Table of contents

Let be the generalized Sato-Levine invariant, that is, the unique Vassiliev invariant of order 3 for two-component links that is equal to zero on double torus links of type . It is proved that

where is the invariant of order 3 proposed by Viro and Polyak in the form of representations of Gauss diagrams and is the linking number.

Evolution equations containing rapidly oscillating terms with respect to the spatial variables or the time variable are considered. The trajectory attractors of these equations are proved to approach the trajectory attractors of the equations whose terms are the averages of the corresponding terms of the original equations. The corresponding Cauchy problems are not assumed here to be uniquely soluble. At the same time if the Cauchy problems for the equations under consideration are uniquely soluble, then they generate semigroups having global attractors. These global attractors also converge to the global attractors of the averaged equations in the corresponding spaces. These results are applied to the following equations and systems of mathematical physics: the 3D and 2D Navier-Stokes systems with rapidly oscillating external forces, reaction-diffusion systems, the complex Ginzburg-Landau equation, the generalized Chafee-Infante equation, and dissipative hyperbolic equations with rapidly oscillating terms and coefficients.

A version of the Poincaré-Hopf theorem is established for multivalued vector fields on submanifolds of a reflexive space. The connection between the critical values and homological characteristics of the Lebesgue sets of Lipschitz functionals is studied. Applications to the theory of operator inclusions with parameters are indicated.

Infinitesimal -th order bendings, , of higher-dimensional surfaces are considered in higher-dimensional flat spaces (for an infinitesimal bending is assumed to be an analytic bending). In terms of the Allendoerfer type number, criteria are established for the -rigidity (in the terminology of Sabitov) of such surfaces. In particular, an -infinitesimal analogue is proved of the classical theorem of Allendoerfer on the unbendability of surfaces with type number  and the class of -rigid fibred surfaces is distinguished.

Natural actions of direct products of general linear groups on tensor products of the corresponding complex linear spaces are considered. Among these actions, all actions with finitely many orbits are distinguished. The main results of the paper are the classification of orbits and the construction of the orbit abutment graphs for all such actions.

Asymptotic equalities are obtained for upper bounds of the deviations of Fourier sums in the classes of convolutions of Poisson kernels and continuous functions with moduli of continuity not exceeding fixed majorants.

A dissipative integro-differential operator L arising in the linearization of Boltzmann's equation in one-speed particle transport theory is considered. Under assumptions ensuring that the point spectrum of L is finite a scalar multiple of the characteristic functions of L is found and a condition for the absence of spectral singularities is indicated. Using the techniques of non-stationary scattering theory and the Sz.-Nagy-Foias functional model direct and inverse wave operators with the completeness property are constructed. The structure of the operator L in the invariant subspace corresponding to its continuous spectrum is studied.