Table of contents

The paper investigates the number of solutions of an algebraic congruence , where is a prime. Under certain conditions for the polynomial the asymptotic formula is obtained by elementary methods.

In this article we derive a new effective estimate for solutions of Thue's equation as a function of the coefficients of the equation.

We investigate the possibility of constructing a solution of the abelian imbedding problem for number fields with prescribed ramification points and local valuations.

We give a classification of irreducible affine algebraic varieties which are quasihomogeneous with respect to a regular action by a connected linear group of automorphisms and are such that the isotropy subgroup of a point in general position contains a maximal unipotent subgroup of the group of transformations. We find criteria for the normality and factoriality of such varieties. We compute the divisor class group and give a complete description of the orbits in such varieties.

It is proved that an arbitrary metabelian -group is imbeddable in a metabelian -group. Varieties consisting of metabelian -groups are described and a number of theorems on the structure of a free metabelian -group are obtained. A certain specially defined ring-an analog of the group ring-plays an essential role in the proofs.

It is shown that for the nodes , where , ; , the following statements hold: 1) The Hermite-Fejér interpolation process for an arbitrary polynomial converges In with rapidity . 2) The process , where are Lagrange fundamental polynomials with nodes , diverges at all points of for every function , .

By means of a resonance theorem we will establish the existence of functions in (where is an -dimensional region) whose expansion in eigenfunctions of the Laplacian is not Riesz-summable of order if .

Let be the unit cube of three-dimensional space , and let , , be mappings of class . We prove that the set of functions on which can be represented in the form

where the are arbitrary continuous functions, , is nowhere dense in .

In this work there are given necessary and sufficient conditions for the absolute continuity and equivalence , , of a Wiener measure and a measure corresponding to a process of diffusion type with differential . The densities (the Radon-Nikodým derivatives) of one measure with respect to the other are found. Questions of the absolute continuity and equivalence of measures and are investigated for the case when is an Itô process. Conditions under which an Itô process is of diffusion type are derived. It is proved that (up to equivalence) every process for which is a process of diffusion type.

It is shown here that sufficiently thin elastic shells of arbitrary convexity and with a mobile hinged support are nonrigid. That is, for such shells, in the absence of external loading, it is proved by an asymptotic method that the boundary-value problem for the corresponding system of nonlinear partial differential equations in the theory of shells has at least one solution besides the trivial one. The former solution corresponds to an equilibrium shape close to the buckled shape obtained from the original shell surface by reflection in the plane containing the supporting contour.