Table of contents

The Kubo-Martin-Schwinger state is constructed for a Hamiltonian dynamical system whose phase space is Hilbert space, with Hamiltonian representable as the sum of two terms: the square of the norm and a function that is smooth on the completion of the original space in the nuclear norm. Bibliography: 5 titles.

The system of equations of elasticity theory

is solved in a homogeneous isotropic medium. Here is a matrix differential operator, is the stress operator, , is a small parameter, is a smooth bounded closed surface of revolution, and is the exterior of . The case where

is considered. The reflected wave satisfies the radiation condition. The asymptotics of is constructed with precision as , where is arbitrary. The formulas obtained are useful everywhere near , including its endpoints, and at a distance. The asymptotics of the scattering amplitudes of the reflected waves is found. Figures: 1. Bibliography: 16 titles.

In this paper properties of the discrete Sturm-Liouville operator are considered, and the scattering problem for this operator is studied using asymptotic formulas for orthogonal polynomials with matrix coefficients. Bibliography: 22 titles.

In this paper the authors establish algebraic conditions necessary and sufficient for given free homotopy classes of loops in two-dimensional manifolds to be representable by simple nonintersecting loops. Bibliography: 9 titles.

For linear differential operations the author studies the connection between the general notion of a correctly posed boundary value problem, given by Hörmander, and its description in terms of the boundary conditions. It is shown that, knowing one correctly posed problem and the kernels of operators that are maximal for the original and adjoint operation, it is possible to describe all correctly posed problems. Examples of explicit realization of this construction are presented. For operators with constant coefficients in a compact domain the author establishes the existence of correctly posed problems with poor regularity properties for the solutions, as well as problems in whose graph there is no dense set of infinitely differentiable functions. Bibliography: 8 titles.

Using the standard theta series of genus , the Hecke rings , for a covering of the symplectic group are constructed. The special role of four subrings of is described, as well as some finitely generated arithmetic subrings . The latter are important in the study of multiplicative properties of the Fourier coefficients of Engel modular forms of half-integral weight. Bibliography: 11 titles.

In this paper the authors establish approximation properties of measurable sets, which are then applied to study the differential properties of sets connected with the notion of points of density. A refinement of the well-known Lebesgue theorem on points of density of measurable sets is obtained. Bibliography: 7 titles.

In the case of characteristic zero, the Engel identity implies nilpotence in the variety generated by simple infinite-dimensional Lie algebras of Cartan type. An analogous result is also true for 2-metabelian Lie algebras (an algebra is called 2-metabelian if every 2-generator subalgebra is metabelian) over a field whose characteristic does not divide 5!, which in this case permits one to prove solvability of the variety of 2-metabelian Lie algebras. Bibliography: 10 titles.

A conjecture of Forsythe on the asymptotic behavior of the s-step method of steepest descent for a quadratic functional is confirmed for the two-step method, and the essential range of the asymptotic rate of convergence is found. Conditions are determined for the eigenvalues of the matrix to be in the asymptotic spectrum of the method. Devices for increasing the efficiency of the s-step method are proposed and justified on the basis of the results obtained. Bibliography: 20 titles.

Qualitative properties of solutions of the Dirichlet problem are studied for a general second order elliptic equation. Bibliography: 16 titles.

Several new nonlinear evolution equations integrable by the inverse problem method are obtained. The method applied in finding these equations is believed to be essentially new. The comparison of that method with other methods for finding nonlinear evolution equations integrable by the inverse problem method is given. In particular, it is shown that the methods using the Heisenberg equation (the so-called Lax representation) are not suitable to obtain the equations studied here. Bibliography: 23 titles.

This paper considers the problem of obtaining the principal term in the deviations of periodic functions in the space , , from their Ces𝑎̀ro means of arbitrary order. Bibliography: 7 titles.

This paper studies quasivarieties of groups, closed under restricted wreath products. It is shown that if a class of groups is closed with respect to restricted wreath products, then the quasivariety generated by is also closed under restricted wreath products. The base rank of a nontrivial quasivariety, closed under restricted wreath products, is found to be two. Conditions are given that ensure that a countable group from a given quasivariety is isomorphically embeddable in a 2-generator group from the same quasivariety. Finally, the cardinality of the set of all quasivarieties that consist of torsionfree groups and are closed with respect to restricted wreath products is computed. Bibliography: 12 titles.

In this paper the structure of a nonsubnormal tightly embedded subgroup is determined. Bibliography: 16 titles.

An integral equation of boundary value problems for the Laplace equation is investigated. Bibliography: 14 titles.

In this paper it is proved that a Lie algebra over a field of characteristic 0 with an algebraic adjoint representation is locally finite dimensional, provided the algebra satisfies a polynomial identity. In particular, a Lie algebra (over a field of characteristic 0) whose adjoint representation is algebraic of bounded degree is locally finite dimensional. Bibliography: 22 titles.

In this paper the author constructs an asymptotic expansion of the resolvent of the operator of the Dirichlet problem for an elliptic equation of divergence form with a power degeneracy on the boundary. To construct the expansion a variant of the technique of pseudodifferential operators (DO's) with operator-valued symbols is used, in combination with the technique of "ordinary" scalar DO's. The difference between the resolvent and the approximation thus obtained is an integral operator whose kernel decreases at infinity faster than any power of the spectral parameter. In a neighborhood of the boundary this operator smooths only in directions tangent to the boundary. Bibliography: 16 titles.