Table of contents

Let be a union of finitely many smooth orientable bounded disjoint surfaces in of various dimensions (between 1 and ), and let be the algebra of functions continuous on () and having discontinuities of homogeneous type on surfaces in . This article includes a description of the algebra of symbols for the algebra generated by all the operators of the form acting in , where and , with and the direct and inverse Fourier transformations, respectively, and with a homogeneous function on of degree zero whose restriction to the unit sphere in is continuous. A criterion for operators in to be Noetherian operators is given. Bibliography: 25 titles.

This article deals with an analysis of the spectral meaning of the determinant of the characteristic function of a nonselfadjoint operator L acting in a separable Hilbert space H. Bibliography: 22 titles.

Assuming a nontrivial displacement of the zeros of Dirichlet -functions with quadratic characters, the author obtains asymptotic formulas for the number of lattice points in regions on the surface (), where is an arbitrary nondegenerate integral quadratic form, , and is a divisor of twice the discriminant of . The cases of an ellipsoid, a two-sheeted hyperboloid, and a one-sheeted hyperboloid are examined in a uniform way. Bibliography: 25 titles.

The author studies some questions about the way in which the convergence of double trigonometric series depends on the properties of the coefficients. Bibliography: 11 titles.

By means of the method of an approximate spectral projection operator the classical asymptotic formula for the distribution function of eigenvalues with an estimate of the remainder is proved both for problems with an unsolvable constraint such as an incompressibility condition (the Navier-Stokes and Maxwell systems) and those with a solvable constraint (an example is the spectral problem of the theory of electroelasticity). Problems in a bounded Lipschitz domain are considered. We note that an estimate of the remainder for the linearized Navier-Stokes system was obtained earlier only for the case of a domain with boundary of class , while for problems with solvable constraints only the leading term of the asymptotics was known; the asymptotics of the spectrum in the problem of the theory of electroelasticity has not been studied previously. Bibliography: 13 titles.

Analogues of Titchmarsh's theorem for transforms with kernels of Mittag-Leffler type are established. Bibliography: 8 titles.

In this paper a maximality condition is established for verbal subalgebras of a relatively free algebra, over a field of cardinality at least , of the variety of Lie algebras with a finite set of free generators. (The variety consists of the algebras obtained by extending nilpotent Lie algebras of class by means of nilpotent Lie algebras of class .) Bibliography: 5 titles.

For any positive integer the author constructs a continuous mapping of the -dimensional Menger compactum onto itself that is universal in the class of mappings between -dimensional compacta, i.e., for any continuous mapping between -dimensional compacta there exist imbeddings of and in such that the restriction of to is homeomorphic to . The mapping plays the same role in the theory of Menger -dimensional manifolds as the projection plays in the theory of -manifolds ( is the Hilbert cube). It can be used to carry over the classical theorems in the theory of -manifolds to the theory of -manifolds: Stabilization Theorem.For any -manifold and any imbedding of in the space is homeomorphic to .Triangulation Theorem.For any -manifold there exists an -dimensional polyhedron such that the space is homeomorphic to for every imbedding of in . Bibliography: 20 titles.

Let be the group ring of a group over a ring with identity. The ring is said to be Lie -nilpotent if for every sequence of elements of there is an index such that the Lie commutator . It is proved that is a Lie -nilpotent ring if and only if is Lie -nilpotent and one of the following conditions is satisfied: 1) is an Abelian group, or 2) is a ring of characteristic ( prime), is a nilpotent group and its commutator subgroup is a finite -group. Bibliography: 3 titles.

A mixed identity in variables over a group is a word (where the coefficients lie in , , and ) taking the value 1 for any values of the variables in . The concept of a mixed variety of groups is introduced as an object corresponding to a certain set of mixed identities and generalizing the concept of a variety of groups; an analogue of Birkhoff's theorem is proved; minimal mixed varieties generated by a finite group are described; the question of whether the mixed identities of a group can be derived from its identities is studied; and for nilpotent and metabelian groups it is established that all their mixed identities with coefficients in a finitely generated subgroup are finitely based, from which the same property is deduced for the identities of such groups with finitely many distinguished points. Bibliography: 16 titles.

The Fredholm property and spectral properties are considered for pseudodifferential operators on with symbol satisfying the estimates

(1)

where is a basic weight function. As follows from (1), differentiation of the symbol does not improve its behavior at infinity. The family of limit operators is introduced for a pseudodifferential operator. A theorem is proved giving necessary and sufficient conditions for the Fredholm property in terms of invertibility of the family of limit operators. Some properties of the spectrum are formulated in the same terms. Examples are given which illustrate the main results. Bibliography: 14 titles.

The author studies the behavior, for large time values , of a nonnegative solution of the second mixed problem for a uniformly parabolic equation

in a cylindrical domain , where is an unbounded domain in . It is shown that for a certain class of unbounded domains , the behavior of the solution of the problem as is determined by the behavior, for large values of the parameter , of the means of the initial function over the sets , , . Bibliography: 8 titles.

Linear meromorphic Pfaffian systems are studied on the complex projective spaces , . Bibliography: 13 titles.

Variational regularizing algorithms are presented and justified for solving ill-posed extremal problems with an approximately given functional. The algorithms use "non a priori" ways of choosing the regularization parameter according to the generalized residual principle, the generalized quasisolution principle, and the generalized smoothing functional principle for extremal problems. Bibliography: 17 titles.

The problem

is considered in the cylindrical region . A criterion for uniform stabilization (with respect to in ) of the mean over of order , , of the solution of this problem is proved for a rather broad class of unbounded domains (determined by conditions of isoperimetric type). Bibliography: 15 titles.

The set of the permutations of degree having only cycles with lengths in a fixed set is investigated. The set is distinguished in the set of all positive integers by imposing certain number-theoretic conditions. The following assertions are proved. 1) If is the cardinality of the finite set , then there exist positive constants and with such that

2) If the uniform probability distribution is introduced on the finite set and if is the number of cycles in a random permutation in , then the random variable is asymptotically normal with parameters 0 and 1 as . Bibliography: 4 titles.

Linear problems of optimal time to the origin of coordinates with constant coefficients and constant convex set giving geometric constraints on the control are considered. It is proved that if the dimension of the phase space is greater than two, then arbitrarily small perturbations of such problems can lead to the situation that any optimal control is a function discontinuous on a set of positive measure in the "perturbed" problem for all initial states in some neighborhood of a given initial state . Figures: 1. Bibliography: 14 titles.

An elliptic equation of order 2m with nonlocal conditions near the boundary is considered. The presence of nonlocal terms in the boundary conditions leads to the appearance of singularities in the solution. The Noether property is proved for this problem in weighted spaces and asymptotics of the solutions are obtained. These results are applied to the study of the solvability and smoothness of solutions of elliptic differential-difference equations. Bibliography: 22 titles.