Table of contents

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A lower bound is given for the number of periodic trajectories inside a strictly convex domain in three dimensional space.

The main result is an analogue of Tikhonov's classical theorem for stochastic differential equations.

Let be the class of sequentially complete locally convex spaces such that an existence theorem holds for the linear Cauchy problem , , with respect to functions . It is proved that if , then for an arbitrary set . It is also proved that a topological product of infinitely many infinite-dimensional Fréchet spaces, each not isomorphic to , does not belong to .

An asymptotic formula for the number of primitive polynomials of the form f (x) + a, a = 1, ..., h, where f (x)∊F_p[x], is obtained, "on the average" over all polynomials f of fixed degree, and an estimate for the number of "sparse" factorable polynomials is also obtained.

It is proved that if a Lie ring L admits an automorphism of prime order p with a finite number m of fixed points and with pL = L, then L has a nilpotent subring of index bounded in terms of p and m and whose nilpotency class is bounded in terms of p. It is also shown that if a nilpotent periodic group admits an automorphism of prime order p which has a finite number m of fixed points, then it has a nilpotent subgroup of finite index bounded in terms of m and p and whose class is bounded in terms of p (this gives a positive answer to Hartley's Question 8.81b in the Kourovka Notebook). From this and results of Fong, Hartley, and Meixner, modulo the classification of finite simple groups the following corollary is obtained: a locally finite group in which there is a finite centralizer of an element of prime order is almost nilpotent (with the same bounds on the index and nilpotency class of the subgroup). The proof makes use of the Higman-Kreknin-Kostrikin theorem on the boundedness of the nilpotency class of a Lie ring which admits an automorphism of prime order with a single (trivial) fixed point.

Weakly nonlinear semi-Hamiltonian systems of n differential equations of hydrodynamic type in Riemann invariants are considered, and the geometry of the (n + 2)-web formed by the characteristics and the level lines of the independent variables are studied. It is shown that the rank of this web on the general solution of the system is equal to n. This result is used to obtain formulas for the general integral of the systems under consideration, with the necessary arbitrariness in n functions of a single argument. Separate consideration is given to the cases n = 3 and n = 4, for which it is possible not only to integrate the corresponding systems, but also to give a complete classification of them to within so-called transformations via a solution (reciprocal transformations). It turns out that for n = 3 they can all be linearized (and are thus equivalent), while for n = 4 there exist exactly five mutually nonequivalent systems, and any other system can be reduced to one of them by a transformation via a solution. There is a discussion of the connection between weakly nonlinear semi-Hamiltonian systems and Dupin cyclides-hypersurfaces of Euclidean space whose principal curvatures are constant along the corresponding principal directions. Some unsolved problems are formulated at the end of the paper.

New results are obtained on the approximation of elements of Sobolev classes W_p^l in the L_q metric by interpolating splines of order 2m - 1 and deficiency 1, defined on nonuniform nets Δ_n. The results are stated in terms of global and local properties of Δ_n, and depend mainly on an integral representation of the error.

Series from perturbation theory are constructed for the Bloch eigenvalues and eigenfunctions for the periodic Schrödinger operator in . An extensive set of quasimomenta on which the series converge is described. It is shown that the series have asymptotic character at high energies. They are infinitely differentiable with respect to the quasimomentum, and preserve their asymptotic character under such differentiation.

For the diffusion equation in the exterior of a closed set , , with Neumann conditions on the boundary,

pointwise stabilization, the central limit theorem, and uniform stabilization are studied. The basic condition on the set is formulated in terms of extension properties. Model examples of sets are indicated which are of interest from the viewpoint of mathematical physics and applied probability theory.

Inverse type theorems are proved for multipoint Padé approximants of functions holomorphic in a neighborhood of the unit disk, where the interpolation knots belong to the unit disk and satisfy a sufficiently general asymptotic condition.

Properties of sheaves of graded Lie algebras associated with a flat mapping of complex spaces are established. In particular, for a minimal versal deformation the tangent algebra of a fiber defines a linearization of the algebra of liftable fields on the base, which in turn enables one to find the discriminant of the deformation and its modular subspace. A criterion is obtained for the nilpotency of the tangent algebra of the germ of a hypersurface with a unique singular point. It is proved that in the algebra of liftable fields on the base of a minimal versal deformation of such a germ there always exists a basis with symmetric coefficient matrix.

A criterion is established for the possibility of approximation by harmonic functions and, in particular, by harmonic polynomials in the -norm on compact subsets of . This criterion, which is in terms of harmonic -capacity in , yields a natural analog to the theorem of Vitushkin on rational approximation in terms of analytic capacity.

The existence of a series of resonances located near the real axis is proved for the problem of scattering on a domain with a small aperture (narrow slit) (a domain of trap type). A method is presented for computing the resonances, based on a model of slits of zero width, based in turn on the theory of selfadjoint extensions of symmetric operators.

We investigate the structure of simple modular Lie algebras over an algebraically closed field of characteristic . Let denote an optimal torus in some -envelope . We prove: If and is a Cartan subalgebra, then is classical. If and distinguishes the roots of on , then is of Cartan type. The methods give new proofs even for the restricted simple Lie algebras.

The purpose of this article is to obtain best possible lower estimates of the errors of quadrature formulas with a fixed number of nodes for classes of functions with bounded mixed derivative. The results obtained here lead to a strengthening of known lower estimates for the deviation of grids.

Theorems are proved that reduce the proof of the Brauer conjecture for finite groups G with a p-soluble centralizer of a p-element to the evaluation of the minimum of a suitable positive definite quadratic form, whose matrix is given in terms of the Cartan matrix of a p-block of a group of simpler structure than G.