Convergence and local equilibrium for the one-dimensional nonzero mean exclusion process

We consider finite-range, nonzero mean, one-dimensional exclusion processes on ℤ. We show that, if the initial configuration has ``density' α, then the process converges in distribution to the product Bernoulli measure with mean density α. From this we deduce the strong form of local equilibrium in the hydrodynamic limit for non-product initial measures.     

Two-Wavelet Localization Operators on $$ L^p(\mathbb{R}^n) $$ for the Weyl-Heisenberg Group

We give results on the boundedness and compactness of localization operators with two admissible wavelets on for the Weyl-Heisenberg group.     

Richman extensions of an integral domain

We introduce the notion of a Richman extension of an integral domain, and show its basic properties. We further show that a flat extension need not be a Richman extension, and a Richman extension need not be an LCM-stable extension.     

Efficient parallel algorithms for shortest paths in planar digraphs

Efficient parallel algorithms are presented, on the CREW PRAM model, for generating a succinct encoding of all pairs shortest path information in a directed planar graphG with real-valued edge costs but no negative cycles. We assume that a planar embedding ofG is given, together with a set ofq faces that cover all the vertices. Then our algorithm runs inO(log² n) time and employsO(nq+M(q)) processors (whereM(t) is the number of processors required to multiply twot×t matrices inO(logt) time). Let us note here that wheneverq<n then our processor bound is better than the best previous one (M(n)).O(log² n) time,n-processor algorithms are presented for various subproblems, including that of generating all pairs shortest path information in a directedouterplanar graph. Our work is based on the fundamental hammock-decomposition technique of G. Frederickson. We achieve this decomposition inO(logn log*n) parallel time by usingO(n) processors. The hammock-decomposition seems to be a fundamental operation that may help in improving efficiency of many parallel (and sequential) graph algorithms.This work was partially supported by the EEC ESPRIT Basic Research Action No. 3075 (ALCOM) and by the Ministry of Industry, Energy and Technology of Greece.     

Sharp probability estimates for generalized Smirnov statistics

We give sharp, uniform estimates for the probability that the empirical distribution function for n uniform-[0,1] random variables stays to one side of a given line. Author’s address: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA     

Equidistribution quantitative des points de petite hauteur sur la droite projective

We introduce a new class of adelic heights on the projective line. We estimate their essential minimum and prove a result of equidistribution (at every place) for points of small height with estimates on the speed of convergence. To each rational function R in one variable and defined over a number field K, is associated a normalized height on the algebraic closure of K. We show that these dynamically defined heights are adelic in our sense, and deduce from this equidistribution results for preimages of points under R at every place of K. Our approach follows that of Bilu, and relies on potential theory in the complex plane, as well as in the Berkovich space associated to the projective line over , for each prime p. Le premier auteur tient à remercier chaleureusement le project MECESUP UCN0202, ainsi que l'ACI ``Systèmes Dynamiques Polynomiaux' qui ont permis son séjour à l'Université Catholique d'Antofagasta. Le deuxième auteur est partiellement soutenu par le projet FONDECYT N 1040683. Enfin, nous remercions le rapporteur pour sa lecture détaillée de l'article.     

Numerical solution for exterior problems

Summary We solve the Helmholtz equation in an exterior domain in the plane. A perfect absorption condition is introduced on a circle which contains the obstacle. This boundary condition is given explicitly by Bessel functions. We use a finite element method in the bounded domain. An explicit formula is used to compute the solution out of the circle. We give an error estimate and we present relevant numerical results.     

Gaussian random fields,infinite dimensional Ornstein-Uhlenbeck processes,and symmetric Markov processes

A general treatment of infinite dimensional Ornstein-Uhlenbeck processes (OUPs) is presented. Emphasis is put on their connection with ordinary Gaussian random fields, and OUPs as symmetric Markov processes. We also discuss the relation to second quantisation and Gaussian Markov random fields.Supported in part by the Swedish Natural Science Research Council, NFR.     

Convergence to equilibrium of critical branching particle systems and superprocesses,and related nonlinear partial differential equations

We consider a class of multitype particle systems in ^d undergoing spatial diffusion and critical stable multitype branching, and their limits known as critical stable multitype Dawson-Watanabe processes, or superprocesses. We show that for large classes of initial states, the particle process and the superprocess converge in distribution towards known equilibrium states as time tends to infinity. As an application we obtain the asymptotic behavior of a system of nonlinear partial differential equations whose solution is related to the distribution of both the particle process and the superprocess.Research partially supported by CONACyT (Mexico), CNRS (France) and BMfWuF (Austria).     

A quasi-Newton method for solving fixed point problems in Hilbert spaces

Summary We study the convergence properties of a projective variant of Newton's method for the approximation of fixed points of completely continuous operators in Hilbert spaces. We then discuss applications to nonlinear integral equations and we produce some numerical examples.     

Spectral Tau approximation of the two-dimensional stokes problem

Summary We analyse the Spectral Tau method for the approximation of the Stokes system on a square with Dirichlet boundary conditions. We provide an error estimate, in the norm of the Sobolev spaceH ^s, for the approximation of a divergence free vector field with polynomial divergence free vector fields. We apply this result to prove some convergence estimates for the solution of the discrete Stokes problem.This work has been partially supported by the U.S. Army through its European Research Office under contract No. DAJA-84-C 0035     

Approximation by finite differences of the propagation of acoustic waves in stratified media

Summary In this paper, we analyze the approximation of acoustic waves in a two layered media by a finite diffrences variational scheme. We examine in particular the approximation of the guided waves. We point out the existence of purely numerical parasitic phenomena and quantify the numerical dispersion relative to guided waves.     

Convergence of nested classical iterative methods for linear systems

Summary Classical iterative methods for the solution of algebraic linear systems of equations proceed by solving at each step a simpler system of equations. When this system is itself solved by an (inner) iterative method, the global method is called a two-stage iterative method. If this process is repeated, then the resulting method is called a nested iterative method. We study the convergence of such methods and present conditions on the splittings corresponding to the iterative methods to guarantee convergence forany number of inner iterations. We also show that under the conditions presented, the spectral radii of the global iteration matrices decrease when the number of inner iterations increases. The proof uses a new comparison theorem for weak regular splittings. We extend our results to larger classes of iterative methods, which include iterative block Gauss-Seidel. We develop a theory for the concatenation of such iterative methods. This concatenation appears when different numbers of inner interations are performed at each outer step. We also analyze block methods, where different numbers of inner iterations are performed for different diagonal blocks.Dedicated to Richard S. Varga on the occasion of his sixtieth birthdayP.J. Lanzkron was supported by Exxon Foundation Educational grant 12663 and the UNISYS Corporation; D.J. Rose was supported by AT&T Bell Laboratories, the Microelectronic Center of North Carolina and the Office of Naval Research under contract number N00014-85-K-0487; D.B. Szyld was supported by the National Science Foundation grant DMS-8807338.     

Spline approximant in Hilbertian subspaces ofC(Ω)

Summary In this paper we give a new approach of approximation by spline functions. We define and study approximant spline functions which can be easly calculated without solving a linear system. We investigate also the error in using approximant spline functions.     

Propagation-Separation Approach for Local Likelihood Estimation

The paper presents a unified approach to local likelihood estimation for a broad class of nonparametric models, including e.g. the regression, density, Poisson and binary response model. The method extends the adaptive weights smoothing (AWS) procedure introduced in Polzehl and Spokoiny (2000) in context of image denoising. The main idea of the method is to describe a greatest possible local neighborhood of every design point X_i in which the local parametric assumption is justified by the data. The method is especially powerful for model functions having large homogeneous regions and sharp discontinuities. The performance of the proposed procedure is illustrated by numerical examples for density estimation and classification. We also establish some remarkable theoretical nonasymptotic results on properties of the new algorithm. This includes the ``propagation' property which particularly yields the root-n consistency of the resulting estimate in the homogeneous case. We also state an ``oracle' result which implies rate optimality of the estimate under usual smoothness conditions and a ``separation' result which explains the sensitivity of the method to structural changes.     

A particle method to solve the Navier-Stokes system

Summary We extend to the case of the two-dimensional Navier-Stokes equations, a particle method introduced in a previous paper to solve linear convection-diffusion equations. The method is based on a viscous splitting of the operator. The particles move under the effect of the velocity field but are not affected by the diffusion which is taken into account by the weights. We prove the stability and the convergence of the method.     

On Carleman formulas for the Dolbeault Cohomology

We discuss the Cauchy problem for the Dolbeault cohomology in a domain of C ⁿ with data on a part of the boundary. In this setting we introduce the concept of a Carleman function which proves useful in the study of uniqueness. Apart from an abstract framework we show explicit Carleman formulas for the Dolbeault cohomology. To the memory of Lamberto Cattabriga     

Linearly implicit extrapolation methods for differential-algebraic systems

Summary We consider the numerical solution of implicit differential equations in which the solution derivative appears multiplied by a solution-dependent singular matrix. We study extrapolation methods based on two linearly implicit Euler discretizations. Their error behaviour is explained by perturbed asymptotic expansions.     

A three-dimensional quadratic nonconforming element

Summary We define a second-degree nonconforming element on tetrahedra. We build a basis for the opproximation space derived from this element. We prove a discrete regularity property similar to the one that holds for the corresponding two-dimensional element.This work was partly supported by NSERC and by the Ministère de l'Education du Québec     

Vector bundles on a three-dimensional neighborhood of a ruled surface

Let S be a ruled surface inside a smooth threefold W and let E be a vector bundle on a formal neighborhood of S. We find minimal conditions under which the local moduli space of E is finite dimensional and smooth. Moreover, we show that E is a flat limit of a flat family of vector bundles whose general element we describe explicitly.