A Glivenko-Cantelli lemma and weak convergence for empirical processes of associated sequences

Summary With the help of an extension of Hoeffding's equality, we develop a way for estimating the covariance structures for empirical functions of associated sequences in terms of covariances of the original random variables. Based on these estimations, a Glivenko-Cantelli lemma for associated sequences and weak convergence for empirical processes of stationary associated sequences are obtained, all under the conditions on covariances of the original random variables.This research has been supported partially by a scholarship from the Faculty of Graduate Studies and Research of Carleton University, Ottawa, and by an NSERC Canada grant of M. Csörg     

The stressed state of glass disks during strength testing

By applying the improved theory of plates, which makes it possible to determine all components of the stress tensor, we study the stressed state of glass disks under axisymmetric bending. We obtain a closed-form solution of the corresponding boundary-value problem for systems of singularly perturbed equations. It is shown that the size of the zone of pure bending-the domain with uniformly stretched surface-differs from those known in the literature. The results obtained make it possible to determine the optimal geometric parameters of the punch-glass disk-support system in strength testing of the glass by the method of axisymmetric bending.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 73–79.     

Designing a regression experiment in Hilbert space

Two forms of a priori information about unknown parameters in observing linear regression with noise in a Hilbert space are considered. In each section one gets best restricted linear unbiased estimates of the unknown parameters generalizing known results about the finite-dimensional case. Problems of choice of optimal designs are solved for bounded closed sets which are natural from the point of view of applications. The results are compared with those which hold without a priori information.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 23–28, 1988.     

Determination of a harmonic function with minimal gradient deviation

We find a function u(x, y) harmonic in the upper half-plane y>0 satisfying the conditions grad u(x₀,y₀) b|₂ min, . The solution of the problem is expressed in terms of the Poisson integral for the upper half-plane.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 55–57.     

Computation of Feynman integrals of functionals that are functions of quadratic functionals

If the arguments of a function G: ^{d 1} are taken as quadratic functional defined on a space C of continuous functions, we obtain a functional G: C ¹.We give a formula for computing analytic Feynman integrals of such functionals. We also propose a method of approximate computation of sequential Feynman integrals based on replacing the kernel of an integral operator by a degenerate kernel.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 58–61.     

Optimization of reliability and longevity of cylindrical shells subject to corrosive wear

The problem of the optimal (with respect to the minimum average rate of weight loss in a structure) design of a smooth cylindrical shell loaded by an axial compressive force and under corrosive wear, the rate of which depends exponentially on the stresses, is solved. The problem is solved analytically using apparatus of the method of Lagrange multipliers.Translated from Matematicheskie Metody i Fiziko-mekhanicheskie Polya, No. 26, pp. 59–63, 1987.