Table of contents

With denoting the error of best uniform rational approximation from to on , we determine the numbers , where each of these numbers was calculated with a precision of at least 200 significant digits. With these numbers, the Richardson extrapolation method was applied to the products , and it appears, to at least 10 significant digits, that

which gives rise to an interesting new conjecture in the theory of rational approximation.

The results presented here are based on the use of approximations of analytic functions in certain questions involving numerical methods for "physical" equations. The concepts of relative closeness and of stability of signals are introduced. A simple method is given for approximate inversion of the Laplace transformation, with an estimate of the relative error. Also, estimates are obtained for the dimensions of spaces in which it is possible to find approximate solutions of multidimensional systems of linear equations, as a function of the accuracy of approximation and the "size" of the spectrum of the operator.

The problem of spectral synthesis in a complex domain for a differential operator with symbol , is reduced to the problem of a local description of the closed submodules of a module (of entire functions of exponential type) over the ring of polynomials of the form , .

It is proved that the Taylor series of a meromorphic function of two variables converges absolutely in the closed unit bidisk if this function satisfies a Hölder condition in with exponent , while for any there exists a rational function with Hölder exponent such that the indicated series diverges. This result solves the problem of stability of two-dimensional recursive digital filters. In its proof the structure of the asymptotic behavior of the Taylor coefficients of a meromorphic function of two variables is investigated.

This is a detailed study of the problem of the existence and characterization of finite-dimensional Chebyshev subspaces of the spaces and on the interval , where is an even nonnegative continuous nondecreasing function on the half-line , and the function is measurable, finite, and positive almost everywhere on . If is an -function, it is characterized as a Chebyshev subspace of the Orlicz spaces with the Luxemburg norm.

The concept of mean dimension is introduced for a broad class of subspaces of , and analogues of the Kolmogorov widths, Bernstein widths, Gel'fand widths, and linear widths are defined. The precise values of these quantities are computed for Sobolev classes of functions on in compatible metrics, and the corresponding extremal spaces and operators are described. A closely related problem of optimal recovery of functions in Sobolev classes is also studied.

The author establishes that, for every function that is analytic inside the unit disk and belongs to the space with , the equation

is satisfied, where and are the minimal deviations of from polynomials of degree at most and from rational functions of order at most . In particular, if and only if can be continued analytically over the disk . There is also a similar proposition for the approximation of functions in the spaces , .

A sufficient condition for algebraicity of complex analytic sets is established.

The Navier-Stokes system is considered in a plane domain that has several exits to infinity having the form of channels of bounded width. It is assumed that the external force decays sufficiently fast at infinity. Solutions are considered that are defined and bounded for all . Such solutions lie on an attractor of the system. An asymptotic expansion as is obtained for these solutions. The presence of this expansion indicates, in particular, that turbulence in this situation does not propagate to infinity.

Random walks in a random environment are considered on the set of integers when the moving particle can go at most steps to the right and at most steps to the left in a unit of time. The transition probabilities for such a random walk from a point are determined by the vector . It is assumed that the sequence is a sequence of independent identically distributed random vectors. Asymptotic properties with probability 1 are investigated for such a random process. An invariance principle and the law of the iterated logarithm for a product of independent random matrices are proved as auxiliary results.

Steady-state and time-dependent problems are studied for the equation

where , is a two-dimensional closed manifold, and is the projection onto the subspace of solenoidal vector fields that admit a single-valued flow function. Existence of steady-state solutions is proved. For the evolution problem Lyapunov stability of the zero solution in Sobolev-Liouville spaces is proved by the method of vanishing viscosity. The existence of generalized weak attractors, an integer, is proved. A -weak attractor is constructed in the phase space for the velocity vortex equation.

The concept of oriented degree is extended to the class of mappings of the form

f – g,

where f is a proper Fredholm mapping of nonnegative index and g a continuous f-compactly restrictable mapping. In the case when f is a Fredholm mapping of zero index and f and g are equivariant with respect to the action of the circle and the torus, formulas are obtained which express the degree of these mappings in terms of invariants of representations of the corresponding groups. An application to the investigation of the global behavior of a bifurcation branch of a certain nonlinear boundary value problem is given.

In a domain we consider the first boundary value problem for a quasilinear parabolic fourth-order equation with a small parameter in the highest derivatives, which degenerates for into a second order equation. It is well known that the semigroup corresponding to this problem has an attractor, that is, an invariant attracting set in the phase space. In this paper we investigate the structure of this attractor by means of an asymptotic expansion in . The dominant term of the asymptotics is the solution of a second-order equation. The asymptotic expansion also contains boundary layer functions, which are responsible for the deterioration of the differential properties of the elements of the attractor near the boundary. The asymptotics constructed in this way (with an estimate of the remainder) enable us to study the differential properties of attractors and their behavior as in any interior subdomain , . For simplicity, the investigation is carried out in the case when is a bounded cylindrical domain. The generalization to does not present any difficulties.

A connection is established between the well-known trace formulas of Gel'fand-Levitan and Kreĭn.

After giving a brief introduction to our new "theory of dual systems of integer vectors", we give the first applications to the theory of positive quadratic forms. We consider the question of enumerating the L-polytopes of lattices, paying particular attention to the case of five-dimensional lattices. The results reported in this paper were announced earlier by the authors in Doklady [1]; here we give the details.

The famous Steiner problem in the Euclidean plane, which is that of investigating minimal nets spanning fixed finite subsets M of points in the plane, is solved when M is extremal, i.e. when M lies on the boundary of its convex hull, and the nets are nondegenerate, i.e. have no vertices of degree 2.

It is shown that if a quasiconformal automorphism of the unit ball in () has coefficient of quasiconformality in the ball of radius with asymptotic growth such that , then it has a radial limit at almost every point of the boundary. This asymptotic growth of is sharp in a certain sense.