url stringlengths 31 38 | title stringlengths 7 229 | abstract stringlengths 44 2.87k | text stringlengths 319 2.51M | meta dict |
|---|---|---|---|---|
https://arxiv.org/abs/1103.1072 | Orthogonal decomposition of Lorentz transformations | The canonical decomposition of a Lorentz algebra element into a sum of orthogonal simple (decomposable) Lorentz bivectors is discussed, as well as the decomposition of a proper orthochronous Lorentz transformation into a product of commuting Lorentz transformations, each of which is the exponential of a simple bivector... | \section{Introduction}
It is common to define a three--dimensional rotation by specifying its axis and rotation angle. However, since the axis of rotation is fixed, we may also define the rotation in terms of the two--plane orthogonal to the axis. This point of view is useful in higher dimensions, where one no longe... | {
"timestamp": "2012-11-27T02:04:14",
"yymm": "1103",
"arxiv_id": "1103.1072",
"language": "en",
"url": "https://arxiv.org/abs/1103.1072",
"abstract": "The canonical decomposition of a Lorentz algebra element into a sum of orthogonal simple (decomposable) Lorentz bivectors is discussed, as well as the decom... |
https://arxiv.org/abs/1803.01290 | Second homotopy and invariant geometry of flag manifolds | We use the Hopf fibration to explicitly compute generators of the second homotopy group of the flag manifolds of a compact Lie group. We show that these $2$-spheres have nice geometrical properties such as being totally geodesic surfaces with respect to any invariant metric on the flag manifold. We characterize when th... | \section*{Introduction}
We consider (generalized) flag manifolds of a compact Lie group $U$, that is, homogeneous manifolds $\mathbb{F}_\Theta=U/U_\Theta$, where the isotropy $U_\Theta$ is a connected subgroup with maximal rank. These omnipresent manifolds have being studied from several points of view: symplectic ge... | {
"timestamp": "2018-03-06T02:09:43",
"yymm": "1803",
"arxiv_id": "1803.01290",
"language": "en",
"url": "https://arxiv.org/abs/1803.01290",
"abstract": "We use the Hopf fibration to explicitly compute generators of the second homotopy group of the flag manifolds of a compact Lie group. We show that these $... |
https://arxiv.org/abs/1302.6632 | Constructive proof of the Carpenter's Theorem | We give a constructive proof of Carpenter's Theorem due to Kadison. Unlike the original proof our approach also yields the real case of this theorem. | \section{Kadison's theorem}
In \cite{k1} and \cite{k2} Kadison gave a complete characterization of the diagonals of orthogonal projections on a Hilbert space $\mathcal H$.
\begin{thm}[Kadison]\label{Kadison} Let $\{d_{i}\}_{i\in I}$ be a sequence in $[0,1]$. Define
\[a=\sum_{d_{i}<1/2}d_{i} \quad\text{and}\quad b=\s... | {
"timestamp": "2013-02-28T02:00:41",
"yymm": "1302",
"arxiv_id": "1302.6632",
"language": "en",
"url": "https://arxiv.org/abs/1302.6632",
"abstract": "We give a constructive proof of Carpenter's Theorem due to Kadison. Unlike the original proof our approach also yields the real case of this theorem.",
"s... |
https://arxiv.org/abs/1110.0994 | A Spectral Sequence Connecting Continuous With Locally Continuous Group Cohomology | We present a spectral sequence connecting the continuous and 'locally continuous' group cohomologies for topological groups. As an application it is shown that for contractible topological groups these cohomology concepts coincide. Similar results for k-groups and smooth cochains on Lie groups are also obtained. | \section*{Introduction}
There exist various cohomology concepts for topological groups $G$ and
topological coefficient groups $V$ which take the topologies of the group and
that of the coefficients into account.
One is obtained by restricting oneself to the complex $C_c^* (G;V)$
continuous group cochains only whose ... | {
"timestamp": "2011-10-06T02:03:28",
"yymm": "1110",
"arxiv_id": "1110.0994",
"language": "en",
"url": "https://arxiv.org/abs/1110.0994",
"abstract": "We present a spectral sequence connecting the continuous and 'locally continuous' group cohomologies for topological groups. As an application it is shown t... |
https://arxiv.org/abs/1305.3587 | Perimeter-minimizing Tilings by Convex and Non-convex Pentagons | We study the presumably unnecessary convexity hypothesis in the theorem of Chung et al. [CFS] on perimeter-minimizing planar tilings by convex pentagons. We prove that the theorem holds without the convexity hypothesis in certain special cases, and we offer direction for future research. | \section{Introduction}
\label{intro}
\subsection{Tilings of the plane by pentagons}
Chung et al. \cite[Thm. 3.5]{pen11} proved that certain ``Cairo'' and ``Prismatic'' pentagons provide least-perimeter tilings by (mixtures of) convex pentagons, and they conjecture that the restriction to convex pentagons is unnecessa... | {
"timestamp": "2013-05-16T02:02:31",
"yymm": "1305",
"arxiv_id": "1305.3587",
"language": "en",
"url": "https://arxiv.org/abs/1305.3587",
"abstract": "We study the presumably unnecessary convexity hypothesis in the theorem of Chung et al. [CFS] on perimeter-minimizing planar tilings by convex pentagons. We... |
https://arxiv.org/abs/1908.11683 | Identities and Properties of Multi-Dimensional Generalized Bessel Functions | The Generalized Bessel Function (GBF) extends the single variable Bessel function to several dimensions and indices in a nontrivial manner. Two-dimensional GBFs have been studied extensively in the literature and have found application in laser physics, crystallography, and electromagnetics. In this article, we documen... | \section{Introduction}
Bessel functions \cite{watson22} are pervasive in mathematics and physics and are particularly important in the study of wave propagation. Bessel functions were first studied in the context of solutions to a second order differential equation known as Bessel's equation:
\begin{equation}
x^2f''(x... | {
"timestamp": "2021-04-29T02:17:59",
"yymm": "1908",
"arxiv_id": "1908.11683",
"language": "en",
"url": "https://arxiv.org/abs/1908.11683",
"abstract": "The Generalized Bessel Function (GBF) extends the single variable Bessel function to several dimensions and indices in a nontrivial manner. Two-dimensiona... |
https://arxiv.org/abs/1110.6851 | Finite dimensional ordered vector spaces with Riesz interpolation and Effros-Shen's unimodularity conjecture | It is shown that, for any field F \subseteq R, any ordered vector space structure of F^n with Riesz interpolation is given by an inductive limit of sequence with finite stages (F^n,\F_{>= 0}^n) (where n does not change). This relates to a conjecture of Effros and Shen, since disproven, which is given by the same statem... | \section{Introduction}
In this article we prove the following.
\begin{thm}
\label{MainResult}
Let $\mathbb{F}$ be a subfield of the real numbers, let $n$ be a natural number, and suppose that $(V,V^+)$ is a ordered directed $n$-dimensional vector space over $\mathbb{F}$ with Riesz interpolation.
Then there exists an i... | {
"timestamp": "2011-11-01T01:07:28",
"yymm": "1110",
"arxiv_id": "1110.6851",
"language": "en",
"url": "https://arxiv.org/abs/1110.6851",
"abstract": "It is shown that, for any field F \\subseteq R, any ordered vector space structure of F^n with Riesz interpolation is given by an inductive limit of sequenc... |
https://arxiv.org/abs/1710.08253 | Differential posets and restriction in critical groups | In recent work, Benkart, Klivans, and Reiner defined the critical group of a faithful representation of a finite group $G$, which is analogous to the critical group of a graph. In this paper we study maps between critical groups induced by injective group homomorphisms and in particular the map induced by restriction o... | \section{Introduction} \label{sec:intro}
The critical group $K(\Gamma)$ is a well-studied abelian group invariant of a finite graph $\Gamma$ which encodes information about the dynamics of a process called \textit{chip firing} on the graph (see \cite{LP} where critical groups are called \textit{sandpile groups}). Rec... | {
"timestamp": "2017-10-24T02:13:47",
"yymm": "1710",
"arxiv_id": "1710.08253",
"language": "en",
"url": "https://arxiv.org/abs/1710.08253",
"abstract": "In recent work, Benkart, Klivans, and Reiner defined the critical group of a faithful representation of a finite group $G$, which is analogous to the crit... |
https://arxiv.org/abs/2206.12269 | Data-driven reduced order models using invariant foliations, manifolds and autoencoders | This paper explores how to identify a reduced order model (ROM) from a physical system. A ROM captures an invariant subset of the observed dynamics. We find that there are four ways a physical system can be related to a mathematical model: invariant foliations, invariant manifolds, autoencoders and equation-free models... | \section{Introduction}
There is a great interest in the scientific community to identify
explainable and/or parsimonious mathematical models from data. In
this paper we classify these methods and identify one concept that
is best suited to accurately calculate reduced order models (ROM)
from off-line data. A ROM must ... | {
"timestamp": "2022-06-27T02:13:05",
"yymm": "2206",
"arxiv_id": "2206.12269",
"language": "en",
"url": "https://arxiv.org/abs/2206.12269",
"abstract": "This paper explores how to identify a reduced order model (ROM) from a physical system. A ROM captures an invariant subset of the observed dynamics. We fi... |
https://arxiv.org/abs/1606.03670 | The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices | The probability that all eigenvalues of a product of $m$ independent $N \times N$ sub-blocks of a Haar distributed random real orthogonal matrix of size $(L_i+N) \times (L_i+N)$, $(i=1,\dots,m)$ are real is calculated as a multi-dimensional integral, and as a determinant. Both involve Meijer G-functions. Evaluation for... | \section{Introduction}
In general an $N \times N$ real matrix may have both real and complex eigenvalues. A natural question is thus to ask for the probability $p_{N,k}^{X}$ that a random real matrix $X$ chosen from a particular distribution has a specific number
$k$ of real eigenvalues. Due to the complex eigenvalues ... | {
"timestamp": "2017-07-06T02:07:35",
"yymm": "1606",
"arxiv_id": "1606.03670",
"language": "en",
"url": "https://arxiv.org/abs/1606.03670",
"abstract": "The probability that all eigenvalues of a product of $m$ independent $N \\times N$ sub-blocks of a Haar distributed random real orthogonal matrix of size ... |
https://arxiv.org/abs/1801.03741 | Double asymptotic for random walks on hypercubes | We consider the sum of the coordinates of a simple random walk on the K-dimensional hypercube, and prove a double asymptotic of this process, as both the time parameter n and the space parameter K tend to infinity. Depending on the asymptotic ratio of the two parameters, they converge towards either a Brownian motion, ... | \section{Introduction}
Many results (like the Law of Large Numbers or the Central Limit Theorem) are already known for the asymptotic behavior in time of an additive functional of a Markov chain (see for instance \cite{MT09}). But the case where we consider a sequence of such processes is only partially studied. Here ... | {
"timestamp": "2019-09-23T02:08:54",
"yymm": "1801",
"arxiv_id": "1801.03741",
"language": "en",
"url": "https://arxiv.org/abs/1801.03741",
"abstract": "We consider the sum of the coordinates of a simple random walk on the K-dimensional hypercube, and prove a double asymptotic of this process, as both the ... |
https://arxiv.org/abs/1410.5032 | Monotonicity of Avoidance Coupling on $K_N$ | Answering a question by Angel, Holroyd, Martin, Wilson and Winkler, we show that the maximal number of non-colliding coupled simple random walks on the complete graph $K_N$, which take turns, moving one at a time, is monotone in $N$. We use this fact to couple $\lceil \frac N4 \rceil$ such walks on $K_N$, improving the... | \section{Introduction}
Let $G=([N],E)$ be a graph whose vertices are the set of integers $[N]=\{1,\dots,N\}$.
A \emph{simple random walk} on this graph is a Markov chain $(X_t)_{t\in\mathbb{Z}}$ of
elements in $[N]$ such that for all $t\in\mathbb{Z}$ the distribution of $X_t$ is
uniform on the neighbors of $X_{t-1}$.
... | {
"timestamp": "2015-05-12T02:08:01",
"yymm": "1410",
"arxiv_id": "1410.5032",
"language": "en",
"url": "https://arxiv.org/abs/1410.5032",
"abstract": "Answering a question by Angel, Holroyd, Martin, Wilson and Winkler, we show that the maximal number of non-colliding coupled simple random walks on the comp... |
https://arxiv.org/abs/1103.1339 | Isotone maps on lattices | Let (L_i : i\in I) be a family of lattices in a nontrivial lattice variety V, and let \phi_i: L_i --> M, for i\in I, be isotone maps (not assumed to be lattice homomorphisms) to a common lattice M (not assumed to lie in V). We show that the maps \phi_i can be extended to an isotone map \phi: L --> M, where L is the fre... | \section{Introduction}\label{S.intro}
By Yu.\,I. Sorkin \cite[Theorem~3]{sorkin},
if $\E L = (L_i \mid i\in I)$ is a family of lattices
and $\gf_i\colon L_i\to M$ are isotone maps of the lattices $L_i$
into a lattice $M$, then there exists an
isotone map $\gf$ from the
free product $\Free\E L$ of the $L_i$ to $M$
that ... | {
"timestamp": "2011-03-08T02:04:09",
"yymm": "1103",
"arxiv_id": "1103.1339",
"language": "en",
"url": "https://arxiv.org/abs/1103.1339",
"abstract": "Let (L_i : i\\in I) be a family of lattices in a nontrivial lattice variety V, and let \\phi_i: L_i --> M, for i\\in I, be isotone maps (not assumed to be l... |
https://arxiv.org/abs/2205.09049 | Random graph embeddings with general edge potentials | In this paper, we study random embeddings of polymer networks distributed according to any potential energy which can be expressed in terms of distances between pairs of monomers. This includes freely jointed chains, steric effects, Lennard-Jones potentials, bending energies, and other physically realistic models.A con... | \section{Introduction}
In the study of network polymers, it is common to represent the polymer topology by a graph and study the spatial distribution of the monomers in terms of the eigenvalues and eigenvectors of the Kirchhoff (or architecture~\cite{Kuchanov88}) matrix. In mathematics, this matrix is usually known as ... | {
"timestamp": "2022-05-19T02:22:16",
"yymm": "2205",
"arxiv_id": "2205.09049",
"language": "en",
"url": "https://arxiv.org/abs/2205.09049",
"abstract": "In this paper, we study random embeddings of polymer networks distributed according to any potential energy which can be expressed in terms of distances b... |
https://arxiv.org/abs/1407.6701 | Uniform growth rate | In an evolutionary system in which the rules of mutation are local in nature, the number of possible outcomes after $m$ mutations is an exponential function of $m$ but with a rate that depends only on the set of rules and not the size of the original object. We apply this principle to find a uniform upper bound for the... | \section{Introduction}
Let $G$ be a group and $S$ be a generating set for $G$. We denote the
word length in $G$ associated to $S$ with $\norm{\param}_S$. Recall that
the growth rate of $G$ (relative to $S$) is defined to be
\[
h_G = \lim_{R \to \infty} \frac{\log \, \# B_R(G)}{R},
\qquad\text{where}\... | {
"timestamp": "2016-05-13T02:06:08",
"yymm": "1407",
"arxiv_id": "1407.6701",
"language": "en",
"url": "https://arxiv.org/abs/1407.6701",
"abstract": "In an evolutionary system in which the rules of mutation are local in nature, the number of possible outcomes after $m$ mutations is an exponential function... |
https://arxiv.org/abs/2105.04684 | An automatic system to detect equivalence between iterative algorithms | When are two algorithms the same? How can we be sure a recently proposed algorithm is novel, and not a minor twist on an existing method? In this paper, we present a framework for reasoning about equivalence between a broad class of iterative algorithms, with a focus on algorithms designed for convex optimization. We p... |
\section{Introduction}\label{intro}
Large-scale optimization problems in machine learning, signal processing, and imaging have fueled ongoing interest in iterative optimization algorithms.
New optimization algorithms are regularly proposed in order to
capture more complicated models, reduce computational burdens,... | {
"timestamp": "2021-05-12T02:06:03",
"yymm": "2105",
"arxiv_id": "2105.04684",
"language": "en",
"url": "https://arxiv.org/abs/2105.04684",
"abstract": "When are two algorithms the same? How can we be sure a recently proposed algorithm is novel, and not a minor twist on an existing method? In this paper, w... |
https://arxiv.org/abs/1405.1003 | From Boltzmann to random matrices and beyond | These expository notes propose to follow, across fields, some aspects of the concept of entropy. Starting from the work of Boltzmann in the kinetic theory of gases, various universes are visited, including Markov processes and their Helmholtz free energy, the Shannon monotonicity problem in the central limit theorem, t... | \section{Ludwig Boltzmann and his H-Theorem}
\smallskip
\subsection{Entropy}
A simple way to introduce the Boltzmann entropy is to use the concept of
combinatorial disorder. More precisely, let us consider a system of $n$
distinguishable particles, each of them being in one of the $r$ possible
states (typically ene... | {
"timestamp": "2015-02-27T02:07:28",
"yymm": "1405",
"arxiv_id": "1405.1003",
"language": "en",
"url": "https://arxiv.org/abs/1405.1003",
"abstract": "These expository notes propose to follow, across fields, some aspects of the concept of entropy. Starting from the work of Boltzmann in the kinetic theory o... |
https://arxiv.org/abs/2112.03758 | Matrix completion and semidefinite matrices | Positive semidefinite Hermitian matrices that are not fully specified can be completed provided their underlying graph is chordal. If the matrix is positive definite the completion can be uniquely characterized as the matrix that maximizes the determinant, or as the matrix whose inverse has zeroes in those places that ... | \section{Introduction}
The question of which partial Hermitian matrices $H$ can be completed to give a fully specified positive definite Hermitian matrix was solved in \cite{grone}. The solution provided in \cite{grone} was constructive but only gave one element of the completion with each step. While giving the correc... | {
"timestamp": "2021-12-08T02:22:36",
"yymm": "2112",
"arxiv_id": "2112.03758",
"language": "en",
"url": "https://arxiv.org/abs/2112.03758",
"abstract": "Positive semidefinite Hermitian matrices that are not fully specified can be completed provided their underlying graph is chordal. If the matrix is positi... |
https://arxiv.org/abs/1107.1638 | Weighted algorithms for compressed sensing and matrix completion | This paper is about iteratively reweighted basis-pursuit algorithms for compressed sensing and matrix completion problems. In a first part, we give a theoretical explanation of the fact that reweighted basis pursuit can improve a lot upon basis pursuit for exact recovery in compressed sensing. We exhibit a condition th... | \section{Introduction}
\label{sec:introduction}
In this paper, we consider the statistical analysis of high
dimensional structured data in two close setups: vectors with small
support and matrices with low rank. In the first setup, known as
Compressed Sensing (CS)
\cite{MR2241189,MR1639094,MR2243152,MR2236170,IEEE-Do... | {
"timestamp": "2011-07-11T02:02:46",
"yymm": "1107",
"arxiv_id": "1107.1638",
"language": "en",
"url": "https://arxiv.org/abs/1107.1638",
"abstract": "This paper is about iteratively reweighted basis-pursuit algorithms for compressed sensing and matrix completion problems. In a first part, we give a theore... |
https://arxiv.org/abs/2204.02933 | Quantitative differentiation and the medial axis | We study the medial axis of a set $K$ in Euclidean space (the set of points in space with more than one closest point in $K$) from a "coarse" and "quantitative" perspective. We show that on "most" balls $B(x,r)$ in the complement of $K$, the set of almost-closest points to $x$ in $K$ takes up a small angle as seen from... | \section{Introduction}
If $K\subseteq \mathbb{R}^k$, the distance from a point $p\in\mathbb{R}^k$ to the set $K$ is
$$ d(p,K) = \inf\{d(p,x) : x\in K\},$$
where $d(p,x)$ denotes the Euclidean distance $|p-x|$. If $K$ is closed and $p\in \mathbb{R}^k$, then there is always a point $x\in K$ such that $d(p,x)=d(p,K)$, ... | {
"timestamp": "2022-04-07T02:25:23",
"yymm": "2204",
"arxiv_id": "2204.02933",
"language": "en",
"url": "https://arxiv.org/abs/2204.02933",
"abstract": "We study the medial axis of a set $K$ in Euclidean space (the set of points in space with more than one closest point in $K$) from a \"coarse\" and \"quan... |
https://arxiv.org/abs/1409.2214 | Approximation of eigenvalues of spot cross volatility matrix with a view toward principal component analysis | In order to study the geometry of interest rates market dynamics, Malliavin, Mancino and Recchioni [A non-parametric calibration of the HJM geometry: an application of Itô calculus to financial statistics, {\it Japanese Journal of Mathematics}, 2, pp.55--77, 2007] introduced a scheme, which is based on the Fourier Seri... | \section{Introduction}
Let $X$ be a $d$-dimensional stochastic process defined on a probability space
$(\Omega, \mathcal{F}, (\mathcal{F})_t, P)$ by
\begin{equation} \label{eqX}
d X (t) = \bA (t, w)dt + \bB (t,w)dW(t), \ 0 \leq t \leq T,
\end{equation}
where $W$ is a $d_1$-dimensional standard Brownian motion,... | {
"timestamp": "2014-09-09T02:12:49",
"yymm": "1409",
"arxiv_id": "1409.2214",
"language": "en",
"url": "https://arxiv.org/abs/1409.2214",
"abstract": "In order to study the geometry of interest rates market dynamics, Malliavin, Mancino and Recchioni [A non-parametric calibration of the HJM geometry: an app... |
https://arxiv.org/abs/1702.00205 | Decomposing Weighted Graphs | We solve the following problem: Can an undirected weighted graph G be parti- tioned into two non-empty induced subgraphs satisfying minimum constraints for the sum of edge weights at vertices of each subgraph? We show that this is possible for all constraints a(x), b(x) satisfying d_G(x) >= a(x) + b(x) + 2W_G(x), for e... | \section{Introduction}
All graphs considered in this paper are finite, undirected and weighted. A weighted graph is a triple $G = (V, E, w)$ such that $(V, E)$ is an undirected simple finite graph and $w : E \mapsto \mathbb{R}_{>0}$ is a weight function. Where $xy \notin E$, we further define $w_{xy} = w_{yx} = 0$.
... | {
"timestamp": "2017-02-02T02:04:28",
"yymm": "1702",
"arxiv_id": "1702.00205",
"language": "en",
"url": "https://arxiv.org/abs/1702.00205",
"abstract": "We solve the following problem: Can an undirected weighted graph G be parti- tioned into two non-empty induced subgraphs satisfying minimum constraints fo... |
https://arxiv.org/abs/1905.01568 | Relations among spheroidal and spherical harmonics | A contragenic function in a domain $\Omega\subseteq\mathbf{R}^3$ is a reduced-quaternion-valued (i.e. the last coordinate function is zero) harmonic function, which is orthogonal in $L^2(\Omega)$ to all monogenic functions and their conjugates. The notion of contragenicity depends on the domain and thus is not a local ... | \section{Introduction}
In certain physical problems in nonspherical domains, it has been
found convenient to replace the classical solid spherical harmonics
with harmonic functions better adapted to the domain in question. For
example, spheroidal harmonics are used in \cite{Hot} for modeling
potential fields ar... | {
"timestamp": "2019-05-07T02:17:20",
"yymm": "1905",
"arxiv_id": "1905.01568",
"language": "en",
"url": "https://arxiv.org/abs/1905.01568",
"abstract": "A contragenic function in a domain $\\Omega\\subseteq\\mathbf{R}^3$ is a reduced-quaternion-valued (i.e. the last coordinate function is zero) harmonic fu... |
https://arxiv.org/abs/2006.02599 | Hamilton Cycles in the Semi-random Graph Process | The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $u$ is presented to the player independently and uniformly at random. The player then adaptively selects a vertex $v$, and adds the edge $uv$ to the graph. For a fixed... | \section{Introduction}
In this paper, we consider the \textbf{semi-random process} introduced recently in~\cite{process1} that can be viewed as a ``one player game''. The process starts from $G_0$, the empty graph on the vertex set $[n]=\{1,\ldots,n\}$. In each step $t$, a vertex $u_t$ is chosen uniformly at random fr... | {
"timestamp": "2020-06-05T02:05:58",
"yymm": "2006",
"arxiv_id": "2006.02599",
"language": "en",
"url": "https://arxiv.org/abs/2006.02599",
"abstract": "The semi-random graph process is a single player game in which the player is initially presented an empty graph on $n$ vertices. In each round, a vertex $... |
https://arxiv.org/abs/1507.02753 | The Density of Shifted and Affine Eisenstein Polynomials | In this paper we provide a complete answer to a question by Heyman and Shparlinski concerning the natural density of polynomials which are irreducible by Eisenstein's criterion after applying some shift. The main tool we use is a local to global principle for density computations over a free $\mathbb Z$-module of finit... | \section{Introduction}
\label{sec:introduction}
Let $\mathbb{Z}$ be the ring of rational integers.
The Eisenstein irreducibility criterion~\cite{bib:eisenstein,bib:schoenemann} is
a very convenient tool to establish that a polynomial in $\mathbb{Z}[x]$ is
irreducible.
It is a well understood fact that the density of i... | {
"timestamp": "2015-07-13T02:04:57",
"yymm": "1507",
"arxiv_id": "1507.02753",
"language": "en",
"url": "https://arxiv.org/abs/1507.02753",
"abstract": "In this paper we provide a complete answer to a question by Heyman and Shparlinski concerning the natural density of polynomials which are irreducible by ... |
https://arxiv.org/abs/1405.2331 | Fixed points of local actions of nilpotent Lie groups on surfaces | Let $G$ be connected nilpotent Lie group acting locally on a real surface $M$. Let $\varphi$ be the local flow on $M$ induced by a $1$-parameter subgroup. Assume $K$ is a compact set of fixed points of $\varphi$ and $U$ is a neighborhood of $K$ containing no other fixed points.Theorem: If the Dold fixed-point index of ... | \section{Introduction} \mylabel{sec:intro}
{\em Notation:}
$M$ is a manifold with tangent bundle $TM$ and boundary $\ensuremath{\partial} M$.
The set of vector fields on $M$ is $\ensuremath{\mathfrak v} (M)$, and the set of vector
fields that are $C^1$ (continously differentiable) is $\ensuremath{\mathfrak v}^1
(M)... | {
"timestamp": "2014-05-12T02:11:42",
"yymm": "1405",
"arxiv_id": "1405.2331",
"language": "en",
"url": "https://arxiv.org/abs/1405.2331",
"abstract": "Let $G$ be connected nilpotent Lie group acting locally on a real surface $M$. Let $\\varphi$ be the local flow on $M$ induced by a $1$-parameter subgroup. ... |
https://arxiv.org/abs/1701.06030 | Fourth-order time-stepping for stiff PDEs on the sphere | We present in this paper algorithms for solving stiff PDEs on the unit sphere with spectral accuracy in space and fourth-order accuracy in time. These are based on a variant of the double Fourier sphere method in coefficient space with multiplication matrices that differ from the usual ones, and implicit-explicit time-... | \section{Introduction}
We are interested in computing smooth solutions of stiff PDEs on the unit sphere of the form
\begin{equation}
u_t = \mathcal{L}u + \mathcal{N}(u), \quad u(t=0,x,y,z)=u_0(x,y,z),
\label{PDE}
\end{equation}
\noindent where $u(t,x,y,z)$ is a function of time $t$ and Cartesian coordinates $(x,y,z)... | {
"timestamp": "2017-12-27T02:02:09",
"yymm": "1701",
"arxiv_id": "1701.06030",
"language": "en",
"url": "https://arxiv.org/abs/1701.06030",
"abstract": "We present in this paper algorithms for solving stiff PDEs on the unit sphere with spectral accuracy in space and fourth-order accuracy in time. These are... |
https://arxiv.org/abs/2205.04211 | Real Algebraic Geometry, Positivity and Convexity | Chapters 1 to 4 are the lecture notes of my course "Real Algebraic Geometry I" from the winter term 2020/2021. Chapters 5 to 8 are the lecture notes of its continuation "Real Algebraic Geometry II" from the summer term 2021. Chapters 9 and 10 are the lecture notes of its further continuation "Geometry of Linear Matrix ... | \chapter{Introduction}
The study of polynomial equations is a canonical subject in mathematics education, as is illustrated by the following examples: Quadratic equations in one variable (high school), systems of linear equations (linear algebra), polynomial equations in one variable and their symmetries (algebra, Gal... | {
"timestamp": "2022-05-10T02:36:33",
"yymm": "2205",
"arxiv_id": "2205.04211",
"language": "en",
"url": "https://arxiv.org/abs/2205.04211",
"abstract": "Chapters 1 to 4 are the lecture notes of my course \"Real Algebraic Geometry I\" from the winter term 2020/2021. Chapters 5 to 8 are the lecture notes of ... |
https://arxiv.org/abs/1706.09093 | On the imaginary parts of chromatic root | While much attention has been directed to the maximum modulus and maximum real part of chromatic roots of graphs of order $n$ (that is, with $n$ vertices), relatively little is known about the maximum imaginary part of such graphs. We prove that the maximum imaginary part can grow linearly in the order of the graph. We... | \section{Introduction}
A (vertex) $k$-colouring of a (finite, undirected, simple) graph $G$ is a function $f:V(G) \rightarrow \{1,\ldots,k\}$ such that no two adjacent vertices receive the same colour, that is, if $uv$ is an edge of $G$, then $f(u) \neq f(v)$. The function $\pi(G,k)$ that for all nonnegative integers $... | {
"timestamp": "2017-06-29T02:02:31",
"yymm": "1706",
"arxiv_id": "1706.09093",
"language": "en",
"url": "https://arxiv.org/abs/1706.09093",
"abstract": "While much attention has been directed to the maximum modulus and maximum real part of chromatic roots of graphs of order $n$ (that is, with $n$ vertices)... |
https://arxiv.org/abs/1312.7538 | A technique for determining the signs of sensitivities of steady states in chemical reaction networks | We present a computational procedure to characterize the signs of sensitivities of steady states to parameter perturbations in chemical reaction networks. | \section{Introduction}
An important question in the mathematical analysis of chemical reaction
networks is the characterization of sensitivities of steady states to
perturbations in parameters. An example of a parameter is the total
concentration of an enzyme in its various activity states. Its value might be
manipu... | {
"timestamp": "2013-12-31T02:07:40",
"yymm": "1312",
"arxiv_id": "1312.7538",
"language": "en",
"url": "https://arxiv.org/abs/1312.7538",
"abstract": "We present a computational procedure to characterize the signs of sensitivities of steady states to parameter perturbations in chemical reaction networks.",... |
https://arxiv.org/abs/0903.3456 | A Groupoid Approach to Discrete Inverse Semigroup Algebras | Let $K$ be a commutative ring with unit and $S$ an inverse semigroup. We show that the semigroup algebra $KS$ can be described as a convolution algebra of functions on the universal étale groupoid associated to $S$ by Paterson. This result is a simultaneous generalization of the author's earlier work on finite inverse ... | \section{Introduction}
It is by now well established in the $C^*$-algebra community that there is a close relationship between inverse semigroup $C^*$-algebras and \'etale groupoid $C^*$-algebras~\cite{Paterson,Exel,Renault,graphinverse,ultragraph,higherrank,resendeetale,strongmorita}. More precisely, Paterson assigne... | {
"timestamp": "2009-03-20T06:33:31",
"yymm": "0903",
"arxiv_id": "0903.3456",
"language": "en",
"url": "https://arxiv.org/abs/0903.3456",
"abstract": "Let $K$ be a commutative ring with unit and $S$ an inverse semigroup. We show that the semigroup algebra $KS$ can be described as a convolution algebra of f... |
https://arxiv.org/abs/0905.0318 | Mod-Poisson convergence in probability and number theory | Building on earlier work introducing the notion of "mod-Gaussian" convergence of sequences of random variables, which arises naturally in Random Matrix Theory and number theory, we discuss the analogue notion of "mod-Poisson" convergence. We show in particular how it occurs naturally in analytic number theory in the cl... | \section{Introduction} \label{section:Intro}
In our earlier paper~\cite{jkn} with J. Jacod,\footnote{\ Although
this new paper is largely self-contained, it is likely to be most
useful for readers who have at least looked at the introduction and
the examples in~\cite{jkn}, especially Section 4} motivated by
res... | {
"timestamp": "2009-12-26T20:47:17",
"yymm": "0905",
"arxiv_id": "0905.0318",
"language": "en",
"url": "https://arxiv.org/abs/0905.0318",
"abstract": "Building on earlier work introducing the notion of \"mod-Gaussian\" convergence of sequences of random variables, which arises naturally in Random Matrix Th... |
https://arxiv.org/abs/1903.00800 | On the correspondence of external rays under renormalization | Let $P$ be a monic polynomial of degree $D \geq 3$ whose filled Julia set $K_P$ has a non-degenerate periodic component $K$ of period $k \geq 1$ and renormalization degree $2 \leq d<D$. Let $I=I_K$ denote the set of angles $\theta$ on the circle ${\mathbb T}={\mathbb R}/{\mathbb Z}$ for which the (smooth or broken) ext... | \section{Introduction}\label{sec:intro}
This paper provides an understanding of the external rays that accumulate (and in particular land) on a non-degenerate periodic component of a disconnected polynomial Julia set. It can be viewed as a complement to the well-studied case of connected Julia sets initiated by Douady... | {
"timestamp": "2019-03-05T02:15:20",
"yymm": "1903",
"arxiv_id": "1903.00800",
"language": "en",
"url": "https://arxiv.org/abs/1903.00800",
"abstract": "Let $P$ be a monic polynomial of degree $D \\geq 3$ whose filled Julia set $K_P$ has a non-degenerate periodic component $K$ of period $k \\geq 1$ and ren... |
https://arxiv.org/abs/1209.2655 | Positivity and Transportation | We prove in this paper that the weighted volume of the set of integral transportation matrices between two integral histograms r and c of equal sum is a positive definite kernel of r and c when the set of considered weights forms a positive definite matrix. The computation of this quantity, despite being the subject of... | \section*{Notes and References}\addcontentsline{toc}{section}{Notes and References}\markright{Notes and References}
\begin{small}\addtolength{\baselineskip}{-1.0pt}\parindent 0pt \parskip 4pt}{\clearpage\end{small}\addtolength{\baselineskip}{1.0pt}}
\newcommand{\nrsection}[1]{\subsection*{#1}}
\newcommand{\nrsubsection... | {
"timestamp": "2012-09-13T02:05:27",
"yymm": "1209",
"arxiv_id": "1209.2655",
"language": "en",
"url": "https://arxiv.org/abs/1209.2655",
"abstract": "We prove in this paper that the weighted volume of the set of integral transportation matrices between two integral histograms r and c of equal sum is a pos... |
https://arxiv.org/abs/1812.01086 | Centroaffine Duality for Spatial Polygons | In this paper, we discuss centroaffine geometry of polygons in $3$-space. For a polygon $X$ that is locally convex with respect to an origin together with a transversal vector field $U$, we define the centroaffine dual pair $(Y,V)$ similarly to [6]. We prove that vertices of $(X,U)$ correspond to flattening points for ... | \section{Introduction}
Affine differential geometry of curves in $3$-space studies differential concepts that are invariant under the affine group. When a distinguished
origin is fixed, the study of such curves becomes part of the centroaffine differential geometry.
We shall consider in this paper centroaffine conce... | {
"timestamp": "2019-05-14T02:32:47",
"yymm": "1812",
"arxiv_id": "1812.01086",
"language": "en",
"url": "https://arxiv.org/abs/1812.01086",
"abstract": "In this paper, we discuss centroaffine geometry of polygons in $3$-space. For a polygon $X$ that is locally convex with respect to an origin together with... |
https://arxiv.org/abs/2203.06829 | Stabilized exponential-SAV schemes preserving energy dissipation law and maximum bound principle for the Allen-Cahn type equations | It is well-known that the Allen-Cahn equation not only satisfies the energy dissipation law but also possesses the maximum bound principle (MBP) in the sense that the absolute value of its solution is pointwise bounded for all time by some specific constant under appropriate initial/boundary conditions. In recent years... | \section{Introduction}
Let us consider a class of reaction-diffusion equations taking the following form
\begin{equation}
\label{AllenCahn}
u_t = \varepsilon^2\Delta u + f(u), \quad t > 0, \ \bm{x} \in \Omega,
\end{equation}
where $\Omega\subset\mathbb{R}^d$ is a spatial domain,
$u=u(t,\bm{x}):[0,\infty)\times\overlin... | {
"timestamp": "2022-03-15T01:32:48",
"yymm": "2203",
"arxiv_id": "2203.06829",
"language": "en",
"url": "https://arxiv.org/abs/2203.06829",
"abstract": "It is well-known that the Allen-Cahn equation not only satisfies the energy dissipation law but also possesses the maximum bound principle (MBP) in the se... |
https://arxiv.org/abs/1003.2821 | Unitary equivalence to a complex symmetric matrix: a modulus criterion | We develop a procedure for determining whether a square complex matrix is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. Our approach has several advantages over existing methods. We discuss these differences and present a number of examples. | \section{Introduction}
Following \cite{Tener}, we say that a matrix $T \in M_n(\mathbb{C})$ is \emph{UECSM} if it is
unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. Our primary motivation for studying this concept
stems from the emerging theory of complex symmetric operators
on Hilbert... | {
"timestamp": "2010-03-16T01:01:47",
"yymm": "1003",
"arxiv_id": "1003.2821",
"language": "en",
"url": "https://arxiv.org/abs/1003.2821",
"abstract": "We develop a procedure for determining whether a square complex matrix is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix. Our ap... |
https://arxiv.org/abs/1409.4510 | Minimum Weight Resolving Sets of Grid Graphs | For a simple graph $G=(V,E)$ and for a pair of vertices $u,v \in V$, we say that a vertex $w \in V$ resolves $u$ and $v$ if the shortest path from $w$ to $u$ is of a different length than the shortest path from $w$ to $v$. A set of vertices ${R \subseteq V}$ is a resolving set if for every pair of vertices $u$ and $v$ ... | \section{Introduction}
Let $G=(V,E)$ be a simple graph, and for each pair of vertices $u,v \in V$, let $d(u,v)$ denote the length of the shortest path from $u$ to $v$, where ${d(u,u)=0}$ ${\forall u \in V}$ and $d(u,v) = \infty$ if $u$ and $v$ are disconnected. For two distinct vertices $u,v \in V$, a vertex $w$ is s... | {
"timestamp": "2014-09-17T02:07:26",
"yymm": "1409",
"arxiv_id": "1409.4510",
"language": "en",
"url": "https://arxiv.org/abs/1409.4510",
"abstract": "For a simple graph $G=(V,E)$ and for a pair of vertices $u,v \\in V$, we say that a vertex $w \\in V$ resolves $u$ and $v$ if the shortest path from $w$ to ... |
https://arxiv.org/abs/0712.1680 | Diamond-$α$ Jensen's Inequality on Time Scales | The theory and applications of dynamic derivatives on time scales has recently received considerable attention. The primary purpose of this paper is to give basic properties of diamond-$\alpha$ derivatives which are a linear combination of delta and nabla dynamic derivatives on time scales. We prove a generalized versi... | \section{Introduction}
Jensen's inequality is of great interest in the theory of
differential and difference equations, as well as other areas of
mathematics. The original Jensen's inequality can be stated as
follows:
\begin{theorem}\emph{\cite{mit}}
\label{thm1} If $g \in C([a, b], (c, d))$ and $f \in C((c, d),
\mat... | {
"timestamp": "2008-04-09T18:02:29",
"yymm": "0712",
"arxiv_id": "0712.1680",
"language": "en",
"url": "https://arxiv.org/abs/0712.1680",
"abstract": "The theory and applications of dynamic derivatives on time scales has recently received considerable attention. The primary purpose of this paper is to give... |
https://arxiv.org/abs/2105.02722 | Mutual Visibility in Graphs | Let $G=(V,E)$ be a graph and $P\subseteq V$ a set of points. Two points are mutually visible if there is a shortest path between them without further points. $P$ is a mutual-visibility set if its points are pairwise mutually visible. The mutual-visibility number of $G$ is the size of any largest mutual-visibility set. ... | \section{Computational complexity}\label{sec:complexity}
To study the computational complexity of finding a maximum mutual-visibility set\xspace in a graph, we introduce the following decision problem.
\begin{definition}
{\sc Mutual-Visibility}\xspace problem: \\
{\sc Instance}: A graph $G=(V,E)$, a positive integer $... | {
"timestamp": "2021-05-07T02:21:52",
"yymm": "2105",
"arxiv_id": "2105.02722",
"language": "en",
"url": "https://arxiv.org/abs/2105.02722",
"abstract": "Let $G=(V,E)$ be a graph and $P\\subseteq V$ a set of points. Two points are mutually visible if there is a shortest path between them without further poi... |
https://arxiv.org/abs/2205.09096 | Extremal arrangements of points on the sphere for weighted cone-volume functionals | Weighted cone-volume functionals are introduced for the convex polytopes in $\mathbb{R}^n$. For these functionals, geometric inequalities are proved and the equality conditions are characterized. A variety of corollaries are derived, including extremal properties of the regular polytopes involving the $L_p$ surface are... | \section{Introduction}
The regular polytopes are a cornerstone of convex and discrete geometry, often arising as the solutions to geometric extremal problems. For example, among all polytopes inscribed in the Euclidean ball which have a fixed number of vertices and are combinatorially equivalent to a regular polytope... | {
"timestamp": "2022-05-19T02:23:49",
"yymm": "2205",
"arxiv_id": "2205.09096",
"language": "en",
"url": "https://arxiv.org/abs/2205.09096",
"abstract": "Weighted cone-volume functionals are introduced for the convex polytopes in $\\mathbb{R}^n$. For these functionals, geometric inequalities are proved and ... |
https://arxiv.org/abs/1601.02177 | Optimal-order bounds on the rate of convergence to normality for maximum likelihood estimators | It is well known that under general regularity conditions the distribution of the maximum likelihood estimator (MLE) is asymptotically normal. Very recently, bounds of the optimal order $O(1/\sqrt n)$ on the closeness of the distribution of the MLE to normality in the so-called bounded Wasserstein distance were obtaine... | \section{Introduction}
\label{intro}
Let us begin with the following quote from Kiefer \cite{kiefer68} of 1968:
\begin{quote}
a second area of what seem to me important problems to work on
has to do with the fact that we do have, in many settings, quite a good large sample
theory, but we don't know how larg... | {
"timestamp": "2016-12-15T02:01:55",
"yymm": "1601",
"arxiv_id": "1601.02177",
"language": "en",
"url": "https://arxiv.org/abs/1601.02177",
"abstract": "It is well known that under general regularity conditions the distribution of the maximum likelihood estimator (MLE) is asymptotically normal. Very recent... |
https://arxiv.org/abs/1504.07406 | On Maximal Unbordered Factors | Given a string $S$ of length $n$, its maximal unbordered factor is the longest factor which does not have a border. In this work we investigate the relationship between $n$ and the length of the maximal unbordered factor of $S$. We prove that for the alphabet of size $\sigma \ge 5$ the expected length of the maximal un... | \section{Introduction}
If a proper prefix of a string is simultaneously its suffix, then it is called a border of the string. Given a string $S$ of length $n$, its maximal unbordered factor is the longest factor which does not have a border. The relationship between $n$ and the length of the maximal unbordered factor o... | {
"timestamp": "2015-04-29T02:08:01",
"yymm": "1504",
"arxiv_id": "1504.07406",
"language": "en",
"url": "https://arxiv.org/abs/1504.07406",
"abstract": "Given a string $S$ of length $n$, its maximal unbordered factor is the longest factor which does not have a border. In this work we investigate the relati... |
https://arxiv.org/abs/0909.1482 | On positive Matrices which have a Positive Smith Normal Form | It is known that any symmetric matrix $M$ with entries in $\R[x]$ and which is positive semi-definite for any substitution of $x\in\R$, has a Smith normal form whose diagonal coefficients are constant sign polynomials in $\R[x]$. We generalize this result by considering a symmetric matrix $M$ with entries in a formally... | \section{Introduction}
Arising in various areas in Mathematics, there is a seminal result (we refer to \cite{Dj}) which says that any $n\times n$-symmetric matrix $M$ with entries in ${\mathbb R}[x]$ which is positive for any substitution of $x\in{\mathbb R}$ is a matricial sum of squares (it can be written $M=\sum_{i=... | {
"timestamp": "2009-09-08T14:49:21",
"yymm": "0909",
"arxiv_id": "0909.1482",
"language": "en",
"url": "https://arxiv.org/abs/0909.1482",
"abstract": "It is known that any symmetric matrix $M$ with entries in $\\R[x]$ and which is positive semi-definite for any substitution of $x\\in\\R$, has a Smith norma... |
https://arxiv.org/abs/2301.10902 | Efficient Hyperdimensional Computing | Hyperdimensional computing (HDC) is a method to perform classification that uses binary vectors with high dimensions and the majority rule. This approach has the potential to be energy-efficient and hence deemed suitable for resource-limited platforms due to its simplicity and massive parallelism. However, in order to ... | \section{Introduction}
{\em Hyperdimensional computing} (HDC) is an emerging learning paradigm inspired by an abstract representation of neuron activity in the human brain using high-dimensional binary vectors. Compared with other well-known training methods like artificial neural networks (ANNs),
HDCs have the advant... | {
"timestamp": "2023-01-27T02:04:36",
"yymm": "2301",
"arxiv_id": "2301.10902",
"language": "en",
"url": "https://arxiv.org/abs/2301.10902",
"abstract": "Hyperdimensional computing (HDC) is a method to perform classification that uses binary vectors with high dimensions and the majority rule. This approach ... |
https://arxiv.org/abs/1804.09191 | On the uniqueness of polynomial embeddings of the real 1-sphere in the plane | This paper considers real forms of closed algebraic $\mathbb{C}^*$-embeddings in $\mathbb{C}^2$. The classification of such embeddings was recently completed by Cassou-Nogues, Koras, Palka and Russell. Based on their classification, this paper shows that, up to an algebraic change of coordinates, there is only one poly... | \section{Introduction}
Let $\field{S}^n$ denote the real $n$-sphere as an algebraic variety over $\field{R}$. Daigle asked whether every polynomial embedding of
$\field{S}^1$ in $\field{R}^2$ is equivalent to the standard embedding.\footnote{D. Daigle, University of Ottawa, private communication, 2013}
Our main result... | {
"timestamp": "2018-04-26T02:00:26",
"yymm": "1804",
"arxiv_id": "1804.09191",
"language": "en",
"url": "https://arxiv.org/abs/1804.09191",
"abstract": "This paper considers real forms of closed algebraic $\\mathbb{C}^*$-embeddings in $\\mathbb{C}^2$. The classification of such embeddings was recently comp... |
https://arxiv.org/abs/1007.5101 | A relative isoperimetric inequality for certain warped product spaces | Given a warped product space $\mathbb{R} \times_{f} N$ with logarithmically convex warping function $f$, we prove a relative isoperimetric inequality for regions bounded between a subset of a vertical fiber and its image under an almost everywhere differentiable mapping in the horizontal direction. In particular, given... | \section{Introduction}\label{S:Intro}
This paper proves a relative isoperimetric inequality for warped product spaces with log convex warping function. The original relative isoperimetric inequality is the solution to the classical Dido problem in the plane, which asks for the greatest amount of area that can be encl... | {
"timestamp": "2010-09-23T02:02:47",
"yymm": "1007",
"arxiv_id": "1007.5101",
"language": "en",
"url": "https://arxiv.org/abs/1007.5101",
"abstract": "Given a warped product space $\\mathbb{R} \\times_{f} N$ with logarithmically convex warping function $f$, we prove a relative isoperimetric inequality for ... |
https://arxiv.org/abs/2105.01575 | A systematic and complete proof of the existence and uniqueness of self-descriptive numbers | All the already known results on self descriptive numbers, together with the demonstration of the uniqueness for bases greater than 6, are here obtained through a systematic scheme of proof and not trial and error. The proof is also complete for all the possible cases had been taken into account. | \section{Introduction}
The subject of \textit{self-descriptive numbers} as it will be defined and discussed in the following has received some interest by mathematicians \cite{libro1}, \cite{libro2},\cite{articolo1},\cite{articolo2}.
A list of these numbers expressed in the base $b=10$ is found on the OEIS \cite... | {
"timestamp": "2021-05-05T02:22:25",
"yymm": "2105",
"arxiv_id": "2105.01575",
"language": "en",
"url": "https://arxiv.org/abs/2105.01575",
"abstract": "All the already known results on self descriptive numbers, together with the demonstration of the uniqueness for bases greater than 6, are here obtained t... |
https://arxiv.org/abs/1608.08665 | A-posteriori snapshot location for POD in optimal control of linear parabolic equations | In this paper we study the approximation of an optimal control problem for linear para\-bolic PDEs with model order reduction based on Proper Orthogonal Decomposition (POD-MOR). POD-MOR is a Galerkin approach where the basis functions are obtained upon information contained in time snapshots of the parabolic PDE relate... | \section{Introduction}
Optimization with PDE constraints is nowadays a well-studied topic motivated by its relevance in industrial applications. We are interested in the numerical approximation of such optimization problems in an efficient and reliable way using surrogate models obtained with POD-MOR. The surrogate m... | {
"timestamp": "2016-09-01T02:00:57",
"yymm": "1608",
"arxiv_id": "1608.08665",
"language": "en",
"url": "https://arxiv.org/abs/1608.08665",
"abstract": "In this paper we study the approximation of an optimal control problem for linear para\\-bolic PDEs with model order reduction based on Proper Orthogonal ... |
https://arxiv.org/abs/1802.03385 | Krein-type theorems and ordered structure for Cauchy-de Branges spaces | We extend some results of M.G. Krein to the class of entire functions which can be represented as ratios of discrete Cauchy transforms in the plane. As an application we obtain new versions of de Branges' Ordering Theorem for nearly invariant subspaces in a class of Hilbert spaces of entire functions. Examples illustra... | \section{Introduction}
\label{int}
\subsection{Krein's theorem.}
M.G. Krein's theorem about the Cartwright class functions
plays a seminal role in entire function theory and its
applications to spectral theory of linear operators.
Recall that an entire function $F$ is said to be of {\it Cartwright class}
if it is of... | {
"timestamp": "2018-04-03T02:09:12",
"yymm": "1802",
"arxiv_id": "1802.03385",
"language": "en",
"url": "https://arxiv.org/abs/1802.03385",
"abstract": "We extend some results of M.G. Krein to the class of entire functions which can be represented as ratios of discrete Cauchy transforms in the plane. As an... |
https://arxiv.org/abs/0802.4377 | Unitary super perfect numbers | We shall show that 9, 165 are all of the odd unitary super perfect numbers. | \section{Introduction}\label{intro}
We denote by $\sigma(N)$ the sum of divisors of $N$.
$N$ is called to be perfect if $\sigma(N)=2N$.
It is a well-known unsolved problem whether or not
an odd perfect number exists. Interest to this problem
has produced many analogous notions.
D. Suryanarayana \cite{Sur} called $N$ ... | {
"timestamp": "2008-02-29T13:36:46",
"yymm": "0802",
"arxiv_id": "0802.4377",
"language": "en",
"url": "https://arxiv.org/abs/0802.4377",
"abstract": "We shall show that 9, 165 are all of the odd unitary super perfect numbers.",
"subjects": "Number Theory (math.NT)",
"title": "Unitary super perfect num... |
https://arxiv.org/abs/2212.06766 | Homomorphism Conjugacy versus Centralizer Actions in the Symmetric Group | We explore when generator-conjugate homomorphisms are conjugate and when element-conjugate homomorphisms are conjugate from abelian or dihedral groups to the symmetric group. We completely determine when such homomorphisms are conjugate in the case where the source group has two generators by studying centralizer actio... | \section{Introduction}
Element-conjugate linear representations of finite degree are conjugate if the underlying field is algebraically closed and of characteristic zero. We are motivated by exploring the relationship between element-conjugacy and homomorphism conjugacy for other target groups. M. Larsen initiated this... | {
"timestamp": "2022-12-14T02:18:09",
"yymm": "2212",
"arxiv_id": "2212.06766",
"language": "en",
"url": "https://arxiv.org/abs/2212.06766",
"abstract": "We explore when generator-conjugate homomorphisms are conjugate and when element-conjugate homomorphisms are conjugate from abelian or dihedral groups to ... |
https://arxiv.org/abs/1403.2125 | Two-orbit convex polytopes and tilings | We classify the convex polytopes whose symmetry groups have two orbits on the flags. These exist only in two or three dimensions, and the only ones whose combinatorial automorphism group is also two-orbit are the cuboctahedron, the icosidodecahedron, and their duals. The combinatorially regular two-orbit convex polytop... |
\section{Introduction}
Here we will classify all convex polytopes, and face-to-face tilings of Euclidean space by convex polytopes, whose flags have two orbits under the action of the symmetry group. First we briefly define these terms.
A \emph{convex polytope} is the convex hull of a finite set of points in $d$-dim... | {
"timestamp": "2014-03-11T01:09:27",
"yymm": "1403",
"arxiv_id": "1403.2125",
"language": "en",
"url": "https://arxiv.org/abs/1403.2125",
"abstract": "We classify the convex polytopes whose symmetry groups have two orbits on the flags. These exist only in two or three dimensions, and the only ones whose co... |
https://arxiv.org/abs/2111.12199 | Jacobi identity in polyhedral products | We show that a relation among minimal non-faces of a fillable complex $K$ yields an identity of iterated (higher) Whitehead products in a polyhedral product over $K$. In particular, for the $(n-1)$-skeleton of a simplicial $n$-sphere, we always have such an identity, and for the $(n-1)$-skeleton of a $(n+1)$-simplex, t... | \section{Introduction}\label{Introduction}
Let $K$ be a simplicial complex with vertex set $[m]=\{1,2,\ldots,m\}$, and let $(\underline{X},\underline{A})=\{(X_i,A_i)\}_{i=1}^m$ be a collection of pairs of spaces indexed by vertices of $K$. If all $(X_i,A_i)$ are a common pair $(X,A)$, then we abbreviate $(\underli... | {
"timestamp": "2021-11-25T02:04:41",
"yymm": "2111",
"arxiv_id": "2111.12199",
"language": "en",
"url": "https://arxiv.org/abs/2111.12199",
"abstract": "We show that a relation among minimal non-faces of a fillable complex $K$ yields an identity of iterated (higher) Whitehead products in a polyhedral produ... |
https://arxiv.org/abs/2007.09765 | On the illumination of centrally symmetric cap bodies in small dimensions | The illumination number $I(K)$ of a convex body $K$ in Euclidean space $\mathbb{E}^d$ is the smallest number of directions that completely illuminate the boundary of a convex body. A cap body $K_c$ of a ball is the convex hull of a Euclidean ball and a countable set of points outside the ball under the condition that e... | \section{Introduction}\label{section-introduction}
\subsection{On the status of the Illumination Conjecture}
Let $\mathbb{E}^d$ denote a $d$-dimensional Euclidean space. The origin point is denoted $\mathbf{o}$, and $\mathbb{S}^{d-1}$ is an origin-centered $(d-1)$-sphere with a unit radius. We say that $K \subset \ma... | {
"timestamp": "2020-07-21T02:23:41",
"yymm": "2007",
"arxiv_id": "2007.09765",
"language": "en",
"url": "https://arxiv.org/abs/2007.09765",
"abstract": "The illumination number $I(K)$ of a convex body $K$ in Euclidean space $\\mathbb{E}^d$ is the smallest number of directions that completely illuminate the... |
https://arxiv.org/abs/1406.7403 | Tautological classes on the moduli space of hyperelliptic curves with rational tails | We study tautological classes on the moduli space of stable $n$-pointed hyperelliptic curves of genus $g$ with rational tails. Our result gives a complete description of tautological relations. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobia... | \section*{Introduction}
In this article we study tautological classes on the moduli space
$\mathcal{H}_{g,n}^{rt}$ of stable
$n$-pointed hyperelliptic curves of genus $g$ with rational tails.
Tautological classes are natural algebraic cycles reflecting the nature of the generic object parameterized by the moduli spa... | {
"timestamp": "2014-07-15T02:13:57",
"yymm": "1406",
"arxiv_id": "1406.7403",
"language": "en",
"url": "https://arxiv.org/abs/1406.7403",
"abstract": "We study tautological classes on the moduli space of stable $n$-pointed hyperelliptic curves of genus $g$ with rational tails. Our result gives a complete d... |
https://arxiv.org/abs/math/0701928 | 2-torus manifolds, cobordism and small covers | Let ${\frak M}_n$ be the set of equivariant unoriented cobordism classes of all $n$-dimensional 2-torus manifolds, where an $n$-dimensional 2-torus manifold $M$ is a smooth closed manifold of dimension $n$ with effective smooth action of a rank $n$ 2-torus group $({\Bbb Z}_2)^n$. Then ${\frak M}_n$ forms an abelian gro... | \section{Introduction}
An $n$-dimensional 2-torus manifold $M$ is a smooth closed manifold
of dimension $n$ with effective smooth action of a rank $n$ 2-torus
group $({\Bbb Z}_2)^n$. Since the action is effective, the fixed
point set of the action is 0-dimensional (i.e., it is formed by
finitely many isolated points) ... | {
"timestamp": "2007-01-31T17:22:02",
"yymm": "0701",
"arxiv_id": "math/0701928",
"language": "en",
"url": "https://arxiv.org/abs/math/0701928",
"abstract": "Let ${\\frak M}_n$ be the set of equivariant unoriented cobordism classes of all $n$-dimensional 2-torus manifolds, where an $n$-dimensional 2-torus m... |
https://arxiv.org/abs/1812.03024 | Dickson's Lemma, Higman's Theorem and Beyond: a survey of some basic results in order theory | We provide proofs for the fact that certain orders have no descending chains and no antichains. | \section{Introduction} \label{sec:intro}
We investigate some finiteness conditions of a partially ordered set $\algop{A}{\le}$.
As usual, we write $a \ge b$ for $b \le a$, and $a < b$ (or $b > a$) for $a \le b$ and $a \neq b$. Furthermore, $a \perp b$ stands for
($a \not\le b$ and $b \not \le a$); in this case, we cal... | {
"timestamp": "2018-12-10T02:15:57",
"yymm": "1812",
"arxiv_id": "1812.03024",
"language": "en",
"url": "https://arxiv.org/abs/1812.03024",
"abstract": "We provide proofs for the fact that certain orders have no descending chains and no antichains.",
"subjects": "Logic (math.LO)",
"title": "Dickson's L... |
https://arxiv.org/abs/2106.00437 | Homological duality for covering groups of reductive $p$-adic groups | In this largely expository paper we extend properties of the homological duality functor $RHom_{\mathcal H}(-,{\mathcal H})$ where ${\mathcal H}$ is the Hecke algebra of a reductive $p$-adic group, to the case where it is the Hecke algebra of a finite central extension of a reductive $p$-adic group. The most important ... | \section{Introduction}
\subsection{}
Let $G$ be a reductive $p$-adic group or a covering group (finite central extension) of such a group.
The homological duality for the abelian category $\mathcal M(G)$ of smooth representations of $G$
is defined at the level of derived categories as
\[ D_h:=\RHom_\mathcal H(-,\... | {
"timestamp": "2021-06-02T02:21:08",
"yymm": "2106",
"arxiv_id": "2106.00437",
"language": "en",
"url": "https://arxiv.org/abs/2106.00437",
"abstract": "In this largely expository paper we extend properties of the homological duality functor $RHom_{\\mathcal H}(-,{\\mathcal H})$ where ${\\mathcal H}$ is th... |
https://arxiv.org/abs/2006.04745 | Generalized golden mean and the efficiency of thermal machines | We investigate generic heat engines and refrigerators operating between two heat reservoirs, for the condition when their efficiencies are equal to each other. It is shown that the corresponding value of efficiency is given as the inverse of the generalized golden mean, $\phi_p$, where the parameter $p$ depends on the ... | \section{Introduction}
Although the golden mean (golden ratio)
has engaged artists, mathematicians and philosophers since antiquity
\cite{Coxeter, Ogilvy1990, Markowsky1992, Livio},
it has been appreciated more recently that it
is not a unique number as far as many of its
algebraic and geometric properties are conc... | {
"timestamp": "2020-06-16T02:08:42",
"yymm": "2006",
"arxiv_id": "2006.04745",
"language": "en",
"url": "https://arxiv.org/abs/2006.04745",
"abstract": "We investigate generic heat engines and refrigerators operating between two heat reservoirs, for the condition when their efficiencies are equal to each o... |
https://arxiv.org/abs/math/0604356 | Long $n$-zero-free sequences in finite cyclic groups | A sequence in the additive group ${\mathbb Z}_n$ of integers modulo $n$ is called $n$-zero-free if it does not contain subsequences with length $n$ and sum zero. The article characterizes the $n$-zero-free sequences in ${\mathbb Z}_n$ of length greater than $3n/2-1$. The structure of these sequences is completely deter... | \section{Introduction}
\label{Intro}
The Erd\H{o}s--Ginzburg--Ziv theorem \cite{ErdosGinzburgZiv}
states that each sequence of length ${2n{-}1}$ in the cyclic group
of order~$n$ has a subsequence of length~$n$ and sum~zero. This
article characterizes all sequences with length greater
than~$3n/2{-}1$ in the same group ... | {
"timestamp": "2006-04-16T21:00:31",
"yymm": "0604",
"arxiv_id": "math/0604356",
"language": "en",
"url": "https://arxiv.org/abs/math/0604356",
"abstract": "A sequence in the additive group ${\\mathbb Z}_n$ of integers modulo $n$ is called $n$-zero-free if it does not contain subsequences with length $n$ a... |
https://arxiv.org/abs/1506.03435 | Nielsen-Schreier implies the finite Axiom of Choice | We present a new proof that the statement 'every subgroup of a free group is free' implies the Axiom of Choice for finite sets. | \section{Introduction}
In 1921, Nielsen \cite{Nielsen1921NS} proved that every subgroup of a finitely generated free group is free. This result was generalised to arbitrary free groups by Schreier \cite{Schreier1927Untergruppen} in 1927, giving us the following result.
\begin{principle}{NS}{Nielsen-Schreier}
... | {
"timestamp": "2015-10-13T02:20:00",
"yymm": "1506",
"arxiv_id": "1506.03435",
"language": "en",
"url": "https://arxiv.org/abs/1506.03435",
"abstract": "We present a new proof that the statement 'every subgroup of a free group is free' implies the Axiom of Choice for finite sets.",
"subjects": "Logic (ma... |
https://arxiv.org/abs/math/0611158 | Simple Homotopy Types and Finite Spaces | We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse of finite spaces induces a simplicial ... | \section{Introduction}
J.H.C. Whitehead's theory of simple homotopy types is inspired by Tietze's theorem in combinatorial group theory, which states that any finite presentation
of a group could be deformed into any other by a finite sequence of elementary moves, which are now called Tietze transformations. Whitehead... | {
"timestamp": "2006-11-08T20:08:55",
"yymm": "0611",
"arxiv_id": "math/0611158",
"language": "en",
"url": "https://arxiv.org/abs/math/0611158",
"abstract": "We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space ... |
https://arxiv.org/abs/1207.3606 | Dual concepts of almost distance-regularity and the spectral excess theorem | Generally speaking, `almost distance-regular' graphs share some, but not necessarily all, of the regularity properties that characterize distance-regular graphs. In this paper we propose two new dual concepts of almost distance-regularity, thus giving a better understanding of the properties of distance-regular graphs.... | \section{Preliminaries}
Almost distance-regular graphs, recently studied in the literature, are graphs
which share some, but not necessarily all, of the regularity properties that
characterize distance-regular graphs. Two examples of the former are partially
distance-regular graphs \cite{p91} and $m$-walk-regular graph... | {
"timestamp": "2012-07-17T02:03:06",
"yymm": "1207",
"arxiv_id": "1207.3606",
"language": "en",
"url": "https://arxiv.org/abs/1207.3606",
"abstract": "Generally speaking, `almost distance-regular' graphs share some, but not necessarily all, of the regularity properties that characterize distance-regular gr... |
https://arxiv.org/abs/1112.0098 | The fast track to Löwner's theorem | The operator monotone functions defined in the positive half-line are of particular importance. We give a version of the theory in which integral representations for these functions can be established directly without invoking Löwner's detailed analysis of matrix monotone functions of a fixed order or the theory of ana... | \section{Introduction and preliminaries}
The functional calculus is defined by the spectral theorem. Since we only deal with matrices the function $ f(x) $ of a hermitian matrix $ x $ is defined for any function $ f $ defined on the spectrum of $ x. $
\begin{definition}
Let $I$ be an interval of any type. A function... | {
"timestamp": "2013-02-05T02:01:52",
"yymm": "1112",
"arxiv_id": "1112.0098",
"language": "en",
"url": "https://arxiv.org/abs/1112.0098",
"abstract": "The operator monotone functions defined in the positive half-line are of particular importance. We give a version of the theory in which integral representa... |
https://arxiv.org/abs/1304.1217 | On the communication complexity of sparse set disjointness and exists-equal problems | In this paper we study the two player randomized communication complexity of the sparse set disjointness and the exists-equal problems and give matching lower and upper bounds (up to constant factors) for any number of rounds for both of these problems. In the sparse set disjointness problem, each player receives a k-s... |
\section{Discussion}
\label{sec:discussion}
The $r$-round protocol we gave in \autoref{sec:upperbound} solves the sparse set
disjointness problem in $O(k\log^{(r)}k)$ total communication. As we proved in
\autoref{sec:lowerbound} this is optimal. The same, however,
cannot be said of the error probability. With the sa... | {
"timestamp": "2013-04-05T02:00:49",
"yymm": "1304",
"arxiv_id": "1304.1217",
"language": "en",
"url": "https://arxiv.org/abs/1304.1217",
"abstract": "In this paper we study the two player randomized communication complexity of the sparse set disjointness and the exists-equal problems and give matching low... |
https://arxiv.org/abs/1206.6367 | A comparison of the discrete Kolmogorov-Smirnov statistic and the Euclidean distance | Goodness-of-fit tests gauge whether a given set of observations is consistent (up to expected random fluctuations) with arising as independent and identically distributed (i.i.d.) draws from a user-specified probability distribution known as the "model." The standard gauges involve the discrepancy between the model and... | \section{Introduction}
\label{intro}
Testing goodness-of-fit is one of the foundations of modern statistics,
as elucidated by~\cite{rao}, for example.
The formulation in the discrete setting involves $n$
independent and identically distributed (i.i.d.)\ draws
from a probability distribution over $m$ bins
(``categorie... | {
"timestamp": "2012-06-28T02:05:59",
"yymm": "1206",
"arxiv_id": "1206.6367",
"language": "en",
"url": "https://arxiv.org/abs/1206.6367",
"abstract": "Goodness-of-fit tests gauge whether a given set of observations is consistent (up to expected random fluctuations) with arising as independent and identical... |
https://arxiv.org/abs/2003.07382 | Slack Ideals in Macaulay2 | Recently Gouveia, Thomas and the authors introduced the slack realization space, a new model for the realization space of a polytope. It represents each polytope by its slack matrix, the matrix obtained by evaluating each facet inequality at each vertex. Unlike the classical model, the slack model naturally mods out pr... | \section{Introduction}
Slack matrices of polytopes are nonnegative real matrices whose entries express the slack of a vertex in a facet inequality. In particular, the zero pattern of a slack matrix encodes the vertex-facet incidence structure of the polytope. Slack matrices have found remarkable use in the theory of... | {
"timestamp": "2020-11-03T02:05:11",
"yymm": "2003",
"arxiv_id": "2003.07382",
"language": "en",
"url": "https://arxiv.org/abs/2003.07382",
"abstract": "Recently Gouveia, Thomas and the authors introduced the slack realization space, a new model for the realization space of a polytope. It represents each p... |
https://arxiv.org/abs/1510.07284 | Random version of Dvoretzky's theorem in $\ell_p^n$ | We study the dependence on $\varepsilon$ in the critical dimension $k(n,p,\varepsilon)$ for which one can find random sections of the $\ell_p^n$-ball which are $(1+\varepsilon)$-spherical. We give lower (and upper) estimates for $k(n,p,\varepsilon)$ for all eligible values $p$ and $\varepsilon$ as $n\to \infty$, which ... | \section{Introduction}
The fundamental theorem of Dvoretzky from \cite{Dvo} in geometric language states that every centrally
symmetric convex body on $\mathbb R^n$ has a central section of large dimension which is almost spherical.
The optimal form of the theorem, which was proved by Milman
in \cite{Mil}, reads as... | {
"timestamp": "2017-03-10T02:03:13",
"yymm": "1510",
"arxiv_id": "1510.07284",
"language": "en",
"url": "https://arxiv.org/abs/1510.07284",
"abstract": "We study the dependence on $\\varepsilon$ in the critical dimension $k(n,p,\\varepsilon)$ for which one can find random sections of the $\\ell_p^n$-ball w... |
https://arxiv.org/abs/1705.01350 | The Samuelson's model as a singular discrete time system | In this paper we revisit the famous classical Samuelson's multiplier-accelerator model for national economy. We reform this model into a singular discrete time system and study its solutions. The advantage of this study gives a better understanding of the structure of the model and more deep and elegant results. | \section{Introduction}
Many authors have studied generalised discrete \& continuous time systems, see [1-19], and their applications especially in cases where the memory effect is needed including generalised discrete \& continuous time systems with delays, see [20-38]. Many of these results have already been extended... | {
"timestamp": "2017-05-04T02:05:00",
"yymm": "1705",
"arxiv_id": "1705.01350",
"language": "en",
"url": "https://arxiv.org/abs/1705.01350",
"abstract": "In this paper we revisit the famous classical Samuelson's multiplier-accelerator model for national economy. We reform this model into a singular discrete... |
https://arxiv.org/abs/1911.03958 | A spanning bandwidth theorem in random graphs | The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $(\frac{k-1}{k}+o(1))n$ contains all $n$-vertex $k$-colourable graphs $H$ with bounded maximum degree and bandwidth $o(n)$. In [arXiv:1612.00661] a random graph analogue of this statement is pro... | \section{Introduction}
One major topic of research in extremal graph theory is to determine
minimum degree conditions on a graph~$G$ which force it to contain copies
of a spanning subgraph $H$. The primal example of such a theorem is Dirac's
theorem~\cite{dirac1952}, which states that if $\delta(G)\ge\tfrac12 v(G)$
the... | {
"timestamp": "2019-11-12T02:16:43",
"yymm": "1911",
"arxiv_id": "1911.03958",
"language": "en",
"url": "https://arxiv.org/abs/1911.03958",
"abstract": "The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $(\\frac{k-1}{k}+o(1))n$ con... |
https://arxiv.org/abs/1604.08574 | Axial compression of a thin elastic cylinder: bounds on the minimum energy scaling law | We consider the axial compression of a thin elastic cylinder placed about a hard cylindrical core. Treating the core as an obstacle, we prove upper and lower bounds on the minimum energy of the cylinder that depend on its relative thickness and the magnitude of axial compression. We focus exclusively on the setting whe... | \section{Introduction}
In many controlled experiments involving the axial compression of
thin elastic cylinders, one observes complex folding patterns (see,
e.g., \cite{donnell1934new,horton1965imperfections,pogorelov1988bendings,seffen2014surface}).
It is natural to wonder if such patterns are required to minimize
el... | {
"timestamp": "2016-05-12T02:00:38",
"yymm": "1604",
"arxiv_id": "1604.08574",
"language": "en",
"url": "https://arxiv.org/abs/1604.08574",
"abstract": "We consider the axial compression of a thin elastic cylinder placed about a hard cylindrical core. Treating the core as an obstacle, we prove upper and lo... |
https://arxiv.org/abs/2012.02125 | On the Impossibility of Convergence of Mixed Strategies with No Regret Learning | We study the limiting behavior of the mixed strategies that result from optimal no-regret learning strategies in a repeated game setting where the stage game is any 2 by 2 competitive game. We consider optimal no-regret algorithms that are mean-based and monotonic in their argument. We show that for any such algorithm,... |
\subsection{Beyond the monotonicity assumption: A conjecture}\label{sec:conjecture}
In Section~\ref{sec: beyond_mean_based}, we showed that our results can be extended beyond exact mean-based strategies, allowing their applicability to a wide range of popular online learning algorithms.
In this section, we ask wheth... | {
"timestamp": "2021-08-03T02:34:11",
"yymm": "2012",
"arxiv_id": "2012.02125",
"language": "en",
"url": "https://arxiv.org/abs/2012.02125",
"abstract": "We study the limiting behavior of the mixed strategies that result from optimal no-regret learning strategies in a repeated game setting where the stage g... |
https://arxiv.org/abs/1812.05833 | Total Colourings - A survey | The smallest integer $k$ needed for the assignment of colors to the elements so that the coloring is proper (vertices and edges) is called the total chromatic number of a graph. Vizing and Behzed conjectured that the total coloring can be done using at most $\Delta(G)+2$ colors, where $\Delta(G)$ is the maximum degree ... | \section{Introduction}
Let $G$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$. An element of $G$ is a vertex or an edge of $G$. The \textit{total coloring} of a graph $G$ is an assignment of colors to the vertices and edges such that no two
incident or adjacent elements receive the same color. The \texti... | {
"timestamp": "2018-12-17T02:09:19",
"yymm": "1812",
"arxiv_id": "1812.05833",
"language": "en",
"url": "https://arxiv.org/abs/1812.05833",
"abstract": "The smallest integer $k$ needed for the assignment of colors to the elements so that the coloring is proper (vertices and edges) is called the total chrom... |
https://arxiv.org/abs/0802.2109 | On slicing invariants of knots | The slicing number of a knot, $u_s(K)$, is the minimum number of crossing changes required to convert $K$ to a slice knot. This invariant is bounded above by the unknotting number and below by the slice genus $g_s(K)$. We show that for many knots, previous bounds on unknotting number obtained by Ozsvath and Szabo and b... | \section{Introduction}
\label{sec:intro}
The unknotting number of a knot is the minimum number of crossing changes required to convert it to an unknot. Ozsv\'{a}th\ and Szab\'{o}\ used Heegaard Floer theory to provide a powerful obstruction to a knot having unknotting number one \cite{osu1}. This obstruction was gene... | {
"timestamp": "2008-02-15T19:52:34",
"yymm": "0802",
"arxiv_id": "0802.2109",
"language": "en",
"url": "https://arxiv.org/abs/0802.2109",
"abstract": "The slicing number of a knot, $u_s(K)$, is the minimum number of crossing changes required to convert $K$ to a slice knot. This invariant is bounded above b... |
https://arxiv.org/abs/1902.06484 | Find Subtrees of Specified Weight and Cycles of Specified Length in Linear Time | We apply the Euler tour technique to find subtrees of specified weight as follows. Let $k, g, N_1, N_2 \in \mathbb{N}$ such that $1 \leq k \leq N_2$, $g + h > 2$ and $2k - 4g - h + 3 \leq N_2 \leq 2k + g + h - 2$, where $h := 2N_1 - N_2$. Let $T$ be a tree of $N_1$ vertices and let $c : V(T) \rightarrow \mathbb{N}$ be ... | \section{Introduction} \label{sec:intro}
Given a tree $T$ and vertex weights $c: V(T) \rightarrow \mathbb{N}$, it is natural to ask subtrees of which specified weight would exist. Let $S$ be a subtree of $T$. We define $c(S) := \sum_{v \in V(S)} c(v)$. Let $k, g \in \mathbb{N}$ with $1 \leq k \leq c(T)$. We aim at find... | {
"timestamp": "2019-10-15T02:06:12",
"yymm": "1902",
"arxiv_id": "1902.06484",
"language": "en",
"url": "https://arxiv.org/abs/1902.06484",
"abstract": "We apply the Euler tour technique to find subtrees of specified weight as follows. Let $k, g, N_1, N_2 \\in \\mathbb{N}$ such that $1 \\leq k \\leq N_2$, ... |
https://arxiv.org/abs/1702.00669 | The variation of the maximal function of a radial function | We study the problem concerning the variation of the Hardy-Littlewood maximal function in higher dimensions. As the main result, we prove that the variation of the non-centered Hardy-Littlewood maximal function of a radial function is comparable to the variation of the function itself. | \section{Introduction}
The non-centered Hardy-Littlewood maximal operator $M$ is defined by
setting for $f\in L^1_{loc}(\mathbb{R}^n)\,$ that
\begin{equation}\label{eq:1}
M f(x)=\sup_{B(z,r)\ni x}
\frac{1}{|B(z,r)|}\int_{B(z,r)}|f(y)|\,dy\,=:\,\sup_{B(z,r)\ni x}\vint_{B(z,r)}|f(y)|\,dy\
\end{equation}
for every $x\in\m... | {
"timestamp": "2017-02-03T02:07:16",
"yymm": "1702",
"arxiv_id": "1702.00669",
"language": "en",
"url": "https://arxiv.org/abs/1702.00669",
"abstract": "We study the problem concerning the variation of the Hardy-Littlewood maximal function in higher dimensions. As the main result, we prove that the variati... |
https://arxiv.org/abs/1905.01366 | Noncommutative cross-ratio and Schwarz derivative | We present here a theory of noncommutative cross-ratio, Schwarz derivative and their connections and relations to the operator cross-ratio. We apply the theory to "noncommutative elementary geometry" and relate it to noncommutative integrable systems. We also provide a noncommutative version of the celebrated "pentagra... | \section{Introduction}
Cross-ratio and Schwarz derivative are some of the most famous
invariants in mathematics (see \cite{Lab}, \cite{OT}, \cite{OT2}).
Different versions of their noncommutative analogs and various
applications of these constructions to integrable systems, control theory and other subjects
were discu... | {
"timestamp": "2019-05-23T02:10:22",
"yymm": "1905",
"arxiv_id": "1905.01366",
"language": "en",
"url": "https://arxiv.org/abs/1905.01366",
"abstract": "We present here a theory of noncommutative cross-ratio, Schwarz derivative and their connections and relations to the operator cross-ratio. We apply the t... |
https://arxiv.org/abs/1912.02846 | The Maximum Wiener Index of Maximal Planar Graphs | The Wiener index of a connected graph is the sum of the distances between all pairs of vertices in the graph. It was conjectured that the Wiener index of an $n$-vertex maximal planar graph is at most $\lfloor\frac{1}{18}(n^3+3n^2)\rfloor$. We prove this conjecture and for every $n$, $n \geq 10$, determine the unique $n... | \subsubsection*{key words}
\section{Introduction}
The Wiener index is a graph invariant based on distances in the graph. For a connected graph $G$, the Wiener index is the sum of distances between all unordered pairs of vertices in the graph and is denoted by $W(G)$. That means,
\begin{equation*}
W(G) = \sum_{\{u,... | {
"timestamp": "2019-12-09T02:00:50",
"yymm": "1912",
"arxiv_id": "1912.02846",
"language": "en",
"url": "https://arxiv.org/abs/1912.02846",
"abstract": "The Wiener index of a connected graph is the sum of the distances between all pairs of vertices in the graph. It was conjectured that the Wiener index of ... |
https://arxiv.org/abs/1907.11094 | Improving the Accuracy of Principal Component Analysis by the Maximum Entropy Method | Classical Principal Component Analysis (PCA) approximates data in terms of projections on a small number of orthogonal vectors. There are simple procedures to efficiently compute various functions of the data from the PCA approximation. The most important function is arguably the Euclidean distance between data items, ... | \subsection{Experiments with Rayleigh Quotients}
\input{tablesrq}
Table~\ref{tab:RQ1} describes the average difference
in evaluating the column and row space Rayleigh quotient.
Smaller mean and standard deviation of $\lvert \text{$r_\text{ent}$}-r \rvert$ indicate
better estimates for the Maximum Entropy Method.
Th... | {
"timestamp": "2019-07-26T02:12:36",
"yymm": "1907",
"arxiv_id": "1907.11094",
"language": "en",
"url": "https://arxiv.org/abs/1907.11094",
"abstract": "Classical Principal Component Analysis (PCA) approximates data in terms of projections on a small number of orthogonal vectors. There are simple procedure... |
https://arxiv.org/abs/1806.01638 | Numerical Integration as an Initial Value Problem | Numerical integration (NI) packages commonly used in scientific research are limited to returning the value of a definite integral at the upper integration limit, also commonly referred to as numerical quadrature. These quadrature algorithms are typically of a fixed accuracy and have only limited ability to adapt to th... | \section{Introduction}
\label{sec:introduction}
Numerical integration(NI) is one of the most useful numerical tools that is
routinely utilized in all scientific and engineering applications. There are
general and specialized methods that efficiently compute definite integrals
even for many pathological cases. Most of... | {
"timestamp": "2018-06-06T02:09:20",
"yymm": "1806",
"arxiv_id": "1806.01638",
"language": "en",
"url": "https://arxiv.org/abs/1806.01638",
"abstract": "Numerical integration (NI) packages commonly used in scientific research are limited to returning the value of a definite integral at the upper integratio... |
https://arxiv.org/abs/0809.2981 | Presenting the cohomology of a Schubert variety | We extend the short presentation due to [Borel '53] of the cohomology ring of a generalized flag manifold to a relatively short presentation of the cohomology of any of its Schubert varieties. Our result is stated in a root-system uniform manner by introducing the essential set of a Coxeter group element, generalizing ... | \section{Introduction}
\label{intro-section}
The cohomology of the generalized flag manifold $G/B$, for any complex
semisimple algebraic group $G$ and Borel subgroup $B$, has a classical
presentation due to Borel \cite{Borel}. Pick a maximal torus
$T\subset B\subset G$, choose a field $\Bbbk$ of characteristic zero,... | {
"timestamp": "2009-09-04T23:19:22",
"yymm": "0809",
"arxiv_id": "0809.2981",
"language": "en",
"url": "https://arxiv.org/abs/0809.2981",
"abstract": "We extend the short presentation due to [Borel '53] of the cohomology ring of a generalized flag manifold to a relatively short presentation of the cohomolo... |
https://arxiv.org/abs/2009.04451 | Dimension of finite free complexes over commutative Noetherian rings | Foxby defined the (Krull) dimension of a complex of modules over a commutative Noetherian ring in terms of the dimension of its homology modules. In this note it is proved that the dimension of a bounded complex of free modules of finite rank can be computed directly from the matrices representing the differentials of ... | \section*{Introduction}
\noindent
This short note concerns certain homological
invariants---specifically, dimension and depth---of complexes of
modules over commutative Noetherian local rings. The concepts of depth
and dimension for modules, introduced by Krull and by Auslander and
Buchsbaum, respectively, need no rec... | {
"timestamp": "2020-09-10T02:21:13",
"yymm": "2009",
"arxiv_id": "2009.04451",
"language": "en",
"url": "https://arxiv.org/abs/2009.04451",
"abstract": "Foxby defined the (Krull) dimension of a complex of modules over a commutative Noetherian ring in terms of the dimension of its homology modules. In this ... |
https://arxiv.org/abs/1710.10674 | Existence and stability of steady compressible Navier-Stokes solutions on a finite interval with noncharacteristic boundary conditions | We study existence and stability of steady solutions of the isentropic compressible Navier-Stokes equations on a finite interval with non characteristic boundary conditions, for general not necessarily small-amplitude data. We show that there exists a unique solution, about which the linearized spatial operator possess... | \section{Introduction}
In this paper, we initiate in the simplest setting of 1D isentropic gas dynamics,
a systematic study of existence and stability of steady solutions of systems of hyperbolic parabolic
equations on a bounded domain, with noncharacteristic inflow or outflow boundary conditions,
and data and solutio... | {
"timestamp": "2019-01-08T02:26:06",
"yymm": "1710",
"arxiv_id": "1710.10674",
"language": "en",
"url": "https://arxiv.org/abs/1710.10674",
"abstract": "We study existence and stability of steady solutions of the isentropic compressible Navier-Stokes equations on a finite interval with non characteristic b... |
https://arxiv.org/abs/1303.3064 | Locality and thermalization in closed quantum systems | We derive a necessary and sufficient condition for the thermalization of a local observable in a closed quantum system which offers an alternative explanation, independent of the eigenstate thermalization hypothesis, for the thermalization process. We also show that this approach is useful to investigate thermalization... | \section{Supplementary Material}
The purpose of this supplementary material is threefold. In
Sec.~\ref{Details} we will first provide some additional information
about the basis rotation required to split an observable into a local
and a non-local part, numerical data testing the ETH,
as well as some technical details ... | {
"timestamp": "2013-11-05T02:10:35",
"yymm": "1303",
"arxiv_id": "1303.3064",
"language": "en",
"url": "https://arxiv.org/abs/1303.3064",
"abstract": "We derive a necessary and sufficient condition for the thermalization of a local observable in a closed quantum system which offers an alternative explanati... |
https://arxiv.org/abs/1804.07494 | Parallel Quicksort without Pairwise Element Exchange | Quicksort is an instructive classroom approach to parallel sorting on distributed memory parallel computers with many opportunities for illustrating specific implementation alternatives and tradeoffs with common communication interfaces like MPI. The (two) standard distributed memory Quicksort implementations exchange ... | \section{Introduction}
Quicksort~\cite{Hoare62} is often used in the classroom as an example
of a sorting algorithm with obvious potential for parallelization on
different types of parallel computers, and with enough obstacles to
make the discussion instructive. Still, distributed memory parallel
Quicksort is practica... | {
"timestamp": "2018-10-25T02:10:45",
"yymm": "1804",
"arxiv_id": "1804.07494",
"language": "en",
"url": "https://arxiv.org/abs/1804.07494",
"abstract": "Quicksort is an instructive classroom approach to parallel sorting on distributed memory parallel computers with many opportunities for illustrating speci... |
https://arxiv.org/abs/2105.12864 | The Maker-Breaker percolation game on the square lattice | We study the $(m,b)$ Maker-Breaker percolation game on $\mathbb{Z}^2$, introduced by Day and Falgas-Ravry. As our first result, we show that Breaker has a winning strategy for the $(m,b)$-game whenever $b \geq (2-\frac{1}{14} + o(1))m$, breaking the ratio $2$ barrier proved by Day and Falgas-Ravry.Addressing further qu... | \section{Introduction}
\label{sec:intro}
\subsection{Background}
The so-called \emph{Maker-Breaker games} are a large and well studied class of positional games.
To define the simplest version of a Maker-Breaker game, we need a finite or infinite set $\Lambda$ and a family $\mathcal{F}$ of subsets of $\Lambda$.
The ... | {
"timestamp": "2021-05-28T02:04:59",
"yymm": "2105",
"arxiv_id": "2105.12864",
"language": "en",
"url": "https://arxiv.org/abs/2105.12864",
"abstract": "We study the $(m,b)$ Maker-Breaker percolation game on $\\mathbb{Z}^2$, introduced by Day and Falgas-Ravry. As our first result, we show that Breaker has ... |
https://arxiv.org/abs/1905.09987 | On diagonals of operators: selfadjoint, normal and other classes | We provide a survey of the current state of the study of diagonals of operators, especially selfadjoint operators. In addition, we provide a few new results made possible by recent work of Müller-Tomilov and Kaftal-Loreaux. This is an expansion of the second author's lecture part II at OT27. | \section{Introduction}
\label{sec:introduction}
By a diagonal of an operator $T \in B(\ensuremath{\mathcal{H}})$ we mean a sequence $(\angles{Te_n,e_n})_{n=1}^{\infty}$ where $\{e_n\}_{n=1}^{\infty}$ is an orthonormal basis of $\ensuremath{\mathcal{H}}$.
The orthonormal basis is not fixed, and so $T$ has many diagonal... | {
"timestamp": "2019-05-27T02:05:37",
"yymm": "1905",
"arxiv_id": "1905.09987",
"language": "en",
"url": "https://arxiv.org/abs/1905.09987",
"abstract": "We provide a survey of the current state of the study of diagonals of operators, especially selfadjoint operators. In addition, we provide a few new resul... |
https://arxiv.org/abs/0803.3846 | Combinatorics of binomial primary decomposition | An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is derived from a characteristic-free combinatorial description of certain primary components of binomial ideals in affine semigroup rings, namely those that are associa... | \section{Introduction
A binomial is a polynomial with at most two terms; a binomial ideal is
an ideal generated by binomials. Binomial ideals abound as the
defining ideals of classical algebraic varieties, particularly because
equivariantly embedded affine or projective toric varieties
correspond to prime binomial ... | {
"timestamp": "2008-03-27T01:26:32",
"yymm": "0803",
"arxiv_id": "0803.3846",
"language": "en",
"url": "https://arxiv.org/abs/0803.3846",
"abstract": "An explicit lattice point realization is provided for the primary components of an arbitrary binomial ideal in characteristic zero. This decomposition is de... |
https://arxiv.org/abs/1612.09108 | Nonstandard Measure Spaces with Values in non-Archimedean Fields | The aim of this contribution is to bring together the areas of $p$-adic analysis and nonstandard analysis. We develop a nonstandard measure theory with values in a complete non-Archimedean valued field $K$, e.g. the $p-$adic numbers $\mathbb{Q}_p$. The corresponding theory for real-valued measures is well known by the ... | \section{Introduction}
Integrals of functions with values in a complete non-Archimedean field are studied in the field of {\em $p$-adic analysis} and a general measure-theoretical approach to $p$-adic integration has been developed by A. van Rooij \cite{rooij}.
$p$-adic measures and integrals are used in number theor... | {
"timestamp": "2016-12-30T02:07:19",
"yymm": "1612",
"arxiv_id": "1612.09108",
"language": "en",
"url": "https://arxiv.org/abs/1612.09108",
"abstract": "The aim of this contribution is to bring together the areas of $p$-adic analysis and nonstandard analysis. We develop a nonstandard measure theory with va... |
https://arxiv.org/abs/0708.0923 | Spherical Nilpotent Orbits in Positive Characteristic | Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we classify all the spherical nilpotent G-orbits in the Lie algebra of G. The classification is the same as in the characteristic zero case obtained by D.I. Pan... | \section{Introduction}
\label{s:intro}
Let $G$ be a connected reductive linear algebraic group
defined over an algebraically closed field $k$
of characteristic $p > 0$. With the exception of Subsection
\ref{sub:bad}, we assume throughout that $p$ is \emph{good} for $G$
(see Subsection \ref{sub:not} for a definition).
... | {
"timestamp": "2008-05-27T15:37:54",
"yymm": "0708",
"arxiv_id": "0708.0923",
"language": "en",
"url": "https://arxiv.org/abs/0708.0923",
"abstract": "Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this n... |
https://arxiv.org/abs/1205.0095 | Dependence of Kolmogorov widths on the ambient space | We study the dependence of the Kolmogorov widths of a compact set on the ambient Banach space. | \section{Introduction}
Let $\mz $ be a subset of a Banach space $\mx $ and $x\in \mx $.
The {\it distance from $x$ to $\mz $} is defined as
\[E(x,\mz )=\inf\{||x-z||:~z\in \mz \}.\]
\begin{definition}
{\rm Let $K$ be a subset of a Banach space $\mx $, $n\in\mathbb{N}\cup\{0\}$.
The {\it Kolmogorov $n$-width} (or {\it ... | {
"timestamp": "2012-05-02T02:01:40",
"yymm": "1205",
"arxiv_id": "1205.0095",
"language": "en",
"url": "https://arxiv.org/abs/1205.0095",
"abstract": "We study the dependence of the Kolmogorov widths of a compact set on the ambient Banach space.",
"subjects": "Functional Analysis (math.FA); Metric Geomet... |
https://arxiv.org/abs/2011.07244 | A Blaschke-Lebesgue Theorem for the Cheeger constant | In this paper we prove a new extremal property of the Reuleaux triangle: it maximizes the Cheeger constant among all bodies of (same) constant width. The proof relies on a fine analysis of the optimality conditions satisfied by an optimal Reuleaux polygon together with an explicit upper bound for the inradius of the op... | \section{Introduction}
Bodies of constant width (also named after L. Euler {\it orbiforms}) have attracted much attention in the mathematical community along the last centuries.
Several surveys have been devoted to these objects, and contain an abundant literature.
We refer notably to a chapter in Bonnesen-Fenchel's
f... | {
"timestamp": "2020-11-17T02:09:14",
"yymm": "2011",
"arxiv_id": "2011.07244",
"language": "en",
"url": "https://arxiv.org/abs/2011.07244",
"abstract": "In this paper we prove a new extremal property of the Reuleaux triangle: it maximizes the Cheeger constant among all bodies of (same) constant width. The ... |
https://arxiv.org/abs/0907.4412 | The Cohomology Ring of the Space of Rational Functions | Let Rat_k be the space of based holomorphic maps from S^2 to itself of degree k. Let beta_k denote the Artin's braid group on k strings and let Bbeta_k be the classifying space of beta_k. Let C_k denote the space of configurations of length less than or equal to k of distinct points in R^2 with labels in S^1. The three... | \section{Introduction}
Let $\beta_k$ denote the Artin's braid group on $k$ strings. Let
$B\beta_k$ be the classifying space of $\beta_k$. Let $C_k({\mathbb{R}}^2,S^1)$
denote the space of configurations of length less than or equal to $k$
of distict points in ${\mathbb{R}}^2$ with labels in $S^1$, with some
identifica... | {
"timestamp": "2009-07-25T13:53:06",
"yymm": "0907",
"arxiv_id": "0907.4412",
"language": "en",
"url": "https://arxiv.org/abs/0907.4412",
"abstract": "Let Rat_k be the space of based holomorphic maps from S^2 to itself of degree k. Let beta_k denote the Artin's braid group on k strings and let Bbeta_k be t... |
https://arxiv.org/abs/math/0511684 | Global residues for sparse polynomial systems | We consider families of sparse Laurent polynomials f_1,...,f_n with a finite set of common zeroes Z_f in the complex algebraic n-torus. The global residue assigns to every Laurent polynomial g the sum of its Grothendieck residues over the set Z_f. We present a new symbolic algorithm for computing the global residue as ... | \section{Introduction}
Let $f_1=\cdots=f_n=0$ be a system of Laurent polynomial equations in $n$
variables whose Newton polytopes are $\Delta_1,\dots,\Delta_n$. Suppose the
solution set $Z_f$ in the algebraic torus ${\mathbb T}=(\mathbb C-\{0\})^n$ is finite.
This will be true if the coefficients of the $f_i$ are gener... | {
"timestamp": "2006-05-30T18:36:58",
"yymm": "0511",
"arxiv_id": "math/0511684",
"language": "en",
"url": "https://arxiv.org/abs/math/0511684",
"abstract": "We consider families of sparse Laurent polynomials f_1,...,f_n with a finite set of common zeroes Z_f in the complex algebraic n-torus. The global res... |
https://arxiv.org/abs/0804.0427 | Fibered orbifolds and crystallographic groups | In this paper, we prove that a normal subgroup N of an n-dimensional crystallographic group G determines a geometric fibered orbifold structure on the flat orbifold E^n/G, and conversely every geometric fibered orbifold structure on E^n/G is determined by a normal subgroup N of G, which is maximal in its commensurabili... | \section{Introduction}
Let $E^n$ be Euclidean $n$-space.
A map $\phi:E^n\to E^n$ is an isometry of $E^n$
if and only if there is an $a\in E^n$ and an $A\in {\rm O}(n)$ such that
$\phi(x) = a + Ax$ for each $x$ in $E^n$.
We shall write $\phi = a+ A$.
In particular, every translation $\tau = a + I$ is an isometry of... | {
"timestamp": "2009-10-20T18:49:03",
"yymm": "0804",
"arxiv_id": "0804.0427",
"language": "en",
"url": "https://arxiv.org/abs/0804.0427",
"abstract": "In this paper, we prove that a normal subgroup N of an n-dimensional crystallographic group G determines a geometric fibered orbifold structure on the flat ... |
https://arxiv.org/abs/2205.03928 | Number of complete subgraphs of Peisert graphs and finite field hypergeometric functions | For a prime $p\equiv 3\pmod{4}$ and a positive integer $t$, let $q=p^{2t}$. Let $g$ be a primitive element of the finite field $\mathbb{F}_q$. The Peisert graph $P^\ast(q)$ is defined as the graph with vertex set $\mathbb{F}_q$ where $ab$ is an edge if and only if $a-b\in\langle g^4\rangle \cup g\langle g^4\rangle$. We... | \part{title}
\usepackage{amsmath,amsthm,amssymb,amscd}
\newcommand{\mathcal E}{\mathcal E}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{result}[theorem]{Result}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\n... | {
"timestamp": "2022-05-10T02:24:02",
"yymm": "2205",
"arxiv_id": "2205.03928",
"language": "en",
"url": "https://arxiv.org/abs/2205.03928",
"abstract": "For a prime $p\\equiv 3\\pmod{4}$ and a positive integer $t$, let $q=p^{2t}$. Let $g$ be a primitive element of the finite field $\\mathbb{F}_q$. The Peis... |
https://arxiv.org/abs/1901.02652 | $d$-Galvin families | The Galvin problem asks for the minimum size of a family $\mathcal{F} \subseteq \binom{[n]}{n/2}$ with the property that, for any set $A$ of size $\frac n 2$, there is a set $S \in \mathcal{F}$ which is balanced on $A$, meaning that $|S \cap A| = |S \cap \overline{A}|$. We consider a generalization of this question tha... |
\section{Introduction}
\label{section:intro}
\subsection{Galvin problem}
The starting point of this paper is a question raised by Galvin in
extremal combinatorics. Given two sets $A$ and $S$, we say that $S$ is
\defin{balanced on $A$} if $|S \cap A| = \frac{|S|}{2}$.
\begin{figure}[h!]
\label{fig:exGalvin}
\ce... | {
"timestamp": "2019-01-10T02:08:33",
"yymm": "1901",
"arxiv_id": "1901.02652",
"language": "en",
"url": "https://arxiv.org/abs/1901.02652",
"abstract": "The Galvin problem asks for the minimum size of a family $\\mathcal{F} \\subseteq \\binom{[n]}{n/2}$ with the property that, for any set $A$ of size $\\fr... |
https://arxiv.org/abs/1511.04720 | On some series formed by values of the Riemann Zeta function | The partial fraction expansion of coth($\pi$z), due to Euler, is generalized to power series having for coefficients the Riemann zeta function evaluated at certain arithmetic sequences. A further generalization using arbitrary Dirichlet series is also proposed. The resulting formulas are new, as far as we know, since t... | \section{Origin of the series}
The starting point for this paper is the known formula \cite[p. 49]{Valiron:1942}:
$$
\suny \frac1{n^2+z^2} = \begin{cases}
\displaystyle \frac{\pi^2}{6} & \text{if }z=0,
\\
\displaystyle \frac{\pi}{2z} \coth (\pi z) - \frac1{2z^2} & \text{if }0<|z|<1.
\end{cases}
$$
... | {
"timestamp": "2015-11-17T02:11:52",
"yymm": "1511",
"arxiv_id": "1511.04720",
"language": "en",
"url": "https://arxiv.org/abs/1511.04720",
"abstract": "The partial fraction expansion of coth($\\pi$z), due to Euler, is generalized to power series having for coefficients the Riemann zeta function evaluated ... |
https://arxiv.org/abs/1809.07398 | On Permutation Weights and $q$-Eulerian Polynomials | Weights of permutations were originally introduced by Dugan, Glennon, Gunnells, and Steingrímsson (Journal of Combinatorial Theory, Series A 164:24-49, 2019) in their study of the combinatorics of tiered trees. Given a permutation $\sigma$ viewed as a sequence of integers, computing the weight of $\sigma$ involves recu... | \section{Introduction}
Dugan, Glennon, Gunnells, and Steingr\'\i msson defined certain weights of permutations in their work on the combinatorics of tiered trees \cite{dugan2019tiered}. Tiered trees are a generalization of maxmin trees, which were originally introduced by Postnikov \cite{postnikov1997intransitive} an... | {
"timestamp": "2019-09-18T02:04:48",
"yymm": "1809",
"arxiv_id": "1809.07398",
"language": "en",
"url": "https://arxiv.org/abs/1809.07398",
"abstract": "Weights of permutations were originally introduced by Dugan, Glennon, Gunnells, and Steingrímsson (Journal of Combinatorial Theory, Series A 164:24-49, 20... |
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