Some rapidly converging series for

For a natural number , the author derives several families of series representations for the Riemann Zeta function . Each of these series representing converges remarkably rapidly with its general term having the order estimate:

Relevant connections of the results presented here with many other known series representations for are also pointed out.

     

Non-parametric Estimation for the M/G/∞ Queue

Given an M/G/ queue with input rate and service-time distribution G, we consider the problem of estimating and G from data on the queue-length process Q = (Q_t). Our motivation is to study departures of G from exponentiality, following recent work of Bingham and Dunham (1997, Ann. Inst. Statist. Math., 49, 667–679).     

On random algebraic polynomials

This paper provides asymptotic estimates for the expected number of real zeros and -level crossings of a random algebraic polynomial of the form , where are independent standard normal random variables and is a constant independent of . It is shown that these asymptotic estimates are much greater than those for algebraic polynomials of the form .

     

菲涅耳轴上衍射场的计算

给出了一种计算菲涅耳衍射的方法,它既能比较精确地计算轴上的衍射场,也能用来近似地估算轴外的衍射场。     

Sharp Spectral Asymptotics and Weyl Formula for Elliptic Operators with Non-smooth Coefficients

The aim of this paper is to give the Weyl formula for eigenvalues of self-adjoint elliptic operators, assuming that first-order derivatives of the coefficients are Lipschitz continuous. The approach is based on the asymptotic formula of Hörmander"s type for the spectral function of pseudodifferential operators having Lipschitz continuous Hamiltonian flow and obtained via a regularization procedure of nonsmooth coefficients.     

Stochastic methods for Dirichlet problems

Using an equivalent expression for solutions of second order Dirichlet problems in terms of Ito type stochastic differential equations, we develop a numerical solution method for Dirichlet boundary value problems. It is possible with this idea to solve for solution values of a partial differential equation at isolated points without having to construct any kind of mesh and without knowing approximations for the solution at any other points. Our method is similar to a recently published approach, but differs primarily in the handling of the boundary. Some numerical examples are presented, applying these techniques to model Laplace and Poisson equations on the unit disk. Visiting Professor, Universidad de Salamanca.     

Contributions of non-extremum critical points to the semi-classical trace formula

We study the semi-classical trace formula at a critical energy level for a h-pseudo-differential operator on whose principal symbol has a totally degenerate critical point for that energy. We compute the contribution to the trace formula of isolated non-extremum critical points under a condition of “real principal type”. The new contribution to the trace formula is valid for all time in a compact subset of but the result is modest since we have restrictions on the dimension.     

Trace formula on the p-adic upper half-plane

This article aims at showing a p-adic analogue of Selberg's trace formula, which describes a duality between the spectrum of a Hilbert-Schmidt operator and the length of prime geodesics appearing in the p-adic upper half-plane associated with a hyperbolic discontinuous subgroup of . Then we construct Markov processes on the fundamental domain relative to such subgroups, to whose transition operators the trace formula applied and a p-adic analogue of prime geodesic theorem is proved.     

Geometrical and Topological Properties of Real Polynomial Fibres

We establish a Gauss—Bonnet type formula for a smooth fibre of a nonproper real polynomial of ⁿ . For this we need to study topological properties of a generic hyperplane section of this fibre.