url stringlengths 31 38 | title stringlengths 7 229 | abstract stringlengths 44 2.87k | text stringlengths 319 2.51M | meta dict |
|---|---|---|---|---|
https://arxiv.org/abs/0909.3354 | The Number of Independent Sets in a Regular Graph | We show that the number of independent sets in an N-vertex, d-regular graph is at most (2^{d+1} - 1)^{N/2d}, where the bound is sharp for a disjoint union of complete d-regular bipartite graphs. This settles a conjecture of Alon in 1991 and Kahn in 2001. Kahn proved the bound when the graph is assumed to be bipartite. ... | \section{Introduction}
Let $G$ be a (simple, finite, undirected) graph. An \emph{independent set} is a subset of the vertices with no two adjacent. Let $\mathcal I(G)$ denote the collection of independent sets of $G$ and let $i(G)$ be its cardinality. We would like to determine an upper bound for $i(G)$ when $G$ is an... | {
"timestamp": "2009-09-18T06:43:21",
"yymm": "0909",
"arxiv_id": "0909.3354",
"language": "en",
"url": "https://arxiv.org/abs/0909.3354",
"abstract": "We show that the number of independent sets in an N-vertex, d-regular graph is at most (2^{d+1} - 1)^{N/2d}, where the bound is sharp for a disjoint union o... |
https://arxiv.org/abs/2003.08058 | Geometric approach to graph magnitude homology | In this paper, we introduce a new method to compute magnitude homology of general graphs. To each direct sum component of magnitude chain complexes, we assign a pair of simplicial complexes whose simplicial chain complex is isomorphic to it. First we states our main theorem specialized to trees, which gives another pro... | \section{Introduction}
Leinster (\cite{Lein}) introduced magnitude of finite metric spaces which measures `` the number of efficient points". Magnitude homology has been invented as a categoryfication of magnitude of a graph which is equipped with a graph metric, by Hepworth-Willerton (\cite{Hep}).
Magnitude homology ... | {
"timestamp": "2020-03-19T01:05:52",
"yymm": "2003",
"arxiv_id": "2003.08058",
"language": "en",
"url": "https://arxiv.org/abs/2003.08058",
"abstract": "In this paper, we introduce a new method to compute magnitude homology of general graphs. To each direct sum component of magnitude chain complexes, we as... |
https://arxiv.org/abs/1602.01126 | A Combinatorial Approach to the Symmetry of $q,t$-Catalan Numbers | The \emph{$q,t$-Catalan numbers} $C_n(q,t)$ are polynomials in $q$ and $t$ that reduce to the ordinary Catalan numbers when $q=t=1$. These polynomials have important connections to representation theory, algebraic geometry, and symmetric functions. Haglund and Haiman discovered combinatorial formulas for $C_n(q,t)$ as ... | \section{Introduction}
\label{sec:intro}
\subsection{Background on $q,t$-Catalan Numbers.}
\label{subsec:background-qtcat}
The \emph{$q,t$-Catalan numbers} were introduced in 1996 by Garsia and
Haiman as part of an ongoing study of Macdonald's symmetric polynomials
and the representation theory of the diagonal harmo... | {
"timestamp": "2016-02-04T02:01:03",
"yymm": "1602",
"arxiv_id": "1602.01126",
"language": "en",
"url": "https://arxiv.org/abs/1602.01126",
"abstract": "The \\emph{$q,t$-Catalan numbers} $C_n(q,t)$ are polynomials in $q$ and $t$ that reduce to the ordinary Catalan numbers when $q=t=1$. These polynomials ha... |
https://arxiv.org/abs/1708.00471 | Limit theorems for random simplices in high dimensions | Let $r=r(n)$ be a sequence of integers such that $r\leq n$ and let $X_1,\ldots,X_{r+1}$ be independent random points distributed according to the Gaussian, the Beta or the spherical distribution on $\mathbb{R}^n$. Limit theorems for the log-volume and the volume of the random convex hull of $X_1,\ldots,X_{r+1}$ are est... | \section{Introduction}
In the last decades, random polytopes have become one of the most central models studied in stochastic geometry. In particular, they have seen numerous applications to other branches of mathematics such as asymptotic geometric analysis, compressed sensing, computational geometry, optimization or... | {
"timestamp": "2017-08-03T02:00:49",
"yymm": "1708",
"arxiv_id": "1708.00471",
"language": "en",
"url": "https://arxiv.org/abs/1708.00471",
"abstract": "Let $r=r(n)$ be a sequence of integers such that $r\\leq n$ and let $X_1,\\ldots,X_{r+1}$ be independent random points distributed according to the Gaussi... |
https://arxiv.org/abs/1412.0011 | On the representation of finite distributive lattices | A simple but elegant result of Rival states that every sublattice $L$ of a finite distributive lattice $\mathcal{P}$ can be constructed from $\mathcal{P}$ by removing a particular family $\mathcal{I}_L$ of its irreducible intervals.Applying this in the case that $\mathcal{P}$ is a product of a finite set $\mathcal{C}$ ... | \section{Introduction}
In this section we give a brief survey of several classical
results about the representations of finite distributive lattices.
We then explain how our results generalize them.
Though we say `generalize', our ideas come very naturally from a result of Rival
which yields a diffe... | {
"timestamp": "2014-12-02T02:00:31",
"yymm": "1412",
"arxiv_id": "1412.0011",
"language": "en",
"url": "https://arxiv.org/abs/1412.0011",
"abstract": "A simple but elegant result of Rival states that every sublattice $L$ of a finite distributive lattice $\\mathcal{P}$ can be constructed from $\\mathcal{P}$... |
https://arxiv.org/abs/2105.00407 | Fractal necklaces with no cut points | The fractal necklaces in R^d (d>1) introduced in this paper are a class of connected fractal sets generated by the so-called necklace IFSs, for which a lot of basic topology questions are interesting. We give two subclasses of fractal necklaces and prove that every necklace in these two classes has no cut points. Also,... | \section{Introduction}
Let $I=\{1,2,\cdots, n\}$. For each $k\in I$ let $f_k:\mathbb{R}^d\to\mathbb{R}^d$ be a contractive map satisfying $$|f_k(x)-f_k(y)|\leq c_k|x-y|$$ for all $x,y\in\mathbb{R}^d$, where $c_k\in(0,1)$. According to Hutchinson \cite{H},
there is a unique nonempty compact subset $F$ of $\mathbb{R}^d... | {
"timestamp": "2022-02-22T02:09:51",
"yymm": "2105",
"arxiv_id": "2105.00407",
"language": "en",
"url": "https://arxiv.org/abs/2105.00407",
"abstract": "The fractal necklaces in R^d (d>1) introduced in this paper are a class of connected fractal sets generated by the so-called necklace IFSs, for which a lo... |
https://arxiv.org/abs/1809.00309 | The Zero Number Diminishing Property under General Boundary Conditions | The so-called {\it zero number diminishing property} (or {\it zero number argument}) is a powerful tool in qualitative studies of one dimensional parabolic equations, which says that, under the zero- or non-zero-Dirichlet boundary conditions, the number of zeroes of the solution $u(x,t)$ of a linear equation is finite,... | \section{Introduction}
Consider the following one dimensional linear parabolic equation:
\begin{equation}\label{linear}
u_t=a(x,t) u_{xx}+b(x,t) u_x+c(x,t) u\quad \mbox{ in } \Omega := \{(x,t) \mid \xi_1 (t) <x < \xi_2 (t),\ t\in (0,T]\},
\end{equation}
where $T>0$ is a constant, $\xi_1$ and $\xi_2$ are continuous... | {
"timestamp": "2018-09-05T02:11:22",
"yymm": "1809",
"arxiv_id": "1809.00309",
"language": "en",
"url": "https://arxiv.org/abs/1809.00309",
"abstract": "The so-called {\\it zero number diminishing property} (or {\\it zero number argument}) is a powerful tool in qualitative studies of one dimensional parabo... |
https://arxiv.org/abs/2008.09001 | A Ramsey Type problem for highly connected subgraphs | Bollobás and Gyárfás conjectured that for any $k, n \in \mathbb{Z}^+$ with $n > 4(k-1)$, every 2-edge-coloring of the complete graph on $n$ vertices leads to a $k$-connected monochromatic subgraph with at least $n-2k+2$ vertices. We find a counterexample with $n = \lfloor 5k-2.5-\sqrt{8k-\frac{31}{4}} \rfloor$, thus di... | \section{Introduction}
Ramsey theory is one of the most important research areas in combinatorics.
For any given integers $s, t$,
the Ramsey number $R(s, t)$ is the smallest integer $n$,
such that for any 2-edge-colored (red/blue) $K_n$,
there must exist a red $K_s$ or a blue $K_t$.
In 1930, Ramsey \cite{Ramsey30... | {
"timestamp": "2020-09-08T02:23:54",
"yymm": "2008",
"arxiv_id": "2008.09001",
"language": "en",
"url": "https://arxiv.org/abs/2008.09001",
"abstract": "Bollobás and Gyárfás conjectured that for any $k, n \\in \\mathbb{Z}^+$ with $n > 4(k-1)$, every 2-edge-coloring of the complete graph on $n$ vertices lea... |
https://arxiv.org/abs/2202.10105 | On the limiting amplitude principle for the wave equation with variable coefficients | In this paper, we prove new results on the validity of the limiting amplitude principle (LAP) for the wave equation with nonconstant coefficients, not necessarily in divergence form. Under suitable assumptions on the coefficients and on the source term, we establish the LAP for space dimensions 2 and 3. This result is ... | \section{Introduction}
\label{sec:intro}
An essential ingredient in connecting time- and frequency-domain wave problems is the limiting amplitude principle (LAP). Originally proposed as one of the tools to select the unique solution of the Helmholtz equation problem in an infinite domain, it has been studied in numero... | {
"timestamp": "2023-01-31T02:25:45",
"yymm": "2202",
"arxiv_id": "2202.10105",
"language": "en",
"url": "https://arxiv.org/abs/2202.10105",
"abstract": "In this paper, we prove new results on the validity of the limiting amplitude principle (LAP) for the wave equation with nonconstant coefficients, not nec... |
https://arxiv.org/abs/2108.03707 | Macaulay bases of modules | We define Macaulay bases of modules, which are a common generalization of Groebner bases and Macaulay $H$-bases to suitably graded modules over a commutative graded $\mathbf{k}$-algebra, where the index sets of the two gradings may differ. This includes Groebner bases of modules as a special case, in contrast to previo... | \section{Introduction}
Many computational tasks involving a module $M$ (or an ideal) over a finitely generated commutative ring $R$ can be solved by first computing a \textit{Groebner basis}, which is a special generating set of $M$ parameterized by a \textit{term order}. Groebner bases were first defined by Buchberger... | {
"timestamp": "2021-08-10T02:22:05",
"yymm": "2108",
"arxiv_id": "2108.03707",
"language": "en",
"url": "https://arxiv.org/abs/2108.03707",
"abstract": "We define Macaulay bases of modules, which are a common generalization of Groebner bases and Macaulay $H$-bases to suitably graded modules over a commutat... |
https://arxiv.org/abs/0805.2042 | Braid ordering and knot genus | The genus of knots is a one of the fundamental invariant and can be seen as a complexity of knots. In this paper, we give a lower bound of genus using Dehornoy floor, which is a measure of complexity of braids in terms of braid ordering. | \section{Introduction}
Let $B_{n}$ be the degree $n$ braid group, defined by the presentation
\[
B_{n} =
\left\langle
\sigma_{1},\sigma_{2},\cdots ,\sigma_{n-1}
\left|
\begin{array}{ll}
\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i} & |i-j|\geq 2 \\
\sigma_{i}\sigma_{j}\sigma_{i}=\sigma_{j}\sigma_{i}\sigma_{j} & |i-j|=... | {
"timestamp": "2009-12-10T12:09:28",
"yymm": "0805",
"arxiv_id": "0805.2042",
"language": "en",
"url": "https://arxiv.org/abs/0805.2042",
"abstract": "The genus of knots is a one of the fundamental invariant and can be seen as a complexity of knots. In this paper, we give a lower bound of genus using Dehor... |
https://arxiv.org/abs/2203.14707 | The Constructor-Blocker Game | We study the following game version of the generalized graph Turán problem. For two fixed graphs $F$ and $H$, two players, Constructor and Blocker, alternately claim unclaimed edges of the complete graph $K_n$. Constructor can only claim edges so that he never claims all edges of any copy of $F$, i.e. his graph must re... | \section{Introduction}
The Tur\'an problem for a set $\cF$ of graphs asks the following: What is the maximum number $ex(n,\cF)$ of edges that a graph on $n$ vertices can have without containing any $F \in \cF$ as a subgraph? When $\cF$ contains a single graph $F$, we simply write $ex(n,F)$. This function has been int... | {
"timestamp": "2022-06-06T02:13:28",
"yymm": "2203",
"arxiv_id": "2203.14707",
"language": "en",
"url": "https://arxiv.org/abs/2203.14707",
"abstract": "We study the following game version of the generalized graph Turán problem. For two fixed graphs $F$ and $H$, two players, Constructor and Blocker, altern... |
https://arxiv.org/abs/2110.08870 | Gallai's path decomposition in planar graphs | In 1968, Gallai conjectured that the edges of any connected graph with $n$ vertices can be partitioned into $\lceil \frac{n}{2} \rceil$ paths. We show that this conjecture is true for every planar graph. More precisely, we show that every connected planar graph except $K_3$ and $K_5^-$ ($K_5$ minus one edge) can be dec... | \section{Introduction}
Given a finite undirected connected graph $G$, a \emph{$k$-path decomposition} of $G$ is a partition of the edges of $G$ into $k$ paths.
In 1968, Gallai stated this simple but surprising conjecture~\cite{Lovasz-covering}: every graph on $n$ vertices admits a $\lceil \frac{n}{2} \rceil$-path decom... | {
"timestamp": "2021-10-19T02:22:21",
"yymm": "2110",
"arxiv_id": "2110.08870",
"language": "en",
"url": "https://arxiv.org/abs/2110.08870",
"abstract": "In 1968, Gallai conjectured that the edges of any connected graph with $n$ vertices can be partitioned into $\\lceil \\frac{n}{2} \\rceil$ paths. We show ... |
https://arxiv.org/abs/1608.08572 | Separated Nets in Nilpotent Groups | In this paper we generalize several results on separated nets in Euclidean space to separated nets in connected simply connected nilpotent Lie groups. We show that every such group $G$ contains separated nets that are not biLipschitz equivalent. We define a class of separated nets in these groups arising from a general... | \section{Introduction}
A subset $Y$ of a metric space $(X,d)$ is a {\it separated net} (or Delone set) if for some $0<c<C$, any two $c$-balls centered at distinct elements of $Y$ are disjoint (i.e. $Y$ is uniformly discrete), and the $C$-neighborhood of $Y$ is all of $X$ (i.e. $Y$ is coarsely dense). Separated nets are... | {
"timestamp": "2016-08-31T02:08:18",
"yymm": "1608",
"arxiv_id": "1608.08572",
"language": "en",
"url": "https://arxiv.org/abs/1608.08572",
"abstract": "In this paper we generalize several results on separated nets in Euclidean space to separated nets in connected simply connected nilpotent Lie groups. We ... |
https://arxiv.org/abs/1705.03271 | Finite Convergence Analysis and Weak Sharp Solutions for Variational Inequalities | In this paper, we study the weak sharpness of the solution set of variational inequality problem (in short, VIP) and the finite convergence property of the sequence generated by some algorithm for finding the solutions of VIP. In particular, we give some characterizations of weak sharpness of the solution set of VIP wi... | \section{Introduction}
Burke and Ferris \cite{BF93} introduced the concept of weak sharp solutions for an optimization problem
in terms of a gap function and gave its characterization in terms of a geometric condition.
Marcotte and Zhu \cite{MZ98} exploited that geometric condition to introduce the concept
of weak s... | {
"timestamp": "2017-05-10T02:06:04",
"yymm": "1705",
"arxiv_id": "1705.03271",
"language": "en",
"url": "https://arxiv.org/abs/1705.03271",
"abstract": "In this paper, we study the weak sharpness of the solution set of variational inequality problem (in short, VIP) and the finite convergence property of th... |
https://arxiv.org/abs/1112.0829 | How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman | Consider a gambling game in which we are allowed to repeatedly bet a portion of our bankroll at favorable odds. We investigate the question of how to minimize the expected number of rounds needed to increase our bankroll to a given target amount.Specifically, we disprove a 50-year old conjecture of L. Breiman, that the... | \section{The Conjecture}
Consider a favorable gambling game,
such as betting at 3:1 odds on the outcome of a fair coin toss.
If we are allowed to play this as many times as we like (decided adaptively),
we can eventually increase our winnings to any desired
target amount, with certainty. For instance, proportional... | {
"timestamp": "2011-12-06T02:04:05",
"yymm": "1112",
"arxiv_id": "1112.0829",
"language": "en",
"url": "https://arxiv.org/abs/1112.0829",
"abstract": "Consider a gambling game in which we are allowed to repeatedly bet a portion of our bankroll at favorable odds. We investigate the question of how to minimi... |
https://arxiv.org/abs/0805.1235 | Schur-Weyl duality over finite fields | We prove a version of Schur--Weyl duality over finite fields. We prove that for any field $k$, if $k$ has at least $r+1$ elements, then Schur--Weyl duality holds for the $r$th tensor power of a finite dimensional vector space $V$. Moreover, if the dimension of $V$ is at least $r+1$, the natural map $k\Sym_r \to End\_{G... | \section{Introduction}\noindent
Let $k$ be a field and let $V$ be an $n$-dimensional vector space over
$k$. We identify $G=GL(V)$ with $GL(n,k)$ and $V$ with $k^n$ as usual,
by fixing a basis $\{v_1, \dots, v_n\}$ of $V$. $G$ acts on $V$ in
the natural way, and thus on the tensor product space $V^{\otimes
r}$. Moreo... | {
"timestamp": "2009-06-30T14:36:07",
"yymm": "0805",
"arxiv_id": "0805.1235",
"language": "en",
"url": "https://arxiv.org/abs/0805.1235",
"abstract": "We prove a version of Schur--Weyl duality over finite fields. We prove that for any field $k$, if $k$ has at least $r+1$ elements, then Schur--Weyl duality ... |
https://arxiv.org/abs/1708.09716 | Milnor and Tjurina numbers for a hypersurface germ with isolated singularity | Assume that $f:(\mathbb{C}^n,0) \to (\mathbb{C},0)$ is an analytic function germ at the origin with only isolated singularity. Let $\mu$ and $\tau$ be the corresponding Milnor and Tjurina numbers. We show that $\dfrac{\mu}{\tau} \leq n$. As an application, we give a lower bound for the Tjurina number in terms of $n$ an... | \section{Main result}
Assume that $f:(\mathbb{C}^{n},0) \to (\mathbb{C},0)$ is an analytic function germ at the origin with only isolated singularity. Set $X=f^{-1}(0)$.
Let $S=\mathbb{C}\lbrace x_{1}, \ldots, x_{n} \rbrace$ denote the formal power series ring. Set $J_{f}=(\partial f/\partial x_{1}, \ldots, \partial ... | {
"timestamp": "2018-07-04T02:10:54",
"yymm": "1708",
"arxiv_id": "1708.09716",
"language": "en",
"url": "https://arxiv.org/abs/1708.09716",
"abstract": "Assume that $f:(\\mathbb{C}^n,0) \\to (\\mathbb{C},0)$ is an analytic function germ at the origin with only isolated singularity. Let $\\mu$ and $\\tau$ b... |
https://arxiv.org/abs/math/0609414 | On the degree two entry of a Gorenstein $h$-vector and a conjecture of Stanley | In this note we establish a (non-trivial) lower bound on the degree two entry $h_2$ of a Gorenstein $h$-vector of any given socle degree $e$ and any codimension $r$.In particular, when $e=4$, that is for Gorenstein $h$-vectors of the form $h=(1,r,h_2,r,1)$, our lower bound allows us to prove a conjecture of Stanley on ... | \section{Introduction}
Gorenstein rings arise in many areas of mathematics - including algebraic geometry, combinatorics, and complexity theory (see, e.g, $[Hu]$, $[Pa]$, $[St4]$, $[At]$, $[St5]$, $[KS]$). Often these rings are standard graded algebras over a field $k$. Then it is a basic problem to understand the vec... | {
"timestamp": "2007-11-27T20:51:27",
"yymm": "0609",
"arxiv_id": "math/0609414",
"language": "en",
"url": "https://arxiv.org/abs/math/0609414",
"abstract": "In this note we establish a (non-trivial) lower bound on the degree two entry $h_2$ of a Gorenstein $h$-vector of any given socle degree $e$ and any c... |
https://arxiv.org/abs/1504.03763 | Mutiscale Mapper: A Framework for Topological Summarization of Data and Maps | Summarizing topological information from datasets and maps defined on them is a central theme in topological data analysis. \textsf{Mapper}, a tool for such summarization, takes as input both a possibly high dimensional dataset and a map defined on the data, and produces a summary of the data by using a cover of the co... | \section{The instability of Mapper}
\label{appendix:instability-mapper}
In this section we briefly discuss how one may perceive that the simplicial complexes produced by Mapper may not admit a simple notion of stability.
As an example consider the situation in Figure \ref{fig:counter-mapper-1}. Consider for each $\d... | {
"timestamp": "2016-01-13T02:11:18",
"yymm": "1504",
"arxiv_id": "1504.03763",
"language": "en",
"url": "https://arxiv.org/abs/1504.03763",
"abstract": "Summarizing topological information from datasets and maps defined on them is a central theme in topological data analysis. \\textsf{Mapper}, a tool for s... |
https://arxiv.org/abs/2003.13095 | On dense subsets of matrices with distinct eigenvalues and distinct singular values | It is well known that the set of all $ n \times n $ matrices with distinct eigenvalues is a dense subset of the set of all real or complex $ n \times n $ matrices. In [Hartfiel, D. J. Dense sets of diagonalizable matrices. Proc. Amer. Math. Soc., 123(6): 1669-1672, 1995.], the author established a necessary and suffici... | \section{Introduction}
It is well known that the set of all $ n \times n $ matrices with distinct eigenvalues is a dense subset of the set of all real or complex $ n \times n $ matrices. An arbitrary subspace of the set of all $n \times n$ matrices may not have a dense subset of matrices with distinct eigenvalues. F... | {
"timestamp": "2020-03-31T02:20:40",
"yymm": "2003",
"arxiv_id": "2003.13095",
"language": "en",
"url": "https://arxiv.org/abs/2003.13095",
"abstract": "It is well known that the set of all $ n \\times n $ matrices with distinct eigenvalues is a dense subset of the set of all real or complex $ n \\times n ... |
https://arxiv.org/abs/2107.00492 | Dyadic John-Nirenberg space | We discuss the dyadic John-Nirenberg space that is a generalization of functions of bounded mean oscillation. A John-Nirenberg inequality, which gives a weak type estimate for the oscillation of a function, is discussed in the setting of medians instead of integral averages. We show that the dyadic maximal operator is ... | \section{Introduction}
The space of functions of bounded mean oscillation ($BMO$) was introduced by John and Nirenberg in~\cite{john_original}.
Let $Q_0$ be a cube with sides parallel to the coordinate axis in $\mathbb R^n$.
A function $f \in L^1(Q_0)$ belongs to $BMO(Q_0)$ if
\begin{equation}\label{bmo}
\sup \Xint-... | {
"timestamp": "2021-10-11T02:04:10",
"yymm": "2107",
"arxiv_id": "2107.00492",
"language": "en",
"url": "https://arxiv.org/abs/2107.00492",
"abstract": "We discuss the dyadic John-Nirenberg space that is a generalization of functions of bounded mean oscillation. A John-Nirenberg inequality, which gives a w... |
https://arxiv.org/abs/1910.14200 | Cops that surround a robber | We introduce the game of Surrounding Cops and Robbers on a graph, as a variant of the original game of Cops and Robbers. In contrast to the original game in which the cops win by occupying the same vertex as the robber, they now win by occupying each of the robber's neighbouring vertices. We denote by $\sigma(G)$ the {... | \section{Introduction}
The vertex-pursuit game of Cops and Robbers was introduced about 40 years ago by
Nowakowski and Winkler~\cite{NW1983} and by Quilliot~\cite{Quilliot}.
Initially only a single cop and a single robber were considered, but
in 1984 Aigner and Fromme generalised the game to include a team of cops as ... | {
"timestamp": "2021-08-06T02:22:53",
"yymm": "1910",
"arxiv_id": "1910.14200",
"language": "en",
"url": "https://arxiv.org/abs/1910.14200",
"abstract": "We introduce the game of Surrounding Cops and Robbers on a graph, as a variant of the original game of Cops and Robbers. In contrast to the original game ... |
https://arxiv.org/abs/2107.01874 | Schubert Eisenstein series and Poisson summation for Schubert varieties | The first author and Bump defined Schubert Eisenstein series by restricting the summation in a degenerate Eisenstein series to a particular Schubert variety. In the case of $\mathrm{GL}_3$ over $\mathbb{Q}$ they proved that these Schubert Eisenstein series have meromorphic continuations in all parameters and conjecture... | \section{Introduction}
In this paper we
prove the Poisson summation conjecture of Braverman-Kazhdan, Lafforgue, Ng\^o, and Sakellaridis for a particular family of varieties related to Schubert varieties (see Theorem \ref{thm:PS:intro}).
We were motivated to prove this case of the conjectures due to the fact that it ... | {
"timestamp": "2021-10-05T02:35:57",
"yymm": "2107",
"arxiv_id": "2107.01874",
"language": "en",
"url": "https://arxiv.org/abs/2107.01874",
"abstract": "The first author and Bump defined Schubert Eisenstein series by restricting the summation in a degenerate Eisenstein series to a particular Schubert varie... |
https://arxiv.org/abs/2005.05379 | Gap Sets for the Spectra of Cubic Graphs | We study gaps in the spectra of the adjacency matrices of large finite cubic graphs. It is known that the gap intervals $(2 \sqrt{2},3)$ and $[-3,-2)$ achieved in cubic Ramanujan graphs and line graphs are maximal. We give constraints on spectra in [-3,3] which are maximally gapped and construct examples which achieve ... | \section{Introduction}\label{sec:intro}
By a cubic graph we mean a finite $3$-regular connected graph with no loops or multiple edges. Denote the set of such graphs by $\bf{X}$ and the subset of $\bf{X}$ which can be realized as planar graphs by $\bf{X}_{Planar}$.
For $Y \in \bf{X}$ we denote the adjacency matrix of $... | {
"timestamp": "2021-01-18T02:15:57",
"yymm": "2005",
"arxiv_id": "2005.05379",
"language": "en",
"url": "https://arxiv.org/abs/2005.05379",
"abstract": "We study gaps in the spectra of the adjacency matrices of large finite cubic graphs. It is known that the gap intervals $(2 \\sqrt{2},3)$ and $[-3,-2)$ ac... |
https://arxiv.org/abs/2206.07085 | Understanding the Generalization Benefit of Normalization Layers: Sharpness Reduction | Normalization layers (e.g., Batch Normalization, Layer Normalization) were introduced to help with optimization difficulties in very deep nets, but they clearly also help generalization, even in not-so-deep nets. Motivated by the long-held belief that flatter minima lead to better generalization, this paper gives mathe... | \section{Experiments} \label{sec:add-exp}
In this section,
we provide experiments on matrix completion and CIFAR-10
to validate the main claim in our theory:
GD+WD\xspace on scale-invariant loss persistently reduces spherical sharpness in the EoS regime
(the regime where $2/\tilde{\eta}_t$ roughly equals to the spheri... | {
"timestamp": "2022-06-16T02:01:03",
"yymm": "2206",
"arxiv_id": "2206.07085",
"language": "en",
"url": "https://arxiv.org/abs/2206.07085",
"abstract": "Normalization layers (e.g., Batch Normalization, Layer Normalization) were introduced to help with optimization difficulties in very deep nets, but they c... |
https://arxiv.org/abs/0812.4653 | Exponential Sums and Distinct Points on Arcs | Suppose that some harmonic analysis arguments have been invoked to show that the indicator function of a set of residue classes modulo some integer has a large Fourier coefficient. To get information about the structure of the set of residue classes, we then need a certain type of complementary result. A solution to th... | \section{Introduction} \label{sec:intro}
In additive combinatorics, and in additive combinatorial number theory in particular, situations of the following type are rather common. Let $A$ be a set of $N$ residue classes modulo an integer $m$. Suppose that some harmonic analysis arguments have been invoked to show that t... | {
"timestamp": "2008-12-26T08:54:37",
"yymm": "0812",
"arxiv_id": "0812.4653",
"language": "en",
"url": "https://arxiv.org/abs/0812.4653",
"abstract": "Suppose that some harmonic analysis arguments have been invoked to show that the indicator function of a set of residue classes modulo some integer has a la... |
https://arxiv.org/abs/1803.02537 | Packing chromatic number of subdivisions of cubic graphs | A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $\chi_p(G)$, of a graph $G$ is the minimum $k$ such that $G$ has a packing $k$-coloring. For a ... | \section{Introduction}
\label{intro}
Your text comes here. Separate text sections with
\section{Section title}
\label{sec:1}
Text with citations \cite{RefB} and \cite{RefJ}.
\subsection{Subsection title}
\label{sec:2}
as required. Don't forget to give each section
and subsection a unique label (see Sect.~\ref{sec:1}).
... | {
"timestamp": "2018-10-09T02:15:24",
"yymm": "1803",
"arxiv_id": "1803.02537",
"language": "en",
"url": "https://arxiv.org/abs/1803.02537",
"abstract": "A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\\ldots,V_k$ such that for each $1\\leq i\\leq k$ the distance between any t... |
https://arxiv.org/abs/1005.1521 | Bi-banded Paths, a Bijection and the Narayana Numbers | We find a bijection between bi-banded paths and peak-counting paths, applying to two classes of lattice paths including Dyck paths. Thus we find a new interpretation of Narayana numbers as coefficients of weight polynomials enumerating bi-banded Dyck paths, which class of paths has arisen naturally in previous literat... | \section{Introduction}
In a paper of Brak and Essam \cite{BE} the class of bi-banded Dyck paths arose in the solution of TASEP (Totally ASymmetric Exclusion Process) - a model for particle-hopping with excluded volume \cite{Bl}. It was observed that the coefficients of the weight polynomial, or partition function, f... | {
"timestamp": "2010-05-11T02:02:11",
"yymm": "1005",
"arxiv_id": "1005.1521",
"language": "en",
"url": "https://arxiv.org/abs/1005.1521",
"abstract": "We find a bijection between bi-banded paths and peak-counting paths, applying to two classes of lattice paths including Dyck paths. Thus we find a new inte... |
https://arxiv.org/abs/2009.08136 | Multidimensional Scaling, Sammon Mapping, and Isomap: Tutorial and Survey | Multidimensional Scaling (MDS) is one of the first fundamental manifold learning methods. It can be categorized into several methods, i.e., classical MDS, kernel classical MDS, metric MDS, and non-metric MDS. Sammon mapping and Isomap can be considered as special cases of metric MDS and kernel classical MDS, respective... | \section{Introduction}
Multidimensional Scaling (MDS) \cite{cox2008multidimensional}, first proposed in \cite{torgerson1952multidimensional}, is one of the earliest proposed manifold learning methods.
It can be used for manifold learning, dimensionality reduction, and feature extraction \cite{ghojogh2019feature}.
Th... | {
"timestamp": "2020-09-18T02:11:12",
"yymm": "2009",
"arxiv_id": "2009.08136",
"language": "en",
"url": "https://arxiv.org/abs/2009.08136",
"abstract": "Multidimensional Scaling (MDS) is one of the first fundamental manifold learning methods. It can be categorized into several methods, i.e., classical MDS,... |
https://arxiv.org/abs/1712.05204 | Inverse of multivector: Beyond p+q=5 threshold | The algorithm of finding inverse multivector (MV) numerically and symbolically is of paramount importance in the applied Clifford geometric algebra (GA) $Cl_{p,q}$. The first general MV inversion algorithm was based on matrix representation of MV. The complexity of calculations and size of the answer in a symbolic form... | \section{Introduction\label{sec:1}}
The knowledge of how to find inverse multivector (MV) in the
Clifford algebra in a symbolic and coordinate-free form is very
important both from practical computational and purely theoretical
point of views. A universal formula for inverse MV would allow to
write down a fast and gen... | {
"timestamp": "2018-03-15T01:07:06",
"yymm": "1712",
"arxiv_id": "1712.05204",
"language": "en",
"url": "https://arxiv.org/abs/1712.05204",
"abstract": "The algorithm of finding inverse multivector (MV) numerically and symbolically is of paramount importance in the applied Clifford geometric algebra (GA) $... |
https://arxiv.org/abs/2209.03930 | Graphs which satisfy a Vizing-like bound for power domination of Cartesian products | Power domination is a two-step observation process that is used to monitor power networks and can be viewed as a combination of domination and zero forcing. Given a graph $G$, a subset $S\subseteq V(G)$ that can observe all vertices of $G$ using this process is known as a power dominating set of $G$, and the power domi... | \section{Introduction} \label{sec:intro}
Electrical power networks may be monitored through a two-step process called power domination. Phase measurement units (PMUs) are used to observe the information; however, since PMUs are expensive, one would like to use the smallest number of PMUs possible to observe the power ... | {
"timestamp": "2022-09-09T02:18:25",
"yymm": "2209",
"arxiv_id": "2209.03930",
"language": "en",
"url": "https://arxiv.org/abs/2209.03930",
"abstract": "Power domination is a two-step observation process that is used to monitor power networks and can be viewed as a combination of domination and zero forcin... |
https://arxiv.org/abs/1607.02086 | Ellipses of minimal eccentricity inscribed in midpoint diagonal quadrilaterals | In an earlier paper of the author, we showed that there is a unique ellipse of minimal eccentricity, $E_I$, inscribed in any convex quadrilateral, $Q$. Using a different approach in this paper, we prove that there is a unique ellipse of minimal eccentricity, $E_I$, inscribed in a midpoint diagonal quadrilateral, $Q$, w... | \section{Introduction}
In \cite{H1} the author proved numerous results about ellipses inscribe
\textbf{\ }in convex quadrilaterals, $Q$. By \textit{inscribed} we mean that
the ellipse lies inside $Q$ and is tangent to each side of $Q$. In
particular, we proved that there exists a unique ellipse, $E_{I}$, of
minim... | {
"timestamp": "2016-09-05T02:06:07",
"yymm": "1607",
"arxiv_id": "1607.02086",
"language": "en",
"url": "https://arxiv.org/abs/1607.02086",
"abstract": "In an earlier paper of the author, we showed that there is a unique ellipse of minimal eccentricity, $E_I$, inscribed in any convex quadrilateral, $Q$. Us... |
https://arxiv.org/abs/1409.4128 | Real roots of random polynomials: expectation and repulsion | Let $P_{n}(x)= \sum_{i=0}^n \xi_i x^i$ be a Kac random polynomial where the coefficients $\xi_i$ are iid copies of a given random variable $\xi$. Our main result is an optimal quantitative bound concerning real roots repulsion. This leads to an optimal bound on the probability that there is a double root.As an applicat... | \section{Introduction} \label{section:intro}
Let $\xi$ be a real random variable having no atom at 0, zero mean and unit variance. Our object of study is the random polynomial
\begin{equation} \label{defP} P_{n}(x) := \sum_{i=0}^n \xi_i x^i \end{equation} where $\xi_i$ are iid copies of $\xi$. This polynomial i... | {
"timestamp": "2014-09-16T02:12:35",
"yymm": "1409",
"arxiv_id": "1409.4128",
"language": "en",
"url": "https://arxiv.org/abs/1409.4128",
"abstract": "Let $P_{n}(x)= \\sum_{i=0}^n \\xi_i x^i$ be a Kac random polynomial where the coefficients $\\xi_i$ are iid copies of a given random variable $\\xi$. Our ma... |
https://arxiv.org/abs/2003.06232 | Bernstein spectral method for quasinormal modes and other eigenvalue problems | Spectral methods are now common in the solution of ordinary differential eigenvalue problems in a wide variety of fields, such as in the computation of black hole quasinormal modes. Most of these spectral codes are based on standard Chebyshev, Fourier, or some other orthogonal basis functions. In this work we highlight... | \section{Introduction}
\label{sec:introduction}
Black holes in general relativity are simple spacetime objects, fully specified by only a handful of constants. When the spacetime around black holes is disturbed by surrounding complex distributions of matter and fields, as they are found in nature, these spacetime dist... | {
"timestamp": "2021-11-08T02:13:00",
"yymm": "2003",
"arxiv_id": "2003.06232",
"language": "en",
"url": "https://arxiv.org/abs/2003.06232",
"abstract": "Spectral methods are now common in the solution of ordinary differential eigenvalue problems in a wide variety of fields, such as in the computation of bl... |
https://arxiv.org/abs/2102.08242 | Positivity-preserving methods for population models | Many important applications are modelled by differential equations with positive solutions. However, it remains an outstanding open problem to develop numerical methods that are both (i) of a high order of accuracy and (ii) capable of preserving positivity. It is known that the two main families of numerical methods, R... | \section{Introduction}
{{}
Numerical integration of mathematical models is an essential step in the implementation and analysis of population models: chemical reactions (see for example \cite{edsberg74ipc,sandu01pni} or \cite{hairer10sod}), biochemical systems \cite{bruggeman07aso}, and the evolution of epidemics... | {
"timestamp": "2022-05-03T02:38:01",
"yymm": "2102",
"arxiv_id": "2102.08242",
"language": "en",
"url": "https://arxiv.org/abs/2102.08242",
"abstract": "Many important applications are modelled by differential equations with positive solutions. However, it remains an outstanding open problem to develop num... |
https://arxiv.org/abs/1108.1708 | Mixing and hitting times for finite Markov chains | Let 0<\alpha<1/2. We show that the mixing time of a continuous-time reversible Markov chain on a finite state space is about as large as the largest expected hitting time of a subset of stationary measure at least \alpha of the state space. Suitably modified results hold in discrete time and/or without the reversibilit... | \section{Introduction}\label{sec:setup}
The present paper is a contribution to the general quantitative theory of finite-state Markov chains that was started in
\cite{Aldous_IneqReversible} and further developed in \cite{AldousLovaszWinkler_IneqGeneral}. The gist of those papers is that the so-called mixing time of a... | {
"timestamp": "2012-06-08T02:00:36",
"yymm": "1108",
"arxiv_id": "1108.1708",
"language": "en",
"url": "https://arxiv.org/abs/1108.1708",
"abstract": "Let 0<\\alpha<1/2. We show that the mixing time of a continuous-time reversible Markov chain on a finite state space is about as large as the largest expect... |
https://arxiv.org/abs/2207.00447 | Prediction of random variables by excursion metric projections | We use the concept of excursions for the prediction of random variables without any moment existence assumptions. To do so, an excursion metric on the space of random variables is defined which appears to be a kind of a weighted $L^1$-distance. Using equivalent forms of this metric and the specific choice of excursion ... | \section{Introduction} \label{sectIntro}
Let $Y:\Omega\to\mathbb R$ be a square integrable random variable defined on a probability space $(\Omega,{\cal F}, \mathbb{P})$, and let ${\cal G}\subset {\cal F}$ be a sub--$\sigma$--algebra generated by a family of random variables $\{Z_j\}$ which are observable.
The classi... | {
"timestamp": "2022-07-04T02:16:06",
"yymm": "2207",
"arxiv_id": "2207.00447",
"language": "en",
"url": "https://arxiv.org/abs/2207.00447",
"abstract": "We use the concept of excursions for the prediction of random variables without any moment existence assumptions. To do so, an excursion metric on the spa... |
https://arxiv.org/abs/0906.5447 | The optimal strategy for symmetric rendezvous search on K3 | In the symmetric rendezvous search game played on Kn (the completely connected graph on n vertices) two players are initially placed at two distinct vertices (called locations). The game is played in discrete steps and at each step each player can either stay where he is or move to a different location. The players sha... | \section{Symmetric rendezvous search on $K_3$}
In the symmetric rendezvous search game played on $K_n$ (the
completely connected graph on $n$ vertices) two players are initially
placed at two distinct vertices (called locations). The game is played
in discrete steps, and at each step each player can either stay where
h... | {
"timestamp": "2009-06-30T11:16:35",
"yymm": "0906",
"arxiv_id": "0906.5447",
"language": "en",
"url": "https://arxiv.org/abs/0906.5447",
"abstract": "In the symmetric rendezvous search game played on Kn (the completely connected graph on n vertices) two players are initially placed at two distinct vertice... |
https://arxiv.org/abs/2109.14108 | Connected domination in grid graphs | Given an undirected simple graph, a subset of the vertices of the graph is a {\em dominating set} if every vertex not in the subset is adjacent to at least one vertex in the subset. A subset of the vertices of the graph is a {\em connected dominating set} if the subset is a dominating set and the subgraph induced by th... | \section{Introduction} \label{sec:intro}
\ifnum \count10 > 0
\fi
\ifnum \count11 > 0
Given an undirected simple graph,
a subset of the vertices of the graph is a {\em dominating set}
if every vertex not in the subset is adjacent to at least one vertex in the subset.
By imposing restrictions on dominating sets... | {
"timestamp": "2021-09-30T02:06:43",
"yymm": "2109",
"arxiv_id": "2109.14108",
"language": "en",
"url": "https://arxiv.org/abs/2109.14108",
"abstract": "Given an undirected simple graph, a subset of the vertices of the graph is a {\\em dominating set} if every vertex not in the subset is adjacent to at lea... |
https://arxiv.org/abs/1801.06914 | Maximal metrics for the first Steklov eigenvalue on surfaces | In recent years, eigenvalue optimization problems have received a lot of attention, in particular, due to their connection with the theory of minimal surfaces. In the present paper we prove that on any orientable surface there exists a smooth metric maximizing the first normalized Steklov eigenvalue. For surfaces of ge... | \section{Introduction}
\subsection{Steklov problem on surfaces}
Let $(M,g)$ be a connected compact Riemannian surface with smooth boundary.
Steklov eigenvalues are defined as numbers $\sigma$ such that there exists a non-zero solution of the following system,
\begin{equation*}
\left\{
\begin{array}{rcl}
\Delta u &=... | {
"timestamp": "2018-01-23T02:10:41",
"yymm": "1801",
"arxiv_id": "1801.06914",
"language": "en",
"url": "https://arxiv.org/abs/1801.06914",
"abstract": "In recent years, eigenvalue optimization problems have received a lot of attention, in particular, due to their connection with the theory of minimal surf... |
https://arxiv.org/abs/1807.09618 | Stability for vertex isoperimetry in the cube | We prove a stability version of Harper's cube vertex isoperimetric inequality, showing that subsets of the cube with vertex boundary close to the minimum possible are close to (generalised) Hamming balls. Furthermore, we obtain a local stability result for ball-like sets that gives a sharp estimate for the vertex bound... | \section{Introduction}
Isoperimetric inequalities have a long history in mathematics,
starting from the classical Euclidean isoperimetric inequality in $\mb{R}^d$
that balls minimise surface area among all sets with given volume.
There is also a rich theory of isoperimetric inequalities in the
discrete setting, whic... | {
"timestamp": "2018-07-26T02:10:55",
"yymm": "1807",
"arxiv_id": "1807.09618",
"language": "en",
"url": "https://arxiv.org/abs/1807.09618",
"abstract": "We prove a stability version of Harper's cube vertex isoperimetric inequality, showing that subsets of the cube with vertex boundary close to the minimum ... |
https://arxiv.org/abs/1607.06161 | Correspondences between convex geometry and complex geometry | We present several analogies between convex geometry and the theory of holomorphic line bundles on smooth projective varieties or Kähler manifolds. We study the relation between positive products and mixed volumes. We define and study a Blaschke addition for divisor classes and mixed divisor classes, and prove new geom... | \section{Introduction}
Many deep results concerning holomorphic line bundles in complex geometry are inspired by results in convex geometry. For example, the Khovanskii-Teissier inequalities for $(1,1)$-classes are inspired by the Alexandrov-Fenchel
inequalities for convex bodies. Such results were first explor... | {
"timestamp": "2017-07-10T02:07:38",
"yymm": "1607",
"arxiv_id": "1607.06161",
"language": "en",
"url": "https://arxiv.org/abs/1607.06161",
"abstract": "We present several analogies between convex geometry and the theory of holomorphic line bundles on smooth projective varieties or Kähler manifolds. We stu... |
https://arxiv.org/abs/0806.0092 | Asymptotically tight bounds on subset sums | For a subset A of a finite abelian group G we define Sigma(A)={sum_{a\in B}a:B\subset A}. In the case that Sigma(A) has trivial stabiliser, one may deduce that the size of Sigma(A) is at least quadratic in |A|; the bound |Sigma(A)|>= |A|^{2}/64 has recently been obtained by De Vos, Goddyn, Mohar and Samal. We improve t... | \section{Introduction}
For a subset $A$ of a finite abelian group $G$ we define, \begin{equation*} \Sigma(A)=\Big\{\sum_{a\in B}a:\, B\subset A\Big\}\end{equation*} the set of all elements which may be expressed as a sum of elements of $A$ (with repetition not allowed). For a subset $S\subset G$ the stabiliser $\stab... | {
"timestamp": "2008-05-31T18:35:34",
"yymm": "0806",
"arxiv_id": "0806.0092",
"language": "en",
"url": "https://arxiv.org/abs/0806.0092",
"abstract": "For a subset A of a finite abelian group G we define Sigma(A)={sum_{a\\in B}a:B\\subset A}. In the case that Sigma(A) has trivial stabiliser, one may deduce... |
https://arxiv.org/abs/1404.3054 | Collatz meets Fibonacci | The Collatz map is defined for a positive even integer as half that integer, and for a positive odd integer as that integer threefold, plus one. The Collatz conjecture states that when the map is iterated the number one is eventually reached. We study permutations that arise as sequences from this iteration. We show th... | \section{Introduction}
\label{sec:introduction}
A well-known conjecture due to Lothar Collatz from 1937 states that when the function
\[
f(x) = \left \{
\begin{array}{ll}
x/2 & \mathrm{if\ } x \equiv 0 \pmod{2} \\
3x+1 & \mathrm{if\ } x \equiv 1 \pmod{2}
\end{array}
\right.
\]
is iterated from an initial p... | {
"timestamp": "2015-01-19T02:00:48",
"yymm": "1404",
"arxiv_id": "1404.3054",
"language": "en",
"url": "https://arxiv.org/abs/1404.3054",
"abstract": "The Collatz map is defined for a positive even integer as half that integer, and for a positive odd integer as that integer threefold, plus one. The Collatz... |
https://arxiv.org/abs/1512.09006 | Distinct distances between points and lines | We show that for $m$ points and $n$ lines in the real plane, the number of distinct distances between the points and the lines is $\Omega(m^{1/5}n^{3/5})$, as long as $m^{1/2}\le n\le m^2$. We also prove that for any $m$ points in the plane, not all on a line, the number of distances between these points and the lines ... | \section{Introduction}\label{sec:intro}
In 1946 Paul Erd{\H o}s~\cite{Er46} posed the following two problems, which later became
extremely influential and central questions in combinatorial geometry:
the so-called \emph{repeated distances} and \emph{distinct distances} problems.
The first problem deals with the maximum... | {
"timestamp": "2015-12-31T02:10:55",
"yymm": "1512",
"arxiv_id": "1512.09006",
"language": "en",
"url": "https://arxiv.org/abs/1512.09006",
"abstract": "We show that for $m$ points and $n$ lines in the real plane, the number of distinct distances between the points and the lines is $\\Omega(m^{1/5}n^{3/5})... |
https://arxiv.org/abs/1407.5559 | Some Liouville theorems for the fractional Laplacian | In this paper, we prove the following result. Let $\alpha$ be any real number between $0$ and $2$. Assume that $u$ is a solution of $$ \left\{\begin{array}{ll} (-\Delta)^{\alpha/2} u(x) = 0 , \;\; x \in \mathbb{R}^n ,\\ \displaystyle\underset{|x| \to \infty}{\underline{\lim}} \frac{u(x)}{|x|^{\gamma}} \geq 0 , \end{arr... | \section{Introduction}
The well-known classical Liouville's Theorem states that
{\em Any harmonic function bounded below in all of $R^n$ is constant.}
One of its important applications is the proof of the Fundamental Theorem of Algebra. It is
also a key ingredient in deriving a priori estimates for solutions ... | {
"timestamp": "2014-11-11T02:02:12",
"yymm": "1407",
"arxiv_id": "1407.5559",
"language": "en",
"url": "https://arxiv.org/abs/1407.5559",
"abstract": "In this paper, we prove the following result. Let $\\alpha$ be any real number between $0$ and $2$. Assume that $u$ is a solution of $$ \\left\\{\\begin{arr... |
https://arxiv.org/abs/1312.1675 | Components of spaces of curves with constrained curvature on flat surfaces | Let $S$ be a complete flat surface, such as the Euclidean plane. We obtain direct characterizations of the connected components of the space of all curves on $S$ which start and end at given points in given directions, and whose curvatures are constrained to lie in a given interval, in terms of all parameters involved.... | \section{Introduction}\label{S:introduction}
To abbreviate the notation, we shall identify $\R^2$ with $\C$ throughout. A curve $\ga\colon [0,1]\to \C$ is called \tdef{regular} if its derivative is continuous and never vanishes. Its \tdef{unit tangent} is then defined as
\begin{equation*
\ta_\ga\colon [0,1]\to \Ss^1,\... | {
"timestamp": "2015-07-30T02:10:59",
"yymm": "1312",
"arxiv_id": "1312.1675",
"language": "en",
"url": "https://arxiv.org/abs/1312.1675",
"abstract": "Let $S$ be a complete flat surface, such as the Euclidean plane. We obtain direct characterizations of the connected components of the space of all curves o... |
https://arxiv.org/abs/1101.2085 | Operator Ideals arising from Generating Sequences | In this note, we will discuss how to relate an operator ideal on Banach spaces to the sequential structures it defines. Concrete examples of ideals of compact, weakly compact, completely continuous, Banach-Saks and weakly Banach-Saks operators will be demonstrated. | \section{Introduction}
Let $T: E\to F$ be a linear operator between Banach spaces.
Let $U_E, U_F$ be the closed unit balls of $E,F$, respectively.
Note that closed unit balls serve simultaneously the basic model for
open sets and bounded sets in Banach spaces.
The usual way to describe $T$ is to state
either... | {
"timestamp": "2011-01-12T02:01:43",
"yymm": "1101",
"arxiv_id": "1101.2085",
"language": "en",
"url": "https://arxiv.org/abs/1101.2085",
"abstract": "In this note, we will discuss how to relate an operator ideal on Banach spaces to the sequential structures it defines. Concrete examples of ideals of compa... |
https://arxiv.org/abs/1601.06322 | On local matchability in groups and vector spaces | In this paper, we define locally matchable subsets of a group which is extracted from the concept of matchings in groups and used as a tool to give alternative proofs for existing results in matching theory. We also give the linear analogue of locally matchability for subspaces in a field extension. Our tools mix addit... | \section{Introduction}
The notion of matchings in groups was used to study an old problem of Wakeford concerning canonical forms for symmetric
tensors \cite{11}. Losonczy in \cite{10} introduced matchings for groups to work on Wakeford's problem
\,Let $G$ be an additive abelian group and $A$ and $B$ be two non-empty... | {
"timestamp": "2016-02-03T02:13:14",
"yymm": "1601",
"arxiv_id": "1601.06322",
"language": "en",
"url": "https://arxiv.org/abs/1601.06322",
"abstract": "In this paper, we define locally matchable subsets of a group which is extracted from the concept of matchings in groups and used as a tool to give altern... |
https://arxiv.org/abs/2012.08786 | Wiener index and graphs, almost half of whose vertices satisfy Šoltés property | The Wiener index $W(G)$ of a connected graph $G$ is a sum of distances between all pairs of vertices of $G$. In 1991, Šoltés formulated the problem of finding all graphs $G$ such that for every vertex $v$ the equation $W(G)=W(G-v)$ holds. The cycle $C_{11}$ is the only known graph with this property. In this paper we c... | \section{Introduction}\label{S:Introduction}
Let $G=(V,E)$ be a simple connected graph. The {\it Wiener index} of a graph $G$ is defined as the sum of distances between all pairs of vertices of $G$:
$$W(G)=\frac{1}{2}\sum_{v,u\in V, v\neq u}{d_G(u,v)},$$
where $d_G(x,y)$ is a usual graph distance, i.e. the length of... | {
"timestamp": "2021-08-17T02:19:08",
"yymm": "2012",
"arxiv_id": "2012.08786",
"language": "en",
"url": "https://arxiv.org/abs/2012.08786",
"abstract": "The Wiener index $W(G)$ of a connected graph $G$ is a sum of distances between all pairs of vertices of $G$. In 1991, Šoltés formulated the problem of fin... |
https://arxiv.org/abs/math/0607089 | Transportation Distance and the Central Limit Theorem | For probability measures on a complete separable metric space, we present sufficient conditions for the existence of a solution to the Kantorovich transportation problem. We also obtain sufficient conditions (which sometimes also become necessary) for the convergence, in transportation, of probability measures when the... | \section{Introduction}
Let $(M,d)$ be a metric space and let $c:M\times M\rightarrow {\bf R}$,
be a non-negative Borel function.
The transportation $c$-distance $T_{c}(\mu , \nu )$ between two probability measures
$\mu $ and $\nu $ defined on the Borel $\sigma $-field $\mathcal B(M)$ is given via
$$T_{c}(\mu ,\nu )=\... | {
"timestamp": "2006-07-04T14:17:59",
"yymm": "0607",
"arxiv_id": "math/0607089",
"language": "en",
"url": "https://arxiv.org/abs/math/0607089",
"abstract": "For probability measures on a complete separable metric space, we present sufficient conditions for the existence of a solution to the Kantorovich tra... |
https://arxiv.org/abs/0803.3205 | Anick's fibration and the odd primary homotopy exponent of spheres | For primes p>=3, Cohen, Moore, and Neisendorfer showed that the exponent of the p-torsion in the homotopy groups of S^2n+1 is p^n. This was obtained as a consequence of a thorough analysis of the homotopy theory of Moore spaces. Anick further developed this for p>=5 by constructing a homotopy fibration S^2n-1 --> T^2n+... | \section{Introduction}
\label{sec:intro}
Cohen, Moore, and Neisendorfer's~\cite{CMN1,CMN2,N2}
determination of the odd primary homotopy exponent of spheres
is a miletone in homotopy theory. They showed that
if $p$ is an odd prime then $p^{n}$ is the least power of $p$
which annihilates the $p$-torsion in $\pi_{\... | {
"timestamp": "2008-03-21T18:46:01",
"yymm": "0803",
"arxiv_id": "0803.3205",
"language": "en",
"url": "https://arxiv.org/abs/0803.3205",
"abstract": "For primes p>=3, Cohen, Moore, and Neisendorfer showed that the exponent of the p-torsion in the homotopy groups of S^2n+1 is p^n. This was obtained as a co... |
https://arxiv.org/abs/2010.08904 | Consecutive Radio Labeling of Hamming Graphs | For a graph $G$, a $k$-radio labeling of $G$ is the assignment of positive integers to the vertices of $G$ such that the closer two vertices are on the graph, the greater the difference in labels is required to be. Specifically, $\vert f(u)-f(v)\vert\geq k + 1 - d(u,v)$ where $f(u)$ is the label on a vertex $u$ in $G$.... | \section{Introduction}
Graph labeling was first introduced by Rosa in 1966 \cite{Rosa}. Since then, numerous types of labeling have been subject to extensive study including vertex coloring, graceful labeling, harmonious labeling, $k$-radio labeling, and more. For a survey of graph labeling, see Gallian \cite{Gallian}.... | {
"timestamp": "2020-10-20T02:17:02",
"yymm": "2010",
"arxiv_id": "2010.08904",
"language": "en",
"url": "https://arxiv.org/abs/2010.08904",
"abstract": "For a graph $G$, a $k$-radio labeling of $G$ is the assignment of positive integers to the vertices of $G$ such that the closer two vertices are on the gr... |
https://arxiv.org/abs/1704.08661 | Expected Number of Distinct Subsequences in Randomly Generated Binary Strings | When considering binary strings, it's natural to wonder how many distinct subsequences might exist in a given string. Given that there is an existing algorithm which provides a straightforward way to compute the number of distinct subsequences in a fixed string, we might next be interested in the expected number of dis... | \section{Introduction}
This paper uses the definitions for string and subsequence provided in \cite{Flaxman}. A binary string of length $n$ is some $A=a_1a_2...a_n \in \{0,1\}^n$, and another string $B$ of length $m \le n$ is a subsequence of $A$ if there exist indices $i_1 < i_2 < ...< i_m$ such that
$$B=a_{i_1}a_{i_... | {
"timestamp": "2018-06-14T02:11:32",
"yymm": "1704",
"arxiv_id": "1704.08661",
"language": "en",
"url": "https://arxiv.org/abs/1704.08661",
"abstract": "When considering binary strings, it's natural to wonder how many distinct subsequences might exist in a given string. Given that there is an existing algo... |
https://arxiv.org/abs/1104.0426 | The Randic index and the diameter of graphs | The {\it Randić index} $R(G)$ of a graph $G$ is defined as the sum of 1/\sqrt{d_ud_v} over all edges $uv$ of $G$, where $d_u$ and $d_v$ are the degrees of vertices $u$ and $v,$ respectively. Let $D(G)$ be the diameter of $G$ when $G$ is connected. Aouchiche-Hansen-Zheng conjectured that among all connected graphs $G$ o... | \section{Introduction}
In 1975, the chemist Milan Randi\'c \cite{Randic} proposed a
topological index $R$ under the name "branching index", suitable
for measuring the extent of branching of the carbon-atom skeleton
of saturated hydrocarbons. The branching index was renamed the
molecular connectivity index and is often... | {
"timestamp": "2011-04-05T02:02:02",
"yymm": "1104",
"arxiv_id": "1104.0426",
"language": "en",
"url": "https://arxiv.org/abs/1104.0426",
"abstract": "The {\\it Randić index} $R(G)$ of a graph $G$ is defined as the sum of 1/\\sqrt{d_ud_v} over all edges $uv$ of $G$, where $d_u$ and $d_v$ are the degrees of... |
https://arxiv.org/abs/1605.08982 | Coordinate Descent Face-Off: Primal or Dual? | Randomized coordinate descent (RCD) methods are state-of-the-art algorithms for training linear predictors via minimizing regularized empirical risk. When the number of examples ($n$) is much larger than the number of features ($d$), a common strategy is to apply RCD to the dual problem. On the other hand, when the num... |
\section{Introduction}
In the last 5 years or so, randomized coordinate descent (RCD) methods \cite{ShalevTewari11, Nesterov:2010RCDM, UCDC, PCDM} have become immensely popular in a variety of machine learning tasks, with supervised learning being a prime example. The main reasons behind the rise of RCD-type me... | {
"timestamp": "2016-05-31T02:09:09",
"yymm": "1605",
"arxiv_id": "1605.08982",
"language": "en",
"url": "https://arxiv.org/abs/1605.08982",
"abstract": "Randomized coordinate descent (RCD) methods are state-of-the-art algorithms for training linear predictors via minimizing regularized empirical risk. When... |
https://arxiv.org/abs/1507.02069 | Random Walks and Evolving Sets: Faster Convergences and Limitations | Analyzing the mixing time of random walks is a well-studied problem with applications in random sampling and more recently in graph partitioning. In this work, we present new analysis of random walks and evolving sets using more combinatorial graph structures, and show some implications in approximating small-set expan... | \section{Introduction}
Analyzing the mixing time of random walks is a fundamental problem with many applications in random sampling~\cite{LPW}.
The evolving set process is an elegant tool introduced by Morris and Peres~\cite{MP} to provide sharp analyses of mixing time (see the survey~\cite{MT}).
Recently, random walk... | {
"timestamp": "2015-07-09T02:06:47",
"yymm": "1507",
"arxiv_id": "1507.02069",
"language": "en",
"url": "https://arxiv.org/abs/1507.02069",
"abstract": "Analyzing the mixing time of random walks is a well-studied problem with applications in random sampling and more recently in graph partitioning. In this ... |
https://arxiv.org/abs/0805.1553 | Pleijel's nodal domain theorem for free membranes | We prove an analogue of Pleijel's nodal domain theorem for piecewise analytic planar domains with Neumann boundary conditions. This confirms a conjecture made by Pleijel in 1956. The proof is a combination of Pleijel's original approach and an estimate due to Toth and Zelditch for the number of boundary zeros of Neuman... | \section{Introduction}
Let $\Omega \subset \mathbb{R}^2$ be a bounded planar domain. Let
$0\le \lambda_1 < \lambda_2\le \dots \le \lambda_k\le \dots$ be the
eigenvalues of the Laplacian on $\Omega$ with either Dirichlet or
Neumann boundary conditions, and $\phi_1,\phi_2,\dots, \phi_k,\dots$
be an orthonormal basis of e... | {
"timestamp": "2008-05-11T20:23:04",
"yymm": "0805",
"arxiv_id": "0805.1553",
"language": "en",
"url": "https://arxiv.org/abs/0805.1553",
"abstract": "We prove an analogue of Pleijel's nodal domain theorem for piecewise analytic planar domains with Neumann boundary conditions. This confirms a conjecture ma... |
https://arxiv.org/abs/2210.01079 | Small order limit of fractional Dirichlet sublinear-type problems | We study the asymptotic behavior of solutions to various Dirichlet sublinear-type problems involving the fractional Laplacian when the fractional parameter s tends to zero. Depending on the type on nonlinearity, positive solutions may converge to a characteristic function or to a positive solution of a limit nonlinear ... | \section{Introduction}
Consider a positive solution of a sublinear-type problem such as
\begin{align*}
(-\Delta)^{s} u_s = f(u_s)\quad \text{ in }\Omega,\qquad u_s=0\quad \text{ on }\mathbb R^N\backslash\Omega,
\end{align*}
where $s\in(0,1)$, $N\geq 1$, $\Omega\subset \mathbb R^N$ is a bounded Lipschitz set, and... | {
"timestamp": "2022-10-04T02:32:48",
"yymm": "2210",
"arxiv_id": "2210.01079",
"language": "en",
"url": "https://arxiv.org/abs/2210.01079",
"abstract": "We study the asymptotic behavior of solutions to various Dirichlet sublinear-type problems involving the fractional Laplacian when the fractional paramete... |
https://arxiv.org/abs/1704.00350 | Tomaszewski's Problem on Randomly Signed Sums: Breaking the 3/8 Barrier | Let $v_1$, $v_2$, ..., $v_n$ be real numbers whose squares add up to 1. Consider the $2^n$ signed sums of the form $S = \sum \pm v_i$. Holzman and Kleitman (1992) proved that at least 3/8 of these sums satisfy $|S| \le 1$. This 3/8 bound seems to be the best their method can achieve. Using a different method, we improv... | \section{Introduction}
Let $v_1$, $v_2$, \dots, $v_n$ be real numbers such that the sum of their squares is at most~$1$.
Consider the $2^n$ signed sums of the form $S = \pm v_1 \pm v_2 \pm \dots \pm v_n$.
In 1986,
B.~Tomaszewski (see Guy~\cite{Guy}) asked the following question:
is it always true that at least $... | {
"timestamp": "2017-09-01T02:06:04",
"yymm": "1704",
"arxiv_id": "1704.00350",
"language": "en",
"url": "https://arxiv.org/abs/1704.00350",
"abstract": "Let $v_1$, $v_2$, ..., $v_n$ be real numbers whose squares add up to 1. Consider the $2^n$ signed sums of the form $S = \\sum \\pm v_i$. Holzman and Kleit... |
https://arxiv.org/abs/1502.03637 | The Mathematical Work of V. K. Patodi | We give a brief survey on aspects of the local index theory as developed from the mathematical works of V. K. Patodi. It is dedicated to the 70th anniversary of Patodi. | \section{Introduction} \label{s0}
Vijay Kumar Patodi was born on March 12, 1945 in Guna, India. He passed away on December 21, 1976, at the age of 31.
It is remarkable that even in such a short period of life, Patodi made quite a number of fundamental contributions to mathematics. These contributions have had deep... | {
"timestamp": "2015-02-13T02:11:49",
"yymm": "1502",
"arxiv_id": "1502.03637",
"language": "en",
"url": "https://arxiv.org/abs/1502.03637",
"abstract": "We give a brief survey on aspects of the local index theory as developed from the mathematical works of V. K. Patodi. It is dedicated to the 70th annivers... |
https://arxiv.org/abs/2106.10559 | The trace-reinforced ants process does not find shortest paths | In this paper, we study a probabilistic reinforcement-learning model for ants searching for the shortest path(s) between their nest and a source of food. In this model, the nest and the source of food are two distinguished nodes $N$ and $F$ in a finite graph $\mathcal G$. The ants perform a sequence of random walks on ... | \section{Introduction and main results}
{\bf Context:}
It is believed that ants are able to find shortest paths between their nest and a source of food
with no other means of communications than the pheromones they lay behind them.
This phenomenon has been observed empirically in the biology literature
(see, e.g.,~\ci... | {
"timestamp": "2021-06-22T02:12:16",
"yymm": "2106",
"arxiv_id": "2106.10559",
"language": "en",
"url": "https://arxiv.org/abs/2106.10559",
"abstract": "In this paper, we study a probabilistic reinforcement-learning model for ants searching for the shortest path(s) between their nest and a source of food. ... |
https://arxiv.org/abs/1808.06657 | Designs over finite fields by difference methods | One of the very first results about designs over finite fields, by S. Thomas, is the existence of a cyclic 2-$(n,3,7)$ design over $\mathbb{F}_{2}$ for every integer $n$ coprime with 6. Here, by means of difference methods, we reprove and improve a little bit this result showing that it is true, more generally, for eve... | \section{Introduction}
In this paper we adapt very well known difference methods to the
construction of \emph{designs over finite fields}. Our main result will
be the existence of a cyclic 2-$(n,3,7)$ design over $\mathbb{F}_{2}$
for every odd positive $n$. It should be noted that in the case
$n\equiv \pm 1$ (mod 6) o... | {
"timestamp": "2019-02-27T02:12:03",
"yymm": "1808",
"arxiv_id": "1808.06657",
"language": "en",
"url": "https://arxiv.org/abs/1808.06657",
"abstract": "One of the very first results about designs over finite fields, by S. Thomas, is the existence of a cyclic 2-$(n,3,7)$ design over $\\mathbb{F}_{2}$ for e... |
https://arxiv.org/abs/1910.02454 | Critical digraphs with few vertices | We show that every k-dichromatic vertex-critical digraph on at most 2k-2 vertices has a disconnected complement. This answers a question of Bang-Jensen et al., and generalises a classical theorem of Gallai on undirected vertex-critical graphs. | \section{Introduction}
Neumann-Lara~\cite{NL82} defined the \emph{dichromatic number} $\chi(G)$ of a
digraph $G$ as the smallest number or colours needed to colour the vertices of
$G$ so that no colour class induces a directed cycle.
A digraph $G$ with $\chi(G)=k$ is said to be \emph{$k$-critical} (resp.\
\emph{$k$-ve... | {
"timestamp": "2019-10-08T02:15:14",
"yymm": "1910",
"arxiv_id": "1910.02454",
"language": "en",
"url": "https://arxiv.org/abs/1910.02454",
"abstract": "We show that every k-dichromatic vertex-critical digraph on at most 2k-2 vertices has a disconnected complement. This answers a question of Bang-Jensen et... |
https://arxiv.org/abs/2109.10634 | Filtered integration rules for finite Hilbert transforms | A product quadrature rule, based on the filtered de la Vallée Poussin polynomial approximation, is proposed for evaluating the finite Hilbert transform in [-1; 1]. Convergence results are stated in weighted uniform norm for functions belonging to suitable Besov type subspaces. Several numerical tests are provided, also... | \section{Introduction}
The numerical computation of the Hilbert transform of a function plays an important role in several fields, since many mathematical models in applied sciences lead to it
(see e.g. \cite{kalandya,king} and the references therein). Depending on the specific application, we may consider bounded or ... | {
"timestamp": "2021-09-23T02:14:40",
"yymm": "2109",
"arxiv_id": "2109.10634",
"language": "en",
"url": "https://arxiv.org/abs/2109.10634",
"abstract": "A product quadrature rule, based on the filtered de la Vallée Poussin polynomial approximation, is proposed for evaluating the finite Hilbert transform in... |
https://arxiv.org/abs/0805.1909 | On universal Lie nilpotent associative algebras | We study the quotient Q_i(A) of a free algebra A by the ideal M_i(A) generated by relation that the i-th commutator of any elements is zero. In particular, we completely describe such quotient for i=4 (for i<=3 this was done previously by Feigin and Shoikhet). We also study properties of the ideals M_i(A), e.g. when M_... | \section{Introduction}
Let $A$ be an associative unital algebra over a field $k$. Let us regard
it as a Lie algebra with bracket $[a,b]=ab-ba$, and consider the
terms of its lower central series $L_i(A)$ defined inductively by
$L_1(A)=A$ and $L_{i+1}(A)=[A,L_i(A)]$. Denote by $M_i(A)$ the
two-sided ideal in $A$ genera... | {
"timestamp": "2008-05-13T21:09:10",
"yymm": "0805",
"arxiv_id": "0805.1909",
"language": "en",
"url": "https://arxiv.org/abs/0805.1909",
"abstract": "We study the quotient Q_i(A) of a free algebra A by the ideal M_i(A) generated by relation that the i-th commutator of any elements is zero. In particular, ... |
https://arxiv.org/abs/0809.3421 | Sub-exponentially localized kernels and frames induced by orthogonal expansions | The aim of this paper is to construct sup-exponentially localized kernels and frames in the context of classical orthogonal expansions, namely, expansions in Jacobi polynomials, spherical harmonics, orthogonal polynomials on the ball and simplex, and Hermite and Laguerre functions. | \section{Introduction}\label{introduction}
\setcounter{equation}{0}
Orthogonal expansions have been recently used for the construction of kernels and frames (needlets)
with localization that is faster than the reciprocal of any polynomial rate
in non-standard settings such as on
the sphere, interval and ball with wei... | {
"timestamp": "2008-09-19T19:02:37",
"yymm": "0809",
"arxiv_id": "0809.3421",
"language": "en",
"url": "https://arxiv.org/abs/0809.3421",
"abstract": "The aim of this paper is to construct sup-exponentially localized kernels and frames in the context of classical orthogonal expansions, namely, expansions i... |
https://arxiv.org/abs/1910.01829 | On the eigenvalue region of permutative doubly stochastic matrices | This paper is devoted to the study of eigenvalue region of the doubly stochastic matrices which are also permutative, that is, each row of such a matrix is a permutation of any other row. We call these matrices as permutative doubly stochastic (PDS) matrices. A method is proposed to obtain symbolic representation of al... | \section{Introduction}\label{sec:1}
Characterization of the eigenvalue region of doubly stochastic (DS) matrices is one of the long-standing open problems in matrix theory. This problem relates to many other problems, including inverse eigenvalue problem and eigenvalue realizability problem. Despite several attempts, ... | {
"timestamp": "2020-04-21T02:08:39",
"yymm": "1910",
"arxiv_id": "1910.01829",
"language": "en",
"url": "https://arxiv.org/abs/1910.01829",
"abstract": "This paper is devoted to the study of eigenvalue region of the doubly stochastic matrices which are also permutative, that is, each row of such a matrix i... |
https://arxiv.org/abs/0812.3707 | Minimal Euclidean representations of graphs | A simple graph G is said to be representable in a real vector space of dimension m if there is an embedding of the vertex set in the vector space such that the Euclidean distance between any two distinct vertices is one of only two distinct values a or b, with distance a if the vertices are adjacent and distance b othe... | \section{Introduction}\label{sec:intro}
Recently, Nguyen Van Th\'e \cite{NguyenVanThe08} revived a problem of representing graphs in Euclidean space, which, according to Pouzet \cite{Pouzet79}, was originally introduced by Specker before 1972.
A simple graph is \defn{representable} in $\re^m$ if there is an embedding... | {
"timestamp": "2009-05-29T23:28:31",
"yymm": "0812",
"arxiv_id": "0812.3707",
"language": "en",
"url": "https://arxiv.org/abs/0812.3707",
"abstract": "A simple graph G is said to be representable in a real vector space of dimension m if there is an embedding of the vertex set in the vector space such that ... |
https://arxiv.org/abs/2007.14716 | Weakly saturated random graphs | As introduced by Bollobás, a graph $G$ is weakly $H$-saturated if the complete graph $K_n$ is obtained by iteratively completing copies of $H$ minus an edge. For all graphs $H$, we obtain an asymptotic lower bound for the critical threshold $p_c$, at which point the Erdős--Rényi graph ${\mathcal G}_{n,p}$ is likely to ... | \section{Introduction}\label{S_intro}
The concept of {\it weak saturation} was introduced
and studied in early work of Bollob{\'a}s \cite{B65}.
Given graphs $G$ and $H$, the graph $\l G\r_H$ is obtained
by iteratively completing copies of $H$ minus an edge, starting with $G$.
Formally, set $G_0=G$, and for $t\ge1$... | {
"timestamp": "2021-03-03T02:05:53",
"yymm": "2007",
"arxiv_id": "2007.14716",
"language": "en",
"url": "https://arxiv.org/abs/2007.14716",
"abstract": "As introduced by Bollobás, a graph $G$ is weakly $H$-saturated if the complete graph $K_n$ is obtained by iteratively completing copies of $H$ minus an ed... |
https://arxiv.org/abs/1811.07481 | Intersection theorems for families of matchings of complete $k$-partite $k$-graphs | The celebrated Erdős-Ko-Rado Theorem and other intersection theorems on families of sets have previously been generalised with respect to families of permutations. We prove that many of these generalised intersection theorems for permutation families and related families naturally extend to families of matchings of com... | \section{Introduction}
For an integer $n$, let $[n]$ denote the set $\{ 1,2,\ldots,n\}$.
The power set of a set $X$ is denoted by $2^{X}$
and the set of subsets of $X$ of size $r$
is denoted by $\binom{X}{r}$.
A family of sets $\mathscr{F}$ is {\em intersecting}
if $A \cap B \neq \emptyset$ for all $A,B \in \mathscr{F... | {
"timestamp": "2018-11-20T02:40:26",
"yymm": "1811",
"arxiv_id": "1811.07481",
"language": "en",
"url": "https://arxiv.org/abs/1811.07481",
"abstract": "The celebrated Erdős-Ko-Rado Theorem and other intersection theorems on families of sets have previously been generalised with respect to families of perm... |
https://arxiv.org/abs/chao-dyn/9802014 | Computing periodic orbits using the anti-integrable limit | Chaotic dynamics can be effectively studied by continuation from an anti-integrable limit. Using the Henon map as an example, we obtain a simple analytical bound on the domain of existence of the horseshoe that is equivalent to the well-known bound of Devaney and Nitecki. We also reformulate the popular method for find... | \section{#1} \label{#2}}
\newcommand{\eq}[1]{(\ref{#1})}
\newcommand{\Th}[1]{theorem (\ref{#1})}
\newcommand{\makev}[2]{\left(\begin{array}{c} #1 \\ #2 \end{array}\right)}
\newcommand{H\'{e}non }{H\'{e}non }
\def {\mathbf{\zeta}}{{\mathbf{\zeta}}}
\def {\mathbf{q}} {{\mathbf{q}}}
\def {\mathbf{s}} {{\mathbf{s}}}
\def ... | {
"timestamp": "1998-02-13T01:09:30",
"yymm": "9802",
"arxiv_id": "chao-dyn/9802014",
"language": "en",
"url": "https://arxiv.org/abs/chao-dyn/9802014",
"abstract": "Chaotic dynamics can be effectively studied by continuation from an anti-integrable limit. Using the Henon map as an example, we obtain a simp... |
https://arxiv.org/abs/2105.02429 | The breadth of Lie poset algebras | The breadth of a Lie algebra $L$ is defined to be the maximal dimension of the image of $ad_x=[x,-]:L\to L$, for $x\in L$. Here, we initiate an investigation into the breadth of three families of Lie algebras defined by posets and provide combinatorial breadth formulas for members of each family. | \section{Introduction}
\textit{Convention:} We assume throughout that all Lie algebras are over an algebraically closed field of characteristic zero, \textbf{k}, which we may take, without any loss of generality, to be the complex numbers.
\medskip
The complete classification of simple Lie algebras, elegantly couch... | {
"timestamp": "2021-06-07T02:04:49",
"yymm": "2105",
"arxiv_id": "2105.02429",
"language": "en",
"url": "https://arxiv.org/abs/2105.02429",
"abstract": "The breadth of a Lie algebra $L$ is defined to be the maximal dimension of the image of $ad_x=[x,-]:L\\to L$, for $x\\in L$. Here, we initiate an investig... |
https://arxiv.org/abs/1506.07540 | Global Optimality in Tensor Factorization, Deep Learning, and Beyond | Techniques involving factorization are found in a wide range of applications and have enjoyed significant empirical success in many fields. However, common to a vast majority of these problems is the significant disadvantage that the associated optimization problems are typically non-convex due to a multilinear form or... |
\section{Discussion and Conclusions}
We begin the discussion of our results with a cautionary note; namely, these results can be challenging to apply in practice. In particular, many algorithms based on alternating minimization can typically only guarantee convergence to a critical point, and with the inherent non-c... | {
"timestamp": "2015-06-26T02:00:48",
"yymm": "1506",
"arxiv_id": "1506.07540",
"language": "en",
"url": "https://arxiv.org/abs/1506.07540",
"abstract": "Techniques involving factorization are found in a wide range of applications and have enjoyed significant empirical success in many fields. However, commo... |
https://arxiv.org/abs/1703.04353 | Nonexistence of positive solutions for Henon equation | We consider the semilinear elliptic equation $$ -\Delta u = |x|^\alpha u^p \quad \hbox{in } \mathbb{R}^N, $$ where $N\ge 3$, $\alpha>-2$ and $p>1$. We show that there are no positive solutions provided that the exponent $p$ additionally verifies $$ 1<p<\frac{N+2\alpha+2}{N-2}. $$ This solves an open problem posed in pr... | \section{Introduction}
\setcounter{section}{1}
\setcounter{equation}{0}
Probably the most well-known nonlinear Liouville theorem in the literature is the one obtained in the celebrated
paper \cite{GS}. There, nonexistence of positive solutions of the elliptic equation
\begin{equation}\label{eq-gs}
-\Delta u = u^p \qu... | {
"timestamp": "2017-03-14T01:12:37",
"yymm": "1703",
"arxiv_id": "1703.04353",
"language": "en",
"url": "https://arxiv.org/abs/1703.04353",
"abstract": "We consider the semilinear elliptic equation $$ -\\Delta u = |x|^\\alpha u^p \\quad \\hbox{in } \\mathbb{R}^N, $$ where $N\\ge 3$, $\\alpha>-2$ and $p>1$.... |
https://arxiv.org/abs/0808.0958 | On the complete integrability and linearization of nonlinear ordinary differential equations - Part III: Coupled first order equations | Continuing our study on the complete integrability of nonlinear ordinary differential equations, in this paper we consider the integrability of a system of coupled first order nonlinear ordinary differential equations (ODEs) of both autonomous and non-autonomous types. For this purpose, we modify the original Prelle-Si... | \section{Introduction}
\label{sec4:1}
In our previous two works (Chandrasekar \textit{et al.} 2005; 2006)
we have studied in some detail the extended modified Prelle-Singer
(PS) procedure (Prelle \& Singer 1983; Duarte \textit{et al.} 2002)
so as to apply it
to a class of second and third order nonlinear ordinary diff... | {
"timestamp": "2008-10-10T09:03:49",
"yymm": "0808",
"arxiv_id": "0808.0958",
"language": "en",
"url": "https://arxiv.org/abs/0808.0958",
"abstract": "Continuing our study on the complete integrability of nonlinear ordinary differential equations, in this paper we consider the integrability of a system of ... |
https://arxiv.org/abs/1912.04889 | Universal and unavoidable graphs | The Turán number $\text{ex}(n,H)$ of a graph $H$ is the maximal number of edges in an $H$-free graph on $n$ vertices. In $1983$ Chung and Erdős asked which graphs $H$ with $e$ edges minimize $\text{ex}(n,H)$. They resolved this question asymptotically for most of the range of $e$ and asked to complete the picture. In t... | \section{Introduction}
\par The following question of Tur\'an dating back to 1941 \cite{Turan1941external} is one of the most classical problems of graph theory. Given a fixed graph $H,$ what is the maximal number of edges one can have in an $n$ vertex graph which does not contain a copy of $H$ as a subgraph? The an... | {
"timestamp": "2019-12-11T02:20:28",
"yymm": "1912",
"arxiv_id": "1912.04889",
"language": "en",
"url": "https://arxiv.org/abs/1912.04889",
"abstract": "The Turán number $\\text{ex}(n,H)$ of a graph $H$ is the maximal number of edges in an $H$-free graph on $n$ vertices. In $1983$ Chung and Erdős asked whi... |
https://arxiv.org/abs/1411.1443 | Boundary Integrals and Approximations of Harmonic Functions | Formulae for the value of a harmonic function at the center of a rectangle are found that involve boundary integrals. The central value of a harmonic function is shown to be well approximated by the mean value of the function on the boundary plus a very small number (often just 1 or 2) of additional boundary integrals.... | \section{Introduction}\label{s1}
One of the best known theorems about solutions of Laplace's equation is the mean value theorem that says the value of a harmonic function $h$ at the center of the ball is the mean
value of the boundary values of $h$ on the boundary.
Epstein \cite{Ep} proved that if this holds for all ... | {
"timestamp": "2015-01-28T02:17:13",
"yymm": "1411",
"arxiv_id": "1411.1443",
"language": "en",
"url": "https://arxiv.org/abs/1411.1443",
"abstract": "Formulae for the value of a harmonic function at the center of a rectangle are found that involve boundary integrals. The central value of a harmonic functi... |
https://arxiv.org/abs/2110.02655 | A new integral equation for Brownian stopping problems with finite time horizon | For classical finite time horizon stopping problems driven by a Brownian motion\[V(t,x) = \sup_{t\leq\tau\leq0}E_{(t,x)}[g(\tau,W_{\tau})],\] we derive a new class of Fredholm type integral equations for the stopping set. For large problem classes of interest, we show by analytical arguments that the equation uniquely ... | \section{Introduction}
Let $W$ be an $n$-dimensional standard Brownian motion started at time $t\leq 0$ in $\textbf{x} \in \rr ^{n}$, $g:\rr_{\leq 0}\times \rr^{n} \to \rr$ a payoff function with properties to be specified later. We consider the stopping problem with finite time horizon
\begin{equation}\label{eq:sp1}
... | {
"timestamp": "2021-10-07T02:19:20",
"yymm": "2110",
"arxiv_id": "2110.02655",
"language": "en",
"url": "https://arxiv.org/abs/2110.02655",
"abstract": "For classical finite time horizon stopping problems driven by a Brownian motion\\[V(t,x) = \\sup_{t\\leq\\tau\\leq0}E_{(t,x)}[g(\\tau,W_{\\tau})],\\] we d... |
https://arxiv.org/abs/1907.07886 | Sparsity of integer solutions in the average case | We examine how sparse feasible solutions of integer programs are, on average. Average case here means that we fix the constraint matrix and vary the right-hand side vectors. For a problem in standard form with m equations, there exist LP feasible solutions with at most m many nonzero entries. We show that under relativ... | \section{Introduction}
Let $m,n \in \mathbb{Z}_{\ge 1}$ and $A \in \mathbb{Z}^{m \times n}$.
We always assume that $A$ has full row rank.
We also view $A$ as a set of its column vectors.
So, $W \subseteq A$ implies that $W$ is a subset of the columns of $A$.
We aim to find a sparse integer vector in the set
\[
P(A,b)... | {
"timestamp": "2019-07-19T02:06:50",
"yymm": "1907",
"arxiv_id": "1907.07886",
"language": "en",
"url": "https://arxiv.org/abs/1907.07886",
"abstract": "We examine how sparse feasible solutions of integer programs are, on average. Average case here means that we fix the constraint matrix and vary the right... |
https://arxiv.org/abs/2110.00993 | On monoid graphs | We investigate Cayley graphs of finite semigroups and monoids. First, we look at semigroup digraphs, i.e., directed Cayley graphs of semigroups, and give a Sabidussi-type characterization in the case of monoids. We then correct a proof of Zelinka from '81 that characterizes semigroup digraphs with outdegree $1$. Furthe... | \section{Introduction}
After being introduced in 1878~\cite{Cayley1878}, Cayley graphs soon became a prominent tool in both graph and group theory. As of today Cayley graphs of groups play a prominent role in (books devoted to) algebraic graph theory, see e.g.~\cite{God-01,Kna-19}. On one hand, they are useful for con... | {
"timestamp": "2021-10-07T02:02:56",
"yymm": "2110",
"arxiv_id": "2110.00993",
"language": "en",
"url": "https://arxiv.org/abs/2110.00993",
"abstract": "We investigate Cayley graphs of finite semigroups and monoids. First, we look at semigroup digraphs, i.e., directed Cayley graphs of semigroups, and give ... |
https://arxiv.org/abs/1212.1925 | Polynomials of small degree evaluated on matrices | A celebrated theorem of Shoda states that over any field K (of characteristic 0), every matrix with trace 0 can be expressed as a commutator AB-BA, or, equivalently, that the set of values of the polynomial f(x,y)=xy-yx on the nxn-matrix K-algebra contains all matrices with trace 0. We generalize Shoda's theorem by sho... | \section{Introduction}
We begin by recalling a theorem, which was originally proved for fields of characteristic $0$ by Shoda~\cite{Shoda} and later extended to all fields by Albert and Muckenhoupt~\cite{AM}.
\begin{theorem}[Shoda/Albert/Muckenhoupt]\label{Shoda}
Let $K$ be any field, $n\geq 2$ an integer, and $M \in... | {
"timestamp": "2012-12-11T02:02:21",
"yymm": "1212",
"arxiv_id": "1212.1925",
"language": "en",
"url": "https://arxiv.org/abs/1212.1925",
"abstract": "A celebrated theorem of Shoda states that over any field K (of characteristic 0), every matrix with trace 0 can be expressed as a commutator AB-BA, or, equi... |
https://arxiv.org/abs/2104.05292 | Computer Algebra in R with caracas | The capability of R to do symbolic mathematics is enhanced by the caracas package. This package uses the Python computer algebra library SymPy as a back-end but caracas is tightly integrated in the R environment, thereby enabling the R user with symbolic mathematics within R. Key components of the caracas package are i... |
\subsection{Keywords}\label{keywords}}
Differentiation, Factor analysis, Hessian matrix, Integration, Lagrange
multiplier, Limit, Linear algebra, Principal component analysis, Score
function, Symbolic mathematics, Taylor expansion, Teaching.
\hypertarget{introduction}{%
\subsection{Introduction}\label{introduction}}... | {
"timestamp": "2021-04-20T02:32:22",
"yymm": "2104",
"arxiv_id": "2104.05292",
"language": "en",
"url": "https://arxiv.org/abs/2104.05292",
"abstract": "The capability of R to do symbolic mathematics is enhanced by the caracas package. This package uses the Python computer algebra library SymPy as a back-e... |
https://arxiv.org/abs/2012.09570 | Level Set Percolation in Two-Dimensional Gaussian Free Field | The nature of level set percolation in the two-dimension Gaussian Free Field has been an elusive question. Using a loop-model mapping, we show that there is a nontrivial percolation transition, and characterize the critical point. In particular, the correlation length diverges exponentially, and the critical clusters a... | \subsection{Numerical Methods}
\textbf{Gaussian Free Fields.}
The numerical tests are all performed on a square lattice of size $L \times L$, with periodic boundary conditions on both directions, $x \equiv x +L$, $y \equiv y + L$. The 2D GFF can be numerically generated by the standard Fourier filter method:
\be... | {
"timestamp": "2021-03-23T01:41:34",
"yymm": "2012",
"arxiv_id": "2012.09570",
"language": "en",
"url": "https://arxiv.org/abs/2012.09570",
"abstract": "The nature of level set percolation in the two-dimension Gaussian Free Field has been an elusive question. Using a loop-model mapping, we show that there ... |
https://arxiv.org/abs/1003.5444 | Roots of Ehrhart polynomials arising from graphs | Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck {\it et al.}\ that all roots $\alpha$ of Ehrhart polynomials of polytopes of dimension $D$ satisfy $-D \le \Re(\alpha) \le D - 1$... | \section{Method of Computation}
\label{sec:metho}
This appendix presents an outline of the procedure used to compute the roots of
the Ehrhart polynomials of edge or symmetric edge polytopes
in Sections~\ref{sec:simple} and~\ref{sec:symmetric}.
Both polytopes are constructed from connected simple graphs.
For each num... | {
"timestamp": "2010-12-30T02:04:07",
"yymm": "1003",
"arxiv_id": "1003.5444",
"language": "en",
"url": "https://arxiv.org/abs/1003.5444",
"abstract": "Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not mere... |
https://arxiv.org/abs/1605.01117 | Interpolation in Algebraic Geometry | This is an expanded version of the two papers "Interpolation of Varieties of Minimal Degree" and "Interpolation Problems: Del Pezzo Surfaces." It is well known that one can find a rational normal curve in $\mathbb P^n$ through $n+3$ general points. More recently, it was shown that one can always find nonspecial curves ... | \chapter[Appendix]{Appendix: Interpolation of del Pezzo surfaces, with Anand Patel}
\label{section:appendix}
\section{Introduction to the appendix}
The main result of this appendix is that del Pezzo surfaces satisfy weak interpolation, over a field of characteristic $0$.
For the remainder of this chapter, we assume ou... | {
"timestamp": "2016-05-05T02:03:10",
"yymm": "1605",
"arxiv_id": "1605.01117",
"language": "en",
"url": "https://arxiv.org/abs/1605.01117",
"abstract": "This is an expanded version of the two papers \"Interpolation of Varieties of Minimal Degree\" and \"Interpolation Problems: Del Pezzo Surfaces.\" It is w... |
https://arxiv.org/abs/1708.07984 | Some Kähler structures on products of 2-spheres | We consider a family of Kähler structures on products of 2-spheres, arising from complex Bott manifolds. These are obtained via iterated $\mathbb P^1$-bundle constructions, generalizing the classical Hirzebruch surfaces. We show that the resulting Kähler structures all have identical Chern classes. We construct Bott di... | \section{Introduction}
In complex geometry, it is interesting to study the class of complex structures (or K\"ahler structures) supported on a fixed smooth
oriented manifold $M$. Since the basic invariants of a complex manifolds are the Chern classes, it is tempting to try and use these
to distinguish complex structur... | {
"timestamp": "2017-08-29T02:05:09",
"yymm": "1708",
"arxiv_id": "1708.07984",
"language": "en",
"url": "https://arxiv.org/abs/1708.07984",
"abstract": "We consider a family of Kähler structures on products of 2-spheres, arising from complex Bott manifolds. These are obtained via iterated $\\mathbb P^1$-bu... |
https://arxiv.org/abs/1805.09185 | Alternating Randomized Block Coordinate Descent | Block-coordinate descent algorithms and alternating minimization methods are fundamental optimization algorithms and an important primitive in large-scale optimization and machine learning. While various block-coordinate-descent-type methods have been studied extensively, only alternating minimization -- which applies ... | \section{Introduction}\label{sec:intro}
First-order methods for minimizing smooth convex functions are a cornerstone of large-scale optimization and machine learning. Given the size and heterogeneity of the data in these applications, there is a particular interest in designing iterative methods that, at each iteratio... | {
"timestamp": "2018-05-24T02:11:49",
"yymm": "1805",
"arxiv_id": "1805.09185",
"language": "en",
"url": "https://arxiv.org/abs/1805.09185",
"abstract": "Block-coordinate descent algorithms and alternating minimization methods are fundamental optimization algorithms and an important primitive in large-scale... |
https://arxiv.org/abs/1703.02120 | Root separation for reducible monic polynomials of odd degree | We study root separation of reducible monic integer polynomials of odd degree. Let h(P) be the naive height, sep(P) the minimal distance between two distinct roots of an integer polynomial P(x) and sep(P)=h(P)^{-e(P)}. Let e_r*(d)=limsup_{deg(P)=d, h(P)-> +infty} e(P), where the limsup is taken over the reducible monic... | \section{Introduction}
All the polynomials that we deal with in this paper have integer coefficients. For any such polynomial,
we can look at how close two of its real or complex roots can be. Since we can always find polynomials
with distinct roots as close as desired, we need to introduce some measure of size for ... | {
"timestamp": "2017-03-08T02:01:28",
"yymm": "1703",
"arxiv_id": "1703.02120",
"language": "en",
"url": "https://arxiv.org/abs/1703.02120",
"abstract": "We study root separation of reducible monic integer polynomials of odd degree. Let h(P) be the naive height, sep(P) the minimal distance between two disti... |
https://arxiv.org/abs/1903.04929 | The Regge symmetry, confocal conics, and the Schläfli formula | The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in mathematical physics. Here we give a simple geometric proof of Regge symmetries in Euclidean, spherical, and hyperbolic geom... | \section{Introduction}
The goal of this article is to give an elementary proof of the following theorem known as the \emph{Regge symmetry}.
\begin{thm}
\label{thm:Regge}
Let $\Delta$ be a spherical, hyperbolic, or Euclidean tetrahedron with edge lengths $x$, $y$, $a$, $b$, $c$, $d$
as shown in Figure \ref{fig:TwoTetra... | {
"timestamp": "2019-03-14T01:14:06",
"yymm": "1903",
"arxiv_id": "1903.04929",
"language": "en",
"url": "https://arxiv.org/abs/1903.04929",
"abstract": "The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as ... |
https://arxiv.org/abs/1011.4247 | Minimal free resolutions for certain affine monomial curve | Given an arbitrary field k and an arithmetic sequence of positive integers m_0<...<m_n, we consider the affine monomial curve parameterized by X_0=t^{m_0},...,X_n=t^{m_n}. In this paper, we conjecture that the Betti numbers of its coordinate ring are completely determined by n and the value of m_0 modulo n. We first sh... | \section*{Introduction}
Let $k$ denote an arbitrary field and $R$ be the polynomial ring
$k[X_0,\ldots,X_n]$. Consider the $k$-algebra homomorphism $\varphi:R\rightarrow k[t]$ given
by $\varphi(X_i)=t^{m_i}$, $i=0,\ldots,n$. Then the ideal
${\mathcal P}:=\ker{\varphi}\subset R$ is the defining ideal of the monomial
cur... | {
"timestamp": "2010-11-19T02:02:22",
"yymm": "1011",
"arxiv_id": "1011.4247",
"language": "en",
"url": "https://arxiv.org/abs/1011.4247",
"abstract": "Given an arbitrary field k and an arithmetic sequence of positive integers m_0<...<m_n, we consider the affine monomial curve parameterized by X_0=t^{m_0},.... |
https://arxiv.org/abs/1303.0443 | Euler elasticae in the plane and the Whitney--Graustein theorem | In this paper, we apply classical energy principles to Euler elasticae, i.e., closed C^2 curves in the plane supplied with the Euler functional U (the integral of the square of the curvature along the curve). We study the critical points of U, find the shapes of the curves corresponding to these critical points and sho... | \section*{Introduction}
The aim of this paper is to test how the energy functional approach works on
the moduli space of all regular $\mathcal C^2$ curves in the plane $\mathbb R^2$
in the case of the Euler functional
$$
U(\gamma) = \int _0^{2\pi}\, \big( \kappa(\gamma(s)) \big)^2 ds,
$$
where $\gamma:\mathbb S^1 \to ... | {
"timestamp": "2013-03-05T02:01:55",
"yymm": "1303",
"arxiv_id": "1303.0443",
"language": "en",
"url": "https://arxiv.org/abs/1303.0443",
"abstract": "In this paper, we apply classical energy principles to Euler elasticae, i.e., closed C^2 curves in the plane supplied with the Euler functional U (the integ... |
https://arxiv.org/abs/2207.09410 | An Elementary Proof of a Theorem of Hardy and Ramanujan | Let $Q(n)$ denote the number of integers $1 \leq q \leq n$ whose prime factorization $q= \prod^{t}_{i=1}p^{a_i}_i$ satisfies $a_1\geq a_2\geq \ldots \geq a_t$. Hardy and Ramanujan proved that $$ \log Q(n) \sim \frac{2\pi}{\sqrt{3}} \sqrt{\frac{\log(n)}{\log\log(n)}}\;. $$ Before proving the above precise asymptotic for... | \section{Introduction}
Let $\ell_k=p_1\cdot p_2 \cdots p_k$ denote the product of the first $k$ prime numbers,
and take $\mathcal{Q}$ to be the set of integers $q$ which can be expressed as $q=\ell_1^{b_1}\cdot \ell_2^{b_2}\cdots \ell_{t}^{b_t}$,
for some $t \geq 1$ and sequence of non-negative integers $b_1,\ldots,b_... | {
"timestamp": "2022-07-20T02:22:44",
"yymm": "2207",
"arxiv_id": "2207.09410",
"language": "en",
"url": "https://arxiv.org/abs/2207.09410",
"abstract": "Let $Q(n)$ denote the number of integers $1 \\leq q \\leq n$ whose prime factorization $q= \\prod^{t}_{i=1}p^{a_i}_i$ satisfies $a_1\\geq a_2\\geq \\ldots... |
https://arxiv.org/abs/1703.08270 | Algebraic properties of toric rings of graphs | Let $G = (V,E)$ be a simple graph. We investigate the Cohen-Macaulayness and algebraic invariants, such as the Castelnuovo-Mumford regularity and the projective dimension, of the toric ring $k[G]$ via those of toric rings associated to induced subgraphs of $G$. | \section{Introduction}
Let $G = (V,E)$ be a simple graph over the vertex set $V$ and with edge set $E \subseteq 2^V$. Let $k$ be an arbitrary field, and identify the vertices and edges of $G$ with the variables in polynomial rings $k[V]$ and $k[E]$, respectively. The \emph{toric ring} associated to $G$, denoted by $k[... | {
"timestamp": "2017-07-13T02:05:44",
"yymm": "1703",
"arxiv_id": "1703.08270",
"language": "en",
"url": "https://arxiv.org/abs/1703.08270",
"abstract": "Let $G = (V,E)$ be a simple graph. We investigate the Cohen-Macaulayness and algebraic invariants, such as the Castelnuovo-Mumford regularity and the proj... |
https://arxiv.org/abs/2001.01212 | Lectures on the Calabi-Yau Landscape | In these lecture notes, we survey the landscape of Calabi-Yau threefolds, and the use of machine learning to explore it. We begin with the compact portion of the landscape, focusing in particular on complete intersection Calabi-Yau varieties (CICYs) and elliptic fibrations. Non-compact Calabi-Yau manifolds are manifest... | \section{Introduction}
Superstring theories demand our spacetime dimension to be 10, which means we should reduce them to an effectively 4-dimensional theory. The standard solution of string compactification, as a generalization of Kaluza-Klein compactification, renders the extra six dimensions Calabi-Yau (CY). Thus, ... | {
"timestamp": "2020-02-05T02:16:48",
"yymm": "2001",
"arxiv_id": "2001.01212",
"language": "en",
"url": "https://arxiv.org/abs/2001.01212",
"abstract": "In these lecture notes, we survey the landscape of Calabi-Yau threefolds, and the use of machine learning to explore it. We begin with the compact portion... |
https://arxiv.org/abs/1810.13312 | What is the probability that two elements of a finite ring have product zero? | In this paper we consider the probability that two elements of a finite ring have product zero. We find bounds of this probability of a finite commutative ring with identity 1. The explicit computations for the ring $\mathbf{Z}_n$, the ring of integers modulo $n$, have been obtained. | \section{Introduction}
The problem of finding the probability $P(G)$ that two elements of a finite group $G$ commute was considered by Gustafson \cite{MR02944871}. He showed that $P(G)\leq 5/8$. For more studies about probability and group theory, see\cite{rusin1979probability, lescot1988degre}.
I. Beck in 1988 \... | {
"timestamp": "2018-11-01T01:12:43",
"yymm": "1810",
"arxiv_id": "1810.13312",
"language": "en",
"url": "https://arxiv.org/abs/1810.13312",
"abstract": "In this paper we consider the probability that two elements of a finite ring have product zero. We find bounds of this probability of a finite commutative... |
https://arxiv.org/abs/1601.01457 | New asymptotic results in principal component analysis | Let $X$ be a mean zero Gaussian random vector in a separable Hilbert space ${\mathbb H}$ with covariance operator $\Sigma:={\mathbb E}(X\otimes X).$ Let $\Sigma=\sum_{r\geq 1}\mu_r P_r$ be the spectral decomposition of $\Sigma$ with distinct eigenvalues $\mu_1>\mu_2> \dots$ and the corresponding spectral projectors $P_... | \section{Introduction}
Let $X,X_1,\dots, X_n, \dots$ be i.i.d. random variables sampled from a Gaussian distribution in
a separable Hilbert space ${\mathbb H}$ with zero mean and covariance operator $\Sigma:={\mathbb E}X\otimes X$
and let $\hat \Sigma=\hat \Sigma_n := n^{-1}\sum_{j=1}^n X_j\otimes X_j$ denote the sam... | {
"timestamp": "2016-01-08T02:06:54",
"yymm": "1601",
"arxiv_id": "1601.01457",
"language": "en",
"url": "https://arxiv.org/abs/1601.01457",
"abstract": "Let $X$ be a mean zero Gaussian random vector in a separable Hilbert space ${\\mathbb H}$ with covariance operator $\\Sigma:={\\mathbb E}(X\\otimes X).$ L... |
https://arxiv.org/abs/2103.15850 | An upper bound on the size of Sidon sets | In this entry point into the subject, combining two elementary proofs, we decrease the gap between the upper and lower bounds by $0.2\%$ in a classical combinatorial number theory problem. We show that the maximum size of a Sidon set of $\{ 1, 2, \ldots, n\}$ is at most $\sqrt{n}+ 0.998n^{1/4}$ for sufficiently large $... | \section{History}
In 1932 S.~Sidon asked a question of a fellow student P. Erd{\H{o}}s.
Their advisor was L.~Fej\'er, an outstanding mathematician (cf.~Fej\'er kernel) working on summability of infinite series, who had a number of outstanding students who contributed to mathematical analysis
(M.~Fekete 1909 [Fekete'... | {
"timestamp": "2021-06-18T02:07:35",
"yymm": "2103",
"arxiv_id": "2103.15850",
"language": "en",
"url": "https://arxiv.org/abs/2103.15850",
"abstract": "In this entry point into the subject, combining two elementary proofs, we decrease the gap between the upper and lower bounds by $0.2\\%$ in a classical c... |
https://arxiv.org/abs/1802.09449 | Maximal Cocliques in $\operatorname{PSL}_2(q)$ | The generating graph of a finite group is a structure which can be used to encode certain information about the group. It was introduced by Liebeck and Shalev and has been further investigated by Lucchini, Maróti, Roney-Dougal and others. We investigate maximal cocliques (totally disconnected induced subgraphs of the g... | \section{Introduction}
It is a well-known result of Steinberg (and later others) that every finite simple group may be generated by just 2 elements, and from \cite{GenProb,KantorLubotzky} we know that if we pick two elements of the group at random then the chance that they will generate the group is high. This then mo... | {
"timestamp": "2019-01-15T02:30:26",
"yymm": "1802",
"arxiv_id": "1802.09449",
"language": "en",
"url": "https://arxiv.org/abs/1802.09449",
"abstract": "The generating graph of a finite group is a structure which can be used to encode certain information about the group. It was introduced by Liebeck and Sh... |
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