Table of contents

Examples are constructed of soluble finitely-based quasi-varieties of groups that generate non-finitely based varieties (with degree of solubility at most 7).

The problem of the decomposition of one-relator products of cyclics into non-trivial free products with amalgamation is considered. Two theorems are proved, one of which is as follows. Let , where , , and is a cyclically reduced word containing in the free group on and . Then is a non-trivial free product with amalgamation. One consequence of this theorem is a proof of the conjecture of Fine, Levin, and Rosenberger that each two-generator one-relator group with torsion is a non-trivial free product with amalgamation.

The case when smooth functions are not dense in a weighted Sobolev space is considered. New examples of the inequality (where is the closure of the space of smooth functions) are presented. We pose the problem of 'viscosity' or 'attainable' spaces (that is, spaces that are in a certain sense limits of weighted Sobolev spaces corresponding to 'well-behaved' weights, which means weights bounded above and away from zero) such that . A precise definition of this property of 'attainability' is given in terms of the convergence of the solutions of the corresponding elliptic equations. It is proved that an attainable space always exists, but does not in general coincide with the extreme spaces and . Examples of strict inclusions are presented.

An equation of convolution type on an interval is considered. A realization of an extension of the corresponding integral operator in the form of an operator of Wiener-Hopf type is obtained. A result on the structure of the eigenspaces of the original operator and a criterion for its invertibility are proved on this basis. A formula enabling one to find the resolvent of the original operator given a factorization of the symbol of the auxiliary Wiener-Hopf operator is obtained.

The problem of the continuation of a real-valued function from a subset of a metric space to the whole of the space is considered. A well-known result of McShane enables one to extend a uniformly continuous function preserving its modulus of continuity. However, some natural questions remain unanswered in the process. A new scheme for the extension of a broad class of functions, including bounded and Lipschitz functions, is proposed. Several properties of these extensions, useful in applications, are proved. They include the preservation of constraints on the increments of a function defined in terms of quasiconcave majorants. This result enables one to refine and generalize well-known results on the problem of the traces of functions with bounded gradient. The extension in question is used in two problems on function approximation. In particular, a direct proof of the density of the class in is given.

The problem of topological trajectory classification of Morse-Smale flows on closed two-dimensional surfaces is considered. Important results in this direction have been obtained by Peixoto and his school. However, the complete solution of this problem has not yet been accurately presented. The new topological invariants constructed in our work have a simpler form than those in the works of Peixoto. In particular, a list of Morse-Smale flows of small complexity is given which has been obtained by the authors by means of the invariants constructed by them.

The asymptotic behaviour (for large values of time) of the solutions of the first mixed boundary-value problem for the wave equation in domains with non-compact, non-star-shaped boundaries is considered. Estimates with respect to the spectral parameter of the solutions of the first boundary-value problem for the Helmholtz equation are obtained.