On a class of differential-difference equations arising in number theory

Supported in part by an NSF grant.     

On a class of algorithms from experimental design theory

The theory of optimal (approximate) linear regression design has produced several iterative methods to solve a special type of convex minimization problems. The present paper gives a unified and extended theoretical treatment of the methods. The emphasis is on the mathematical structures relevant for the optimization process, rather than on the statistical background of experimental design. So the main body of the paper can be read independently from the experimental design context. Applications are given to a special class of extremum problems arising in statistics. The numerical results obtained indicate that the methods are of practical interest     

On the Lusternik-Schnirelman theory of a real cohomology class

Farber developed a Lusternik-Schnirelman theory for finite CW-complexes X and cohomology classes H ¹ (X;). This theory has similar properties as the classical Lusternik-Schnirelman theory. In particular in [7] Farber defines a homotopy invariant cat(X,) as a generalization of the Lusternik-Schnirelman category. If X is a closed smooth manifold this invariant relates to the number of zeros of a closed 1-form representing . Namely, a closed 1-form representing which admits a gradient-like vector field with no homoclinic cycles has at least cat(X,) zeros. In this paper we define an invariant F(X,) for closed smooth manifolds X which gives the least number of zeros a closed 1-form representing can have such that it admits a gradient-like vector field without homoclinic cycles and give estimations for this number. Mathematics Subject Classification (2000): Primary 37C29; Secondary 58E05     

On an infinite series of Abel occurring in the theory of interpolation

The purpose of this paper is to show that for a certain class of functions f which are analytic in the complex plane possibly minus (−∞, −1], the Abel series f(0) + Σ_{n = 1}^∞ f⁽ⁿ⁾(nβ) z(z − nβ)^{n − 1}/n! is convergent for all β>0. Its sum is an entire function of exponential type and can be evaluated in terms of f. Furthermore, it is shown that the Abel series of f for small β>0 approximates f uniformly in half-planes of the form Re(z) − 1 + δ, δ>0. At the end of the paper some special cases are discussed.     

Some asymptotic results in hydrodynamic lubrication theory

Exact and asymptotic solutions to Reynolds equation for one-dimensional hydrodynamic lubrication are obtained for the case of rigid bearing surfaces and pressure-dependent viscosity.     

An optimization problem in hydrodynamic lubrication theory

If an arbitrary film profile is given, then we show how to find a film profile which will support more weight than the given profile. We do this for a discrete problem-our methods may be used on a digital computer-and we establish convergence results. The problem of finding a film profile which will support more weight than any other profile is also discussed.     

An L refined projection approximate solution of the radiation transfer equation in stellar atmospheres

This paper deals with the numerical approximation of the solution of a weakly singular integral equation of the second kind which appears in Astrophysics. The reference space is the complex Banach space of Lebesgue integrable functions on a bounded interval whose amplitude represents the optical thickness of the atmosphere. The kernel of the integral operator is defined through the first exponential-integral function and depends on the albedo of the media. The numerical approximation is based on a sequence of piecewise constant projections along the common annihilator of the corresponding local means. In order to produce high precision solutions without solving large scale linear systems, we develop an iterative refinement technique of a low order approximation. For this scheme, parallelization of matrix computations is suitable.     

On a class of stochastic differential equations arising from the stochastic approximation theory

The paper dealt with generalized stochastic approximation procedures of Robbins-Monro type. We consider these procedures as strong solutions of some stochastic differential equations with respect to semimartingales and investigate their almost sure convergence and mean square convergence     

On boundary value problem solvability theory for a class of high-order operator-differential equations

In this note, we establish sufficient conditions for the correct and unique solvability of various boundary value problems for a class of even-order operator-differential equations on the half-axis. These conditions are unimprovable in terms of operator coefficients of the equation. We note that the principal part of the equation under study suffers a discontinuity.     

Some analytical semigroups occurring in probability theory

One-parameter semigroups occurring in operator-limit distributions are investigated. The topological-algebraic background of the relevant monoids is discussed and Lie semigroup theory is applied to the Urbanik Decomposability Semigroup.     

Statistical theory of nonlinear hydrodynamic fluctuations in ionic systems

A system of equations for time correlation functions in ionic systems is obtained and solved on the basis of the Fokker-Planck equation for the distribution function of coarse-grained dynamic variables such as mass density, charge density, momentum density, current density, and total energy. Statistical weight, thermodynamic forces and hydrodynamic velocities are calculated both in Gaussian and higher approximations. Expressions for generalized transfer coefficients depending onk andw, with separate fluctuation contributions, are found.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 3, pp. 448–464, September, 1996.     

The Barnett approximation in the theory of hydrodynamic fluctuations

The Langevin dynamics and fluctuational-dissipative relationships for the hydrodynamic fluctuations for systems which are described in the third Barnett order with respect to the gradients of the hydrodynamic variables are generalized on the basis of a kinetic approach.     

On a singularity occurring in a self-correcting point process model

In a self-correcting point process model a boundary point of the parameter set is shown to be singular. This means a local behavior of the model which is qualitatively different from the LAN (or LAMN) condition satisfied at the other parameter points. As a consequence we obtain a nonnormal limiting distribution of the ML-estimator normalized with the random Fisher information.Work supported by a Heisenberg grant of the Deutsche Forschungsgemeinschaft.     

Boundary value problems occurring in plate deflection theory

The paper deals with several iterative methods for solving some boundary value problems occurring in plate deflection theory, making it possible to avoid the lengthy problem preparation which is required to solve these problems by finite difference methods.