url stringlengths 31 38 | title stringlengths 7 229 | abstract stringlengths 44 2.87k | text stringlengths 319 2.51M | meta dict |
|---|---|---|---|---|
https://arxiv.org/abs/1205.5306 | Polytopes of Minimum Positive Semidefinite Rank | The positive semidefinite (psd) rank of a polytope is the smallest $k$ for which the cone of $k \times k$ real symmetric psd matrices admits an affine slice that projects onto the polytope. In this paper we show that the psd rank of a polytope is at least the dimension of the polytope plus one, and we characterize thos... | \section{Introduction}
Efficient representations of polytopes are of fundamental importance in contexts such as linear optimization
where the complexity of many algorithms depends on the size of the representation. A standard idea to find a compact description of a complicated polytope $P \subset \mathbb{R}^n$ is to l... | {
"timestamp": "2013-08-01T02:07:55",
"yymm": "1205",
"arxiv_id": "1205.5306",
"language": "en",
"url": "https://arxiv.org/abs/1205.5306",
"abstract": "The positive semidefinite (psd) rank of a polytope is the smallest $k$ for which the cone of $k \\times k$ real symmetric psd matrices admits an affine slic... |
https://arxiv.org/abs/2203.10509 | Stability Of Matrix Polynomials In One And Several Variables | The paper presents methods of eigenvalue localisation of regular matrix polynomials, in particular, stability of matrix polynomials is investigated. For this aim a stronger notion of hyperstability is introduced and widely discussed. Matrix versions of the Gauss-Lucas theorem and Szász inequality are shown. Further, to... | \section{Introduction}
Given a matrix polynomial $P(\lambda)=\lambda^d A_d+ \lambda^{d-1} A_{d-1} + \dots + A_0$ ($A_0, A_1, \dots , A_d \in \mathbb{C}^{n, n}$) it is a natural question to ask under which conditions all eigenvalues are located outside a given set $D$. By eigenvalues we mean zeros of the function ... | {
"timestamp": "2022-05-18T02:25:50",
"yymm": "2203",
"arxiv_id": "2203.10509",
"language": "en",
"url": "https://arxiv.org/abs/2203.10509",
"abstract": "The paper presents methods of eigenvalue localisation of regular matrix polynomials, in particular, stability of matrix polynomials is investigated. For t... |
https://arxiv.org/abs/0711.3544 | Alternative Method for Determining the Feynman Propagator of a Non-Relativistic Quantum Mechanical Problem | A direct procedure for determining the propagator associated with a quantum mechanical problem was given by the Path Integration Procedure of Feynman. The Green function, which is the Fourier Transform with respect to the time variable of the propagator, can be derived later. In our approach, with the help of a Laplace... | \section{Introduction}
It is well know that quantum mechanics acquired its f\/inal
formulation in 1925--1926 through fundamental papers of Schr\"odinger
and Heisenberg. Originally these papers appeared as two independent
views of the structure of quantum mechanics, but in 1927
Schr\"odinger established their equivalen... | {
"timestamp": "2007-12-06T22:18:51",
"yymm": "0711",
"arxiv_id": "0711.3544",
"language": "en",
"url": "https://arxiv.org/abs/0711.3544",
"abstract": "A direct procedure for determining the propagator associated with a quantum mechanical problem was given by the Path Integration Procedure of Feynman. The G... |
https://arxiv.org/abs/1610.07497 | Analyzing the structure of multidimensional compressed sensing problems through coherence | Recently it has been established that asymptotic incoherence can be used to facilitate subsampling, in order to optimize reconstruction quality, in a variety of continuous compressed sensing problems, and the coherence structure of certain one-dimensional Fourier sampling problems was determined. This paper extends the... | \section{Introduction}
Exploiting additional structure has always been central to the success of compressed sensing, ever since it was introduced by Cand\`es, Romberg \& Tao \cite{CandesRombergTao} and Donoho \cite{donohoCS}. Sparsity and incoherence has allowed us to recover signals and images from uniformly subsamp... | {
"timestamp": "2016-10-25T02:12:04",
"yymm": "1610",
"arxiv_id": "1610.07497",
"language": "en",
"url": "https://arxiv.org/abs/1610.07497",
"abstract": "Recently it has been established that asymptotic incoherence can be used to facilitate subsampling, in order to optimize reconstruction quality, in a vari... |
https://arxiv.org/abs/1012.5904 | Sutured Floer homology distinguishes between Seifert surfaces | We exhibit the first example of a knot in the three-sphere with a pair of minimal genus Seifert surfaces that can be distinguished using the sutured Floer homology of their complementary manifolds together with the Spin^c-grading. This answers a question of Juhász. More precisely, we show that the Euler characteristic ... | \section{Introduction}
Let $K$ be an oriented knot in the three-sphere $S^3$. Then $K$ is the oriented boundary of at least one connected compact oriented surface in $S^3$ called a Seifert surface for $K$. Two Seifert surfaces $R_1$ and $R_2$ of a knot are considered to be {\it equivalent} if they are ambient isot... | {
"timestamp": "2011-04-07T02:01:22",
"yymm": "1012",
"arxiv_id": "1012.5904",
"language": "en",
"url": "https://arxiv.org/abs/1012.5904",
"abstract": "We exhibit the first example of a knot in the three-sphere with a pair of minimal genus Seifert surfaces that can be distinguished using the sutured Floer h... |
https://arxiv.org/abs/2207.11042 | Strong c-concavity and stability in optimal transport | The stability of solutions to optimal transport problems under variation of the measures is fundamental from a mathematical viewpoint: it is closely related to the convergence of numerical approaches to solve optimal transport problems and justifies many of the applications of optimal transport. In this article, we int... | \section{Introduction}
The theory of optimal transport has had an important impact in applied
mathematics, with applications in inverse problems, in variational
modeling of evolution PDEs
\cite{villani2003topics,santambrogio2015optimal}, and in machine
learning \cite{peyre2019computational} to name but a few. Numerical... | {
"timestamp": "2022-07-25T02:13:02",
"yymm": "2207",
"arxiv_id": "2207.11042",
"language": "en",
"url": "https://arxiv.org/abs/2207.11042",
"abstract": "The stability of solutions to optimal transport problems under variation of the measures is fundamental from a mathematical viewpoint: it is closely relat... |
https://arxiv.org/abs/2005.01045 | Locally testable codes via high-dimensional expanders | Locally testable codes (LTC) are error-correcting codes that have a local tester which can distinguish valid codewords from words that are "far" from all codewords by probing a given word only at a very few (sublinear, typically constant) number of locations. Such codes form the combinatorial backbone of PCPs. A major ... | \section{Local Testability in Vector Spaces} \label{sec:applications}
In this section we demonstrate how the main theorem fits in with, and generalizes, the known results on testability of Reed-Muller codes. In this case the MAS is the Grassmannian complex MAS described in \pref{lem:Grassmann-is-MAS} for \(V=\mathbb{F... | {
"timestamp": "2020-05-05T02:18:51",
"yymm": "2005",
"arxiv_id": "2005.01045",
"language": "en",
"url": "https://arxiv.org/abs/2005.01045",
"abstract": "Locally testable codes (LTC) are error-correcting codes that have a local tester which can distinguish valid codewords from words that are \"far\" from al... |
https://arxiv.org/abs/1809.03158 | Minimum Eccentric Connectivity Index for Graphs with Fixed Order and Fixed Number of Pending Vertices | The eccentric connectivity index of a connected graph $G$ is the sum over all vertices $v$ of the product $d_{G}(v) e_{G}(v)$, where $d_{G}(v)$ is the degree of $v$ in $G$ and $e_{G}(v)$ is the maximum distance between $v$ and any other vertex of $G$. This index is helpful for the prediction of biological activities of... | \section{Introduction}
A chemical graph is a representation of the structural formula of a chemical compound in terms of graph theory where atoms are represented by vertices and chemical bonds by edges. Arthur Cayley \cite{Cayley1874} was probably the first to publish results that consider chemical graphs.
In an a... | {
"timestamp": "2018-09-11T02:15:21",
"yymm": "1809",
"arxiv_id": "1809.03158",
"language": "en",
"url": "https://arxiv.org/abs/1809.03158",
"abstract": "The eccentric connectivity index of a connected graph $G$ is the sum over all vertices $v$ of the product $d_{G}(v) e_{G}(v)$, where $d_{G}(v)$ is the deg... |
https://arxiv.org/abs/1206.3210 | A refined and unified version of the inverse scattering method for the Ablowitz-Ladik lattice and derivative NLS lattices | We refine and develop the inverse scattering theory on a lattice in such a way that the Ablowitz-Ladik lattice and derivative NLS lattices as well as their matrix analogs can be solved in a unified way. The inverse scattering method for the (matrix analog of the) Ablowitz-Ladik lattice is simplified to the same level a... | \section{Introduction}
The cubic nonlinear Schr\"odinger (NLS) equation~\cite{ZS1,ZS2}
is probably the most prominent
example of an
integrable
partial differential equation
in
\mbox{$1+1$}
space-time
dimensions.
The
inverse scattering
method
for the
NLS
equation
devised
by Zakharov and Shabat~\cite{ZS1}
... | {
"timestamp": "2012-07-13T02:06:39",
"yymm": "1206",
"arxiv_id": "1206.3210",
"language": "en",
"url": "https://arxiv.org/abs/1206.3210",
"abstract": "We refine and develop the inverse scattering theory on a lattice in such a way that the Ablowitz-Ladik lattice and derivative NLS lattices as well as their ... |
https://arxiv.org/abs/2108.00592 | Smith Normal Form and the Generalized Spectral Characterization of Graphs | Spectral characterization of graphs is an important topic in spectral graph theory, which has received a lot of attention from researchers in recent years. It is generally very hard to show a given graph to be determined by its spectrum. Recently, Wang [10] gave a simple arithmetic condition for graphs being determined... | \section{Introduction}
The spectrum of a graph encodes a lot of combinatorial information about the given graph and thus has long been a powerful tool in dealing with various problems in graph theory.
A long-standing unsolved question in spectral graph theory is ``Which graphs are determined by their spectra (DS for... | {
"timestamp": "2021-08-03T02:27:11",
"yymm": "2108",
"arxiv_id": "2108.00592",
"language": "en",
"url": "https://arxiv.org/abs/2108.00592",
"abstract": "Spectral characterization of graphs is an important topic in spectral graph theory, which has received a lot of attention from researchers in recent years... |
https://arxiv.org/abs/1612.02722 | Area bounds for minimal surfaces in geodesic ball of hyperbolic space | In hyperbolic space $H^n$ we set a geodesic ball of radius $\rho$. Consider a $k$ dimensional minimal submanifold passing through the origin of the geodesic ball with boundary lies on the boundary of that geodesic ball. We prove that its area is no less than the totally geodesic $k$ dimensional submanifold passing thro... | \section{Introduction}
This article mainly proves a sharp lower bound of area for $k$ dimensional minimal surfaces passing through the origin of the geodesic ball $B_p(\rho_0) $ in standard hyperbolic space $H^n$ with boundary on $ \partial B_p(\rho_0) $
Fix a point $p \in H^n$ and a positive number $\rho_0$. Denot... | {
"timestamp": "2016-12-09T02:06:36",
"yymm": "1612",
"arxiv_id": "1612.02722",
"language": "en",
"url": "https://arxiv.org/abs/1612.02722",
"abstract": "In hyperbolic space $H^n$ we set a geodesic ball of radius $\\rho$. Consider a $k$ dimensional minimal submanifold passing through the origin of the geode... |
https://arxiv.org/abs/2103.01471 | On the Connectivity and Giant Component Size of Random K-out Graphs Under Randomly Deleted Nodes | Random K-out graphs, denoted $\mathbb{H}(n;K)$, are generated by each of the $n$ nodes drawing $K$ out-edges towards $K$ distinct nodes selected uniformly at random, and then ignoring the orientation of the arcs. Recently, random K-out graphs have been used in applications as diverse as random (pairwise) key predistrib... | \section{Introduction}
\label{sec:introduction}
Random graphs are widely used in modeling and analysis of diverse real-world networks including social networks~\cite{newman2002random}, economic networks~\cite{kakade2005economic}, and communication networks~\cite{goldenberg2010survey}. In recent years, a random graph ... | {
"timestamp": "2021-03-03T02:14:57",
"yymm": "2103",
"arxiv_id": "2103.01471",
"language": "en",
"url": "https://arxiv.org/abs/2103.01471",
"abstract": "Random K-out graphs, denoted $\\mathbb{H}(n;K)$, are generated by each of the $n$ nodes drawing $K$ out-edges towards $K$ distinct nodes selected uniforml... |
https://arxiv.org/abs/1502.00366 | Partitions into a small number of part sizes | We study $\nu_k(n)$, the number of partitions of $n$ into $k$ part sizes, and find numerous arithmetic progressions where $\nu_2$ and $\nu_3$ take on values divisible by 2 and 4. Expanding earlier work, we show $\nu_2(An+B) \equiv 0 \pmod{4}$ for (A,B) = (36,30), (72,42), (252,114), (196,70), and likely many other prog... | \section{Introduction}
Denote the number of partitions of $n$ in which exactly $k$ sizes of part appear by $\nu_k(n)$. For instance, $\nu_2(5) = 5$, counting $$4+1, 3+2, 3+1+1, 2+2+1, \text{ and } 2+1+1+1.$$ This easily stated function has been studied by Major P. A. MacMahon \cite{MacMahon}, George Andrews \cite{GE... | {
"timestamp": "2016-05-05T02:12:38",
"yymm": "1502",
"arxiv_id": "1502.00366",
"language": "en",
"url": "https://arxiv.org/abs/1502.00366",
"abstract": "We study $\\nu_k(n)$, the number of partitions of $n$ into $k$ part sizes, and find numerous arithmetic progressions where $\\nu_2$ and $\\nu_3$ take on v... |
https://arxiv.org/abs/2207.08985 | Representing systems of reproducing kernels in spaces of analytic functions | We give an elementary construction of representing systems of the Cauchy kernels in the Hardy spaces $H^p$, $1 \le p <\infty$, as well as of representing systems of reproducing kernels in weighted Hardy spaces. | \section{Introduction and main results}
\label{section1}
A system $\{x_n\}_{n\ge 1}$ in a separable infinite-dimensional Banach space $X$
is said to be a {\it representing system for $X$} if, for every
element $x\in X$, there exists a sequence of complex numbers $\{c_n\}_{n\ge 1}$ such that
$$
x=\sum_{n\ge 1} ... | {
"timestamp": "2022-07-20T02:05:21",
"yymm": "2207",
"arxiv_id": "2207.08985",
"language": "en",
"url": "https://arxiv.org/abs/2207.08985",
"abstract": "We give an elementary construction of representing systems of the Cauchy kernels in the Hardy spaces $H^p$, $1 \\le p <\\infty$, as well as of representin... |
https://arxiv.org/abs/1310.6019 | Graph Clustering with Surprise: Complexity and Exact Solutions | Clustering graphs based on a comparison of the number of links within clusters and the expected value of this quantity in a random graph has gained a lot of attention and popularity in the last decade. Recently, Aldecoa and Marin proposed a related, but slightly different approach leading to the quality measure surpris... | \section{Introduction}
\label{sec:introduction}
\emph{Graph clustering}, i.e., the partitioning of the entities of a network into densely connected groups, has received growing attention in the literature of the last decade, with applications ranging from the analysis of social networks to recommendation systems and bi... | {
"timestamp": "2013-10-23T02:09:49",
"yymm": "1310",
"arxiv_id": "1310.6019",
"language": "en",
"url": "https://arxiv.org/abs/1310.6019",
"abstract": "Clustering graphs based on a comparison of the number of links within clusters and the expected value of this quantity in a random graph has gained a lot of... |
https://arxiv.org/abs/2006.13677 | Accurately approximating extreme value statistics | We consider the extreme value statistics of $N$ independent and identically distributed random variables, which is a classic problem in probability theory. When $N\to\infty$, fluctuations around the maximum of the variables are described by the Fisher-Tippett-Gnedenko theorem, which states that the distribution of maxi... | \section{Introduction}
Extreme value (EV) statistics \cite{Gumbel,Leadbetter,Majumdar,Hansen} is an important subfield of probability theory. Given a random variable $\chi$ which describes the magnitude of a recurring event, the focus is on the statistical properties of the maximal value of a set of $N$ such events. E... | {
"timestamp": "2021-07-08T02:04:01",
"yymm": "2006",
"arxiv_id": "2006.13677",
"language": "en",
"url": "https://arxiv.org/abs/2006.13677",
"abstract": "We consider the extreme value statistics of $N$ independent and identically distributed random variables, which is a classic problem in probability theory... |
https://arxiv.org/abs/1207.6344 | A new symmetry criterion based on the distance function and applications to PDE's | We prove that, if $\Omega\subset \mathbb{R}^n$ is an open bounded starshaped domain of class $C^2$, the constancy over $\partial \Omega$ of the function $$\varphi(y) = \int_0^{\lambda(y)} \prod_{j=1}^{n-1}[1-t \kappa_j(y)]\, dt$$ implies that $\Omega$ is a ball. Here $k_j(y)$ and $\lambda(y)$ denote respectively the pr... | \section{Introduction}
Characterizing special classes of hypersurfaces in a metric space, in particular spheres, in terms of some
properties of their principal curvatures, is a classical and challenging problem in Differential Geometry.
A fundamental result by Alexandrov states that a bounded smooth domain in the Eucl... | {
"timestamp": "2012-07-27T02:06:02",
"yymm": "1207",
"arxiv_id": "1207.6344",
"language": "en",
"url": "https://arxiv.org/abs/1207.6344",
"abstract": "We prove that, if $\\Omega\\subset \\mathbb{R}^n$ is an open bounded starshaped domain of class $C^2$, the constancy over $\\partial \\Omega$ of the functio... |
https://arxiv.org/abs/1804.09820 | A Nonlinear Spectral Method for Core--Periphery Detection in Networks | We derive and analyse a new iterative algorithm for detecting network core--periphery structure. Using techniques in nonlinear Perron-Frobenius theory, we prove global convergence to the unique solution of a relaxed version of a natural discrete optimization problem. On sparse networks, the cost of each iteration scale... | \section{Motivation}
\label{sec:mot}
Large, complex networks record pairwise interactions between components in a system.
In many circumstances, we wish to
summarize this wealth of information by
extracting high-level information or visualizing key features.
Two of the most important and well-studied tasks are
\begin... | {
"timestamp": "2019-02-12T02:27:35",
"yymm": "1804",
"arxiv_id": "1804.09820",
"language": "en",
"url": "https://arxiv.org/abs/1804.09820",
"abstract": "We derive and analyse a new iterative algorithm for detecting network core--periphery structure. Using techniques in nonlinear Perron-Frobenius theory, we... |
https://arxiv.org/abs/1810.12421 | On the forces that cable webs under tension can support and how to design cable webs to channel stresses | In many applications of Structural Engineering the following question arises: given a set of forces $\mathbf{f}_1,\mathbf{f}_2,\dots,\mathbf{f}_N$ applied at prescribed points $\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_N$, under what constraints on the forces does there exist a truss structure (or wire web) with all e... | \section{Introduction}\label{Introduction}
One of the main goals of Structural Engineering is to find performing structures when one incorporates into the design a specific type of material or substructure. Many materials behave quite differently under tension or compression: concrete and masonry structures are two ... | {
"timestamp": "2019-03-04T02:03:26",
"yymm": "1810",
"arxiv_id": "1810.12421",
"language": "en",
"url": "https://arxiv.org/abs/1810.12421",
"abstract": "In many applications of Structural Engineering the following question arises: given a set of forces $\\mathbf{f}_1,\\mathbf{f}_2,\\dots,\\mathbf{f}_N$ app... |
https://arxiv.org/abs/0912.1411 | Three notions of tropical rank for symmetric matrices | We introduce and study three different notions of tropical rank for symmetric and dissimilarity matrices in terms of minimal decompositions into rank 1 symmetric matrices, star tree matrices, and tree matrices. Our results provide a close study of the tropical secant sets of certain nice tropical varieties, including t... | \section{Introduction}
In this paper, we study tropical secant sets and rank for symmetric matrices.
Our setting is
the \emph{tropical semiring} $(\mathbb{R}\cup\{ \infty\}, \oplus , \odot )$, where
tropical addition is given by $ x \oplus y =\min(x,y)$ and tropical
multiplication is given by $x \odot y = x+y$.
The ... | {
"timestamp": "2009-12-08T07:12:04",
"yymm": "0912",
"arxiv_id": "0912.1411",
"language": "en",
"url": "https://arxiv.org/abs/0912.1411",
"abstract": "We introduce and study three different notions of tropical rank for symmetric and dissimilarity matrices in terms of minimal decompositions into rank 1 symm... |
https://arxiv.org/abs/1609.06171 | Coincidences among skew dual stable Grothendieck polynomials | The question of when two skew Young diagrams produce the same skew Schur function has been well-studied. We investigate the same question in the case of stable Grothendieck polynomials, which are the K-theoretic analogues of the Schur functions. We prove a necessary condition for two skew shapes to give rise to the sam... | \section{Introduction}
It is well known that the Schur functions indexed by the set of partitions $\{s_{\lambda}\}$ form a linear basis for the ring of symmetric functions over $\mathbb{Z}$. However, for general skew shapes $\lambda / \mu$, the corresponding Schur functions are no longer linearly independent. In fact, ... | {
"timestamp": "2016-09-21T02:05:54",
"yymm": "1609",
"arxiv_id": "1609.06171",
"language": "en",
"url": "https://arxiv.org/abs/1609.06171",
"abstract": "The question of when two skew Young diagrams produce the same skew Schur function has been well-studied. We investigate the same question in the case of s... |
https://arxiv.org/abs/1809.01131 | On Banach space projective tensor product of $C^*$-algebras | We analyze certain algebraic structures of the Banach space projective tensor product of $C^*$-algebras which are comparable with their known counterparts or the Haagerup tensor product and the operator space projective tensor product of $C^*$-algebras. Highlights of this analysis include (a) injectivity of the Banach ... | \section{ Introduction}
Around 1960s, Gelbaum initiated the study of certain spaces of ideals
of the Banach space projective tensor product $A \otimes^\gamma B$ of Banach
algebras $A$ and $B$, who was then followed by Laursen and Tomiyama -
see \cite{gelbaum1, gelbaum2, gelbaum3, laur, tom} and the references
therein.... | {
"timestamp": "2018-09-06T02:00:10",
"yymm": "1809",
"arxiv_id": "1809.01131",
"language": "en",
"url": "https://arxiv.org/abs/1809.01131",
"abstract": "We analyze certain algebraic structures of the Banach space projective tensor product of $C^*$-algebras which are comparable with their known counterparts... |
https://arxiv.org/abs/1006.0491 | Multiple recurrence and the structure of probability-preserving systems | In 1975 Szemerédi proved the long-standing conjecture of Erdős and Turán that any subset of $\bbZ$ having positive upper Banach density contains arbitrarily long arithmetic progressions. Szemerédi's proof was entirely combinatorial, but two years later Furstenberg gave a quite different proof of Szemerédi's Theorem by ... | \chapter*{Preface}
In 1975 Szemer\'edi proved the long-standing conjecture of Erd\H{o}s and Tur\'an that any subset of $\mathbb{Z}$ having positive upper Banach density contains arbitrarily long arithmetic progressions. Szemer\'edi's proof was entirely combinatorial, but two years later Furstenberg gave a quite diffe... | {
"timestamp": "2010-06-09T02:02:30",
"yymm": "1006",
"arxiv_id": "1006.0491",
"language": "en",
"url": "https://arxiv.org/abs/1006.0491",
"abstract": "In 1975 Szemerédi proved the long-standing conjecture of Erdős and Turán that any subset of $\\bbZ$ having positive upper Banach density contains arbitraril... |
https://arxiv.org/abs/1406.4895 | Computing The Extension Complexities of All 4-Dimensional 0/1-Polytopes | We present slight refinements of known general lower and upper bounds on sizes of extended formulations for polytopes. With these observations we are able to compute the extension complexities of all 0/1-polytopes up to dimension 4. We provide a complete list of our results including geometric constructions of minimum ... | \section{Introduction}
\noindent
The theory of extended formulations is a fast-developing research field that adresses the problem of writing a polytope
as the projection of a preferably simpler polyhedron.
More precisely, given a polytope $ P \in \mathbb{R}^p $, a polyhedron $ Q \in \mathbb{R}^q $ together with a lin... | {
"timestamp": "2014-06-20T02:01:53",
"yymm": "1406",
"arxiv_id": "1406.4895",
"language": "en",
"url": "https://arxiv.org/abs/1406.4895",
"abstract": "We present slight refinements of known general lower and upper bounds on sizes of extended formulations for polytopes. With these observations we are able t... |
https://arxiv.org/abs/2001.02753 | Locating conical degeneracies in the spectra of parametric self-adjoint matrices | A simple iterative scheme is proposed for locating the parameter values for which a 2-parameter family of real symmetric matrices has a double eigenvalue. The convergence is proved to be quadratic. An extension of the scheme to complex Hermitian matrices (with 3 parameters) and to location of triple eigenvalues (5 para... | \section{Introduction}
A theorem of von Neumann and Wigner states that, generically, a
two-parameter family of real symmetric matrices has multiple
eigenvalues at isolated points \cite{WigVon}. In other words, the
matrices with multiple eigenvalues have co-dimension 2 in the manifold
of real symmetric matrices \cite[A... | {
"timestamp": "2020-12-14T02:27:34",
"yymm": "2001",
"arxiv_id": "2001.02753",
"language": "en",
"url": "https://arxiv.org/abs/2001.02753",
"abstract": "A simple iterative scheme is proposed for locating the parameter values for which a 2-parameter family of real symmetric matrices has a double eigenvalue.... |
https://arxiv.org/abs/1712.09891 | Fractional Sturm-Liouville eigenvalue problems, I | We introduce and present the general solution of three two-term fractional differential equations of mixed Caputo/Riemann Liouville type. We then solve a Dirichlet type Sturm-Liouville eigenvalue problem for a fractional differential equation derived from a special composition of a Caputo and a Riemann-Liouville operat... | \section{Introduction}
Fractional Sturm-Liouville Problems (FSLP) generalize classical SLP in that the ordinary derivatives are replaced by {\it fractional derivatives}, or derivatives of fractional order. As an introduction the interested reader may wish to consult the great variety of works on the subject, starting ... | {
"timestamp": "2017-12-29T02:11:23",
"yymm": "1712",
"arxiv_id": "1712.09891",
"language": "en",
"url": "https://arxiv.org/abs/1712.09891",
"abstract": "We introduce and present the general solution of three two-term fractional differential equations of mixed Caputo/Riemann Liouville type. We then solve a ... |
https://arxiv.org/abs/2209.01171 | Aperiodicity of positive operators that increase the support of functions | Consider a positive operator $T$ on an $L^p$-space (or, more generally, a Banach lattice) which increases the support of functions in the sense that $supp(Tf) \supseteq supp{f}$ for every function $f \ge 0$. We show that this implies, under mild assumptions, that $T$ has no unimodular eigenvalues except for possibly th... | \section{Introduction}
\subsection*{Motivation}
If a matrix $T \in \mathbb{R}^{d \times d}$ with spectral radius $\spr(T) = 1$ has only entries $\ge 0$ and all diagonal entries of $T$ are non-zero, then it follows from classical Perron--Frobenius theory that $1$ is the only eigenvalue of $T$ in the complex unit circl... | {
"timestamp": "2022-09-05T02:20:55",
"yymm": "2209",
"arxiv_id": "2209.01171",
"language": "en",
"url": "https://arxiv.org/abs/2209.01171",
"abstract": "Consider a positive operator $T$ on an $L^p$-space (or, more generally, a Banach lattice) which increases the support of functions in the sense that $supp... |
https://arxiv.org/abs/1604.07005 | Continuous homomorphisms between algebras of iterated Laurent series over a ring | We study continuous homomorphisms between algebras of iterated Laurent series over a commutative ring. We give a full description of such homomorphisms in terms of a discrete data determined by the images of parameters. In similar terms, we give a criterion of invertibility of an endomorphism and provide an explicit fo... | \section{Introduction}
In this paper, we study continuous homomorphisms between algebras of iterated Laurent series in many variables over a commutative ring. Let us start with a previously known case of one variable. Let $A$ be a commutative ring and let $A((t))$ be the ring of Laurent series over $A$. For simplici... | {
"timestamp": "2016-09-05T02:03:51",
"yymm": "1604",
"arxiv_id": "1604.07005",
"language": "en",
"url": "https://arxiv.org/abs/1604.07005",
"abstract": "We study continuous homomorphisms between algebras of iterated Laurent series over a commutative ring. We give a full description of such homomorphisms in... |
https://arxiv.org/abs/2201.09861 | The sharp form of the Kolmogorov--Rogozin inequality and a conjecture of Leader--Radcliffe | Let $X$ be a random variable and define its concentration function by $$\mathcal{Q}_{h}(X)=\sup_{x\in \mathbb{R}}\mathbb{P}(X\in (x,x+h]).$$ For a sum $S_n=X_1+\cdots+X_n$ of independent real-valued random variables the Kolmogorov-Rogozin inequality states that $$\mathcal{Q}_{h}(S_n)\leq C\left(\sum_{i=1}^{n}(1-\mathca... | \section{Introduction}
Define the concentration function ${\mathcal{Q}}_{h}(X)$ of a random variable $X$ is defined to be the quantity
\begin{equation*}
{\mathcal{Q}}_{h}(X)=\sup_{x\in {\mathbb{R}}}{\mathbb{P}}(X\in (x,x+h]).
\end{equation*}
Let $S_{n}=X_{1}+\cdots+ X_{n}$ be a sum of independent random variables. Doe... | {
"timestamp": "2022-01-25T02:47:33",
"yymm": "2201",
"arxiv_id": "2201.09861",
"language": "en",
"url": "https://arxiv.org/abs/2201.09861",
"abstract": "Let $X$ be a random variable and define its concentration function by $$\\mathcal{Q}_{h}(X)=\\sup_{x\\in \\mathbb{R}}\\mathbb{P}(X\\in (x,x+h]).$$ For a s... |
https://arxiv.org/abs/1101.1172 | The existence of k-radius sequences | Let $n$ and $k$ be positive integers, and let $F$ be an alphabet of size $n$. A sequence over $F$ of length $m$ is a \emph{$k$-radius sequence} if any two distinct elements of $F$ occur within distance $k$ of each other somewhere in the sequence. These sequences were introduced by Jaromczyk and Lonc in 2004, in order t... | \section{Introduction}
\label{sec:introduction}
Let $n$ and $k$ be positive integers, and let $F$ be an alphabet of
size $n$. A sequence $a_1,a_2,\ldots,a_{m}$ over $F$ of length
$m$ is a \emph{$k$-radius sequence} if for all $x,y\in F$ there
exists $i,j\in\{1,2,\ldots,m\}$ such that $a_i=x$, $a_j=y$ and
$|i-j|\leq k... | {
"timestamp": "2011-08-08T02:01:49",
"yymm": "1101",
"arxiv_id": "1101.1172",
"language": "en",
"url": "https://arxiv.org/abs/1101.1172",
"abstract": "Let $n$ and $k$ be positive integers, and let $F$ be an alphabet of size $n$. A sequence over $F$ of length $m$ is a \\emph{$k$-radius sequence} if any two ... |
https://arxiv.org/abs/1406.7339 | Block Kaczmarz Method with Inequalities | The randomized Kaczmarz method is an iterative algorithm that solves overdetermined systems of linear equations. Recently, the method was extended to systems of equalities and inequalities by Leventhal and Lewis. Even more recently, Needell and Tropp provided an analysis of a block version of the method for systems of ... | \section{Introduction}
The Kaczmarz method~\cite{Kac37:Angenaeherte-Aufloesung} is an iterative algorithm for solving linear systems of equations. It is usually applied to large-scale overdetermined systems because of its simplicity and speed (but also converges in the underdetermined case to the least-norm solutio... | {
"timestamp": "2014-09-04T02:03:14",
"yymm": "1406",
"arxiv_id": "1406.7339",
"language": "en",
"url": "https://arxiv.org/abs/1406.7339",
"abstract": "The randomized Kaczmarz method is an iterative algorithm that solves overdetermined systems of linear equations. Recently, the method was extended to system... |
https://arxiv.org/abs/1003.2809 | Minimal paths in the commuting graphs of semigroups | Let $S$ be a finite non-commutative semigroup. The commuting graph of $S$, denoted $\cg(S)$, is the graph whose vertices are the non-central elements of $S$ and whose edges are the sets $\{a,b\}$ of vertices such that $a\ne b$ and $ab=ba$. Denote by $T(X)$ the semigroup of full transformations on a finite set $X$. Let ... | \section{Introduction}
\setcounter{equation}{0}
The commuting graph of a finite non-abelian group $G$ is a simple graph whose
vertices are all non-central elements of $G$ and two distinct vertices $x,y$ are adjacent if $xy=yx$.
Commuting graphs of various groups have been studied in terms of their properties (such ... | {
"timestamp": "2010-09-09T02:00:38",
"yymm": "1003",
"arxiv_id": "1003.2809",
"language": "en",
"url": "https://arxiv.org/abs/1003.2809",
"abstract": "Let $S$ be a finite non-commutative semigroup. The commuting graph of $S$, denoted $\\cg(S)$, is the graph whose vertices are the non-central elements of $S... |
https://arxiv.org/abs/1802.03579 | On the weighted safe set problem on paths and cycles | Let $G$ be a graph, and let $w: V(G) \to \mathbb{R}$ be a weight function on the vertices of $G$. For every subset $X$ of $V(G)$, let $w(X)=\sum_{v \in X} w(v).$ A non-empty subset $S \subset V(G)$ is a weighted safe set of $(G,w)$ if, for every component $C$ of the subgraph induced by $S$ and every component $D$ of $G... | \section{Introduction}
We start with a question about number sequences in combinatorial number theory.
For a sequence $a_1, \ldots, a_n$ of positive integers and a segment $I$
consisting
of a
subsequence $a_i, a_{i+1},\ldots, a_{i+|I|-1}$, let $s(I)=\sum_{j=i}^{j=i+|I|-1}a_j$.
We consider partitioning $a_1,\ldo... | {
"timestamp": "2018-05-31T02:06:36",
"yymm": "1802",
"arxiv_id": "1802.03579",
"language": "en",
"url": "https://arxiv.org/abs/1802.03579",
"abstract": "Let $G$ be a graph, and let $w: V(G) \\to \\mathbb{R}$ be a weight function on the vertices of $G$. For every subset $X$ of $V(G)$, let $w(X)=\\sum_{v \\i... |
https://arxiv.org/abs/1605.04890 | Product of simplices and sets of positive upper density in $\mathbb{R}^d$ | We establish that any subset of $\mathbb{R}^d$ of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed two-dimensional rectangle provided $d\geq4$.We further present an extension of this result to configurations that are the product of two non-degenerate si... | \section{Introduction}\label{intro}
\subsection{Background}
Recall that the \emph{upper Banach density} of a measurable set $A\subseteq\mathbb R^d$ is defined by
\begin{equation}\label{BD}
\delta^*(A)=\lim_{N\rightarrow\infty}\sup_{t\in\mathbb R^d}\frac{|A\cap(t+Q_N)|}{|Q_N|},\end{equation}
where $|\cdot|$ denotes L... | {
"timestamp": "2017-01-24T02:06:45",
"yymm": "1605",
"arxiv_id": "1605.04890",
"language": "en",
"url": "https://arxiv.org/abs/1605.04890",
"abstract": "We establish that any subset of $\\mathbb{R}^d$ of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates ... |
https://arxiv.org/abs/1302.3192 | Some properties of finite rings | A well-known theorem of Wedderburn asserts that a finite division ring is commutative. In a division ring the group of invertible elements is as large as possible. Here we will be particularly interested in the case where this group is as small as possible, namely reduced to 1. We will show that, if this is the case, t... | \section{Boolean rings} A ring $R$ is boolean if all its elements are
idempotent, i.e., $x^2=x$ for all $x\in R$. A simple example of a boolean ring
is ${\mathbb Z} _2$. Products of boolean rings are also boolean, so we may construct a
large class of such rings.
\begin{proposition} If $R$ is a boolean ring, then $c... | {
"timestamp": "2013-02-14T02:03:29",
"yymm": "1302",
"arxiv_id": "1302.3192",
"language": "en",
"url": "https://arxiv.org/abs/1302.3192",
"abstract": "A well-known theorem of Wedderburn asserts that a finite division ring is commutative. In a division ring the group of invertible elements is as large as po... |
https://arxiv.org/abs/1108.3737 | Representing integers as linear combinations of powers | At a conference in Debrecen in October 2010 Nathanson announced some results concerning the arithmetic diameters of certain sets. He proposed some related results on the representation of integers by sums or differences of powers of 2 and 3. In this note we prove some results on this problem and the more general proble... | \section{Introduction}
Let $P$ be a nonempty finite set of prime numbers, and let $T$ be the set of positive integers that are products of powers of primes in $P$. Put $T_P=T\cup (-T)$. Then there does not exist an integer $k$ such that every positive integer can be represented as a sum of at most $k$ elements of $T_P... | {
"timestamp": "2011-08-19T02:03:06",
"yymm": "1108",
"arxiv_id": "1108.3737",
"language": "en",
"url": "https://arxiv.org/abs/1108.3737",
"abstract": "At a conference in Debrecen in October 2010 Nathanson announced some results concerning the arithmetic diameters of certain sets. He proposed some related r... |
https://arxiv.org/abs/2008.08905 | Linear algebra and quantum algorithm | In mathematical aspect, we introduce quantum algorithm and the mathematical structure of quantum computer. Quantum algorithm is expressed by linear algebra on a finite dimensional complex inner product space. The mathematical formulations of quantum mechanics had been established in around 1930, by von Neumann. The for... | \section{Introduction}
As quantum computer hardware production, which seemed a long way off, has made some progresses recently, much attention is also being paid to the study of quantum algorithm. The class of decision problems which solvable by a quantum computer in polynomial time is called BQP(bounded error quantum... | {
"timestamp": "2020-08-21T02:12:53",
"yymm": "2008",
"arxiv_id": "2008.08905",
"language": "en",
"url": "https://arxiv.org/abs/2008.08905",
"abstract": "In mathematical aspect, we introduce quantum algorithm and the mathematical structure of quantum computer. Quantum algorithm is expressed by linear algebr... |
https://arxiv.org/abs/2206.00601 | Quantitative invertibility of non-Hermitian random matrices | The problem of estimating the smallest singular value of random square matrices is important in connection with matrix computations and analysis of the spectral distribution. In this survey, we consider recent developments in the study of quantitative invertibility in the non-Hermitian setting, and review some applicat... | \section{Introduction}
Given an $N\times n$ ($N\geq n$) matrix $A$, its singular values are defined as square roots of the eigenvalues
of the positive semidefinite $n\times n$ matrix $A^* A$:
\[
s_i(A):=\sqrt{\lambda_i(A^* A)}, \quad i=1,2,\dots,n,
\]
where we assume the non-increasing ordering $\lambda_1(A^* A)\geq \... | {
"timestamp": "2022-06-02T02:24:54",
"yymm": "2206",
"arxiv_id": "2206.00601",
"language": "en",
"url": "https://arxiv.org/abs/2206.00601",
"abstract": "The problem of estimating the smallest singular value of random square matrices is important in connection with matrix computations and analysis of the sp... |
https://arxiv.org/abs/1506.00928 | A Curved Brunn-Minkowski Inequality for the Symmetric Group | In this paper, we construct an injection $A \times B \rightarrow M \times M$ from the product of any two nonempty subsets of the symmetric group into the square of their midpoint set, where the metric is that corresponding to the conjugacy class of transpositions. If $A$ and $B$ are disjoint, our construction allows to... | \section{Introduction}
The classical Brunn-Minkowski inequality
may be formulated as follows: given two compact nonempty
sets $A,B \subset \mathbb{R}^n$, one has
\begin{equation*}
\label{ineq:ClassicalBM}
\log |M_t| \geq (1-t) \log |A| + t \log |B|
\end{equation*}
\noindent
for any $0 \leq t \leq 1$, where
... | {
"timestamp": "2015-06-03T02:11:53",
"yymm": "1506",
"arxiv_id": "1506.00928",
"language": "en",
"url": "https://arxiv.org/abs/1506.00928",
"abstract": "In this paper, we construct an injection $A \\times B \\rightarrow M \\times M$ from the product of any two nonempty subsets of the symmetric group into t... |
https://arxiv.org/abs/1403.3479 | On the boundary of weighted numerical ranges | In this article, we are going to introduce the weighted numerical range which is a further generalization both the c-numerical range and the rank k numerical range. If the boundaries of weighted numerical ranges of two matrices (possibly of different sizes) overlap at sufficiently many points, then the two matrices sha... | \section{Introduction}
Let $M_n$ denote the space of all $n\times n$ complex matrices and $\IR^n$ the set of all real $n$-tuples. For any $A\in M_n$, we denote $\lambda_1(A), \ldots, \lambda_n(A)$ the $n$ eigenvalues of $A$. In the case that $A$ is hermitian, we assume that $\lambda_1(A)\ge \lambda_2(A)\ge\cdots\g... | {
"timestamp": "2015-06-16T02:09:27",
"yymm": "1403",
"arxiv_id": "1403.3479",
"language": "en",
"url": "https://arxiv.org/abs/1403.3479",
"abstract": "In this article, we are going to introduce the weighted numerical range which is a further generalization both the c-numerical range and the rank k numerica... |
https://arxiv.org/abs/1101.2949 | Perfect powers in elliptic divisibility sequences | It is shown that there are finitely many perfect powers in an elliptic divisibility sequence whose first term is divisible by 2 or 3. For Mordell curves the same conclusion is shown to hold if the first term is greater than 1. Examples of Mordell curves and families of congruent number curves are given with correspondi... | \section{Introduction}
Using modular techniques inspired by the proof of Fermat's Last Theorem, it was finally shown in \cite{MR2215137} that the only perfect powers in the Fibonnaci sequence are $1$, $8$ and $144$. Fibonnaci is just one example of an infinte sequence $(h_m)$ of integers
\[
\ldots ,h_{-2}, h_{-1... | {
"timestamp": "2011-01-20T02:00:49",
"yymm": "1101",
"arxiv_id": "1101.2949",
"language": "en",
"url": "https://arxiv.org/abs/1101.2949",
"abstract": "It is shown that there are finitely many perfect powers in an elliptic divisibility sequence whose first term is divisible by 2 or 3. For Mordell curves the... |
https://arxiv.org/abs/1605.04207 | Functional lower bounds for arithmetic circuits and connections to boolean circuit complexity | We say that a circuit $C$ over a field $F$ functionally computes an $n$-variate polynomial $P$ if for every $x \in \{0,1\}^n$ we have that $C(x) = P(x)$. This is in contrast to syntactically computing $P$, when $C \equiv P$ as formal polynomials. In this paper, we study the question of proving lower bounds for homogene... |
\section{Introduction}
Arithmetic circuits are one of the most natural models of computation for studying computation with multivariate polynomials. One of the most fundamental questions in this area of research is to show that there are low degree polynomials which cannot be efficiently computed by \emph{small sized... | {
"timestamp": "2016-05-16T02:10:43",
"yymm": "1605",
"arxiv_id": "1605.04207",
"language": "en",
"url": "https://arxiv.org/abs/1605.04207",
"abstract": "We say that a circuit $C$ over a field $F$ functionally computes an $n$-variate polynomial $P$ if for every $x \\in \\{0,1\\}^n$ we have that $C(x) = P(x)... |
https://arxiv.org/abs/math/0508024 | Volume preserving codimension one Anosov flows in dimensions greater than three are suspensions | We show that every volume preserving codimension one Anosov flow on a closed Riemannian manifold of dimension greater than three admits a global cross section and is therefore topologically conjugate to a suspension of a linear toral automorphism. This proves a conjecture of Verjovsky from the 1970's in the volume pres... | \section{Introduction}
The theory of hyperbolic dynamical systems, despite its long history,
still abounds with open fundamental problems. Among these is the
following
\begin{vconj}
Every codimension one Anosov flow on a closed Riemannian manifold of
dimension greater than three admits a global cross section.
... | {
"timestamp": "2007-03-22T01:11:11",
"yymm": "0508",
"arxiv_id": "math/0508024",
"language": "en",
"url": "https://arxiv.org/abs/math/0508024",
"abstract": "We show that every volume preserving codimension one Anosov flow on a closed Riemannian manifold of dimension greater than three admits a global cross... |
https://arxiv.org/abs/1105.6042 | Weighted Integral Means of Mixed Areas and Lengths under Holomorphic Mappings | This note addresses monotonic growths and logarithmic convexities of the weighted ($(1-t^2)^\alpha dt^2$, $-\infty<\alpha<\infty$, $0<t<1$) integral means $\mathsf{A}_{\alpha,\beta}(f,\cdot)$ and $\mathsf{L}_{\alpha,\beta}(f,\cdot)$ of the mixed area $(\pi r^2)^{-\beta}A(f,r)$ and the mixed length $(2\pi r)^{-\beta}L(f... | \section{Introduction}
From now on, $\mathbb D$ represents the unit disk in the finite
complex plane $\mathbb C$, $H(\mathbb D)$ denotes the space of
holomorphic mappings $f: \mathbb D\to\mathbb C$, and $U(\mathbb D)$
stands for all univalent functions in $H(\mathbb D)$. For any real
number $\alpha$, positive number $r... | {
"timestamp": "2011-05-31T02:05:13",
"yymm": "1105",
"arxiv_id": "1105.6042",
"language": "en",
"url": "https://arxiv.org/abs/1105.6042",
"abstract": "This note addresses monotonic growths and logarithmic convexities of the weighted ($(1-t^2)^\\alpha dt^2$, $-\\infty<\\alpha<\\infty$, $0<t<1$) integral mea... |
https://arxiv.org/abs/1512.01832 | Variable selection with Hamming loss | We derive non-asymptotic bounds for the minimax risk of variable selection under expected Hamming loss in the Gaussian mean model in $\mathbb{R}^d$ for classes of $s$-sparse vectors separated from 0 by a constant $a > 0$. In some cases, we get exact expressions for the nonasymptotic minimax risk as a function of $d, s,... | \section{Introduction}
In recent years, the problem of variable selection in high-dimensional regression models has been
extensively studied from the theoretical and computational viewpoints.
In making effective high-dimensional inference, sparsity plays a key role.
With regard to variable selection in sparse high-... | {
"timestamp": "2017-03-13T01:06:33",
"yymm": "1512",
"arxiv_id": "1512.01832",
"language": "en",
"url": "https://arxiv.org/abs/1512.01832",
"abstract": "We derive non-asymptotic bounds for the minimax risk of variable selection under expected Hamming loss in the Gaussian mean model in $\\mathbb{R}^d$ for c... |
https://arxiv.org/abs/1511.05522 | On the Classification of Pointed Fusion Categories up to weak Morita Equivalence | A pointed fusion category is a rigid tensor category with finitely many isomorphism classes of simple objects which moreover are invertible. Two tensor categories $C$ and $D$ are weakly Morita equivalent if there exists an indecomposable right module category $M$ over $C$ such that $Fun_C(M,M)$ and $D$ are tensor equiv... | \section*{Introduction}
Pointed fusion categories are rigid tensor categories with finitely many isomorphism classes of simple objects with the property
that all simple objects are invertible. Any pointed fusion category $\mathcal{C}$ is equivalent to the fusion category $Vect(G,\omega)$ of
complex vector spaces grade... | {
"timestamp": "2017-03-14T01:02:23",
"yymm": "1511",
"arxiv_id": "1511.05522",
"language": "en",
"url": "https://arxiv.org/abs/1511.05522",
"abstract": "A pointed fusion category is a rigid tensor category with finitely many isomorphism classes of simple objects which moreover are invertible. Two tensor ca... |
https://arxiv.org/abs/1312.6466 | Optimal Confidence Bands for Shape-Restricted Curves | Let $Y$ be a stochastic process on $[0,1]$ satisfying $dY(t) = n^{1/2} f(t) dt + dW(t)$, where $n \ge 1$ is a given scale parameter (``sample size''), $W$ is standard Brownian motion and $f$ is an unknown function. Utilizing suitable multiscale tests we construct confidence bands for $f$ with guaranteed given coverage ... | \section{Introduction}
\label{Introduction}
Nonparametric statistical models often involve some unknown function $f$ defined on a real interval $J$. For instance $f$ might be the probability density of some distribution or a regression function. Nonparametric point estimators for such a curve $f$ are abundant. The av... | {
"timestamp": "2013-12-24T02:12:11",
"yymm": "1312",
"arxiv_id": "1312.6466",
"language": "en",
"url": "https://arxiv.org/abs/1312.6466",
"abstract": "Let $Y$ be a stochastic process on $[0,1]$ satisfying $dY(t) = n^{1/2} f(t) dt + dW(t)$, where $n \\ge 1$ is a given scale parameter (``sample size''), $W$ ... |
https://arxiv.org/abs/1908.03197 | Supertrees | A $k$-universal permutation, or $k$-superpermutation, is a permutation that contains all permutations of length $k$ as patterns. The problem of finding the minimum length of a $k$-superpermutation has recently received significant attention in the field of permutation patterns. One can ask analogous questions for other... | \section{Introduction}
\subsection{Background}\label{subsec:background}
Let $S_n$ denote the set of permutations of the set $[n]=\{1,\ldots,n\}$. We write permutations as words in one-line notation. Given $\mu\in S_m$, we say that the permutation $\sigma=\sigma_1\cdots\sigma_n\in S_n$ \emph{contains the pattern} $\... | {
"timestamp": "2020-05-19T02:03:47",
"yymm": "1908",
"arxiv_id": "1908.03197",
"language": "en",
"url": "https://arxiv.org/abs/1908.03197",
"abstract": "A $k$-universal permutation, or $k$-superpermutation, is a permutation that contains all permutations of length $k$ as patterns. The problem of finding th... |
https://arxiv.org/abs/2104.01362 | On infinitely many foliations by caustics in strictly convex open billiards | Reflection in strictly convex bounded planar billiard acts on the space of oriented lines and preserves a standard area form. A caustic is a curve $C$ whose tangent lines are reflected by the billiard to lines tangent to $C$. The famous Birkhoff Conjecture states that the only strictly convex billiards with a foliation... | \section{Introduction and main results}
The billiard reflection from a strictly convex smooth planar curve $\gamma\subset\mathbb R^2$
(parametrized by either a circle, or an interval) is a map $T$ acting on a
subset in the space of oriented lines: on the so-called {\it phase cylinder} consisting of
those lines th... | {
"timestamp": "2022-01-13T02:00:32",
"yymm": "2104",
"arxiv_id": "2104.01362",
"language": "en",
"url": "https://arxiv.org/abs/2104.01362",
"abstract": "Reflection in strictly convex bounded planar billiard acts on the space of oriented lines and preserves a standard area form. A caustic is a curve $C$ who... |
https://arxiv.org/abs/1904.02313 | Counting self-conjugate (s,s+1,s+2)-core partitions | We are concerned with counting self-conjugate $(s,s+1,s+2)$-core partitions. A Motzkin path of length $n$ is a path from $(0,0)$ to $(n,0)$ which stays above the $x$-axis and consists of the up $U=(1,1)$, down $D=(1,-1)$, and flat $F=(1,0)$ steps. We say that a Motzkin path of length $n$ is symmetric if its reflection ... | \section{Introduction}
Let $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_{\ell})$ be a partition of a positive integer $n$. The {\it Young diagram} of $\lambda$ is a collection of $n$ boxes in $\ell$ rows with $\lambda_i$ boxes in row $i$. For example, the Young diagram for $\lambda=(5,4,2)$ is below.
\begin{center}
\t... | {
"timestamp": "2019-04-05T02:07:54",
"yymm": "1904",
"arxiv_id": "1904.02313",
"language": "en",
"url": "https://arxiv.org/abs/1904.02313",
"abstract": "We are concerned with counting self-conjugate $(s,s+1,s+2)$-core partitions. A Motzkin path of length $n$ is a path from $(0,0)$ to $(n,0)$ which stays ab... |
https://arxiv.org/abs/1208.5521 | Small sets of reals through the prism of fractal dimensions | A separable metric space X is an H-null set if any uniformly continuous image of X has Hausdorff dimension zero. upper H-null, directed P-null and P-null sets are defined likewise, with other fractal dimensions in place of Hausdorff dimension. We investigate these sets and show that in 2^\omega{} they coincide, respect... | \section{Introduction}
\label{sec:intro}
\subsection*{Strong measure zero and Hausdorff dimension}
By the definition due to Borel, a metric space $X$ has
\emph{strong measure zero} (\smz) if for any sequence $\seq{\eps_n}$
of positive numbers there is a cover $\{U_n\}$ of $X$
such that $\diam U_n\leq\eps_n$ f... | {
"timestamp": "2012-08-29T02:00:35",
"yymm": "1208",
"arxiv_id": "1208.5521",
"language": "en",
"url": "https://arxiv.org/abs/1208.5521",
"abstract": "A separable metric space X is an H-null set if any uniformly continuous image of X has Hausdorff dimension zero. upper H-null, directed P-null and P-null se... |
https://arxiv.org/abs/2207.11505 | Monotone Subsequences in Locally Uniform Random Permutations | A locally uniform random permutation is generated by sampling $n$ points independently from some absolutely continuous distribution $\rho$ on the plane and interpreting them as a permutation by the rule that $i$ maps to $j$ if the $i$th point from the left is the $j$th point from below. As $n$ tends to infinity, decrea... | \section{Introduction}\label{sec:introduction}
It has been known since the 1970s that
the longest decreasing (or increasing) subsequence of a random
permutation of $\{1,2,\dotsc,n\}$ has length approximately
$2\sqrt{n}$ for large $n$. More generally, the (scaled) limit
of the cardinality of the largest union of
$\lfloo... | {
"timestamp": "2022-07-26T02:08:49",
"yymm": "2207",
"arxiv_id": "2207.11505",
"language": "en",
"url": "https://arxiv.org/abs/2207.11505",
"abstract": "A locally uniform random permutation is generated by sampling $n$ points independently from some absolutely continuous distribution $\\rho$ on the plane a... |
https://arxiv.org/abs/1211.0328 | Some lower bounds for the $L$-intersection number of graphs | For a set of non-negative integers $L$, the $L$-intersection number of a graph is the smallest number $l$ for which there is an assignment on the vertices to subsets $A_v \subseteq \{1,\dots, l\}$, such that every two vertices $u,v$ are adjacent if and only if $|A_u \cap A_v|\in L$. The bipartite $L$-intersection numbe... | \section{Introduction}
A {\it graph representation} is an assignment on the vertices of graph to a family of objects satisfying certain conditions and a rule which determines from the objects whether or not two vertices are adjacent. In the literature, different types of graph representations such as the set intersect... | {
"timestamp": "2013-08-22T02:06:02",
"yymm": "1211",
"arxiv_id": "1211.0328",
"language": "en",
"url": "https://arxiv.org/abs/1211.0328",
"abstract": "For a set of non-negative integers $L$, the $L$-intersection number of a graph is the smallest number $l$ for which there is an assignment on the vertices t... |
https://arxiv.org/abs/1512.05044 | The Drift Laplacian and Hermitian Geometry | Let $(M^n, h)$ be a compact Hermitian manifold. Suppose $\lambda$ is the lowest eigenvalue of the complex Laplacian on $M$. We prove that $\lambda \geq C$ where $C$ depends only on the dimension $n$, the diameter $d$, the Ricci curvature of the Levi-Civita connection on $M$, and a norm, expressed in curvature, that det... | \section{Introduction}
This preprint's main goal is to obtain a lower bound on the spectrum of the complex Laplacian on a compact Hermitian manifold. To do this, we need two seemingly unrelated results. First, we derive an estimate for the principal eigenvalue of a Laplacian with drift. Second, using recent results f... | {
"timestamp": "2017-02-28T02:08:03",
"yymm": "1512",
"arxiv_id": "1512.05044",
"language": "en",
"url": "https://arxiv.org/abs/1512.05044",
"abstract": "Let $(M^n, h)$ be a compact Hermitian manifold. Suppose $\\lambda$ is the lowest eigenvalue of the complex Laplacian on $M$. We prove that $\\lambda \\geq... |
https://arxiv.org/abs/1906.05914 | Uniqueness of bubbling solutions of mean field equations with non-quantized singularities | For singular mean field equations defined on a compact Riemann surface, we prove the uniqueness of bubbling solutions if some blowup points coincide with bubbling sources. If the strength of the bubbling sources at blowup points are not multiple of $4\pi$ we prove that bubbling solutions are unique under non-degeneracy... | \section{Introduction}
The main goal of this article is to study the uniqueness property of the following mean field equations with singularities:
\begin{equation}\label{m-equ}
\Delta_g v+\rho\bigg(\frac{he^v}{\int_M h e^v{\rm d}\mu}-\frac{1}{vol_g(M)}\bigg)=\sum_{j=1}^N 4\pi \alpha_j (\delta_{q_j}-\frac{1}{vol_... | {
"timestamp": "2019-06-17T02:01:48",
"yymm": "1906",
"arxiv_id": "1906.05914",
"language": "en",
"url": "https://arxiv.org/abs/1906.05914",
"abstract": "For singular mean field equations defined on a compact Riemann surface, we prove the uniqueness of bubbling solutions if some blowup points coincide with ... |
https://arxiv.org/abs/1101.5450 | Quasi-Monte Carlo rules for numerical integration over the unit sphere $\mathbb{S}^2$ | We study numerical integration on the unit sphere $\mathbb{S}^2 \subset \mathbb{R}^3$ using equal weight quadrature rules, where the weights are such that constant functions are integrated exactly.The quadrature points are constructed by lifting a $(0,m,2)$-net given in the unit square $[0,1]^2$ to the sphere $\mathbb{... | \section{Introduction}
We consider the unit sphere $\mathbb{S}^2 = \{\boldsymbol{z} =
(z_1,z_2,z_3) \in \mathbb{R}^3: \|\boldsymbol{z}\| =
\sqrt{z_1^2+z_2^2+z_3^2} = 1\}$. Let $f:\mathbb{S}^2\to\mathbb{R}$ be integrable. Then we estimate the integral $\int_{\mathbb{S}^2} f \, \mathrm{d}\sigma_2$, where $\sigma_2$ is t... | {
"timestamp": "2011-08-01T02:01:18",
"yymm": "1101",
"arxiv_id": "1101.5450",
"language": "en",
"url": "https://arxiv.org/abs/1101.5450",
"abstract": "We study numerical integration on the unit sphere $\\mathbb{S}^2 \\subset \\mathbb{R}^3$ using equal weight quadrature rules, where the weights are such tha... |
https://arxiv.org/abs/1409.4490 | Rigidity and tolerance for perturbed lattices | A perturbed lattice is a point process $\Pi=\{x+Y_x:x\in \mathbb{Z}^d\}$ where the lattice points in $\mathbb{Z}^d$ are perturbed by i.i.d.\ random variables $\{Y_x\}_{x\in \mathbb{Z}^d}$. A random point process $\Pi$ is said to be rigid if $|\Pi\cap B_0(1)|$, the number of points in a ball, can be exactly determined g... | \section{Introduction}\label{sec:intro}
Let $\Pi=\{x+Y_x:x\in \mathbb{Z}^d\}$ denote the lattice $\mathbbm{Z}^d$ perturbed by independent and identically distributed random variables $\{Y_x\}_{x\in \mathbb{Z}^d}$ taking values in $\R^d$. In this paper we address the questions of rigidity and deletion tolerance... | {
"timestamp": "2014-09-17T02:06:56",
"yymm": "1409",
"arxiv_id": "1409.4490",
"language": "en",
"url": "https://arxiv.org/abs/1409.4490",
"abstract": "A perturbed lattice is a point process $\\Pi=\\{x+Y_x:x\\in \\mathbb{Z}^d\\}$ where the lattice points in $\\mathbb{Z}^d$ are perturbed by i.i.d.\\ random v... |
https://arxiv.org/abs/1805.02225 | An Upper Bound for the Moments of a G.C.D. related to Lucas Sequences | Let $(u_n)_{n \geq 0}$ be a non-degenerate Lucas sequence, given by the relation $u_n=a_1 u_{n-1}+a_2 u_{n-2}$. Let $\ell_u(m)=lcm(m, z_u(m))$, for $(m,a_2)=1$, where $z_u(m)$ is the rank of appearance of $m$ in $u_n$. We prove that $$\sum_{\substack{m>x\\ (m,a_2)=1}}\frac{1}{\ell_u(m)}\leq \exp(-(1/\sqrt{6}-\varepsilo... | \section{Introduction}
\label{section 1}
Let $(u_n)_{n\geq 0}$ be an integral linear recurrence, that is, $(u_n)_{n\geq 0}$ is a sequence of integers and there exist $a_1, \dots, a_k\in\mathbb{Z}$, with $a_k\neq 0$, such that
$$u_{n}=a_{1}u_{n-1}+\cdots+a_{k}u_{n-k},$$
for all integers $n\geq k$, with $k$ a fixed posit... | {
"timestamp": "2019-01-08T02:27:07",
"yymm": "1805",
"arxiv_id": "1805.02225",
"language": "en",
"url": "https://arxiv.org/abs/1805.02225",
"abstract": "Let $(u_n)_{n \\geq 0}$ be a non-degenerate Lucas sequence, given by the relation $u_n=a_1 u_{n-1}+a_2 u_{n-2}$. Let $\\ell_u(m)=lcm(m, z_u(m))$, for $(m,... |
https://arxiv.org/abs/2205.01456 | Schur properties of randomly perturbed sets | A set $A$ of integers is said to be Schur if any two-colouring of $A$ results in monochromatic $x,y$ and $z$ with $x+y=z$. We study the following problem: how many random integers from $[n]$ need to be added to some $A\subseteq [n]$ to ensure with high probability that the resulting set is Schur? Hu showed in 1980 that... | \section{Introduction} \label{sec:intro}
A \emph{Schur triple} in a set $A\subseteq \mb{N}$ is a triple $(x,y,z)\in A^3$ such that $x+y=z$, and we say a set $A \subseteq \mb{N}$ is \emph{$r$-Schur} if any $r$-colouring of the elements in $A$ results in a monochromatic Schur triple.
Note that the property of $A$ being... | {
"timestamp": "2022-05-04T02:20:03",
"yymm": "2205",
"arxiv_id": "2205.01456",
"language": "en",
"url": "https://arxiv.org/abs/2205.01456",
"abstract": "A set $A$ of integers is said to be Schur if any two-colouring of $A$ results in monochromatic $x,y$ and $z$ with $x+y=z$. We study the following problem:... |
https://arxiv.org/abs/2112.06755 | Equilateral Chains and Cyclic Central Configurations of the Planar 5-body Problem | Central configurations and relative equilibria are an important facet of the study of the $N$-body problem, but become very difficult to rigorously analyze for $N>3$. In this paper we focus on a particular but interesting class of configurations of the 5-body problem: the equilateral pentagonal configurations, which ha... | \section{Introduction}
In this work we consider some particular classes of relative equilibria of a planar $N$-body problem, in which $N$ point particles with non-negative masses $m_i$ interact through a central potential $U$:
$$m_i \ddot{q}_{i;j} = \frac{\partial U}{\partial q_{i;j}}, \ \ i \in \{0,\ldots N-1\}, $$... | {
"timestamp": "2021-12-14T02:40:39",
"yymm": "2112",
"arxiv_id": "2112.06755",
"language": "en",
"url": "https://arxiv.org/abs/2112.06755",
"abstract": "Central configurations and relative equilibria are an important facet of the study of the $N$-body problem, but become very difficult to rigorously analyz... |
https://arxiv.org/abs/2012.08807 | Mean-field and graph limits for collective dynamics models with time-varying weights | In this paper, we study a model for opinion dynamics where the influence weights of agents evolve in time via an equation which is coupled with the opinions' evolution. We explore the natural question of the large population limit with two approaches: the now classical mean-field limit and the more recent graph limit. ... | \section{Introduction}
Over the past few years, there has been a soaring interest for the study of multi-agent collective behavior models.
Indeed, they can be applied in many different areas: biology with the study of collective flocks and swarms \cite{MR2310046,MR2205679,MR2064375,1582249}, aviation \cite{MR1617559}... | {
"timestamp": "2020-12-17T02:12:57",
"yymm": "2012",
"arxiv_id": "2012.08807",
"language": "en",
"url": "https://arxiv.org/abs/2012.08807",
"abstract": "In this paper, we study a model for opinion dynamics where the influence weights of agents evolve in time via an equation which is coupled with the opinio... |
https://arxiv.org/abs/1007.1791 | Fredman's reciprocity, invariants of abelian groups, and the permanent of the Cayley table | Let $R$ be the regular representation of a finite abelian group $G$ and let $C_n$ denote the cyclic group of order $n$. For $G=C_n$, we compute the Poincare series of all $C_n$-isotypic components in $S^{\cdot} R\otimes \wedge^{\cdot} R$ (the symmetric tensor exterior algebra of $R$). From this we derive a general reci... | \section{Introduction}
\noindent
In the beginning of the 1970s, M.~Fredman \cite{fred} considered the
problem of computing the number of vectors
$(\lambda_0,\lambda_1,\dots,\lambda_{n-1})$ with non-negative integer components that satisfy
\begin{equation} \label{eq:fred}
\lambda_0+\dots +\lambda_{n-1}=m \quad \text... | {
"timestamp": "2010-07-13T02:01:43",
"yymm": "1007",
"arxiv_id": "1007.1791",
"language": "en",
"url": "https://arxiv.org/abs/1007.1791",
"abstract": "Let $R$ be the regular representation of a finite abelian group $G$ and let $C_n$ denote the cyclic group of order $n$. For $G=C_n$, we compute the Poincare... |
https://arxiv.org/abs/1704.01251 | On collisions times of self-sorting interacting particles in one-dimension with random initial positions and velocities | We investigate a one-dimensional system of $N$ particles, initially distributed with random positions and velocities, interacting through binary collisions. The collision rule is such that there is a time after which the $N$ particles do not interact and become sorted according to their velocities. When the collisions ... | \section{Introduction}
\seth{\ay{\jld{We c}\seth{onsider} a collection of $N$ `identical' point-particles, with equal mass, moving \seth{on $\mathbb{R}$} and interacting through \jld{a linear} binary collision \ayd{rule} \jld{given by}
\begin{equation}
\begin{split}
v_i' &= (1-\epsilon)v_j \\
v_j' &= (1-\epsilon)v_i .... | {
"timestamp": "2017-04-06T02:02:38",
"yymm": "1704",
"arxiv_id": "1704.01251",
"language": "en",
"url": "https://arxiv.org/abs/1704.01251",
"abstract": "We investigate a one-dimensional system of $N$ particles, initially distributed with random positions and velocities, interacting through binary collision... |
https://arxiv.org/abs/1605.03040 | A note on the statistical view of matrix completion | A very simple interpretation of matrix completion problem is introduced based on statistical models. Combined with the well-known results from missing data analysis, such interpretation indicates that matrix completion is still a valid and principled estimation procedure even without the missing completely at random (M... | \section*{Acknowledgements}%
\addtocontents{toc}{\protect\vspace{6pt}}%
\addcontentsline{toc}{section}{Acknowledgements}%
}
\setlength{\textwidth}{15.3 truecm} \setlength{\textheight}{23.9
truecm}
\newcommand{\nonumber \\}{\nonumber \\}
\def\pr{\textsf{P}}
\def\ep{\textsf{E}}
\def\Cov{\textsf{Cov}}
\def\Var{\texts... | {
"timestamp": "2016-05-11T02:11:27",
"yymm": "1605",
"arxiv_id": "1605.03040",
"language": "en",
"url": "https://arxiv.org/abs/1605.03040",
"abstract": "A very simple interpretation of matrix completion problem is introduced based on statistical models. Combined with the well-known results from missing dat... |
https://arxiv.org/abs/1703.08057 | PageRank in Undirected Random Graphs | PageRank has numerous applications in information retrieval, reputation systems, machine learning, and graph partitioning. In this paper, we study PageRank in undirected random graphs with an expansion property. The Chung-Lu random graph is an example of such a graph. We show that in the limit, as the size of the graph... | \section*{\nomname
\renewcommand{\nomname}{List of Symbols}
\usepackage{etoolbox,ragged2e,siunitx}
\newcommand{\DescrCol}[1]{\hfill\parbox[t]{12em}{#1}\ignorespaces}
\newcommand{\nomsubtitle}[1]{\item[\large\bfseries #1]}
\renewcommand\nomgroup[1]{\def\csname nomstart#1\endcsname}\nomtemp{\csname nomstart#1\endcsnam... | {
"timestamp": "2017-03-24T01:05:46",
"yymm": "1703",
"arxiv_id": "1703.08057",
"language": "en",
"url": "https://arxiv.org/abs/1703.08057",
"abstract": "PageRank has numerous applications in information retrieval, reputation systems, machine learning, and graph partitioning. In this paper, we study PageRan... |
https://arxiv.org/abs/cond-mat/0701707 | Relaxation dynamics in strained fiber bundles | Under an applied external load the global load-sharing fiber bundle model, with individual fiber strength thresholds sampled randomly from a probability distribution, will relax to an equilibrium state, or to complete bundle breakdown. The relaxation can be viewed as taking place in a sequence of steps. In the first st... | \section{Introduction}
Bundles of fibers, with a statistical distribution of breakdown thresholds
for the individual fibers, are simple and interesting models of failure
processes in materials. They can be analyzed to an extent that is
not possible for most materials (for reviews, see \cite{Herrmann,Chakrabarti,Sahimi... | {
"timestamp": "2007-01-29T12:12:46",
"yymm": "0701",
"arxiv_id": "cond-mat/0701707",
"language": "en",
"url": "https://arxiv.org/abs/cond-mat/0701707",
"abstract": "Under an applied external load the global load-sharing fiber bundle model, with individual fiber strength thresholds sampled randomly from a p... |
https://arxiv.org/abs/1701.03378 | Linearizing the Word Problem in (some) Free Fields | We describe a solution of the word problem in free fields (coming from non-commutative polynomials over a commutative field) using elementary linear algebra, provided that the elements are given by minimal linear representations. It relies on the normal form of Cohn and Reutenauer and can be used more generally to (pos... | \section{Representing Elements}\label{sec:wp.rep}
Although there are several ways for representing elements
in (a subset of) the free field (linear representation
\cite{Cohn1999a
, linearization
\cite{Cohn1985a
,
realization
\cite{Helton2006a
, proper linear system
\cite{Salomaa1978a
, etc.)
the concept of
a \emph{li... | {
"timestamp": "2017-01-13T02:06:28",
"yymm": "1701",
"arxiv_id": "1701.03378",
"language": "en",
"url": "https://arxiv.org/abs/1701.03378",
"abstract": "We describe a solution of the word problem in free fields (coming from non-commutative polynomials over a commutative field) using elementary linear algeb... |
https://arxiv.org/abs/1701.02191 | Actuator design for parabolic distributed parameter systems with the moment method | In this paper, we model and solve the problem of designing in an optimal way actuators for parabolic partial differential equations settled on a bounded open connected subset $\Omega$ of IR n. We optimize not only the location but also the shape of actuators, by finding what is the optimal distribution of actuators in ... | \section{Introduction and modeling of the problem}\label{secintro}
In this article, we model and solve the problem of finding the optimal shape and location of internal controllers for parabolic equations with (mainly) Dirichlet boundary conditions and (mainly) in the one-dimensional case $\Omega=(0,\pi)$. Such questi... | {
"timestamp": "2017-01-10T02:11:00",
"yymm": "1701",
"arxiv_id": "1701.02191",
"language": "en",
"url": "https://arxiv.org/abs/1701.02191",
"abstract": "In this paper, we model and solve the problem of designing in an optimal way actuators for parabolic partial differential equations settled on a bounded o... |
https://arxiv.org/abs/1807.08672 | A Bound on the Rate of Convergence in the Central Limit Theorem for Renewal Processes under Second Moment Conditions | A famous result in renewal theory is the Central Limit Theorem for renewal processes. As in applications usually only observations from a finite time interval are available, a bound on the Kolmogorov distance to the normal distribution is desirable. Here we provide an explicit non-uniform bound for the Renewal Central ... | \section{Introduction}
Let $Z, Z_i, i=1, 2, \ldots$ be i.i.d. non-negative random variables with positive mean $\mu$ and finite variance $\sigma^2$, and let
$$X_t = \max \{ n: \sum_{i=1}^n Z_i \le t\}.$$
Then $(X_t, t \ge 0)$ is a classical renewal process.
Renewal processes are a cornerstone in applied probabi... | {
"timestamp": "2018-07-24T02:22:52",
"yymm": "1807",
"arxiv_id": "1807.08672",
"language": "en",
"url": "https://arxiv.org/abs/1807.08672",
"abstract": "A famous result in renewal theory is the Central Limit Theorem for renewal processes. As in applications usually only observations from a finite time inte... |
https://arxiv.org/abs/1910.06922 | Gradient penalty from a maximum margin perspective | A popular heuristic for improved performance in Generative adversarial networks (GANs) is to use some form of gradient penalty on the discriminator. This gradient penalty was originally motivated by a Wasserstein distance formulation. However, the use of gradient penalty in other GAN formulations is not well motivated.... | \subsubsection*{\bibname}}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{url}
\usepackage{booktabs}
\usepackage{amsfonts}
\usepackage{nicefrac}
\usepackage{microtype}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{algpseudocode,algori... | {
"timestamp": "2019-10-16T02:18:58",
"yymm": "1910",
"arxiv_id": "1910.06922",
"language": "en",
"url": "https://arxiv.org/abs/1910.06922",
"abstract": "A popular heuristic for improved performance in Generative adversarial networks (GANs) is to use some form of gradient penalty on the discriminator. This ... |
https://arxiv.org/abs/1005.2988 | $L^p$ spectrum and heat dynamics of locally symmetric spaces of higher rank | The aim of this paper is to study the spectrum of the $L^p$ Laplacian and the dynamics of the $L^p$ heat semigroup on non-compact locally symmetric spaces of higher rank. Our work here generalizes previously obtained results in the setting of locally symmetric spaces of rank one to higher rank spaces. Similarly as in t... | \section{Introduction}
The aim of this paper is to study the spectrum of the $L^p$ Laplacian $\DMp$ and the dynamics of the $L^p$ heat semigroup $e^{-t\DMp}: L^p(M)\to L^p(M)$ on non-compact locally symmetric spaces
$M=\Gamma\backslash X$ of higher rank with finite volume.
More precisely, $X$ is a symmetric space of... | {
"timestamp": "2010-05-18T02:03:04",
"yymm": "1005",
"arxiv_id": "1005.2988",
"language": "en",
"url": "https://arxiv.org/abs/1005.2988",
"abstract": "The aim of this paper is to study the spectrum of the $L^p$ Laplacian and the dynamics of the $L^p$ heat semigroup on non-compact locally symmetric spaces o... |
https://arxiv.org/abs/1903.06968 | Creation of discontinuities in circle maps | Circle maps frequently arise in mathematical models of physical or biological systems. Motivated by Cherry flows and `threshold' systems such as integrate and fire neuronal models, models of cardiac arrhythmias, and models of sleep/wake regulation, we consider how structural transitions in circle maps occur. In particu... | \section{Introduction}\label{section:intro}
Degree one circle maps $f:{\mathbb S}^1\to {\mathbb S}^1$ are
described by real functions $F:{\mathbb R}\to {\mathbb R}$ with
$F(x+1)=F(x)+1$ and $f(x)=F(x)$ modulo 1. These maps arise naturally
in many situations and $F$ may be injective (or not) and continuous
(or not) lead... | {
"timestamp": "2019-03-19T01:14:49",
"yymm": "1903",
"arxiv_id": "1903.06968",
"language": "en",
"url": "https://arxiv.org/abs/1903.06968",
"abstract": "Circle maps frequently arise in mathematical models of physical or biological systems. Motivated by Cherry flows and `threshold' systems such as integrate... |
https://arxiv.org/abs/1203.5845 | Accuracy and Stability of Filters for Dissipative PDEs | Data assimilation methodologies are designed to incorporate noisy observations of a physical system into an underlying model in order to infer the properties of the state of the system. Filters refer to a class of data assimilation algorithms designed to update the estimation of the state as data is acquired sequential... | \section{Introduction}
\label{sec:intro}
Assimilating large
data sets into mathematical models of time-evolving
systems presents a major challenge in a wide range of
applications. Since the data and the model are often
uncertain, a natural overarching framework
for the formulation of such problems is that of Bayes... | {
"timestamp": "2013-08-06T02:07:10",
"yymm": "1203",
"arxiv_id": "1203.5845",
"language": "en",
"url": "https://arxiv.org/abs/1203.5845",
"abstract": "Data assimilation methodologies are designed to incorporate noisy observations of a physical system into an underlying model in order to infer the propertie... |
https://arxiv.org/abs/1912.10175 | Quantitative results on the multi-parameters Proximal Point Algorithm | We give a quantitative analysis of a theorem due to Fenghui Wang and Huanhuan Cui concerning the convergence of a multi-parametric version of the proximal point algorithm. Wang and Cui's result ensures the convergence of the algorithm to a zero of the operator. Our quantitative analysis provides explicit bounds on the ... | \section{Introduction}
In this paper we give a quantitative analysis of a theorem due to Fenghui Wang and Huanhuan Cui concerning the strong convergence of a multi-parametric version of the proximal point algorithm in Hilbert spaces.
The \emph{proximal point algorithm} $(\mathsf{PPA})$ is recognized as a powerful and... | {
"timestamp": "2019-12-24T02:04:09",
"yymm": "1912",
"arxiv_id": "1912.10175",
"language": "en",
"url": "https://arxiv.org/abs/1912.10175",
"abstract": "We give a quantitative analysis of a theorem due to Fenghui Wang and Huanhuan Cui concerning the convergence of a multi-parametric version of the proximal... |
https://arxiv.org/abs/1911.09084 | Breakdown of Liesegang precipitation bands in a simplified fast reaction limit of the Keller-Rubinow model | We study solutions to the integral equation \[ \omega(x) = \Gamma - x^2 \int_{0}^1 K(\theta) \, H(\omega(x\theta)) \, \mathrm d \theta \] where $\Gamma>0$, $K$ is a weakly degenerate kernel satisfying, among other properties, $K(\theta) \sim k \, (1-\theta)^\sigma$ as $\theta \to 1$ for constants $k>0$ and $\sigma \in ... | \section{Introduction}
Reaction-diffusion equations with discontinuous hysteresis occur in a
range of modeling problems \cite{BrokateS:1996:HysteresisPT,
KrasnoselskiiP:1989:SystemsH, Mayergoyz:1991:MathematicalMH,
Visintin:1994:DifferentialMH, Visintin:2014:TenIH}. We are
particularly interested in non-ideal relays-... | {
"timestamp": "2020-10-29T01:23:52",
"yymm": "1911",
"arxiv_id": "1911.09084",
"language": "en",
"url": "https://arxiv.org/abs/1911.09084",
"abstract": "We study solutions to the integral equation \\[ \\omega(x) = \\Gamma - x^2 \\int_{0}^1 K(\\theta) \\, H(\\omega(x\\theta)) \\, \\mathrm d \\theta \\] wher... |
https://arxiv.org/abs/1207.7209 | Concentration inequalities for order statistics | This note describes non-asymptotic variance and tail bounds for order statistics of samples of independent identically distributed random variables. Those bounds are checked to be asymptotically tight when the sampling distribution belongs to a maximum domain of attraction. If the sampling distribution has non-decreasi... | \section{Introduction}
\label{sec:introduction}
The purpose of this note is to develop non-asymptotic variance and tail bounds for order statistics.
In the sequel, $X_1,\ldots,X_n$ are independent random variables, distributed according to some probability distribution $F$, and
$X_{(1)}\geq X_{(2)} \geq \ldots \g... | {
"timestamp": "2012-08-01T02:03:07",
"yymm": "1207",
"arxiv_id": "1207.7209",
"language": "en",
"url": "https://arxiv.org/abs/1207.7209",
"abstract": "This note describes non-asymptotic variance and tail bounds for order statistics of samples of independent identically distributed random variables. Those b... |
https://arxiv.org/abs/1108.3909 | Relative Commutator Theory in Semi-Abelian Categories | Basing ourselves on the concept of double central extension from categorical Galois theory, we study a notion of commutator which is defined relative to a Birkhoff subcategory B of a semi-abelian category A. This commutator characterises Janelidze and Kelly's B-central extensions; when the subcategory B is determined b... | \section{Introduction}
The aim of this article is to fill in the question mark in the diagram
\[
\xymatrix@!0@C=6em@R=3em{& \fbox{\small ?} \ar@{-}[dl] \ar@{-}[dd] \ar@{-}[rd] \\
\fbox{\small Janelidze \& Kelly} \ar@{-}[dd] && \fbox{\txt{\small Huq}} \ar@{-}[dd]\\
& \fbox{\small Everaert} \ar@{-}[dl] \ar@{-}[rd] \\
\f... | {
"timestamp": "2011-08-22T02:01:01",
"yymm": "1108",
"arxiv_id": "1108.3909",
"language": "en",
"url": "https://arxiv.org/abs/1108.3909",
"abstract": "Basing ourselves on the concept of double central extension from categorical Galois theory, we study a notion of commutator which is defined relative to a B... |
https://arxiv.org/abs/2211.07862 | Improved expected $L_2$-discrepancy formulas on jittered sampling | We study the expected $ L_2-$discrepancy under two classes of partitions, explicit and exact formulas are derived respectively. These results attain better expected $L_2-$discrepancy formulas than jittered sampling. | \section{Introduction}\label{intro}
It is well known that classical jittered sampling (JS) patterns perform better than traditional Monte Carlo (MC) patterns in terms of convergence order, see \cite{jittsamp,ZD2016,KP2}. This means stratified sampling is the refinement of the traditional Monte Carlo method, which invo... | {
"timestamp": "2022-11-16T02:07:41",
"yymm": "2211",
"arxiv_id": "2211.07862",
"language": "en",
"url": "https://arxiv.org/abs/2211.07862",
"abstract": "We study the expected $ L_2-$discrepancy under two classes of partitions, explicit and exact formulas are derived respectively. These results attain bette... |
https://arxiv.org/abs/2212.05617 | Decomposition of the Leinster-Cobbold Diversity Index | The Leinster and Cobbold diversity index possesses a number of merits; in particular, it generalises many existing indices and defines an effective number. We present a scheme to quantify the contribution of richness, evenness, and taxonomic similarity to this index. Compared to the work of van Dam (2019), our approach... | \section{Introduction}
Measuring biodiversity is a difficult task due to sampling issues and
accounting for missing data, but also as there is no one universally
accepted definition of what biodiversity is \citep{Daly2018}. In
ecological practice, definitions of biodiversity can include
contributions from multiple ch... | {
"timestamp": "2022-12-13T02:16:03",
"yymm": "2212",
"arxiv_id": "2212.05617",
"language": "en",
"url": "https://arxiv.org/abs/2212.05617",
"abstract": "The Leinster and Cobbold diversity index possesses a number of merits; in particular, it generalises many existing indices and defines an effective number... |
https://arxiv.org/abs/2010.14137 | The leapfrog algorithm as nonlinear Gauss-Seidel | Several applications in optimization, image, and signal processing deal with data that belong to the Stiefel manifold St(n,p), that is, the set of n-by-p matrices with orthonormal columns. Some applications, like the Riemannian center of mass, require evaluating the Riemannian distance between two arbitrary points on S... | \section{Introduction} \label{sec:geodesic_exp_log}
The object of study in this paper is the compact Stiefel manifold, i.e., the set of orthonormal $n$-by-$p$ matrices
\[
\Stnp = \lbrace X \in \R^{n \times p}: \ X\tr\! X = I_p \rbrace.
\]
Here, we are concerned with computing the Riemannian distance between two poin... | {
"timestamp": "2022-09-13T02:15:55",
"yymm": "2010",
"arxiv_id": "2010.14137",
"language": "en",
"url": "https://arxiv.org/abs/2010.14137",
"abstract": "Several applications in optimization, image, and signal processing deal with data that belong to the Stiefel manifold St(n,p), that is, the set of n-by-p ... |
https://arxiv.org/abs/1608.00032 | On the Power of Likelihood Ratio Tests in Dimension-Restricted Submodels | Likelihood ratio tests are widely used to test statistical hypotheses about parametric families of probability distributions. If interest is restricted to a subfamily of distributions, then it is natural to inquire if the restricted LRT is superior to the unrestricted LRT. Marden's general LRT conjecture posits that an... | \section{Introduction}
\label{intro}
We compare restricted and unrestricted likelihood ratio tests
in situations where the restriction decreases the dimension of the alternative. The issues that concern us are motivated by an elementary example.
\subparagraph{Basic Example} Suppose that $X=(X_1,X_2)$ has a bivariat... | {
"timestamp": "2016-08-02T02:01:29",
"yymm": "1608",
"arxiv_id": "1608.00032",
"language": "en",
"url": "https://arxiv.org/abs/1608.00032",
"abstract": "Likelihood ratio tests are widely used to test statistical hypotheses about parametric families of probability distributions. If interest is restricted to... |
https://arxiv.org/abs/2105.05645 | Homotopy Comomentum Maps in Multisymplectic Geometry | Homotopy comomentum maps are a higher generalization of the notion of moment map introduced to extend the concept of Hamiltonian actions to the framework of multisymplectic geometry. Loosely speaking, higher means passing from considering symplectic 2-form to consider differential forms in higher degrees. The goal of t... | \chapter{Abstract} \label{ch:abstract}
\vspace{-3em}
\Momaps are a higher generalization of the notion of moment map introduced to extend the concept of Hamiltonian actions to the framework of multisymplectic geometry.
Loosely speaking, higher means passing from considering symplectic $2... | {
"timestamp": "2021-05-13T02:18:07",
"yymm": "2105",
"arxiv_id": "2105.05645",
"language": "en",
"url": "https://arxiv.org/abs/2105.05645",
"abstract": "Homotopy comomentum maps are a higher generalization of the notion of moment map introduced to extend the concept of Hamiltonian actions to the framework ... |
https://arxiv.org/abs/1810.03494 | k-price auctions and Combination auctions | We provide an exact analytical solution of the Nash equilibrium for $k$- price auctions. We also introduce a new type of auction and demonstrate that it has fair solutions other than the second price auctions, therefore paving the way for replacing second price auctions. | \section{Introduction}
Second price auctions, also known as Vickrey auctions \cite{vickrey1961counterspeculation}, are well known and largely used as examples in online or Government auctions because it gives bidders an incentive to bid their true value. Nevertheless, second price auctions have reached some limits bec... | {
"timestamp": "2019-03-06T02:19:28",
"yymm": "1810",
"arxiv_id": "1810.03494",
"language": "en",
"url": "https://arxiv.org/abs/1810.03494",
"abstract": "We provide an exact analytical solution of the Nash equilibrium for $k$- price auctions. We also introduce a new type of auction and demonstrate that it h... |
https://arxiv.org/abs/1606.05459 | Plant complexes and homological stability for Hurwitz spaces | We study Hurwitz spaces with regard to homological stabilization. By a Hurwitz space, we mean a moduli space of branched, not necessarily connected coverings of a disk with fixed structure group and number of branch points. We choose a sequence of subspaces of Hurwitz spaces which is suitable for our investigations.In ... | \section{Introduction}
Understanding the topology of moduli spaces is a key aspect in order to grasp the behavior in families of the parametrized objects. Now, moduli spaces often come in sequences, such as the moduli spaces $\mathcal{M}_{g}$ of Riemann surfaces of genus~$g$.
In some cases, such sequences satisfy \em... | {
"timestamp": "2016-06-24T02:10:16",
"yymm": "1606",
"arxiv_id": "1606.05459",
"language": "en",
"url": "https://arxiv.org/abs/1606.05459",
"abstract": "We study Hurwitz spaces with regard to homological stabilization. By a Hurwitz space, we mean a moduli space of branched, not necessarily connected coveri... |
https://arxiv.org/abs/1904.07587 | Depth functions of powers of homogeneous ideals | We settle a conjecture of Herzog and Hibi, which states that the function depth $S/Q^n$, $n \ge 1$, where $Q$ is a homogeneous ideal in a polynomial ring $S$, can be any convergent numerical function. We also give a positive answer to a long-standing open question of Ratliff on the associated primes of powers of ideals... | \section{Introduction}
Let $S$ be a standard graded algebra over a field $k$.
For a homogeneous ideal $Q \subseteq S$, we call the function $\depth S/Q^n$, $n \ge 1$
the \emph{depth function} of $Q$.
The goal of this paper is to prove the following conjecture of Herzog and Hibi in \cite{HH} (see also \cite[Problem 3... | {
"timestamp": "2019-04-17T02:26:04",
"yymm": "1904",
"arxiv_id": "1904.07587",
"language": "en",
"url": "https://arxiv.org/abs/1904.07587",
"abstract": "We settle a conjecture of Herzog and Hibi, which states that the function depth $S/Q^n$, $n \\ge 1$, where $Q$ is a homogeneous ideal in a polynomial ring... |
https://arxiv.org/abs/1601.02245 | On parallel solution of ordinary differential equations | In this paper the performance of a parallel iterated Runge-Kutta method is compared versus those of the serial fouth order Runge-Kutta and Dormand-Prince methods. It was found that, typically, the runtime for the parallel method is comparable to that of the serial versions, thought it uses considerably more computation... | \section{Introduction} \label{sec:intro}
The numerical solution of an initial value problem given as a system of
ordinary differential equations (ODEs) is often required in engineering and
applied sciences, and is less common, but not unusual in pure sciences.
For precisely estimating asymptotic properties of the so... | {
"timestamp": "2016-01-12T02:08:26",
"yymm": "1601",
"arxiv_id": "1601.02245",
"language": "en",
"url": "https://arxiv.org/abs/1601.02245",
"abstract": "In this paper the performance of a parallel iterated Runge-Kutta method is compared versus those of the serial fouth order Runge-Kutta and Dormand-Prince ... |
https://arxiv.org/abs/1808.10038 | Theoretical Linear Convergence of Unfolded ISTA and its Practical Weights and Thresholds | In recent years, unfolding iterative algorithms as neural networks has become an empirical success in solving sparse recovery problems. However, its theoretical understanding is still immature, which prevents us from fully utilizing the power of neural networks. In this work, we study unfolded ISTA (Iterative Shrinkage... | \section{Introduction}
\vspace{-0.5em}
This paper aims to recover a sparse vector $x^\ast$ from its noisy linear
measurements:
\begin{equation}
\label{eq:linear_model}
b = A x^\ast + \varepsilon,
\end{equation}
where $ b\in\mathbb{R}^m $, $ x\in\mathbb{R}^n $,
$ A\in\mathbb{R}^{m \times n} $, $ \varepsilon\in\mathbb{... | {
"timestamp": "2018-11-06T02:18:51",
"yymm": "1808",
"arxiv_id": "1808.10038",
"language": "en",
"url": "https://arxiv.org/abs/1808.10038",
"abstract": "In recent years, unfolding iterative algorithms as neural networks has become an empirical success in solving sparse recovery problems. However, its theor... |
https://arxiv.org/abs/2105.00266 | Data-driven discovery of Green's functions with human-understandable deep learning | There is an opportunity for deep learning to revolutionize science and technology by revealing its findings in a human interpretable manner. To do this, we develop a novel data-driven approach for creating a human-machine partnership to accelerate scientific discovery. By collecting physical system responses under exci... | \section*{Results}
\subsection*{Deep learning Green's functions.}
\begin{figure*}[ht!]
\centering
\vspace{0.5cm}
\begin{overpic}[width=\textwidth, trim=0 0 0 0,clip]{Figure/figure1-final.pdf}
\put(15,50){\textbf{A}}
\put(16,33){\textbf{B}}
\put(16,17){\textbf{C}}
\put(46,47){\textbf{D}}
\put(51,38){\textbf{E}}
\put(51... | {
"timestamp": "2022-03-14T01:16:04",
"yymm": "2105",
"arxiv_id": "2105.00266",
"language": "en",
"url": "https://arxiv.org/abs/2105.00266",
"abstract": "There is an opportunity for deep learning to revolutionize science and technology by revealing its findings in a human interpretable manner. To do this, w... |
https://arxiv.org/abs/2006.07458 | Projection Robust Wasserstein Distance and Riemannian Optimization | Projection robust Wasserstein (PRW) distance, or Wasserstein projection pursuit (WPP), is a robust variant of the Wasserstein distance. Recent work suggests that this quantity is more robust than the standard Wasserstein distance, in particular when comparing probability measures in high-dimensions. However, it is rule... | \section{Acknowledgements}
We would like to thank Minhui Huang for very helpful discussion with the proof of Lemma 3.4 and Theorem 3.7. This work is supported in part by the Mathematical Data Science program of the Office of Naval Research under grant number N00014-18-1-2764.
\bibliographystyle{plainnat}
\section{Con... | {
"timestamp": "2021-02-09T02:08:15",
"yymm": "2006",
"arxiv_id": "2006.07458",
"language": "en",
"url": "https://arxiv.org/abs/2006.07458",
"abstract": "Projection robust Wasserstein (PRW) distance, or Wasserstein projection pursuit (WPP), is a robust variant of the Wasserstein distance. Recent work sugges... |
https://arxiv.org/abs/2203.11338 | Fast Toeplitz eigenvalue computations, joining interpolation-extrapolation matrix-less algorithms and simple-loop conjectures: the preconditioned setting | Under appropriate technical assumptions, the simple-loop theory allows to deduce various types of asymptotic expansions for the eigenvalues of Toeplitz matrices $T_{n}(f)$ generated by a function $f$, unfortunately, such a theory is not available in the preconditioning setting, that is for matrices of the form $T_{n}^{... | \section{Introduction}
In this work we design and test fast procedures for the computation of all the eigenvalues of large preconditioned Toeplitz matrices of the form $X_{n}\equiv T_{n}^{-1}(g)T_{n}(l)$ with $g,l$ even, real-valued on $Q\equiv(-\pi,\pi)$, $g>0$ on $(0,\pi)$ such that $f\equiv\frac{l}{g}$ is monotone ... | {
"timestamp": "2022-03-23T01:06:11",
"yymm": "2203",
"arxiv_id": "2203.11338",
"language": "en",
"url": "https://arxiv.org/abs/2203.11338",
"abstract": "Under appropriate technical assumptions, the simple-loop theory allows to deduce various types of asymptotic expansions for the eigenvalues of Toeplitz ma... |
https://arxiv.org/abs/2006.14552 | Practical Trade-Offs for the Prefix-Sum Problem | Given an integer array A, the prefix-sum problem is to answer sum(i) queries that return the sum of the elements in A[0..i], knowing that the integers in A can be changed. It is a classic problem in data structure design with a wide range of applications in computing from coding to databases. In this work, we propose a... |
\section{The $b$-ary Fenwick-Tree}\label{sec:fentree_bary}
The classic {\method{Fenwick-Tree}} we described in Section~\ref{sec:fentree}
exploits the base-2 representation of a number in $[0,n)$
to support {\code{sum}} and {\code{update}} in time $O(\log n)$.
If we change the base of the representation to a value $b... | {
"timestamp": "2020-10-08T02:02:51",
"yymm": "2006",
"arxiv_id": "2006.14552",
"language": "en",
"url": "https://arxiv.org/abs/2006.14552",
"abstract": "Given an integer array A, the prefix-sum problem is to answer sum(i) queries that return the sum of the elements in A[0..i], knowing that the integers in ... |
https://arxiv.org/abs/1702.05659 | On Loss Functions for Deep Neural Networks in Classification | Deep neural networks are currently among the most commonly used classifiers. Despite easily achieving very good performance, one of the best selling points of these models is their modular design - one can conveniently adapt their architecture to specific needs, change connectivity patterns, attach specialised layers, ... | \section{Introduction}
For the last few years the Deep Learning (DL) research has been rapidly developing.
It evolved from tricky pretraining routines~\cite{larochelle2009exploring} to a highly modular, customisable framework for building
machine learning systems for various problems, spanning from image recognition~\... | {
"timestamp": "2017-02-21T02:05:11",
"yymm": "1702",
"arxiv_id": "1702.05659",
"language": "en",
"url": "https://arxiv.org/abs/1702.05659",
"abstract": "Deep neural networks are currently among the most commonly used classifiers. Despite easily achieving very good performance, one of the best selling point... |
https://arxiv.org/abs/2112.07184 | Calibrated and Sharp Uncertainties in Deep Learning via Density Estimation | Accurate probabilistic predictions can be characterized by two properties -- calibration and sharpness. However, standard maximum likelihood training yields models that are poorly calibrated and thus inaccurate -- a 90% confidence interval typically does not contain the true outcome 90% of the time. This paper argues t... |
\section{Obtaining Calibrated And Sharp Uncertainties in Practice}\label{sec:algorithm}
\section{Enforcing Distribution Calibration via Density Estimation}\label{sec:algorithm}
This section introduces algorithms that ensure the distirbution calibration of any predictive machine learning model while main... | {
"timestamp": "2021-12-15T02:11:11",
"yymm": "2112",
"arxiv_id": "2112.07184",
"language": "en",
"url": "https://arxiv.org/abs/2112.07184",
"abstract": "Accurate probabilistic predictions can be characterized by two properties -- calibration and sharpness. However, standard maximum likelihood training yiel... |
https://arxiv.org/abs/2302.09729 | Embedding theorems for random graphs with specified degrees | Given an $n\times n$ symmetric matrix $W\in [0,1]^{[n]\times [n]}$, let $\mathcal{G}(n,W)$ be the random graph obtained by independently including each edge $jk$ with probability $W_{jk}$. Given a degree sequence ${\bf d}=(d_1,\ldots, d_n)$, let $\mathcal{G}(n,{\bf d})$ denote a uniformly random graph with degree seque... | \section{Introduction}
Given a degree sequence ${\bf d} = (d_1, \ldots, d_n)$ where $\norm{{\bf d}}_1=\sum_{i=1}^n d_i$ is even, let ${\mathcal G}(n,{\bf d})$ denote a random graph chosen uniformly from the set of graphs on $[n]$ where vertex $i$ has degree $d_i$.
Random graphs with a specified degree sequence are a ... | {
"timestamp": "2023-02-21T02:20:09",
"yymm": "2302",
"arxiv_id": "2302.09729",
"language": "en",
"url": "https://arxiv.org/abs/2302.09729",
"abstract": "Given an $n\\times n$ symmetric matrix $W\\in [0,1]^{[n]\\times [n]}$, let $\\mathcal{G}(n,W)$ be the random graph obtained by independently including eac... |
https://arxiv.org/abs/1702.02918 | The Geometry of Strong Koszul Algebras | Koszul algebras with quadratic Groebner bases, called strong Koszul algebras, are studied. We introduce affine algebraic varieties whose points are in one-to-one correspondence with certain strong Koszul algebras and we investigate the connection between the varieties and the algebras. | \section{Introduction}\label{sec-ntro}The connection between affine algebraic
varieties and commutative rings, especially quotients of commutative polynomial
rings over a field, is well established. In this paper, we introduce a new connection
between affine algebraic varieties and a class of Koszul algebra which a... | {
"timestamp": "2017-02-10T02:06:56",
"yymm": "1702",
"arxiv_id": "1702.02918",
"language": "en",
"url": "https://arxiv.org/abs/1702.02918",
"abstract": "Koszul algebras with quadratic Groebner bases, called strong Koszul algebras, are studied. We introduce affine algebraic varieties whose points are in one... |
https://arxiv.org/abs/2002.03899 | K-bMOM: a robust Lloyd-type clustering algorithm based on bootstrap Median-of-Means | We propose a new clustering algorithm that is robust to the presence of outliers in the dataset. We perform Lloyd-type iterations with robust estimates of the centroids. More precisely, we build on the idea of median-of-means statistics to estimate the centroids, but allow for replacement while constructing the blocks.... | \section{Introduction}
\sloppypar Data scientists have nowadays to deal with massive and
complex datasets, that are often corrupted by outliers. Classical
data mining procedures such as K-means or more general EM algorithms
for instance are however sensitive to the presence of outliers, which
can induce a time consumi... | {
"timestamp": "2020-02-11T02:30:29",
"yymm": "2002",
"arxiv_id": "2002.03899",
"language": "en",
"url": "https://arxiv.org/abs/2002.03899",
"abstract": "We propose a new clustering algorithm that is robust to the presence of outliers in the dataset. We perform Lloyd-type iterations with robust estimates of... |
https://arxiv.org/abs/2012.13769 | Population Quasi-Monte Carlo | Monte Carlo methods are widely used for approximating complicated, multidimensional integrals for Bayesian inference. Population Monte Carlo (PMC) is an important class of Monte Carlo methods, which utilizes a population of proposals to generate weighted samples that approximate the target distribution. The generic PMC... | \section{Introduction}
\label{sec:introduction}
A fundamental challenge in Bayesian inference is the evaluation of integrals involving some multi-dimensional posterior distribution $\pi$. Generally, closed-form analytical solutions are not feasible, and Monte Carlo (MC) methods are often used for approximation. Of suc... | {
"timestamp": "2020-12-29T02:13:31",
"yymm": "2012",
"arxiv_id": "2012.13769",
"language": "en",
"url": "https://arxiv.org/abs/2012.13769",
"abstract": "Monte Carlo methods are widely used for approximating complicated, multidimensional integrals for Bayesian inference. Population Monte Carlo (PMC) is an i... |
https://arxiv.org/abs/2011.12245 | Effect of barren plateaus on gradient-free optimization | Barren plateau landscapes correspond to gradients that vanish exponentially in the number of qubits. Such landscapes have been demonstrated for variational quantum algorithms and quantum neural networks with either deep circuits or global cost functions. For obvious reasons, it is expected that gradient-based optimizer... | \section{Introduction}
Parameterized quantum circuits offer a flexible paradigm for programming Noisy Intermediate Scale Quantum (NISQ) computers. These circuits are utilized in both Variational Quantum Algorithms (VQAs)~\cite{cerezo2020variationalreview,bharti2021noisy,peruzzo2014variational,mcclean2016theory,farhi2... | {
"timestamp": "2021-10-04T02:06:11",
"yymm": "2011",
"arxiv_id": "2011.12245",
"language": "en",
"url": "https://arxiv.org/abs/2011.12245",
"abstract": "Barren plateau landscapes correspond to gradients that vanish exponentially in the number of qubits. Such landscapes have been demonstrated for variationa... |
https://arxiv.org/abs/2208.06151 | Unifying local and global model explanations by functional decomposition of low dimensional structures | We consider a global representation of a regression or classification function by decomposing it into the sum of main and interaction components of arbitrary order. We propose a new identification constraint that allows for the extraction of interventional SHAP values and partial dependence plots, thereby unifying loca... | \section{Introduction}
In the early years of machine learning interpretability research, the focus was mostly on global feature importance methods that assign a single importance value to each feature. More recently, the attention has shifted towards local interpretability methods, which provide explanations for indiv... | {
"timestamp": "2022-08-15T02:07:03",
"yymm": "2208",
"arxiv_id": "2208.06151",
"language": "en",
"url": "https://arxiv.org/abs/2208.06151",
"abstract": "We consider a global representation of a regression or classification function by decomposing it into the sum of main and interaction components of arbitr... |
https://arxiv.org/abs/2004.04728 | Hyperbolic metrics on open subsests of Ptolemaic spaces with sharp parameter bounds | It is shown that a construction of Z. Zhang and Y. Xiao on open subsets of Ptolemaic spaces yields, when the subset has boundary containing at least two points, metrics that are Gromov hyperbolic with parameter $\log 2$ and strongly hyperbolic with parameter $1$ with no further conditions on the open set. A class of ex... | \section*{Introduction}
\blfootnote{2010 Mathematics Subject Classification: 51M10, 53C23.}
\blfootnote{keywords: Ptolemaic, Gromov hyperbolic, strongly hyperbolic,
metric space.}
In this paper a construction in \cite{ZX} is applied to produce a metric on
an open subset of a Ptolemaic space. When the open set has bo... | {
"timestamp": "2020-07-14T02:18:41",
"yymm": "2004",
"arxiv_id": "2004.04728",
"language": "en",
"url": "https://arxiv.org/abs/2004.04728",
"abstract": "It is shown that a construction of Z. Zhang and Y. Xiao on open subsets of Ptolemaic spaces yields, when the subset has boundary containing at least two p... |
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