Table of contents

The basis for most of the results in this paper is a theorem that a perturbed operator with disjoint parts of the spectrum is similar to an operator for which the subspaces constructed from the isolated parts of the unperturbed operator are invariant. In particular, estimates are obtained for the eigenvalues and projections of the perturbed operators, results about equiconvergence of spectral decompositions are obtained, and convergence questions for the eigenvalues are investigated with the use of projection methods.

Bibliography: 15 titles.

The problem of describing a commuting pair of differential operators in terms of its Burchnall–Chaundy curve and the holomorphic bundle over it is considered. A characteristic of the matrix case is the occurrence of vector rank, a bundle having various dimensions over various components of the Burchnall–Chaundy curve. A complete, independent system which determines the pair of operators uniquely is chosen. In the last section, a proof is given of S. P. Novikov's criterion for an operator with periodic coefficients to be an operator of a nontrivial commuting pair.

Bibliography: 25 titles.

It is proved that, if G is a finite group representable as a product of an abelian group A and a group B, then the center of B is contained in a solvable normal subgroup of G. The disposition of in the upper p-series of a finite solvable group G having a factorization of this sort is determined. A corollary for locally finite groups is provided.

Bibliography: 33 titles.

For general degenerate Poisson brackets, analogues are constructed of invariant vector fields, invariant forms, Haar measure and adjoint representation. A pseudogroup operation is defined that corresponds to nonlinear Poisson brackets, and analogues are obtained for the three classical theorems of Lie. The problem of constructing global pseudogroups is examined.

Bibliography: 49 titles.

Conditions are found for the equality of various topologies in inductive limits of spaces of functions defined on an arbitrary set. A new class of sufficient sets is introduced and its connection with classes studied previously is determined. The general dependence between weakly sufficient sets and absolutely representing systems is indicated.

Bibliography: 32 titles.

Global integral representations are constructed for differential forms on domains in complex projective space CPⁿ.

Consequences of these representations are the following: first, criteria for the solvability of the inhomogeneous Cauchy-Riemann equations on q-pseudoconvex and q-pseudoconcave domains in an algebraic manifold; second, explicit formulas and bounds for solutions of these equations; and third, a description of the kernel and image and an inversion formula for the Radon-Penrose transform of (0,q)-forms on q-linearly concave domains in CPⁿ.

Bibliography: 23 titles.

A criterion for admissibility of rules in the modal system Grz is found. On the basis of it an algorithm is constructed that recognizes the admissibility of rules in Grz. The decidability of admissibility in Grz, proved in the paper, yields as a corollary a positive solution of the Kuznetsov-Friedman problem of algorithmic decidability of the admissibility problem in intuitionistic propositional logic. Algebraic analogues of the results obtained here are the decidability of the universal theories of a free pseudo-Boolean algebra and a free topo-Boolean algebra in the variety of algebras corresponding to the system Grz. The elementary theories of these free algebras are hereditarily undecidable.

Bibliography: 15 titles.

A well-known result due to V. I. Arnol'd on the reducibility of a weakly perturbed system of differential equations on a finite-dimensional torus is generalized first to the case when the number of equations is infinite, and, second, to the case when the perturbation is an almost periodic function of time. The reduction is effected by Kolmogorov's method of successive substitutions. Conditions are obtained for the convergence of the method for this problem. It is shown that almost all (in a certain sense) bases of frequencies satisfy the requisite condition.

Bibliography: 10 titles.