Table of contents

In this article known results of A. D. Aleksandrov concerning mixed discriminants, connected with positive definite quadratic forms, are carried over to the case of positive semidefinite quadratic forms. In particular, the analogue of the Frobenius-Koenig theorem is proved for mixed discriminants, and the main result is the derivation of a necessary and sufficient condition for equality to hold in Aleksandrov's inequality. Bibliography: 8 titles.

A nonlinear equation of parabolic type with functions taking values in a Banach space is studied. A family of solutions called a fundamental family is constructed in a neighborhood of a bifurcation point. It is shown that as the fundamental solutions tend either to zero or to some steady-state solution of the nonlinear equation. Conditions are investigated under which the solutions of Cauchy problems behave like fundamental solutions. Bibliography: 15 titles.

It is proved that the free pseudoboolean algebra and the free topoboolean algebra do not have bases of quasi-identities in a finite number of variables. A corollary is that the intuitionistic propositional logic and the modal system do not have finite bases of admissible rules. Infinite recursive bases of quasi-identities are found for and . This implies that the problem of admissibility of rules in the logics and is algorithmically decidable. Bibliography: 14 titles.

The concept of hypoellipticity weight, which generalizes Hörmander's concept of index of hypoellipticity, is introduced for linear differential operators with constant coefficients. Exact formulas are derived for the hypoellipticity weight of regular hypoelliptic operators. These formulas are applied to determine more exactly the Gevrey classes to which the solutions of a regular hypoelliptic equation belong. The results are shown to be unimprovable in terms of Gevrey classes. Bibliography: 23 titles.

A Riemannian manifold is said to be parabolic if there does no exist a positive fundamental solution of the Laplace equation on it. The purpose of this article is to obtain geometric conditions, both necessary and sufficient, for a manifold to be parabolic. Bibliography: 11 titles.

The analyticity of functions that satisfy the Cauchy-Riemann conditions and have summable modulus is established. Thus the Looman-Men'shov and Tolstov theorems are generalized. The theorem of Lindelöf is generalized (from the class of bounded functions to the class ) for certain kinds of domains. Sufficient criteria for continuity on the boundary for some classes of analytic functions are investigated. Bibliography: 21 titles.

The author constructs an infinite series of finite rings , , , which are not embeddable in rings of matrices over commutative rings, and describes their bases of identities and critical rings of the varieties they generate. He shows that finite rings from the ring varieties , , , , are either representable by matrices over commutative rings or generate the respective varieties. Under a supplementary restriction on a variety with exponent it is shown that every finite ring from is representable by matrices over a commutative ring if and only if does not contain any of the rings , , . Bibliography: 14 titles.

The solvability of the problem

in , , is proved, where is the sum of all the principal minors of order of the Hessian , is a bounded strictly convex region in , , with boundary of class , for , under certain restrictions on the occurrence of and as arguments in . Bibliography: 21 titles.

In the indicated class a criterion is given for the unique recovery of the function of rotation number on the basis of a countable collection of polynomial conservation laws. Bibliography: 14 titles.

Generalized solutions of the Dirichlet problem for a fourth order elliptic equation in two independent variables are investigated. Unimprovable estimates are obtained for the modulus of the generalized solution and its first derivatives in the neighborhood of a boundary point; it is also proved that the generalized solutions belong to a Hölder space with an unimprovable index depending on the geometry of the domain. Bibliography: 16 titles.

The general boundary value problem is studied for a parabolic equation in spaces of insufficiently smooth and generalized functions. Starting from Green's formula, the generalized solution of a boundary value problem is defined, and two families (scales) of spaces are constructed in which the boundary value problem is studied: the spaces of solutions , and the spaces of right-hand sides . It is proved that the closure with respect to continuity of the boundary value problem operator establishes an isomorphism of the spaces and for . Bibliography: 35 titles.

Complete asymptotic expansions are obtained for the integrated state density and the spectral function of a Hill operator with smooth potential. These expansions can be differentiated any number of times outside small neighborhoods of forbidden zones. Bibliography: 18 titles.

Let be a function of bounded variation, , and the Weyl kernel of order , i.e. , . Denote by and the classes of functions represented by the corresponding formulas

The conjugate classes of functions and are also considered; they are convolutions of conjugate Weyl kernels with functions of bounded variation. The following main result is proved:

where is the best uniform approximation by trigonometric rational functions of order at most , and is one of the classes

Bibliography: 13 titles.

The microlocal structure of pseudodifferential operators is studied in a neighborhood of a degenerate singular point, and the normal forms of the complete symbols are determined. Bibliography: 18 titles.

The initial-boundary value problem in , , , is solved, where is a three-dimensional rectangular parallelepiped. Two-level methods of second-order approximation are considered: families of projection and finite-difference schemes with a splitting operator as well as Crank-Nicolson schemes. Error estimates in of order for all are derived. It is shown that the inclusion of values yields sharpened estimates when is discontinuous. Accuracy of the estimates with respect to order - and in the case of Crank-Nicolson schemes their unimprovability - is proved. It is found that for difference schemes with splitting operator when , must have in not only order smoothness with respect to (as in the case of Crank-Nicolson schemes) but also order smoothness (in a certain weak sense) in the space variables. Only one scheme with splitting operator out of each family constitutes an important exception, a scheme equivalent to one proposed by J. Douglas and its projective analogue, and that only for . The situation described is qualitatively different from those studied previously in the literature. Bibliography: 17 titles.

Estimates are established for a measure of the algebraic independence of the values of the exponential function and certain other functions, and a theorem is proved on the number of algebraically independent quantities among a series of values of the exponential function. Bibliography: 11 titles.