url stringlengths 31 38 | title stringlengths 7 229 | abstract stringlengths 44 2.87k | text stringlengths 319 2.51M | meta dict |
|---|---|---|---|---|
https://arxiv.org/abs/2007.02590 | Angle sums of random polytopes | For two families of random polytopes we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any given dimension of the polytope under consideration. As special cases, we compute the expected sums of internal and external angles at all faces of any fixed dimension... | \section{Introduction}
\subsection{Angles and face numbers}
For a convex polytope $P\subset\mathbb{R}^d$ denote by $\mathcal{F}(P)$ the set of its faces including $P$ itself. The classical Euler relation (see, e.g.,~\cite[Chapter~8]{bG03}) states that for every polytope $P$,
\begin{align}\label{916}
\sum_{F\in\mat... | {
"timestamp": "2020-07-16T02:19:26",
"yymm": "2007",
"arxiv_id": "2007.02590",
"language": "en",
"url": "https://arxiv.org/abs/2007.02590",
"abstract": "For two families of random polytopes we compute explicitly the expected sums of the conic intrinsic volumes and the Grassmann angles at all faces of any g... |
https://arxiv.org/abs/2005.00566 | Relationships between the number of inputs and other complexity measures of Boolean functions | We generalize and extend the ideas in a recent paper of Chiarelli, Hatami and Saks to prove new bounds on the number of relevant variables for boolean functions in terms of a variety of complexity measures. Our approach unifies and refines all previously known bounds of this type. We also improve Nisan and Szegedy's we... | \section{Introduction}
Is a boolean function $f: \{0,1\}^n \to \{0,1\}$ necessarily ``complex'' simply because it takes many input variables? (Of course one has to count only the \textit{relevant} inputs, ignoring any dummy variables which $f$ does not actually need.) In 1983, Simon \cite{Simon} answered this question... | {
"timestamp": "2020-05-05T02:01:04",
"yymm": "2005",
"arxiv_id": "2005.00566",
"language": "en",
"url": "https://arxiv.org/abs/2005.00566",
"abstract": "We generalize and extend the ideas in a recent paper of Chiarelli, Hatami and Saks to prove new bounds on the number of relevant variables for boolean fun... |
https://arxiv.org/abs/1709.01415 | A fractal shape optimization problem in branched transport | We investigate the following question: what is the set of unit volume which can be best irrigated starting from a single source at the origin, in the sense of branched transport? We may formulate this question as a shape optimization problem and prove existence of solutions, which can be considered as a sort of "unit b... | \section*{Introduction}
Given two probability measures $\mu,\nu$ on $\mathbb{R}^d$, a classical optimization problem amounts to finding a connection between the two measures which has minimal cost. In branched transport, such a connection will be performed along a $1$-dimensional structure such that the cost for movin... | {
"timestamp": "2018-01-01T02:04:00",
"yymm": "1709",
"arxiv_id": "1709.01415",
"language": "en",
"url": "https://arxiv.org/abs/1709.01415",
"abstract": "We investigate the following question: what is the set of unit volume which can be best irrigated starting from a single source at the origin, in the sens... |
https://arxiv.org/abs/2107.02724 | The proportion of derangements characterizes the symmetric and alternating groups | Let $G$ be a subgroup of the symmetric group $S_n$. If the proportion of fixed-point-free elements in $G$ (or a coset) equals the proportion of fixed-point-free elements in $S_n$, then $G=S_n$. The analogue for $A_n$ holds if $n \ge 7$. We give an application to monodromy groups. | \section{Introduction}
\subsection{Derangements in permutation groups}
Motivated by an application to monodromy groups, we prove the following.
\begin{theorem}
\label{T:main}
Let $G$ be a subgroup of the symmetric group $S_n$ for some $n \ge 1$.
Let $C$ be a coset of $G$ in $S_n$.
If
\begin{equation}
\label{E:C}
\fr... | {
"timestamp": "2021-10-20T02:02:19",
"yymm": "2107",
"arxiv_id": "2107.02724",
"language": "en",
"url": "https://arxiv.org/abs/2107.02724",
"abstract": "Let $G$ be a subgroup of the symmetric group $S_n$. If the proportion of fixed-point-free elements in $G$ (or a coset) equals the proportion of fixed-poin... |
https://arxiv.org/abs/0904.2115 | Colorful Strips | Given a planar point set and an integer $k$, we wish to color the points with $k$ colors so that any axis-aligned strip containing enough points contains all colors. The goal is to bound the necessary size of such a strip, as a function of $k$. We show that if the strip size is at least $2k{-}1$, such a coloring can al... | \section{Introduction}
In this paper, we are interested in coloring finite point sets in $\Re^d$ so that any region
bounded by two parallel axis-aligned hyperplanes, that contains at least
some fixed number of points, also contains a point of each color.
To rephrase, we study the following problem:
What is the minimu... | {
"timestamp": "2009-04-14T15:00:42",
"yymm": "0904",
"arxiv_id": "0904.2115",
"language": "en",
"url": "https://arxiv.org/abs/0904.2115",
"abstract": "Given a planar point set and an integer $k$, we wish to color the points with $k$ colors so that any axis-aligned strip containing enough points contains al... |
https://arxiv.org/abs/2008.09107 | Greedoids from flames | A digraph $ D $ with $ r\in V(D) $ is an $ r $-flame if for every $ {v\in V(D)-r} $, the in-degree of $ v $ is equal to the local edge-connectivity $ \lambda_D(r,v) $. We show that for every digraph $ D $ and $ r\in V(D) $, the edge sets of the $ r $-flame subgraphs of $ D $ form a greedoid. Our method yields a new pro... | \section{Introduction}
Subgraphs preserving some connectivity properties while having as few edges as possible have been a subject of
interest since the beginning of graph theory. Suppose that $ D $ is a digraph with $ r\in V(D) $ and let us denote the local
edge-connectivity\footnote{The local edge-connectivity fr... | {
"timestamp": "2021-04-01T02:25:17",
"yymm": "2008",
"arxiv_id": "2008.09107",
"language": "en",
"url": "https://arxiv.org/abs/2008.09107",
"abstract": "A digraph $ D $ with $ r\\in V(D) $ is an $ r $-flame if for every $ {v\\in V(D)-r} $, the in-degree of $ v $ is equal to the local edge-connectivity $ \\... |
https://arxiv.org/abs/0807.3415 | On the spectral gap of the Kac walk and other binary collision processes | We give a new and elementary computation of the spectral gap of the Kac walk on the N-sphere. The result is obtained as a by-product of a more general observation which allows to reduce the analysis of the spectral gap of an N-component system to that of the same system for N=3. The method applies to a number of random... | \section{Introduction}
The following model for energy preserving binary collisions
was introduced by M.\ Kac in \cite{Kac}.
Let $\nu$ denote the uniform probability measure on the sphere
$$
S^{N-1} = \{\eta\in\bbR^N\,:\;\sum_{i=1}^N\eta_i^2=1\}\,,
$$
and consider the $\nu$--reversible
Markov process on $S^{N-1}$ with... | {
"timestamp": "2008-07-22T10:39:40",
"yymm": "0807",
"arxiv_id": "0807.3415",
"language": "en",
"url": "https://arxiv.org/abs/0807.3415",
"abstract": "We give a new and elementary computation of the spectral gap of the Kac walk on the N-sphere. The result is obtained as a by-product of a more general obser... |
https://arxiv.org/abs/1707.04018 | Hardy's inequality in a limiting case on general bounded domains | In this paper, we study Hardy's inequality in a limiting case: $$\int_{\Omega} |\nabla u |^N dx \ge C_N(\Omega) \int_{\Omega} \frac{|u(x)|^N}{|x|^N \left(\log \frac{R}{|x|} \right)^N} dx $$ for functions $u \in W^{1,N}_0(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with $R = \sup_{x \in \Omega} |x|$. ... | \section{Introduction}
The classical Hardy inequality in one space dimension states that
\begin{equation}
\label{Hardy_1D}
\int_0^{\infty} |u'(t)|^p \, dt \ge \( \frac{p-1}{p} \)^p \int_0^{\infty} \frac{|u(t)|^p}{t^p} \, dt
\end{equation}
holds for all $u \in W^{1,p}_0(0, +\infty)$ where $1 < p < \infty$.
Thi... | {
"timestamp": "2018-03-09T02:04:56",
"yymm": "1707",
"arxiv_id": "1707.04018",
"language": "en",
"url": "https://arxiv.org/abs/1707.04018",
"abstract": "In this paper, we study Hardy's inequality in a limiting case: $$\\int_{\\Omega} |\\nabla u |^N dx \\ge C_N(\\Omega) \\int_{\\Omega} \\frac{|u(x)|^N}{|x|^... |
https://arxiv.org/abs/2205.14260 | A Note on the Fibonacci Sequence and Schreier-type Sets | A set $A$ of positive integers is said to be Schreier if either $A = \emptyset$ or $\min A\ge |A|$. We give a bijective map to prove the recurrence of the sequence $(|\mathcal{K}_{n, p, q}|)_{n=1}^\infty$ (for fixed $p\ge 1$ and $q\ge 2$), where $$\mathcal{K}_{n, p, q} \ = \ \{A\subset \{1, \ldots, n\}\,:\, \mbox{eithe... | \section{Introduction}
The Fibonacci sequence is defined as follows: $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n\ge 3$.
There has been many nice combinatorial interpretations of $(F_n)_{n=1}^\infty$ (see \seqnum{A000045} for a quick summary), and the Fibonacci sequence often appears unexpectedly and satisfyin... | {
"timestamp": "2022-05-31T02:03:31",
"yymm": "2205",
"arxiv_id": "2205.14260",
"language": "en",
"url": "https://arxiv.org/abs/2205.14260",
"abstract": "A set $A$ of positive integers is said to be Schreier if either $A = \\emptyset$ or $\\min A\\ge |A|$. We give a bijective map to prove the recurrence of ... |
https://arxiv.org/abs/2107.04554 | Whitney's Extension Theorem and the finiteness principle for curves in the Heisenberg group | Consider the sub-Riemannian Heisenberg group $\mathbb{H}$. In this paper, we answer the following question: given a compact set $K \subseteq \mathbb{R}$ and a continuous map $f:K \to \mathbb{H}$, when is there a horizontal $C^m$ curve $F:\mathbb{R} \to \mathbb{H}$ such that $F|_K = f$? Whitney originally answered this ... | \section{Introduction}
Fix a compact set $K \subseteq \mathbb{R}^n$ and a continuous function $f:K \to \mathbb{R}$,
and consider the following question:
\smallskip
\noindent
\textbf{Whitney's question.} When is there some $F \in C^m(\mathbb{R}^n)$ such that $F|_K = f$?
\smallskip
Whenever such an $F$ exists, we w... | {
"timestamp": "2021-07-12T02:22:50",
"yymm": "2107",
"arxiv_id": "2107.04554",
"language": "en",
"url": "https://arxiv.org/abs/2107.04554",
"abstract": "Consider the sub-Riemannian Heisenberg group $\\mathbb{H}$. In this paper, we answer the following question: given a compact set $K \\subseteq \\mathbb{R}... |
https://arxiv.org/abs/2302.03715 | Decompositions and Terracini loci of cubic forms of low rank | We study Waring rank decompositions for cubic forms of rank $n+2$ in $n+1$ variables. In this setting, we prove that if a concise form has more than one non-redundant decomposition of length $n+2$, then all such decompositions share at least $n-3$ elements, and the remaining elements lie in a special configuration. Fol... | \section{Introduction}
For an integer $n$, let $V$ be a complex vector space of dimension $n+1$ and let $\mathbb{P}^n = \mathbb{P} V$ be the projective space of lines in $V$. Identify $V$ with the space of linear forms on $V^*$ and let $S^d V$ be the space of homogeneous polynomials of degree $d$ on $V^*$. The Waring ... | {
"timestamp": "2023-02-09T02:00:27",
"yymm": "2302",
"arxiv_id": "2302.03715",
"language": "en",
"url": "https://arxiv.org/abs/2302.03715",
"abstract": "We study Waring rank decompositions for cubic forms of rank $n+2$ in $n+1$ variables. In this setting, we prove that if a concise form has more than one n... |
https://arxiv.org/abs/1010.4101 | Boundary-twisted normal form and the number of elementary moves to unknot | Suppose $K$ is an unknot lying in the 1-skeleton of a triangulated 3-manifold with $t$ tetrahedra. Hass and Lagarias showed there is an upper bound, depending only on $t$, for the minimal number of elementary moves to untangle $K$. We give a simpler proof, utilizing a normal form for surfaces whose boundary is containe... | \section{Introduction}
Suppose $M$ is a triangulated, compact 3-manifold with $t$ tetrahedra and $K$ is an unknot in the 1-skeleton. Recall that $K$ can be isotoped in $M$ using polygonal moves across triangles called \emph{elementary moves}. J. Hass and J. Lagarias obtained an upper bound of $2^{10^7t}$ on the mini... | {
"timestamp": "2010-10-21T02:00:59",
"yymm": "1010",
"arxiv_id": "1010.4101",
"language": "en",
"url": "https://arxiv.org/abs/1010.4101",
"abstract": "Suppose $K$ is an unknot lying in the 1-skeleton of a triangulated 3-manifold with $t$ tetrahedra. Hass and Lagarias showed there is an upper bound, dependi... |
https://arxiv.org/abs/1910.02856 | Combinatorial considerations on the invariant measure of a stochastic matrix | The invariant measure is a fundamental object in the theory of Markov processes. In finite dimensions a Markov process is defined by transition rates of the corresponding stochastic matrix. The Markov tree theorem provides an explicit representation of the invariant measure of a stochastic matrix. In this note, we give... | \section{A stochastic matrix and its invariant measure}
We consider a finite state space $Z:=\{1, \dots, N\}$ where the number of species $N\in\mathbb N$ is fixed. A stochastic matrix $M=(m_{ij})_{i,j=1, \dots N}$ (also called Markov operator) on $\mathbb{R}^N$ is a real matrix with non-negative entries and which sati... | {
"timestamp": "2019-10-08T02:28:16",
"yymm": "1910",
"arxiv_id": "1910.02856",
"language": "en",
"url": "https://arxiv.org/abs/1910.02856",
"abstract": "The invariant measure is a fundamental object in the theory of Markov processes. In finite dimensions a Markov process is defined by transition rates of t... |
https://arxiv.org/abs/1403.8139 | A Generalization of Tokuyama's Formula to the Hall-Littlewood Polynomials | A theorem due to Tokuyama expresses Schur polynomials in terms of Gelfand-Tsetlin patterns, providing a deformation of the Weyl character formula and two other classical results, Stanley's formula for the Schur $q$-polynomials and Gelfand's parametrization for the Schur polynomial. We generalize Tokuyama's formula to t... |
\section{Introduction}
\indent Schur polynomials, a special class of symmetric polynomials, play an important role in representation theory. They encode the characters of irreducible representations of general linear groups, which may be computed via the Weyl character formula. Tokuyama \cite{tokuyama} gave a deforma... | {
"timestamp": "2014-09-26T02:04:21",
"yymm": "1403",
"arxiv_id": "1403.8139",
"language": "en",
"url": "https://arxiv.org/abs/1403.8139",
"abstract": "A theorem due to Tokuyama expresses Schur polynomials in terms of Gelfand-Tsetlin patterns, providing a deformation of the Weyl character formula and two ot... |
https://arxiv.org/abs/2006.16562 | Nonlinear Matrix Concentration via Semigroup Methods | Matrix concentration inequalities provide information about the probability that a random matrix is close to its expectation with respect to the $l_2$ operator norm. This paper uses semigroup methods to derive sharp nonlinear matrix inequalities. In particular, it is shown that the classic Bakry-Émery curvature criteri... | \section{Motivation}
Matrix concentration inequalities describe the probability that a random matrix is close to its expectation,
with deviations measured in the $\ell_2$ operator norm.
The basic models---sums of independent random matrices
and matrix-valued martingales---have been studied extensively,
and they admit ... | {
"timestamp": "2021-01-08T02:00:57",
"yymm": "2006",
"arxiv_id": "2006.16562",
"language": "en",
"url": "https://arxiv.org/abs/2006.16562",
"abstract": "Matrix concentration inequalities provide information about the probability that a random matrix is close to its expectation with respect to the $l_2$ ope... |
https://arxiv.org/abs/1908.02834 | Curves orthogonal to a vector field in Euclidean spaces | A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [Chen 2017, Tamkang J. Math. 48, 209] to any space dimension: we prove that rectifying curves are geodesics on hypercones. We later use this association to characterize rectifying curve... | \section{Introduction}
In Euclidean space we may ask ``When does the position vector of a regular curve always lie orthogonal to a vector field?''. In other words, the problem consists in characterizing the curves $\alpha:I\to\mathbb{E}^{m+2}$ for which $\langle\alpha-p,\mathbf{V}\rangle=0$ in $I$, where $p$ is a cons... | {
"timestamp": "2020-04-17T02:04:37",
"yymm": "1908",
"arxiv_id": "1908.02834",
"language": "en",
"url": "https://arxiv.org/abs/1908.02834",
"abstract": "A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [Chen 2017, Tamkang... |
https://arxiv.org/abs/2002.02021 | A dichotomy for bounded degree graph homomorphisms with nonnegative weights | We consider the complexity of counting weighted graph homomorphisms defined by a symmetric matrix $A$. Each symmetric matrix $A$ defines a graph homomorphism function $Z_A(\cdot)$, also known as the partition function. Dyer and Greenhill [10] established a complexity dichotomy of $Z_A(\cdot)$ for symmetric $\{0, 1\}$-m... | \section{Hardness for $Z_A(\cdot)$ on simple graphs for real symmetric $A$}\label{sec:Goldberg-et-al-2010-dichotomy}
There is a more direct approach to prove
the \#P-hardness part of the Bulatov-Grohe dichotomy
(Theorem~\ref{thm:Bulatov-Grohe}) for simple graphs.
Although this method does not handle degree-boundedne... | {
"timestamp": "2020-02-07T02:02:28",
"yymm": "2002",
"arxiv_id": "2002.02021",
"language": "en",
"url": "https://arxiv.org/abs/2002.02021",
"abstract": "We consider the complexity of counting weighted graph homomorphisms defined by a symmetric matrix $A$. Each symmetric matrix $A$ defines a graph homomorph... |
https://arxiv.org/abs/2202.12045 | Pushing Blocks by Sweeping Lines | We investigate the reconfiguration of $n$ blocks, or "tokens", in the square grid using "line pushes". A line push is performed from one of the four cardinal directions and pushes all tokens that are maximum in that direction to the opposite direction. Tokens that are in the way of other tokens are displaced in the sam... |
\subsection{All Feasible Permutations Are Even}\label{s:4.1}
In this section, we will prove Theorem~\ref{thm:even}, which states that only \emph {even} permutations are possible in the Permutation Puzzle.
For a labeled canonical configuration $C$, let $C'$ be an extension of the labeling where also the empty cell... | {
"timestamp": "2022-03-29T02:54:00",
"yymm": "2202",
"arxiv_id": "2202.12045",
"language": "en",
"url": "https://arxiv.org/abs/2202.12045",
"abstract": "We investigate the reconfiguration of $n$ blocks, or \"tokens\", in the square grid using \"line pushes\". A line push is performed from one of the four c... |
https://arxiv.org/abs/2207.04261 | Fuzzy Clustering by Hyperbolic Smoothing | We propose a novel method for building fuzzy clusters of large data sets, using a smoothing numerical approach. The usual sum-of-squares criterion is relaxed so the search for good fuzzy partitions is made on a continuous space, rather than a combinatorial space as in classical methods \cite{Hartigan}. The smoothing al... | \section{Introduction}
Methods for making groups from data sets are usually based on the idea of disjoint sets, such as the classical crisp clustering. The most well known are hierarchical and $k$-means \cite{Hartigan}, whose resulting clusters are sets will no intersection. However, this restriction may not be natura... | {
"timestamp": "2022-07-12T02:09:01",
"yymm": "2207",
"arxiv_id": "2207.04261",
"language": "en",
"url": "https://arxiv.org/abs/2207.04261",
"abstract": "We propose a novel method for building fuzzy clusters of large data sets, using a smoothing numerical approach. The usual sum-of-squares criterion is rela... |
https://arxiv.org/abs/2211.17055 | Latitudinal regionalization of rotating spherical shell convection | Convection occurs ubiquitously on and in rotating geophysical and astrophysical bodies. Prior spherical shell studies have shown that the convection dynamics in polar regions can differ significantly from the lower latitude, equatorial dynamics. Yet most spherical shell convective scaling laws use globally-averaged qua... | \section{Introduction}
It has long been known that spherical shell rotating convection significantly differs between the low latitudes
\citep[e.g.,][]{Busse77, Gillet06} situated outside the
axially-aligned cylinder that circumscribes the inner spherical
shell boundary (the tangent cylinder, TC)
and the higher lati... | {
"timestamp": "2022-12-01T02:17:07",
"yymm": "2211",
"arxiv_id": "2211.17055",
"language": "en",
"url": "https://arxiv.org/abs/2211.17055",
"abstract": "Convection occurs ubiquitously on and in rotating geophysical and astrophysical bodies. Prior spherical shell studies have shown that the convection dynam... |
https://arxiv.org/abs/2006.07013 | A Unified Analysis of Stochastic Gradient Methods for Nonconvex Federated Optimization | In this paper, we study the performance of a large family of SGD variants in the smooth nonconvex regime. To this end, we propose a generic and flexible assumption capable of accurate modeling of the second moment of the stochastic gradient. Our assumption is satisfied by a large number of specific variants of SGD in t... | \section{Introduction}
\label{sec:intro}
In this paper, we develop a general framework for studying and designing SGD-type methods for solving {\em nonconvex distributed/federated optimization problems} \cite{khirirat2018distributed, FEDLEARN, kairouz2019advances}. Given $m$ machines/workers/devices, each having acc... | {
"timestamp": "2020-06-15T02:11:45",
"yymm": "2006",
"arxiv_id": "2006.07013",
"language": "en",
"url": "https://arxiv.org/abs/2006.07013",
"abstract": "In this paper, we study the performance of a large family of SGD variants in the smooth nonconvex regime. To this end, we propose a generic and flexible a... |
https://arxiv.org/abs/1004.1855 | Fast Correlation Greeks by Adjoint Algorithmic Differentiation | We show how Adjoint Algorithmic Differentiation (AAD) allows an extremely efficient calculation of correlation Risk of option prices computed with Monte Carlo simulations. A key point in the construction is the use of binning to simultaneously achieve computational efficiency and accurate confidence intervals. We illus... | \section*{Forward and Adjoint Algorithmic Differentiation}
Both the Forward and Adjoint mode of AD aim at calculating the derivatives of
a computer implemented function. They differ by the direction of
propagation of the chain rule through the composition of instructions representing the function.
To illustrate this... | {
"timestamp": "2010-04-13T02:02:04",
"yymm": "1004",
"arxiv_id": "1004.1855",
"language": "en",
"url": "https://arxiv.org/abs/1004.1855",
"abstract": "We show how Adjoint Algorithmic Differentiation (AAD) allows an extremely efficient calculation of correlation Risk of option prices computed with Monte Car... |
https://arxiv.org/abs/1511.02838 | Boundness of $b_2$ for hyperkähler manifolds with vanishing odd-Betti numbers | We prove that $b_2$ is bounded for hyperkähler manifolds with vanishing odd-Betti numbers. The explicit upper boundary is conjectured. Following the method described by Sawon we prove that $b_2$ is bounded in dimension eight and ten in the case of vanishing odd-Betti numbers by 24 and 25 respectively. | \section{Introduction}
A Riemannian manifold $(M, g)$ is called hyperk\"ahler if it admits a
triple of a complex structures $I, J, K$ satisfying quaternionic relations and
K\"ahler with a respect to $g$. A hyperk\"ahler
manifold is always holomorphically symplectic. By the Yau Theorem \cite{Y}, a
hyperk\"ahler structu... | {
"timestamp": "2015-11-10T02:27:22",
"yymm": "1511",
"arxiv_id": "1511.02838",
"language": "en",
"url": "https://arxiv.org/abs/1511.02838",
"abstract": "We prove that $b_2$ is bounded for hyperkähler manifolds with vanishing odd-Betti numbers. The explicit upper boundary is conjectured. Following the metho... |
https://arxiv.org/abs/2212.12199 | On splitting of the normalizer of a maximal torus in finite groups of Lie type | Let $G$ be a finite group of Lie type and $T$ a maximal torus of $G$. In this paper we complete the study of the question of the existence of a complement for the torus $T$ in its algebraic normalizer $N(G,T)$. It is proved that every maximal torus of the group $G\in\{G_2(q), {}^2G_2(q), {}^3D_4(q)\}$ has a complement ... | \section{Introduction}
The splitting problem for the normalizer of a maximal torus was first formulated by J.~Tits~\cite{Tits}. Let $\overline{G}$ be a simple connected linear algebraic group over the algebraic closure of the prime field of characteristic $p$. Let $\sigma$ be a Steinberg endomorphism and $\overline{T}... | {
"timestamp": "2022-12-26T02:07:40",
"yymm": "2212",
"arxiv_id": "2212.12199",
"language": "en",
"url": "https://arxiv.org/abs/2212.12199",
"abstract": "Let $G$ be a finite group of Lie type and $T$ a maximal torus of $G$. In this paper we complete the study of the question of the existence of a complement... |
https://arxiv.org/abs/1908.04365 | On $q$-deformed real numbers | We associate a formal power series with integer coefficients to a positive real number, we interpret this series as a "$q$-analogue of a real." The construction is based on the notion of $q$-deformed rational number introduced inarXiv:1812.00170. Extending the construction to negative real numbers, we obtain certain La... | \section{Introduction}\label{IntSec}
We take a new and experimental route to introduce a certain version of ``$q$-deformed real numbers'',
extending $q$-deformations of rationals introduced in~\cite{MGOqR}.
Given a real number~$x\geq0$, we will construct a formal power series with integer coefficients associated with~... | {
"timestamp": "2019-10-08T02:21:28",
"yymm": "1908",
"arxiv_id": "1908.04365",
"language": "en",
"url": "https://arxiv.org/abs/1908.04365",
"abstract": "We associate a formal power series with integer coefficients to a positive real number, we interpret this series as a \"$q$-analogue of a real.\" The cons... |
https://arxiv.org/abs/1806.08844 | A linear state feedback switching rule for global stabilization of switched nonlinear systems about a nonequilibrium point | A switched equilibrium of a switched system of two subsystems is a such a point where the vector fields of the two subsystems point strictly towards one another. Using the concept of stable convex combination that was developed by Wicks-Peleties-DeCarlo (1998) for linear systems, Bolzern-Spinelli (2004) offered a desig... | \section{Introduction}\label{sec:int}
Using the concept of stable convex combination that was developed by Wicks et al \cite{eur} for linear systems, Bolzern-Spinelli \cite{bol} offered a design of a state feedback switching rule that is capable to stabilize an affine switched system\footnote{Bolzern-Spinelli \cite{bo... | {
"timestamp": "2018-06-26T02:01:47",
"yymm": "1806",
"arxiv_id": "1806.08844",
"language": "en",
"url": "https://arxiv.org/abs/1806.08844",
"abstract": "A switched equilibrium of a switched system of two subsystems is a such a point where the vector fields of the two subsystems point strictly towards one a... |
https://arxiv.org/abs/1301.5691 | The Dupire derivatives and Fréchet derivatives on continuous pathes | In this paper, we study the relation between Fréchet derivatives and Dupire derivatives, in which the latter are recently introduced by Dupire [4]. After introducing the definition of Fréchet derivatives for non-anticipative functionals, we prove that the Dupire derivatives and the extended Fréchet derivatives are cohe... | \section{Introduction}
Recently Dupire {\normalsize \cite{Dupire.B} introduced the functional
It\^{o}'s calculus, which was further developed in Cont and Fourni
\cite{Cont-1}-\cite{Cont-3}. }The key idea of Dupire
{\normalsize \cite{Dupire.B} is to introduce the new "local" derivatives,
i.e., horizontal derivati... | {
"timestamp": "2013-01-25T02:00:44",
"yymm": "1301",
"arxiv_id": "1301.5691",
"language": "en",
"url": "https://arxiv.org/abs/1301.5691",
"abstract": "In this paper, we study the relation between Fréchet derivatives and Dupire derivatives, in which the latter are recently introduced by Dupire [4]. After in... |
https://arxiv.org/abs/2006.12689 | Bounds for Combinatorial Types of Non-Attacking Riders | Given q non-attacking riders with r moves, the number of combinatorial types has not been found for r greater than 2 and q greater than 3. This paper aims to create upper and lower bound functions which can be applied to any q and r, regardless of size. | \section{Introduction/Background}
\large{\null\quad Consider first an n by n infinite chessboard upon which is placed a queen. Now, if a second queen is placed, we dictate that these queens cannot be in line of fire of each other. With this assumption, it is clear that the second piece can be placed in 8 different loca... | {
"timestamp": "2020-06-25T02:20:45",
"yymm": "2006",
"arxiv_id": "2006.12689",
"language": "en",
"url": "https://arxiv.org/abs/2006.12689",
"abstract": "Given q non-attacking riders with r moves, the number of combinatorial types has not been found for r greater than 2 and q greater than 3. This paper aims... |
https://arxiv.org/abs/1304.6782 | Minimal Residual Methods for Complex Symmetric, Skew Symmetric, and Skew Hermitian Systems | While there is no lack of efficient Krylov subspace solvers for Hermitian systems, there are few for complex symmetric, skew symmetric, or skew Hermitian systems, which are increasingly important in modern applications including quantum dynamics, electromagnetics, and power systems. For a large consistent complex symme... | \section{Introduction} \label{sec:intro}
Krylov subspace methods \red{for linear systems} are generally divided
into two classes: \red{those} for Hermitian matrices
(e.g.\red{,}\ \CG~\cite{HS52}, \MINRES~\cite{PS75},
\SYMMLQ~\cite{PS75}, \MINRES-\QLP~\cite{CPS11,CS12,CS12b,C06}) and
those for general matrices without... | {
"timestamp": "2014-01-14T02:17:18",
"yymm": "1304",
"arxiv_id": "1304.6782",
"language": "en",
"url": "https://arxiv.org/abs/1304.6782",
"abstract": "While there is no lack of efficient Krylov subspace solvers for Hermitian systems, there are few for complex symmetric, skew symmetric, or skew Hermitian sy... |
https://arxiv.org/abs/1905.11255 | Kernel Conditional Density Operators | We introduce a novel conditional density estimation model termed the conditional density operator (CDO). It naturally captures multivariate, multimodal output densities and shows performance that is competitive with recent neural conditional density models and Gaussian processes. The proposed model is based on a novel ... |
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"timestamp": "2019-10-30T01:15:45",
"yymm": "1905",
"arxiv_id": "1905.11255",
"language": "en",
"url": "https://arxiv.org/abs/1905.11255",
"abstract": "We introduce a novel conditional density estimation model termed the conditional density operator (CDO). It naturally captures multivariate, multimodal ou... |
https://arxiv.org/abs/1806.06403 | Geometric mean extension for data sets with zeros | There are numerous examples in different research fields where the use of the geometric mean is more appropriate than the arithmetic mean. However, the geometric mean has a serious limitation in comparison with the arithmetic mean. Means are used to summarize the information in a large set of values in a single number;... | \section{Introduction}
Increasingly, research generates large amounts of information. Yet it is hard to work with large data sets directly, hence it is necessary to reduce their complexity whilst maintaining essential information. The most common way in which data sets are summarised is the use of the statistical quan... | {
"timestamp": "2019-04-05T02:19:48",
"yymm": "1806",
"arxiv_id": "1806.06403",
"language": "en",
"url": "https://arxiv.org/abs/1806.06403",
"abstract": "There are numerous examples in different research fields where the use of the geometric mean is more appropriate than the arithmetic mean. However, the ge... |
https://arxiv.org/abs/1910.09901 | Parallel Stochastic Optimization Framework for Large-Scale Non-Convex Stochastic Problems | In this paper, we consider the problem of stochastic optimization, where the objective function is in terms of the expectation of a (possibly non-convex) cost function that is parametrized by a random variable. While the convergence speed is critical for many emerging applications, most existing stochastic optimization... | \chapter{Parallel Stochastic Optimization Framework}
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\usepa... | {
"timestamp": "2019-10-23T02:15:13",
"yymm": "1910",
"arxiv_id": "1910.09901",
"language": "en",
"url": "https://arxiv.org/abs/1910.09901",
"abstract": "In this paper, we consider the problem of stochastic optimization, where the objective function is in terms of the expectation of a (possibly non-convex) ... |
https://arxiv.org/abs/1406.5104 | On the Theory and Algorithm for rigorous discretization in applications of Information Theory | We identify fundamental issues with discretization when estimating information-theoretic quantities in the analysis of data. These difficulties are theoretical in nature and arise with discrete datasets carrying significant implications for the corresponding claims and results. Here we describe the origins of the metho... | \section{Results}
Mathematically, Shannon entropy of a variable $X$, defined as $H(X) = \sum_{i} -p_i \log (p_i)$ \cite{Shan}, is a well-defined quantity when $X$ is discrete, taking distinct values $i$ with probabilities $p_i$. This is not the case when $X$ is sampled from a continuous probability distribution where a... | {
"timestamp": "2014-06-24T02:13:23",
"yymm": "1406",
"arxiv_id": "1406.5104",
"language": "en",
"url": "https://arxiv.org/abs/1406.5104",
"abstract": "We identify fundamental issues with discretization when estimating information-theoretic quantities in the analysis of data. These difficulties are theoreti... |
https://arxiv.org/abs/1704.03892 | Approximating the Largest Root and Applications to Interlacing Families | We study the problem of approximating the largest root of a real-rooted polynomial of degree $n$ using its top $k$ coefficients and give nearly matching upper and lower bounds. We present algorithms with running time polynomial in $k$ that use the top $k$ coefficients to approximate the maximum root within a factor of ... | \section{Introduction}
For a non-negative vector ${{\bm{\mu}}}=(\mu_1,\dots,\mu_n)\in\mathbb{R}_+^n$, let $\chi_{{{\bm{\mu}}}}$ denote the unique monic polynomial with roots $\mu_1,\dots,\mu_n$:
\[ \chi_{{{\bm{\mu}}}}(x):=\prod_{i=1}^n(x-\mu_i). \]
Suppose that we do not know ${{\bm{\mu}}}$, but rather know the top ... | {
"timestamp": "2017-04-14T02:00:36",
"yymm": "1704",
"arxiv_id": "1704.03892",
"language": "en",
"url": "https://arxiv.org/abs/1704.03892",
"abstract": "We study the problem of approximating the largest root of a real-rooted polynomial of degree $n$ using its top $k$ coefficients and give nearly matching u... |
https://arxiv.org/abs/1510.07711 | Primes and polynomials with restricted digits | Let $q$ be a sufficiently large integer, and $a_0\in\{0,\dots,q-1\}$. We show there are infinitely many prime numbers which do not have the digit $a_0$ in their base $q$ expansion. Similar results are obtained for values of a polynomial (satisfying the necessary local conditions) and if multiple digits are excluded.Our... | \section{Introduction}\label{sec:Introduction}
Let $a_0\in\{0,\dots,q-1\}$ and let
\[\mathcal{A}=\Bigl\{\sum_{i\ge 0}n_i q^i: n_i\in\{0,\dots,q-1\}\backslash\{a_0\}\Bigr\}\]
be the set of numbers which have no digit equal to $a_0$ when written in base $q$. For fixed $q$, the number of elements of $\mathcal{A}$ which a... | {
"timestamp": "2015-10-28T01:04:49",
"yymm": "1510",
"arxiv_id": "1510.07711",
"language": "en",
"url": "https://arxiv.org/abs/1510.07711",
"abstract": "Let $q$ be a sufficiently large integer, and $a_0\\in\\{0,\\dots,q-1\\}$. We show there are infinitely many prime numbers which do not have the digit $a_0... |
https://arxiv.org/abs/2011.10130 | Binary Discrete Fourier Transform and its Inversion | A binary vector of length $N$ has elements that are either 0 or 1. We investigate the question of whether and how a binary vector of known length can be reconstructed from a limited set of its discrete Fourier transform (DFT) coefficients. A priori information that the vector is binary provides a powerful constraint. W... | \section{Introduction}
\label{sec:intro}
Super-resolution in imaging and signal processing is a subject of significant current importance. Fundamentally, the resolution limit is related to experimental inaccessibility of the spatial Fourier harmonics of an object beyond some band limit~\cite{maznev_2017_1}. The result... | {
"timestamp": "2021-10-05T02:24:15",
"yymm": "2011",
"arxiv_id": "2011.10130",
"language": "en",
"url": "https://arxiv.org/abs/2011.10130",
"abstract": "A binary vector of length $N$ has elements that are either 0 or 1. We investigate the question of whether and how a binary vector of known length can be r... |
https://arxiv.org/abs/0912.3814 | A Family of Recompositions of the Penrose Aperiodic Protoset and Its Dynamic Properties | This paper describes a recomposition of the rhombic Penrose aperiodic protoset due to Robert Ammann. We show that the three prototiles that result from the recomposition form an aperiodic protoset in their own right without adjacency rules. An interation process is defined on the space of Ammann tilings that produces a... | \section{Introduction}
The Penrose aperiodic tiles have been well-studied by deBruijn \cite{deB}, Lunnon and Pleasants \cite{LP}, and others. See Senechal \cite{Senechal} for a survey. This paper describes a recomposition of the rhombic Penrose aperiodic protoset defined by Robert Ammann in Gr\"unbaum and Shep... | {
"timestamp": "2009-12-18T22:36:06",
"yymm": "0912",
"arxiv_id": "0912.3814",
"language": "en",
"url": "https://arxiv.org/abs/0912.3814",
"abstract": "This paper describes a recomposition of the rhombic Penrose aperiodic protoset due to Robert Ammann. We show that the three prototiles that result from the ... |
https://arxiv.org/abs/1412.5398 | Nowhere-zero 5-flows on cubic graphs with oddness 4 | Tutte's 5-Flow Conjecture from 1954 states that every bridgeless graph has a nowhere-zero 5-flow. In 2004, Kochol proved that the conjecture is equivalent to its restriction on cyclically 6-edge connected cubic graphs. We prove that every cyclically 6-edge-connected cubic graph with oddness at most 4 has a nowhere-zero... | \section[]{Introduction}
An integer nowhere-zero $k$-flow on a graph $G$ is an assignment of a direction and a value of $\{1, \dots, (k-1)\}$ to each edge of $G$ such that the Kirchhoff's law is satisfied at every vertex of $G$. This is the most restrictive definition of a nowhere-zero $k$-flow. But it is equivale... | {
"timestamp": "2014-12-18T02:12:36",
"yymm": "1412",
"arxiv_id": "1412.5398",
"language": "en",
"url": "https://arxiv.org/abs/1412.5398",
"abstract": "Tutte's 5-Flow Conjecture from 1954 states that every bridgeless graph has a nowhere-zero 5-flow. In 2004, Kochol proved that the conjecture is equivalent t... |
https://arxiv.org/abs/1709.00748 | Recovery of the singularities of a potential from backscattering data in general dimension | We prove that in dimension $n\ge 2$ the main singularities of a complex potential $q$ having a certain a priori regularity are contained in the Born approximation $q_{B}$ constructed from backscattering data. This is archived using a new explicit formula for the multiple dispersion operators in the Fourier transform si... | \section{Introduction and main theorems}
The central problem in inverse scattering for the Schrödinger equation is to recover a potential $q(x)$, $x \in \RR^n$, from the scattering data, the so called scattering amplitude $u_\infty$. The scattering amplitude measures the far field response of the Hamiltonian $H:=-\Del... | {
"timestamp": "2019-01-18T02:01:06",
"yymm": "1709",
"arxiv_id": "1709.00748",
"language": "en",
"url": "https://arxiv.org/abs/1709.00748",
"abstract": "We prove that in dimension $n\\ge 2$ the main singularities of a complex potential $q$ having a certain a priori regularity are contained in the Born appr... |
https://arxiv.org/abs/1111.4866 | A strong form of the Quantitative Isoperimetric inequality | We give a refinement of the quantitative isoperimetric inequality. We prove that the isoperimetric gap controls not only the Fraenkel asymmetry but also the oscillation of the boundary. | \section{Introduction and statement of the results}
In recent years there has been a growing interest in the study of the stability of a large class of geometric and functional inequalities, such as the isoperimetric and the Sobolev inequality. After some early work going back to the beginning of last century the firs... | {
"timestamp": "2011-11-22T02:05:21",
"yymm": "1111",
"arxiv_id": "1111.4866",
"language": "en",
"url": "https://arxiv.org/abs/1111.4866",
"abstract": "We give a refinement of the quantitative isoperimetric inequality. We prove that the isoperimetric gap controls not only the Fraenkel asymmetry but also the... |
https://arxiv.org/abs/0710.1782 | Linear and nonlinear tails I: general results and perturbation theory | For nonlinear wave equations with a potential term we prove pointwise space-time decay estimates and develop a perturbation theory for small initial data. We show that the perturbation series has a positive convergence radius by a method which reduces the wave equation to an algebraic one. We demonstrate that already f... | \subsection{Proof{#1}}}
\newcommand{\textit{Proof:}\\}[1][]{\subsection{#1}}
\def\refa#1{(\ref{#1})}
\def$\rightarrow$ {$\rightarrow$ }
\def\rightarrow{\rightarrow}
\def\Rightarrow{\Rightarrow}
\def\mathcal{C}{\mathcal{C}}
\def\mathbb{N}{\mathbb{N}}
\def\mathbb{R}{\mathbb{R}}
\def\R_+^{1+3}{\mathbb{R}_+^{1+3}}
\def{... | {
"timestamp": "2009-03-04T12:58:55",
"yymm": "0710",
"arxiv_id": "0710.1782",
"language": "en",
"url": "https://arxiv.org/abs/0710.1782",
"abstract": "For nonlinear wave equations with a potential term we prove pointwise space-time decay estimates and develop a perturbation theory for small initial data. W... |
https://arxiv.org/abs/1901.08957 | Minimizing lattice structures for Morse potential energy in two and three dimensions | We investigate the local and global optimality of the triangular, square, simple cubic, face-centred-cubic (FCC), body-centred-cubic (BCC) lattices and the hexagonal-close-packing (HCP) structure for a potential energy per point generated by a Morse potential with parameters $(\alpha,r_0)$. The optimality of the triang... | \section{Introduction and main results}
A fundamental question in Mathematical Physics that has been actively investigated recently is the following ``Crystal Problem" (also called ``The crystallization conjecture" see e.g. \cite{RadinLowT,BlancLewin-2015}): Why are solids crystalline? Answering this question in a rig... | {
"timestamp": "2019-02-05T02:19:24",
"yymm": "1901",
"arxiv_id": "1901.08957",
"language": "en",
"url": "https://arxiv.org/abs/1901.08957",
"abstract": "We investigate the local and global optimality of the triangular, square, simple cubic, face-centred-cubic (FCC), body-centred-cubic (BCC) lattices and th... |
https://arxiv.org/abs/2210.03124 | Learning Transfer Operators by Kernel Density Estimation | Inference of transfer operators from data is often formulated as a classical problem that hinges on the Ulam method. The conventional description, known as the Ulam-Galerkin method, involves projecting onto basis functions represented as characteristic functions supported over a fine grid of rectangles. From this persp... |
\section{Introduction}
Transfer operators play a vital role in the global analysis of dynamical systems. Abundant data from dynamical systems make those operators popular in data-driven analysis methods in complex systems. Hence, numerically estimating the transfer operators through data is key to success in global an... | {
"timestamp": "2022-10-10T02:00:10",
"yymm": "2210",
"arxiv_id": "2210.03124",
"language": "en",
"url": "https://arxiv.org/abs/2210.03124",
"abstract": "Inference of transfer operators from data is often formulated as a classical problem that hinges on the Ulam method. The conventional description, known a... |
https://arxiv.org/abs/1507.07765 | Percolation and isoperimetry on roughly transitive graphs | In this paper we study percolation on a roughly transitive graph G with polynomial growth and isoperimetric dimension larger than one. For these graphs we are able to prove that p_c < 1, or in other words, that there exists a percolation phase. The main results of the article work for both dependent and independent per... | \section{Introduction}
Since its introduction by Broadbent and Hammersley in \cite{PSP:2048852}, the model of independent percolation has received major attention from the physical and mathematical communities.
From the perspective of applications, it has the potential to model several different systems, from the flow... | {
"timestamp": "2016-02-17T02:09:07",
"yymm": "1507",
"arxiv_id": "1507.07765",
"language": "en",
"url": "https://arxiv.org/abs/1507.07765",
"abstract": "In this paper we study percolation on a roughly transitive graph G with polynomial growth and isoperimetric dimension larger than one. For these graphs we... |
https://arxiv.org/abs/1912.04901 | Closed graphs and open maps | We offer a new perspective on the closed graph theorem and the open mapping theorem for separated barrelled spaces and fully complete spaces. | \section*{Introduction}
\medbreak
The `classical' closed graph theorem asserts that a linear function between two Banach spaces is continuous if its graph is closed; the `classical' open mapping theorem asserts that a surjective continuous linear function between Banach spaces is open. Although these two theore... | {
"timestamp": "2019-12-12T02:00:21",
"yymm": "1912",
"arxiv_id": "1912.04901",
"language": "en",
"url": "https://arxiv.org/abs/1912.04901",
"abstract": "We offer a new perspective on the closed graph theorem and the open mapping theorem for separated barrelled spaces and fully complete spaces.",
"subject... |
https://arxiv.org/abs/1610.06760 | Generalized Zalcman conjecture for some classes of analytic functions | For functions $f(z)= z+ a_2 z^2 + a_3 z^3 + \cdots$ in various subclasses of normalized analytic functions, we consider the problem of estimating the generalized Zalcman coefficient functional $\phi(f,n,m;\lambda):=|\lambda a_n a_m -a_{n+m-1}|$. For all real parameters $\lambda$ and $ \beta<1$, we provide the sharp upp... | \section{Introduction and Preliminaries}
Let $\mathcal{A}$ be the class of all normalized analytic functions of the form $f(z)= z+ a_2 z^2 + a_3 z^3 + \cdots$ defined on the open unit disc $\mathbb{D}$. The subclass of $\mathcal{A}$ consisting of univalent functions is denoted by $\mathcal{S}$. Let $\mathcal{S}_{\ma... | {
"timestamp": "2016-11-10T02:02:29",
"yymm": "1610",
"arxiv_id": "1610.06760",
"language": "en",
"url": "https://arxiv.org/abs/1610.06760",
"abstract": "For functions $f(z)= z+ a_2 z^2 + a_3 z^3 + \\cdots$ in various subclasses of normalized analytic functions, we consider the problem of estimating the gen... |
https://arxiv.org/abs/2204.14182 | On non-counital Frobenius algebras | A Frobenius algebra is a finite-dimensional algebra $A$ which comes equipped with a coassociative, counital comultiplication map $\Delta$ that is an $A$-bimodule map. Here, we examine comultiplication maps for generalizations of Frobenius algebras: finite-dimensional self-injective (quasi-Frobenius) algebras. We show t... | \section{Introduction} \label{sec:intro}
All algebraic structures in this work are over a field $\Bbbk$ of characteristic 0. Moreover, all algebras are finite-dimensional as $\Bbbk$-vector spaces, and are unital and associative.
This work is motivated by the various characterizations of Frobenius algebras in the lite... | {
"timestamp": "2022-05-02T02:20:57",
"yymm": "2204",
"arxiv_id": "2204.14182",
"language": "en",
"url": "https://arxiv.org/abs/2204.14182",
"abstract": "A Frobenius algebra is a finite-dimensional algebra $A$ which comes equipped with a coassociative, counital comultiplication map $\\Delta$ that is an $A$-... |
https://arxiv.org/abs/1310.5493 | Brooks' theorem on powers of graphs | We prove that for $k\geq 3$, the bound given by Brooks' theorem on the chromatic number of $k$-th powers of graphs of maximum degree $\Delta \geq 3$ can be lowered by 1, even in the case of online list coloring. | \section{Introduction}
\label{sec:intro}
A graph $G=(V,E)$ is \textit{$k$-colorable} if there is a way to color each vertex with an element of $\{1,\cdots,k\}$ so that no two adjacent vertices receive distinct colors. A generalization of $k$-colorability is \textit{list $k$-colorability} (or \textit{$k$-choosability}),... | {
"timestamp": "2013-10-22T02:11:57",
"yymm": "1310",
"arxiv_id": "1310.5493",
"language": "en",
"url": "https://arxiv.org/abs/1310.5493",
"abstract": "We prove that for $k\\geq 3$, the bound given by Brooks' theorem on the chromatic number of $k$-th powers of graphs of maximum degree $\\Delta \\geq 3$ can ... |
https://arxiv.org/abs/0803.0045 | Direct limits of infinite-dimensional Lie groups | Many infinite-dimensional Lie groups of interest can be expressed as a union of an ascending sequence of (finite- or infinite-dimensional) Lie groups. In this survey article, we compile general results concerning such ascending unions, describe the main classes of examples, and explain what the general theory tells us ... | \section{\,Introduction}
Many infinite-dimensional
Lie groups $G$ can be expressed as the union
$G=\bigcup_{n\in {\mathbb N}}\,G_n$
of a sequence $G_1\subseteq G_2\subseteq \cdots$
of (finite- or infinite-dimensional) Lie groups,
such that the inclusion maps $j_n\colon G_n\to G$
and $j_{m,n}\colon G_n\to G_m$ (for $n\l... | {
"timestamp": "2008-04-02T00:30:47",
"yymm": "0803",
"arxiv_id": "0803.0045",
"language": "en",
"url": "https://arxiv.org/abs/0803.0045",
"abstract": "Many infinite-dimensional Lie groups of interest can be expressed as a union of an ascending sequence of (finite- or infinite-dimensional) Lie groups. In th... |
https://arxiv.org/abs/2205.13656 | Finite difference schemes for the parabolic $p$-Laplace equation | We propose a new finite difference scheme for the degenerate parabolic equation \[ \partial_t u - \mbox{div}(|\nabla u|^{p-2}\nabla u) =f, \quad p\geq 2. \] Under the assumption that the data is Hölder continuous, we establish the convergence of the explicit-in-time scheme for the Cauchy problem provided a suitable sta... | \section{Introduction} \label{sec:intro}
Recently, a new monotone finite difference discretization of the $p$-Laplacian was introduced by the authors in \cite{dTLi22}. It is based on the mean value property presented in \cite{dTLi20, BS18}. The aim of this paper is to propose an explicit-in-time finite difference nume... | {
"timestamp": "2022-05-30T02:03:46",
"yymm": "2205",
"arxiv_id": "2205.13656",
"language": "en",
"url": "https://arxiv.org/abs/2205.13656",
"abstract": "We propose a new finite difference scheme for the degenerate parabolic equation \\[ \\partial_t u - \\mbox{div}(|\\nabla u|^{p-2}\\nabla u) =f, \\quad p\\... |
https://arxiv.org/abs/1507.01080 | More bounds for the Grundy number of graphs | A coloring of a graph $G=(V,E)$ is a partition $\{V_1, V_2, \ldots, V_k\}$ of $V$ into independent sets or color classes. A vertex $v\in V_i$ is a Grundy vertex if it is adjacent to at least one vertex in each color class $V_j$ for every $j<i$. A coloring is a Grundy coloring if every vertex is a Grundy vertex, and the... | \subsection{\large Clique number}
Zaker \cite{Zak06} showed that for a graph $G$, $\Gamma(G)=2$ if and
only if $G$ is a complete bipartite (see also the page 351 in \cite{
Cha}). Zaker and Soltani \cite{Zak15} showed that for any integer
$k\geq 2$, the smallest triangle-free graph of Grundy number $k$ has
$2k-2$... | {
"timestamp": "2015-12-10T02:02:43",
"yymm": "1507",
"arxiv_id": "1507.01080",
"language": "en",
"url": "https://arxiv.org/abs/1507.01080",
"abstract": "A coloring of a graph $G=(V,E)$ is a partition $\\{V_1, V_2, \\ldots, V_k\\}$ of $V$ into independent sets or color classes. A vertex $v\\in V_i$ is a Gru... |
https://arxiv.org/abs/2102.07873 | Eigenvalue bounds for the Paneitz operator and its associated third-order boundary operator on locally conformally flat manifolds | In this paper we study bounds for the first eigenvalue of the Paneitz operator $P$ and its associated third-order boundary operator $B^3$ on four-manifolds. We restrict to orientable, simply connected, locally confomally flat manifolds that have at most two umbilic boundary components. The proof is based on showing tha... | \section{Introduction}
Let $(M^4,g)$ be a 4-dimensional Riemannian manifold and denote by $\Ric$ and $W$ the Ricci and Weyl tensors of $g$, respectively. Define $J$ to be the trace of the Schouten tensor $A = \frac{1}{2}(\Ric-Jg)$ (actually $J$ is a multiple of the scalar curvature $R$, this is, $J=\frac{1}{6}R... | {
"timestamp": "2021-08-10T02:33:20",
"yymm": "2102",
"arxiv_id": "2102.07873",
"language": "en",
"url": "https://arxiv.org/abs/2102.07873",
"abstract": "In this paper we study bounds for the first eigenvalue of the Paneitz operator $P$ and its associated third-order boundary operator $B^3$ on four-manifold... |
https://arxiv.org/abs/2211.01541 | Factorization conditions for nonlinear second-order differential equations | For the case of nonlinear second-order differential equations with a constant coefficient of the first derivative term and polynomial nonlinearities, the factorization conditions of Rosu and Cornejo-Perez are approached in two ways: (i) by commuting the subindices of the factorization functions in the two factorization... | \section{Introduction}
Many dynamical systems in mechanics and in physics in general are described by non linear second order differential equations or evolve under the action of internal forces with small non-linear components, especially during external forcing or along the relaxing stage after the forcing has been c... | {
"timestamp": "2022-11-04T01:05:34",
"yymm": "2211",
"arxiv_id": "2211.01541",
"language": "en",
"url": "https://arxiv.org/abs/2211.01541",
"abstract": "For the case of nonlinear second-order differential equations with a constant coefficient of the first derivative term and polynomial nonlinearities, the ... |
https://arxiv.org/abs/math/0701014 | A lower bound for the size of the largest critical sets in Latin squares | A critical set in an $n \times n$ array is a set $C$ of given entries, such that there exists a unique extension of $C$ to an $n\times n$ Latin square and no proper subset of $C$ has this property. The cardinality of the largest critical set in any Latin square of order $n$ is denoted by $\lcs{n}$. We give a lower boun... | \section{Introduction}
A {\sf Latin square} of order $n$ is an $n$ $\times$ $n$ array of
integers chosen from the set $X = \{1,2, \ldots, n\}$ such that
each element of $X$ occurs exactly once in each row and exactly
once in each column. A Latin square can also be written as a set
of ordered triples $\{ (i,j;k) ... | {
"timestamp": "2006-12-31T05:22:04",
"yymm": "0701",
"arxiv_id": "math/0701014",
"language": "en",
"url": "https://arxiv.org/abs/math/0701014",
"abstract": "A critical set in an $n \\times n$ array is a set $C$ of given entries, such that there exists a unique extension of $C$ to an $n\\times n$ Latin squa... |
https://arxiv.org/abs/2001.09000 | Optimal error estimate of the finite element approximation of second order semilinear non-autonomous parabolic PDEs | In this work, we investigate the numerical approximation of the second order non-autonomous semilnear parabolic partial differential equation (PDE) using the finite element method. To the best of our knowledge, only the linear case is investigated in the literature. Using an approach based on evolution operator dependi... | \section{Introduction}
\label{intro}
Nonlinear partial differential equations are powerful tools in modelling real-world phenomena in many fields such as in geo-engineering. For instance processes such as oil
and gas recovery from hydrocarbon reservoirs and mining heat from geothermal reservoirs can be modelled by non... | {
"timestamp": "2020-01-27T02:10:16",
"yymm": "2001",
"arxiv_id": "2001.09000",
"language": "en",
"url": "https://arxiv.org/abs/2001.09000",
"abstract": "In this work, we investigate the numerical approximation of the second order non-autonomous semilnear parabolic partial differential equation (PDE) using ... |
https://arxiv.org/abs/2105.02715 | Generalized tournament matrices with the same principal minors | A generalized tournament matrix $M$ is a nonnegative matrix that satisfies $M+M^{t}=J-I$, where $J$ is the all ones matrix and $I$ is the identity matrix. In this paper, a characterization of generalized tournament matrices with the same principal minors of orders $2$, $3$, and $4$ is given. In particular, it is proven... | \section{Introduction}
Let $M=(m_{ij})$ be an $n\times n$ matrix. With each nonempty subset $X
\subseteq \{1,\ldots,n\}$, we associate the \emph{principal submatrix} $M[X]$
of $M$ whose rows and columns are indexed by the elements of $X$. A
\emph{principal minor} of $M$ is the determinant of a principal submatri... | {
"timestamp": "2021-05-07T02:21:37",
"yymm": "2105",
"arxiv_id": "2105.02715",
"language": "en",
"url": "https://arxiv.org/abs/2105.02715",
"abstract": "A generalized tournament matrix $M$ is a nonnegative matrix that satisfies $M+M^{t}=J-I$, where $J$ is the all ones matrix and $I$ is the identity matrix.... |
https://arxiv.org/abs/2201.12391 | Efficient optimization-based quadrature for variational discretization of nonlocal problems | Casting nonlocal problems in variational form and discretizing them with the finite element (FE) method facilitates the use of nonlocal vector calculus to prove well-posedeness, convergence, and stability of such schemes. Employing an FE method also facilitates meshing of complicated domain geometries and coupling with... | \section{Introduction}
\label{sec:Introduction}
Nonlocal models have become viable alternatives to partial differential equations (PDEs) for applications where small-scale effects affect the global behavior of a system or when discontinuities in the quantity of interest make it impractical to use differential operators... | {
"timestamp": "2022-02-01T02:01:47",
"yymm": "2201",
"arxiv_id": "2201.12391",
"language": "en",
"url": "https://arxiv.org/abs/2201.12391",
"abstract": "Casting nonlocal problems in variational form and discretizing them with the finite element (FE) method facilitates the use of nonlocal vector calculus to... |
https://arxiv.org/abs/2207.01320 | City products of right-angled buildings and their universal groups | We introduce the notion of city products of right-angled buildings that produces a new right-angled building out of smaller ones.More precisely, if $M$ is a right-angled Coxeter diagram of rank $n$ and $\Delta_1,\dots,\Delta_n$ are right-angled buildings, then we construct a new right-angled building $\Delta := \mathrm... | \section{Introduction}
A building is called \emph{right-angled} if its Coxeter group is right-angled, which means that the only values occurring in its Coxeter matrix are $1$, $2$ and $\infty$.
The prototypical example is the case where the Coxeter matrix has rank $2$ with a label $\infty$, in which case the building ... | {
"timestamp": "2022-07-05T02:25:54",
"yymm": "2207",
"arxiv_id": "2207.01320",
"language": "en",
"url": "https://arxiv.org/abs/2207.01320",
"abstract": "We introduce the notion of city products of right-angled buildings that produces a new right-angled building out of smaller ones.More precisely, if $M$ is... |
https://arxiv.org/abs/1909.06022 | Error Analysis of Supremizer Pressure Recovery for POD based Reduced Order Models of the time-dependent Navier-Stokes Equations | For incompressible flow models, the pressure term serves as a Lagrange multiplier to ensure that the incompressibility constraint is satisfied. In engineering applications, the pressure term is necessary for calculating important quantities based on stresses like the lift and drag. For reduced order models generated vi... | \section{Introduction}
Let $\Omega \subset \mathbb{R}^{d}$, $d=2,3$ be a regular open domain with Lipschitz continuous boundary $\Gamma$. We consider the Navier-Stokes equations (NSE) with no-slip boundary conditions:
\begin{equation}\label{eqn:nse-1}
\begin{aligned}
&u_t + u\cdot\nabla u + \nabla p - \nu\Delta u = f,... | {
"timestamp": "2019-09-16T02:05:34",
"yymm": "1909",
"arxiv_id": "1909.06022",
"language": "en",
"url": "https://arxiv.org/abs/1909.06022",
"abstract": "For incompressible flow models, the pressure term serves as a Lagrange multiplier to ensure that the incompressibility constraint is satisfied. In enginee... |
https://arxiv.org/abs/1907.07866 | On the equality of domination number and $ 2 $-domination number | The 2-domination number $\gamma_2(G)$ of a graph $G$ is the minimum cardinality of a set $ D \subseteq V(G) $ for which every vertex outside $ D $ is adjacent to at least two vertices in $ D $. Clearly, $ \gamma_2(G) $ cannot be smaller than the domination number $ \gamma(G) $. We consider a large class of graphs and c... | \section{Introduction}
In this paper, we continue to expand on the study of graphs that satisfy the equality $\gamma(G) = \gamma_2(G)$, where $\gamma(G)$ and $\gamma_2(G)$ stand for the domination number and the $ 2 $-domination number of a graph $ G $, respectively. If $\gamma(G) = \gamma_2(G)$ holds for a graph $G$, ... | {
"timestamp": "2021-01-05T02:24:19",
"yymm": "1907",
"arxiv_id": "1907.07866",
"language": "en",
"url": "https://arxiv.org/abs/1907.07866",
"abstract": "The 2-domination number $\\gamma_2(G)$ of a graph $G$ is the minimum cardinality of a set $ D \\subseteq V(G) $ for which every vertex outside $ D $ is ad... |
https://arxiv.org/abs/1610.04161 | Why Deep Neural Networks for Function Approximation? | Recently there has been much interest in understanding why deep neural networks are preferred to shallow networks. We show that, for a large class of piecewise smooth functions, the number of neurons needed by a shallow network to approximate a function is exponentially larger than the corresponding number of neurons n... | \section{Introduction}
Neural networks have drawn significant interest from the machine learning community, especially due to their recent empirical successes (see the surveys \citep{bengio2009learning}). Neural networks are used to build state-of-art systems in various applications such as image recognition, speech re... | {
"timestamp": "2017-03-07T02:01:08",
"yymm": "1610",
"arxiv_id": "1610.04161",
"language": "en",
"url": "https://arxiv.org/abs/1610.04161",
"abstract": "Recently there has been much interest in understanding why deep neural networks are preferred to shallow networks. We show that, for a large class of piec... |
https://arxiv.org/abs/2211.10607 | Bounds for the collapsibility number of a simplicial complex and non-cover complexes of hypergraphs | The collapsibility number of simplicial complexes was introduced by Wegner in order to understand the intersection patterns of convex sets. This number also plays an important role in a variety of Helly type results. There are only a few upper bounds for the collapsibility number of complexes available in literature. I... | \section{Introduction}
Let $X$ be a (finite) simplicial complex. Let $\gamma,\sigma \in X$ be such that $|\gamma|\leq d$ and $\sigma \in X$ is the only maximal simplex that contains $\gamma$. Then, $(\gamma, \sigma)$ is called a {\it free pair} and $\gamma$ is called a {\it free face} of $\sigma$ in $X$. An {\it el... | {
"timestamp": "2022-11-22T02:05:34",
"yymm": "2211",
"arxiv_id": "2211.10607",
"language": "en",
"url": "https://arxiv.org/abs/2211.10607",
"abstract": "The collapsibility number of simplicial complexes was introduced by Wegner in order to understand the intersection patterns of convex sets. This number al... |
https://arxiv.org/abs/2005.11135 | Almost sure behavior of linearly edge-reinforced random walks on the half-line | We study linearly edge-reinforced random walks on $\mathbb{Z}_+$, where each edge $\{x,x+1\}$ has the initial weight $x^{\alpha} \vee 1$, and each time an edge is traversed, its weight is increased by $\Delta$. It is known that the walk is recurrent if and only if $\alpha \leq 1$. The aim of this paper is to study the ... | \section{Linearly edge-reinforced random walks on the half-line}
\section{Introduction}
Reinforced random walks (RRWs), introduced by Coppersmith and Diaconis, are a class of self-interacting random walks that have attracted many researchers for three decades or more. Quoting from Diaconis \cite{Diaconis88},
\begin{q... | {
"timestamp": "2020-07-28T02:06:32",
"yymm": "2005",
"arxiv_id": "2005.11135",
"language": "en",
"url": "https://arxiv.org/abs/2005.11135",
"abstract": "We study linearly edge-reinforced random walks on $\\mathbb{Z}_+$, where each edge $\\{x,x+1\\}$ has the initial weight $x^{\\alpha} \\vee 1$, and each ti... |
https://arxiv.org/abs/1907.10398 | Medians in median graphs and their cube complexes in linear time | The median of a set of vertices $P$ of a graph $G$ is the set of all vertices $x$ of $G$ minimizing the sum of distances from $x$ to all vertices of $P$. In this paper, we present a linear time algorithm to compute medians in median graphs, improving over the existing quadratic time algorithm. We also present a linear ... | \section{Introduction}
The median problem (also called the Fermat-Torricelli problem or the
Weber problem) is one of the oldest optimization problems in Euclidean
geometry~\cite{LoMoWe}. The \emph{median problem} can be defined for
any metric space $(X,d)$: given a finite set $P\subset X$ of points
with positive wei... | {
"timestamp": "2020-07-24T02:18:43",
"yymm": "1907",
"arxiv_id": "1907.10398",
"language": "en",
"url": "https://arxiv.org/abs/1907.10398",
"abstract": "The median of a set of vertices $P$ of a graph $G$ is the set of all vertices $x$ of $G$ minimizing the sum of distances from $x$ to all vertices of $P$. ... |
https://arxiv.org/abs/1201.5909 | Brownian approximation to counting graphs | Let C(n,k) denote the number of connected graphs with n labeled vertices and n+k-1 edges. For any sequence (k_n), the limit of C(n,k_n) as n tends to infinity is known. It has been observed that, if k_n=o(\sqrt{n}), this limit is asymptotically equal to the $k_n$th moment of the area under the standard Brownian excursi... | \section{Introduction} Let $C(n,k)$ denote the number of connected graphs with $n$ labeled vertices and $n+k-1$ edges. For example, $C(n,0)$ is the number of labeled trees on $n$ vertices and is equal to $n^{n-2}$ by Cayley's theorem. There is a rich history of the study of the asymptotics of the sequence $C(n,k)$. Wri... | {
"timestamp": "2012-06-04T02:05:29",
"yymm": "1201",
"arxiv_id": "1201.5909",
"language": "en",
"url": "https://arxiv.org/abs/1201.5909",
"abstract": "Let C(n,k) denote the number of connected graphs with n labeled vertices and n+k-1 edges. For any sequence (k_n), the limit of C(n,k_n) as n tends to infini... |
https://arxiv.org/abs/1711.01194 | New Bounds on the Biplanar Crossing Number of Low-dimensional Hypercubes | In this note we provide an improved upper bound on the biplanar crossing number of the 8-dimensional hypercube. The $k$-planar crossing number of a graph $cr_k(G)$ is the number of crossings required when every edge of $G$ must be drawn in one of $k$ distinct planes. It was shown in Czabarka et al. that $cr_2(Q_8) \leq... | \section{Introduction}
The traditional \textit{crossing number} of a graph $G=(V,E)$, denoted by $cr(G)$, is the minimum number of edge crossings required to draw $G$ in the 2-dimensional Euclidean plane. To study printed circuit boards, Owens \cite{Owe} generalized the question: what is the minimum number of edge cro... | {
"timestamp": "2017-11-06T02:10:09",
"yymm": "1711",
"arxiv_id": "1711.01194",
"language": "en",
"url": "https://arxiv.org/abs/1711.01194",
"abstract": "In this note we provide an improved upper bound on the biplanar crossing number of the 8-dimensional hypercube. The $k$-planar crossing number of a graph ... |
https://arxiv.org/abs/1403.3127 | An asymptotic relationship between coupling methods for stochastically modeled population processes | This paper is concerned with elucidating a relationship between two common coupling methods for the continuous time Markov chain models utilized in the cell biology literature. The couplings considered here are primarily used in a computational framework by providing reductions in variance for different Monte Carlo est... | \section{Introduction}
\label{sec:intro}
Models of biochemical reaction networks with stochastic dynamics have become increasingly popular in the science literature over the previous fifteen years where they are often studied via computational methods and, in particular, Monte Carlo methods. These computational method... | {
"timestamp": "2014-08-05T02:02:04",
"yymm": "1403",
"arxiv_id": "1403.3127",
"language": "en",
"url": "https://arxiv.org/abs/1403.3127",
"abstract": "This paper is concerned with elucidating a relationship between two common coupling methods for the continuous time Markov chain models utilized in the cell... |
https://arxiv.org/abs/1005.5492 | Matroid automorphisms of the H_4 root system | We study the rank 4 linear matroid $M(H_4)$ associated with the 4-dimensional root system $H_4$. This root system coincides with the vertices of the 600-cell, a 4-dimensional regular solid. We determine the automorphism group of this matroid, showing half of the 14,400 automorphisms are geometric and half are not. We p... | \section{Introduction} Regular polytopes in 4-dimensions are notoriously difficult to understand geometrically. Coxeter's classic text \cite{cox} is an excellent resource, concentrating on both the metric properties and the symmetry groups of regular polytopes. Another approach to understanding these polytopes is... | {
"timestamp": "2010-10-28T02:00:46",
"yymm": "1005",
"arxiv_id": "1005.5492",
"language": "en",
"url": "https://arxiv.org/abs/1005.5492",
"abstract": "We study the rank 4 linear matroid $M(H_4)$ associated with the 4-dimensional root system $H_4$. This root system coincides with the vertices of the 600-cel... |
https://arxiv.org/abs/1901.06974 | A variational scheme for hyperbolic obstacle problems | We consider an obstacle problem for (possibly non-local) wave equations, and we prove existence of weak solutions through a convex minimization approach based on a time discrete approximation scheme. We provide the corresponding numerical implementation and raise some open questions. | \section{Introduction}
Obstacle type problems are nowadays a well established subject with many dedicated contributions in the recent literature. Obstacle problems for the minimizers of classical energies and regularity of the arising free boundary have been extensively studied, both for local operators (see, e.g. \ci... | {
"timestamp": "2019-01-24T02:12:20",
"yymm": "1901",
"arxiv_id": "1901.06974",
"language": "en",
"url": "https://arxiv.org/abs/1901.06974",
"abstract": "We consider an obstacle problem for (possibly non-local) wave equations, and we prove existence of weak solutions through a convex minimization approach b... |
https://arxiv.org/abs/2203.08815 | QUBOs for Sorting Lists and Building Trees | We show that the fundamental tasks of sorting lists and building search trees or heaps can be modeled as quadratic unconstrained binary optimization problems (QUBOs). The idea is to understand these tasks as permutation problems and to devise QUBOs whose solutions represent appropriate permutation matrices. We discuss ... | \section{Introduction}
In this paper, we are concerned with quadratic unconstrained binary optimization problems (QUBOs) of the form
\begin{equation}
\label{eq:QUBO}
\vec{z}_* = \amin{\vec{z} \in \{ 0, 1 \}^N} \, \trn{\vec{z}} \mat{R} \, \vec{z} + \ipt{\vec{r}}{\vec{z}}
\end{equation}
where the objective is to find an... | {
"timestamp": "2022-03-18T01:00:22",
"yymm": "2203",
"arxiv_id": "2203.08815",
"language": "en",
"url": "https://arxiv.org/abs/2203.08815",
"abstract": "We show that the fundamental tasks of sorting lists and building search trees or heaps can be modeled as quadratic unconstrained binary optimization probl... |
https://arxiv.org/abs/1202.4656 | Scoring Play Combinatorial Games Under Different Operators | Scoring play games were first studied by Fraser Stewart for his PhD thesis. He showed that under the disjunctive sum, scoring play games are partially ordered, but do not have the same "nice" structure of normal play games. In this paper I will be considering scoring play games under three different operators given by ... | \section{Introduction}
Until very recently scoring play games have not received the kind of treatment or analysis that normal and mis\`ere play games have. The general definition of a scoring play game is given below, for further reading on the general structure of scoring play games see \cite{FS} and \cite{FSP}.
In... | {
"timestamp": "2012-02-22T02:03:57",
"yymm": "1202",
"arxiv_id": "1202.4656",
"language": "en",
"url": "https://arxiv.org/abs/1202.4656",
"abstract": "Scoring play games were first studied by Fraser Stewart for his PhD thesis. He showed that under the disjunctive sum, scoring play games are partially order... |
https://arxiv.org/abs/2212.10942 | Strain topological metamaterials | Topological physics has revolutionised materials science, introducing topological insulators and superconductors with applications from smart materials to quantum computing. Bulk-boundary correspondence (BBC) is a core concept therein, where the non-trivial topology of a material's bulk predicts localized topological s... | \subsection*{Mass Dimer}
We begin with the mass dimer shown in Fig.~\ref{fig1}\textbf{(b)}. This system is a periodic 1D mass-spring chain with two alternating masses, $m_{1}$ and $m_{2}$, connected with a spring of stiffness $k$. The equations of motion of the particle displacements ($u_{A|B,n}$) in the $n_{th}$ unit ... | {
"timestamp": "2022-12-22T02:12:05",
"yymm": "2212",
"arxiv_id": "2212.10942",
"language": "en",
"url": "https://arxiv.org/abs/2212.10942",
"abstract": "Topological physics has revolutionised materials science, introducing topological insulators and superconductors with applications from smart materials to... |
https://arxiv.org/abs/2106.01313 | A modal description of paraxial structured light propagation | Here we outline a description of paraxial light propagation from a modal perspective. By decomposing the initial transverse field into a spatial basis whose elements have known and analytical propagation characteristics, we are able to analytically propagate any desired field, making the calculation fast and easy. By s... | \section{Introduction}
\noindent Our understanding of the propagation of light has refined over the centuries, starting with geometric approaches that have their foundation in concepts outlined more than 400 years ago, through to a wave description that was given a firm theoretical footing nearly 200 years later \cite{... | {
"timestamp": "2021-06-03T02:31:02",
"yymm": "2106",
"arxiv_id": "2106.01313",
"language": "en",
"url": "https://arxiv.org/abs/2106.01313",
"abstract": "Here we outline a description of paraxial light propagation from a modal perspective. By decomposing the initial transverse field into a spatial basis who... |
https://arxiv.org/abs/2202.13697 | Metric, Schauder and Operator-Valued Frames | Notion of frames and Bessel sequences for metric spaces have been introduced. This notion is related with the notion of Lipschitz free Banach spaces. \ It is proved that every separable metric space admits a metric $\mathcal{M}_d$-frame. Through Lipschitz-free Banach spaces it is showed that there is a correspondence b... | \chapter*{}
\par~
\begin{center}
\textbf{{\fontsize{16}{1em}\selectfont DECLARATION}} \\
\textit{By the Ph.D. Research Scholar}
\end{center}
\par I hereby declare that the research thesis entitled
\textbf{METRIC, SCHAUDER AND OPERATOR-VALUED
FRAMES} which is being
submitted to the \textbf{National Institute of Tech... | {
"timestamp": "2022-03-01T02:44:57",
"yymm": "2202",
"arxiv_id": "2202.13697",
"language": "en",
"url": "https://arxiv.org/abs/2202.13697",
"abstract": "Notion of frames and Bessel sequences for metric spaces have been introduced. This notion is related with the notion of Lipschitz free Banach spaces. \\ I... |
https://arxiv.org/abs/2007.00167 | The Integers as a Higher Inductive Type | We consider the problem of defining the integers in Homotopy Type Theory (HoTT). We can define the type of integers as signed natural numbers (i.e., using a coproduct), but its induction principle is very inconvenient to work with, since it leads to an explosion of cases. An alternative is to use set-quotients, but her... | \section{Introduction}
\label{sec:introduction}
How to define the integers in Homotopy Type Theory? This can sound like a trivial
question. The first answer is as signed natural numbers:
\begin{definition}\label{Z-w}
Let $\mathbb{Z}_w$ be the inductive type generated by the following constructors:
\begin{itemize}
... | {
"timestamp": "2020-07-02T02:08:27",
"yymm": "2007",
"arxiv_id": "2007.00167",
"language": "en",
"url": "https://arxiv.org/abs/2007.00167",
"abstract": "We consider the problem of defining the integers in Homotopy Type Theory (HoTT). We can define the type of integers as signed natural numbers (i.e., using... |
https://arxiv.org/abs/1001.0708 | On the so-called Boy or Girl Paradox | A quite old problem has been recently revitalized by Leonard Mlodinow's book The Drunkard's Walk, where it is presented in a way that has definitely confused several people, that wonder why the prevalence of the name of one daughter among the population should change the probability that the other child is a girl too. ... | \section{Introduction}
A classical series of problems in elementary probability
theory is about the gender combinations
($m$-$m$, $m$-$f$, $f$-$m$ and $f$-$f$)
in a family of two children. Being this an academic exercise
(in the bad sense of the term), usually one does not
attempt to assess how much one believes that ... | {
"timestamp": "2010-01-05T15:29:10",
"yymm": "1001",
"arxiv_id": "1001.0708",
"language": "en",
"url": "https://arxiv.org/abs/1001.0708",
"abstract": "A quite old problem has been recently revitalized by Leonard Mlodinow's book The Drunkard's Walk, where it is presented in a way that has definitely confuse... |
https://arxiv.org/abs/1711.04965 | Near-optimal sample complexity for convex tensor completion | We analyze low rank tensor completion (TC) using noisy measurements of a subset of the tensor. Assuming a rank-$r$, order-$d$, $N \times N \times \cdots \times N$ tensor where $r=O(1)$, the best sampling complexity that was achieved is $O(N^{\frac{d}{2}})$, which is obtained by solving a tensor nuclear-norm minimizatio... | \section{Introduction}\label{introduction}
Representing data as multi-dimensional arrays, i.e., tensors, arises naturally in many modern applications such as interpolating large scale seismic data \cite{kreimer2013tensor,da2015optimization}, medical images \cite{mocks1988topographic}, data mining \cite{acar2005modeling... | {
"timestamp": "2017-11-15T02:40:32",
"yymm": "1711",
"arxiv_id": "1711.04965",
"language": "en",
"url": "https://arxiv.org/abs/1711.04965",
"abstract": "We analyze low rank tensor completion (TC) using noisy measurements of a subset of the tensor. Assuming a rank-$r$, order-$d$, $N \\times N \\times \\cdot... |
https://arxiv.org/abs/1708.00502 | Estimation of the covariance structure of heavy-tailed distributions | We propose and analyze a new estimator of the covariance matrix that admits strong theoretical guarantees under weak assumptions on the underlying distribution, such as existence of moments of only low order. While estimation of covariance matrices corresponding to sub-Gaussian distributions is well-understood, much le... | \section{Introduction}
Estimation of the covariance matrix is one of the fundamental problems in data analysis: many important statistical tools, such as Principal Component Analysis(PCA) \cite{hotelling1933analysis} and regression analysis, involve covariance estimation as a crucial step.
For instance, PCA has immed... | {
"timestamp": "2018-01-17T02:04:38",
"yymm": "1708",
"arxiv_id": "1708.00502",
"language": "en",
"url": "https://arxiv.org/abs/1708.00502",
"abstract": "We propose and analyze a new estimator of the covariance matrix that admits strong theoretical guarantees under weak assumptions on the underlying distrib... |
https://arxiv.org/abs/1106.3109 | Stretched exponential behavior and random walks on diluted hypercubic lattices | Diffusion on a diluted hypercube has been proposed as a model for glassy relaxation and is an example of the more general class of stochastic processes on graphs. In this article we determine numerically through large scale simulations the eigenvalue spectra for this stochastic process and calculate explicitly the time... | \section*{Introduction}
In 1854 R. Kohlrausch used a phenomenological expression
\begin{equation}
\label{kohl}
q_{K}(t)=\exp(-(t/\tau)^\beta)
\end{equation}
to parametrize the non-exponential decay of the electric polarization
of Leyden jars (primitive capacitors)\cite{RK}; his son F. Kohlrausch
later used the sam... | {
"timestamp": "2011-06-17T02:00:43",
"yymm": "1106",
"arxiv_id": "1106.3109",
"language": "en",
"url": "https://arxiv.org/abs/1106.3109",
"abstract": "Diffusion on a diluted hypercube has been proposed as a model for glassy relaxation and is an example of the more general class of stochastic processes on g... |
https://arxiv.org/abs/1608.02342 | Recent Progress on Definability of Henselian Valuations | Although the study of the definability of henselian valuations has a long history starting with J. Robinson, most of the results in this area were proven during the last few years. We survey these results which address the definability of concrete henselian valuations, the existence of definable henselian valuations on... | \section{Definability of a given henselian valuation}
The first definability results of henselian valuations were shown 50 years ago. \label{sec:1}
The main aim was to reduce the decidability of some field to the decidability
of a subring or subfield. Julia Robinson showed the following theorem which implies
that the ... | {
"timestamp": "2016-08-09T02:11:03",
"yymm": "1608",
"arxiv_id": "1608.02342",
"language": "en",
"url": "https://arxiv.org/abs/1608.02342",
"abstract": "Although the study of the definability of henselian valuations has a long history starting with J. Robinson, most of the results in this area were proven ... |
https://arxiv.org/abs/1409.6366 | A direct proof for Lovett's bound on the communication complexity of low rank matrices | The log-rank conjecture in communication complexity suggests that the deterministic communication complexity of any Boolean rank-r function is bounded by polylog(r). Recently, major progress was made by Lovett who proved that the communication complexity is bounded by O(r^1/2 * log r). Lovett's proof is based on known ... | \section{Introduction}
In the classical \emph{communication complexity} setting, we imagine to have two players, Alice and Bob
and a function $f : X \times Y \to \{ \pm 1\}$. The players agree on a communication protocol beforehand; then
Alice is given an input $x \in X$ and Bob is presented an input $y \in Y$. Then t... | {
"timestamp": "2014-09-24T02:04:46",
"yymm": "1409",
"arxiv_id": "1409.6366",
"language": "en",
"url": "https://arxiv.org/abs/1409.6366",
"abstract": "The log-rank conjecture in communication complexity suggests that the deterministic communication complexity of any Boolean rank-r function is bounded by po... |
https://arxiv.org/abs/2206.01795 | Robust Topological Inference in the Presence of Outliers | The distance function to a compact set plays a crucial role in the paradigm of topological data analysis. In particular, the sublevel sets of the distance function are used in the computation of persistent homology -- a backbone of the topological data analysis pipeline. Despite its stability to perturbations in the Ha... | \section*{Acknowledgements}
BKS is supported by National Science Foundation (NSF) CAREER Award DMS-1945396. SK is partially supported by JSPS KAKENHI Grant Number 21H03403. SV, KF, and SK were supported by JST, CREST Grant Number JPMJCR15D3, Japan.
\section{Conclusion \& Discussion}
\label{sec:discussion}
In this p... | {
"timestamp": "2022-06-07T02:02:54",
"yymm": "2206",
"arxiv_id": "2206.01795",
"language": "en",
"url": "https://arxiv.org/abs/2206.01795",
"abstract": "The distance function to a compact set plays a crucial role in the paradigm of topological data analysis. In particular, the sublevel sets of the distance... |
https://arxiv.org/abs/1512.06547 | Kumjian-Pask algebras of finitely-aligned higher-rank graphs | We extend the the definition of Kumjian-Pask algebras to include algebras associated to finitely aligned higher-rank graphs. We show that these Kumjian-Pask algebras are universally defined and have a graded uniqueness theorem. We also prove the Cuntz-Kreiger uniqueness theorem; to do this, we use a groupoid approach. ... | \section{Introduction}
In the 1990s, $C^{\ast }$-algebras of row-finite directed graphs were
introduced in \cite{BPRS00,KPR98,KPRR97}. Since their first appearance,
these $C^{\ast }$-algebras have been intensively studied (for example, see
\cite{R08}). Some of the earliest results about these algebras include the... | {
"timestamp": "2015-12-22T02:19:40",
"yymm": "1512",
"arxiv_id": "1512.06547",
"language": "en",
"url": "https://arxiv.org/abs/1512.06547",
"abstract": "We extend the the definition of Kumjian-Pask algebras to include algebras associated to finitely aligned higher-rank graphs. We show that these Kumjian-Pa... |
https://arxiv.org/abs/1512.01700 | Stabilizing the unstable output of persistent homology computations | We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persistent homology algorithm is usually viewed as a procedure which starts with a filtered complex and ends with a persistence diagram. This procedure is stable (at least t... | \section{Introduction} \label{sec:intro}
Persistence diagrams, also called barcodes, are one of the main tools in topological data analysis (TDA) \cite{Carlsson2009,FrosLand99,Edelsbrunner2010,ghrist:survey}. In combination with machine-learning and statistical techniques, they have been used in a wide variety of real... | {
"timestamp": "2017-05-01T02:02:13",
"yymm": "1512",
"arxiv_id": "1512.01700",
"language": "en",
"url": "https://arxiv.org/abs/1512.01700",
"abstract": "We propose a general technique for extracting a larger set of stable information from persistent homology computations than is currently done. The persist... |
https://arxiv.org/abs/2212.03104 | On LC-subgroup of a periodic group | As a natural continuation of study $LCM$-groups, we explore other properties of $LCM$-groups and $LC$-series. We obtain some characterizations of finite groups which are not LCM-groups but all proper sections are $LCM$-groups. Also, for a $p$-group $G$, we prove that $G$ is a $LC$-nilpotent group and we obtain a bound ... | \section{Introduction}
Let $G$ be a periodic group, and let
$LCM(G)$ be the set of all $x \in G$ such that
$o(x^ny)$ divides the least common multiple of $o(x^n)$ and $o(y)$ for all $y \in G$ and all integers $n$. The subgroup generated by $LCM(G)$ of $G$ is denoted by $LC(G)$. A group $G$ is said a LCM-group if... | {
"timestamp": "2022-12-07T02:16:31",
"yymm": "2212",
"arxiv_id": "2212.03104",
"language": "en",
"url": "https://arxiv.org/abs/2212.03104",
"abstract": "As a natural continuation of study $LCM$-groups, we explore other properties of $LCM$-groups and $LC$-series. We obtain some characterizations of finite g... |
https://arxiv.org/abs/2102.03277 | Minimum projective linearizations of trees in linear time | The Minimum Linear Arrangement problem (MLA) consists of finding a mapping $\pi$ from vertices of a graph to distinct integers that minimizes $\sum_{\{u,v\}\in E}|\pi(u) - \pi(v)|$. In that setting, vertices are often assumed to lie on a horizontal line and edges are drawn as semicircles above said line. For trees, var... | \section{Introduction}
\label{sec:introduction}
A linear arrangement $\arr$ of a graph $G=(V,E)$ is a linear ordering of its vertices (it can also be seen as a permutation), i.e., vertices lie on a horizontal line. In such arrangement, the distance $d(u,v)$ between two vertices $u,v$ can be defined as $d(u,v)=|\arr(u)... | {
"timestamp": "2022-05-04T02:16:29",
"yymm": "2102",
"arxiv_id": "2102.03277",
"language": "en",
"url": "https://arxiv.org/abs/2102.03277",
"abstract": "The Minimum Linear Arrangement problem (MLA) consists of finding a mapping $\\pi$ from vertices of a graph to distinct integers that minimizes $\\sum_{\\{... |
https://arxiv.org/abs/2011.08880 | Artifacts of Quantization in Distance Transforms | Distance transforms are a central tool in shape analysis, morphometry, and curve evolution problems. This work describes and investigates an artifact present in distance maps computed from sampled signals. Namely, sampling reflects through the distance transform causing quantization in the resulting distance map. Gradi... | \section{Introduction}
The distance transform is a fundamental tool of image processing, used extensively in shape analysis~\cite{blum1967transformation,kimmel1995skeletonization,siddiqi2002hamilton,niblack1992generating}, morphometry~\cite{hildebrand1997new}, and curve evolution~\cite{osher1988fronts, caselles1993geo... | {
"timestamp": "2020-11-19T02:01:03",
"yymm": "2011",
"arxiv_id": "2011.08880",
"language": "en",
"url": "https://arxiv.org/abs/2011.08880",
"abstract": "Distance transforms are a central tool in shape analysis, morphometry, and curve evolution problems. This work describes and investigates an artifact pres... |
https://arxiv.org/abs/2007.11133 | Unsupervised Learning of Solutions to Differential Equations with Generative Adversarial Networks | Solutions to differential equations are of significant scientific and engineering relevance. Recently, there has been a growing interest in solving differential equations with neural networks. This work develops a novel method for solving differential equations with unsupervised neural networks that applies Generative ... | \section*{Appendix}
\label{appendix}
\subsection*{Description of Experiments}
\label{appendix:experiments}
A plot of the various classical loss functions is provided in Figure \ref{fig:huber_loss_l1_l2_simple}.
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth]{figures/huber_l2_l1.png}
\c... | {
"timestamp": "2020-07-23T02:06:16",
"yymm": "2007",
"arxiv_id": "2007.11133",
"language": "en",
"url": "https://arxiv.org/abs/2007.11133",
"abstract": "Solutions to differential equations are of significant scientific and engineering relevance. Recently, there has been a growing interest in solving differ... |
https://arxiv.org/abs/2201.00450 | On randomized sketching algorithms and the Tracy-Widom law | There is an increasing body of work exploring the integration of random projection into algorithms for numerical linear algebra. The primary motivation is to reduce the overall computational cost of processing large datasets. A suitably chosen random projection can be used to embed the original dataset in a lower-dimen... | \section{Introduction}
Sketching is a probabilistic data compression technique that makes use of random projection \citep{cormode_sketch_2011, mahoney_randomized_2011, woodruff_sketching_2014}. Suppose interest lies in a $n \times d$ dataset $\mat{A}$. When $n$ and or $d$ are large, typical data analysis tasks will inv... | {
"timestamp": "2022-01-04T02:19:05",
"yymm": "2201",
"arxiv_id": "2201.00450",
"language": "en",
"url": "https://arxiv.org/abs/2201.00450",
"abstract": "There is an increasing body of work exploring the integration of random projection into algorithms for numerical linear algebra. The primary motivation is... |
https://arxiv.org/abs/math/0411166 | A context-free and a 1-counter geodesic language for a Baumslag-Solitar group | We give a language of unique geodesic normal forms for the Baumslag-Solitar group BS(1,2) that is context-free and 1-counter. We discuss the classes of context-free, 1-counter and counter languages, and explain how they are inter-related. | \section{Introduction}
In this article we give a simple combinatorial description of a language\
of normal form s for the solvable Baumslag-Solitar\ group BS$(1,2)$\ with the standard generating set,
such that each normal form\ word is geodesic, each group element has a unique normal form\
representative, and the langu... | {
"timestamp": "2004-11-08T16:07:59",
"yymm": "0411",
"arxiv_id": "math/0411166",
"language": "en",
"url": "https://arxiv.org/abs/math/0411166",
"abstract": "We give a language of unique geodesic normal forms for the Baumslag-Solitar group BS(1,2) that is context-free and 1-counter. We discuss the classes o... |
https://arxiv.org/abs/1702.08232 | Hajós-like theorem for signed graphs | The paper designs five graph operations, and proves that every signed graph with chromatic number $q$ can be obtained from all-positive complete graphs $(K_q,+)$ by repeatedly applying these operations. This result gives a signed version of the Hajós theorem, emphasizing the role of all-positive complete graphs played ... | \section{Introduction}
We consider a graph to be finite and simple, i.e., with no loops or multiple edges.
Let $G$ be a graph and $\sigma\colon\ E(G)\rightarrow \{1,-1\}$ be a mapping. The pair $(G,\sigma)$ is called a \emph{signed graph}. We say that $G$ is the \emph{underlying graph} of $(G,\sigma)$ and $\sigma$ is... | {
"timestamp": "2017-02-28T02:11:24",
"yymm": "1702",
"arxiv_id": "1702.08232",
"language": "en",
"url": "https://arxiv.org/abs/1702.08232",
"abstract": "The paper designs five graph operations, and proves that every signed graph with chromatic number $q$ can be obtained from all-positive complete graphs $(... |
https://arxiv.org/abs/2009.14226 | Union-Find Decoders For Homological Product Codes | Homological product codes are a class of codes that can have improved distance while retaining relatively low stabilizer weight. We show how to build union-find decoders for these codes, using a union-find decoder for one of the codes in the product and a brute force decoder for the other code. We apply this constructi... | \section{Review of Homological Product}
\label{review}
In this section, we review the homological product. The product we use is a ``multiple sector" product rather than the ``single sector" product used in \cite{bravyi2014homological}.
Throughout this paper, we consider CSS codes over qubits, meaning that all the vec... | {
"timestamp": "2021-03-09T02:47:38",
"yymm": "2009",
"arxiv_id": "2009.14226",
"language": "en",
"url": "https://arxiv.org/abs/2009.14226",
"abstract": "Homological product codes are a class of codes that can have improved distance while retaining relatively low stabilizer weight. We show how to build unio... |
https://arxiv.org/abs/2211.12204 | Online size Ramsey numbers: Path vs $C_4$ | Given two graphs $G$ and $H$, a size Ramsey game is played on the edge set of $K_\mathbb{N}$. In every round, Builder selects an edge and Painter colours it red or blue. Builder's goal is to force Painter to create a red copy of $G$ or a blue copy of $H$ as soon as possible. The online (size) Ramsey number $\tilde r(G,... | \section{Introduction}
Let $G$ and $H$ be finite graphs. Consider the following game $\tilde R(G,H)$ played on the infinite board $K_\mathbb N$ (i.e. the board is a complete graph with the vertex set $\mathbb N$). In every round, Builder chooses a previously unselected edge of $K_\mathbb N$ and Painter colours it red... | {
"timestamp": "2022-11-23T02:14:00",
"yymm": "2211",
"arxiv_id": "2211.12204",
"language": "en",
"url": "https://arxiv.org/abs/2211.12204",
"abstract": "Given two graphs $G$ and $H$, a size Ramsey game is played on the edge set of $K_\\mathbb{N}$. In every round, Builder selects an edge and Painter colours... |
https://arxiv.org/abs/solv-int/9605010 | A new integrable system related to the Toda lattice | A new integrable lattice system is introduced, and its integrable discretizations are obtained. A Bäcklund transformation between this new system and the Toda lattice, as well as between their discretizations, is established. | \section{Introduction}
We want to introduce in this paper a new integrable lattice system:
\begin{equation}\label{new}
\ddot{x}_k=\dot{x}_k\Big(\exp(x_{k+1}-x_k)-\exp(x_k-x_{k-1})\Big),
\end{equation}
along with two integrable discretizations thereof. In the difference equations
below $x_k=x_k(t)$ are supposed to be ... | {
"timestamp": "1996-06-05T18:59:13",
"yymm": "9605",
"arxiv_id": "solv-int/9605010",
"language": "en",
"url": "https://arxiv.org/abs/solv-int/9605010",
"abstract": "A new integrable lattice system is introduced, and its integrable discretizations are obtained. A Bäcklund transformation between this new sys... |
https://arxiv.org/abs/0711.4325 | On Three Different Notions of Monotone Subsequences | We review how the monotone pattern compares to other patterns in terms of enumerative results on pattern avoiding permutations. We consider three natural definitions of pattern avoidance, give an overview of classic and recent formulas, and provide some new results related to limiting distributions. | \section{Introduction}
Monotone subsequences in a permutation $p=p_1p_2\cdots p_n$ has been the
subject of vigorous research for over sixty years. In this paper, we will
review three different lines of work. In all of them, we will consider
increasing subsequences of a permutation of length $n$ that have a
{\em fixed... | {
"timestamp": "2007-11-27T19:42:28",
"yymm": "0711",
"arxiv_id": "0711.4325",
"language": "en",
"url": "https://arxiv.org/abs/0711.4325",
"abstract": "We review how the monotone pattern compares to other patterns in terms of enumerative results on pattern avoiding permutations. We consider three natural de... |
https://arxiv.org/abs/1704.00451 | Characterization of minimizers of an anisotropic variant of the Rudin-Osher-Fatemi functional with $L^1$ fidelity term | In this paper we study an anisotropic variant of the Rudin-Osher-Fatemi functional with $L^1$ fidelity term of the form \[ E(u) = \int_{\mathbb{R}^n} \phi(\nabla u) + \lambda \| u -f \|_{L^1(\mathbb{R}^n)}. \] We will characterize the minimizers of $E$ in terms of the Wulff shape of $\phi$ and the dual anisotropy. In p... | \section{Introduction}
We analyze an anisotropic variant of the Rudin-Osher-Fatemi functional (anisotropic $\TV$-$L^1$ energy in the following)
\[
E(u) = \int_{\R^n} \phi(\nabla u) + \lambda \| u -f \|_{L^1(\R^n)},
\]
which has been proposed by Choksi et al.~in \cite{Choksi2011} for the special case of $\phi... | {
"timestamp": "2017-04-04T02:10:58",
"yymm": "1704",
"arxiv_id": "1704.00451",
"language": "en",
"url": "https://arxiv.org/abs/1704.00451",
"abstract": "In this paper we study an anisotropic variant of the Rudin-Osher-Fatemi functional with $L^1$ fidelity term of the form \\[ E(u) = \\int_{\\mathbb{R}^n} \... |
https://arxiv.org/abs/1305.2584 | On Borsuk's conjecture for two-distance sets | In this paper we answer Larman's question on Borsuk's conjecture for two-distance sets. We find a two-distance set consisting of 416 points on the unit sphere in the dimension 65 which cannot be partitioned into 83 parts of smaller diameter. This also reduces the smallest dimension in which Borsuk's conjecture is known... | \section{Introduction}
\label{intro}
For each $n\in{\mathbb N}$ the Borsuk number $b(n)$ is the minimal number such that any bounded set in
$\mathbb{R}^{n}$ consisting of at least 2 points can be partitioned into $b(n)$ parts of smaller diameter.
In 1933 Karol Borsuk~\cite{Bor} conjectured that $b(n)=n+1$. The conjectu... | {
"timestamp": "2013-08-30T02:07:12",
"yymm": "1305",
"arxiv_id": "1305.2584",
"language": "en",
"url": "https://arxiv.org/abs/1305.2584",
"abstract": "In this paper we answer Larman's question on Borsuk's conjecture for two-distance sets. We find a two-distance set consisting of 416 points on the unit sphe... |
https://arxiv.org/abs/1503.03188 | Optimal prediction for sparse linear models? Lower bounds for coordinate-separable M-estimators | For the problem of high-dimensional sparse linear regression, it is known that an $\ell_0$-based estimator can achieve a $1/n$ "fast" rate on the prediction error without any conditions on the design matrix, whereas in absence of restrictive conditions on the design matrix, popular polynomial-time methods only guarante... | \section{Introduction}
The classical notion of minimax risk, which plays a central role in
decision theory, allows for the statistician to implement any possible
estimator, regardless of its computational cost. For many problems,
there are a variety of estimators, which can be ordered in terms of
their computational... | {
"timestamp": "2015-12-01T02:14:28",
"yymm": "1503",
"arxiv_id": "1503.03188",
"language": "en",
"url": "https://arxiv.org/abs/1503.03188",
"abstract": "For the problem of high-dimensional sparse linear regression, it is known that an $\\ell_0$-based estimator can achieve a $1/n$ \"fast\" rate on the predi... |
https://arxiv.org/abs/1109.0322 | Bayesian nonparametric multivariate convex regression | In many applications, such as economics, operations research and reinforcement learning, one often needs to estimate a multivariate regression function f subject to a convexity constraint. For example, in sequential decision processes the value of a state under optimal subsequent decisions may be known to be convex or ... | \section{Introduction}
Consider the problem of estimating the function $f$ for the model
$$y = f(\mathbf{x}) + \epsilon,$$
where $\mathbf{x} \in \mathcal{X} \subset \mathbb{R}^p$, $y \in \mathbb{R}$, $f:\mathbb{R}^p \rightarrow \mathbb{R}$ is a mean regression function and $\epsilon\sim N(0,\sigma^2).$ Given the observ... | {
"timestamp": "2011-09-05T02:00:26",
"yymm": "1109",
"arxiv_id": "1109.0322",
"language": "en",
"url": "https://arxiv.org/abs/1109.0322",
"abstract": "In many applications, such as economics, operations research and reinforcement learning, one often needs to estimate a multivariate regression function f su... |
https://arxiv.org/abs/1610.09587 | A Counting Lemma for Binary Matroids and Applications to Extremal Problems | In graph theory, the Szemerédi regularity lemma gives a decomposition of the indicator function for any graph $G$ into a structured component, a uniform part, and a small error. This result, in conjunction with a counting lemma that guarantees many copies of a subgraph $H$ provided a copy of $H$ appears in the structur... | \section{Introduction}
In this paper, the term \emph{matroid} refers to a simple binary matroid. A \emph{simple binary matroid} $M$ is, for our purposes, a full-rank subset of $\mathbb{F}_2^r\setminus \{0\}$ for some positive integer $r=r(M)$ called the \emph{rank} of $M$. The \emph{critical number} $\chi(M)$, another ... | {
"timestamp": "2016-11-01T01:04:57",
"yymm": "1610",
"arxiv_id": "1610.09587",
"language": "en",
"url": "https://arxiv.org/abs/1610.09587",
"abstract": "In graph theory, the Szemerédi regularity lemma gives a decomposition of the indicator function for any graph $G$ into a structured component, a uniform p... |
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