Distribution results for lattices in SL(2,ℚ p )

In this paper we study the ergodic properties of the linear action of lattices Γ of SL(2,ℚ_p) on ℚ_p × ℚ_p and distribution results for orbits of Γ. Following Serre, one can define a “geodesic flow” for an associated tree (actually associated to GL(2,ℚ_p)). The approach we use is based on an extension of this approach to “frame flows” which are a natural compact group extension of the geodesic flow.     

The parameter uniform numerical method for singularly perturbed parabolic reaction–diffusion problems on equidistributed grids

This article presents the study of singularly perturbed parabolic reaction–diffusion problems with boundary layers. To solve these problems, we use a modified backward Euler finite difference scheme on layer adapted nonuniform meshes at each time level. The nonuniform meshes are obtained by equidistribution of a positive monitor function, which involves the second-order spatial derivative of the singular component of the solution. The equidistributing monitor function at each time level allows us to use this technique to non-linear parabolic problems. The truncation error and the stability analysis are obtained. Parameter–uniform error estimates are derived for the numerical solution. To support the theoretical results, numerical experiments are carried out.     

Sur les entiers dont la somme des chiffres est moyenne

The goal of this paper is to study sets of integers with an average sum of digits. More precisely, let g be a fixed integer, s(n) be the sum of the digits of n in basis g. Let f:N→N such that, in any interval [g^ν,g^ν+1[, f(n) is constant and near from (g-1)ν/2. We give an asymptotic for the number of integers n<x such that s(n)=f(n) and we prove that for every irrational α the sequence (αn) is equidistributed mod 1, for n satisfying s(n)=f(n).     

等分布度与伪随机性检验

本文从熵的角度探讨了伪随机性检验问题，提出了有关序列的等分布度ｒｎ（Ｋ）概念，并证明：一序列｛Ｘｎ｝（０≤ｘｎ≤１）为等分布的充分必要条件是：Ｋ＞１，ｎ→∞ｌｉｍｒｎＫ＝１。利用这一结果对几个典型伪随机序列的等分布度进行了比较，给出了它们的伪随机性优劣排序。     

A local adaptive grid procedure for incompressible flows with multigridding and equidistribution concepts

An adaptive grid solution procedure is developed for incompressible flow problems in which grid refinement based on an equidistribution law is performed in high-error-estimate regions that are flagged from a preliminary coarse grid solution. Solutions on the locally refined and equidistributed meshes are obtained using boundary conditions interpolated from the preliminary coarse grid solution, and solutions on both the refined and coarse grid regions are successively improved using a multigrid approach. For this purpose, suitable correction terms for the coarse grid equations are derived for all variables in the flagged regions. This procedure with Local Adaptation, Multigridding and Equidistribution (LAME) concepts is applied to various flow problems to demonstrate the accuracy improvements obtained using this method.     

Equidistribution of Rational Matrices in their Conjugacy Classes

Let G be a connected simply connected almost -simple algebraic group with non-compact and a cocompact congruence subgroup. For any homogeneous manifold of finite volume, and a , we show that the Hecke orbit T _a (x ₀ H) is equidistributed on as , provided H is a non-compact commutative reductive subgroup of G. As a corollary, we generalize the equidistribution result of Hecke points ([COU], [EO1]) to homogeneous spaces G/H. As a concrete application, we describe the equidistribution result in the rational matrices with a given characteristic polynomial. The second author partially supported by DMS 0333397. Received: May 2005 Revision: March 2006 Accepted: June 2006     

Hermite collocation solution of near-singular problems using numerical coordinate transformations based on adaptivity

A coordinate transformation approach is described that enables Hermite collocation methods to be applied efficiently in one space dimension to steady and unsteady differential problems with steep solutions. The work is an extension of earlier work by Mulholland et al. (J. Comput. Phys. 131 (1997) 280). A coarse grid is generated by an adaptive finite difference method and this grid is used to construct a steady or unsteady coordinate transformation that is based on monotonic cubic spline approximation. An uneven grid is generated by means of the coordinate transformation and the differential problem is solved on this grid using Hermite collocation. Numerical results are presented for steady and unsteady problems.