Table of contents

In this article we present a new method of studying the generalized Stieltjes moment problem that is connected with the application of convolution transforms. Using this method, we consider the existence and uniqueness of solutions to the problem and we prove a number of new results. Moreover, the question of uniqueness is considered under less restrictive conditions than in other investigations.

A class G of discrete groups of the Lobačevskiĭ plane with compact fundamental domain, which are extendible to discrete groups of Lobačevskiĭ space, is considered herein. It is the class of symmetry groups of normal regular partitions of the Lobačevskiĭ plane into equal polygons which meet in equal angles at the vertices of the partition and in which a circle can be inscribed. It is shown that for any finite set of groups in the class G there is a countable class of discrete groups of Lobačevskiĭ space, every member of which contains all groups of the given set as subgroups.

A two-variable system of first-order partial differential equations is investigated which has, in the region under consideration, one family of real characteristics and two families of imaginary characteristics. A general linear boundary value problem for the system is studied. It is proved that if a certain condition is imposed on the coefficients in the boundary conditions, there is only a finite number of linearly independent solutions of the homogeneous problem and of the adjoint homogeneous problem. A formula for the index of the above problem is derived and a necessary and sufficient condition for the solvability of the inhomogeneous problem is obtained in terms of the homogeneous adjoint problem.

A Noether theory is constructed for a class of integro-functional equations, and the exact limits of its applicability are indicated.

We consider potentials whose kernels are generalizations of the well-known functions of Levi for general boundary problems. We ascertain in which Hölder spaces the operators act which are generated by these potentials. The results obtained are applied to the construction of Green's functions.

Equivalence is verified between the Helly problem in the theory of moments and the problem of best approximation by elements of subspaces of finite defect. The existence and uniqueness conditions for the solution of these problems in a space of continuous functions are investigated.

This paper contains some results on the well-known problem of O. Ju. Šmidt and its generalization to groups of arbitrary cardinality. The existence of infinite abelian subgroups in infinite periodic groups is also studied.

In this paper polynilpotent verbal products of groups are regarded as polyverbal in the sense of O. N. Golovin. It is proved that among them only the nilpotent products have a finite basis of polyidentities.

We solve the problem of the solvability of a connected algebraic group, over an arbitrary field, which has an almost-regular automorphism. The results are used to prove existence theorems for invariant maximal tori and unipotent subgroups.

An elementary method is advanced for studying the asymptotic behavior of sums of multiplicative functions.

It is proven that two-dimensional algebraic tori are rational over their field of definition.

An arithmetic formula is obtained for the class number of fields of complex multiplication with odd conductor.