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url stringlengths 31 38	title stringlengths 7 229	abstract stringlengths 44 2.87k	text stringlengths 319 2.51M	meta dict
https://arxiv.org/abs/1207.1228	1D analysis of 2D isotropic random walks	Many stochastic systems in physics and biology are investigated by recording the two-dimensional (2D) positions of a moving test particle in regular time intervals. The resulting sample trajectories are then used to induce the properties of the underlying stochastic process. Often, it can be assumed a priori that the u...	\section*{Quantifying 2D trajectories} We consider a measured trajectory (Fig.1) that consists of $N+1$ discrete two-dimensional points $\vec{R}_t=(x_t,y_t)$ with $t=0\ldots N$, sampled in equal time intervals $\delta t_{rec}$. In the following, it is implicitly assumed that all absolute times $t$ and lag times $\Delt...	{ "timestamp": "2012-08-27T02:01:24", "yymm": "1207", "arxiv_id": "1207.1228", "language": "en", "url": "https://arxiv.org/abs/1207.1228", "abstract": "Many stochastic systems in physics and biology are investigated by recording the two-dimensional (2D) positions of a moving test particle in regular time in...
https://arxiv.org/abs/gr-qc/9510013	Normal frames for non-Riemannian connections	The principal properties of geodesic normal coordinates are the vanishing of the connection components and first derivatives of the metric components at some point. It is well-known that these hold only at points where the connection has vanishing torsion and non-metricity. However, it is shown that normal frames, poss...	\section{Introduction} Almost the defining property of general relativity is its coordinate and frame independence, a feature which becomes obvious when tensor notation is used. Nevertheless, it is frequently convenient to single out certain coordinate and frame choices in order to simplify particular calculations or...	{ "timestamp": "1995-10-09T10:44:18", "yymm": "9510", "arxiv_id": "gr-qc/9510013", "language": "en", "url": "https://arxiv.org/abs/gr-qc/9510013", "abstract": "The principal properties of geodesic normal coordinates are the vanishing of the connection components and first derivatives of the metric component...
https://arxiv.org/abs/1105.1158	Regularity properties of nonlocal minimal surfaces via limiting arguments	We prove an improvement of flatness result for nonlocal minimal surfaces which is independent of the fractional parameter $s$ when $s\rightarrow 1^-$.As a consequence, we obtain that all the nonlocal minimal cones are flat and that all the nonlocal minimal surfaces are smooth when the dimension of the ambient space is ...	\section{Notation}\label{notation} A point~$x\in {\mathbb R}^n$ will be often written in coordinates as~$x=(x',x_n)\in{\mathbb R}^{n-1} \times{\mathbb R}$. The complement of a set~$\Omega\subseteq{\mathbb R}^n$ will be denoted by~$\CC \Omega:={\mathbb R}^n\setminus\Omega$. For any $P\in{\mathbb R}^n$ and $\rho>0$, we...	{ "timestamp": "2013-02-07T02:02:20", "yymm": "1105", "arxiv_id": "1105.1158", "language": "en", "url": "https://arxiv.org/abs/1105.1158", "abstract": "We prove an improvement of flatness result for nonlocal minimal surfaces which is independent of the fractional parameter $s$ when $s\\rightarrow 1^-$.As a ...
https://arxiv.org/abs/1808.00223	The power spectrum indicator: A new, efficient method for the early detection of chaos	To determine the regular or chaotic nature of the orbits in dynamical systems can be quite an issue. In this article, following Vozikis et al. (2000), we propose a new tool, namely, the Power Spectrum Indicator (PSI), $\psi^2$, that enables us to determine, as early as posible, whether an orbit of a two-dimensional map...	\section{Introduction} A major issue in studying non-integrable dynamical systems is the determination, as early as posible, of the orbits' (chaotic or not) nature. In the pioneering work of H\'enon and Heiles (1964), when the related research was limited to two-dimensional $(2D)$ systems, the study of the orbits' natu...	{ "timestamp": "2018-08-02T02:07:22", "yymm": "1808", "arxiv_id": "1808.00223", "language": "en", "url": "https://arxiv.org/abs/1808.00223", "abstract": "To determine the regular or chaotic nature of the orbits in dynamical systems can be quite an issue. In this article, following Vozikis et al. (2000), we ...
https://arxiv.org/abs/2007.05844	Weak normality properties in $Ψ$-spaces	Almost disjoint families of true cardinality $\mathfrak{c}$ are used to produce an example of a mildly-normal not partly-normal $\Psi$-space and a quasi-normal not almost-normal $\Psi$-space. This is related with a problem posed by Lufti Kalantan where he asks whether there exists a mad family so that the related Mrówk...	\section{Introduction} \noindent Mr\'owka-Isbell $\Psi$-spaces give a number of interesting counterexamples in many areas of topology including normality and related covering properties (\cite{Mr}, \cite{GJ}). $\Psi$-spaces associated to maximal almost disjoint families are never normal. Weakenings of normality have ...	{ "timestamp": "2020-07-14T02:11:47", "yymm": "2007", "arxiv_id": "2007.05844", "language": "en", "url": "https://arxiv.org/abs/2007.05844", "abstract": "Almost disjoint families of true cardinality $\\mathfrak{c}$ are used to produce an example of a mildly-normal not partly-normal $\\Psi$-space and a quasi...
https://arxiv.org/abs/1501.04918	A density property for fractional weighted Sobolev spaces	In this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev space can be approximated by a smooth function with compact support.The additional difficulty in this nonlocal setting is caused by the fact that the weights are not nece...	\section{Introduction} Goal of this paper is to provide an approximation result by smooth and compactly supported functions for a fractional Sobolev space with weights that are not necessarily translation invariant. The functional framework is the following. Given $s\in(0,1)$, $p\in(1,+\infty)$ and \begin{equation}...	{ "timestamp": "2015-01-21T02:14:20", "yymm": "1501", "arxiv_id": "1501.04918", "language": "en", "url": "https://arxiv.org/abs/1501.04918", "abstract": "In this paper we show a density property for fractional weighted Sobolev spaces. That is, we prove that any function in a fractional weighted Sobolev spac...
https://arxiv.org/abs/2008.02238	On the equilibrium shape of a crystal	A solution is given to a long-standing open problem posed by Almgren.	\section{Introduction} According to thermodynamics, the equilibrium shape of a small drop of water or a small crystal minimizes the free energy under a mass constraint. The phenomenon was independently discovered by W. Gibbs in 1878 \cite{G} and P. Curie in 1885 \cite{Crist}. Assuming the gravitational effect is negl...	{ "timestamp": "2022-01-03T02:16:52", "yymm": "2008", "arxiv_id": "2008.02238", "language": "en", "url": "https://arxiv.org/abs/2008.02238", "abstract": "A solution is given to a long-standing open problem posed by Almgren.", "subjects": "Analysis of PDEs (math.AP)", "title": "On the equilibrium shape o...
https://arxiv.org/abs/1508.00381	Trojan dynamics well approximated by a new Hamiltonian normal form	We revisit a classical perturbative approach to the Hamiltonian related to the motions of Trojan bodies, in the framework of the Planar Circular Restricted Three-Body Problem (PCRTBP), by introducing a number of key new ideas in the formulation. In some sense, we adapt the approach of Garfinkel (1977) to the context of...	\section{Introduction}\label{sec:intro} Series expansions in terms of small physical parameters are a common way of dealing with dynamical models in Celestial Mechanics. One case where this approach has been extensively used is the problem of Trojan motion. The so-called Trojan stability problem, i.e. the study of the...	{ "timestamp": "2015-08-04T02:15:00", "yymm": "1508", "arxiv_id": "1508.00381", "language": "en", "url": "https://arxiv.org/abs/1508.00381", "abstract": "We revisit a classical perturbative approach to the Hamiltonian related to the motions of Trojan bodies, in the framework of the Planar Circular Restricte...
https://arxiv.org/abs/2011.10498	Weighted automata are compact and actively learnable	We show that weighted automata over the field of two elements can be exponentially more compact than non-deterministic finite state automata. To show this, we combine ideas from automata theory and communication complexity. However, weighted automata are also efficiently learnable in Angluin's minimal adequate teacher ...	\section{Introduction} Weighted automata (WAs) are an alternative model of finite state machines and a natural way to represent monoids. They have received a lot of interest in the the learning community because they provide an interesting way to represent and analyze sequence data, such as music~\cite{MMW09} or text ...	{ "timestamp": "2020-11-23T02:17:30", "yymm": "2011", "arxiv_id": "2011.10498", "language": "en", "url": "https://arxiv.org/abs/2011.10498", "abstract": "We show that weighted automata over the field of two elements can be exponentially more compact than non-deterministic finite state automata. To show this...
https://arxiv.org/abs/2202.06482	Splitting numerical integration for matrix completion	Low rank matrix approximation is a popular topic in machine learning. In this paper, we propose a new algorithm for this topic by minimizing the least-squares estimation over the Riemannian manifold of fixed-rank matrices. The algorithm is an adaptation of classical gradient descent within the framework of optimization...	\section{Introduction} Computing an efficient and reliable low-rank approximation of a given matrix is a fundamental task in many machine learning problems, such as principal component analysis~\cite{jolliffe2002principal}, face recognition~\cite{muller2004singular} and large scale data compression~\cite{drineas2006s...	{ "timestamp": "2022-02-15T02:34:56", "yymm": "2202", "arxiv_id": "2202.06482", "language": "en", "url": "https://arxiv.org/abs/2202.06482", "abstract": "Low rank matrix approximation is a popular topic in machine learning. In this paper, we propose a new algorithm for this topic by minimizing the least-squ...
https://arxiv.org/abs/2002.02657	Optimization of Structural Similarity in Mathematical Imaging	It is now generally accepted that Euclidean-based metrics may not always adequately represent the subjective judgement of a human observer. As a result, many image processing methodologies have been recently extended to take advantage of alternative visual quality measures, the most prominent of which is the Structural...	\section{Introduction} Image denoising, deblurring, and inpainting are only a few examples of standard image processing tasks which are traditionally solved through numerical optimization. In most cases, the objective function associated with such problems is expressed as the sum of a {\em data fidelity term} $f$ and a...	{ "timestamp": "2020-02-10T02:06:28", "yymm": "2002", "arxiv_id": "2002.02657", "language": "en", "url": "https://arxiv.org/abs/2002.02657", "abstract": "It is now generally accepted that Euclidean-based metrics may not always adequately represent the subjective judgement of a human observer. As a result, m...
https://arxiv.org/abs/0801.4500	On the motion under focal attraction in a rotating medium	New results are established here on the phase portraits and bifurcations of the kinematic model in a system of ODE's, first presented by H.K. Wilson in his 1971 book, and by him attributed to L. Markus (unpublished). A new, self-sufficient, study which extends Wilson's result and allows an essential conclusion for the ...	\section{Introduction}\label{sec:1} Consider the following family of planar differential equations depending on three real parameters $(\omega, v, R)$, with $\omega \geq 0, \; v > 0, \; R > 0$, \begin{equation} \label{eq:01} \begin{array}{lclr} \dot{x} & = & -\omega y + v\frac{R - x}{\sqrt{(R - x)^2 + y^2}}, ...	{ "timestamp": "2008-01-29T15:21:16", "yymm": "0801", "arxiv_id": "0801.4500", "language": "en", "url": "https://arxiv.org/abs/0801.4500", "abstract": "New results are established here on the phase portraits and bifurcations of the kinematic model in a system of ODE's, first presented by H.K. Wilson in his ...
https://arxiv.org/abs/1908.08420	When is the Sum of Two Closed Subgroups Closed in a Locally Compact Abelian Group	Locally compact abelian groups are classified in which the sum of any two closed subgroups is itself closed. This amounts to reproving and extending results by Yu.~N.~Mukhin from 1970. Namely we contribute a complete classification of all totally disconnected \lca\ groups with $X+Y$ closed for any closed subgroups $X$ ...	\section{Introduction and Main Results} \label{s:intro-ab} It was {\sc R. Dedekind}, who in 1877 proved the modular law for the subgroup lattice of a certain abelian group (see \cite{Dedekind77}). However, his proof works for {\em any} abelian group. In 1970 Yu.~N.~Mukhin investigated the analogous property for \...	{ "timestamp": "2019-08-23T02:13:08", "yymm": "1908", "arxiv_id": "1908.08420", "language": "en", "url": "https://arxiv.org/abs/1908.08420", "abstract": "Locally compact abelian groups are classified in which the sum of any two closed subgroups is itself closed. This amounts to reproving and extending resul...
https://arxiv.org/abs/1102.5464	Lattices of subalgebras of Leibniz algebras	I describe the lattice of subalgebras of a one-generator Leibniz algebra. Using this, I show that, apart from one special case, a lattice isomorphism between Leibniz algebras L, L' maps the Leibniz kernel of L to that of L'.	\section{Introduction} \label{sec-intro} A lot is known about the lattices of subalgebras of Lie algebras. See for example, Barnes \cite{LIsos}, \cite{Lautos}, Goto \cite{Goto}, Towers \cite{T-dimpres}, \cite{T-Lautos}, \cite{smod}. In this paper, I look at some basic results needed to extend these results to Leibniz...	{ "timestamp": "2011-03-01T02:01:15", "yymm": "1102", "arxiv_id": "1102.5464", "language": "en", "url": "https://arxiv.org/abs/1102.5464", "abstract": "I describe the lattice of subalgebras of a one-generator Leibniz algebra. Using this, I show that, apart from one special case, a lattice isomorphism betwee...
https://arxiv.org/abs/1611.08253	A geometric proof of Lück's vanishing theorem for the first $L^2$-Betti number of the total space of a fibration	A significant theorem of Lück says that the first $L^2$-Betti number of the total space of a fibration vanishes under some conditions on the fundamental groups. The proof is based on constructions on chain complexes. In the present paper, we translate the proof into the world of CW-complexes to make it more accessible.	\section{Introduction} In \cite{Lueck}, Lück proved the following significant theorem: \begin{theorem}[{\cite[Theorem 3.1]{Lueck}}] Let $F\xrightarrow{i} E\xrightarrow{p}B$ be a fibration of connected CW-complexes such that $F$ and $B$ have finite $2$-skeletons. Then $E$ has finite $2$-skeleton up to homotopy. If the ...	{ "timestamp": "2016-11-28T02:07:16", "yymm": "1611", "arxiv_id": "1611.08253", "language": "en", "url": "https://arxiv.org/abs/1611.08253", "abstract": "A significant theorem of Lück says that the first $L^2$-Betti number of the total space of a fibration vanishes under some conditions on the fundamental g...
https://arxiv.org/abs/1111.2730	A statistical and computational theory for robust and sparse Kalman smoothing	Kalman smoothers reconstruct the state of a dynamical system starting from noisy output samples. While the classical estimator relies on quadratic penalization of process deviations and measurement errors, extensions that exploit Piecewise Linear Quadratic (PLQ) penalties have been recently proposed in the literature. ...	\section{Introduction} Consider the following discrete-time linear state-space model \begin{equation} \label{LinearGaussModel} \begin{array}{rcl} x_1&=&x_0 + w_1 \\ x_k & = & G_k x_{k-1} + w_k, \qquad k=2,3,\ldots,N \\ z_k & = & H_k x_k + v_k, \quad \qquad k=1,2,\ldots,N \end{array} \end{...	{ "timestamp": "2011-11-14T02:02:13", "yymm": "1111", "arxiv_id": "1111.2730", "language": "en", "url": "https://arxiv.org/abs/1111.2730", "abstract": "Kalman smoothers reconstruct the state of a dynamical system starting from noisy output samples. While the classical estimator relies on quadratic penalizat...
https://arxiv.org/abs/2102.00906	On upper bounds for the count of elite primes	We look at upper bounds for the count of certain primes related to the Fermat numbers $F_n=2^{2^n}+1$ called elite primes. We first note an oversight in a result of Krizek, Luca and Somer and give the corrected, slightly weaker upper bound. We then assume the Generalized Riemann Hypothesis for Dirichlet L functions and...	\section{\@startsection {section}{1}{\z@} {-30pt \@plus -1ex \@minus -.2ex} {2.3ex \@plus.2ex} {\normalfont\normalsize\bfseries\boldmath}} \renewcommand\subsection{\@startsection{subsection}{2}{\z@} {-3.25ex\@plus -1ex \@minus -.2ex} {1.5ex \@plus .2ex} {\normalfont\normalsize\bfseries\boldmath}} \renewcommand{\@secc...	{ "timestamp": "2021-02-02T02:36:37", "yymm": "2102", "arxiv_id": "2102.00906", "language": "en", "url": "https://arxiv.org/abs/2102.00906", "abstract": "We look at upper bounds for the count of certain primes related to the Fermat numbers $F_n=2^{2^n}+1$ called elite primes. We first note an oversight in a...
https://arxiv.org/abs/2005.03219	$L^{q}$-error estimates for approximation of irregular functionals of random vectors	Avikainen showed that, for any $p,q \in [1,\infty)$, and any function $f$ of bounded variation in $\mathbb{R}$, it holds that $\mathbb{E}[\|f(X)-f(\widehat{X})\|^{q}] \leq C(p,q) \mathbb{E}[\|X-\widehat{X}\|^{p}]^{\frac{1}{p+1}}$, where $X$ is a one-dimensional random variable with a bounded density, and $\widehat{X}$ is a...	\section{Introduction}\label{sec_1} Numerical analysis for stochastic differential equations (SDEs) is a significant issue in stochastic calculus, from both theory and practical points of view. In order to approximate the solution of the SDE $\mathrm{d} X(t)=b(X(t)) \mathrm{d} t+\sigma(X(t))\mathrm{d} B(t)$ driven by...	{ "timestamp": "2020-11-30T02:06:16", "yymm": "2005", "arxiv_id": "2005.03219", "language": "en", "url": "https://arxiv.org/abs/2005.03219", "abstract": "Avikainen showed that, for any $p,q \\in [1,\\infty)$, and any function $f$ of bounded variation in $\\mathbb{R}$, it holds that $\\mathbb{E}[\|f(X)-f(\\wi...
https://arxiv.org/abs/1703.00102	SARAH: A Novel Method for Machine Learning Problems Using Stochastic Recursive Gradient	In this paper, we propose a StochAstic Recursive grAdient algoritHm (SARAH), as well as its practical variant SARAH+, as a novel approach to the finite-sum minimization problems. Different from the vanilla SGD and other modern stochastic methods such as SVRG, S2GD, SAG and SAGA, SARAH admits a simple recursive framewor...	\section{Introduction} \label{introduction} We are interested in solving a problem of the form \begin{gather}\label{eq:problem} \min_{w\in\R^d} \left\{ \ P(w) \stackrel{\text{def}}{=} \frac{1}{n} \sum_{i\in[n]} f_i(w)\right\}, \end{gather} where each $f_i$, $i \in [n]\stackrel{\text{def}}{=}\{1,\dots,n\}$, is convex w...	{ "timestamp": "2017-06-06T02:03:32", "yymm": "1703", "arxiv_id": "1703.00102", "language": "en", "url": "https://arxiv.org/abs/1703.00102", "abstract": "In this paper, we propose a StochAstic Recursive grAdient algoritHm (SARAH), as well as its practical variant SARAH+, as a novel approach to the finite-su...
https://arxiv.org/abs/2302.06812	Scalable Optimal Multiway-Split Decision Trees with Constraints	There has been a surge of interest in learning optimal decision trees using mixed-integer programs (MIP) in recent years, as heuristic-based methods do not guarantee optimality and find it challenging to incorporate constraints that are critical for many practical applications. However, existing MIP methods that build ...	\section{Additional Details for Section 3: Problem Formulation} \subsection{Proof for Proposition 1} \begin{proof} Given a $k$-dimensional input data $x^k$, denote $\eta_f$ as the number of unique feature values associated with the $f^{th}$ feature. The number of paths from the source to the sink is given by $\|\mathc...	{ "timestamp": "2023-02-15T02:06:07", "yymm": "2302", "arxiv_id": "2302.06812", "language": "en", "url": "https://arxiv.org/abs/2302.06812", "abstract": "There has been a surge of interest in learning optimal decision trees using mixed-integer programs (MIP) in recent years, as heuristic-based methods do no...
https://arxiv.org/abs/2203.08978	Flooding in weighted sparse random graphs of active and passive nodes	This paper discusses first passage percolation and flooding on large weighted sparse random graphs with two types of nodes: active and passive nodes. In mathematical physics passive nodes can be interpreted as closed gates where fluid flow or water cannot pass through and active nodes can be interpreted as open gates w...	\section{Introduction} First passage percolation is one of the classical models in probability theory and mathematical physics introduced by Hammersley and Welsh \cite{HW} in $1965$ as a generalization of Bernoulli percolation \cite{BH,DC}. Its large interest due to the model simplicity and its various applications fro...	{ "timestamp": "2022-03-18T01:08:51", "yymm": "2203", "arxiv_id": "2203.08978", "language": "en", "url": "https://arxiv.org/abs/2203.08978", "abstract": "This paper discusses first passage percolation and flooding on large weighted sparse random graphs with two types of nodes: active and passive nodes. In m...
https://arxiv.org/abs/0909.5388	A Universal Crease Pattern for Folding Orthogonal Shapes	We present a universal crease pattern--known in geometry as the tetrakis tiling and in origami as box pleating--that can fold into any object made up of unit cubes joined face-to-face (polycubes). More precisely, there is one universal finite crease pattern for each number n of unit cubes that need to be folded. This r...	\section{Introduction} An early result in computational origami is that every polyhedral surface can be folded from a large enough square of paper \cite{Demaine-Demaine-Mitchell-2000}. But each such folding uses a different crease pattern. Into how many different shapes can a single crease pattern fold? Our motivatio...	{ "timestamp": "2009-09-29T21:32:58", "yymm": "0909", "arxiv_id": "0909.5388", "language": "en", "url": "https://arxiv.org/abs/0909.5388", "abstract": "We present a universal crease pattern--known in geometry as the tetrakis tiling and in origami as box pleating--that can fold into any object made up of uni...
https://arxiv.org/abs/1506.07779	On phase separation in systems of coupled elliptic equations: asymptotic analysis and geometric aspects	We consider a family of positive solutions to the system of $k$ components \[-\Delta u_{i,\beta} = f(x, u_{i,\beta}) - \beta u_{i,\beta} \sum_{j \neq i} a_{ij} u_{j,\beta}^2 \qquad \text{in $\Omega$}, \] where $\Omega \subset \mathbb{R}^N$ with $N \ge 2$. It is known that uniform bounds in $L^\infty$ of $\{\mathbf{u}_{...	\section{Introduction} The aim of this paper is to prove qualitative properties of positive solutions to competing systems with variational interaction, whose prototype is the coupled Gross-Pitaevskii equation \[ \begin{cases} -\Delta u_{i,\beta} +\lambda_{i,\beta} u_{i,\beta} = \mu_i u_{i,\beta}^3 -\beta u_{i,\beta...	{ "timestamp": "2015-09-04T02:08:54", "yymm": "1506", "arxiv_id": "1506.07779", "language": "en", "url": "https://arxiv.org/abs/1506.07779", "abstract": "We consider a family of positive solutions to the system of $k$ components \\[-\\Delta u_{i,\\beta} = f(x, u_{i,\\beta}) - \\beta u_{i,\\beta} \\sum_{j \\...
https://arxiv.org/abs/2212.09919	Generating Functions for Asymmetric Random Walk Processes With Double Absorbing Barriers	Generating functions for asymmetric step-size paths restricted by two absorbing barriers are derived. The method begins by applying the Lagrange inversion formula to arbitrary powers of roots of the characteristic equation, that being a trinomial, which produces generating function as function (z) of the conditional pr...	\section{Introduction} The one step forward, one step back process is well understood. Examples are given by Feller (1957), Lengyel, T. (2009), Krattenthaler (2000), etc. Because the characteristic function can be expressed as a quadratic, it's possible to generate closed-form, analytic expressions as either sums of ...	{ "timestamp": "2022-12-21T02:04:00", "yymm": "2212", "arxiv_id": "2212.09919", "language": "en", "url": "https://arxiv.org/abs/2212.09919", "abstract": "Generating functions for asymmetric step-size paths restricted by two absorbing barriers are derived. The method begins by applying the Lagrange inversion...
https://arxiv.org/abs/0712.2337	Mould expansions for the saddle-node and resurgence monomials	This article is an introduction to some aspects of Écalle's mould calculus, a powerful combinatorial tool which yields surprisingly explicit formulas for the normalising series attached to an analytic germ of singular vector field or of map. This is illustrated on the case of the saddle-node, a two-dimensional vector f...	\section{Introduction} Mould calculus was developed by J.~\'Ecalle in relation with his Resurgence theory almost thirty years ago (\cite{Eca81}, \cite{dulac}, \cite{Eca93}). The primary goal of this text is to give an introduction to mould calculus, together with an exposition of the way it can be applied to a spec...	{ "timestamp": "2007-12-14T13:35:59", "yymm": "0712", "arxiv_id": "0712.2337", "language": "en", "url": "https://arxiv.org/abs/0712.2337", "abstract": "This article is an introduction to some aspects of Écalle's mould calculus, a powerful combinatorial tool which yields surprisingly explicit formulas for th...
https://arxiv.org/abs/1805.03778	Threshold functions for substructures in random subsets of finite vector spaces	The study of substructures in random objects has a long history, beginning with Erdős and Rényi's work on subgraphs of random graphs. We study the existence of certain substructures in random subsets of vector spaces over finite fields. First we provide a general framework which can be applied to establish coarse thres...	\section{Introduction}\label{sec:introduction} Let $\mathbb{F}_{q}$ denote the finite field with $q$ elements, where $q$ is a prime power, and let $\mathbb{F}_{q}^{n}$ be the $n$-dimensional vector space over this field. Our goal is to establish threshold functions for the existence of certain structures, which we cal...	{ "timestamp": "2018-05-15T02:13:10", "yymm": "1805", "arxiv_id": "1805.03778", "language": "en", "url": "https://arxiv.org/abs/1805.03778", "abstract": "The study of substructures in random objects has a long history, beginning with Erdős and Rényi's work on subgraphs of random graphs. We study the existen...
https://arxiv.org/abs/2207.11299	Rank-constrained Hyperbolic Programming	We extend rank-constrained optimization to general hyperbolic programs (HP) using the notion of matroid rank. For LP and SDP respectively, this reduces to sparsity-constrained LP and rank-constrained SDP that are already well-studied. But for QCQP and SOCP, we obtain new interesting optimization problems. For example, ...	\section{Introduction} In this paper, we study rank-constrained and sparsity-constrained hyperbolic programming (HP). Specifically, we consider four types of HP: linear programming (LP), quadratically constrained quadratic program (QCQP), second order cone programming (SOCP), and semidefinite programming (SDP). Rank-c...	{ "timestamp": "2022-07-26T02:01:09", "yymm": "2207", "arxiv_id": "2207.11299", "language": "en", "url": "https://arxiv.org/abs/2207.11299", "abstract": "We extend rank-constrained optimization to general hyperbolic programs (HP) using the notion of matroid rank. For LP and SDP respectively, this reduces to...
https://arxiv.org/abs/2111.15567	Distribution-free tests of multivariate independence based on center-outward quadrant, Spearman, Kendall, and van der Waerden statistics	Due to the lack of a canonical ordering in ${\mathbb R}^d$ for $d>1$, defining multivariate generalizations of the classical univariate ranks has been a long-standing open problem in statistics. Optimal transport has been shown to offer a solution in which multivariate ranks are obtained by transporting data points to ...	\section{Introduction} The problem of testing for independence between two random variables with unspecified densities has been among the very first applications of rank-based methods in statistical inference. Spearman's correlation coefficient was proposed in the early 1900s \citep{Spearman1904}, and Kendall's ra...	{ "timestamp": "2021-12-01T02:30:38", "yymm": "2111", "arxiv_id": "2111.15567", "language": "en", "url": "https://arxiv.org/abs/2111.15567", "abstract": "Due to the lack of a canonical ordering in ${\\mathbb R}^d$ for $d>1$, defining multivariate generalizations of the classical univariate ranks has been a ...
https://arxiv.org/abs/1806.00230	Invariant property for discontinuous mean-type mappings	It is known that if $M,\,N$ are continuous two-variable means such that $\|M(x,y)-N(x,y)\| < \|x-y\|$ for every $x,\ y$ with $x\ne y$, then there exists a unique invariant mean (which is continuous too).We are looking for invariant means for pairs satisfying the inequality above, but continuity of means is not assumed.In t...	\section{Introduction} The idea of invariant means was first introduced by Gauss \cite{Gau18} who considered so-called arithmetic-geometric means. It was obtained as a limit in the iteration process \Eq{*}{ x_{n+1}=\frac{x_n+y_n}{2},\qquad y_{n+1}=\sqrt{x_ny_n} \qquad (n \in \N_+\cup\{0\}), } where $x_0,\,y_0$ are two...	{ "timestamp": "2018-06-04T02:08:13", "yymm": "1806", "arxiv_id": "1806.00230", "language": "en", "url": "https://arxiv.org/abs/1806.00230", "abstract": "It is known that if $M,\\,N$ are continuous two-variable means such that $\|M(x,y)-N(x,y)\| < \|x-y\|$ for every $x,\\ y$ with $x\\ne y$, then there exists a ...
https://arxiv.org/abs/1606.00176	Large time monotonicity of solutions of reaction-diffusion equations in R^N	In this paper, we consider nonnegative solutions of spatially heterogeneous Fisher-KPP type reaction-diffusion equations in the whole space. Under some assumptions on the initial conditions, including in particular the case of compactly supported initial conditions, we show that, above any arbitrary positive value, the...	\subsubsection*{Framework and main assumptions} The initial condition $u_0$ is in $L^{\infty}(\mathbb{R}^N)$ with $0\le u_0(x)\le 1$ a.e. in $\mathbb{R}^N$ and $u_0$ is non-trivial, in the sense that $\\|u_0\\|_{L^{\infty}(\mathbb{R}^N)}>0$. We also assume that either there exists $\beta>0$ such that \begin{equation}\la...	{ "timestamp": "2016-06-02T02:06:46", "yymm": "1606", "arxiv_id": "1606.00176", "language": "en", "url": "https://arxiv.org/abs/1606.00176", "abstract": "In this paper, we consider nonnegative solutions of spatially heterogeneous Fisher-KPP type reaction-diffusion equations in the whole space. Under some as...
https://arxiv.org/abs/1807.04634	On the Decomposition of Forces	We show that any continuously differentiable force is decomposed into the sum of a Rayleigh force and a gyroscopic force. We also extend this result to piecewise continuously differentiable forces. Our result improves the result on the decomposition of forces in a book by David Merkin and further extends it to piecewis...	\section{Introduction} Generally forces depend on both position variables and velocity variables. Among those forces, gyroscopic forces receive special attention since they do not contribute to any change in the total energy of a mechanical system. Another special type of force is a Rayleigh force, which can be rep...	{ "timestamp": "2018-07-13T02:09:08", "yymm": "1807", "arxiv_id": "1807.04634", "language": "en", "url": "https://arxiv.org/abs/1807.04634", "abstract": "We show that any continuously differentiable force is decomposed into the sum of a Rayleigh force and a gyroscopic force. We also extend this result to pi...
https://arxiv.org/abs/1112.3413	Potential scattering and the continuity of phase-shifts	Let $S(k)$ be the scattering matrix for a Schrödinger operator (Laplacian plus potential) on $\RR^n$ with compactly supported smooth potential. It is well known that $S(k)$ is unitary and that the spectrum of $S(k)$ accumulates on the unit circle only at 1; moreover, $S(k)$ depends analytically on $k$ and therefore its...	\section{Introduction} In this article, we consider scattering in $\RR^n$ due to a nonpositive potential function, which we call a potential well. We denote the potential $-V(x)$, $V \geq 0$, and assume for simplicity that $V$ is smooth and compactly supported. Recall that the Schr\"odinger operator $H = \Delta - V$, ...	{ "timestamp": "2012-03-16T01:01:34", "yymm": "1112", "arxiv_id": "1112.3413", "language": "en", "url": "https://arxiv.org/abs/1112.3413", "abstract": "Let $S(k)$ be the scattering matrix for a Schrödinger operator (Laplacian plus potential) on $\\RR^n$ with compactly supported smooth potential. It is well ...
https://arxiv.org/abs/1509.01420	Anti-Urysohn spaces	All spaces are assumed to be infinite Hausdorff spaces. We call a space "anti-Urysohn" $($AU in short$)$ iff any two non-emty regular closed sets in it intersect. We prove that$\bullet$ for every infinite cardinal ${\kappa}$ there is a space of size ${\kappa}$ in which fewer than $cf({\kappa})$ many non-empty regular c...	\section{Introduction} In this paper ``space'' means ``infinite Hausdorff topological space''. The space $X$ is called {\em anti-Urysohn} ({\em AU}, in short) iff $A\cap B\ne \emptyset$ for any $A,B\in \operatorname{RC}^+(X)$, where $\operatorname{RC}^+(X)$ denotes the family of non-empty regular closed sets in $X...	{ "timestamp": "2015-09-07T02:08:05", "yymm": "1509", "arxiv_id": "1509.01420", "language": "en", "url": "https://arxiv.org/abs/1509.01420", "abstract": "All spaces are assumed to be infinite Hausdorff spaces. We call a space \"anti-Urysohn\" $($AU in short$)$ iff any two non-emty regular closed sets in it ...
https://arxiv.org/abs/1901.06480	Classifying uniformly generated groups	A finite group $G$ is called uniformly generated, if whenever there is a (strictly ascending) chain of subgroups $1<\langle x_1\rangle<\langle x_1,x_2\rangle <\cdots<\langle x_1,x_2,\dots,x_d\rangle=G$, then $d$ is the minimal number of generators of $G$. Our main result classifies the uniformly generated groups with...	\section{Introduction}\label{S1} Let $G$ be a finite group. A chain $1=G_0<G_1<\cdots<G_d=G$ of subgroups of a $G$ is called \emph{unrefinable} if $G_i$ is maximal in $G_{i+1}$ for each $i$. The \emph{length} of $G$, denoted $\ell(G)$, is the maximum length of an unrefinable chain, and the \emph{depth} of $G$, den...	{ "timestamp": "2019-05-31T02:10:43", "yymm": "1901", "arxiv_id": "1901.06480", "language": "en", "url": "https://arxiv.org/abs/1901.06480", "abstract": "A finite group $G$ is called uniformly generated, if whenever there is a (strictly ascending) chain of subgroups $1<\\langle x_1\\rangle<\\langle x_1,x_...
https://arxiv.org/abs/1701.04444	The phases of large networks with edge and triangle constraints	Based on numerical simulation and local stability analysis we describe the structure of the phase space of the edge/triangle model of random graphs. We support simulation evidence with mathematical proof of continuity and discontinuity for many of the phase transitions. All but one of themany phase transitions in this ...	\section{Introduction} \label{SEC:Intro} We use the variational formalism of equilibrium statistical mechanics to analyze large random graphs. More specifically we analyze ``emergent phases", which represent the (nonrandom) large-scale structure of typical large graphs under global constraints on subgraph densities. ...	{ "timestamp": "2017-02-09T02:07:05", "yymm": "1701", "arxiv_id": "1701.04444", "language": "en", "url": "https://arxiv.org/abs/1701.04444", "abstract": "Based on numerical simulation and local stability analysis we describe the structure of the phase space of the edge/triangle model of random graphs. We su...
https://arxiv.org/abs/2008.02143	On the correctness of monadic backward induction	In control theory, to solve a finite-horizon sequential decision problem (SDP) commonly means to find a list of decision rules that result in an optimal expected total reward (or cost) when taking a given number of decision steps. SDPs are routinely solved using Bellman's backward induction. Textbook authors (e.g. Bert...	\section{Introduction} \label{section:introduction} Backward induction is a method introduced by \cite{bellman1957} that is routinely used to solve \emph{finite-horizon sequential decision problems (SDP)}. Such problems lie at the core of many applications in economics, logistics, and computer science \citep{finus...	{ "timestamp": "2021-09-13T02:19:25", "yymm": "2008", "arxiv_id": "2008.02143", "language": "en", "url": "https://arxiv.org/abs/2008.02143", "abstract": "In control theory, to solve a finite-horizon sequential decision problem (SDP) commonly means to find a list of decision rules that result in an optimal e...
https://arxiv.org/abs/2111.00630	Long time decay and asymptotics for the complex mKdV equation	We study the asymptotics of the complex modified Korteweg-de Vries equation\begin{equation} \partial_t u + \partial_x^3 u = \pm \|u\|^2 \partial_x u \end{equation} In the real valued case, it is known that solutions with small, localized initial data exhibit modified scattering for $\|x\| \geq t^{1/3}$, and behave self-s...	\section{Introduction} We study the complex modified Korteweg-de Vries (mKdV) equation \begin{equation}\label{eqn:cmkdv} \partial_t u + \partial_x^3 u = \pm \|u\|^2 \partial_x u \end{equation} This equation appears as a model in nonlinear optics, where it models higher order corrections for waves travelling in a non...	{ "timestamp": "2022-07-29T02:05:50", "yymm": "2111", "arxiv_id": "2111.00630", "language": "en", "url": "https://arxiv.org/abs/2111.00630", "abstract": "We study the asymptotics of the complex modified Korteweg-de Vries equation\\begin{equation*} \\partial_t u + \\partial_x^3 u = \\pm \|u\|^2 \\partial_x u \...
https://arxiv.org/abs/math/9307210	Using sums of squares to prove that certain entire functions have only real zeros	It is shown how sums of squares of real valued functions can be used to give new proofs of the reality of the zeros of the Bessel functions $J_\alpha (z)$ when $\alpha \ge -1,$ confluent hypergeometric functions ${}_0F_1(c\/; z)$ when $c>0$ or $0>c>-1$, Laguerre polynomials $L_n^\alpha(z)$ when $\alpha \ge -2,$ and Jac...	\section{Introduction} In a 1975 survey paper \cite{gg75} on positivity and special functions it was shown how sums of squares of special functions could be used to prove the nonnegativity of the Fej\'er kernel, the positivity of integrals of Bessel functions \cite{gg75B} and of the Cotes' numbers for some Jacobi absc...	{ "timestamp": "1998-09-01T22:23:58", "yymm": "9307", "arxiv_id": "math/9307210", "language": "en", "url": "https://arxiv.org/abs/math/9307210", "abstract": "It is shown how sums of squares of real valued functions can be used to give new proofs of the reality of the zeros of the Bessel functions $J_\\alpha...
https://arxiv.org/abs/math-ph/9911038	Continuous analog of Gauss-Newton method	A continuous analog of Gauss-Newton method for solving nonlinear ill-posed problems is proposed. Its converegence is proved. A numerical example is presented to demonstrate efficiency of the propsed method.	\section{Introduction} \vspace*{-0.5pt} \noindent \setcounter{equation}{0} \renewcommand{\theequation}{1.\arabic{equation}} Let $H_1$ and $H_2$ be real Hilbert spaces and $\varphi:H_1 \to H_2$ a nonlinear operator. Let us consider the equation: \begin{equation}\label{eq1} \varphi(x)=0.\end{equation} We assume ...	{ "timestamp": "1999-11-26T18:46:39", "yymm": "9911", "arxiv_id": "math-ph/9911038", "language": "en", "url": "https://arxiv.org/abs/math-ph/9911038", "abstract": "A continuous analog of Gauss-Newton method for solving nonlinear ill-posed problems is proposed. Its converegence is proved. A numerical example...
https://arxiv.org/abs/2006.02685	Phase transitions in non-linear urns with interacting types	We investigate reinforced non-linear urns with interacting types, and show that where there are three interacting types there are phenomena which do not occur with two types. In a model with three types where the interactions between the types are symmetric, we show the existence of a double phase transition with three...	\section{Introduction and definitions} In general terms, an urn model is a system containing a number of particles of different types (often regarded as balls of different colours, for ease of visualisation). At each time step, a set of particles is sampled from the system, whose contents are then altered depending o...	{ "timestamp": "2020-06-05T02:09:39", "yymm": "2006", "arxiv_id": "2006.02685", "language": "en", "url": "https://arxiv.org/abs/2006.02685", "abstract": "We investigate reinforced non-linear urns with interacting types, and show that where there are three interacting types there are phenomena which do not o...