relation to the zeros of a polynomial. For a holomorphic function $f:\mathbb{C} \rightarrow \mathbb{C}$ a point $z \in \mathbb{C}$ is called a... | {
"timestamp": "2016-02-18T02:05:51",
"yymm": "1602",
"arxiv_id": "1602.05298",
"language": "en",
"url": "https://arxiv.org/abs/1602.05298",
"abstract": "In the first part we study critical points of random polynomials. We choose two deterministic sequences of complex numbers,whose empirical measures conver... |
https://arxiv.org/abs/1603.08651 | Parkable convex sets and finite-dimensional Hilbert spaces | A subset of a convex body $B$ containing the origin in a Euclidean space is {\it parkable in $B$} if it can be translated inside $B$ in such a manner that the translate the origin. We provide characterizations of ellipsoids and of centrally symmetric convex bodies in Euclidean spaces of dimension $\ge 3$ based on the n... | \section{#1}\setcounter{lemma}{0}}
\title{Parkable convex sets and finite-dimensional Hilbert spaces}
\author{Alexandru Chirvasitu\footnote{University of Washington, \url{chirva@uw.edu}}}
\begin{document}
\date{}
\maketitle
\begin{abstract}
A subset of a convex body $B$ containing the origin in a Euclidean spa... | {
"timestamp": "2016-03-30T02:06:11",
"yymm": "1603",
"arxiv_id": "1603.08651",
"language": "en",
"url": "https://arxiv.org/abs/1603.08651",
"abstract": "A subset of a convex body $B$ containing the origin in a Euclidean space is {\\it parkable in $B$} if it can be translated inside $B$ in such a manner tha... |
https://arxiv.org/abs/1806.03774 | Counting subgroups of fixed order in finite abelian groups | We use recurrence relations to derive explicit formulas for counting the number of subgroups of given order (or index) in rank 3 finite abelian p-groups and use these to derive similar formulas in few cases for rank 4. As a consequence, we answer some questions by M. T$\ddot{a}$rn$\ddot{a}$uceanu in \cite{MT} and L. T$... | \section{Introduction}
The subject of counting various kinds of subgroups of finite abelian groups has a long and rich history. For instance, in the early 1900's, Miller\cite{MI04} determined the number of cyclic subgroups of prime power order in a finite abelian p-group $G$, where p is a prime number. In about the sa... | {
"timestamp": "2018-06-18T02:04:29",
"yymm": "1806",
"arxiv_id": "1806.03774",
"language": "en",
"url": "https://arxiv.org/abs/1806.03774",
"abstract": "We use recurrence relations to derive explicit formulas for counting the number of subgroups of given order (or index) in rank 3 finite abelian p-groups a... |
https://arxiv.org/abs/2103.13727 | Conway's spiral and a discrete Gömböc with 21 point masses | We show an explicit construction in 3 dimensions for a convex, mono-monostatic polyhedron (i.e., having exactly one stable and one unstable equilibrium) with 21 vertices and 21 faces. This polyhedron is a 0-skeleton, with equal masses located at each vertex. The above construction serves as an upper bound for the minim... | \section{Introduction}\label{sec:intro}
\subsection{Mono-stability and homogeneous polyhedra}
If a rigid body has one single stable position then we call it \emph{mono-stable}, and this property was probably first explored by Archimedes as he developed his famous design for ships \cite{Archimedes}. Mono-stability mi... | {
"timestamp": "2021-10-14T02:23:03",
"yymm": "2103",
"arxiv_id": "2103.13727",
"language": "en",
"url": "https://arxiv.org/abs/2103.13727",
"abstract": "We show an explicit construction in 3 dimensions for a convex, mono-monostatic polyhedron (i.e., having exactly one stable and one unstable equilibrium) w... |
https://arxiv.org/abs/1401.6619 | On cycles in intersection graph of rings | Let $R$ be a commutative ring with non-zero identity. We describe all $C_3$- and $C_4$-free intersection graph of non-trivial ideals of $R$ as well as $C_n$-free intersection graph when $R$ is a reduced ring. Also, we shall describe all complete, regular and $n$-claw-free intersection graphs. Finally, we shall prove th... | \section{Introduction}
If $S=\{S_1,\ldots,S_n\}$ is a family of sets, then the intersection graph of $S$, is the graph having $S$ as its vertex set with $S_i$ adjacent to $S_j$ if $i\neq j$ and $S_i\cap S_j\neq\emptyset$. A well-know theorem due to Marczewski \cite{tam-frm} states that all graphs are intersection graph... | {
"timestamp": "2014-01-28T02:08:50",
"yymm": "1401",
"arxiv_id": "1401.6619",
"language": "en",
"url": "https://arxiv.org/abs/1401.6619",
"abstract": "Let $R$ be a commutative ring with non-zero identity. We describe all $C_3$- and $C_4$-free intersection graph of non-trivial ideals of $R$ as well as $C_n$... |
https://arxiv.org/abs/1911.12336 | Synchronization of Kuramoto Oscillators in Dense Networks | We study synchronization properties of systems of Kuramoto oscillators. The problem can also be understood as a question about the properties of an energy landscape created by a graph. More formally, let $G=(V,E)$ be a connected graph and $(a_{ij})_{i,j=1}^{n}$ denotes its adjacency matrix. Let the function $f:\mathbb{... | \section{Introduction}
We study a simple problem that can be understood from a variety of
perspectives. Perhaps its simplest formulation is as follows: let
$G=(V,E)$ be a connected graph and $(a_{ij})_{i,j=1}^{n}$ denotes its
adjacency matrix. We assume the graph is simple, and thus
$a_{ii} = 0$ for $i = 1, \ldots, n... | {
"timestamp": "2020-04-20T02:14:12",
"yymm": "1911",
"arxiv_id": "1911.12336",
"language": "en",
"url": "https://arxiv.org/abs/1911.12336",
"abstract": "We study synchronization properties of systems of Kuramoto oscillators. The problem can also be understood as a question about the properties of an energy... |
https://arxiv.org/abs/2006.14482 | A metric on directed graphs and Markov chains based on hitting probabilities | The shortest-path, commute time, and diffusion distances on undirected graphs have been widely employed in applications such as dimensionality reduction, link prediction, and trip planning. Increasingly, there is interest in using asymmetric structure of data derived from Markov chains and directed graphs, but few metr... | \section{Introduction}
\subsection{Motivation}
Many finite spaces can be endowed with meaningful metrics. For undirected graphs, the geodesic (shortest path), commute time (effective resistance), and diffusion distance~\cite{lafon04diffusion,coifman05geometric,coifman06diffusion} metrics are widely applied~\cite{coifm... | {
"timestamp": "2021-01-19T02:43:57",
"yymm": "2006",
"arxiv_id": "2006.14482",
"language": "en",
"url": "https://arxiv.org/abs/2006.14482",
"abstract": "The shortest-path, commute time, and diffusion distances on undirected graphs have been widely employed in applications such as dimensionality reduction, ... |
https://arxiv.org/abs/1602.03246 | Reliability Polynomials of Simple Graphs having Arbitrarily many Inflection Points | In this paper we show that for each $n$, there exists a simple graph whose reliability polynomial has at least $n$ inflection points. | \section{Introduction}
The reliability of a graph $G$ is the probability that the graph remains connected when each edge is included, or ``functions", with independent probability $p$. Equivalently, we can say that each edge fails with probability $q = 1-p$. This function can be written as a polynomial in either $p$ ... | {
"timestamp": "2016-02-11T02:02:54",
"yymm": "1602",
"arxiv_id": "1602.03246",
"language": "en",
"url": "https://arxiv.org/abs/1602.03246",
"abstract": "In this paper we show that for each $n$, there exists a simple graph whose reliability polynomial has at least $n$ inflection points.",
"subjects": "Com... |
https://arxiv.org/abs/1007.4222 | Strict inequality in the box-counting dimension product formulas | It is known that the upper box-counting dimension of a Cartesian product satisfies the inequality $\dim_{B}\left(F\times G\right)\leq \dim_{B}\left(F\right) + \dim_{B}\left(G\right)$ whilst the lower box-counting dimension satisfies the inequality $\dim_{LB}\left(F\times G\right)\geq \dim_{LB}\left(F\right) + \dim_{LB}... | \part{Use this type of header for very long papers only}
\section{Preliminaries}
\label{intro}
\noindent
In a metric space $X$ the Hausdorff dimension of a compact set $F\subset X$ is defined as the supremum of the $d\geq 0$ such that the $d$-dimensional Hausdorff measure $\mathcal{H}^{d}\left(F\right)$ of $F$ is infi... | {
"timestamp": "2010-07-27T02:00:26",
"yymm": "1007",
"arxiv_id": "1007.4222",
"language": "en",
"url": "https://arxiv.org/abs/1007.4222",
"abstract": "It is known that the upper box-counting dimension of a Cartesian product satisfies the inequality $\\dim_{B}\\left(F\\times G\\right)\\leq \\dim_{B}\\left(F... |
https://arxiv.org/abs/2107.09450 | Monochromatic Edges in Complete Multipartite Hypergraphs | Consider the following problem. In a school with three classes containing $n$ students each, given that their genders are unknown, find the minimum possible number of triples of same-gender students not all of which are from the same class. Muaengwaeng asked this question and conjectured that the minimum scenario occur... | \section{Introduction}
A \textit{hypergraph} is a pair $(V, E)$ where $V$ is a finite set of vertices and $E$ is a collection of subsets of $V$. Each subset in $E$ is called an \textit{edge}. An \textit{$r$-uniform} hypergraph contains only edges of size $r$ and if it contains all possible edges of size $r$, this $r$-... | {
"timestamp": "2021-07-21T02:18:35",
"yymm": "2107",
"arxiv_id": "2107.09450",
"language": "en",
"url": "https://arxiv.org/abs/2107.09450",
"abstract": "Consider the following problem. In a school with three classes containing $n$ students each, given that their genders are unknown, find the minimum possib... |
https://arxiv.org/abs/1807.01492 | The minimal $k$-dispersion of point sets in high-dimensions | In this manuscript we introduce and study an extended version of the minimal dispersion of point sets, which has recently attracted considerable attention. Given a set $\mathscr P_n=\{x_1,\dots,x_n\}\subset [0,1]^d$ and $k\in\{0,1,\dots,n\}$, we define the $k$-dispersion to be the volume of the largest box amidst a poi... | \section{Introduction and main results}
A classical problem in computational geometry and complexity asks for the size of the largest empty and axis-parallel box given a point configuration in the cube $[0,1]^2$. From a complexity point of view this \emph{maximum empty rectangle problem} was already studied by Naamad,... | {
"timestamp": "2018-07-05T02:06:07",
"yymm": "1807",
"arxiv_id": "1807.01492",
"language": "en",
"url": "https://arxiv.org/abs/1807.01492",
"abstract": "In this manuscript we introduce and study an extended version of the minimal dispersion of point sets, which has recently attracted considerable attention... |
https://arxiv.org/abs/1804.09697 | Electrostatic Interpretation of Zeros of Orthogonal Polynomials | We study the differential equation $ - (p(x) y')' + q(x) y' = \lambda y,$ where $p(x)$ is a polynomial of degree at most 2 and $q(x)$ is a polynomial of degree at most 1. This includes the classical Jacobi polynomials, Hermite polynomials, Legendre polynomials, Chebychev polynomials and Laguerre polynomials. We provide... | \section{Introduction}
\subsection{Introduction.}
We start by describing an 1885 result of Stieltjes for Jacobi polynomials \cite{stieltjes}. Jacobi polynomials $P_n^{\alpha, \beta}(x)$, for real $\alpha, \beta > -1$, are the unique (up to a constant factor) solutions of the equation
$$ (1 - x^2 ) y''(x) - \left( \beta... | {
"timestamp": "2018-05-17T02:08:36",
"yymm": "1804",
"arxiv_id": "1804.09697",
"language": "en",
"url": "https://arxiv.org/abs/1804.09697",
"abstract": "We study the differential equation $ - (p(x) y')' + q(x) y' = \\lambda y,$ where $p(x)$ is a polynomial of degree at most 2 and $q(x)$ is a polynomial of ... |
https://arxiv.org/abs/1011.3486 | Computing the $\sin_{p}$ function via the inverse power method | In this paper, we discuss a new iterative method for computing $\sin_{p}$. This function was introduced by Lindqvist in connection with the unidimensional nonlinear Dirichlet eigenvalue problem for the $p$-Laplacian. The iterative technique was inspired by the inverse power method in finite dimensional linear algebra a... | \section{Introduction}
In this paper we present a new method to compute the function $\sin_{p}$,
inspired by recent work done by the authors in \cite{BEM}, where an iterative
algorithm based on the inverse power method of linear algebra was introduced
for the computation of the first eigenvalue and first eigenfun... | {
"timestamp": "2010-11-16T02:04:25",
"yymm": "1011",
"arxiv_id": "1011.3486",
"language": "en",
"url": "https://arxiv.org/abs/1011.3486",
"abstract": "In this paper, we discuss a new iterative method for computing $\\sin_{p}$. This function was introduced by Lindqvist in connection with the unidimensional ... |
https://arxiv.org/abs/2209.02588 | A generalization of Chu-Vandermonde's Identity | We present and prove a general form of Vandermonde's identity and use it as an alternative solution to a classic probability problem. | \section{Introduction}
Vandermonde's identity, its diverse proofs, and applications have been the spot of research throughout the centuries. Nowadays, it is a classic identity whose proof is available in most combinatorics-related books~\cite{knuth94}. However, there are still some efforts to give new proofs for this w... | {
"timestamp": "2022-09-07T02:51:52",
"yymm": "2209",
"arxiv_id": "2209.02588",
"language": "en",
"url": "https://arxiv.org/abs/2209.02588",
"abstract": "We present and prove a general form of Vandermonde's identity and use it as an alternative solution to a classic probability problem.",
"subjects": "Gen... |
https://arxiv.org/abs/1811.03008 | Limit points of normalized prime gaps | We show that at least 1/3 of positive real numbers are in the set of limit points of normalized prime gaps. More precisely, if $p_n$ denotes the $n$th prime and $\mathbb{L}$ is the set of limit points of the sequence $\{(p_{n+1}-p_n)/\log p_n\}_{n=1}^\infty,$ then for all $T\geq 0$ the Lebesque measure of $\mathbb{L} \... | \section{Introduction and main results}
The Prime Number Theorem tells us that the gap $p_{n+1}-p_n$ between consecutive primes is asymptotically $\log p_n$ on average ($p_n$ denotes the $n$th prime). It is therefore reasonable to consider the distribution of the normalized prime gaps $(p_{n+1}-p_n)/\log p_n;$ by heuri... | {
"timestamp": "2020-11-03T02:39:38",
"yymm": "1811",
"arxiv_id": "1811.03008",
"language": "en",
"url": "https://arxiv.org/abs/1811.03008",
"abstract": "We show that at least 1/3 of positive real numbers are in the set of limit points of normalized prime gaps. More precisely, if $p_n$ denotes the $n$th pri... |
https://arxiv.org/abs/2204.04729 | On dually-CPT and strong-CPT posets | A poset is a containment of paths in a tree (CPT) if it admits a representation by containment where each element of the poset is represented by a path in a tree and two elements are comparable in the poset if and only if the corresponding paths are related by the inclusion relation. Recently Alcón, Gudiño and Gutierre... | \section{Introduction}
A poset is called a containment order of paths in a tree (CPT for short)
if it admits a representation by containment where each element of the poset
corresponds to a path in a tree and for two elements $x$ and $y$, we have
$x < y$ in the poset if and only if the path corresponding to $x$ is pr... | {
"timestamp": "2022-04-12T02:25:31",
"yymm": "2204",
"arxiv_id": "2204.04729",
"language": "en",
"url": "https://arxiv.org/abs/2204.04729",
"abstract": "A poset is a containment of paths in a tree (CPT) if it admits a representation by containment where each element of the poset is represented by a path in... |
https://arxiv.org/abs/2111.13342 | On connected components with many edges | We prove that if $H$ is a subgraph of a complete multipartite graph $G$, then $H$ contains a connected component $H'$ satisfying $|E(H')||E(G)|\geq |E(H)|^2$. We use this to prove that every three-coloring of the edges of a complete graph contains a monochromatic connected subgraph with at least $1/6$ of the edges. We ... | \section{Introduction}
A classical observation of Erd\H{o}s and Rado is that in any two-coloring of the edges of the complete graph $K_n$, one of the color classes forms a connected graph. In \cite{gyarfas1977partition}, Gyárfás proves the following generalization of this observation: For any $k\geq 2$, in every $k$-c... | {
"timestamp": "2022-08-30T02:11:27",
"yymm": "2111",
"arxiv_id": "2111.13342",
"language": "en",
"url": "https://arxiv.org/abs/2111.13342",
"abstract": "We prove that if $H$ is a subgraph of a complete multipartite graph $G$, then $H$ contains a connected component $H'$ satisfying $|E(H')||E(G)|\\geq |E(H)... |
https://arxiv.org/abs/1010.5005 | Error Estimates for Generalized Barycentric Interpolation | We prove the optimal convergence estimate for first order interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson approach uses Vorono... |
\section{Introduction}
While a rich theory of finite element error estimation exists for meshes made of triangular or quadrilateral elements, relatively little attention has been paid to meshes constructed from arbitrary polygonal elements. Many quality-controlled domain meshing schemes could be simplified if polygo... | {
"timestamp": "2011-04-19T02:00:24",
"yymm": "1010",
"arxiv_id": "1010.5005",
"language": "en",
"url": "https://arxiv.org/abs/1010.5005",
"abstract": "We prove the optimal convergence estimate for first order interpolants used in finite element methods based on three major approaches for generalizing baryc... |
https://arxiv.org/abs/2105.00247 | Extension of tetration to real and complex heights | The continuous tetrational function ${^x}r=\tau(r,x)$, the unique solution of equation $\tau(r,x)=r^{\tau(r,x-1)}$ and its differential equation $\tau'(r,x) =q \tau(r,x) \tau'(r,x-1)$, is given explicitly as ${^x}r=\exp_{r}^{\lfloor x \rfloor+1}[\{x\}]_q$, where $x$ is a real variable called height, $r$ is a real const... | \section{Introduction}
Tetration, i.e., iterated exponentiation, is the fourth hyperoperation after addition, multiplication, and exponentiation \cite{Goodstein1947}. Tetration is defined as
\[^{n}r\colonequals \underset{n}{\underbrace{r^{r^{\udots^{r}}}}},\]
meaning that copies of $r$ are exponentiated \(n\) times in... | {
"timestamp": "2021-09-01T02:13:58",
"yymm": "2105",
"arxiv_id": "2105.00247",
"language": "en",
"url": "https://arxiv.org/abs/2105.00247",
"abstract": "The continuous tetrational function ${^x}r=\\tau(r,x)$, the unique solution of equation $\\tau(r,x)=r^{\\tau(r,x-1)}$ and its differential equation $\\tau... |
https://arxiv.org/abs/1812.00610 | Weak discrete maximum principle and $L^\infty$ analysis of the DG method for the Poisson equation on a polygonal domain | We derive several $L^\infty$ error estimates for the symmetric interior penalty (SIP) discontinuous Galerkin (DG) method applied to the Poisson equation in a two-dimensional polygonal domain. Both local and global estimates are examined. The weak maximum principle (WMP) for the discrete harmonic function is also establ... | \section{Introduction}
The discontinuous Galerkin (DG) method, which was proposed originally by Reed and Hill \cite{osti_4491151} in 1973, is a powerful method for solving numerically a wide range of partial differential equations (PDEs). We use a discontinuous function which is a polynomial on each element and introd... | {
"timestamp": "2018-12-04T02:25:23",
"yymm": "1812",
"arxiv_id": "1812.00610",
"language": "en",
"url": "https://arxiv.org/abs/1812.00610",
"abstract": "We derive several $L^\\infty$ error estimates for the symmetric interior penalty (SIP) discontinuous Galerkin (DG) method applied to the Poisson equation ... |
https://arxiv.org/abs/1306.5872 | Geometric properties of inverse polynomial images | Given a polynomial $\T_n$ of degree $n$, consider the inverse image of $\R$ and $[-1,1]$, denoted by $\T_n^{-1}(\R)$ and $\T_n^{-1}([-1,1])$, respectively. It is well known that $\T_n^{-1}(\R)$ consists of $n$ analytic Jordan arcs moving from $\infty$ to $\infty$. In this paper, we give a necessary and sufficient condi... | \section{Introduction}
Let $\PP_n$ be the set of all polynomials of degree $n$ with complex coefficients. For a polynomial $\T_n\in\PP_n$, consider the inverse images $\T_n^{-1}(\R)$ and $\T_n^{-1}([-1,1])$, defined by
\begin{equation}
\T_n^{-1}(\R):=\bigl\{z\in\C:\T_n(z)\in\R\bigr\}
\end{equation}
and
\begin... | {
"timestamp": "2013-06-26T02:01:40",
"yymm": "1306",
"arxiv_id": "1306.5872",
"language": "en",
"url": "https://arxiv.org/abs/1306.5872",
"abstract": "Given a polynomial $\\T_n$ of degree $n$, consider the inverse image of $\\R$ and $[-1,1]$, denoted by $\\T_n^{-1}(\\R)$ and $\\T_n^{-1}([-1,1])$, respectiv... |
https://arxiv.org/abs/1203.5829 | Ensemble estimators for multivariate entropy estimation | The problem of estimation of density functionals like entropy and mutual information has received much attention in the statistics and information theory communities. A large class of estimators of functionals of the probability density suffer from the curse of dimensionality, wherein the mean squared error (MSE) decay... | \section{Introduction}
Non-linear functionals of probability densities $f$ of the form $G(f) = \int g(f(x),x) f(x) dx$ arise in applications of information theory, machine learning, signal processing and statistical estimation. Important examples of such functionals include Shannon $g(f,x)=-\log(f)$ and R\'enyi $g(f,x... | {
"timestamp": "2013-03-05T02:01:58",
"yymm": "1203",
"arxiv_id": "1203.5829",
"language": "en",
"url": "https://arxiv.org/abs/1203.5829",
"abstract": "The problem of estimation of density functionals like entropy and mutual information has received much attention in the statistics and information theory co... |
https://arxiv.org/abs/1302.7252 | Stability results for some fully nonlinear eigenvalue estimates | In this paper, we give some stability estimates for the Faber-Krahn inequality relative to the eigenvalues of Hessian operators | \section{Introduction}
In this paper we prove some stability estimates for the eigenvalue
$\lambda_k(\Omega)$ of the $k$-Hessian operator, that has the
variational characterization
\begin{equation*}
\lambda_k(\Omega) = \min\left\{\int_\Omega (-u)
S_k(D^2u)\,dx,\; u\in \Phi_k^2(\Omega) \text{ and }
\int_\Omega (... | {
"timestamp": "2013-03-27T01:01:43",
"yymm": "1302",
"arxiv_id": "1302.7252",
"language": "en",
"url": "https://arxiv.org/abs/1302.7252",
"abstract": "In this paper, we give some stability estimates for the Faber-Krahn inequality relative to the eigenvalues of Hessian operators",
"subjects": "Analysis of... |
https://arxiv.org/abs/1709.07290 | Comparing the Switch and Curveball Markov Chains for Sampling Binary Matrices with Fixed Marginals | The Curveball algorithm is a variation on well-known switch-based Markov chain approaches for uniformly sampling binary matrices with fixed row and column sums. Instead of a switch, the Curveball algorithm performs a so-called binomial trade in every iteration of the algorithm. Intuitively, this could lead to a better ... | \section{Introduction}
The problem of uniformly sampling binary matrices with fixed row and column sums (marginals) has received a lot of attention, see, e.g., \cite{Rao1996,Kannan1999,Erdos2013,Erdos2015,Erdos2016}.
Equivalent formulations for this problem are the uniform sampling of undirected bipartite graphs, or th... | {
"timestamp": "2017-10-19T02:07:27",
"yymm": "1709",
"arxiv_id": "1709.07290",
"language": "en",
"url": "https://arxiv.org/abs/1709.07290",
"abstract": "The Curveball algorithm is a variation on well-known switch-based Markov chain approaches for uniformly sampling binary matrices with fixed row and column... |
https://arxiv.org/abs/math/9805084 | Coloring Distance Graphs on the Integers | Given a set D of positive integers, the associated distance graph on the integers is the graph with the integers as vertices and an edge between distinct vertices if their difference lies in D. We investigate the chromatic numbers of distance graphs. We show that, if $D = {d_1,d_2,d_3,...}$, with $d_n | d_{n+1}$ for al... | \section{Introduction} \label{S:intro}
What is the least number of classes into which the integers
can be partitioned, so that no two members of the same class
differ by a square? What if ``square'' is replaced by ``factorial''?
Questions like these can be formulated as graph coloring problems.
Given a set $D$ of pos... | {
"timestamp": "1998-05-19T22:08:10",
"yymm": "9805",
"arxiv_id": "math/9805084",
"language": "en",
"url": "https://arxiv.org/abs/math/9805084",
"abstract": "Given a set D of positive integers, the associated distance graph on the integers is the graph with the integers as vertices and an edge between disti... |
https://arxiv.org/abs/1810.02281 | A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks | We analyze speed of convergence to global optimum for gradient descent training a deep linear neural network (parameterized as $x \mapsto W_N W_{N-1} \cdots W_1 x$) by minimizing the $\ell_2$ loss over whitened data. Convergence at a linear rate is guaranteed when the following hold: (i) dimensions of hidden layers are... |
\section{Approximate Balancedness and Deficiency Margin Under Customary Initialization} \label{app:balance_margin_init}
Two assumptions concerning initialization facilitate our main convergence result (Theorem~\ref{theorem:converge}):
\emph{(i)}~the initial weights $W_1(0),\ldots,W_N(0)$ are approximately balanced (s... | {
"timestamp": "2019-10-29T01:06:05",
"yymm": "1810",
"arxiv_id": "1810.02281",
"language": "en",
"url": "https://arxiv.org/abs/1810.02281",
"abstract": "We analyze speed of convergence to global optimum for gradient descent training a deep linear neural network (parameterized as $x \\mapsto W_N W_{N-1} \\c... |
https://arxiv.org/abs/1610.07539 | Origami Constructions of Rings of Integers of Imaginary Quadratic Fields | In the making of origami, one starts with a piece of paper, and through a series of folds along seed points one constructs complicated three-dimensional shapes. Mathematically, one can think of the complex numbers as representing the piece of paper, and the seed points and folds as a way to generate a subset of the com... | \section{Introduction}
In origami, the artist uses intersections of folds as reference points to make new folds. This kind of construction can be extended to points on the complex plane. In \cite{Buhler2010}, the authors define one such mathematical construction. In this construction, one can think of the complex plan... | {
"timestamp": "2016-10-25T02:12:43",
"yymm": "1610",
"arxiv_id": "1610.07539",
"language": "en",
"url": "https://arxiv.org/abs/1610.07539",
"abstract": "In the making of origami, one starts with a piece of paper, and through a series of folds along seed points one constructs complicated three-dimensional s... |
https://arxiv.org/abs/1106.5444 | An Extension of Young's Inequality | Young's inequality is extended to the context of absolutely continuous measures. Several applications are included. | \section{Introduction}
Young's inequality \cite{Y1912} asserts that every strictly increasing
continuous function $f:\left[ 0,\infty\right) \longrightarrow\left[
0,\infty\right) $ with $f\left( 0\right) =0$ and $\underset{x\rightarrow
\infty}{\lim}f\left( x\right) =\infty$ verifies an inequality of the
fo... | {
"timestamp": "2011-06-28T02:06:47",
"yymm": "1106",
"arxiv_id": "1106.5444",
"language": "en",
"url": "https://arxiv.org/abs/1106.5444",
"abstract": "Young's inequality is extended to the context of absolutely continuous measures. Several applications are included.",
"subjects": "Classical Analysis and ... |
https://arxiv.org/abs/2010.00315 | Exact hyperplane covers for subsets of the hypercube | Alon and Füredi (1993) showed that the number of hyperplanes required to cover $\{0,1\}^n\setminus \{0\}$ without covering $0$ is $n$. We initiate the study of such exact hyperplane covers of the hypercube for other subsets of the hypercube. In particular, we provide exact solutions for covering $\{0,1\}^n$ while missi... | \section{Introduction}
A vector $v\in \mathbb{R}^n$ and a scalar $\alpha\in \mathbb{R}$ determine the hyperplane
\[
\{x\in \mathbb{R}^n:\langle v,x\rangle \coloneqq v_1x_1+\dots+v_nx_n=\alpha\}
\]
in $\mathbb{R}^n$. How many hyperplanes are needed to cover $\{0,1\}^n$? Only two are required; for instance, $\{x:x_1=0\}... | {
"timestamp": "2021-07-02T02:25:57",
"yymm": "2010",
"arxiv_id": "2010.00315",
"language": "en",
"url": "https://arxiv.org/abs/2010.00315",
"abstract": "Alon and Füredi (1993) showed that the number of hyperplanes required to cover $\\{0,1\\}^n\\setminus \\{0\\}$ without covering $0$ is $n$. We initiate th... |
https://arxiv.org/abs/2301.11638 | One-dimensional integral Rellich type inequalities | The motive of this note is twofold. Inspired by the recent development of a new kind of Hardy inequality, here we discuss the corresponding Hardy-Rellich and Rellich inequality versions in the integral form. The obtained sharp Hardy-Rellich type inequality improves the previously known result. Meanwhile, the establishe... | \section{Introduction}
In the celebrated paper \cite{hardy}, Godfrey H. Hardy first stated the famous inequality which reads as: let $1<p<\infty$ and $f$ be a $p$-integrable function on $(0, \infty)$, which vanishes at zero, then the function $r\longmapsto \frac{1}{r}\int_{0}^{r} f(t) \:{\rm d}t$ is $p$-integrable over... | {
"timestamp": "2023-01-30T02:10:02",
"yymm": "2301",
"arxiv_id": "2301.11638",
"language": "en",
"url": "https://arxiv.org/abs/2301.11638",
"abstract": "The motive of this note is twofold. Inspired by the recent development of a new kind of Hardy inequality, here we discuss the corresponding Hardy-Rellich ... |
https://arxiv.org/abs/1602.03445 | Davenport constant for commutative rings | The Davenport constant is one measure for how "large" a finite abelian group is. In particular, the Davenport constant of an abelian group is the smallest $k$ such that any sequence of length $k$ is reducible. This definition extends naturally to commutative semigroups, and has been studied in certain finite commutativ... | \section{Introduction}
The Davenport constant is an important concept in additive number theory. In
particular, it measures the largest zero-free sequence of an abelian group.
The Davenport constant was introduced by Davenport in 1966 \cite{Davenport}, but
was actually studied prior to that in 1963 by Rogers \cite{Ro... | {
"timestamp": "2016-02-11T02:10:05",
"yymm": "1602",
"arxiv_id": "1602.03445",
"language": "en",
"url": "https://arxiv.org/abs/1602.03445",
"abstract": "The Davenport constant is one measure for how \"large\" a finite abelian group is. In particular, the Davenport constant of an abelian group is the smalle... |
https://arxiv.org/abs/2010.05840 | Cohomology fractals, Cannon-Thurston maps, and the geodesic flow | Cohomology fractals are images naturally associated to cohomology classes in hyperbolic three-manifolds. We generate these images for cusped, incomplete, and closed hyperbolic three-manifolds in real-time by ray-tracing to a fixed visual radius. We discovered cohomology fractals while attempting to illustrate Cannon-Th... | \section{Introduction}
\begin{figure}[htb]
\centering
\subfloat[Cannon--Thurston map.]{
\label{Fig:CTVector}
\includegraphics[width=0.47\textwidth]{Figures/match_up/two_colour2_rot/two_colour_Cannon-Thurston_match_up_vector_1000}
}
\thinspace
\subfloat[Cohomology fractal.]{
\label{Fig:CTPixelColour}
\includegraphics[w... | {
"timestamp": "2020-10-13T02:40:56",
"yymm": "2010",
"arxiv_id": "2010.05840",
"language": "en",
"url": "https://arxiv.org/abs/2010.05840",
"abstract": "Cohomology fractals are images naturally associated to cohomology classes in hyperbolic three-manifolds. We generate these images for cusped, incomplete, ... |
https://arxiv.org/abs/2007.01985 | Limits of almost homogeneous spaces and their fundamental groups | We say that a sequence of proper geodesic spaces $X_i$ consists of \textit{almost homogeneous spaces} if there is a sequence of discrete groups of isometries $G_i \leq Iso(X_i)$ such that diam$(X_i/G_i)\to 0$ as $i \to \infty$.We show that if a sequence $X_i$ of almost homogeneous spaces converges in the pointed Gromov... | \section{Introduction}
We say that a sequence of proper geodesic spaces $X_i$ consists of \textit{almost homogeneous spaces} if there is a sequence of discrete groups of isometries $G_i \leq Iso(X_i)$ such that diam$(X_i/G_i)\to 0$ as $i \to \infty$.
\begin{remark}
\rm A sequence of homogeneous spaces $X_i$ does not... | {
"timestamp": "2021-12-01T02:13:58",
"yymm": "2007",
"arxiv_id": "2007.01985",
"language": "en",
"url": "https://arxiv.org/abs/2007.01985",
"abstract": "We say that a sequence of proper geodesic spaces $X_i$ consists of \\textit{almost homogeneous spaces} if there is a sequence of discrete groups of isomet... |
https://arxiv.org/abs/1311.6176 | Inverse questions for the large sieve | Suppose that an infinite set $A$ occupies at most $\frac{1}{2}(p+1)$ residue classes modulo $p$, for every sufficiently large prime $p$. The squares, or more generally the integer values of any quadratic, are an example of such a set. By the large sieve inequality the number of elements of $A$ that are at most $X$ is $... | \subsection[#1]{\sc #1}}
\newcommand\E{\mathbb{E}}
\newcommand\Z{\mathbb{Z}}
\newcommand\R{\mathbb{R}}
\newcommand\T{\mathbb{T}}
\newcommand\C{\mathbb{C}}
\newcommand\N{\mathbb{N}}
\newcommand\A{\mathscr{A}}
\newcommand\B{\mathscr{B}}
\newcommand\G{\mathbf{G}}
\newcommand\SL{\operatorname{SL}}
\newcommand\Upp{\operator... | {
"timestamp": "2013-11-26T02:10:27",
"yymm": "1311",
"arxiv_id": "1311.6176",
"language": "en",
"url": "https://arxiv.org/abs/1311.6176",
"abstract": "Suppose that an infinite set $A$ occupies at most $\\frac{1}{2}(p+1)$ residue classes modulo $p$, for every sufficiently large prime $p$. The squares, or mo... |
https://arxiv.org/abs/2106.10388 | Upper bounds for critical probabilities in Bernoulli Percolation models | We consider bond and site Bernoulli Percolation in both the oriented and the non-oriented cases on $\mathbb{Z}^d$ and obtain rigorous upper bounds for the critical points in those models for every dimension $d \geq 3$. | \section{Introduction}
The study of Bernoulli percolation on $\mathbb{Z}^d$ is more than 60 years old and the existence of a non-trivial phase transition for $d\geq 2$ is well established for the model and several of its variants, but the exact value of the critical parameter $p_c$ is seldom known. A celebrated r... | {
"timestamp": "2021-06-22T02:04:44",
"yymm": "2106",
"arxiv_id": "2106.10388",
"language": "en",
"url": "https://arxiv.org/abs/2106.10388",
"abstract": "We consider bond and site Bernoulli Percolation in both the oriented and the non-oriented cases on $\\mathbb{Z}^d$ and obtain rigorous upper bounds for th... |
https://arxiv.org/abs/2103.11443 | Bipartite biregular Moore graphs | A bipartite graph $G=(V,E)$ with $V=V_1\cup V_2$ is biregular if all the vertices of a stable set $V_i$ have the same degree $r_i$ for $i=1,2$. In this paper, we give an improved new Moore bound for an infinite family of such graphs with odd diameter. This problem was introduced in 1983 by Yebra, Fiol, and Fàbrega.\\ B... | \section{Introduction}
The {\emph{degree/diameter problem}} for graphs consists in finding the largest order of a graph with prescribed degree and diameter. We call this number the \emph{Moore bound}, and a graph whose order coincides with this bound is called a {\emph{Moore graph}}.
There is a lot of work related ... | {
"timestamp": "2021-03-23T01:21:11",
"yymm": "2103",
"arxiv_id": "2103.11443",
"language": "en",
"url": "https://arxiv.org/abs/2103.11443",
"abstract": "A bipartite graph $G=(V,E)$ with $V=V_1\\cup V_2$ is biregular if all the vertices of a stable set $V_i$ have the same degree $r_i$ for $i=1,2$. In this p... |
https://arxiv.org/abs/2002.01597 | Berge cycles in non-uniform hypergraphs | We consider two extremal problems for set systems without long Berge cycles. First we give Dirac-type minimum degree conditions that force long Berge cycles. Next we give an upper bound for the number of hyperedges in a hypergraph with bounded circumference. Both results are best possible in infinitely many cases. | \section{Introduction}
\subsection{Classical results on longest cycles in graphs}
The {\em circumference} $c(G)$ of a graph $G$ is the length of its longest cycle.
In particular, if a graph has a cycle $C$ which covers all of its vertices, $V(C)=V(G)$, we say it is {\em hamiltonian}.
A classical result of Dirac st... | {
"timestamp": "2020-02-06T02:04:47",
"yymm": "2002",
"arxiv_id": "2002.01597",
"language": "en",
"url": "https://arxiv.org/abs/2002.01597",
"abstract": "We consider two extremal problems for set systems without long Berge cycles. First we give Dirac-type minimum degree conditions that force long Berge cycl... |
https://arxiv.org/abs/1206.4740 | Leinartas's partial fraction decomposition | These notes describe Leinartas's algorithm for multivariate partial fraction decompositions and employ an implementation thereof in Sage. | \section{Introduction}
In \cite{Lein1978}, Le{\u\i}nartas\ gave an algorithm for decomposing multivariate rational expressions into partial fractions.
In these notes I re-present Le{\u\i}nartas's\ algorithm, because it is not well-known, because its English translation \cite{Lein1978} is difficult to find, and becau... | {
"timestamp": "2012-06-26T02:07:37",
"yymm": "1206",
"arxiv_id": "1206.4740",
"language": "en",
"url": "https://arxiv.org/abs/1206.4740",
"abstract": "These notes describe Leinartas's algorithm for multivariate partial fraction decompositions and employ an implementation thereof in Sage.",
"subjects": "C... |
https://arxiv.org/abs/1001.2949 | A Property of the Frobenius Map of a Polynomial Ring | Let R be a ring of polynomials in a finite number of variables over a perfect field k of characteristic p>0 and let F:R\to R be the Frobenius map of R, i.e. F(r)=r^p. We explicitly describe an R-module isomorphism Hom_R(F_*(M),N)\cong Hom_R(M,F^*(N)) for all R-modules M and N. Some recent and potential applications are... | \section{Introduction}
The main result of this paper is Theorem \ref{main} which is a type of adjointness property for the Frobenius map of a polynomial ring over a perfect field. The interest in this fairly elementary result comes from its striking recent applications (see \cite[Sections 5 and 6]{WZ} and \cite{YZ}) an... | {
"timestamp": "2010-01-18T04:36:20",
"yymm": "1001",
"arxiv_id": "1001.2949",
"language": "en",
"url": "https://arxiv.org/abs/1001.2949",
"abstract": "Let R be a ring of polynomials in a finite number of variables over a perfect field k of characteristic p>0 and let F:R\\to R be the Frobenius map of R, i.e... |
https://arxiv.org/abs/1706.06630 | Improved upper bounds in the moving sofa problem | The moving sofa problem, posed by L. Moser in 1966, asks for the planar shape of maximal area that can move around a right-angled corner in a hallway of unit width. It is known that a maximal area shape exists, and that its area is at least 2.2195... - the area of an explicit construction found by Gerver in 1992 - and ... | \section{Introduction}
The \textbf{moving sofa problem} is a well-known unsolved problem in geometry, first posed by Leo Moser in 1966 \cite{unsolved-problems, moser}. It asks:
\begin{quote}
\textit{What is the planar shape of maximal area that can be moved around a right-angled corner in a hallway of unit width?}
\... | {
"timestamp": "2018-01-08T02:15:09",
"yymm": "1706",
"arxiv_id": "1706.06630",
"language": "en",
"url": "https://arxiv.org/abs/1706.06630",
"abstract": "The moving sofa problem, posed by L. Moser in 1966, asks for the planar shape of maximal area that can move around a right-angled corner in a hallway of u... |
https://arxiv.org/abs/math/0606320 | Remarks on the Cayley Representation of Orthogonal Matrices and on Perturbing the Diagonal of a Matrix to Make it Invertible | This note contains two remarks. The first remark concerns the extension of the well-known Cayley representation of rotation matrices by skew symmetric matrices to rotation matrices admitting -1 as an eigenvalue and then to all orthogonal matrices. We review a method due to Hermann Weyl and another method involving mult... |
\section{The Cayley Representation of Orthogonal Matrices}
\label{sec1}
Given any rotation matrix, $R\in \mathbf{SO}(n)$,
if $R$ does not admit $-1$ as an eigenvalue, then
there is a unique skew symmetric matrix, $S$,
($\transpos{S} = -S$) so that
\[
R = (I - S)(I + S)^{-1}.
\]
This is a classical result of Cayley \... | {
"timestamp": "2006-06-13T23:47:20",
"yymm": "0606",
"arxiv_id": "math/0606320",
"language": "en",
"url": "https://arxiv.org/abs/math/0606320",
"abstract": "This note contains two remarks. The first remark concerns the extension of the well-known Cayley representation of rotation matrices by skew symmetric... |
https://arxiv.org/abs/0911.4204 | Maximal independent sets and separating covers | In 1973, Katona raised the problem of determining the maximum number of subsets in a separating cover on n elements. The answer to Katona's question turns out to be the inverse to the answer to a much simpler question: what is the largest integer which is the product of positive integers with sum n? We give a combinato... | \section{Introduction}
We begin with a simply stated problem, which has made numerous appearances in mathematics competitions:%
\footnote{In particular, the 1976 IMO asked for the $n=1976$ case, the 1979 Putnam asked for the $n=1979$ case, and on April 23rd 2002, the 3rd Community College of Philadelphia Colonial Math... | {
"timestamp": "2010-08-30T02:00:22",
"yymm": "0911",
"arxiv_id": "0911.4204",
"language": "en",
"url": "https://arxiv.org/abs/0911.4204",
"abstract": "In 1973, Katona raised the problem of determining the maximum number of subsets in a separating cover on n elements. The answer to Katona's question turns o... |
https://arxiv.org/abs/2211.06484 | A regular $n$-gon spiral | We construct a polygonal spiral by arranging a sequence of regular $n$-gons such that each $n$-gon shares a specified side and vertex with the $(n+1)$-gon in the construction. By offering flexibility for determining the size of each $n$-gon in the spiral, we show that a number of different analytical and asymptotic beh... | \section{Introduction}
Spirals are pervasive in fluid motion, biological structures, and engineering, affording numerous applications of mathematical spirals in modeling real-world phenomena \cite[ch.~11-14]{Thompson} \cite{Davis, Hammer}. Polygonal spirals are loosely defined as spirals that can be constructed using ... | {
"timestamp": "2022-11-15T02:01:39",
"yymm": "2211",
"arxiv_id": "2211.06484",
"language": "en",
"url": "https://arxiv.org/abs/2211.06484",
"abstract": "We construct a polygonal spiral by arranging a sequence of regular $n$-gons such that each $n$-gon shares a specified side and vertex with the $(n+1)$-gon... |
https://arxiv.org/abs/1305.5394 | A note on the Waring ranks of reducible cubic forms | Let $W_3(n)$ be the set of Waring ranks of reducible cubic forms in $n+1$ variables. We prove that $W_3(n)\subseteq \lbrace 1,..., 2n+1\rbrace$. | \section{Introduction}
\indent
Let $K$ be an algebraically closed field of characteristic zero, let $V$ be a $(n+1)$-dimensional $K$-vector space and $F\in S^d V$, namely a homogeneous polynomial of degree $d$ in $n+1$ indeterminates. The {\bf Waring problem for polynomials} asks for the least value $s$ such that the... | {
"timestamp": "2014-12-18T02:12:34",
"yymm": "1305",
"arxiv_id": "1305.5394",
"language": "en",
"url": "https://arxiv.org/abs/1305.5394",
"abstract": "Let $W_3(n)$ be the set of Waring ranks of reducible cubic forms in $n+1$ variables. We prove that $W_3(n)\\subseteq \\lbrace 1,..., 2n+1\\rbrace$.",
"sub... |
https://arxiv.org/abs/1809.05769 | Differentiation Matrices for Univariate Polynomials | We collect here elementary properties of differentiation matrices for univariate polynomials expressed in various bases, including orthogonal polynomial bases and non-degree-graded bases such as Bernstein bases and Lagrange \& Hermite interpolational bases. | \section{Introduction}
The transformation of the (possibly infinite) vector of coefficients $\mathbf{a}={\{a_k\}}_{k\geq0}$ in the expansion
\begin{equation}
f(x)=\sum_{k\geq0} a_k\phi_k(x)
\end{equation}
\noindent to the vector of coefficients $\mathbf{b} = \{b_k\}_{k\geq0}$ in the expansion
\begin{equation}
f'(x)... | {
"timestamp": "2018-09-18T02:07:36",
"yymm": "1809",
"arxiv_id": "1809.05769",
"language": "en",
"url": "https://arxiv.org/abs/1809.05769",
"abstract": "We collect here elementary properties of differentiation matrices for univariate polynomials expressed in various bases, including orthogonal polynomial b... |
https://arxiv.org/abs/1704.03913 | Higher-order clustering in networks | A fundamental property of complex networks is the tendency for edges to cluster. The extent of the clustering is typically quantified by the clustering coefficient, which is the probability that a length-2 path is closed, i.e., induces a triangle in the network. However, higher-order cliques beyond triangles are crucia... |
\section{Derivation of higher-order clustering coefficients}
In this section, we derive our higher-order clustering coefficients and some of their
basic properties.
We first present an alternative interpretation of the classical clustering coefficient
and then show how this novel interpretation seamlessly general... | {
"timestamp": "2018-01-08T02:13:21",
"yymm": "1704",
"arxiv_id": "1704.03913",
"language": "en",
"url": "https://arxiv.org/abs/1704.03913",
"abstract": "A fundamental property of complex networks is the tendency for edges to cluster. The extent of the clustering is typically quantified by the clustering co... |
https://arxiv.org/abs/1006.4176 | Unknotting Unknots | A knot is an an embedding of a circle into three-dimensional space. We say that a knot is unknotted if there is an ambient isotopy of the embedding to a standard circle. By representing knots via planar diagrams, we discuss the problem of unknotting a knot diagram when we know that it is unknotted. This problem is surp... | \section{Introduction}
When one first delves into the theory of knots, one learns that knots are typically studied using their diagrams. The first question that arises when considering these knot diagrams is: how can we tell if two knot diagrams represent the same knot? Fortunately, we have a partial answer to this que... | {
"timestamp": "2011-11-08T02:00:36",
"yymm": "1006",
"arxiv_id": "1006.4176",
"language": "en",
"url": "https://arxiv.org/abs/1006.4176",
"abstract": "A knot is an an embedding of a circle into three-dimensional space. We say that a knot is unknotted if there is an ambient isotopy of the embedding to a sta... |
https://arxiv.org/abs/math/0601638 | Upper bounds for edge-antipodal and subequilateral polytopes | A polytope in a finite-dimensional normed space is subequilateral if the length in the norm of each of its edges equals its diameter. Subequilateral polytopes occur in the study of two unrelated subjects: surface energy minimizing cones and edge-antipodal polytopes. We show that the number of vertices of a subequilater... | \section{Notation}
Denote the $d$-dimensional real linear space by $\numbersystem{R}^d$, a norm on $\numbersystem{R}^d$ by $\norm{\cdot}$,
its unit ball by $B$, and the ball with centre $x$ and radius $r$ by $B(x,r)$.
Denote the diameter of a set $C\subseteq \numbersystem{R}^d$ by $\diam(C)$, and (if it is measurable) ... | {
"timestamp": "2006-01-26T13:51:21",
"yymm": "0601",
"arxiv_id": "math/0601638",
"language": "en",
"url": "https://arxiv.org/abs/math/0601638",
"abstract": "A polytope in a finite-dimensional normed space is subequilateral if the length in the norm of each of its edges equals its diameter. Subequilateral p... |
https://arxiv.org/abs/0912.5031 | On two and three periodic Lyness difference equations | We describe the sequences {x_n}_n given by the non-autonomous second order Lyness difference equations x_{n+2}=(a_n+x_{n+1})/x_n, where {a_n}_n is either a 2-periodic or a 3-periodic sequence of positive values and the initial conditions x_1,x_2 are as well positive. We also show an interesting phenomenon of the discre... | \section{Introduction and main result}
This paper fully describes the sequences given by the non-autonomous
second order Lyness difference equations
\begin{equation}\label{eq}
x_{n+2}\,=\,\frac{a_n+x_{n+1}}{x_n},
\end{equation}
where $\{a_n\}_n$ is a $k$-periodic sequence taking positive values,
$k=2,3,$ and the initi... | {
"timestamp": "2009-12-26T18:31:35",
"yymm": "0912",
"arxiv_id": "0912.5031",
"language": "en",
"url": "https://arxiv.org/abs/0912.5031",
"abstract": "We describe the sequences {x_n}_n given by the non-autonomous second order Lyness difference equations x_{n+2}=(a_n+x_{n+1})/x_n, where {a_n}_n is either a ... |
https://arxiv.org/abs/1107.0326 | The Monty Hall Problem in the Game Theory Class | The basic Monty Hall problem is explored to introduce into the fundamental concepts of the game theory and to give a complete Bayesian and a (noncooperative) game-theoretic analysis of the situation. Simple combinatorial arguments are used to exclude the holding action and to find minimax solutions. | \section{Introduction}
\begin{itemize}
\item[]
{\it Suppose you're on a game show, and you're given the choice of three doors:
Behind one door is a car; behind the others, goats.
You pick a door, say No. 1, and the host, who knows what's behind the doors,
opens another door, say No. 3, which has a goat. He... | {
"timestamp": "2011-07-05T02:00:12",
"yymm": "1107",
"arxiv_id": "1107.0326",
"language": "en",
"url": "https://arxiv.org/abs/1107.0326",
"abstract": "The basic Monty Hall problem is explored to introduce into the fundamental concepts of the game theory and to give a complete Bayesian and a (noncooperative... |
https://arxiv.org/abs/math/0503745 | Pseudo-random graphs | Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake, they serve as essential instruments in proving an enormous number of combinatorial statements, making their role quite h... | \section{Introduction}
Random graphs have proven to be one of the most important and fruitful
concepts in modern Combinatorics and Theoretical Computer Science.
Besides being a fascinating study subject for their own sake, they
serve as essential instruments in proving an enormous number of
combinatorial statements, m... | {
"timestamp": "2005-03-31T18:59:47",
"yymm": "0503",
"arxiv_id": "math/0503745",
"language": "en",
"url": "https://arxiv.org/abs/math/0503745",
"abstract": "Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides bein... |
https://arxiv.org/abs/1306.2385 | Linear groups - Malcev's theorem and Selberg's lemma | An account of two fundamental facts concerning finitely generated linear groups: Malcev's theorem on residual finiteness, and Selberg's lemma on virtual torsion-freeness. | \section*{Introduction}
A group is \textbf{linear} if it is (isomorphic to) a subgroup of $\mathrm{GL}_n(K)$, where $K$ is a field. If we want to specify the field, we say that the group is linear over $K$. The following theorems are fundamental, at least from the perspective of combinatorial group theory.
\begin{thm... | {
"timestamp": "2013-06-12T02:00:49",
"yymm": "1306",
"arxiv_id": "1306.2385",
"language": "en",
"url": "https://arxiv.org/abs/1306.2385",
"abstract": "An account of two fundamental facts concerning finitely generated linear groups: Malcev's theorem on residual finiteness, and Selberg's lemma on virtual tor... |
https://arxiv.org/abs/2005.14125 | Notes on ridge functions and neural networks | These notes are about ridge functions. Recent years have witnessed a flurry of interest in these functions. Ridge functions appear in various fields and under various guises. They appear in fields as diverse as partial differential equations (where they are called plane waves), computerized tomography and statistics. T... | \chapter*{Preface}
These notes are about \textit{ridge functions}. Recent years have
witnessed a flurry of interest in these functions. Ridge functions
appear in various fields and under various guises. They appear in fields as diverse as
partial differential equations (where they are called \textit{plane waves}),
com... | {
"timestamp": "2020-09-01T02:06:22",
"yymm": "2005",
"arxiv_id": "2005.14125",
"language": "en",
"url": "https://arxiv.org/abs/2005.14125",
"abstract": "These notes are about ridge functions. Recent years have witnessed a flurry of interest in these functions. Ridge functions appear in various fields and u... |
https://arxiv.org/abs/1805.12119 | A combinatorial characterization of finite groups of prime exponent | The power graph of a group $G$ is a simple and undirected graph with vertex set $G$ and two distinct vertices are adjacent if one is a power of the other. In this article, we characterize (non-cyclic) finite groups of prime exponent and finite elementary abelian $2$-groups (of rank at least $2$) in terms of their power... | \section{Introduction}
Let $G$ be a group. The \emph{power graph} $\mathcal{G}(G)$ of $G$ is a simple and undirected graph whose vertex set is $G$ and distinct vertices $u$ and $v$ are adjacent if $v=u^n$ for some $n \in \mathbb{N}$ or $u=v^m$ for some $m \in \mathbb{N}$. The notion of (directed) power graph of a grou... | {
"timestamp": "2019-03-20T01:26:12",
"yymm": "1805",
"arxiv_id": "1805.12119",
"language": "en",
"url": "https://arxiv.org/abs/1805.12119",
"abstract": "The power graph of a group $G$ is a simple and undirected graph with vertex set $G$ and two distinct vertices are adjacent if one is a power of the other.... |
https://arxiv.org/abs/1911.08067 | Sets in $\mathbb{R}^d$ determining $k$ taxicab distances | We address an analog of a problem introduced by Erdős and Fishburn, itself an inverse formulation of the famous Erdős distance problem, in which the usual Euclidean distance is replaced with the metric induced by the $\ell^1$-norm, commonly referred to as the $\textit{taxicab metric}$. Specifically, we investigate the ... | \section{Introduction}
In 1946, Erd\H{o}s \cite{Erdos} asked a now famous question: given $n\in \mathbb{N}$, what is the minimum number of distinct distances determined by $n$ points in a plane? Denoting this minimum by $f(n)$, he proved via an elementary counting argument that $f(n)=\Omega(\sqrt{n})$, and he conjectu... | {
"timestamp": "2020-05-26T02:08:51",
"yymm": "1911",
"arxiv_id": "1911.08067",
"language": "en",
"url": "https://arxiv.org/abs/1911.08067",
"abstract": "We address an analog of a problem introduced by Erdős and Fishburn, itself an inverse formulation of the famous Erdős distance problem, in which the usual... |
https://arxiv.org/abs/1901.06759 | Submultiplicativity of the numerical radius of commuting matrices of order two | Denote by $w(T)$ the numerical radius of a matrix $T$. An elementary proof is given to the fact that $w(AB) \leq w(A)w(B)$ for a pair of commuting matrices of order two, and characterization is given for the matrix pairs that attain the quality. | \section{Introduction}
Let $M_n$ be the set of $n\times n$ matrices. The numerical range and numerical radius
of $A \in M_n$ are defined by
$$W(A) = \{x^*Ax: x \in {\mathbb C}^n, x^*x = 1\} \qquad \hbox{ and } \qquad
w(A) = \max\{|\mu|: \mu \in W(A)\},$$
respectively.
The numerical range and numerical radius are usefu... | {
"timestamp": "2019-03-01T02:02:57",
"yymm": "1901",
"arxiv_id": "1901.06759",
"language": "en",
"url": "https://arxiv.org/abs/1901.06759",
"abstract": "Denote by $w(T)$ the numerical radius of a matrix $T$. An elementary proof is given to the fact that $w(AB) \\leq w(A)w(B)$ for a pair of commuting matric... |
https://arxiv.org/abs/2010.01539 | A discussion on the approximate solutions of first order systems of non-linear ordinary equations | We develop a one step matrix method in order to obtain approximate solutions of first order systems and non-linear ordinary differential equations, reducible to first order systems. We find a sequence of such solutions that converge to the exact solution. We study the precision, in terms of the local error, of the meth... | \section{Introduction}
This paper pretends to be a contribution to methods to find the approximate solutions of nonlinear first order equations (or systems) with given initial values of the form
\begin{equation}\label{1}
{\mathbf y}'(t) = {\mathbf f}({\mathbf y}(t))\,,\qquad {\mathbf y}(t_0)={\mathbf y}_0\,,
\end{equ... | {
"timestamp": "2021-03-12T02:07:33",
"yymm": "2010",
"arxiv_id": "2010.01539",
"language": "en",
"url": "https://arxiv.org/abs/2010.01539",
"abstract": "We develop a one step matrix method in order to obtain approximate solutions of first order systems and non-linear ordinary differential equations, reduci... |
https://arxiv.org/abs/1701.07963 | Negative (and Positive) Circles in Signed Graphs: A Problem Collection | A signed graph is a graph whose edges are labelled positive or negative. The sign of a circle (cycle, circuit) is the product of the signs of its edges. Most of the essential properties of a signed graph depend on the signs of its circles. Here I describe several questions regarding negative circles and their cousins t... | \section*{Introduction}\label{intro}
A signed graph is a graph with a \emph{signature} that assigns to each edge a positive or negative sign. To me the most important thing about a signed graph is the signs of its circles,\footnote{A circle is a connected, 2-regular graph. The common name ``cycle'' has too many othe... | {
"timestamp": "2018-01-17T02:01:47",
"yymm": "1701",
"arxiv_id": "1701.07963",
"language": "en",
"url": "https://arxiv.org/abs/1701.07963",
"abstract": "A signed graph is a graph whose edges are labelled positive or negative. The sign of a circle (cycle, circuit) is the product of the signs of its edges. M... |
https://arxiv.org/abs/1612.07904 | On Garland's vanishing theorem for $\mathrm{SL}_n$ | This is an expository paper on Garland's vanishing theorem specialized to the case when the linear algebraic group is $\mathrm{SL}_n$. Garland's theorem can be stated as a vanishing of the cohomology groups of certain finite simplicial complexes. The method of the proof is quite interesting on its own. It relates the v... | \section{Introduction}
\subsection{Statement of the theorem}
This is an expository paper on Howard Garland's work \cite{Garland} specialized to the case when the
linear algebraic group is $\mathrm{SL}_n$. Reading \cite{Garland} requires
knowledge of the theory of buildings. On the other hand, the
ideas in \cite{Garlan... | {
"timestamp": "2016-12-26T02:02:27",
"yymm": "1612",
"arxiv_id": "1612.07904",
"language": "en",
"url": "https://arxiv.org/abs/1612.07904",
"abstract": "This is an expository paper on Garland's vanishing theorem specialized to the case when the linear algebraic group is $\\mathrm{SL}_n$. Garland's theorem ... |
https://arxiv.org/abs/1903.10668 | On the Weakly Prime-Additive Numbers with Length 4 | In 1992, Erd$ő$s and Hegyv$á$ri showed that for any prime p, there exist infinitely many length 3 weakly prime-additive numbers divisible by p. In 2018, Fang and Chen showed that for any positive integer m, there exists infinitely many length 3 weakly prime-additive numbers divisible by m if and only if 8 does not divi... | \section{Introduction}
A number $n$ with at least 2 distinct prime divisors is called \textit{prime-additive} if $n=\sum_{p|n}p^{a_p}$ for some $a_p>0$. If additionally $p^{a_p}<n\leq p^{a_p+1}$ for all $p|n$, then $n$ is called \textit{strongly prime-additive}. In 1992, Erd\H{o}s and Hegyv\'{a}ri \cite{Erdos} stated ... | {
"timestamp": "2019-03-27T01:08:12",
"yymm": "1903",
"arxiv_id": "1903.10668",
"language": "en",
"url": "https://arxiv.org/abs/1903.10668",
"abstract": "In 1992, Erd$ő$s and Hegyv$á$ri showed that for any prime p, there exist infinitely many length 3 weakly prime-additive numbers divisible by p. In 2018, F... |
https://arxiv.org/abs/1906.02781 | Tutte Polynomial Activities | Unlike Whitney's definition of the corank-nullity generating function $T(G;x+1,y+1)$, Tutte's definition of his now eponymous polynomial $T(G;x,y)$ requires a total order on the edges of which the polynomial is a posteriori independent. Tutte presented his definition in terms of internal and external activities of maxi... | \chapter*{}
\chapterauthor{Spencer Backman}{Einstein Institute for Mathematics\\
Edmond J. Safra Campus\\
The Hebrew University of Jerusalem\\
Givat Ram. Jerusalem, 9190401, Israel\\
\texttt{spencer.backman@mail.huji.ac.il}\\}
\chapter{Tutte Polynomial Activities}
\section{Synopsis}
Activities are certai... | {
"timestamp": "2019-06-11T02:27:43",
"yymm": "1906",
"arxiv_id": "1906.02781",
"language": "en",
"url": "https://arxiv.org/abs/1906.02781",
"abstract": "Unlike Whitney's definition of the corank-nullity generating function $T(G;x+1,y+1)$, Tutte's definition of his now eponymous polynomial $T(G;x,y)$ requir... |
https://arxiv.org/abs/0706.4112 | Induced Ramsey-type theorems | We present a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earlier results of Rodl, Erdos-Hajnal, Promel-Rodl, Nikiforov, Chung-Graham, and Luczak-Rodl. The proofs are based on a simple lemma (generalizing one by Graham, Rodl, a... | \section{Background and Introduction}
Ramsey theory refers to a large body of deep results in mathematics
concerning partitions of large structures. Its underlying philosophy
is captured succinctly by the statement that ``In a large system,
complete disorder is impossible.'' This is an area in which a great
variety of... | {
"timestamp": "2007-12-27T22:18:10",
"yymm": "0706",
"arxiv_id": "0706.4112",
"language": "en",
"url": "https://arxiv.org/abs/0706.4112",
"abstract": "We present a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earli... |
https://arxiv.org/abs/2109.03749 | Lifting methods in mass partition problems | Many results in mass partitions are proved by lifting $\mathbb{R}^d$ to a higher-dimensional space and dividing the higher-dimensional space into pieces. We extend such methods to use lifting arguments to polyhedral surfaces. Among other results, we prove the existence of equipartitions of $d+1$ measures in $\mathbb{R}... | \section{Introduction}
In a standard mass partition problem, we are given measures or finite families of points in a Euclidean space and we seek to partition the ambient space into regions that meet certain conditions. Some conditions determine how we split the measures and the sets of points. For instance, in an e... | {
"timestamp": "2021-09-09T02:24:48",
"yymm": "2109",
"arxiv_id": "2109.03749",
"language": "en",
"url": "https://arxiv.org/abs/2109.03749",
"abstract": "Many results in mass partitions are proved by lifting $\\mathbb{R}^d$ to a higher-dimensional space and dividing the higher-dimensional space into pieces.... |
https://arxiv.org/abs/1812.10850 | Decomposition of Gaussian processes, and factorization of positive definite kernels | We establish a duality for two factorization questions, one for general positive definite (p.d) kernels $K$, and the other for Gaussian processes, say $V$. The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization for p.d. kernels is intuitively motivated by matrix factoriza... | \section{Introduction}
We give an integrated approach to positive definite (p.d.) kernels
and Gaussian processes, with an emphasis on factorizations, and their
applications. Positive definite kernels serve as powerful tools in
such diverse areas as Fourier analysis, probability theory, stochastic
processes, boundary t... | {
"timestamp": "2018-12-31T02:13:18",
"yymm": "1812",
"arxiv_id": "1812.10850",
"language": "en",
"url": "https://arxiv.org/abs/1812.10850",
"abstract": "We establish a duality for two factorization questions, one for general positive definite (p.d) kernels $K$, and the other for Gaussian processes, say $V$... |
https://arxiv.org/abs/2007.09958 | Monodromy of general hypersurfaces | Let $X$ be a general complex projective hypersurface in $\mathbb{P}^{n+1}$ of degree $d>1$. A point $P$ not in $X$ is called uniform if the monodromy group of the projection of $X$ from $P$ is isomorphic to the symmetric group. We prove that all the points in $\mathbb{P}^{n+1}$ are uniform for $X$, generalizing a resul... | \section{Introduction}
The monodromy group of linear projections of irreducible complex projective varieties has been intensively studied. Fixed an irreducible and reduced projective hypersurface $X \subset \mathbb{P}^{n+1}$, consider its linear projections from a point $P \in \mathbb{P}^{n+1}$. We want to look at thos... | {
"timestamp": "2020-07-21T02:29:47",
"yymm": "2007",
"arxiv_id": "2007.09958",
"language": "en",
"url": "https://arxiv.org/abs/2007.09958",
"abstract": "Let $X$ be a general complex projective hypersurface in $\\mathbb{P}^{n+1}$ of degree $d>1$. A point $P$ not in $X$ is called uniform if the monodromy gro... |
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