url stringlengths 31 38 | title stringlengths 7 229 | abstract stringlengths 44 2.87k | text stringlengths 319 2.51M | meta dict |
|---|---|---|---|---|
https://arxiv.org/abs/1907.03669 | The Weyl formula for planar annuli | We study the zeros of cross-product of Bessel functions and obtain their approximations, based on which we reduce the eigenvalue counting problem for the Dirichlet Laplacian associated with a planar annulus to a lattice point counting problem associated with a special domain in $\mathbb{R}^2$. Unlike other lattice poin... | \section{Introduction}
Let $D \subset \mathbb R^2$ be a bounded domain with piecewise smooth boundary, and let
\[0< \mu^2_1 < \mu^2_2 \le \mu_3^2 \le \cdots \]
be the eigenvalues (counting multiplicity) of the Dirichlet Laplacian associated with $D$. In his seminal work \cite{weyl11:1912}, H. Weyl initiated the stud... | {
"timestamp": "2019-07-09T02:26:47",
"yymm": "1907",
"arxiv_id": "1907.03669",
"language": "en",
"url": "https://arxiv.org/abs/1907.03669",
"abstract": "We study the zeros of cross-product of Bessel functions and obtain their approximations, based on which we reduce the eigenvalue counting problem for the ... |
https://arxiv.org/abs/1701.02420 | Polynomially Interpolated Legendre Multiplier Sequences | We prove that every multiplier sequence for the Legendre basis which can be interpolated by a polynomial has the form $\{h(k^2+k)\}_{k=0}^{\infty}$, where $h\in\mathbb{R}[x]$. We also prove that a non-trivial collection of polynomials of a certain form interpolate multiplier sequences for the Legendre basis, and we sta... | \section{Introduction}\label{s:Introduction}
Over the past decade, there has been an effort to characterize multiplier sequences acting on various orthogonal bases for $\mathbb{R}[x]$ (see \cite{bates}, \cite{bdfu}, \cite{bc}, \cite{bo}, \cite{nreup}, \cite{fhms}, \cite{FP}, \cite{P}). The present work focuses on the L... | {
"timestamp": "2017-01-11T02:02:37",
"yymm": "1701",
"arxiv_id": "1701.02420",
"language": "en",
"url": "https://arxiv.org/abs/1701.02420",
"abstract": "We prove that every multiplier sequence for the Legendre basis which can be interpolated by a polynomial has the form $\\{h(k^2+k)\\}_{k=0}^{\\infty}$, wh... |
https://arxiv.org/abs/1806.02028 | Determining the Generalized Hamming Weight Hierarchy of the Binary Projective Reed-Muller Code | Projective Reed-Muller codes correspond to subcodes of the Reed-Muller code in which the polynomials being evaluated to yield codewords, are restricted to be homogeneous. The Generalized Hamming Weights (GHW) of a code ${\cal C}$, identify for each dimension $\nu$, the smallest size of the support of a subcode of ${\ca... | \section{Introduction}
The notion of Generalized Hamming Weights (GHW), introduced by Wei in \cite{Wei}, is a generalization of minimum Hamming weight of a linear code. In \cite{Wei} , the basic properties of GHW are studied and the weight hierarchy for Hamming code, Reed-Solomon codes, binary Reed-Muller code etc ... | {
"timestamp": "2018-06-07T02:08:03",
"yymm": "1806",
"arxiv_id": "1806.02028",
"language": "en",
"url": "https://arxiv.org/abs/1806.02028",
"abstract": "Projective Reed-Muller codes correspond to subcodes of the Reed-Muller code in which the polynomials being evaluated to yield codewords, are restricted to... |
https://arxiv.org/abs/1509.07385 | Provable approximation properties for deep neural networks | We discuss approximation of functions using deep neural nets. Given a function $f$ on a $d$-dimensional manifold $\Gamma \subset \mathbb{R}^m$, we construct a sparsely-connected depth-4 neural network and bound its error in approximating $f$. The size of the network depends on dimension and curvature of the manifold $\... | \section{Introduction} \label{sec:intro}
In the last decade, deep learning algorithms achieved unprecedented success and state-of-the-art results in various machine learning and artificial intelligence tasks, most notably image recognition, speech recognition, text analysis and Natural Language Processing~\cite{lecun20... | {
"timestamp": "2016-03-29T02:14:31",
"yymm": "1509",
"arxiv_id": "1509.07385",
"language": "en",
"url": "https://arxiv.org/abs/1509.07385",
"abstract": "We discuss approximation of functions using deep neural nets. Given a function $f$ on a $d$-dimensional manifold $\\Gamma \\subset \\mathbb{R}^m$, we cons... |
https://arxiv.org/abs/1701.00665 | A note on split extensions of bialgebras | We prove a universal characterization of Hopf algebras among cocommutative bialgebras over a field: a cocommutative bialgebra is a Hopf algebra precisely when every split extension over it admits a join decomposition. We also explain why this result cannot be extended to a non-cocommutative setting. | \section{Introduction}
An elementary result in the theory of modules says that in any short exact sequence
\[
\xymatrix{0 \ar[r] & K \ar[r]^-{k} & X \ar@<.5ex>[r]^-{f} & Y \ar@{->}@<.5ex>[l]^-{s} \ar[r]& 0}
\qquad\qquad
f\circ s=1_{Y}
\]
where the cokernel $f$ admits a section $s$, the middle object $X$ decomposes as ... | {
"timestamp": "2017-03-14T01:11:46",
"yymm": "1701",
"arxiv_id": "1701.00665",
"language": "en",
"url": "https://arxiv.org/abs/1701.00665",
"abstract": "We prove a universal characterization of Hopf algebras among cocommutative bialgebras over a field: a cocommutative bialgebra is a Hopf algebra precisely ... |
https://arxiv.org/abs/2012.02335 | Tight Chang's-lemma-type bounds for Boolean functions | Chang's lemma (Duke Mathematical Journal, 2002) is a classical result with applications across several areas in mathematics and computer science. For a Boolean function $f$ that takes values in {-1,1} let $r(f)$ denote its Fourier rank. For each positive threshold $t$, Chang's lemma provides a lower bound on $wt(f):=\P... | \section{Introduction}
Chang's lemma \cite{Chang02, Green04} is a classical result in additive combinatorics. Informally, the lemma states that all the large Fourier coefficients of the indicator function of a large subset of an Abelian group reside in a low dimensional subspace. The discovery of this lemma was mot... | {
"timestamp": "2021-05-25T02:13:15",
"yymm": "2012",
"arxiv_id": "2012.02335",
"language": "en",
"url": "https://arxiv.org/abs/2012.02335",
"abstract": "Chang's lemma (Duke Mathematical Journal, 2002) is a classical result with applications across several areas in mathematics and computer science. For a Bo... |
https://arxiv.org/abs/2009.06746 | Orbital stability of KdV multisolitons in $H^{-1}$ | We prove that multisoliton solutions of the Korteweg--de Vries equation are orbitally stable in $H^{-1}(\mathbb{R})$. We introduce a variational characterization of multisolitons that remains meaningful at such low regularity and show that all optimizing sequences converge to the manifold of multisolitons. The proximit... | \section{Introduction}
The history of the Korteweg--de Vries equation
\begin{align}\label{KdV}\tag{KdV}
\frac{d\ }{dt} q = - q''' + 6qq'
\end{align}
is profoundly intertwined with the notion of solitary waves. Indeed, the very goal of Korteweg and de~Vries \cite{KdV1895} was to explain the empirical observation of ... | {
"timestamp": "2020-09-16T02:04:03",
"yymm": "2009",
"arxiv_id": "2009.06746",
"language": "en",
"url": "https://arxiv.org/abs/2009.06746",
"abstract": "We prove that multisoliton solutions of the Korteweg--de Vries equation are orbitally stable in $H^{-1}(\\mathbb{R})$. We introduce a variational characte... |
https://arxiv.org/abs/1611.08850 | Every 4-regular 4-uniform hypergraph has a 2-coloring with a free vertex | In this paper, we continue the study of $2$-colorings in hypergraphs. A hypergraph is $2$-colorable if there is a $2$-coloring of the vertices with no monochromatic hyperedge. It is known (see Thomassen [J. Amer. Math. Soc. 5 (1992), 217--229]) that every $4$-uniform $4$-regular hypergraph is $2$-colorable. Our main re... | \section{Introduction}
In this paper, we continue the study of $2$-colorings in hypergraphs. We adopt the notation and terminology from~\cite{HeYe13,HeYe15}. A \emph{hypergraph} $H = (V,E)$ is a finite set
$V = V(H)$ of elements, called \emph{vertices}, together with a finite multiset $E = E(H)$ of arbitrary subs... | {
"timestamp": "2016-11-29T02:06:02",
"yymm": "1611",
"arxiv_id": "1611.08850",
"language": "en",
"url": "https://arxiv.org/abs/1611.08850",
"abstract": "In this paper, we continue the study of $2$-colorings in hypergraphs. A hypergraph is $2$-colorable if there is a $2$-coloring of the vertices with no mon... |
https://arxiv.org/abs/1110.2235 | On distance, geodesic and arc transitivity of graphs | We compare three transitivity properties of finite graphs, namely, for a positive integer $s$, $s$-distance transitivity, $s$-geodesic transitivity and $s$-arc transitivity. It is known that if a finite graph is $s$-arc transitive but not $(s+1)$-arc transitive then $s\leq 7$ and $s\neq 6$. We show that there are infin... | \section{Introduction}
The study of finite distance-transitive graphs goes back to
Higman's paper \cite{DGH-1} in which ``groups of maximal diameter"
were introduced. These are permutation groups which act distance
transitively on some graph. The family of distance transitive graphs
includes many interesting a... | {
"timestamp": "2011-10-12T02:01:39",
"yymm": "1110",
"arxiv_id": "1110.2235",
"language": "en",
"url": "https://arxiv.org/abs/1110.2235",
"abstract": "We compare three transitivity properties of finite graphs, namely, for a positive integer $s$, $s$-distance transitivity, $s$-geodesic transitivity and $s$-... |
https://arxiv.org/abs/1801.05471 | Minimum saturated families of sets | We call a family $\mathcal{F}$ of subsets of $[n]$ $s$-saturated if it contains no $s$ pairwise disjoint sets, and moreover no set can be added to $\mathcal{F}$ while preserving this property (here $[n] = \{1,\ldots,n\}$).More than 40 years ago, Erdős and Kleitman conjectured that an $s$-saturated family of subsets of ... | \section{Introduction}
\begingroup
\renewcommand{\thefootnote}{}
\footnotetext{
MSC2010 Classification: ~
Primary: 05D05 ~ Secondary: 60C05
}
\endgroup
In extremal set theory, one studies how large, or how small, a family
$\mathcal{F}$ can be, if $\mathcal{F}$ consists of su... | {
"timestamp": "2018-04-26T02:02:56",
"yymm": "1801",
"arxiv_id": "1801.05471",
"language": "en",
"url": "https://arxiv.org/abs/1801.05471",
"abstract": "We call a family $\\mathcal{F}$ of subsets of $[n]$ $s$-saturated if it contains no $s$ pairwise disjoint sets, and moreover no set can be added to $\\mat... |
https://arxiv.org/abs/1001.4582 | More Colourful Simplices | We show that any point in the convex hull of each of (d+1) sets of (d+1) points in general position in \R^d is contained in at least (d+1)^2/2 simplices with one vertex from each set. This improves the known lower bounds for all d >= 4. | \section{Introduction}\label{se:intro}
A point $p \in R^d$ has {\it simplicial depth} $k$
relative to a set $S$ if it is contained in $k$ closed simplices
generated by $(d+1)$ sets of $S$.
This was introduced by Liu \cite{Liu90} as a statistical
measure of how representative $p$ is of $S$, and is
a source of ... | {
"timestamp": "2010-06-01T02:01:49",
"yymm": "1001",
"arxiv_id": "1001.4582",
"language": "en",
"url": "https://arxiv.org/abs/1001.4582",
"abstract": "We show that any point in the convex hull of each of (d+1) sets of (d+1) points in general position in \\R^d is contained in at least (d+1)^2/2 simplices wi... |
https://arxiv.org/abs/1607.08511 | Differential geometry of rectifying submanifolds | A space curve in a Euclidean 3-space $\mathbb E^3$ is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf 110} (2003), no. 2, 147-152]. In this present article, we introduce and study the n... | \section{Introduction}
Let $\mathbb E^3$ denote Euclidean 3-space with its inner product $\left<\;\,,\;\right>$. Consider a unit-speed space curve $x : I\to \mathbb E^3$, where $I=(\alpha,\beta)$ is a real interval.
Let ${\bf x}$ denote the position vector field of $x$ and ${\bf x}'$ be denoted by {\bf t}.
It ... | {
"timestamp": "2016-07-29T02:09:15",
"yymm": "1607",
"arxiv_id": "1607.08511",
"language": "en",
"url": "https://arxiv.org/abs/1607.08511",
"abstract": "A space curve in a Euclidean 3-space $\\mathbb E^3$ is called a rectifying curve if its position vector field always lies in its rectifying plane. This no... |
https://arxiv.org/abs/1207.0566 | Any order superconvergence finite volume schemes for 1D general elliptic equations | We present and analyze a finite volume scheme of arbitrary order for elliptic equations in the one-dimensional setting. In this scheme, the control volumes are constructed by using the Gauss points in subintervals of the underlying mesh. We provide a unified proof for the inf-sup condition, and show that our finite vol... | \section{Introduction}
\setcounter{equation}{0}
The {\it finite volume method} (FVM) attracted a lot of attentions during the past
several decades, we refer to \cite{Bank.R;Rose.D1987,Barth.T;Ohlberger2004, Cai.Z1991, Cai.Z_Park.M2003, ChenWuXu2011, Ewing.R;Lin.T;Lin.Y2002, EymardGallouetHerbin2000, Li.R2000, Olliv... | {
"timestamp": "2012-07-04T02:01:42",
"yymm": "1207",
"arxiv_id": "1207.0566",
"language": "en",
"url": "https://arxiv.org/abs/1207.0566",
"abstract": "We present and analyze a finite volume scheme of arbitrary order for elliptic equations in the one-dimensional setting. In this scheme, the control volumes ... |
https://arxiv.org/abs/2104.06694 | Line graph characterization of power graphs of finite nilpotent groups | This paper deals with the classification of groups $G$ such that power graphs and proper power graphs of $G$ are line graphs. In fact, we classify all finite nilpotent groups whose power graphs are line graphs. Also, we categorize all finite nilpotent groups (except non-abelian $2$-groups) whose proper power graphs are... | \section{Introduction}
\label{sec:intro}
The investigation of graph representations is one of the interesting and popular research topic in algebraic graph theory, as graphs like these enrich both algebra and graph theory. Moreover, they have important applications (see, for example,
\cite{surveypwrgraphkac1, cayleygr... | {
"timestamp": "2021-04-15T02:12:56",
"yymm": "2104",
"arxiv_id": "2104.06694",
"language": "en",
"url": "https://arxiv.org/abs/2104.06694",
"abstract": "This paper deals with the classification of groups $G$ such that power graphs and proper power graphs of $G$ are line graphs. In fact, we classify all fin... |
https://arxiv.org/abs/quant-ph/0508101 | Qdensity - a Mathematica Quantum Computer Simulation | This Mathematica 5.2 package~\footnote{QDENSITY is available atthis http URL} is a simulation of a Quantum Computer. The program provides a modular, instructive approach for generating the basic elements that make up a quantum circuit. The main emphasis is on using the density matrix, although an approach using state v... | \section{INTRODUCTION}
There is already a rich Quantum Computing (QC) literature~\cite{Nielsen} which
holds forth the promise of using quantum interference and superposition
to solve otherwise intractable problems. The field has reached
the point that experimental realizations are of paramount importance
and theoretic... | {
"timestamp": "2006-03-10T19:44:56",
"yymm": "0508",
"arxiv_id": "quant-ph/0508101",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0508101",
"abstract": "This Mathematica 5.2 package~\\footnote{QDENSITY is available atthis http URL} is a simulation of a Quantum Computer. The program provides a m... |
https://arxiv.org/abs/1702.03051 | Density Functional Estimators with k-Nearest Neighbor Bandwidths | Estimating expected polynomials of density functions from samples is a basic problem with numerous applications in statistics and information theory. Although kernel density estimators are widely used in practice for such functional estimation problems, practitioners are left on their own to choose an appropriate bandw... |
\section{Introduction}
\input{introduction}
\section{Kernel Density Estimator with $k$-NN Bandwidth}
\label{sec:kde}
\input{section2}
\section{Local Likelihood Density Estimator with $k$-NN Bandwidth}
\label{sec:klnn}
\input{section3}
\section{Simulations}
\label{sec:simul}
\input{section4}
\section{Discussion... | {
"timestamp": "2017-02-13T02:02:32",
"yymm": "1702",
"arxiv_id": "1702.03051",
"language": "en",
"url": "https://arxiv.org/abs/1702.03051",
"abstract": "Estimating expected polynomials of density functions from samples is a basic problem with numerous applications in statistics and information theory. Alth... |
https://arxiv.org/abs/1506.06172 | Stepwise Methods in Optimal Control Problems | We introduce a new method, stepwise method for solving optimal con- trol problems. Our first motivation for new approach emanate from limi- tations on continuous time control functions in PMP. Practically in most of the real world models, we are not able to change control value for every time such as in drug dose calcu... | \section{Introduction}
Optimal control theory is an effective tool in real world modelling such as physical, biological, economical and other models. Diverse examples are studied in\cite{sethi} and \cite{lenhart} . Optimal control theory is used in chemotherapy of cancer\cite{fisterpan}. Several papers are studied abo... | {
"timestamp": "2015-06-26T02:07:02",
"yymm": "1506",
"arxiv_id": "1506.06172",
"language": "en",
"url": "https://arxiv.org/abs/1506.06172",
"abstract": "We introduce a new method, stepwise method for solving optimal con- trol problems. Our first motivation for new approach emanate from limi- tations on con... |
https://arxiv.org/abs/1902.00092 | Image reconstruction enhancement via masked regularization | Image reconstruction based on an edge-sparsity assumption has become popular in recent years. Many methods of this type are capable of reconstructing nearly perfect edge-sparse images using limited data. In this paper, we present a method to improve the accuracy of a suboptimal image resulting from an edge-sparsity ima... | \section{Introduction}
\label{sec:intro}
A goal in the imaging science community is to be able to reconstruct images from a small amount of data. Compressed sensing algorithms, e.g. \cite{candes2006robust}, use edge-sparsity based reconstruction methods to accomplish this task. Theoretical exact reconstruction guarante... | {
"timestamp": "2019-02-04T02:03:22",
"yymm": "1902",
"arxiv_id": "1902.00092",
"language": "en",
"url": "https://arxiv.org/abs/1902.00092",
"abstract": "Image reconstruction based on an edge-sparsity assumption has become popular in recent years. Many methods of this type are capable of reconstructing near... |
https://arxiv.org/abs/2007.14002 | Equilibrium Behaviors in Repeated Games | We examine a patient player's behavior when he can build reputations in front of a sequence of myopic opponents. With positive probability, the patient player is a commitment type who plays his Stackelberg action in every period. We characterize the patient player's action frequencies in equilibrium. Our results clarif... |
\section{Introduction}\label{sec1}
Economists have long recognized that individuals, firms, and governments can benefit from good reputations. As shown in the seminal work of \citet{FL-89}, a patient player can guarantee himself a high payoff when his opponents believe that he might be committed to play a particular a... | {
"timestamp": "2021-02-11T02:28:08",
"yymm": "2007",
"arxiv_id": "2007.14002",
"language": "en",
"url": "https://arxiv.org/abs/2007.14002",
"abstract": "We examine a patient player's behavior when he can build reputations in front of a sequence of myopic opponents. With positive probability, the patient pl... |
https://arxiv.org/abs/1512.04988 | Large deviations for random projections of $\ell^p$ balls | Let $p\in[1,\infty]$. Consider the projection of a uniform random vector from a suitably normalized $\ell^p$ ball in $\mathbb{R}^n$ onto an independent random vector from the unit sphere. We show that sequences of such random projections, when suitably normalized, satisfy a large deviation principle (LDP) as the dimens... | \section{Introduction}\label{sec-intro}
Consider the projection of a random $n$-dimensional vector $X^{(n)}$ onto some lower dimensional subspace. Our broad goal is to understand and analyze distributional properties of the projections of high-dimensional random vectors (i.e., large $n$), given certain natural assump... | {
"timestamp": "2015-12-17T02:01:49",
"yymm": "1512",
"arxiv_id": "1512.04988",
"language": "en",
"url": "https://arxiv.org/abs/1512.04988",
"abstract": "Let $p\\in[1,\\infty]$. Consider the projection of a uniform random vector from a suitably normalized $\\ell^p$ ball in $\\mathbb{R}^n$ onto an independen... |
https://arxiv.org/abs/1501.06165 | The multiplicity of eigenvalues of the Hodge Laplacian on 5-dimensional compact manifolds | We study multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics. We prove that generically the Hodge Laplacian, restricted to the subspace of co-exact two-forms, has nonzero eigenvalues of multiplicity two. The proof is based on t... | \section{Statement of the problem and results}\label{sec:introduction}
\setcounter{equation}{0}
The multiplicity of the $L^2$-eigenvalues of the Laplacian $\Delta_g \geq 0$ on a smooth compact manifold $(M,g)$ is linked with the symmetry of the manifold. Generally speaking, the multiplicity of an eigenvalue is redu... | {
"timestamp": "2015-01-27T02:11:32",
"yymm": "1501",
"arxiv_id": "1501.06165",
"language": "en",
"url": "https://arxiv.org/abs/1501.06165",
"abstract": "We study multiplicity of the eigenvalues of the Hodge Laplacian on smooth, compact Riemannian manifolds of dimension five for generic families of metrics.... |
https://arxiv.org/abs/2203.13960 | A Relation of the Allen-Cahn equations and the Euler equations and applications of the Equipartition | We will prove that solutions of the Allen-Cahn equations that satisfy the equipartition can be transformed into solutions of the Euler equations with constant pressure. As a consequence, we obtain De Giorgi type results, that is, the level sets of entire solutions are hyperplanes. In addition, we obtain some examples o... | \section{Introduction}
As it is well known, De Giorgi in 1978 \cite{DeGi} suggested a stricking analogy of the Allen Cahn equation ($ \Delta u =f(u) $) with minimal surface theory that led to significant developments in Partial Differential equations and the Calculus of Variations, by stating the following conjecture... | {
"timestamp": "2022-03-29T02:08:03",
"yymm": "2203",
"arxiv_id": "2203.13960",
"language": "en",
"url": "https://arxiv.org/abs/2203.13960",
"abstract": "We will prove that solutions of the Allen-Cahn equations that satisfy the equipartition can be transformed into solutions of the Euler equations with cons... |
https://arxiv.org/abs/2109.00609 | On a Partition Identity of Lehmer | Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck conjectured and Andrews proved the following companion to Euler's identity: the excess of the number of parts in all partitions of n into odd parts over the number o... | \section{Introduction and statement of results}\label{sec_intro}
Many results in the theory of partitions concern identities asserting that the set $\mathcal P_{X}(n)$ of partitions of $n$ satisfying condition $X$ and the set $\mathcal P_{Y}(n)$ of partitions of $n$ satisfying condition $Y$ are equinumerous. Likely ... | {
"timestamp": "2021-09-03T02:03:22",
"yymm": "2109",
"arxiv_id": "2109.00609",
"language": "en",
"url": "https://arxiv.org/abs/2109.00609",
"abstract": "Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck... |
https://arxiv.org/abs/1311.0267 | Sectional curvature for Riemannian manifolds with density | In this paper we introduce two new notions of sectional curvature for Riemannian manifolds with density. Under both notions of curvature we classify the constant curvature manifolds. We also prove generalizations of the theorems of Cartan-Hadamard, Synge, and Bonnet-Myers as well as a generalization of the (non-smooth)... | \section{Introduction}
In this paper we are interested in the geometry of a Riemannian manifold $(M,g)$ with a smooth positive density function, $e^{-f}$. A theory of Ricci curvature for these spaces goes back to Lichnerowicz \cite{Lich1, Lich2} and was later developed by Bakry-Emery \cite{BE} and many others. It ha... | {
"timestamp": "2015-01-27T02:08:59",
"yymm": "1311",
"arxiv_id": "1311.0267",
"language": "en",
"url": "https://arxiv.org/abs/1311.0267",
"abstract": "In this paper we introduce two new notions of sectional curvature for Riemannian manifolds with density. Under both notions of curvature we classify the con... |
https://arxiv.org/abs/0810.1425 | Hodge polynomials and birational types of moduli spaces of coherent systems on elliptic curves | In this paper we consider moduli spaces of coherent systems on an elliptic curve. We compute their Hodge polynomials and determine their birational types in some cases. Moreover we prove that certain moduli spaces of coherent systems are isomorphic. This last result uses the Fourier-Mukai transform of coherent systems ... | \section{Introduction}
\noindent
A {\it coherent system of type $(n,d,k)$} on a smooth projective curve $C$ over an algebraically closed field is by
definition a pair $(E,V)$ consisting of
a vector bundle $E$ of rank $n$ and degree $d$ over $C$ and a vector subspace $V \subset H^0(E)$ of dimension $k$.
For any real ... | {
"timestamp": "2009-04-29T09:03:52",
"yymm": "0810",
"arxiv_id": "0810.1425",
"language": "en",
"url": "https://arxiv.org/abs/0810.1425",
"abstract": "In this paper we consider moduli spaces of coherent systems on an elliptic curve. We compute their Hodge polynomials and determine their birational types in... |
https://arxiv.org/abs/1211.7264 | Macaulay-like marked bases | We define marked sets and bases over a quasi-stable ideal $\mathfrak j$ in a polynomial ring on a Noetherian $K$-algebra, with $K$ a field of any characteristic. The involved polynomials may be non-homogeneous, but their degree is bounded from above by the maximum among the degrees of the terms in the Pommaret basis of... | \section{Introduction}
In this paper,
we are interested in computing the family of ideals $\mathfrak i \subset R=K[x_1,\ldots,x_n]$ whose quotients $R/\mathfrak i$ share the same affine Hilbert polynomial and the same monomial $K$-vector basis, that we choose to be the sous-escalier $\mathcal N(\mathfrak j)$ of a str... | {
"timestamp": "2012-12-03T02:02:16",
"yymm": "1211",
"arxiv_id": "1211.7264",
"language": "en",
"url": "https://arxiv.org/abs/1211.7264",
"abstract": "We define marked sets and bases over a quasi-stable ideal $\\mathfrak j$ in a polynomial ring on a Noetherian $K$-algebra, with $K$ a field of any character... |
https://arxiv.org/abs/1910.05726 | Norm attaining operators which satisfy a Bollobás type theorem | In this paper, we are interested in studying the set $\mathcal{A}_{\|\cdot\|}(X, Y)$ of all norm-attaining operators $T$ from $X$ into $Y$ satisfying the following: given $\epsilon>0$, there exists $\eta$ such that if $\|Tx\| > 1 - \eta$, then there is $x_0$ such that $\| x_0 - x\| < \epsilon$ and $T$ itself attains it... | \section{Introduction and Motivation}
The famous theorem due to Bollob\'as on functionals which attain their norms states that if $x^*$ is a norm one functional which almost attains its norm at some element $x$, in the sense that $x^*(x) > 1 - \eta$ for some $\eta > 0$, then there exist a new functional $x_0^*$ and a ... | {
"timestamp": "2020-09-01T02:30:56",
"yymm": "1910",
"arxiv_id": "1910.05726",
"language": "en",
"url": "https://arxiv.org/abs/1910.05726",
"abstract": "In this paper, we are interested in studying the set $\\mathcal{A}_{\\|\\cdot\\|}(X, Y)$ of all norm-attaining operators $T$ from $X$ into $Y$ satisfying ... |
https://arxiv.org/abs/1609.06433 | Counting Finite Index Subrings of $\mathbb{Z}^n$ | We count subrings of small index of $\mathbb{Z}^n$, where the addition and multiplication are defined componentwise. Let $f_n(k)$ denote the number of subrings of index $k$. For any $n$, we give a formula for this quantity for all integers $k$ that are not divisible by a 9th power of a prime, extending a result of Liu. | \section{Introduction}\label{sec1}
The main goal of this paper is to study the number of subrings of $\mathbb{Z}^n$ of given index. We begin by reviewing an easier problem, counting subgroups of $\mathbb{Z}^n$ of given index.
\subsection{Counting Subgroups of $\mathbb{Z}^n$}\label{sec_counting_subgroups}
The \emph{... | {
"timestamp": "2018-02-13T02:14:19",
"yymm": "1609",
"arxiv_id": "1609.06433",
"language": "en",
"url": "https://arxiv.org/abs/1609.06433",
"abstract": "We count subrings of small index of $\\mathbb{Z}^n$, where the addition and multiplication are defined componentwise. Let $f_n(k)$ denote the number of su... |
https://arxiv.org/abs/1511.01359 | Measures of irrationality for hypersurfaces of large degree | We study various measures of irrationality for hypersurfaces of large degree in projective space and other varieties. These include the least degree of a rational covering of projective space, and the minimal gonality of a covering family of curves. The theme is that positivity properties of canonical bundles lead to l... | \section*{Introduction}
The purpose of this paper is to study various measures of irrationality for hypersurfaces of large degree in projective space and other varieties. The theme is that positivity properties of canonical bundles lead to lower bounds for these invariants. In particular, we prove the conjecture... | {
"timestamp": "2015-11-05T02:11:37",
"yymm": "1511",
"arxiv_id": "1511.01359",
"language": "en",
"url": "https://arxiv.org/abs/1511.01359",
"abstract": "We study various measures of irrationality for hypersurfaces of large degree in projective space and other varieties. These include the least degree of a ... |
https://arxiv.org/abs/1912.05775 | On the Locating Chromatic Number of Trees | Some coloring algorithms gives an upper bound for the locating chromatic number of trees with all the vertices not in an end-path colored by only two colors. That means, a better coloring algorithm could be achieved by optimizing the number of colors used in the end-paths. We provide an estimation of the locating chrom... | \section{Introduction}
Let $G=(V,E)$ be a simple connected graph. For any $u \in V$ and $S \subseteq V$, the distance from vertex $u$ to $S$ is defined by $d(u,S)=\min\{d(u,v) \mid v\in S\}$. A set of vertices $S$ {\em resolves} two vertices $u$ and $v$ if $d(u,S)\ne d(v,S)$. Let $c:V\to\{1,2,\cdots, k\}$ be a $k$-colo... | {
"timestamp": "2020-11-18T02:17:47",
"yymm": "1912",
"arxiv_id": "1912.05775",
"language": "en",
"url": "https://arxiv.org/abs/1912.05775",
"abstract": "Some coloring algorithms gives an upper bound for the locating chromatic number of trees with all the vertices not in an end-path colored by only two colo... |
https://arxiv.org/abs/2005.09205 | Distance matrices of subsets of the Hamming cube | Graham and Winkler derived a formula for the determinant of the distance matrix of a full-dimensional set of $n + 1$ points $\{ x_{0}, x_{1}, \ldots , x_{n} \}$ in the Hamming cube $H_{n} = ( \{ 0,1 \}^{n}, \ell_{1} )$. In this article we derive a formula for the determinant of the distance matrix $D$ of an arbitrary s... | \section{Introduction: Distance and Gram matrices}\label{Sec 1}
The global geometry of a finite metric space $(\{x_{0}, x_{1}, \ldots , x_{m} \}, d)$
is completely encoded within its \textit{distance matrix} $D = (d(x_{i}, x_{j}))_{i,j = 0}^{m}$.
The distance matrices we focus on in this article correspond to metric su... | {
"timestamp": "2020-08-03T02:07:43",
"yymm": "2005",
"arxiv_id": "2005.09205",
"language": "en",
"url": "https://arxiv.org/abs/2005.09205",
"abstract": "Graham and Winkler derived a formula for the determinant of the distance matrix of a full-dimensional set of $n + 1$ points $\\{ x_{0}, x_{1}, \\ldots , x... |
https://arxiv.org/abs/1004.3321 | On the Sandpile group of the cone of a graph | In this article, we give a partial description of the sandpile group of the cone of the cartesian product of graphs in function of the sandpile group of the cone of their factors. Also, we introduce the concept of uniform homomorphism of graphs and prove that every surjective uniform homomorphism of graphs induces an i... | \section{Introduction}
The {\it sandpile models} were firstly introduced by Bak, Tang and Wiesenfeld in~\cite{bak87} and~\cite{bak88},
and have been studied under several names in statistical physics, theoretical computer science, algebraic graph theory, and combinatorics.
The {\it abelian sandpile model} of a graph... | {
"timestamp": "2011-10-13T02:03:41",
"yymm": "1004",
"arxiv_id": "1004.3321",
"language": "en",
"url": "https://arxiv.org/abs/1004.3321",
"abstract": "In this article, we give a partial description of the sandpile group of the cone of the cartesian product of graphs in function of the sandpile group of the... |
https://arxiv.org/abs/1708.00094 | On facial unique-maximum (edge-)coloring | A facial unique-maximum coloring of a plane graph is a vertex coloring where on each face $\alpha$ the maximal color appears exactly once on the vertices of $\alpha$. If the coloring is required to be proper, then the upper bound for the minimal number of colors required for such a coloring is set to $5$. Fabrici and G... | \section{Introduction}
In this paper, we consider simple graphs only.
We call a graph \emph{planar} if it can be embedded in the plane without crossing edges
and we call it \emph{plane} if it is already embedded in this way.
A \emph{coloring} of a graph is an assignment of colors to vertices.
If in a coloring adjacent... | {
"timestamp": "2017-11-28T02:17:53",
"yymm": "1708",
"arxiv_id": "1708.00094",
"language": "en",
"url": "https://arxiv.org/abs/1708.00094",
"abstract": "A facial unique-maximum coloring of a plane graph is a vertex coloring where on each face $\\alpha$ the maximal color appears exactly once on the vertices... |
https://arxiv.org/abs/2110.01703 | Affine dimers from characteristic polygons | Recent work by Forsgård indicates that not every convex lattice polygon arises as the characteristic polygon of an affine dimer or, equivalently, an admissible oriented line arrangement on the torus in general position. We begin the classification of convex lattice polygons arising as characteristic polygons of affine ... | \section{Introduction}
\label{sec:intro}
A dimer model is an embedded bipartite graph on the torus $\mathbb{T}^2$ or, depending on the application, any surface $\Sigma$.
They were originally introduced in statistical mechanics to model molecular interactions.\todo{(B1) Explain where dimers arise in other areas... | {
"timestamp": "2022-02-16T02:03:05",
"yymm": "2110",
"arxiv_id": "2110.01703",
"language": "en",
"url": "https://arxiv.org/abs/2110.01703",
"abstract": "Recent work by Forsgård indicates that not every convex lattice polygon arises as the characteristic polygon of an affine dimer or, equivalently, an admis... |
https://arxiv.org/abs/2012.08731 | The random walk on upper triangular matrices over $\mathbb{Z}/m \mathbb{Z}$ | We study a natural random walk on the $n \times n$ upper triangular matrices, with entries in $\mathbb{Z}/m \mathbb{Z}$, generated by steps which add or subtract a uniformly random row to the row above. We show that the mixing time of this random walk is $O(m^2n \log n+ n^2 m^{o(1)})$. This answers a question of Stong ... | \section{Introduction}
Let $n \geq 2$ and $m \geq 2$ be two integers, and let $G_n(m)$ denote the group of $n\times n$ upper triangular matrices with entries in $\mathbb{Z}/m \mathbb{Z}$ and ones along the diagonal. We number the rows of each matrix in $G_n(m)$ from top to bottom. We consider the following Markov chain... | {
"timestamp": "2020-12-17T02:08:59",
"yymm": "2012",
"arxiv_id": "2012.08731",
"language": "en",
"url": "https://arxiv.org/abs/2012.08731",
"abstract": "We study a natural random walk on the $n \\times n$ upper triangular matrices, with entries in $\\mathbb{Z}/m \\mathbb{Z}$, generated by steps which add o... |
https://arxiv.org/abs/1601.07280 | On Pure Derived Categories | We investigate the properties of pure derived categories of module categories, and show that pure derived categories share many nice properties of classical derived categories. In particular, we show that bounded pure derived categories can be realized as certain homotopy categories. We introduce the pure projective (r... | \section{ Introduction}
\setcounter{equation}{0}
\vspace{0.2cm}
Let $(\mathcal{A},\mathcal{E})$ be an exact category in the sense of [Q] and $\mathbf{K}(\mathcal{A})$
its homotopy category. Then one can consider the triangulated quotient of $\mathbf{K}(\mathcal{A})$
by $\mathcal{E}$, called the derived category ... | {
"timestamp": "2016-01-28T02:06:08",
"yymm": "1601",
"arxiv_id": "1601.07280",
"language": "en",
"url": "https://arxiv.org/abs/1601.07280",
"abstract": "We investigate the properties of pure derived categories of module categories, and show that pure derived categories share many nice properties of classic... |
https://arxiv.org/abs/2210.04489 | An algorithmic approach based on generating trees for enumerating pattern-avoiding inversion sequences | We introduce an algorithmic approach based on generating tree method for enumerating the inversion sequences with various pattern-avoidance restrictions. For a given set of patterns, we propose an algorithm that outputs either an accurate description of the succession rules of the corresponding generating tree or an an... | \section{Introduction}
An {\em inversion sequence} of length $n$ is an integer sequence $e=e_0e_1\cdots e_n$ such that $0\leq e_i\leq i$ for each $0\leq i\leq n$. We denote by $I_n$ the set of inversion sequences of length $n$. There is a bijection between $I_n$ and $S_{n+1}$, the set of permutations of length $n+1$.... | {
"timestamp": "2022-10-11T02:24:32",
"yymm": "2210",
"arxiv_id": "2210.04489",
"language": "en",
"url": "https://arxiv.org/abs/2210.04489",
"abstract": "We introduce an algorithmic approach based on generating tree method for enumerating the inversion sequences with various pattern-avoidance restrictions. ... |
https://arxiv.org/abs/1403.7460 | On Expansion of a Solution of General Non-autonomous Polynomial Differential Equation | We give a recursive formula for an expansion of a solution of a general non-autonomous polynomial differential equation. The formula is given on the algebraic level with a use of shuffle product. This approach minimizes the number of integrations on each order of expansion. Using combinatorics of trees we estimate the ... | \section{Introduction}
\label{sec:Introduction}
Consider a non-autonomous polynomial differential equation, known also as a generalized Abel differential equation
\begin{align}
\label{eq:PreMain}
\dot x(t) &= u_0(t) + u_1(t)x(t) +\cdots+ u_i(t)x^i(t) + \cdots + u_n(t) x^n(t), \\
\nonumber
x(0) &= x_0,
\end... | {
"timestamp": "2014-03-31T02:09:33",
"yymm": "1403",
"arxiv_id": "1403.7460",
"language": "en",
"url": "https://arxiv.org/abs/1403.7460",
"abstract": "We give a recursive formula for an expansion of a solution of a general non-autonomous polynomial differential equation. The formula is given on the algebra... |
https://arxiv.org/abs/2005.05195 | Solving Large-Scale Sparse PCA to Certifiable (Near) Optimality | Sparse principal component analysis (PCA) is a popular dimensionality reduction technique for obtaining principal components which are linear combinations of a small subset of the original features. Existing approaches cannot supply certifiably optimal principal components with more than $p=100s$ of variables. By refor... | \section{Introduction}
In the era of big data, interpretable methods for compressing a high-dimensional dataset into a lower dimensional set which shares the same essential characteristics are imperative. Since the work of \citet{hotelling1933analysis}, principal component analysis (PCA) has been one of the most popula... | {
"timestamp": "2021-08-26T02:19:11",
"yymm": "2005",
"arxiv_id": "2005.05195",
"language": "en",
"url": "https://arxiv.org/abs/2005.05195",
"abstract": "Sparse principal component analysis (PCA) is a popular dimensionality reduction technique for obtaining principal components which are linear combinations... |
https://arxiv.org/abs/1809.03729 | The Maximum Number of Three Term Arithmetic Progressions, and Triangles in Cayley Graphs | Let $G$ be a finite Abelian group. For a subset $S \subseteq G$, let $T_3(S)$ denote the number of length three arithemtic progressions in $S$ and Prob[$S$] $= \frac{1}{|S|^2}\sum_{x,y \in S} 1_S(x+y)$. For any $q \ge 1$ and $\alpha \in [0,1]$, and any $S \subseteq G$ with $|S| = \frac{|G|}{q+\alpha}$, we show $\frac{T... | \section{Introduction}
The study of arithmetic progressions in subsets of integers and general Abelian groups is a central topic in additive combinatorics and has led to the development of many fascinating areas of mathematics. A famous result on three term arithmetic progressions (3APs) is Roth's theorem, which, in... | {
"timestamp": "2018-09-12T02:07:49",
"yymm": "1809",
"arxiv_id": "1809.03729",
"language": "en",
"url": "https://arxiv.org/abs/1809.03729",
"abstract": "Let $G$ be a finite Abelian group. For a subset $S \\subseteq G$, let $T_3(S)$ denote the number of length three arithemtic progressions in $S$ and Prob[$... |
https://arxiv.org/abs/1304.1826 | Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order | Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Latała we provide a concentration inequality for non-necessarily Lipschitz functions $f\colon \R^n \to \R$ with bounded derivatives of higher orders, which hold when the underlying measure satisfies a family... | \section{Introduction}
Concentration of measure inequalities are one of the basic tools in modern probability theory (see the monograph \cite{LedouxConcBook}). The prototypic result for all concentration theorems is arguably the Gaussian concentration inequality \cite{BorellGaussianConc,SCGaussianConc}, which asserts t... | {
"timestamp": "2013-04-09T02:00:21",
"yymm": "1304",
"arxiv_id": "1304.1826",
"language": "en",
"url": "https://arxiv.org/abs/1304.1826",
"abstract": "Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Latała we provide a concentration inequalit... |
https://arxiv.org/abs/1701.04369 | Arithmetic degrees and dynamical degrees of endomorphisms on surfaces | For a dominant rational self-map on a smooth projective variety defined over a number field, Kawaguchi and Silverman conjectured that the (first) dynamical degree is equal to the arithmetic degree at a rational point whose forward orbit is well-defined and Zariski dense. We prove this conjecture for surjective endomorp... | \section{Introduction}\label{intro}
Let $k$ be a number field, $X$ a smooth projective variety over $\overline{k}$,
and $f\colon X\dashrightarrow X$ a dominant rational self-map on $X$ over $\overline{k}$.
Let $I_f \subset X$ be the indeterminacy locus of $f$.
Let $X_f (\overline{k})$ be the set of $\overline{k}$-r... | {
"timestamp": "2017-01-27T02:05:07",
"yymm": "1701",
"arxiv_id": "1701.04369",
"language": "en",
"url": "https://arxiv.org/abs/1701.04369",
"abstract": "For a dominant rational self-map on a smooth projective variety defined over a number field, Kawaguchi and Silverman conjectured that the (first) dynamica... |
https://arxiv.org/abs/1403.0023 | Superspecial rank of supersingular abelian varieties and Jacobians | An abelian variety defined over an algebraically closed field k of positive characteristic is supersingular if it is isogenous to a product of supersingular elliptic curves and is superspecial if it is isomorphic to a product of supersingular elliptic curves. In this paper, the superspecial condition is generalized by ... | \section{Introduction}
If $A$ is a principally polarized abelian variety of dimension $g$ defined over an algebraically closed field $k$ of positive characteristic $p$,
then the multiplication-by-$p$ morphism $[p]=\ver \circ \frob$ is inseparable.
Typically, $A$ is {\it ordinary} in that the Verschiebung morphism $... | {
"timestamp": "2015-10-20T02:12:57",
"yymm": "1403",
"arxiv_id": "1403.0023",
"language": "en",
"url": "https://arxiv.org/abs/1403.0023",
"abstract": "An abelian variety defined over an algebraically closed field k of positive characteristic is supersingular if it is isogenous to a product of supersingular... |
https://arxiv.org/abs/1904.07761 | Trace operators of the bi-Laplacian and applications | We study several trace operators and spaces that are related to the bi-Laplacian. They are motivated by the development of ultraweak formulations for the bi-Laplace equation with homogeneous Dirichlet condition, but are also relevant to describe conformity of mixed approximations.Our aim is to have well-posed (ultrawea... |
\section{Introduction}
The bi-Laplace operator and biharmonic functions have generated sustained interest in the mathematics
community until today. Just in numerical analysis, MathSciNet reports well beyond 500 publications
with these key words in their titles. An early overview of numerical methods for the Dirichlet... | {
"timestamp": "2019-04-17T02:32:26",
"yymm": "1904",
"arxiv_id": "1904.07761",
"language": "en",
"url": "https://arxiv.org/abs/1904.07761",
"abstract": "We study several trace operators and spaces that are related to the bi-Laplacian. They are motivated by the development of ultraweak formulations for the ... |
https://arxiv.org/abs/1812.11466 | Exact Guarantees on the Absence of Spurious Local Minima for Non-negative Rank-1 Robust Principal Component Analysis | This work is concerned with the non-negative rank-1 robust principal component analysis (RPCA), where the goal is to recover the dominant non-negative principal components of a data matrix precisely, where a number of measurements could be grossly corrupted with sparse and arbitrary large noise. Most of the known techn... | \section{Introduction}
\begin{sloppypar}
The principal component analysis (PCA) is perhaps the most widely-used dimension-reduction method that reveals the components with maximum variability in high-dimensional datasets. In particular, given the data matrix $X\in\mathbb{R}^{m\times n}$, where each row corresponds t... | {
"timestamp": "2019-09-05T02:04:45",
"yymm": "1812",
"arxiv_id": "1812.11466",
"language": "en",
"url": "https://arxiv.org/abs/1812.11466",
"abstract": "This work is concerned with the non-negative rank-1 robust principal component analysis (RPCA), where the goal is to recover the dominant non-negative pri... |
https://arxiv.org/abs/1806.11566 | Analysis and preconditioning of parameter-robust finite element methods for Biot's consolidation model | In this paper we consider a three-field formulation of the Biot model which has the displacement, the total pressure, and the pore pressure as unknowns. For parameter-robust stability analysis, we first show a priori estimates of the continuous problem with parameter-dependent norms. Then we study finite element discre... |
\section{Introduction}
In poroelastic media saturated by fluids, the behaviors of porous medium and the saturating fluid flow are described by Biot's consolidation model \cite{MR0066874}.
Poroelasticity models are widely used in geophysics and petrolium engineering applications, so development of finite element met... | {
"timestamp": "2018-07-02T02:12:14",
"yymm": "1806",
"arxiv_id": "1806.11566",
"language": "en",
"url": "https://arxiv.org/abs/1806.11566",
"abstract": "In this paper we consider a three-field formulation of the Biot model which has the displacement, the total pressure, and the pore pressure as unknowns. F... |
https://arxiv.org/abs/1108.1888 | 3-manifolds with nonnegative Ricci curvature | For a noncompact 3-manifold with nonnegative Ricci curvature, we prove that either it is diffeomorphic to $\mathbb{R}^3$ or the universal cover splits. As a corollary, it confirms a conjecture of Milnor in dimension 3. | \section{\bf{Introduction}}
Let $M$ be a complete manifold with nonnegative Ricci curvature, then it is a fundamental question in geometry to find the restriction of the topology on $M$. Recall in 2-dimensional case, Ricci curvature is the same as Gaussian curvature $K$. It is a well known result that if $K \geq 0$,... | {
"timestamp": "2012-10-08T02:00:51",
"yymm": "1108",
"arxiv_id": "1108.1888",
"language": "en",
"url": "https://arxiv.org/abs/1108.1888",
"abstract": "For a noncompact 3-manifold with nonnegative Ricci curvature, we prove that either it is diffeomorphic to $\\mathbb{R}^3$ or the universal cover splits. As ... |
https://arxiv.org/abs/2003.02754 | Generalizations of the Ruzsa-Szemerédi and rainbow Turán problems for cliques | Considering a natural generalization of the Ruzsa-Szemerédi problem, we prove that for any fixed positive integers $r,s$ with $r<s$, there are graphs on $n$ vertices containing $n^{r}e^{-O(\sqrt{\log{n}})}=n^{r-o(1)}$ copies of $K_s$ such that any $K_r$ is contained in at most one $K_s$. We also give bounds for the gen... | \section{Introduction}\label{sec.intrKrKsrainbow}
The famous Ruzsa--Szemerédi or $(6,3)$-problem
is to determine how many edges there can be in a 3-uniform hypergraph on $n$ vertices if no six vertices span three or more edges. This rather specific-sounding problem turns out to have several equivalent formulations and... | {
"timestamp": "2020-03-20T01:01:10",
"yymm": "2003",
"arxiv_id": "2003.02754",
"language": "en",
"url": "https://arxiv.org/abs/2003.02754",
"abstract": "Considering a natural generalization of the Ruzsa-Szemerédi problem, we prove that for any fixed positive integers $r,s$ with $r<s$, there are graphs on $... |
https://arxiv.org/abs/1708.04869 | Martingale Benamou--Brenier: a probabilistic perspective | In classical optimal transport, the contributions of Benamou-Brenier and McCann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas.We suggest a Benamou-Brenier type formulation of the martingale transport problem... | \section{Introduction}
The roots of optimal transport as a mathematical field go back to Monge \cite{Mo81} and Kantorovich \cite{Ka42} who established its modern formulation. Important triggers for its steep development in the last decades were the seminal results of
Benamou, Brenier, and McCann \cite{Br87, Br91, B... | {
"timestamp": "2019-01-16T02:21:57",
"yymm": "1708",
"arxiv_id": "1708.04869",
"language": "en",
"url": "https://arxiv.org/abs/1708.04869",
"abstract": "In classical optimal transport, the contributions of Benamou-Brenier and McCann regarding the time-dependent version of the problem are cornerstones of th... |
https://arxiv.org/abs/1308.4361 | Inequalities with angular integrability and applications | We prove an extension of the Stein-Weiss weighted estimates for fractional integrals, in the context of Lp spaces with different integrability properties in the radial and the angular direction. In this way, the classical estimates can be unified with their improved radial versions. A number of consequences are obtaine... | \chapter*{Introduction}
We study the improvements due to the angular regularity in the context of Sobolev embeddings and PDEs. It is well known that many fundamental inequalities in mathematical analysis get improvements under some additional symmetry assumptions. Such improvements are related to the geometric nature o... | {
"timestamp": "2013-11-21T02:09:43",
"yymm": "1308",
"arxiv_id": "1308.4361",
"language": "en",
"url": "https://arxiv.org/abs/1308.4361",
"abstract": "We prove an extension of the Stein-Weiss weighted estimates for fractional integrals, in the context of Lp spaces with different integrability properties in... |
https://arxiv.org/abs/1709.01982 | Stabilizing Weighted Graphs | An edge-weighted graph $G=(V,E)$ is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stable graphs play an important role in some interesting game theory problems, such as network bargaining games and cooperative matching games, because they characterize ... | \section{Introduction}
Several interesting game theory problems are defined on networks, where the vertices
represent players and the edges model the way players
can interact with each other. In many such games, the structure of the underlying graph that describes the interactions among players is essential in deter... | {
"timestamp": "2017-11-28T02:03:02",
"yymm": "1709",
"arxiv_id": "1709.01982",
"language": "en",
"url": "https://arxiv.org/abs/1709.01982",
"abstract": "An edge-weighted graph $G=(V,E)$ is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stab... |
https://arxiv.org/abs/1810.11882 | Knotting Probability of Equilateral Hexagons | For a positive integer $n\ge 3$, the collection of $n$-sided polygons embedded in $3$-space defines the space of geometric knots. We will consider the subspace of equilateral knots, consisting of embedded $n$-sided polygons with unit length edges. Paths in this space determine isotopies of polygons, so path-components ... | \section{Introduction}
Classically a knot can be defined as a closed, non self-intersecting smooth curve embedded in Euclidean $3$-space. Two knots are considered to be equivalent if one can be smoothly deformed into another. The question of whether or not two given knots are equivalent proves to be a difficult proble... | {
"timestamp": "2018-10-30T01:16:56",
"yymm": "1810",
"arxiv_id": "1810.11882",
"language": "en",
"url": "https://arxiv.org/abs/1810.11882",
"abstract": "For a positive integer $n\\ge 3$, the collection of $n$-sided polygons embedded in $3$-space defines the space of geometric knots. We will consider the su... |
https://arxiv.org/abs/1512.04797 | Pontryagin maximum principle for optimal sampled-data control problems | In this short communication, we first recall a version of the Pontryagin maximum principle for general finite-dimensional nonlinear optimal sampled-data control problems. This result was recently obtained in [L. Bourdin and E. Tr{é}lat , Optimal sampled-data control, and generalizations on time scales,arXiv:1501.07361,... | \section{Introduction}
Optimal control theory is concerned with the analysis of controlled dynamical systems, where one aims at steering such a system from a given configuration to some desired target by minimizing some criterion. The Pontryagin maximum principle (in short, PMP), established at the end of the 50's for ... | {
"timestamp": "2015-12-16T02:08:34",
"yymm": "1512",
"arxiv_id": "1512.04797",
"language": "en",
"url": "https://arxiv.org/abs/1512.04797",
"abstract": "In this short communication, we first recall a version of the Pontryagin maximum principle for general finite-dimensional nonlinear optimal sampled-data c... |
https://arxiv.org/abs/1910.02530 | On the Hausdorff dimension of Riemann's non-differentiable function | Recent findings show that the classical Riemann's non-differentiable function has a physical and geometric nature as the irregular trajectory of a polygonal vortex filament driven by the binormal flow. In this article, we give an upper estimate of its Hausdorff dimension. We also adapt this result to the multifractal s... | \section{Introduction}
\subsection{Riemann's non-differentiable function}
In a lecture in the Royal Prussian Academy of Sciences in 1872, in Berlin, Weierstrass \cite{Weierstrass1872} explained against the belief of the time that a continuous function need not have a well-defined derivative, proposing the famous Wei... | {
"timestamp": "2021-07-19T02:13:35",
"yymm": "1910",
"arxiv_id": "1910.02530",
"language": "en",
"url": "https://arxiv.org/abs/1910.02530",
"abstract": "Recent findings show that the classical Riemann's non-differentiable function has a physical and geometric nature as the irregular trajectory of a polygon... |
https://arxiv.org/abs/1707.03281 | Characterizations of Ideal Cluster Points | Given an ideal $\mathcal{I}$ on $\omega$, we prove that a sequence in a topological space $X$ is $\mathcal{I}$-convergent if and only if there exists a ``big'' $\mathcal{I}$-convergent subsequence. Then, we study several properties and show two characterizations of the set of $\mathcal{I}$-cluster points as classical c... | \section{Introduction}\label{sec:introduction}
Following the concept of statistical convergence as a generalization of the ordinary convergence, Fridy \cite{MR1181163} introduced the statistical limit points and statistical cluster points of a real sequence $(x_n)$ as generalizations of accumulation points.
A rea... | {
"timestamp": "2019-02-19T02:15:50",
"yymm": "1707",
"arxiv_id": "1707.03281",
"language": "en",
"url": "https://arxiv.org/abs/1707.03281",
"abstract": "Given an ideal $\\mathcal{I}$ on $\\omega$, we prove that a sequence in a topological space $X$ is $\\mathcal{I}$-convergent if and only if there exists a... |
https://arxiv.org/abs/1503.09092 | Efficiently decoding Reed-Muller codes from random errors | Reed-Muller codes encode an $m$-variate polynomial of degree $r$ by evaluating it on all points in $\{0,1\}^m$. We denote this code by $RM(m,r)$. The minimal distance of $RM(m,r)$ is $2^{m-r}$ and so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better re... |
\section{Introduction}
Consider the following challenge:
\begin{quote}
Given the truth table of a polynomial $f(\mathbf{x}) \in \mathbb{F}_2[x_1,\dots, x_m]$ of degree at most $r$, in which $1/2-o(1)$ fraction of the locations were flipped (that is, given the evaluations of $f$ over $\mathbb{F}_2^m$ with nearly h... | {
"timestamp": "2015-08-28T02:08:14",
"yymm": "1503",
"arxiv_id": "1503.09092",
"language": "en",
"url": "https://arxiv.org/abs/1503.09092",
"abstract": "Reed-Muller codes encode an $m$-variate polynomial of degree $r$ by evaluating it on all points in $\\{0,1\\}^m$. We denote this code by $RM(m,r)$. The mi... |
https://arxiv.org/abs/math/0504195 | The Eulerian Distribution on Involutions is Indeed Unimodal | Let I_{n,k} (resp. J_{n,k}) be the number of involutions (resp. fixed-point free involutions) of {1,...,n} with k descents. Motivated by Brenti's conjecture which states that the sequence I_{n,0}, I_{n,1},..., I_{n,n-1} is log-concave, we prove that the two sequences I_{n,k} and J_{2n,k} are unimodal in k, for all n. F... | \section{Introduction}
A sequence $a_0,a_1,\ldots,a_n$ of real numbers
is said to be {\it unimodal} if for some $0\leq j\leq n$ we have
$a_0\leq a_1\leq\cdots\leq a_j\geq a_{j+1}\geq\cdots \geq a_n$,
and is said to be {\it log-concave} if
$a_i^2\geq a_{i-1}a_{i+1}$ for all $1\leq i\leq n-1$. Clearly a log-concave
seque... | {
"timestamp": "2005-10-19T18:36:05",
"yymm": "0504",
"arxiv_id": "math/0504195",
"language": "en",
"url": "https://arxiv.org/abs/math/0504195",
"abstract": "Let I_{n,k} (resp. J_{n,k}) be the number of involutions (resp. fixed-point free involutions) of {1,...,n} with k descents. Motivated by Brenti's conj... |
https://arxiv.org/abs/1708.06682 | Weighted isoperimetric inequalities in warped product manifolds | We prove some isoperimetric type inequalities in warped product manifolds, or more generally, multiply warped product manifolds. We then relate them to inequalities involving the higher order mean-curvature integrals. We also apply our results to obtain sharp eigenvalue estimates and some sharp geometric inequalities i... | \section{Introduction}\label{sec: intro}
The classical isoperimetric inequality on the plane states that for a simple closed curve on $\mathbb R^2$, we have $L^2\ge 4\pi A$, where $L$ is the length of the curve and $A$ is the area of the region enclosed by it. The equality holds if and only if the curve is a circle.... | {
"timestamp": "2017-08-23T02:07:47",
"yymm": "1708",
"arxiv_id": "1708.06682",
"language": "en",
"url": "https://arxiv.org/abs/1708.06682",
"abstract": "We prove some isoperimetric type inequalities in warped product manifolds, or more generally, multiply warped product manifolds. We then relate them to in... |
https://arxiv.org/abs/1306.5351 | Divisors on graphs, binomial and monomial ideals, and cellular resolutions | We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of ... | \section{Introduction}
This work is concerned with the development of new connections between the theory of divisors on graphs, potential theory, the theory of lattices, Delaunay decompositions, and commutative algebra.
\subsection{Divisors on graphs}
Let $G$ be a graph. Let $\Div(G)$ be the free abelian group gen... | {
"timestamp": "2013-06-25T02:01:40",
"yymm": "1306",
"arxiv_id": "1306.5351",
"language": "en",
"url": "https://arxiv.org/abs/1306.5351",
"abstract": "We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on... |
https://arxiv.org/abs/1706.09374 | Yet again on polynomial convergence for SDEs with a gradient-type drift | Bounds on convergence rate to the invariant distribution for a class of stochastic differential equations (SDEs) with a gradient-type drift are obtained. | \section{Introduction}
Let us consider a stochastic differential equation in $R^d$
\begin{equation}\label{eq1}
dX_t = dB_t - \nabla U (X_t)\,dt
\end{equation}
with initial data
\begin{equation}\label{eq2}
X_0 = x.
\end{equation}
Here $B_t, \,\, t\ge... | {
"timestamp": "2017-07-25T02:09:09",
"yymm": "1706",
"arxiv_id": "1706.09374",
"language": "en",
"url": "https://arxiv.org/abs/1706.09374",
"abstract": "Bounds on convergence rate to the invariant distribution for a class of stochastic differential equations (SDEs) with a gradient-type drift are obtained."... |
https://arxiv.org/abs/1203.1400 | Isoperimetric inequality under Kähler Ricci flow | Let $({\M}, g(t))$ be a Kähler Ricci flow with positive first Chern class. We prove a uniform isoperimetric inequality for all time. In the process we also prove a Cheng-Yau type log gradient bound for positive harmonic functions on $({\M}, g(t))$, and a Poincaré inequality without assuming the Ricci curvature is bound... | \section{Introduction}
In this paper we study K\"ahler Ricci flows
\begin{equation}
\label{krf}
\partial_t g_{i\bar j} = - R_{i\bar j} + g_{i\bar{j}} = \partial_i \partial_{\bar j} u, \quad t>0,
\end{equation} on a compact, K\"ahler manifold ${\bf M}$ of complex dimension $m=n/2$, with positive first Chern class.
... | {
"timestamp": "2013-07-11T02:05:36",
"yymm": "1203",
"arxiv_id": "1203.1400",
"language": "en",
"url": "https://arxiv.org/abs/1203.1400",
"abstract": "Let $({\\M}, g(t))$ be a Kähler Ricci flow with positive first Chern class. We prove a uniform isoperimetric inequality for all time. In the process we also... |
https://arxiv.org/abs/2212.11216 | Optimal cycles enclosing all the nodes of a $k$-dimensional hypercube | We solve the general problem of visiting all the $2^k$ nodes of a $k$-dimensional hypercube by using a polygonal chain that has minimum link-length, and we show that this optimal value is given by $h(2,k):=3 \cdot 2^{k-2}$ if and only if $k \in \mathbb{N}-\{0,1\}$. Furthermore, for any $k$ above one, we constructively ... | \section{Introduction} \label{sec:Intr}
Given $k \in \mathbb{N}-\{0,1\}$, let the finite subset of $2^k$ points belonging to the Euclidean space $\textit{E} \subset \mathbb{R}^k$ be defined as $H(2,k):=\{\{0,1\} \times \{0,1\} \times \dots \times \{0,1\}\} \subset \mathbb{R}^k$, where the symbol ``$\times$'' denotes th... | {
"timestamp": "2022-12-22T02:18:20",
"yymm": "2212",
"arxiv_id": "2212.11216",
"language": "en",
"url": "https://arxiv.org/abs/2212.11216",
"abstract": "We solve the general problem of visiting all the $2^k$ nodes of a $k$-dimensional hypercube by using a polygonal chain that has minimum link-length, and w... |
https://arxiv.org/abs/1602.05908 | Efficient approaches for escaping higher order saddle points in non-convex optimization | Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have degenerate saddle points su... | \section{Algorithm for Finding Third Order Optimal Points}
\label{sec:alg}
We design an algorithm that is guaranteed to converge to a third order local minimum. Throughout this section we assume both Assumptions~\ref{assump:lipschitzhessian} and \ref{assump:lipthird} \footnote{Note that we actually only cares about a ... | {
"timestamp": "2016-02-19T02:13:20",
"yymm": "1602",
"arxiv_id": "1602.05908",
"language": "en",
"url": "https://arxiv.org/abs/1602.05908",
"abstract": "Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algori... |
https://arxiv.org/abs/0909.5021 | Properties of Translating Solutions to Mean Curvature Flow | In this paper, we study the convexity, interior gradient estimate, Liouville type theorem and asymptotic behavior at infinity of translating solutions to mean curvature flow as well as the nonlinear flow by powers of the mean curvature. | \section*{1. Introduction and Main Results}
\setcounter{section}{1} \setcounter{equation}{0}
In this paper, we study the convexity, interior gradient estimate,
Liouville type theorem and asymptotic behavior at infinity of the
solutions to equation
\begin{eqnarray} \label {1.1}
a_{ij}u_{ij}:=(\delta_{ij}-\frac{u_iu... | {
"timestamp": "2009-09-28T08:59:42",
"yymm": "0909",
"arxiv_id": "0909.5021",
"language": "en",
"url": "https://arxiv.org/abs/0909.5021",
"abstract": "In this paper, we study the convexity, interior gradient estimate, Liouville type theorem and asymptotic behavior at infinity of translating solutions to me... |
https://arxiv.org/abs/2002.11291 | Fitting the trajectories of particles in the equatorial plane of a magnetic dipole with epicycloids | In this paper we discuss epicycloid approximation of the trajectories of charged particles in axisymmetric magnetic fields. Epicycloid trajectories are natural in the Guiding Center approximation and we study in detail the errors arising in this approach. We also discuss how using exact results for particle motion in t... | \section{Introduction}
The study of the trajectory of charged particle in magnetic dipole field is an agelong electromagnetism problem.
There are many applications in this research, such as auroral theory and plasma physics.
Although the motion of particle in magnetic field has been studied in some literatures\cite... | {
"timestamp": "2020-05-12T02:07:08",
"yymm": "2002",
"arxiv_id": "2002.11291",
"language": "en",
"url": "https://arxiv.org/abs/2002.11291",
"abstract": "In this paper we discuss epicycloid approximation of the trajectories of charged particles in axisymmetric magnetic fields. Epicycloid trajectories are na... |
https://arxiv.org/abs/2202.05246 | Monotone Learning | The amount of training-data is one of the key factors which determines the generalization capacity of learning algorithms. Intuitively, one expects the error rate to decrease as the amount of training-data increases. Perhaps surprisingly, natural attempts to formalize this intuition give rise to interesting and challen... | \section{Introduction}
In this work we study the following fundamental question.
Pick some standard learning algorithm~$A$, and consider training it
for some natural task using a data-set of $n$ examples.
\begin{center}
{Does feeding $A$ with more training data \emph{provably} reduces its population loss?}
\en... | {
"timestamp": "2022-02-11T02:23:04",
"yymm": "2202",
"arxiv_id": "2202.05246",
"language": "en",
"url": "https://arxiv.org/abs/2202.05246",
"abstract": "The amount of training-data is one of the key factors which determines the generalization capacity of learning algorithms. Intuitively, one expects the er... |
https://arxiv.org/abs/1103.1873 | Characterizing finitary functions over non-archimedean RCFs via a topological definition of OVF-integrality | When $R$ is a non-archimedean real closed field we say that a function $f\in R(\bar{X})$ is finitary at a point $\bar{b}\in R^n$ if on some neighborhood of $\bar{b}$ the defined values of $f$ are in the finite part of $R$. In this note we give a characterization of rational functions which are finitary on a set defined... | \subsection{Topological statement of OVF-integrality for RCVFs}
Recall that a {\em real closed valued field} (or RCVF) is an OVF which is real closed. The theory RCVF is the model companion of the theory OVF. In this section we give a topological property which is equivalent to OVF-integrality over an RCVF.
\begin{... | {
"timestamp": "2011-03-15T01:02:20",
"yymm": "1103",
"arxiv_id": "1103.1873",
"language": "en",
"url": "https://arxiv.org/abs/1103.1873",
"abstract": "When $R$ is a non-archimedean real closed field we say that a function $f\\in R(\\bar{X})$ is finitary at a point $\\bar{b}\\in R^n$ if on some neighborhood... |
https://arxiv.org/abs/2208.02147 | Weighted composition operators from the Bloch space to weighted Banach spaces on bounded homogeneous domains | We study the bounded and the compact weighted composition operators from the Bloch space into the weighted Banach spaces of holomorphic functions on bounded homogeneous domains, with particular attention to the unit polydisk. For bounded homogeneous domains, we characterize the bounded weighted composition operators an... | \section{Introduction}
Let $X$ and $Y$ be Banach spaces of holomorphic functions on a domain $\Omega \subset \mathbb{C}^n$. For holomorphic functions $\psi:\Omega \to \mathbb{C}$ and $\varphi:\Omega\to\Omega$, the \textit{weighted composition operator} from $X$ to $Y$ with symbols $\psi$ and $\varphi$ is defined as $$... | {
"timestamp": "2022-08-04T02:20:25",
"yymm": "2208",
"arxiv_id": "2208.02147",
"language": "en",
"url": "https://arxiv.org/abs/2208.02147",
"abstract": "We study the bounded and the compact weighted composition operators from the Bloch space into the weighted Banach spaces of holomorphic functions on bound... |
https://arxiv.org/abs/2212.09893 | Diagonals separating the square of a continuum | A metric continuum $X$ is indecomposable if it cannot be put as the union of two of its proper subcontinua. A subset $R$ of $X$ is said to be continuumwise connected provided that for each pair of points $p,q\in R$, there exists a subcontinuum $M$ of $X$ such that $\{p,q\}\subset M\subset R$. Let $X^{2}$ denote the Car... | \section{Introduction}
A {\it continuum} is a compact connected non-degenerate metric space. A subcontinuum
of a continuum $X$ is a nonempty connected closed subset of $X$, so singletons
are subcontinua.
An {\it arc} is a continuum homeomorphic to the interval $[0,1]$. Let $X^{2}$ denote the Cartesian square of $X$ a... | {
"timestamp": "2022-12-21T02:03:27",
"yymm": "2212",
"arxiv_id": "2212.09893",
"language": "en",
"url": "https://arxiv.org/abs/2212.09893",
"abstract": "A metric continuum $X$ is indecomposable if it cannot be put as the union of two of its proper subcontinua. A subset $R$ of $X$ is said to be continuumwis... |
https://arxiv.org/abs/1506.03043 | Disconjugacy characterization by means of spectral of $(k,n-k)$ problems | This paper is devoted to the description of the interval of parameters for which the general linear $n^{\rm th}$-order equation\begin{equation}\label{e-Ln}T_n[M]\,u(t) \equiv u^{(n)}(t)+a_1(t)\, u^{(n-1)}(t)+\cdots +a_{n-1}(t)\, u'(t)+(a_{n}(t)+M)\,u(t)=0 \,,\quad t\in I\equiv[a,b],\end{equation}with $a_i\in C^{n-i}(I)... | \section{Introduction}
There are a huge number of works related to the disconjugacy and its properties, (see \cite{Cop, Lev, Sha} for details). From the last half of the past century till now this subject of investigation has attracted important researchers who have stablished very interesting criteria to ensure such ... | {
"timestamp": "2015-06-15T02:07:45",
"yymm": "1506",
"arxiv_id": "1506.03043",
"language": "en",
"url": "https://arxiv.org/abs/1506.03043",
"abstract": "This paper is devoted to the description of the interval of parameters for which the general linear $n^{\\rm th}$-order equation\\begin{equation}\\label{e... |
https://arxiv.org/abs/1510.01670 | Sketching for Simultaneously Sparse and Low-Rank Covariance Matrices | We introduce a technique for estimating a structured covariance matrix from observations of a random vector which have been sketched. Each observed random vector $\boldsymbol{x}_t$ is reduced to a single number by taking its inner product against one of a number of pre-selected vector $\boldsymbol{a}_\ell$. These obser... |
\section{Introduction}
We introduce and analyze a technique for estimating the covariance matrix $\mtx{\varSigma}$ of $\sN$-dimensional random vector $\vct{x}$ from samples $\vct{x}_1,\vct{x}_2,\ldots,\vct{x}_{Q}$. We will show that these quantities can be estimated accurately from {\em sketches} or {\em compressed ... | {
"timestamp": "2015-10-09T02:12:22",
"yymm": "1510",
"arxiv_id": "1510.01670",
"language": "en",
"url": "https://arxiv.org/abs/1510.01670",
"abstract": "We introduce a technique for estimating a structured covariance matrix from observations of a random vector which have been sketched. Each observed random... |
https://arxiv.org/abs/2005.05779 | Instability of Defection in the Prisoner's Dilemma Under Best Experienced Payoff Dynamics | We study population dynamics under which each revising agent tests each strategy k times, with each trial being against a newly drawn opponent, and chooses the strategy whose mean payoff was highest. When k = 1, defection is globally stable in the prisoner`s dilemma. By contrast, when k > 1 we show that there exists a ... | \section{Introduction}
The standard approach in game theory assumes that players
form beliefs about the various uncertainties they face and then best respond to these beliefs. In equilibrium, the beliefs will be correct and the players will play a Nash equilibrium. However, in some
economic environments where the play... | {
"timestamp": "2021-01-05T02:05:18",
"yymm": "2005",
"arxiv_id": "2005.05779",
"language": "en",
"url": "https://arxiv.org/abs/2005.05779",
"abstract": "We study population dynamics under which each revising agent tests each strategy k times, with each trial being against a newly drawn opponent, and choose... |
https://arxiv.org/abs/hep-th/9311094 | The differential geometry of Fedosov's quantization | B. Fedosov has given a simple and very natural construction of a deformation quantization for any symplectic manifold, using a flat connection on the bundle of formal Weyl algebras associated to the tangent bundle of a symplectic manifold. The connection is obtained by affinizing, nonlinearizing, and iteratively flatte... | \section{Introduction}
\label{sec-intro}
In a remarkable paper \cite{fe:simple}, B. Fedosov
presented a
simple and very natural construction\footnote{Actually, this
construction appeared in several earlier papers, such as
\cite{fe:formal} and \cite{fe:index}.} of a deformation quantization for
any symplectic manifold. ... | {
"timestamp": "1999-10-01T20:45:00",
"yymm": "9311",
"arxiv_id": "hep-th/9311094",
"language": "en",
"url": "https://arxiv.org/abs/hep-th/9311094",
"abstract": "B. Fedosov has given a simple and very natural construction of a deformation quantization for any symplectic manifold, using a flat connection on ... |
https://arxiv.org/abs/2008.00358 | Relational Algorithms for k-means Clustering | This paper gives a k-means approximation algorithm that is efficient in the relational algorithms model. This is an algorithm that operates directly on a relational database without performing a join to convert it to a matrix whose rows represent the data points. The running time is potentially exponentially smaller th... | \section{Analysis of the Weighting Algorithm}
\label{sec:weight_algo_analysis}
The goal in this subsection is to prove
Theorem \ref{thm:coreset_kmeans} which states
that the alternative weights form an $O(1)$-approximate coreset with high probability. Throughout our analysis,
``with high probability'' means that fo... | {
"timestamp": "2021-05-24T02:05:29",
"yymm": "2008",
"arxiv_id": "2008.00358",
"language": "en",
"url": "https://arxiv.org/abs/2008.00358",
"abstract": "This paper gives a k-means approximation algorithm that is efficient in the relational algorithms model. This is an algorithm that operates directly on a ... |
https://arxiv.org/abs/1805.00918 | Some Estimates of Virtual Element Methods for Fourth Order Problems | In this paper, we employ the techniques developed for second order operators to obtain the new estimates of Virtual Element Method for fourth order operators. The analysis is based on elements with proper shape regularity. Estimates for projection and interpolation operators are derived. Also, the biharmonic problem is... | \section{Introduction}
Virtual Element methods are designed to use polygonal/polyhedral meshes. It gives us the flexibility to generate general meshes which is a great advantage especially in computational mechanics. Virtual Element Methods for second order problems are well studied in \cite{Beirao13}--\cite{Beir16-3},... | {
"timestamp": "2018-05-03T02:12:28",
"yymm": "1805",
"arxiv_id": "1805.00918",
"language": "en",
"url": "https://arxiv.org/abs/1805.00918",
"abstract": "In this paper, we employ the techniques developed for second order operators to obtain the new estimates of Virtual Element Method for fourth order operat... |
https://arxiv.org/abs/2003.10015 | General Optimal Polynomial Approximants, Stabilization, and Projections of Unity | For various Hilbert spaces of analytic functions on the unit disk, we characterize when a function $f$ has optimal polynomial approximants given by truncations of a single power series. We also introduce a generalized notion of optimal approximant and use this to explicitly compute orthogonal projections of 1 onto cert... | \section{Background, Introduction, and Notation}
Throughout this paper $\mathcal{H}$ will be a reproducing kernel Hilbert space of analytic functions on the unit disk $\mathbb{D}$. We will denote the reproducing kernel for $\mathcal{H}$ as $k_\lambda(z) = k(z,\lambda)$ and the normalized reproducing kernel as $\hat{k}_... | {
"timestamp": "2020-03-24T01:19:12",
"yymm": "2003",
"arxiv_id": "2003.10015",
"language": "en",
"url": "https://arxiv.org/abs/2003.10015",
"abstract": "For various Hilbert spaces of analytic functions on the unit disk, we characterize when a function $f$ has optimal polynomial approximants given by trunca... |
https://arxiv.org/abs/1401.0153 | The Minimum Number of Rotations About Two Axes for Constructing an Arbitrary Fixed Rotation | For any pair of three-dimensional real unit vectors $\hat{m}$ and $\hat{n}$ with $|\hat{m}^{\rm T} \hat{n}| < 1$ and any rotation $U$, let $N_{\hat{m},\hat{n}}(U)$ denote the least value of a positive integer $k$ such that $U$ can be decomposed into a product of $k$ rotations about either $\hat{m}$ or $\hat{n}$. This w... | \section{#1}}
\newcommand{\refroyalsub}[2]{\ref{#2}}
\newcommand{\refappparen}[1]{(Appendix~\ref{#1})}
\newcommand{\mtilde}[1]{#1}
\newcommand{k}{k}
\begin{document}
\title[The Minimum Number of Rotations About Two Axes]{The Minimum Number of Rotations About Two Axes for Constructing an Arbitrarily Fixed Rotat... | {
"timestamp": "2014-07-01T02:16:53",
"yymm": "1401",
"arxiv_id": "1401.0153",
"language": "en",
"url": "https://arxiv.org/abs/1401.0153",
"abstract": "For any pair of three-dimensional real unit vectors $\\hat{m}$ and $\\hat{n}$ with $|\\hat{m}^{\\rm T} \\hat{n}| < 1$ and any rotation $U$, let $N_{\\hat{m}... |
https://arxiv.org/abs/math/0603122 | Introduction to Partially Ordered Patterns | We review selected known results on partially ordered patterns (POPs) that include co-unimodal, multi- and shuffle patterns, peaks and valleys ((modified) maxima and minima) in permutations, the Horse permutations and others. We provide several (new) results on a class of POPs built on an arbitrary flat poset, obtainin... | \section{Introduction and background}\label{intro}
An occurrence of a \emph{pattern} $\tau$ in a permutation $\pi$ is defined as a subsequence
in $\pi$ (of the same length as $\tau$) whose letters are in the same relative order as those in $\tau$.
For example, the permutation $31425$ has three occurrences ... | {
"timestamp": "2006-09-28T17:47:28",
"yymm": "0603",
"arxiv_id": "math/0603122",
"language": "en",
"url": "https://arxiv.org/abs/math/0603122",
"abstract": "We review selected known results on partially ordered patterns (POPs) that include co-unimodal, multi- and shuffle patterns, peaks and valleys ((modif... |
https://arxiv.org/abs/math/0602050 | Rough Path Analysis Via Fractional Calculus | Using fractional calculus we define integrals of the form $% \int_{a}^{b}f(x_{t})dy_{t}$, where $x$ and $y$ are vector-valued Hölder continuous functions of order $\displaystyle \beta \in (\frac13, \frac12)$ and $f$ is a continuously differentiable function such that $f'$ is $\lambda$-Höldr continuous for some $\lambda... | \section{Introduction}
The theory of rough path analysis has been developed from the seminal paper
by Lyons \ \cite{lyons}. The purpose of this theory is to analyze dynamical
systems $dx_{t}=f(x_{t})dy_{t}$, where the control function $y$ is not
differentiable. If the rough control $y$ has finite $p$-variation on boun... | {
"timestamp": "2006-02-02T19:01:34",
"yymm": "0602",
"arxiv_id": "math/0602050",
"language": "en",
"url": "https://arxiv.org/abs/math/0602050",
"abstract": "Using fractional calculus we define integrals of the form $% \\int_{a}^{b}f(x_{t})dy_{t}$, where $x$ and $y$ are vector-valued Hölder continuous funct... |
https://arxiv.org/abs/1612.00184 | Remarks on Kawamata's effective non-vanishing conjecture for manifolds with trivial first Chern classes | Kawamata proposed a conjecture predicting that every nef and big line bundle on a smooth projective variety with trivial first Chern class has nontrivial global sections. We verify this conjecture for several cases, including (i) all hyperkähler varieties of dimension $\leq 6$; (ii) all known hyperkähler varieties exce... | \section{Introduction}
Kodaira vanishing theorem asserts that $H^{i}(X,K_{X}\otimes L)=0$ for $i>0$ and any ample line bundle $L$ on a smooth projective variety\footnote{Throughout this paper, we work over the complex number field ${\mathbb C}$.}\,$X$. It is natural to ask when $H^0(X, K_{X}\otimes L)$ does not vanish.... | {
"timestamp": "2016-12-06T02:03:55",
"yymm": "1612",
"arxiv_id": "1612.00184",
"language": "en",
"url": "https://arxiv.org/abs/1612.00184",
"abstract": "Kawamata proposed a conjecture predicting that every nef and big line bundle on a smooth projective variety with trivial first Chern class has nontrivial ... |
https://arxiv.org/abs/2007.11730 | Nonclosedness of Sets of Neural Networks in Sobolev Spaces | We examine the closedness of sets of realized neural networks of a fixed architecture in Sobolev spaces. For an exactly $m$-times differentiable activation function $\rho$, we construct a sequence of neural networks $(\Phi_n)_{n \in \mathbb{N}}$ whose realizations converge in order-$(m-1)$ Sobolev norm to a function th... | \section{Introduction} \label{sec:intro}
From an approximation theory perspective, neural networks use observed training data to approximate an unknown target function.
Studying topological properties of sets of neural networks will reveal what kinds of functions can be approximated by neural networks. In particula... | {
"timestamp": "2021-01-29T02:02:57",
"yymm": "2007",
"arxiv_id": "2007.11730",
"language": "en",
"url": "https://arxiv.org/abs/2007.11730",
"abstract": "We examine the closedness of sets of realized neural networks of a fixed architecture in Sobolev spaces. For an exactly $m$-times differentiable activatio... |
https://arxiv.org/abs/1306.4208 | Convex Polytopes from Nested Posets | Motivated by the graph associahedron KG, a polytope whose face poset is based on connected subgraphs of G, we consider the notion of associativity and tubes on posets. This leads to a new family of simple convex polytopes obtained by iterated truncations. These generalize graph associahedra and nestohedra, even encompa... | \section{Background}
\subsection{}
Given a finite graph $G$, the graph associahedron $\KG$ is a polytope whose face poset is based on the connected subgraphs of $G$ \cite{cdf}. For special examples of graphs, $\KG$ becomes well-known, sometimes classical: when $G$ is a path, a cycle, or a complete graph, $\KG$ resul... | {
"timestamp": "2013-06-19T02:02:38",
"yymm": "1306",
"arxiv_id": "1306.4208",
"language": "en",
"url": "https://arxiv.org/abs/1306.4208",
"abstract": "Motivated by the graph associahedron KG, a polytope whose face poset is based on connected subgraphs of G, we consider the notion of associativity and tubes... |
https://arxiv.org/abs/1907.01823 | Spectral gaps, symmetries and log-concave perturbations | We discuss situations where perturbing a probability measure on $\mathbb{R}^n$ does not deteriorate its Poincaré constant by much. A particular example is the symmetric exponential measure in $\mathbb{R}^n$, even log-concave perturbations of which have Poincaré constants that grow at most logarithmically with the dimen... | \section{Introduction}
This work was partly motivated by the study
of a family of probability measures on $\mathbb R^n$ which
naturally appear when considering statistical questions pertaining to sparse linear modeling:
\[ d\nu^{n,Q}(x)=\frac1Z e^{-\|x\|_1-Q(x)}\, dx, \]
where $Q$ is a nonnegative quadratic form, $\|... | {
"timestamp": "2019-07-11T02:00:46",
"yymm": "1907",
"arxiv_id": "1907.01823",
"language": "en",
"url": "https://arxiv.org/abs/1907.01823",
"abstract": "We discuss situations where perturbing a probability measure on $\\mathbb{R}^n$ does not deteriorate its Poincaré constant by much. A particular example i... |
https://arxiv.org/abs/2101.08644 | Statistics for $S_n$ acting on $k$-sets | We study the natural action of $S_n$ on the set of $k$-subsets of the set $\{1,\dots, n\}$ when $1\leq k \leq \frac{n}{2}$. For this action we calculate the maximum size of a minimal base, the height and the maximum length of an irredundant base.Here a "base" is a set with trivial pointwise stabilizer, "height" is the ... | \section{Introduction}\label{s: intro}
In this note we study three statistics pertaining to primitive permutation groups. Our main theorem gives the value of these three statistics for the permutation groups $S_n$ acting (in the natural way) on the set of $k$-subsets of the set $\{1,\dots, n\}$.
Before we state our m... | {
"timestamp": "2021-09-13T02:20:04",
"yymm": "2101",
"arxiv_id": "2101.08644",
"language": "en",
"url": "https://arxiv.org/abs/2101.08644",
"abstract": "We study the natural action of $S_n$ on the set of $k$-subsets of the set $\\{1,\\dots, n\\}$ when $1\\leq k \\leq \\frac{n}{2}$. For this action we calcu... |
https://arxiv.org/abs/0706.0346 | Complex Ratios of Cubic Polynomials | Let $p(w)=(w-w_{1})(w-w_{2})(w-w_{3}),$with $\func{Re}w_{1}<\func{Re}w_{2}<\func{Re}w_{3}$. Assume that if the critical points of $p$ are not identical, then they cannot have equal real parts. Define the ratios $\sigma_{1}=\dfrac{z_{1}-w_{1}}{w_{2}-w_{1}}$ and $\sigma _{2}=\dfrac{z_{2}-w_{2}}{w_{3}-w_{2}}$. $(\sigma_{1... | \section{1. Introduction}
\qquad Let $p(x)$ be a polynomial of degree $n\geq 2$ with $n$ distinct real
roots $r_{1}<r_{2}<\cdots <r_{n}$. Such a polynomial is called hyperbolic.
Let $x_{1}<x_{2}<\cdots <x_{n-1}$ be the critical points of $p$, and define
the ratios $\sigma _{k}=\dfrac{x_{k}-r_{k}}{r_{k+1}-r_{k}},k=1,2,... | {
"timestamp": "2007-06-03T22:08:17",
"yymm": "0706",
"arxiv_id": "0706.0346",
"language": "en",
"url": "https://arxiv.org/abs/0706.0346",
"abstract": "Let $p(w)=(w-w_{1})(w-w_{2})(w-w_{3}),$with $\\func{Re}w_{1}<\\func{Re}w_{2}<\\func{Re}w_{3}$. Assume that if the critical points of $p$ are not identical, ... |
https://arxiv.org/abs/1809.10762 | An Approach to Duality in Nonlinear Filtering | This paper revisits the question of duality between minimum variance estimation and optimal control first described for the linear Gaussian case in the celebrated paper of Kalman and Bucy. A duality result is established for nonlinear filtering, mirroring closely the original Kalman-Bucy duality of control and estimati... |
\section{Introduction}
In Kalman's celebrated paper with Bucy,
it is shown that the problem of optimal estimation is dual
to an optimal control problem~\cite{kalman1961}. A
striking example of the dual relationship is that, with the time arrow reversed,
the dynamic Riccati equation (DRE) of the optimal control is... | {
"timestamp": "2019-03-28T01:04:36",
"yymm": "1809",
"arxiv_id": "1809.10762",
"language": "en",
"url": "https://arxiv.org/abs/1809.10762",
"abstract": "This paper revisits the question of duality between minimum variance estimation and optimal control first described for the linear Gaussian case in the ce... |
https://arxiv.org/abs/2102.12872 | Digital almost nets | Digital nets (in base $2$) are the subsets of $[0,1]^d$ that contain the expected number of points in every not-too-small dyadic box. We construct sets that contain almost the expected number of points in every such box, but which are exponentially smaller than the digital nets. We also establish a lower bound on the s... | \section{Introduction}
We call a subinterval of $[0,1]$ \emph{basic (in base $q$)} if it is of the form $\bigl[\frac{a}{q^k},\frac{a+1}{q^k}\bigr)$,
for nonnegative integers $a$ and~$k$.
A \emph{basic box} is a product of basic intervals, i.e.,
a set of the form $\prod_{i=1}^d\bigl[\frac{a_i}{q^{k_i}},\frac{a_i+1}... | {
"timestamp": "2022-09-28T02:19:38",
"yymm": "2102",
"arxiv_id": "2102.12872",
"language": "en",
"url": "https://arxiv.org/abs/2102.12872",
"abstract": "Digital nets (in base $2$) are the subsets of $[0,1]^d$ that contain the expected number of points in every not-too-small dyadic box. We construct sets th... |
https://arxiv.org/abs/2210.13071 | Normal split submanifolds of rational homogeneous spaces | Let $M \subset X$ be a submanifold of a rational homogeneous space $X$ such that the normal sequence splits. We prove that $M$ is also rational homogeneous. | \section{Introduction}
Let $M \subset \ensuremath{\mathbb{P}}^n$ be a projective manifold such that the normal sequence
$$
0 \rightarrow T_M \rightarrow T_{\ensuremath{\mathbb{P}}^n} \otimes \sO_M \rightarrow N_{M/\ensuremath{\mathbb{P}}^n} \rightarrow 0
$$
splits. Since the tangent bundle of the projective space is... | {
"timestamp": "2022-10-25T02:25:33",
"yymm": "2210",
"arxiv_id": "2210.13071",
"language": "en",
"url": "https://arxiv.org/abs/2210.13071",
"abstract": "Let $M \\subset X$ be a submanifold of a rational homogeneous space $X$ such that the normal sequence splits. We prove that $M$ is also rational homogeneo... |
https://arxiv.org/abs/2010.00119 | On the size of $A+λA$ for algebraic $λ$ | For a finite set $A\subset \mathbb{R}$ and real $\lambda$, let $A+\lambda A:=\{a+\lambda b :\, a,b\in A\}$. Combining a structural theorem of Freiman on sets with small doubling constants together with a discrete analogue of Prékopa--Leindler inequality we prove a lower bound $|A+\sqrt{2} A|\geq (1+\sqrt{2})^2|A|-O({|A... | \section{Introduction}
Let $A$ be a finite non-empty set with elements in a commutative ring $R$ and let $\lambda$ be
an element of $R$. A number of papers
are devoted to bounding
$|A+\lambda A|=|\{x+\lambda y:x,y\in A\}|$
in terms of $|A|$ and various generalizations
of this problem.
In particular, the sums of sever... | {
"timestamp": "2020-10-02T02:04:38",
"yymm": "2010",
"arxiv_id": "2010.00119",
"language": "en",
"url": "https://arxiv.org/abs/2010.00119",
"abstract": "For a finite set $A\\subset \\mathbb{R}$ and real $\\lambda$, let $A+\\lambda A:=\\{a+\\lambda b :\\, a,b\\in A\\}$. Combining a structural theorem of Fre... |
https://arxiv.org/abs/1907.04844 | Minimum k-critical bipartite graphs | We study the problem of Minimum $k$-Critical Bipartite Graph of order $(n,m)$ - M$k$CBG-$(n,m)$: to find a bipartite $G=(U,V;E)$, with $|U|=n$, $|V|=m$, and $n>m>1$, which is $k$-critical bipartite, and the tuple $(|E|, \Delta_U, \Delta_V)$, where $\Delta_U$ and $\Delta_V$ denote the maximum degree in $U$ and $V$, resp... | \section{Introduction}\label{sec:introduction}
\subsection{Design of fault tolerant networks}
Let $\Pi$ be a graph property that is {\em monotone}, i.e., it is preserved when adding edges to a graph. A graph $G$ is fault-tolerant with respect to $\Pi$ if the graph obtained from $G$ by removing a certain set of vertice... | {
"timestamp": "2020-12-08T02:22:07",
"yymm": "1907",
"arxiv_id": "1907.04844",
"language": "en",
"url": "https://arxiv.org/abs/1907.04844",
"abstract": "We study the problem of Minimum $k$-Critical Bipartite Graph of order $(n,m)$ - M$k$CBG-$(n,m)$: to find a bipartite $G=(U,V;E)$, with $|U|=n$, $|V|=m$, a... |
https://arxiv.org/abs/0909.1434 | Homotopy, Delta-equivalence and concordance for knots in the complement of a trivial link | Link-homotopy and self Delta-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Delta-equivalent) to a trivial link. We study link-homotopy and self Delta-equivalence on a certain componen... | \section{Introduction}
For an ordered and oriented $n$-component link $L$, the {\em Milnor invariant}
$\overline{\mu}_L(I)$ is defined for each multi-index $I=i_1i_2...i_m$ with
entries from $\{1,...,n\}$ \cite{Milnor,Milnor2}. Here $m$ is called the
{\em length} of $\overline{\mu}_L(I)$ and denoted by $|I|$. Let $r... | {
"timestamp": "2009-09-08T11:16:42",
"yymm": "0909",
"arxiv_id": "0909.1434",
"language": "en",
"url": "https://arxiv.org/abs/0909.1434",
"abstract": "Link-homotopy and self Delta-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determ... |
https://arxiv.org/abs/1404.3391 | Adjacency labeling schemes and induced-universal graphs | We describe a way of assigning labels to the vertices of any undirected graph on up to $n$ vertices, each composed of $n/2+O(1)$ bits, such that given the labels of two vertices, and no other information regarding the graph, it is possible to decide whether or not the vertices are adjacent in the graph. This is optimal... | \section{Introduction}
An \emph{adjacency labeling scheme} for a given family of graphs is a way of assigning \emph{labels} to the vertices of each graph from the family such that given the labels of two vertices in the graph, and no other information, it is possible to determine whether or not the vertices are adjace... | {
"timestamp": "2014-04-15T02:08:54",
"yymm": "1404",
"arxiv_id": "1404.3391",
"language": "en",
"url": "https://arxiv.org/abs/1404.3391",
"abstract": "We describe a way of assigning labels to the vertices of any undirected graph on up to $n$ vertices, each composed of $n/2+O(1)$ bits, such that given the l... |
https://arxiv.org/abs/2109.05474 | Combinatorial models for topological Reeb spaces | There are two rather distinct approaches to Morse theory nowadays: smooth and discrete. We propose to study a real valued function by assembling all associated sections in a topological category. From this point of view, Reeb functions on stratified spaces are introduced, including both smooth and combinatorial example... | \section{Introduction}
Let~$X$ be a topological space and~$f\colon X \rightarrow {\mathbb{R}}$ a continuous function on it. A \emph{section}~$\sigma$ of~$f$ is a map~\hbox{$ [a,b] \rightarrow X$}, for some real numbers $a\leq b$, subject to~$f\circ \sigma (c)=c$. Two sections~$\sigma\colon [a,b]\rightarrow X$ and~$\rh... | {
"timestamp": "2021-09-14T02:20:55",
"yymm": "2109",
"arxiv_id": "2109.05474",
"language": "en",
"url": "https://arxiv.org/abs/2109.05474",
"abstract": "There are two rather distinct approaches to Morse theory nowadays: smooth and discrete. We propose to study a real valued function by assembling all assoc... |
https://arxiv.org/abs/1003.4436 | Knots and tropical curves | Using elementary ideas from Tropical Geometry, we assign a a tropical curve to every $q$-holonomic sequence of rational functions. In particular, we assign a tropical curve to every knot which is determined by the Jones polynomial of the knot and its parallels. The topical curve explains the relation between the AJ Con... | \section{Introduction}
\lbl{sec.intro}
\subsection{What is a $q$-holonomic sequence?}
\lbl{sub.qholo}
A sequence of rational functions $f_n(q) \in \mathbb Q(q)$
in a variable $q$ is $q$-{\em holonomic} if it
satisfies a linear recursion with coefficients polynomials in $q$ and $q^n$.
In other words, we have
\begin{... | {
"timestamp": "2010-06-17T02:02:14",
"yymm": "1003",
"arxiv_id": "1003.4436",
"language": "en",
"url": "https://arxiv.org/abs/1003.4436",
"abstract": "Using elementary ideas from Tropical Geometry, we assign a a tropical curve to every $q$-holonomic sequence of rational functions. In particular, we assign ... |
https://arxiv.org/abs/1404.4959 | One half of almost symmetric numerical semigroups | Let $S,T$ be two numerical semigroups. We study when $S$ is one half of $T$, with $T$ almost symmetric. If we assume that the type of $T$, $t(T)$, is odd, then for any $S$ there exist infinitely many such $T$ and we prove that $1 \leq t(T) \leq 2t(S)+1$. On the other hand, if $t(T)$ is even, there exists such $T$ if an... | \section{Introduction}
A numerical semigroup $S$ is a submonoid of $\mathbb{N}$ such that
$\mathbb{N} \setminus S$ is finite. Numerical semigroups arise in
several contexts, such as commutative algebra, algebraic geometry,
coding theory, number theory and combinatorics. Many classes of
numerical semigroups are defined... | {
"timestamp": "2014-04-22T02:05:02",
"yymm": "1404",
"arxiv_id": "1404.4959",
"language": "en",
"url": "https://arxiv.org/abs/1404.4959",
"abstract": "Let $S,T$ be two numerical semigroups. We study when $S$ is one half of $T$, with $T$ almost symmetric. If we assume that the type of $T$, $t(T)$, is odd, t... |
https://arxiv.org/abs/1611.00932 | Natural Partial Order on Rings with Involution | In this paper, we introduce a partial order on rings with involution, which is a generalization of the partial order on the set of projections in a Rickart *-ring. We prove that a *-ring with the natural partial order form a sectionally semi-complemented poset. It is proved that every interval [0,x] forms an orthomodul... | \section{introduction} An \textit{involution} $*$ on an associative ring $R$ is a mapping such that $(a+b)^*=a^*+b^*$,
$(ab)^*=b^*a^*$ and $(a^*)^*=a$, for all $ a,b\in R$. A ring with
involution $*$ is called a {\it $*$-ring}. Clearly, identity mapping is an
involution if and only if the ring is commutative. An el... | {
"timestamp": "2016-11-04T01:03:57",
"yymm": "1611",
"arxiv_id": "1611.00932",
"language": "en",
"url": "https://arxiv.org/abs/1611.00932",
"abstract": "In this paper, we introduce a partial order on rings with involution, which is a generalization of the partial order on the set of projections in a Rickar... |
https://arxiv.org/abs/1909.06589 | On Schur multiplier and projective representations of Heisenberg groups | In this article, we study the Schur mutiplier of the discrete as well as the finite Heisenberg groups and their t-variants. We describe the representation groups of these Heisenberg groups and through these give a construction of their finite dimensional complex projective irreducible representations. | \section{Introduction}
The theory of projective representations involves understanding homomorphisms from a group into the projective linear groups. Schur~\cite{IS4, IS7, IS11} extensively studied it. These representations appear naturally in the study of ordinary representations of groups and are known to have many a... | {
"timestamp": "2020-12-24T02:11:24",
"yymm": "1909",
"arxiv_id": "1909.06589",
"language": "en",
"url": "https://arxiv.org/abs/1909.06589",
"abstract": "In this article, we study the Schur mutiplier of the discrete as well as the finite Heisenberg groups and their t-variants. We describe the representation... |
https://arxiv.org/abs/1006.2562 | On a notion of "Galois closure" for extensions of rings | We introduce a notion of "Galois closure" for extensions of rings. We show that the notion agrees with the usual notion of Galois closure in the case of an S_n degree n extension of fields. Moreover, we prove a number of properties of this construction; for example, we show that it is functorial and respects base chang... | \section{Introduction}
Let $A$ be any {ring of rank $n$ over a base ring $B$}, i.e., a
$B$-algebra that is free of rank $n$ as a $B$-module. In this
article, we investigate a natural definition for the ``Galois
closure'' $G(A/B)$ of the ring $A$ as an extension of
$B$.\footnote{All rings are assumed to be commutative... | {
"timestamp": "2010-06-15T02:01:34",
"yymm": "1006",
"arxiv_id": "1006.2562",
"language": "en",
"url": "https://arxiv.org/abs/1006.2562",
"abstract": "We introduce a notion of \"Galois closure\" for extensions of rings. We show that the notion agrees with the usual notion of Galois closure in the case of a... |
https://arxiv.org/abs/1509.07457 | A simplicial complex is uniquely determined by its set of discrete Morse functions | We prove that a connected simplicial complex is uniquely determined by its complex of discrete Morse functions. This settles a question raised by Chari and Joswig. In the 1-dimensional case, this implies that the complex of rooted forests of a connected graph G completely determines G. | \section{Introduction}
The \emph{complex of discrete Morse functions} $\mathfrak{M}(K)$ of a finite simplicial complex $K$ was introduced by Chari and Joswig in \cite{ChJo} to study the topology of simplicial complexes in terms of their sets of discrete deformations. Despite the potential utility of this complex, ve... | {
"timestamp": "2015-09-25T02:13:50",
"yymm": "1509",
"arxiv_id": "1509.07457",
"language": "en",
"url": "https://arxiv.org/abs/1509.07457",
"abstract": "We prove that a connected simplicial complex is uniquely determined by its complex of discrete Morse functions. This settles a question raised by Chari an... |
https://arxiv.org/abs/2107.09962 | Cone Types, Automata, and Regular Partitions in Coxeter Groups | In this article we introduce the notion of a \textit{regular partition} of a Coxeter group. We develop the theory of these partitions, and show that the class of regular partitions is essentially equivalent to the class of automata (not necessarily finite state) recognising the language of reduced words in the Coxeter ... | \section*{Introduction}
In \cite{BH:93}, Brink and Howlett showed that every finitely generated Coxeter system $(W,S)$ is automatic by providing an explicit construction of a finite state automaton $\mathcal{A}_0$ recognising the language $\mathcal{L}(W,S)$ of reduced words of $(W,S)$. A key insight in~\cite{BH:93} ... | {
"timestamp": "2021-12-14T02:27:11",
"yymm": "2107",
"arxiv_id": "2107.09962",
"language": "en",
"url": "https://arxiv.org/abs/2107.09962",
"abstract": "In this article we introduce the notion of a \\textit{regular partition} of a Coxeter group. We develop the theory of these partitions, and show that the ... |
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