The limit spaces of two-dimensional manifolds with uniformly bounded integral curvature

We study the class of closed -dimensional Riemannian manifolds with uniformly bounded diameter and total absolute curvature. Our first theorem states that this class of manifolds is precompact with respect to the Gromov-Hausdorff distance. Our goal in this paper is to completely characterize the topological structure of all the limit spaces of the class of manifolds, which are, in general, not topological manifolds and even may not be locally -connected. We also study the limit of -manifolds with -curvature bound for .

     

The hitting time for a Cox risk process

This paper investigates the hitting time of a Cox risk process. The relationship between the hitting time of the Cox risk process and the classical risk process is established and an explicit expression of the Laplace–Stieltjes transform of the hitting time is derived by the probability method. Similarly, we derive the explicit expression of the Laplace–Stieltjes transform of the last exit time. Further, we study the situation when the intensity process is an n$n$-state Markov process.     

A limit theorem for particle numbers in bounded domains of a branching diffusion process

We shall study the asymptotic behavior of the particle numbers in bounded domains of a binary splitting one-dimensional branching diffusion process. We shall give a Yaglom type limit theorem in the so-called locally subcritical case, and almost sure convergence of the normalized particle number in the locally supercritical case.     

A method for generating uniformly distributed points onN-dimensional spheres

Summary In this note a computer procedure to transform uniform random variables into random points uniformly distributed on an N-dimensional sphere is presented. The procedure is much simpler than the ones thus far published.     

A uniformly convergent scheme for a system of reaction–diffusion equations

In this work a system of two parabolic singularly perturbed equations of reaction–diffusion type is considered. The asymptotic behaviour of the solution and its partial derivatives is given. A decomposition of the solution in its regular and singular parts has been used for the asymptotic analysis of the spatial derivatives. To approximate the solution we consider the implicit Euler method for time stepping and the central difference scheme for spatial discretization on a special piecewise uniform Shishkin mesh. We prove that this scheme is uniformly convergent, with respect to the diffusion parameters, having first-order convergence in time and almost second-order convergence in space, in the discrete maximum norm. Numerical experiments illustrate the order of convergence proved theoretically.     

A von Neumann theorem for uniformly distributed sequences of partitions

In this paper we consider permutations of sequences of partitions, obtaining a result which parallels von Neumann’s theorem on permutations of dense sequences and uniformly distributed sequences of points.     

The threshold contact process: A continuum limit

Summary In the threshold contact process on thed-dimensional integer lattice with ranger, healthy sites become infected at rate if they have at least one infectedr-neighbour, and recover at rate 1. We show that the critical value _c (r) is asymptotic tor ^–d _c asr, where _c is the critical value of the birth rate for a continuum threshold contact process which may be described in terms of an oriented continuous percolation model driven by a Poisson process of rate ind+1 dimensions. We have bounds of 0.7320 < _c < 3 ford=1.     

A limit theorem for a supercritical branching process

The limit distribution if calculated for the time at which a supercritical branching process reaches the level x, when x.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 585–592, May, 1971.     

An empirical process central limit theorem for dependent non-identically distributed random variables

This paper establishes a central limit theorem (CLT) for empirical processes indexed by smooth functions. The underlying random variables may be temporally dependent and non-identically distributed. In particular, the CLT holds for near epoch dependent (i.e., functions of mixing processes) triangular arrays, which include strong mixing arrays, among others. The results apply to classes of functions that have series expansions. The proof of the CLT is particularly simple; no chaining argument is required. The results can be used to establish the asymptotic normality of semiparametric estimators in time series contexts. An example is provided.     

A numerical scheme for regularized anisotropic curve shortening flow

Realistic interfacial energy densities are often non-convex, which results in backward parabolic behavior of the corresponding anisotropic curve shortening flow, thereby inducing phenomena such as the formation of corners and facets. Adding a term that is quadratic in the curvature to the interfacial energy yields a regularized evolution equation for the interface, which is fourth-order parabolic. Using a semi-implicit time discretization, we present a variational formulation of this equation, which allows the use of linear finite elements. The resulting linear system is shown to be uniquely solvable. We also present numerical examples.     

A finite volume scheme for anisotropic diffusion problems

A new finite volume for the discretization of anisotropic diffusion problems on general unstructured meshes in any space dimension is presented. The convergence of the approximate solution and its discrete gradient is proven. The efficiency of the scheme is illustrated by numerical results. To cite this article: R. Eymard et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

A diffusion approximation for limit order book models

This paper derives a diffusion approximation for a sequence of discrete-time one-sided limit order book models with non-linear state dependent order arrival and cancellation dynamics. The discrete time sequences are specified in terms of an ${\mathbb{R}}_{+}$-valued best bid price process and an $L_{loc}^2$-valued volume process. It is shown that under suitable assumptions the sequence of interpolated discrete time models is relatively compact in a localized sense and that any limit point satisfies a certain infinite dimensional SDE. Under additional assumptions on the dependence structure we construct two classes of models, which fit in the general framework, such that the limiting SDE admits a unique solution and thus the discrete dynamics converge to a diffusion limit in a localized sense.     

Hydrodynamical limit for a nongradient system: The generalized symmetric exclusion process

We consider a symmetric simple exclusion process where at most two particles per site are permitted. This model turns out to be nongradient. We prove that the particles' densities, under a diffusive rescaling of space and time, converge to the solution of a diffusion equation. We give a variational characterization of the diffusion coefficent. We also prove, for the generator of the process in finite volume, a lower bound on the spectral gap uniform in the volume. © 1994 John Wiley & Sons, Inc.     

Fractional diffusion limit for a stochastic kinetic equation

We study the stochastic fractional diffusive limit of a kinetic equation involving a small parameter and perturbed by a smooth random term. Generalizing the method of perturbed test functions, under an appropriate scaling for the small parameter, and with the moment method used in the deterministic case, we show the convergence in law to a stochastic fluid limit involving a fractional Laplacian.     

A limit theorem for branching one-dimensional periodic diffusion processes

We give almost sure convergence of appropriately normalized particle numbers in bounded domains of locally supercritical branching diffusion processes with one-dimensional periodic diffusions as their non-branching part processes. Some spectral properties of periodic diffusion operators including Hill's ones are also studied.