Table of contents

Let be a complete connected Riemannian manifold of dimension and let be a second order elliptic operator on that has a representation in local coordinates, where , for some , and the matrix is non-singular. The aim of the paper is the study of the uniqueness of a solution of the elliptic equation for probability measures , which is understood in the weak sense: for all . In addition, the uniqueness of invariant probability measures for the corresponding semigroups generated by the operator is investigated. It is proved that if a probability measure on satisfies the equation and is dense in , then is a unique solution of this equation in the class of probability measures. Examples are presented (even with and smooth ) in which the equation has more than one solution in the class of probability measures. Finally, it is shown that if , then the semigroup generated by has at most one invariant probability measure.

A singularly perturbed boundary-value problem for the eigenvalues of the Laplace operator in a cylinder with a frequent change of the type of boundary conditions on the lateral surface is considered. The case when the homogenized problem involves the second or the third boundary condition on the lateral surface is studied. For a circular cylinder complete two-parameter asymptotic power series for the eigenvalues and the eigenfunctions of the perturbed problem are constructed. In the case when the section of the cylinder is an arbitrary bounded simply connected domain with smooth boundary, the leading terms of asymptotic formulae for eigenvalues convergent to simple limiting eigenvalues, and the leading terms of asymptotic formulae for the corresponding eigenfunctions are found.

The Koenigs function arises as the limit of an appropriately normalized sequence of iterates of holomorphic functions. On the other hand it is a solution of a certain functional equation and can be used for the definition of iterates of the original function. A description of the class of Koenigs functions corresponding to probability generating functions embeddable in a one-parameter group of fractional iterates is provided. The results obtained can be regarded as a test for the embeddability of a Galton-Watson process in a homogeneous Markov branching process.

A new method for the symmetric approximation of the non-stationary Navier-Stokes equations by a Cauchy-Kovalevskaya-type system is proposed. Properties of the modified problem are studied. In particular, the convergence as of the solutions of the modified problem to the solutions of the original problem on an infinite interval is established.

This paper is an investigation of necessary and sufficient conditions for embeddings of the function classes in classes of functions of bounded generalized variation. Theorems of a general character are obtained, along with embedding theorems under certain additional conditions imposed on the modulus of continuity.

The integral

is calculated and the system of orthogonal polynomials with weight equal to the corresponding integrand is constructed. This weight decreases polynomially, therefore only finitely many of its moments converge. As a result the system of orthogonal polynomials is finite. Systems of orthogonal polynomials related to -Dougall's formula and the Askey integral is also constructed. All the three systems consist of Wilson polynomials outside the domain of positiveness of the usual weight.

Divisorial contractions to singularities of the forms with and are classified in the Mori category.