Table of contents

Let be some maps of a set onto sets , . Approximations of real function on by sums are considered, where the  are real function on . Under certain constraints on the the existence of the best possible approximation is proved in three cases. In the first case the function and the approximating sums are bounded, but the functions can be unbounded. In the second case and the  are bounded. In the third case and the  are continuous, and the  are compact sets with metrics, and the maps are continuous.

We consider two-dimensional Navier-Stokes equations and a damped non-linear hyperbolic equation. We suppose that the right-hand sides of these equations have the form , . We suppose also that has an average. The main result of the paper is proof of a global averaging theorem on the convergence of attractors of non-autonomous equations to the attractor of the average autonomous equation as .

A study is made of analytic invertible systems with two degrees of freedom on a fixed three-dimensional manifold of level of the energy integral. It is assumed that the manifold in question is compact and has no singular points (equilibria of the initial system). The natural projection of the energy manifold onto the two-dimensional configuration space is called the domain of possible motion. In the orientable case it is sphere with holes and attached handles. It is well known that for and , the system possesses no non-constant analytic integrals on the corresponding level of the energy integral. The situation in the case of domains of possible motions with a boundary turns out to be very different. The main result can be stated as follows: there are examples of analytically integrable systems with arbitrary values of and .

It is known that the dimension subgroup problem can be solved affirmatively in the class of groups whose lower central series quotients are torsion-free. In this paper it is proved that the same result is true for an arbitrary extension of a group belonging to this class by an Abelian group.

Asymptotic formulae representing for large time the solution of the Cauchy problem are obtained for the generalized Korteweg-de Vries equation with non-linear term to an integer power greater than three. The error terms are estimated. The method is based on the perturbation theory with respect to a parameter characterizing the smallness of the initial data.

Mahler has obtained an inequality for the products of zeros of an algebraic polynomial and its derivative lying outside the unit disc. In this paper a converse inequality with best possible constant is established.

A class of multivalued maps with non-convex non-closed decomposable values is distinguished, and theorems are proved on the existence of continuous selections for such maps. This class contains multivalued maps whose values are extreme points of continuous multivalued maps with closed convex decomposable values in a Banach space of Bochner-integrable functions. The proofs are based on the Baire category theorem. It is known that the set of extreme points of a closed convex set is in general not closed. Hence the results or paper answer the question of the existence of continuous selections for multivalued maps with non-convex non-closed values.

The de la Vallé-Poussin theorem states that if a trigonometric series converges to a finite integrable function f everywhere outside a countable set E, then it is the Fourier series of f. In this paper the theorem is shown to hold also if the exceptional set E is a union of finitely many H-sets.