url stringlengths 31 38 | title stringlengths 7 229 | abstract stringlengths 44 2.87k | text stringlengths 319 2.51M | meta dict |
|---|---|---|---|---|
https://arxiv.org/abs/1809.01509 | Maxwell eigenmodes in product domains | This paper is devoted to Maxwell modes in three-dimensional bounded electromagnetic cavities that have the form of a product of lower dimensional domains in some system of coordinates. The boundary conditions are those of the perfectly conducting or perfectly insulating body. The main case of interest is products in Ca... | \section{Introduction}
\label{S1}
A domain $\Omega$ of $\R^n$ is called a product domain if for a choice of Cartesian coordinates $\bx=(\by,\bz)$ in $\R^n$, the domain $\Omega$ coincides with the product $\cY\times\cZ$ in the sense that
\[
\bx\in\Omega\quad\Longleftrightarrow\quad\by\in\cY \ \ \mbox{and} \ \ \bz\in\... | {
"timestamp": "2018-10-12T02:06:52",
"yymm": "1809",
"arxiv_id": "1809.01509",
"language": "en",
"url": "https://arxiv.org/abs/1809.01509",
"abstract": "This paper is devoted to Maxwell modes in three-dimensional bounded electromagnetic cavities that have the form of a product of lower dimensional domains ... |
https://arxiv.org/abs/1412.5071 | Probabilistic analysis of Wiedemann's algorithm for minimal polynomial computation | Blackbox algorithms for linear algebra problems start with projection of the sequence of powers of a matrix to a sequence of vectors (Lanczos), a sequence of scalars (Wiedemann) or a sequence of smaller matrices (block methods). Such algorithms usually depend on the minimal polynomial of the resulting sequence being th... | \section{Introduction}
\label{introduction}
The minimal polynomial of a $n\times n$ matrix $A$ may be viewed as the minimal scalar generating polynomial of the linearly recurrent sequence of powers of $\bar{A} = (A^0, A^1, A^2, A^3, \ldots)$.
Wiedemann's algorithm \citep{Wiedemann86} projects the matrix sequence
to a ... | {
"timestamp": "2015-06-18T02:14:19",
"yymm": "1412",
"arxiv_id": "1412.5071",
"language": "en",
"url": "https://arxiv.org/abs/1412.5071",
"abstract": "Blackbox algorithms for linear algebra problems start with projection of the sequence of powers of a matrix to a sequence of vectors (Lanczos), a sequence o... |
https://arxiv.org/abs/1609.08842 | Analysis of Carrier's problem | A computational and asymptotic analysis of the solutions of Carrier's problem is presented. The computations reveal a striking and beautiful bifurcation diagram, with an infinite sequence of alternating pitchfork and fold bifurcations as the bifurcation parameter tends to zero. The method of Kuzmak is then applied to c... | \section{Introduction}
\label{intro}
In 1970, G.~F.~Carrier \cite[eq.~(3.5)]{carrier1970} introduced
the following singular perturbation problem
\begin{equation}
\epsilon^2 y'' + 2(1-x^2) y + y^2 = 1, \qquad y(-1) = y(1) = 0,\label{maineqn}
\end{equation}
where $0< \epsilon \ll 1$, and a prime represents $\fdd{}{x}$.
... | {
"timestamp": "2016-09-29T02:04:18",
"yymm": "1609",
"arxiv_id": "1609.08842",
"language": "en",
"url": "https://arxiv.org/abs/1609.08842",
"abstract": "A computational and asymptotic analysis of the solutions of Carrier's problem is presented. The computations reveal a striking and beautiful bifurcation d... |
https://arxiv.org/abs/1702.05634 | Explicit expressions for the moments of the size of an (n, dn-1)-core partition with distinct parts | In a previous paper (arXiv:1608.02262), we used computer-assisted methods to find explicit expressions for the moments of the size of a uniform random (n,n+1)-core partition with distinct parts. In particular, we conjectured that the distribution is asymptotically normal. However, our analysis hinged on a characterizat... | \section{Introduction}\label{sec:intro}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=.5\textwidth]{young.png}
\caption{Young diagram of the partition $9=4+3+1+1,$ showing the hook lengths of each box.}
\label{fig:young}
\end{center}
\end{figure}
\subsection{$(s,t)$-core partitions}
Recall that the \emph... | {
"timestamp": "2017-10-17T02:10:59",
"yymm": "1702",
"arxiv_id": "1702.05634",
"language": "en",
"url": "https://arxiv.org/abs/1702.05634",
"abstract": "In a previous paper (arXiv:1608.02262), we used computer-assisted methods to find explicit expressions for the moments of the size of a uniform random (n,... |
https://arxiv.org/abs/1501.01030 | A note on commuting graphs of matrix rings over fields | We will give a short proof of the fact that if the algebraic closure of a field $\mathbb F$ is a finite extension, then for $n\geq 3$ the commuting graph $\Gamma(M_n(\mathbb F))$ is connected and its diameter is four. | \section{Introduction}
Let $\mathds F$ be a field and $M_n(\mathds F)$ be the algebra of all $n\times n$ matrices over $\mathds F$. The commuting graph of $M_n(\mathds F)$, denoted by $\Gamma(M_n(\mathds F))$,
is the graph whose vertices are all non-central matrices and two distinct vertices $A$ and $B$ are adjacent ... | {
"timestamp": "2015-03-03T02:12:56",
"yymm": "1501",
"arxiv_id": "1501.01030",
"language": "en",
"url": "https://arxiv.org/abs/1501.01030",
"abstract": "We will give a short proof of the fact that if the algebraic closure of a field $\\mathbb F$ is a finite extension, then for $n\\geq 3$ the commuting grap... |
https://arxiv.org/abs/1605.00853 | A primitive associated to the Cantor-Bendixson derivative on the real line | We consider the class of compact countable subsets of the real numbers $\mathbb{R}$. By using an appropriate partition, up to homeomorphism, of this class we give a detailed proof of a result shown by S. Mazurkiewicz and W. Sierpinski related to the cardinality of this partition. Furthermore, for any compact subset of ... | \section{Introduction}
The earliest ideas of limit point and derived set in the space of the real numbers
were both introduced and investigated by Georg Cantor since 1872
(see also \cite{Cantor1872, Cantor1879, Cantor1880, Cantor1882, Cantor1883})
to analyze the convergence set of a trigonometric series. These ... | {
"timestamp": "2016-05-04T02:09:30",
"yymm": "1605",
"arxiv_id": "1605.00853",
"language": "en",
"url": "https://arxiv.org/abs/1605.00853",
"abstract": "We consider the class of compact countable subsets of the real numbers $\\mathbb{R}$. By using an appropriate partition, up to homeomorphism, of this clas... |
https://arxiv.org/abs/1404.2629 | Position Vectors of Numerical Semigroups | We provide a new way to represent numerical semigroups by showing that the position of every Apéry set of a numerical semigroup $S$ in the enumeration of the elements of $S$ is unique, and that $S$ can be re-constructed from this "position vector." We extend the discussion to more general objects called numerical sets,... | \section{Introduction}\label{sec: intro}
We let $\mathbb{N}$ and $\mathbb{N}_0$ denote the positive and nonnegative integers, respectively. A {\em numerical semigroup} $S$ is a subsemigroup of $\mathbb{N}_0$ that contains 0 and has finite complement in $\mathbb{N}_0$. For two elements $u$ and $u'$ in $S$, $u \preceq_S ... | {
"timestamp": "2014-07-16T02:00:44",
"yymm": "1404",
"arxiv_id": "1404.2629",
"language": "en",
"url": "https://arxiv.org/abs/1404.2629",
"abstract": "We provide a new way to represent numerical semigroups by showing that the position of every Apéry set of a numerical semigroup $S$ in the enumeration of th... |
https://arxiv.org/abs/1807.06465 | Invertibility of adjacency matrices for random $d$-regular graphs | Let $d\geq 3$ be a fixed integer and $A$ be the adjacency matrix of a random $d$-regular directed or undirected graph on $n$ vertices. We show there exist constants $\mathfrak d>0$, \begin{align*} {\mathbb P}(\text{$A$ is singular in $\mathbb R$})\leq n^{-\mathfrak{d}}, \end{align*} for $n$ sufficiently large. This ans... | \section{Introduction}
The most famous combinatorial problem concerning random matrices is perhaps the ``singularity'' problem. In a standard setting, when the entries of the $n\times n$ matrix are i.i.d. Bernoulli random variables (taking values $\pm1$ with probability $1/2$), this problem was first done by Koml{\'o}s... | {
"timestamp": "2019-01-01T02:15:33",
"yymm": "1807",
"arxiv_id": "1807.06465",
"language": "en",
"url": "https://arxiv.org/abs/1807.06465",
"abstract": "Let $d\\geq 3$ be a fixed integer and $A$ be the adjacency matrix of a random $d$-regular directed or undirected graph on $n$ vertices. We show there exis... |
https://arxiv.org/abs/1210.4959 | Halving Lines and Their Underlying Graphs | In this paper we study underlying graphs corresponding to a set of halving lines. We establish many properties of such graphs. In addition, we tighten the upper bound for the number of halving lines. | \section{Introduction}
Halving lines have been an interesting object of study for a long time. Given $n$ points in general position on a plane the minimum number of halving lines is $n/2$. The maximum number of halving lines is unknown. The current upper bound of $O(n^{4/3})$ is proven by Dey \cite{Dey98}. The current... | {
"timestamp": "2012-10-19T02:01:05",
"yymm": "1210",
"arxiv_id": "1210.4959",
"language": "en",
"url": "https://arxiv.org/abs/1210.4959",
"abstract": "In this paper we study underlying graphs corresponding to a set of halving lines. We establish many properties of such graphs. In addition, we tighten the u... |
https://arxiv.org/abs/math/0507469 | Distances between the winning numbers in Lottery | We prove an interesting fact about Lottery: the winning 6 numbers (out of 49) in the game of the Lottery contain two consecutive numbers with a surprisingly high probability (almost 50%). | \section{Introduction}
The game of lottery exists and has been run in many countries (such as the UK, the US, Germany, France, Ireland, Australia, Greece, Spain, etc.) for a number of years. In this game, the player chooses $m$ numbers from among the numbers $1,\ldots,n>m$, the order of the choice being unimportant an... | {
"timestamp": "2005-07-22T12:37:50",
"yymm": "0507",
"arxiv_id": "math/0507469",
"language": "en",
"url": "https://arxiv.org/abs/math/0507469",
"abstract": "We prove an interesting fact about Lottery: the winning 6 numbers (out of 49) in the game of the Lottery contain two consecutive numbers with a surpri... |
https://arxiv.org/abs/1307.2403 | Normal Forms for Symplectic Matrices | We give a self contained and elementary description of normal forms for symplectic matrices, based on geometrical considerations. The normal forms in question are expressed in terms of elementary Jordan matrices and integers with values in $\{-1,0,1\}$ related to signatures of quadratic forms naturally associated to th... | \section*{Introduction}
Let $V$ be a real vector space of dimension $2n$ with a non degenerate skewsymmetric bilinear form $\Omega$.
The symplectic group $\operatorname{Sp}(V,\Omega)$ is the set of linear transformations of $V$ which preserve $\Omega$:
\[
\operatorname{Sp}(V,\Omega) = \left\{ \, A: V \rightarrow V\,... | {
"timestamp": "2014-03-20T01:09:54",
"yymm": "1307",
"arxiv_id": "1307.2403",
"language": "en",
"url": "https://arxiv.org/abs/1307.2403",
"abstract": "We give a self contained and elementary description of normal forms for symplectic matrices, based on geometrical considerations. The normal forms in questi... |
https://arxiv.org/abs/2210.12465 | Direction-Critical Configurations in Noncentral General Position | In 1982, Ungar proved that the connecting lines of a set of $n$ noncollinear points in the plane determine at least $2\lfloor n/2 \rfloor$ directions (slopes). Sets achieving this minimum for $n$ odd (even) are called \emph{direction-(near)-critical} and their full classification is still open. To date, there are four ... | \section{Introduction}
In 1970, Scott \cite{S70} proposed the problem of finding the least number of directions (slopes) determined by a set of $n$ points in the plane, not all collinear. He conjectured that the minimum number of different slopes determined by the connecting lines of a set of $n$ noncollinear points in... | {
"timestamp": "2022-10-25T02:10:17",
"yymm": "2210",
"arxiv_id": "2210.12465",
"language": "en",
"url": "https://arxiv.org/abs/2210.12465",
"abstract": "In 1982, Ungar proved that the connecting lines of a set of $n$ noncollinear points in the plane determine at least $2\\lfloor n/2 \\rfloor$ directions (s... |
https://arxiv.org/abs/1405.7256 | Weakly Symmetrically Continuous Functions | We extend the definition of weak symmetric continuity to be applicable for functions defined on any nonempty subset of $\R$. Then we investigate basic properties of weakly symmetrically continuous functions and compare them with those of symmetrically continuous functions and weakly continuous functions. Several exampl... | \section{Introduction and Preliminaries}
There are many types of generalized continuities, three of which will be discussed in this article. Throughout, let $A$ be a nonempty subset of $\R$ and $a \in A$.
\begin{itemize}
\item[(1)] A function $f:A\rightarrow\R$ is said to be symmetrically continuous at $a$ if
\be... | {
"timestamp": "2014-05-29T02:08:36",
"yymm": "1405",
"arxiv_id": "1405.7256",
"language": "en",
"url": "https://arxiv.org/abs/1405.7256",
"abstract": "We extend the definition of weak symmetric continuity to be applicable for functions defined on any nonempty subset of $\\R$. Then we investigate basic prop... |
https://arxiv.org/abs/1508.01507 | Graph Homology and Stability of Coupled Oscillator Networks | There are a number of models of coupled oscillator networks where the question of the stability of fixed points reduces to calculating the index of a graph Laplacian. Some examples of such models include the Kuramoto and Kuramoto--Sakaguchi equations as well as the swing equations, which govern the behavior of generato... | \section{Introduction}
Let $G = (V,E,\Gamma = \{\gamma_{v,w}\}_{v,w\in V})$ be a weighted
graph. The Laplacian matrix\footnote{Our Laplacian is the negative of
the standard definition, but the reasons for this will be clear
below} of $G$ is the $|V|\times|V|$ matrix whose components are
\begin{equation*}
(\L_G)... | {
"timestamp": "2015-08-07T02:11:48",
"yymm": "1508",
"arxiv_id": "1508.01507",
"language": "en",
"url": "https://arxiv.org/abs/1508.01507",
"abstract": "There are a number of models of coupled oscillator networks where the question of the stability of fixed points reduces to calculating the index of a grap... |
https://arxiv.org/abs/2107.05272 | Convexification with bounded gap for randomly projected quadratic optimization | Random projection techniques based on Johnson-Lindenstrauss lemma are used for randomly aggregating the constraints or variables of optimization problems while approximately preserving their optimal values, that leads to smaller-scale optimization problems. D'Ambrosio et al. have applied random projection to a quadrati... | \section{An example appendix}
\bibliographystyle{siamplain}
\section{Conclusions}\label{sec:conclusions}
Random projections have been applied to solve optimization problems in suitable lower-dimensional spaces
in various existing works.
However, to the best of our knowledge, it is the first time they are use... | {
"timestamp": "2021-07-13T02:32:32",
"yymm": "2107",
"arxiv_id": "2107.05272",
"language": "en",
"url": "https://arxiv.org/abs/2107.05272",
"abstract": "Random projection techniques based on Johnson-Lindenstrauss lemma are used for randomly aggregating the constraints or variables of optimization problems ... |
https://arxiv.org/abs/1512.01721 | 2-Cycles on Higher Fano Hypersurfaces | Let F(X_d) be a smooth Fano variety of lines of a hypersurface X_d of degree d. In this paper, we prove the Griffiths group Griff_1(F(X_d)) is trivial if the hypersurface X_d is of 2-Fano type. As a result, we give a positive answer to a question of Professor Voisin about the first Griffiths groups of Fano varieties in... | \section{Introduction}
A fundamental question about cycles on a smooth projective variety $X$ over complex numbers is to determine the Griffiths groups of $X$. Let us recall the definition of the first Griffiths group \[\mathrm{Griff}_1(X)=\frac{\mathrm{CH}_1(X)_{hom}}{\mathrm{CH}_1(X)_{alg}}.\] Professor Voisin asks t... | {
"timestamp": "2016-10-14T02:00:26",
"yymm": "1512",
"arxiv_id": "1512.01721",
"language": "en",
"url": "https://arxiv.org/abs/1512.01721",
"abstract": "Let F(X_d) be a smooth Fano variety of lines of a hypersurface X_d of degree d. In this paper, we prove the Griffiths group Griff_1(F(X_d)) is trivial if ... |
https://arxiv.org/abs/0712.2866 | On Ext-indices of ring extensions | In this paper we are concerned with the finiteness property of Ext-indices of several ring extensions. In this direction, we introduce some conjectures and discuss the relationship of them. Also we give affirmative answers to these conjectures in some special cases. Furthermore, we prove that the trivial extension of a... | \section{introduction}
Throughout the paper, all rings are assumed to be commutative Noetherian rings with unity.
Let $R$ be a ring.
According to \cite{AY}, given nonzero $R$-modules $M$ and $N$, we define $p^R(M,N)$ by the following equality:
$$
p^R (M,N)= \sup \{ i \in \Bbb N \ | \ \mathrm{Ext}_R^i(M,N) \neq... | {
"timestamp": "2007-12-18T03:57:21",
"yymm": "0712",
"arxiv_id": "0712.2866",
"language": "en",
"url": "https://arxiv.org/abs/0712.2866",
"abstract": "In this paper we are concerned with the finiteness property of Ext-indices of several ring extensions. In this direction, we introduce some conjectures and ... |
https://arxiv.org/abs/1604.07099 | On Guarding Orthogonal Polygons with Sliding Cameras | A sliding camera inside an orthogonal polygon $P$ is a point guard that travels back and forth along an orthogonal line segment $\gamma$ in $P$. The sliding camera $g$ can see a point $p$ in $P$ if the perpendicular from $p$ onto $\gamma$ is inside $P$. In this paper, we give the first constant-factor approximation alg... | \section{Art-Gallery-Type Results}
\label{sec:artGalleryTheorem}
We now consider art-gallery-type problems for the MSC and MHSC problem; that is, we give tight bounds, depending on $n$, on the number of sliding cameras needed to guard an orthogonal polygon $P$ with $n$ vertices.
Recall that Aggarwal showed a tight bo... | {
"timestamp": "2016-04-26T02:11:44",
"yymm": "1604",
"arxiv_id": "1604.07099",
"language": "en",
"url": "https://arxiv.org/abs/1604.07099",
"abstract": "A sliding camera inside an orthogonal polygon $P$ is a point guard that travels back and forth along an orthogonal line segment $\\gamma$ in $P$. The slid... |
https://arxiv.org/abs/1903.08560 | Surprises in High-Dimensional Ridgeless Least Squares Interpolation | Interpolators -- estimators that achieve zero training error -- have attracted growing attention in machine learning, mainly because state-of-the art neural networks appear to be models of this type. In this paper, we study minimum $\ell_2$ norm ("ridgeless") interpolation in high-dimensional least squares regression. ... |
\section{Introduction}
Modern deep learning models involve a huge number of parameters. In many
applications, current practice suggests that we should design
the network to be sufficiently complex so that the model (as trained, typically,
by gradient descent) interpolates the data, i.e., achieves zero training
error... | {
"timestamp": "2020-12-08T02:45:17",
"yymm": "1903",
"arxiv_id": "1903.08560",
"language": "en",
"url": "https://arxiv.org/abs/1903.08560",
"abstract": "Interpolators -- estimators that achieve zero training error -- have attracted growing attention in machine learning, mainly because state-of-the art neur... |
https://arxiv.org/abs/math/0607447 | The D_4 root system is not universally optimal | We prove that the D_4 root system (equivalently, the set of vertices of the regular 24-cell) is not a universally optimal spherical code. We further conjecture that there is no universally optimal spherical code of 24 points in S^3, based on numerical computations suggesting that every 5-design consisting of 24 points ... | \section{Introduction}\label{sec1}
In \cite{CK} the authors (building on work by Yudin, Kolushov,
and Andreev in \cite{Y,KY1,KY2,A1,A2}) introduce the notion of a
{\em universally optimal code} in $S^{n-1}$, the unit sphere in
$\mathbb{R}^n$. For a function $f\colon [-1,1) \rightarrow \mathbb{R}$ and a finite set $C
\... | {
"timestamp": "2008-09-09T03:40:56",
"yymm": "0607",
"arxiv_id": "math/0607447",
"language": "en",
"url": "https://arxiv.org/abs/math/0607447",
"abstract": "We prove that the D_4 root system (equivalently, the set of vertices of the regular 24-cell) is not a universally optimal spherical code. We further c... |
https://arxiv.org/abs/1604.00960 | Counting Blanks in Polygonal Arrangements | Inside a two dimensional region ("cake"), there are $m$ non-overlapping tiles of a certain kind ("toppings"). We want to expand the toppings while keeping them non-overlapping, and possibly add some blank pieces of the same "certain kind", such that the entire cake is covered. How many blanks must we add?We study this ... | \section*{Introduction}
Consider a two-dimensional cake $C$ with $m$ pairwise-interior-disjoint
toppings $Z_1,Z_2,\ldots,Z_m$. We
would like to partition the entire cake to pairwise-interior-disjoint pieces $Z_1',Z_2',\ldots,$ such that each topping is contained in a piece and each piece contains at most a single toppi... | {
"timestamp": "2018-01-31T02:08:54",
"yymm": "1604",
"arxiv_id": "1604.00960",
"language": "en",
"url": "https://arxiv.org/abs/1604.00960",
"abstract": "Inside a two dimensional region (\"cake\"), there are $m$ non-overlapping tiles of a certain kind (\"toppings\"). We want to expand the toppings while kee... |
https://arxiv.org/abs/1202.2340 | A Poncelet theorem for lines | Our aim is to prove a Poncelet type theorem for a line configuration on the complex projective. More precisely, we say that a polygon with 2n sides joining 2n vertices A1, A2,..., A2n is well inscribed in a configuration Ln of n lines if each line of the configuration contains exactly two points among A1, A2, ..., A2n.... | \section{Introduction}
Let $\mathcal{L}_n$ be a configuration of $n$ lines $L_1, \cdots, L_n$ in the complex projective plane $\p^2$
and $D$ be a smooth conic
in the same plane. We assume that $\mathcal{L}_n\cap D$ consists in $2n$ distinct points. From a point $A_1$ on $L_1$ (not being
on the other lines neither o... | {
"timestamp": "2012-02-13T02:04:32",
"yymm": "1202",
"arxiv_id": "1202.2340",
"language": "en",
"url": "https://arxiv.org/abs/1202.2340",
"abstract": "Our aim is to prove a Poncelet type theorem for a line configuration on the complex projective. More precisely, we say that a polygon with 2n sides joining ... |
https://arxiv.org/abs/1710.11352 | Variations of the cop and robber game on graphs | We prove new theoretical results about several variations of the cop and robber game on graphs. First, we consider a variation of the cop and robber game which is more symmetric called the cop and killer game. We prove for all $c < 1$ that almost all random graphs are stalemate for the cop and killer game, where each e... | \section{Introduction}
The game of cop and robber on a graph is a simple model of the process of pursuing an adversary. Nowakowski and Winkler \cite{NW} and Quilliot \cite{Q} independently defined the game on a given graph $G$ and identified the graphs on which the cop has a winning strategy, assuming that both players... | {
"timestamp": "2017-11-01T01:05:50",
"yymm": "1710",
"arxiv_id": "1710.11352",
"language": "en",
"url": "https://arxiv.org/abs/1710.11352",
"abstract": "We prove new theoretical results about several variations of the cop and robber game on graphs. First, we consider a variation of the cop and robber game ... |
https://arxiv.org/abs/1603.05525 | Quantitative Tverberg theorems over lattices and other discrete sets | This paper presents a new variation of Tverberg's theorem. Given a discrete set $S$ of $R^d$, we study the number of points of $S$ needed to guarantee the existence of an $m$-partition of the points such that the intersection of the $m$ convex hulls of the parts contains at least $k$ points of $S$. The proofs of the ma... | \section{Discrete quantitative Helly numbers}\label{section-helly}
Helly's theorem and its numerous extensions are of central importance in discrete and computational geometry (see
\cite{amenta2015helly,DGKsurvey63,Eckhoffsurvey93,Wen1997}). Doignon was the first to calculate the $L$-Helly
number for an arbitrary la... | {
"timestamp": "2016-03-21T01:10:34",
"yymm": "1603",
"arxiv_id": "1603.05525",
"language": "en",
"url": "https://arxiv.org/abs/1603.05525",
"abstract": "This paper presents a new variation of Tverberg's theorem. Given a discrete set $S$ of $R^d$, we study the number of points of $S$ needed to guarantee the... |
https://arxiv.org/abs/1906.07598 | Zeros of holomorphic one-forms and topology of Kähler manifolds | A conjecture of Kotschick predicts that a compact Kähler manifold $X$ fibres smoothly over the circle if and only if it admits a holomorphic one-form without zeros. In this paper we develop an approach to this conjecture and verify it in dimension two. In a joint paper with Hao, we use our approach to prove Kotschick's... | \section{Introduction}
This paper is motivated by the following conjecture of Kotschick \cite{Ko}.
\begin{conjecture} \label{conj1}
For a compact K\"ahler manifold $X$, the following are equivalent.
\begin{enumerate}[(A)]
\item $X$ admits a holomorphic one-form without zeros; \label{item:conj1:hol-1-form}
\item \la... | {
"timestamp": "2019-11-11T02:08:11",
"yymm": "1906",
"arxiv_id": "1906.07598",
"language": "en",
"url": "https://arxiv.org/abs/1906.07598",
"abstract": "A conjecture of Kotschick predicts that a compact Kähler manifold $X$ fibres smoothly over the circle if and only if it admits a holomorphic one-form with... |
https://arxiv.org/abs/math/0506200 | On packing spheres into containers (about Kepler's finite sphere packing problem) | In an Euclidean $d$-space, the container problem asks to pack $n$ equally sized spheres into a minimal dilate of a fixed container. If the container is a smooth convex body and $d\geq 2$ we show that solutions to the container problem can not have a ``simple structure'' for large $n$. By this we in particular find that... | \section{Introduction}
How many equally sized spheres
can be packed into a given container?
In 1611, {\sc Kepler} discussed this question
in his booklet
\cite{kepler-1611} and came to the following conclusion:
\begin{quote}
``Coaptatio fiet arctissima, ut nullo praeterea ordine plures
\underline{globuli} \underline... | {
"timestamp": "2006-09-09T22:39:32",
"yymm": "0506",
"arxiv_id": "math/0506200",
"language": "en",
"url": "https://arxiv.org/abs/math/0506200",
"abstract": "In an Euclidean $d$-space, the container problem asks to pack $n$ equally sized spheres into a minimal dilate of a fixed container. If the container i... |
https://arxiv.org/abs/1606.01131 | Absolute real root separation | While the separation (the minimal nonzero distance) between roots of a polynomial is a classical topic, its absolute counterpart (the minimal nonzero distance between their absolute values) does not seem to have been studied much. We present the general context and give tight bounds for the case of real roots. | \section{Separation and absolute separation}
The polynomials $14x^3+17x^2-13x+2 $ and $17x^3-9x^2-7x+8$ hold records in the set of polynomials with integer coefficients in $\{-20,\dots,20\}$ and degree at most~3. The first one has two roots $\alpha_1,\alpha_2$ with
\[0<|\alpha_1-\alpha_2|<0.005, \]
while the second ... | {
"timestamp": "2016-06-06T02:11:08",
"yymm": "1606",
"arxiv_id": "1606.01131",
"language": "en",
"url": "https://arxiv.org/abs/1606.01131",
"abstract": "While the separation (the minimal nonzero distance) between roots of a polynomial is a classical topic, its absolute counterpart (the minimal nonzero dist... |
https://arxiv.org/abs/2111.01099 | Improved lower bounds for van der Waerden numbers | Recently, Ben Green proved that the two-color van der Waerden number $w(3,k)$ is bounded from below by $k^{b_0(k)}$ where $b_0(k) = c_0\left(\frac{\log k }{\log \log k}\right)^{1/3}$. We prove a new lower bound of $k^{b(k)}$ with $b(k) = \frac{c\log k}{\log \log k}$. This is done by modifying Green's argument, replacin... | \section{Introduction}
In this paper, we will be concerned with bounding the two-color van der Waerden numbers $w(3,k)$, defined as follows. For any $k\ge 3$, we let $w(3,k)$ denote the smallest $N$ such that for any blue-red coloring of $[N]:= \{1,\dots,N\}$, there either exists a blue arithmetic progression of lengt... | {
"timestamp": "2022-08-23T02:19:07",
"yymm": "2111",
"arxiv_id": "2111.01099",
"language": "en",
"url": "https://arxiv.org/abs/2111.01099",
"abstract": "Recently, Ben Green proved that the two-color van der Waerden number $w(3,k)$ is bounded from below by $k^{b_0(k)}$ where $b_0(k) = c_0\\left(\\frac{\\log... |
https://arxiv.org/abs/math/0505528 | De Bruijn Covering Codes for Rooted Hypergraphs | What is the length of the shortest sequence $S$ of reals so that the set of consecutive $n$-words in $S$ form a covering code for permutations on $\{1,2, >..., n\}$ of radius $R$ ? (The distance between two $n$-words is the number of transpositions needed to have the same order type.) The above problem can be viewed as... | \section{Introduction}
Suppose $G$ is a graph whose vertex set consists of some subset of all $n$-tuples ${\cal X}^n$ over a finite alphabet ${\cal X}$. The natural distance metric $d(\cdot,\cdot)$ on this graph allows us, for each nonnegative integer $R$, to define a ``ball of radius $R$ centered at $x \in {\cal X}... | {
"timestamp": "2005-05-25T08:17:58",
"yymm": "0505",
"arxiv_id": "math/0505528",
"language": "en",
"url": "https://arxiv.org/abs/math/0505528",
"abstract": "What is the length of the shortest sequence $S$ of reals so that the set of consecutive $n$-words in $S$ form a covering code for permutations on $\\{... |
https://arxiv.org/abs/1705.04768 | Dykstra's Algorithm, ADMM, and Coordinate Descent: Connections, Insights, and Extensions | We study connections between Dykstra's algorithm for projecting onto an intersection of convex sets, the augmented Lagrangian method of multipliers or ADMM, and block coordinate descent. We prove that coordinate descent for a regularized regression problem, in which the (separable) penalty functions are seminorms, is e... | \section{Introduction}
\label{sec:intro}
In this paper, we study two seemingly unrelated but closely connected
convex optimization problems, and associated algorithms.
The first is the {\it best approximation problem}: given
closed, convex sets $C_1,\ldots,C_d \subseteq \mathbb{R}^n$ and $y \in
\mathbb{R}^n$, we see... | {
"timestamp": "2017-05-16T02:02:08",
"yymm": "1705",
"arxiv_id": "1705.04768",
"language": "en",
"url": "https://arxiv.org/abs/1705.04768",
"abstract": "We study connections between Dykstra's algorithm for projecting onto an intersection of convex sets, the augmented Lagrangian method of multipliers or ADM... |
https://arxiv.org/abs/1905.05340 | Multivariate Ranks and Quantiles using Optimal Transport: Consistency, Rates, and Nonparametric Testing | In this paper we study multivariate ranks and quantiles, defined using the theory of optimal transport, and build on the work of Chernozhukov et al.(2017) and Hallin et al.(2021). We study the characterization, computation and properties of the multivariate rank and quantile functions and their empirical counterparts. ... | \section{Introduction}\label{sec:Intro}
Suppose that $X$ is a random vector in $\mathbb{R}^d$, for $d \ge 1$, with distribution $\nu$. When $d=1$, the rank and quantile functions of $X$ are defined as $F$ and $F^{-1}$ (the inverse\footnote{$F^{-1}(p) := \inf \left\{x\in {\mathbb {R}}:p\leq F(x)\right\}$.} of $F$), res... | {
"timestamp": "2019-09-10T02:10:44",
"yymm": "1905",
"arxiv_id": "1905.05340",
"language": "en",
"url": "https://arxiv.org/abs/1905.05340",
"abstract": "In this paper we study multivariate ranks and quantiles, defined using the theory of optimal transport, and build on the work of Chernozhukov et al.(2017)... |
https://arxiv.org/abs/2106.11955 | Large sets of generating tuples for Lie groups | We prove that for a connected, semisimple linear Lie group $G$ the spaces of generating pairs of elements or subgroups are well-behaved in a number of ways: the set of pairs of elements generating a dense subgroup is Zariski-open in the compact case, Euclidean-open in general, and always dense. Similarly, for sufficien... | \section{#1}\setcounter{lemma}{0}}
\title{Large sets of generating tuples for Lie groups}
\author{Alexandru Chirvasitu}
\begin{document}
\date{}
\newcommand{\Addresses}{
\bigskip
\footnotesize
\textsc{Department of Mathematics, University at Buffalo, Buffalo,
NY 14260-2900, USA}\par\nopagebreak \tex... | {
"timestamp": "2021-06-23T02:28:14",
"yymm": "2106",
"arxiv_id": "2106.11955",
"language": "en",
"url": "https://arxiv.org/abs/2106.11955",
"abstract": "We prove that for a connected, semisimple linear Lie group $G$ the spaces of generating pairs of elements or subgroups are well-behaved in a number of way... |
https://arxiv.org/abs/1405.4409 | An Improved Lower Bound for Arithmetic Regularity | The arithmetic regularity lemma due to Green [GAFA 2005] is an analogue of the famous Szemer{é}di regularity lemma in graph theory. It shows that for any abelian group $G$ and any bounded function $f:G \to [0,1]$, there exists a subgroup $H \le G$ of bounded index such that, when restricted to most cosets of $H$, the f... | \section{Introduction}
As an analogue of Szemer\'edi's regularity lemma in graph theory~\cite{Szemeredi78},
Green~\cite{Green05} proposed an arithmetic regularity lemma for abelian groups.
Given an abelian group $G$ and a bounded function $f:G \to [0,1]$, Green showed that one can find a subgroup $H \le G$ of bounde... | {
"timestamp": "2014-05-20T02:08:25",
"yymm": "1405",
"arxiv_id": "1405.4409",
"language": "en",
"url": "https://arxiv.org/abs/1405.4409",
"abstract": "The arithmetic regularity lemma due to Green [GAFA 2005] is an analogue of the famous Szemer{é}di regularity lemma in graph theory. It shows that for any ab... |
https://arxiv.org/abs/1804.02473 | New Perspectives on Neighborhood-Prime Labelings of Graphs | Neighborhood-prime labeling is a variation of prime labeling. A labeling $f:V(G) \to [|V(G)|]$ is a neighborhood-prime labeling if for each vertex $v\in V(G)$ with degree greater than $1$, the greatest common divisor of the set of labels in the neighborhood of $v$ is $1$. In this paper, we introduce techniques for find... | \section{Introduction}
Prime labeling is a type of graph labeling developed by Roger Entriger that was first formally introduced by Tout, Dabboucy, and Howalla~\cite{TDH}. We define $[n]:=\{1,\ldots,n\}$ where $n$ is a positive integer.
Given a simple graph~$G$ of order $n$, a \textit{prime labeling} consists of label... | {
"timestamp": "2018-04-10T02:03:12",
"yymm": "1804",
"arxiv_id": "1804.02473",
"language": "en",
"url": "https://arxiv.org/abs/1804.02473",
"abstract": "Neighborhood-prime labeling is a variation of prime labeling. A labeling $f:V(G) \\to [|V(G)|]$ is a neighborhood-prime labeling if for each vertex $v\\in... |
https://arxiv.org/abs/1904.07498 | A Turán-type theorem for large-distance graphs in Euclidean spaces, and related isodiametric problems | Given a measurable set $A\subset \mathbb R^d$ we consider the "large-distance graph" $\mathcal{G}_A$, on the ground set $A$, in which each pair of points from $A$ whose distance is bigger than 2 forms an edge. We consider the problems of maximizing the $2d$-dimensional Lebesgue measure of the edge set as well as the $d... | \section{Prologue, related work and main results}
Let us begin with a folklore result of Tur\'an which pertains to graphs
that contain no complete graph
(also called a \emph{clique})
on $k$ vertices. Given a graph $G=(V,E)$, we denote by $|V|$ and $|E|$ the number of its vertices and edges, respectively.
... | {
"timestamp": "2019-04-17T02:21:25",
"yymm": "1904",
"arxiv_id": "1904.07498",
"language": "en",
"url": "https://arxiv.org/abs/1904.07498",
"abstract": "Given a measurable set $A\\subset \\mathbb R^d$ we consider the \"large-distance graph\" $\\mathcal{G}_A$, on the ground set $A$, in which each pair of po... |
https://arxiv.org/abs/1912.09282 | Universal Hardy-Sobolev inequalities on hypersurfaces of Euclidean space | In this paper we study Hardy-Sobolev inequalities on hypersurfaces of $\mathbb{R}^{n+1}$, all of them involving a mean curvature term and having universal constants independent of the hypersurface. We first consider the celebrated Sobolev inequality of Michael-Simon and Allard, in our codimension one framework. Using t... | \section{Introduction}
In this article we establish some new Hardy inequalities on hypersurfaces of Euclidean space.
As the one of Carron~\cite{Car} --- for which we find an improved version --- all of them involve
a mean curvature term
and have universal constants. Our inequalities have
their origin in the rec... | {
"timestamp": "2020-03-02T02:09:09",
"yymm": "1912",
"arxiv_id": "1912.09282",
"language": "en",
"url": "https://arxiv.org/abs/1912.09282",
"abstract": "In this paper we study Hardy-Sobolev inequalities on hypersurfaces of $\\mathbb{R}^{n+1}$, all of them involving a mean curvature term and having universa... |
https://arxiv.org/abs/math/9807163 | A bilinear approach to the restriction and Kakeya conjectures | Bilinear restriction estimates have been appeared in work of Bourgain, Klainerman, and Machedon. In this paper we develop the theory of these estimates (together with the analogues for Kakeya estimates). As a consequence we improve the $(L^p,L^p)$ spherical restriction theorem of Wolff from $p > 42/11$ to $p > 34/9$, a... | \section{Introduction}
The purpose of this paper is to investigate bilinear variants of the
restriction and Kakeya conjectures, to relate them to the standard
formulations of these conjectures, and to give applications of this
bilinear approach to existing conjectures. The methods used are
based on several observatio... | {
"timestamp": "1998-07-29T12:22:33",
"yymm": "9807",
"arxiv_id": "math/9807163",
"language": "en",
"url": "https://arxiv.org/abs/math/9807163",
"abstract": "Bilinear restriction estimates have been appeared in work of Bourgain, Klainerman, and Machedon. In this paper we develop the theory of these estimate... |
https://arxiv.org/abs/1508.02432 | Zero-Divisor Graphs of Quotient Rings | The compressed zero-divisor graph $\Gamma_C(R)$ associated with a commutative ring $R$ has vertex set equal to the set of equivalence classes $\{ [r] \mid r \in Z(R), r \neq 0 \}$ where $r \sim s$ whenever $ann(r) = ann(s)$. Distinct classes $[r],[s]$ are adjacent in $\Gamma_C(R)$ if and only if $xy = 0$ for all $x \in... | \section{Introduction}
In \emph{Coloring of Commutative Rings} \cite{beck}, I. Beck introduces the \emph{zero-divisor graph} $\Gamma(R)$ associated with the zero-divisor set of a commutative ring, whose vertex set is the set of zero-divisors. Two distinct zero-divisors $x,y$ are adjacent in $\Gamma(R)$ if and only if... | {
"timestamp": "2015-08-12T02:01:24",
"yymm": "1508",
"arxiv_id": "1508.02432",
"language": "en",
"url": "https://arxiv.org/abs/1508.02432",
"abstract": "The compressed zero-divisor graph $\\Gamma_C(R)$ associated with a commutative ring $R$ has vertex set equal to the set of equivalence classes $\\{ [r] \\... |
https://arxiv.org/abs/1501.01571 | An Introduction to Matrix Concentration Inequalities | In recent years, random matrices have come to play a major role in computational mathematics, but most of the classical areas of random matrix theory remain the province of experts. Over the last decade, with the advent of matrix concentration inequalities, research has advanced to the point where we can conquer many (... | \chapter{Preface}
In recent years, random matrices have come to play a major role in computational mathematics, but most of the classical areas of random matrix theory remain the province of experts. Over the last decade, with the advent of matrix concentration inequalities, research has advanced to the point wher... | {
"timestamp": "2015-01-08T02:13:23",
"yymm": "1501",
"arxiv_id": "1501.01571",
"language": "en",
"url": "https://arxiv.org/abs/1501.01571",
"abstract": "In recent years, random matrices have come to play a major role in computational mathematics, but most of the classical areas of random matrix theory rema... |
https://arxiv.org/abs/1703.05549 | Minimum Perimeter-Sum Partitions in the Plane | Let $P$ be a set of $n$ points in the plane. We consider the problem of partitioning $P$ into two subsets $P_1$ and $P_2$ such that the sum of the perimeters of $\text{CH}(P_1)$ and $\text{CH}(P_2)$ is minimized, where $\text{CH}(P_i)$ denotes the convex hull of $P_i$. The problem was first studied by Mitchell and Wynt... | \section{Introduction}
The clustering problem is to partition a given data set into clusters (that is, subsets)
according to some measure of optimality. We are interested in clustering problems where the
data set is a set~$P$ of points in Euclidean space. Most of these clustering problems fall into
one of two categorie... | {
"timestamp": "2017-03-17T01:04:23",
"yymm": "1703",
"arxiv_id": "1703.05549",
"language": "en",
"url": "https://arxiv.org/abs/1703.05549",
"abstract": "Let $P$ be a set of $n$ points in the plane. We consider the problem of partitioning $P$ into two subsets $P_1$ and $P_2$ such that the sum of the perimet... |
https://arxiv.org/abs/1605.07784 | Fast Algorithms for Robust PCA via Gradient Descent | We consider the problem of Robust PCA in the fully and partially observed settings. Without corruptions, this is the well-known matrix completion problem. From a statistical standpoint this problem has been recently well-studied, and conditions on when recovery is possible (how many observations do we need, how many co... |
\section{Introduction}
{\em Principal component analysis} (PCA) aims to find a low rank subspace that best-approximates a data matrix $Y \in \ensuremath{\mathbb{R}}^{d_1 \times d_2}$. The simple and standard method of PCA by {\em singular value decomposition} (SVD) fails in many modern data problems due to missing and... | {
"timestamp": "2016-09-20T02:12:17",
"yymm": "1605",
"arxiv_id": "1605.07784",
"language": "en",
"url": "https://arxiv.org/abs/1605.07784",
"abstract": "We consider the problem of Robust PCA in the fully and partially observed settings. Without corruptions, this is the well-known matrix completion problem.... |
https://arxiv.org/abs/1510.07926 | Weighted de Bruijn Graphs for the Menage Problem and Its Generalizations | We address the problem of enumeration of seating arrangements of married couples around a circular table such that no spouses sit next to each other and no k consecutive persons are of the same gender. While the case of k=2 corresponds to the classical problème des ménages with a well-studied solution, no closed-form e... | \section{Introduction}
The famous \emph{menage problem} (\emph{probl\`eme des m\'enages}) asks for the number $M_n$ of seating arrangements of $n$ married couples of opposite sex around a circular table
such that
\begin{enumerate}
\item no spouses sit next to each other;
\item females and males alternate.
\end{enume... | {
"timestamp": "2016-08-10T02:09:39",
"yymm": "1510",
"arxiv_id": "1510.07926",
"language": "en",
"url": "https://arxiv.org/abs/1510.07926",
"abstract": "We address the problem of enumeration of seating arrangements of married couples around a circular table such that no spouses sit next to each other and n... |
https://arxiv.org/abs/1912.11315 | Constant index expectation curvature for graphs or Riemannian manifolds | An integral geometric curvature is defined as the index expectation K(x) = E[i(x)] if a probability measure m is given on vector fields on a Riemannian manifold or on a finite simple graph. Such curvatures are local, satisfy Gauss-Bonnet and are independent of any embedding in an ambient space. While realizing constant... | \section{In a nutshell}
\paragraph{}
If a probability distribution $p_x$ is given on the vertex set $V_x$ of
every complete sub-graph $x \in G$ of a finite simple graph $(V,E)$,
one obtains a curvature $K(v) = \sum_{x \in G} p_x(v) \omega(x)$ with
$\omega(x)=(-1)^{{\rm dim}(x)}$.
Such a curvature $K$ satisfies the... | {
"timestamp": "2019-12-25T02:08:41",
"yymm": "1912",
"arxiv_id": "1912.11315",
"language": "en",
"url": "https://arxiv.org/abs/1912.11315",
"abstract": "An integral geometric curvature is defined as the index expectation K(x) = E[i(x)] if a probability measure m is given on vector fields on a Riemannian ma... |
https://arxiv.org/abs/1404.2641 | Formal Fibers of Prime Ideals in Polynomial Rings | Let (R,m) be a Noetherian local domain of dimension n that is essentially finitely generated over a field and let R^ denote the m-adic completion of R. Matsumura has shown that n-1 is the maximal height possible for prime ideals of R^ in the generic formal fiber of R. In this article we prove that every prime ideal of ... | \section{Introduction}\label{intro}
Let $(R,{\bf m})$ be a Noetherian local domain and let $\widehat R$ be the
${\bf m}$-adic completion of $R$. The {\it generic formal fiber ring} of $R$ is
the localization $(R \setminus (0))^{-1}\widehat R$ of $\widehat R$ with respect
to the multiplicatively closed set of nonzero ... | {
"timestamp": "2014-04-11T02:02:08",
"yymm": "1404",
"arxiv_id": "1404.2641",
"language": "en",
"url": "https://arxiv.org/abs/1404.2641",
"abstract": "Let (R,m) be a Noetherian local domain of dimension n that is essentially finitely generated over a field and let R^ denote the m-adic completion of R. Mats... |
https://arxiv.org/abs/2112.00490 | Univariate Rational Sums of Squares | Given rational univariate polynomials f and g such that gcd(f, g) and f / gcd(f, g) are relatively prime, we show that g is non-negative on all the real roots of f if and only if g is a sum of squares of rational polynomials modulo f. We complete our study by exhibiting an algorithm that produces a certificate that a p... | \section{Introduction}
It is a classical result that a real univariate polynomial is {\em non-negative on all ${\mathbb R}$} if and only if it is a sum of squares of real polynomials (and in fact, 2 polynomials are enough). It was then proved by Landau in 1905, see \cite{landau}, that every univariate polynom... | {
"timestamp": "2022-04-13T02:14:54",
"yymm": "2112",
"arxiv_id": "2112.00490",
"language": "en",
"url": "https://arxiv.org/abs/2112.00490",
"abstract": "Given rational univariate polynomials f and g such that gcd(f, g) and f / gcd(f, g) are relatively prime, we show that g is non-negative on all the real r... |
https://arxiv.org/abs/1204.6527 | An upper bound on Euclidean embeddings of rigid graphs with 8 vertices | A graph is called (generically) rigid in R^d if, for any choice of sufficiently generic edge lengths, it can be embedded in R^d in a finite number of distinct ways, modulo rigid transformations. Here, we deal with the problem of determining the maximum number of planar Euclidean embeddings of minimally rigid graphs wit... | \section{Definitions}
Given a graph $G=(V,E)$ with $|V|=n$ and a collection of edge lengths $d_{ij} \in \mathbb{R}^{+}$, a Euclidean embedding of $G$ in $\mathbb{R}^d$ is a mapping of its vertices to a set of points $p_1,\ldots, p_n$, such that $d_{ij} = \Vert p_i - p_j \Vert$, for all $\{i,j\} \in E$.
We call a graph ... | {
"timestamp": "2012-05-01T02:04:39",
"yymm": "1204",
"arxiv_id": "1204.6527",
"language": "en",
"url": "https://arxiv.org/abs/1204.6527",
"abstract": "A graph is called (generically) rigid in R^d if, for any choice of sufficiently generic edge lengths, it can be embedded in R^d in a finite number of distin... |
https://arxiv.org/abs/1810.02673 | Direct and Inverse Theorems on Signed Sumsets of Integers | Let $G$ be an additive abelian group and $h$ be a positive integer. For a nonempty finite subset $A=\{a_0, a_1,\ldots, a_{k-1}\}$ of $G$, we let \[h_{\underline{+}}A:=\{\Sigma_{i=0}^{k-1}\lambda_{i} a_{i}: (\lambda_{0}, \ldots, \lambda_{k-1}) \in \mathbb{Z}^{k},~ \Sigma_{i=0}^{k-1}|\lambda_{i}|=h \},\] be the {\it sign... | \section{Introduction}\label{intro}
Let $G$ be an additive abelian group and $A=\{a_0, a_1,\ldots, a_{k-1}\}$ be a nonempty finite subset of $G$. Let $h$ be a positive integer. The $h$-fold sumset $hA$ of $A$ is the set of all sums of $h$ elements of $A$, that is,
\[hA=\{\Sigma_{i=0}^{k-1}\lambda_{i} a_{i}: (\lambda_{... | {
"timestamp": "2018-10-08T02:11:04",
"yymm": "1810",
"arxiv_id": "1810.02673",
"language": "en",
"url": "https://arxiv.org/abs/1810.02673",
"abstract": "Let $G$ be an additive abelian group and $h$ be a positive integer. For a nonempty finite subset $A=\\{a_0, a_1,\\ldots, a_{k-1}\\}$ of $G$, we let \\[h_{... |
https://arxiv.org/abs/2109.00144 | Hitting distribution of a correlated planar Brownian motion in a disk | In this paper we study the hitting probability of a circumference $C_R$ for a correlated Brownian motion $\underline{B}(t)=\left(B_1(t), B_2(t)\right)$, $\rho$ being the correlation coefficient. The analysis starts by first mapping the circle $C_R$ into an ellipse $E$ with semiaxes depending on $\rho$ and transforming ... | \section{Introduction}
The problem of finding the distribution of the hitting place of the boundary of simply connected sets (in $\mathbb{R}^d$, $d\ge2$) by a Brownian motion has been considered by different authors and from different points of view. Recently (\citeauthor{betsakos} \citeyear{betsakos}) the problem of h... | {
"timestamp": "2021-09-02T02:06:36",
"yymm": "2109",
"arxiv_id": "2109.00144",
"language": "en",
"url": "https://arxiv.org/abs/2109.00144",
"abstract": "In this paper we study the hitting probability of a circumference $C_R$ for a correlated Brownian motion $\\underline{B}(t)=\\left(B_1(t), B_2(t)\\right)$... |
https://arxiv.org/abs/2002.02550 | Nullities for a class of skew-symmetric Toeplitz band matrices | For all $n > k \ge 1$, we give formulas for the nullity $N(n,k)$ of the $n \times n$ skew-symmetric Toeplitz band matrix whose first $k$ superdiagonals have all entries $1$ and whose remaining superdiagonals have all entries $0$. This is accomplished by counting the number of cycles in certain directed graphs. As an ap... | \section{Introduction}
For $n > k \ge 1$ and $x \in \mathbb{R}$ ,
let $A(n,k,x)$ denote the $n \times n$ skew-symmetric
Toeplitz matrix whose first $k$ superdiagonals have all entries $1$,
and whose remaining superdiagonal entries are all $-x$.
For example, $A(6,2,x)$ is the matrix
\[
\left[
\begin{array}
[c]{cccc... | {
"timestamp": "2020-02-10T02:03:14",
"yymm": "2002",
"arxiv_id": "2002.02550",
"language": "en",
"url": "https://arxiv.org/abs/2002.02550",
"abstract": "For all $n > k \\ge 1$, we give formulas for the nullity $N(n,k)$ of the $n \\times n$ skew-symmetric Toeplitz band matrix whose first $k$ superdiagonals ... |
https://arxiv.org/abs/1704.08136 | Sudoku Rectangle Completion | Over the last decade, Sudoku, a combinatorial number-placement puzzle, has become a favorite pastimes of many all around the world. In this puzzle, the task is to complete a partially filled $9 \times 9$ square with numbers 1 through 9, subject to the constraint that each number must appear once in each row, each colum... | \section{Introduction and preliminaries}
\label{sec:definitions}
A Latin square is an $n\times n$ matrix with entries in $1,\ldots,n$ such that each of the numbers $1$ to $n$ appears exactly once in each row and in each column. Latin squares are heavily studied combinatorial objects that date back to the time of Euler... | {
"timestamp": "2017-04-27T02:06:37",
"yymm": "1704",
"arxiv_id": "1704.08136",
"language": "en",
"url": "https://arxiv.org/abs/1704.08136",
"abstract": "Over the last decade, Sudoku, a combinatorial number-placement puzzle, has become a favorite pastimes of many all around the world. In this puzzle, the ta... |
https://arxiv.org/abs/1302.2751 | On the existence of orthonormal geodesic bases for Lie algebras | We show that every unimodular Lie algebra, of dimension at most 4, equipped with an inner product, possesses an orthonormal basis comprised of geodesic elements. On the other hand, we give an example of a solvable unimodular Lie algebra of dimension 5 that has no orthonormal geodesic basis, for any inner product. | \section{Introduction}
Let $\mathfrak g $ be a Lie algebra equipped with an inner product $\langle\cdot,\cdot\rangle$. Consider the corresponding simply connected Lie group $G$ equipped with the left invariant Riemannian metric determined by $\langle\cdot,\cdot\rangle$.
A nonzero element $Y\in\mathfrak g $ is said ... | {
"timestamp": "2013-02-13T02:01:36",
"yymm": "1302",
"arxiv_id": "1302.2751",
"language": "en",
"url": "https://arxiv.org/abs/1302.2751",
"abstract": "We show that every unimodular Lie algebra, of dimension at most 4, equipped with an inner product, possesses an orthonormal basis comprised of geodesic elem... |
https://arxiv.org/abs/1509.04575 | Caratheodory's Theorem in Depth | Let $X$ be a finite set of points in $\mathbb{R}^d$. The Tukey depth of a point $q$ with respect to $X$ is the minimum number $\tau_X(q)$ of points of $X$ in a halfspace containing $q$. In this paper we prove a depth version of Carathéodory's theorem. In particular, we prove that there exists a constant $c$ (that depen... | \section{Introduction}
Carath\'eodory's theorem was proven by Carath\'edory in 1907; it is one of the fundamental results in convex geometry.
For sets of points in $\mathbb{R}^d$, it states the following.
\begin{theorem}[\textbf{Carath\'edory's theorem~\cite{caratheodory}}]
Let $X$ be a set of points in $\mathbb{R}^d$... | {
"timestamp": "2017-04-06T02:07:54",
"yymm": "1509",
"arxiv_id": "1509.04575",
"language": "en",
"url": "https://arxiv.org/abs/1509.04575",
"abstract": "Let $X$ be a finite set of points in $\\mathbb{R}^d$. The Tukey depth of a point $q$ with respect to $X$ is the minimum number $\\tau_X(q)$ of points of $... |
https://arxiv.org/abs/2011.03962 | Completely bounded homomorphisms of the Fourier algebra revisited | Let $A(G)$ and $B(H)$ be the Fourier and Fourier-Stieltjes algebras of locally compact groups $G$ and $H$, respectively. Ilie and Spronk have shown that continuous piecewise affine maps $\alpha: Y \subseteq H\rightarrow G$ induce completely bounded homomorphisms $\Phi:A(G)\rightarrow B(H)$, and that when $G$ is amenabl... | \section{Introduction}
Cohen in \cite{cohen} classified all bounded homomorphisms from the group algebra $L^1(G)$ to the measure algebra $M(H)$, for locally compact abelian groups $G,H$; this was later expounded with different proofs by Rudin in \cite{rudinbook}. The characterisation given was in terms of Pontryagin ... | {
"timestamp": "2021-09-15T02:21:52",
"yymm": "2011",
"arxiv_id": "2011.03962",
"language": "en",
"url": "https://arxiv.org/abs/2011.03962",
"abstract": "Let $A(G)$ and $B(H)$ be the Fourier and Fourier-Stieltjes algebras of locally compact groups $G$ and $H$, respectively. Ilie and Spronk have shown that c... |
https://arxiv.org/abs/1910.00744 | Reverse-Engineering Deep ReLU Networks | It has been widely assumed that a neural network cannot be recovered from its outputs, as the network depends on its parameters in a highly nonlinear way. Here, we prove that in fact it is often possible to identify the architecture, weights, and biases of an unknown deep ReLU network by observing only its output. Ever... | \section{Introduction}
A deep neural network computes a function from inputs to outputs, where the structure and parameters of the network control the function that is expressed. While each parameter influences the overall function, this influence is highly nonlinear, and the effects of different neurons within the net... | {
"timestamp": "2020-02-25T02:10:24",
"yymm": "1910",
"arxiv_id": "1910.00744",
"language": "en",
"url": "https://arxiv.org/abs/1910.00744",
"abstract": "It has been widely assumed that a neural network cannot be recovered from its outputs, as the network depends on its parameters in a highly nonlinear way.... |
https://arxiv.org/abs/1510.06785 | Maximizing Algebraic Connectivity in Interconnected Networks | Algebraic connectivity, the second eigenvalue of the Laplacian matrix, is a measure of node and link connectivity on networks. When studying interconnected networks it is useful to consider a multiplex model, where the component networks operate together with inter-layer links among them. In order to have a well-connec... | \section{Introduction}
Real-world networks are often connected together and therefore influence each other \cite{Arenas2014multilayer}. Robust design of interdependent networks is critical to allow uninterrupted flow of information, power, and goods in spite of possible errors and attacks \cite{Buldyrev2010}.
The se... | {
"timestamp": "2015-10-26T01:02:52",
"yymm": "1510",
"arxiv_id": "1510.06785",
"language": "en",
"url": "https://arxiv.org/abs/1510.06785",
"abstract": "Algebraic connectivity, the second eigenvalue of the Laplacian matrix, is a measure of node and link connectivity on networks. When studying interconnecte... |
https://arxiv.org/abs/1705.08446 | Symmetries of the equations of motion that are not shared by the Lagrangian | We show that if a Lagrangian is invariant under a transformation (with the invariance defined in the standard manner), then the equations of motion obtained from it maintain their form under the transformation. We also show that the converse is not true, giving examples of equations of motion that are form-invariant un... | \section{Introduction}
The role of the symmetries is widely recognized in physics and, especially, in the theory of fields, to the extent that in many cases the symmetries are used as a guide to establish the basic equations of a theory or a model. For instance, in the framework of the special relativity, it is postul... | {
"timestamp": "2017-05-25T02:00:07",
"yymm": "1705",
"arxiv_id": "1705.08446",
"language": "en",
"url": "https://arxiv.org/abs/1705.08446",
"abstract": "We show that if a Lagrangian is invariant under a transformation (with the invariance defined in the standard manner), then the equations of motion obtain... |
https://arxiv.org/abs/1601.03981 | A point-line incidence identity in finite fields, and applications | Let $E \subseteq \mathbb{F}_q^2$ be a set in the 2-dimensional vector space over a finite field with $q$ elements. We prove an identity for the second moment of its incidence function and deduce a variety of existing results from the literature, not all naturally associated with lines in $\mathbb{F}_q^2$, in a unified ... | \section[Introduction]{Introduction}
\label{Introduction}
\let\thefootnote\relax\footnotetext{The second author is supported by the USA NSF DMS Grant 1500984.}
The first lemma in the breakthrough paper of Bourgain, Katz, and Tao \cite{BKT2004} on the sum-product phenomenon in finite fields states that if $A, B \subse... | {
"timestamp": "2016-02-23T02:03:11",
"yymm": "1601",
"arxiv_id": "1601.03981",
"language": "en",
"url": "https://arxiv.org/abs/1601.03981",
"abstract": "Let $E \\subseteq \\mathbb{F}_q^2$ be a set in the 2-dimensional vector space over a finite field with $q$ elements. We prove an identity for the second m... |
https://arxiv.org/abs/2002.07989 | Implicit bias with Ritz-Galerkin method in understanding deep learning for solving PDEs | This paper aims at studying the difference between Ritz-Galerkin (R-G) method and deep neural network (DNN) method in solving partial differential equations (PDEs) to better understand deep learning. To this end, we consider solving a particular Poisson problem, where the information of the right-hand side of the equat... | \section*{Acknowledgments}
\begin{abstract}
This paper aims at studying the difference between Ritz-Galerkin (R-G) method and deep neural network (DNN) method in solving partial differential equations (PDEs) to better understand deep learning. To this end, we consider solving a particular Poisson problem, where the... | {
"timestamp": "2020-12-10T02:08:35",
"yymm": "2002",
"arxiv_id": "2002.07989",
"language": "en",
"url": "https://arxiv.org/abs/2002.07989",
"abstract": "This paper aims at studying the difference between Ritz-Galerkin (R-G) method and deep neural network (DNN) method in solving partial differential equatio... |
https://arxiv.org/abs/2201.13282 | On Tusi's Classification of Cubic Equations and its Connections to Cardano's Formula and Khayyam's Geometric Solution | Omar Khayyam's studies on cubic equations inspired the 12th century Persian mathematician Sharaf al-Din Tusi to investigate the number of positive roots. According to the noted mathematical historian Rashed, Tusi analyzed the problem for five different types of equations. In fact all cubic equations are reducible to a ... | \section{Introduction} \label{sec1} In this article we examine the problem of solving for the real roots of a cubic equation
but in the context of the work of the 12th century Persian mathematician Sharaf al-Din Tusi. In the process we offer novel insights, including canonical representation of cubic equations, strate... | {
"timestamp": "2022-02-01T02:46:53",
"yymm": "2201",
"arxiv_id": "2201.13282",
"language": "en",
"url": "https://arxiv.org/abs/2201.13282",
"abstract": "Omar Khayyam's studies on cubic equations inspired the 12th century Persian mathematician Sharaf al-Din Tusi to investigate the number of positive roots. ... |
https://arxiv.org/abs/1401.0770 | The Expected Shape of Random Doubly Alternating Baxter Permutations | Guibert and Linusson introduced the family of doubly alternating Baxter permutations, i.e. Baxter permutations $\sigma \in S_n$, such that $\sigma$ and $\sigma^{-1}$ are alternating. They proved that the number of such permutations in $S_{2n}$ and $S_{2n+1}$ is the Catalan number $C_n$. In this paper we explore the exp... | \section{Introduction}
\noindent
A \emph{Catalan structure} is a family of combinatorial objects whose number is the Catalan number
$$
C_n \, = \, \frac{1}{n+1}\hskip.02cm \binom{2n}{n}\..
$$
There is a staggering amount of literature on various Catalan structures,
the list of which is ever growing (see~\cite{Gould,Pa... | {
"timestamp": "2014-01-07T02:01:54",
"yymm": "1401",
"arxiv_id": "1401.0770",
"language": "en",
"url": "https://arxiv.org/abs/1401.0770",
"abstract": "Guibert and Linusson introduced the family of doubly alternating Baxter permutations, i.e. Baxter permutations $\\sigma \\in S_n$, such that $\\sigma$ and $... |
https://arxiv.org/abs/1707.02461 | Subspace Clustering with Missing and Corrupted Data | Given full or partial information about a collection of points that lie close to a union of several subspaces, subspace clustering refers to the process of clustering the points according to their subspace and identifying the subspaces. One popular approach, sparse subspace clustering (SSC), represents each sample as a... | \section{Introduction}
In many applications, including image compression \cite{hong2006multiscale, yang2008unsupervised}, network estimation \cite{eriksson2011domain}, video segmentation \cite{costeira1998multibody, kanatani2001}, and recommender systems \cite{zhang2012guess}, what is ostensibly high-dimensional data... | {
"timestamp": "2018-01-16T02:16:52",
"yymm": "1707",
"arxiv_id": "1707.02461",
"language": "en",
"url": "https://arxiv.org/abs/1707.02461",
"abstract": "Given full or partial information about a collection of points that lie close to a union of several subspaces, subspace clustering refers to the process o... |
https://arxiv.org/abs/1801.00070 | Sum of squares certificates for stability of planar, homogeneous, and switched systems | We show that existence of a global polynomial Lyapunov function for a homogeneous polynomial vector field or a planar polynomial vector field (under a mild condition) implies existence of a polynomial Lyapunov function that is a sum of squares (sos) and that the negative of its derivative is also a sum of squares. This... | \section{Introduction}
Consider a continuous time dynamical system
\begin{equation}\label{eq:CT.dynamics}
\dot{x}=f(x),
\end{equation}
where $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ is a polynomial and
has an equilibrium at the origin, i.e., $f(0)=0$. When a
polynomial function $V(x):\mathbb{R}^n\rightarrow\mathbb{R}$ i... | {
"timestamp": "2018-01-03T02:02:20",
"yymm": "1801",
"arxiv_id": "1801.00070",
"language": "en",
"url": "https://arxiv.org/abs/1801.00070",
"abstract": "We show that existence of a global polynomial Lyapunov function for a homogeneous polynomial vector field or a planar polynomial vector field (under a mil... |
https://arxiv.org/abs/2110.06420 | Where are the logs? | The commonly quoted error rates for QMC integration with an infinite low discrepancy sequence is $O(n^{-1}\log(n)^r)$ with $r=d$ for extensible sequences and $r=d-1$ otherwise. Such rates hold uniformly over all $d$ dimensional integrands of Hardy-Krause variation one when using $n$ evaluation points. Implicit in those... |
\section{Introduction}
In this article, we study the asymptotic
error rates for integration by quasi-Monte Carlo (QMC) as $n\to\infty$
while $f$ is fixed.
Most of the error upper bounds in QMC are based on fooling functions
$f_n$ that, given $n$ integration points, are poorly integrated.
By contrast, most of th... | {
"timestamp": "2022-01-19T02:17:34",
"yymm": "2110",
"arxiv_id": "2110.06420",
"language": "en",
"url": "https://arxiv.org/abs/2110.06420",
"abstract": "The commonly quoted error rates for QMC integration with an infinite low discrepancy sequence is $O(n^{-1}\\log(n)^r)$ with $r=d$ for extensible sequences... |
https://arxiv.org/abs/1507.03245 | A Geometric Approach for Bounding Average Stopping Time | We propose a geometric approach for bounding average stopping times for stopped random walks in discrete and continuous time. We consider stopping times in the hyperspace of time indexes and stochastic processes. Our techniques relies on exploring geometric properties of continuity or stopping regions. Especially, we m... | \section{Stopping Times and Convex Sets}
In this section, we shall propose to investigate stopping times with their geometric representations. We shall also establish a connection
between stopping times and convex sets. A stochastic characterization of convex sets is developed.
\subsection{Geometric Representations... | {
"timestamp": "2016-07-19T02:09:24",
"yymm": "1507",
"arxiv_id": "1507.03245",
"language": "en",
"url": "https://arxiv.org/abs/1507.03245",
"abstract": "We propose a geometric approach for bounding average stopping times for stopped random walks in discrete and continuous time. We consider stopping times i... |
https://arxiv.org/abs/math/9906016 | On periodic sequences for algebraic numbers | For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to a new iteration of the triangle. Cubic irrationals that are roots of x^3 + k x^... | \section{Introduction}
In 1848 Hermite \cite{Herm} posed to Jacobi the problem of
generalizing continued fractions so that periodic expansions of a
number reflect its algebraic properties. We state this as:
\noindent{\bf The Hermite Problem}: {\it Find methods for writing
numbers that reflect special algebrai... | {
"timestamp": "1999-08-13T19:37:40",
"yymm": "9906",
"arxiv_id": "math/9906016",
"language": "en",
"url": "https://arxiv.org/abs/math/9906016",
"abstract": "For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n... |
https://arxiv.org/abs/1605.00859 | Are monochromatic Pythagorean triples unavoidable under morphic colorings ? | A Pythagorean triple is a triple of positive integers a, b, c $\in$ N${}^{+}$ satisfying a${}^2$ + b${}^2$ = c${}^2$. Is it true that, for any finite coloring of N${}^{+}$ , at least one Pythagorean triple must be monochromatic? In other words, is the Dio-phantine equation X${}^2$+ Y${}^2$ = Z${}^2$ regular? This probl... | \section{Introduction}
A triple $(a,b,c)$ of positive integers is a \emph{Pythagorean triple} if it satisfies $a^2+b^2=c^2$, as $(3,4,5)$ for instance. Is it true that, for any finite coloring of the set of positive integers, monochromatic Pythagorean triples are unavoidable? While this typical Ramsey-type question h... | {
"timestamp": "2017-04-25T02:09:42",
"yymm": "1605",
"arxiv_id": "1605.00859",
"language": "en",
"url": "https://arxiv.org/abs/1605.00859",
"abstract": "A Pythagorean triple is a triple of positive integers a, b, c $\\in$ N${}^{+}$ satisfying a${}^2$ + b${}^2$ = c${}^2$. Is it true that, for any finite col... |
https://arxiv.org/abs/1609.03669 | Linear Stability of Hyperbolic Moment Models for Boltzmann Equation | Grad's moment models for Boltzmann equation were recently regularized to globally hyperbolic systems, and thus the regularized models attain local well-posedness for Cauchy data. The hyperbolic regularization is only related to the convection term in Boltzmann equation. We in this paper studied the regularized models w... | \section{Conclusion} \label{sec:conclusion}
The linear stability at the local equilibrium of both HME and OHME has
been proved with commonly used approximate collision terms, and
particularly with Boltzmann's binary collision model. Since HME
and OHME contain almost all hyperbolic regularized Grad's moment
system, the ... | {
"timestamp": "2017-03-24T01:02:30",
"yymm": "1609",
"arxiv_id": "1609.03669",
"language": "en",
"url": "https://arxiv.org/abs/1609.03669",
"abstract": "Grad's moment models for Boltzmann equation were recently regularized to globally hyperbolic systems, and thus the regularized models attain local well-po... |
https://arxiv.org/abs/1710.00408 | On the validity of the local Fourier analysis | Local Fourier analysis (LFA) is a useful tool in predicting the convergence factors of geometric multigrid methods (GMG). As is well known, on rectangular domains with periodic boundary conditions this analysis gives the exact convergence factors of such methods. In this work, using the Fourier method, we extend these ... | \section{Introduction}
\label{sec:intro}
The local Fourier analysis (LFA) introduced by A. Brandt \cite{Bra77},
is a tool which provides realistic quantitative estimates of the
asymptotic convergence factors of the GMG algorithms. For
discretizations of partial differential equations, the traditional LFA
is based on a... | {
"timestamp": "2017-10-10T02:07:19",
"yymm": "1710",
"arxiv_id": "1710.00408",
"language": "en",
"url": "https://arxiv.org/abs/1710.00408",
"abstract": "Local Fourier analysis (LFA) is a useful tool in predicting the convergence factors of geometric multigrid methods (GMG). As is well known, on rectangular... |
https://arxiv.org/abs/1509.07116 | The complex Brownian motion as a strong limit of processes constructed from a Poisson process | We construct a family of processes, from a single Poisson process, that converges in law to a complex Brownian motion. Moreover, we find realizations of these processes that converge almost surely to the complex Brownian motion, uniformly on the unit time interval. Finally the rate of convergence is derived. | \section{Introduction}
Kac \cite{K} in 1956 to
obtain a solution from a Poisson of the telegraph equation
\begin{equation}\label{tele} \frac1v \frac{\partial^2 F}{\partial
t^2}=v\frac{\partial^2 F}{\partial x^2}-\frac{2a}{v}\frac{\partial
F}{\partial t},\end{equation} with $a,v>0$, introduced the processes
$$x(t)=v\... | {
"timestamp": "2015-09-25T02:00:27",
"yymm": "1509",
"arxiv_id": "1509.07116",
"language": "en",
"url": "https://arxiv.org/abs/1509.07116",
"abstract": "We construct a family of processes, from a single Poisson process, that converges in law to a complex Brownian motion. Moreover, we find realizations of t... |
https://arxiv.org/abs/1305.2632 | Multiple lattice tiles and Riesz bases of exponentials | Suppose $\Omega\subseteq\RR^d$ is a bounded and measurable set and $\Lambda \subseteq \RR^d$ is a lattice. Suppose also that $\Omega$ tiles multiply, at level $k$, when translated at the locations $\Lambda$. This means that the $\Lambda$-translates of $\Omega$ cover almost every point of $\RR^d$ exactly $k$ times. We s... | \section{Introduction}\label{sec:intro}
\subsection{Riesz bases}\label{sec;riesz-bases}
In this paper we deal with the question of existence of a Riesz (unconditional) basis of exponentials
$$
e_t(x) := e(t \cdot x)= e^{2\pi i t \cdot x},\ \ t \in L,
$$
for the space $L^2(\Omega)$, where $\Omega\subseteq{\mathbb R}^d... | {
"timestamp": "2013-05-14T02:02:08",
"yymm": "1305",
"arxiv_id": "1305.2632",
"language": "en",
"url": "https://arxiv.org/abs/1305.2632",
"abstract": "Suppose $\\Omega\\subseteq\\RR^d$ is a bounded and measurable set and $\\Lambda \\subseteq \\RR^d$ is a lattice. Suppose also that $\\Omega$ tiles multiply,... |
https://arxiv.org/abs/2107.07847 | On the Shroer-Sauer-Ott-Yorke predictability conjecture for time-delay embeddings | Shroer, Sauer, Ott and Yorke conjectured in 1998 that the Takens delay embedding theorem can be improved in a probabilistic context. More precisely, their conjecture states that if $\mu$ is a natural measure for a smooth diffeomorphism of a Riemannian manifold and $k$ is greater than the information dimension of $\mu$,... | \section{Introduction}\label{sec:intro}
\subsection{General background}
This paper concerns probabilistic aspects of the \emph{Takens delay embedding theorem}, dealing with the problem of reconstructing a dynamical system from a sequence of measurements of a one-dimensional observable. More precisely, let $T\colon X \... | {
"timestamp": "2021-07-19T02:14:36",
"yymm": "2107",
"arxiv_id": "2107.07847",
"language": "en",
"url": "https://arxiv.org/abs/2107.07847",
"abstract": "Shroer, Sauer, Ott and Yorke conjectured in 1998 that the Takens delay embedding theorem can be improved in a probabilistic context. More precisely, their... |
https://arxiv.org/abs/2106.06096 | Universality of nodal count distribution in large metric graphs | An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graph's non-trivial topology. This number, called the nodal surplus, is an integer between 0 and the graph's first Betti number $\beta$. We study the distribution of the nodal surplus values in the countably infinite ... | \section{Introduction}
Denoting by $\nu_n$ the number of nodal domains of the $n$-th
Laplacian eigenfunction, Courant's theorem
\cite{Cou_ngwg23,CourantHilbert_volume1} establishes the bound
$\nu_n \leq n$. Here a ``nodal domain'' is a maximal connected
component of the underlying physical space where the eigenfuncti... | {
"timestamp": "2022-05-03T02:01:47",
"yymm": "2106",
"arxiv_id": "2106.06096",
"language": "en",
"url": "https://arxiv.org/abs/2106.06096",
"abstract": "An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graph's non-trivial topology. This number, called t... |
https://arxiv.org/abs/1108.0096 | Mean Value Theorems for Binary Egyptian Fractions | In this paper, we establish two mean value theorems for the number of solutions of the Diophantine equation $\frac{a}{n}=\frac{1}{x}+\frac{1}{y}$, in the case when $a$ is fixed and $n$ varies and in the case when both $a$ and $n$ vary. | \section{#1} \setcounter{equation}{0}}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{cor}{Corollary}
\newtheorem*{conj}{Conjecture}
\theoremstyle{remark}
\newtheorem*{rem}{Remark}
\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
... | {
"timestamp": "2011-08-02T02:01:51",
"yymm": "1108",
"arxiv_id": "1108.0096",
"language": "en",
"url": "https://arxiv.org/abs/1108.0096",
"abstract": "In this paper, we establish two mean value theorems for the number of solutions of the Diophantine equation $\\frac{a}{n}=\\frac{1}{x}+\\frac{1}{y}$, in the... |
https://arxiv.org/abs/2302.05824 | Pointwise error estimates and local superconvergence of Jacobi expansions | As one myth of polynomial interpolation and quadrature, Trefethen [30] revealed that the Chebyshev interpolation of $|x-a|$ (with $|a|<1 $) at the Clenshaw-Curtis points exhibited a much smaller error than the best polynomial approximation (in the maximum norm) in about $95\%$ range of $[-1,1]$ except for a small neigh... | \section{Introduction}
\label{introduction}
Approximation by polynomials plays a fundamental role in algorithm development and numerical analysis of many computational methods. It is known that for a given continuous function $f(x)$ defined on $[-1,1]$, the best polynomial approximation of $f(x)$ in the maximum no... | {
"timestamp": "2023-02-14T02:13:10",
"yymm": "2302",
"arxiv_id": "2302.05824",
"language": "en",
"url": "https://arxiv.org/abs/2302.05824",
"abstract": "As one myth of polynomial interpolation and quadrature, Trefethen [30] revealed that the Chebyshev interpolation of $|x-a|$ (with $|a|<1 $) at the Clensha... |
https://arxiv.org/abs/1408.4444 | Rate of convergence of the mean for sub-additive ergodic sequences | For sub-additive ergodic processes $\{X_{m,n}\}$ with weak dependence, we analyze the rate of convergence of $\mathbb{E}X_{0,n}/n$ to its limit $g$. We define an exponent $\gamma$ given roughly by $\mathbb{E}X_{0,n} \sim ng + n^\gamma$, and, assuming existence of a fluctuation exponent $\chi$ that gives $\mathrm{Var}~X... | \section{Introduction}
\subsection{Subadditive ergodic theorem}
Sub-additive ergodic theory plays a major role in modern mathematics. Its development began in probability with the work of Hammersley-Welsh \cite{HW} and with the seminal paper of Kingman \cite{Kingman}. Substantial discussion with many examples and ap... | {
"timestamp": "2014-10-22T02:14:39",
"yymm": "1408",
"arxiv_id": "1408.4444",
"language": "en",
"url": "https://arxiv.org/abs/1408.4444",
"abstract": "For sub-additive ergodic processes $\\{X_{m,n}\\}$ with weak dependence, we analyze the rate of convergence of $\\mathbb{E}X_{0,n}/n$ to its limit $g$. We d... |
https://arxiv.org/abs/2107.01029 | Words in Random Binary Sequences I | When flipping a fair coin, let $W = L_1L_2...L_N$ with $L_i\in\{H,T\}$ be a binary word of length $N=2$ or $N=3$. In this paper, we establish second- and third-order linear recurrence relations and their generating functions to discuss the probabilities $p_{W}(n)$ that binary words $W$ appear for the first time after $... | \section{Introduction}
\noindent
Consider a game in which one flips a balanced (fair) coin until one gets a given word like $HH$ (two heads in a row) or $HT$ (a head followed by a tail). The less flips it takes, the more one will win. Which word would be better to bet on: $HH$ or $HT$?
\smallskip
\noindent
More gen... | {
"timestamp": "2021-07-05T02:16:51",
"yymm": "2107",
"arxiv_id": "2107.01029",
"language": "en",
"url": "https://arxiv.org/abs/2107.01029",
"abstract": "When flipping a fair coin, let $W = L_1L_2...L_N$ with $L_i\\in\\{H,T\\}$ be a binary word of length $N=2$ or $N=3$. In this paper, we establish second- a... |
https://arxiv.org/abs/2203.15507 | Computation of Centroidal Voronoi Tessellations in High Dimensional spaces | Owing to the natural interpretation and various desirable mathematical properties, centroidal Voronoi tessellations (CVT) have found a wide range of applications and correspondingly a vast development in their literature. However the computation of CVT in higher dimensional spaces still remains difficult. In this paper... | \section{Introduction}
\label{sec::introduction}
Voronoi diagram is a partition of a set into subsets containing elements that are close to each other according to a certain metric. Even though they date centuries, Voronoi tessellations have been found immensely helpful in various applications ranging from health to c... | {
"timestamp": "2022-03-30T02:33:06",
"yymm": "2203",
"arxiv_id": "2203.15507",
"language": "en",
"url": "https://arxiv.org/abs/2203.15507",
"abstract": "Owing to the natural interpretation and various desirable mathematical properties, centroidal Voronoi tessellations (CVT) have found a wide range of appli... |
https://arxiv.org/abs/2012.00988 | Hamilton decompositions of line graphs | It is proved that if a graph is regular of even degree and contains a Hamilton cycle, or regular of odd degree and contains a Hamiltonian $3$-factor, then its line graph is Hamilton decomposable. This result partially extends Kotzig's result that a $3$-regular graph is Hamiltonian if and only if its line graph is Hamil... | \section{Introduction}
Hamilton decomposability of line graphs has been studied extensively. The {\em line graph} of a graph $G$, denoted by $L(G)$, is the graph with a vertex corresponding to each edge of $G$, and in which two vertices are adjacent if and only if their corresponding edges are adjacent in $G$. A {\em ... | {
"timestamp": "2020-12-03T02:13:02",
"yymm": "2012",
"arxiv_id": "2012.00988",
"language": "en",
"url": "https://arxiv.org/abs/2012.00988",
"abstract": "It is proved that if a graph is regular of even degree and contains a Hamilton cycle, or regular of odd degree and contains a Hamiltonian $3$-factor, then... |
https://arxiv.org/abs/1703.00747 | On the location of eigenvalues of matrix polynomials | A number $\lambda \in \mathbb C $ is called an {\it eigenvalue} of the matrix polynomial $P(z)$ if there exists a nonzero vector $x \in \mathbb C^n$ such that $P(\lambda)x = 0$. Note that each finite eigenvalue of $P(z)$ is a zero of the characteristic polynomial $\det(P(z))$. In this paper we establish some (upper and... | \section{Introduction}
Let $\mathbb C^{n\times n}$ be the set of all $n\times n$ matrices whose entries are in $\mathbb C$. For a \textit{matrix polynomial} we mean the matrix-valued function of a complex variable of the form
\begin{equation}\label{mp}
P(z)= A_m z^m + \cdots + A_1 z + A_0,
\end{equation}
where $A_i\... | {
"timestamp": "2019-02-19T02:09:06",
"yymm": "1703",
"arxiv_id": "1703.00747",
"language": "en",
"url": "https://arxiv.org/abs/1703.00747",
"abstract": "A number $\\lambda \\in \\mathbb C $ is called an {\\it eigenvalue} of the matrix polynomial $P(z)$ if there exists a nonzero vector $x \\in \\mathbb C^n$... |
https://arxiv.org/abs/1809.05069 | Cwikel's bound reloaded | There are a couple of proofs by now for the famous Cwikel--Lieb--Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schrödinger operator, proven in the 1970s. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblu... | \section{Introduction}\label{introduction}
We want to find natural bounds, with the right semi-classical behavior, for the number of negative eigenvalues of Schr\"odinger operators $P^2+V$ with $P=-i\nabla$, the momentum operator, or more general operators like the polyharmonic Schr\"odinger operators $|P|^{2\alpha}+V... | {
"timestamp": "2018-09-14T02:12:47",
"yymm": "1809",
"arxiv_id": "1809.05069",
"language": "en",
"url": "https://arxiv.org/abs/1809.05069",
"abstract": "There are a couple of proofs by now for the famous Cwikel--Lieb--Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a... |
https://arxiv.org/abs/1012.2552 | New approximations for the cone of copositive matrices and its dual | We provide convergent hierarchies for the cone C of copositive matrices and its dual, the cone of completely positive matrices. In both cases the corresponding hierarchy consists of nested spectrahedra and provide outer (resp. inner) approximations for C (resp. for its dual), thus complementing previous inner (resp. ou... | \section{Introduction}
In recent years the convex cone $\mathcal{C}$ of {\it copositive} matrices
and its dual cone $\mathcal{C}^*$ of {\it completely positive} matrices
have attracted a lot of attention,
in part because several interesting NP-hard problems can be modelled as convex conic optimization
problems over t... | {
"timestamp": "2012-01-20T02:04:11",
"yymm": "1012",
"arxiv_id": "1012.2552",
"language": "en",
"url": "https://arxiv.org/abs/1012.2552",
"abstract": "We provide convergent hierarchies for the cone C of copositive matrices and its dual, the cone of completely positive matrices. In both cases the correspond... |
https://arxiv.org/abs/1609.02258 | Tighter bound of Sketched Generalized Matrix Approximation | Generalized matrix approximation plays a fundamental role in many machine learning problems, such as CUR decomposition, kernel approximation, and matrix low rank approximation. Especially with today's applications involved in larger and larger dataset, more and more efficient generalized matrix approximation algorithem... | \section{Introduction}
Matrix manipulations are the basis of modern data analysis. As the datasets becomes larger and larger, it is much more difficult to perform exact matrix multiplication, inversion, and decomposition. Consequently, matrix approximation techniques have been extensively studied, including approxima... | {
"timestamp": "2016-09-09T02:02:17",
"yymm": "1609",
"arxiv_id": "1609.02258",
"language": "en",
"url": "https://arxiv.org/abs/1609.02258",
"abstract": "Generalized matrix approximation plays a fundamental role in many machine learning problems, such as CUR decomposition, kernel approximation, and matrix l... |
https://arxiv.org/abs/2009.04079 | Random Covering Sets in Metric Space with Exponentially Mixing Property | Let $\{B(\xi_n,r_n)\}_{n\ge1}$ be a sequence of random balls whose centers $\{\xi_n\}_{n\ge1}$ is a stationary process, and $\{r_n\}_{n\ge1}$ is a sequence of positive numbers decreasing to 0. Our object is the random covering set $E=\limsup\limits_{n\to\infty}B(\xi_n,r_n)$, that is, the points covered by $B(\xi_n,r_n)... | \section{Introduction}
Let $(X,d)$ be a complete metric space. Given a sequence of points $\{x_n\}_{n\ge1}$ in $X$, let $\{r_n\}_{n\ge1}$ be a sequence of positive numbers decreasing to 0. A general covering problem concerns the sets
\[\limsup\limits_{n\to\infty}B(x_n,r_n)=\bigcap_{k=1}^{\infty}\bigcup_{n=k}^{\infty}... | {
"timestamp": "2020-09-10T02:07:23",
"yymm": "2009",
"arxiv_id": "2009.04079",
"language": "en",
"url": "https://arxiv.org/abs/2009.04079",
"abstract": "Let $\\{B(\\xi_n,r_n)\\}_{n\\ge1}$ be a sequence of random balls whose centers $\\{\\xi_n\\}_{n\\ge1}$ is a stationary process, and $\\{r_n\\}_{n\\ge1}$ i... |
https://arxiv.org/abs/2106.04116 | Discrete-to-Continuous Extensions: piecewise multilinear extension, min-max theory and spectral theory | We introduce the homogeneous and piecewise multilinear extensions and the eigenvalue problem for locally Lipschitz function pairs, in order to develop a systematic framework for relating discrete and continuous min-max problems. This also enables us to investigate spectral properties for pairs of $p$-homogeneous functi... | \section{Introduction}\label{sec:introduction}
In his millennium paper \cite{Lovasz00}, Lov\'asz wrote: \textit{Connections
between discrete and continuous may be the subject of mathematical study on
their own right.} In fact, over the last few decades, many firm
bridges between the discrete data world and th... | {
"timestamp": "2021-06-09T02:12:11",
"yymm": "2106",
"arxiv_id": "2106.04116",
"language": "en",
"url": "https://arxiv.org/abs/2106.04116",
"abstract": "We introduce the homogeneous and piecewise multilinear extensions and the eigenvalue problem for locally Lipschitz function pairs, in order to develop a s... |
https://arxiv.org/abs/2209.01149 | Families of Young Functions and Limits of Orlicz Norms | Given a $\sigma$-finite measure space $(X,\mu)$, a Young function $\Phi$, and a one-parameter family of Young functions $\{\Psi_q\}$, we find necessary and sufficient conditions for the associated Orlicz norms of any function $f\in L^\Phi(X,\mu)$ to satisfy \[\lim_{q\rightarrow \infty}\|f\|_{L^{\Psi_q}(X,\mu)}=C\|f\|_{... | \section{Introduction}
Let \((X,\mu)\) be a measure space. A well-known result in classical analysis, see \cite{royden} and \cite{rudin}, is that if \(f:X\rightarrow\mathbb{R}\) is a \(\mu\)-measurable function such that \(f\in L^r(X,\mu)\) for some \(r\geq 1\), then
\begin{equation}\label{basic}
\lim_{p\rightar... | {
"timestamp": "2022-09-05T02:20:00",
"yymm": "2209",
"arxiv_id": "2209.01149",
"language": "en",
"url": "https://arxiv.org/abs/2209.01149",
"abstract": "Given a $\\sigma$-finite measure space $(X,\\mu)$, a Young function $\\Phi$, and a one-parameter family of Young functions $\\{\\Psi_q\\}$, we find necess... |
https://arxiv.org/abs/1811.03357 | Enumeration of lattice polytopes by their volume | A well known result by Lagarias and Ziegler states that there are finitely many equivalence classes of d-dimensional lattice polytopes having volume at most K, for fixed constants d and K. We describe an algorithm for the complete enumeration of such equivalence classes for arbitrary constants d and K. The algorithm, w... | \section{Introduction}
\label{sec:intro}
Finiteness results are not uncommon in the study of lattice polytopes. Most of
these are proven by fixing the dimension, showing an upper bound for the volume
and then using the following result by Lagarias and Ziegler.
\begin{thm}[{\cite[Theorem~2]{LZ91}}]\label{thm:LZ}
... | {
"timestamp": "2018-11-09T02:12:19",
"yymm": "1811",
"arxiv_id": "1811.03357",
"language": "en",
"url": "https://arxiv.org/abs/1811.03357",
"abstract": "A well known result by Lagarias and Ziegler states that there are finitely many equivalence classes of d-dimensional lattice polytopes having volume at mo... |
https://arxiv.org/abs/2209.00594 | Strengthening Hadwiger's conjecture for $4$- and $5$-chromatic graphs | Hadwiger's famous coloring conjecture states that every $t$-chromatic graph contains a $K_t$-minor. Holroyd [Bull. London Math. Soc. 29, (1997), pp. 139--144] conjectured the following strengthening of Hadwiger's conjecture: If $G$ is a $t$-chromatic graph and $S \subseteq V(G)$ takes all colors in every $t$-coloring o... | \section{Introduction}
Given a graph $G$ and a number $t \in \mathbb{N}$, a \emph{$K_t$-minor} in $G$ is a collection $(B_i)_{i=1}^{t}$ of pairwise disjoint non-empty subsets of $V(G)$ such that $G[B_i]$ is connected for every $i \in [t]$ and for every distinct $i, j \in [t]$ the sets $B_i$ and $B_j$ are adjacent\foot... | {
"timestamp": "2022-09-02T02:20:49",
"yymm": "2209",
"arxiv_id": "2209.00594",
"language": "en",
"url": "https://arxiv.org/abs/2209.00594",
"abstract": "Hadwiger's famous coloring conjecture states that every $t$-chromatic graph contains a $K_t$-minor. Holroyd [Bull. London Math. Soc. 29, (1997), pp. 139--... |
https://arxiv.org/abs/1408.4777 | Every point in a Riemmanian manifold is critical | We show that for any point $p$ in a closed Riemannian manifold $M$, there exists at least one point $q\in M$ such that $p$ is critical for the distance function from $q$. We also show that such a point $q$ cannot always be reached with geodesic loops based at $q$ with midpoint $p$. | \section{Main results}
Critical point theory has been of central importance in many areas of mathematics. In Riemannian geometry, however, the most natural functions are distance functions and, due to their possible lack of differentiability at the cut locus, it was not clear for some time what a critical point should ... | {
"timestamp": "2015-03-18T01:09:19",
"yymm": "1408",
"arxiv_id": "1408.4777",
"language": "en",
"url": "https://arxiv.org/abs/1408.4777",
"abstract": "We show that for any point $p$ in a closed Riemannian manifold $M$, there exists at least one point $q\\in M$ such that $p$ is critical for the distance fun... |
https://arxiv.org/abs/1906.10061 | Isoperimetric relations between Dirichlet and Neumann eigenvalues | Inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian have received much attention in the literature, but open problems abound. Here, we study the number of Neumann eigenvalues no greater than the first Dirichlet eigenvalue. Based on a combination of analytical and numerical results, we conjecture... | \section{Introduction}
Let $\Omega \subset \bbR^n$ be a bounded domain with sufficiently smooth boundary. Denote the Dirichlet and Neumann eigenvalues of $-\Delta$ by
\[
0 < \lambda_1 < \lambda_2 \leq \lambda_3 \leq \cdots
\]
and
\[
0 = \mu_1 < \mu_2 \leq \mu_3 \leq \cdots
\]
respectively. In the one-dimensional case... | {
"timestamp": "2019-06-25T02:35:14",
"yymm": "1906",
"arxiv_id": "1906.10061",
"language": "en",
"url": "https://arxiv.org/abs/1906.10061",
"abstract": "Inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian have received much attention in the literature, but open problems abound. Here... |
https://arxiv.org/abs/1208.6247 | Solving Quadratic Equations via PhaseLift when There Are About As Many Equations As Unknowns | This note shows that we can recover a complex vector x in C^n exactly from on the order of n quadratic equations of the form |<a_i, x>|^2 = b_i, i = 1, ..., m, by using a semidefinite program known as PhaseLift. This improves upon earlier bounds in [3], which required the number of equations to be at least on the order... | \section{Introduction}
\label{sec:intro}
Suppose we wish to solve quadratic equations of the form
\begin{equation}
\label{eq:quadratic}
|\<\vct{a}_i, \vct{x}_0\>|^2 = b_i, \quad i = 1, \ldots, m,
\end{equation}
where $\vct{x}_0 \in \mathbb{C}^n$ is unknown and $\vct{a}_i \in
\mathbb{C}^n$ and $b_i \in \mathbb{R... | {
"timestamp": "2012-09-18T02:06:18",
"yymm": "1208",
"arxiv_id": "1208.6247",
"language": "en",
"url": "https://arxiv.org/abs/1208.6247",
"abstract": "This note shows that we can recover a complex vector x in C^n exactly from on the order of n quadratic equations of the form |<a_i, x>|^2 = b_i, i = 1, ...,... |
https://arxiv.org/abs/2004.12521 | Convex hulls of polynomial Julia sets | We prove P. Alexandersson's conjecture that for every complex polynomial $p$ of degree $d \geq 2$ the convex hull $H_p$ of the Julia set $J_p$ of $p$ satisfies $p^{-1}(H_p) \subset H_p$. We further prove that the equality $p^{-1}(H_p) = H_p$ is achieved only if $p$ is affinely conjugated to the Chebyshev polynomial $T_... | \section{Introduction}
Let $d$ be a positive integer greater than or equal to $2$ and let $a_0,a_1,...,a_d$ be complex numbers such that $a_d \neq 0$. Then $p(z)=a_dz^d+...+a_1z+a_0, z \in \mathbb{C}$ is a polynomial of degree $d$. In particular,
\[
\lim_{|z| \to \infty} \frac{|p(z)|}{|z|^d} >0.
\]
This pro... | {
"timestamp": "2020-04-28T02:22:16",
"yymm": "2004",
"arxiv_id": "2004.12521",
"language": "en",
"url": "https://arxiv.org/abs/2004.12521",
"abstract": "We prove P. Alexandersson's conjecture that for every complex polynomial $p$ of degree $d \\geq 2$ the convex hull $H_p$ of the Julia set $J_p$ of $p$ sat... |
https://arxiv.org/abs/1704.03062 | Controlling Lipschitz functions | Given any positive integers $m$ and $d$, we say the a sequence of points $(x_i)_{i\in I}$ in $\mathbb R^m$ is {\em Lipschitz-$d$-controlling} if one can select suitable values $y_i\; (i\in I)$ such that for every Lipschitz function $f:\mathbb R^m\rightarrow \mathbb R^d$ there exists $i$ with $|f(x_i)-y_i|<1$. We conjec... | \section{Introduction}
The following question, in some sense dual to Tarski's famous plank problem~\cite{Ta32,McMS14,Mo32}, was raised by L\'aszl\'o Fejes T\'oth~\cite{FT74}: What is the ``sparsest'' sequence of points in the plane with the property that every straight line $\ell$ comes closer than $1$ to at least one... | {
"timestamp": "2018-04-12T02:09:08",
"yymm": "1704",
"arxiv_id": "1704.03062",
"language": "en",
"url": "https://arxiv.org/abs/1704.03062",
"abstract": "Given any positive integers $m$ and $d$, we say the a sequence of points $(x_i)_{i\\in I}$ in $\\mathbb R^m$ is {\\em Lipschitz-$d$-controlling} if one ca... |
https://arxiv.org/abs/1312.3182 | On The Center Sets and Center Numbers of Some Graph Classes | For a set $S$ of vertices and the vertex $v$ in a connected graph $G$, $\displaystyle\max_{x \in S}d(x,v)$ is called the $S$-eccentricity of $v$ in $G$. The set of vertices with minimum $S$-eccentricity is called the $S$-center of $G$. Any set $A$ of vertices of $G$ such that $A$ is an $S$-center for some set $S$ of ve... | \section{Introduction}
centrality is one of the fundamental notions in graph theory which has established close connection between graph theory and various other areas like Social networks, Flow networks, Facility location problems etc. The main objective of any facility location problem is to identify the location o... | {
"timestamp": "2013-12-12T02:09:33",
"yymm": "1312",
"arxiv_id": "1312.3182",
"language": "en",
"url": "https://arxiv.org/abs/1312.3182",
"abstract": "For a set $S$ of vertices and the vertex $v$ in a connected graph $G$, $\\displaystyle\\max_{x \\in S}d(x,v)$ is called the $S$-eccentricity of $v$ in $G$. ... |
https://arxiv.org/abs/2112.14615 | Orderable groups and semigroup compactifications | Our aim is to find some new links between linear (circular) orderability of groups and topological dynamics. We suggest natural analogs of the concept of algebraic orderability for topological groups involving order-preserving actions on compact spaces and the corresponding enveloping semigroups in the sense of R. Elli... | \section{Introduction}
Orderability properties of groups is an active research direction since a long time.
This theory is closely related to topology. See, for example, \cite{CR-b,DNR,Calegari04,BS}.
Ordered dynamical systems were studied in several recent publications concerning tame systems, Sturmian like ... | {
"timestamp": "2022-09-29T02:01:41",
"yymm": "2112",
"arxiv_id": "2112.14615",
"language": "en",
"url": "https://arxiv.org/abs/2112.14615",
"abstract": "Our aim is to find some new links between linear (circular) orderability of groups and topological dynamics. We suggest natural analogs of the concept of ... |
https://arxiv.org/abs/1904.07416 | Distribution and correlation free two-sample test of high-dimensional means | We propose a two-sample test for high-dimensional means that requires neither distributional nor correlational assumptions, besides some weak conditions on the moments and tail properties of the elements in the random vectors. This two-sample test based on a nontrivial extension of the one-sample central limit theorem ... | \section{Introduction}\label{sec:intro}
Two-sample test of high dimensional means as one of the key issues has attracted a great deal of attention due to its importance in various applications, including \cite{bai:96},
\cite{chen:10:1}, \cite{sri:13:1}, \cite{cai:14:1}, \cite{yama:15:1}, \cite{feng:15:1}, \cite{greg:1... | {
"timestamp": "2019-04-17T02:17:46",
"yymm": "1904",
"arxiv_id": "1904.07416",
"language": "en",
"url": "https://arxiv.org/abs/1904.07416",
"abstract": "We propose a two-sample test for high-dimensional means that requires neither distributional nor correlational assumptions, besides some weak conditions o... |
https://arxiv.org/abs/1709.03451 | Lattice Size and Generalized Basis Reduction in Dimension 3 | The lattice size of a lattice polytope $P$ was defined and studied by Schicho, and Castryck and Cools. They provided an "onion skins" algorithm for computing the lattice size of a lattice polygon $P$ in $\mathbb{R}^2$ based on passing successively to the convex hull of the interior lattice points of $P$.We explain the ... | \section*{Introduction}
The {\it lattice size} $\operatorname{ls}_X(P)$ of a non-empty lattice polytope $P\subset\mathbb R^n$ with respect to a set $X$ of positive Jordan measure was defined in \cite{CasCools} as the smallest $l$ such that $T(P)$ is contained in the $l$-dilate of $X$ for some unimodular transformati... | {
"timestamp": "2017-10-30T01:01:34",
"yymm": "1709",
"arxiv_id": "1709.03451",
"language": "en",
"url": "https://arxiv.org/abs/1709.03451",
"abstract": "The lattice size of a lattice polytope $P$ was defined and studied by Schicho, and Castryck and Cools. They provided an \"onion skins\" algorithm for comp... |
https://arxiv.org/abs/1108.0710 | Chain-making games in grid-like posets | We study the Maker-Breaker game on the hypergraph of chains of fixed size in a poset. In a product of chains, the maximum size of a chain that Maker can guarantee building is $k-\lfloor r/2\rfloor$, where $k$ is the maximum size of a chain in the product, and $r$ is the maximum size of a factor chain. We also study a v... | \section{Introduction}
The {\it Maker-Breaker} game on a hypergraph $\mathcal{H}$ is played by Maker and
Breaker, who alternate turns (beginning with Maker). Each player moves by
choosing a previously unchosen vertex of $\mathcal{H}$. Maker wins by acquiring all
vertices of some edge of $\mathcal{H}$; Breaker wins if... | {
"timestamp": "2011-08-04T02:00:48",
"yymm": "1108",
"arxiv_id": "1108.0710",
"language": "en",
"url": "https://arxiv.org/abs/1108.0710",
"abstract": "We study the Maker-Breaker game on the hypergraph of chains of fixed size in a poset. In a product of chains, the maximum size of a chain that Maker can gua... |
https://arxiv.org/abs/0809.2643 | Loop-erased random walk and Poisson kernel on planar graphs | Lawler, Schramm and Werner showed that the scaling limit of the loop-erased random walk on $\mathbb{Z}^2$ is $\mathrm{SLE}_2$. We consider scaling limits of the loop-erasure of random walks on other planar graphs (graphs embedded into $\mathbb{C}$ so that edges do not cross one another). We show that if the scaling lim... | \section{Introduction}
Let $G$ be a graph. The \emph{loop-erased random walk} or LERW on
$G$ is obtained by performing a random walk on $G$,
and then erasing the loops in the random walk path in chronological order. The
resulting path is a self-avoiding path in the graph $G$,
starting and ending at the same points a... | {
"timestamp": "2008-09-16T08:24:51",
"yymm": "0809",
"arxiv_id": "0809.2643",
"language": "en",
"url": "https://arxiv.org/abs/0809.2643",
"abstract": "Lawler, Schramm and Werner showed that the scaling limit of the loop-erased random walk on $\\mathbb{Z}^2$ is $\\mathrm{SLE}_2$. We consider scaling limits ... |
https://arxiv.org/abs/1612.00160 | Maximum likelihood drift estimation for Gaussian process with stationary increments | The paper deals with the regression model $X_t = \theta t + B_t$, $t\in[0, T ]$, where $B=\{B_t, t\geq 0\}$ is a centered Gaussian process with stationary increments. We study the estimation of the unknown parameter $\theta$ and establish the formula for the likelihood function in terms of a solution to an integral equ... | \section{Introduction}
We study the problem of the drift parameter estimation for the stochastic process
\begin{equation}\label{eq:proc}
X_t = \theta t + B_t,
\end{equation}
where $\theta \in \mathbb{R}$ is an unknown parameter, and $B=\{B_t, t\geq 0\}$ is a centered Gaussian process with stationary increments,
$B_0 =... | {
"timestamp": "2016-12-02T02:02:58",
"yymm": "1612",
"arxiv_id": "1612.00160",
"language": "en",
"url": "https://arxiv.org/abs/1612.00160",
"abstract": "The paper deals with the regression model $X_t = \\theta t + B_t$, $t\\in[0, T ]$, where $B=\\{B_t, t\\geq 0\\}$ is a centered Gaussian process with stati... |
https://arxiv.org/abs/math/0501104 | Asymptotic cohomological functions of toric divisors | We study functions on the class group of a toric variety measuring the rates of growth of the cohomology groups of multiples of divisors. We show that these functions are piecewise polynomial with respect to finite polyhedral chamber decompositions. As applications, we express the self-intersection number of a T-Cartie... | \section{Introduction}
Suppose $D$ is an ample divisor on an $n$-dimensional algebraic
variety. The sheaf cohomology of $\ensuremath{\mathcal{O}}(D)$ does not necessarily reflect
the positivity of $D$; $\ensuremath{\mathcal{O}}(D)$ may have few global sections and its
higher cohomology groups may not vanish. However... | {
"timestamp": "2006-01-12T14:40:19",
"yymm": "0501",
"arxiv_id": "math/0501104",
"language": "en",
"url": "https://arxiv.org/abs/math/0501104",
"abstract": "We study functions on the class group of a toric variety measuring the rates of growth of the cohomology groups of multiples of divisors. We show that... |
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