Table of contents

The phenomenon of instantaneous shrinking of the support in the Cauchy problem for non-linear parabolic equations with a positive initial function that is infinitesimal as is considered. Exact local estimates for the boundary of the support of the solutions are proved. For example, the exact asymptotic formula

holds for the solution of the equation , , , where and .

A study is made of flows with finitely many equilibrium states and of foliations with finitely many singularities of saddle type with integer and half-integer index on closed surfaces, and for a metric of constant curvature the role of geodesics is established in the asymptotic behaviour of semitrajectories of flows and semileaves of foliations upon lifting to the unbranched or branched universal covering.

A class of functional-differential equations of neutral type with coefficients that are functions taking values in the set of (in general, unbounded) operators in separable Hilbert space is considered. For such equations results on the well-posed solubility of initial-boundary-value problems on the semiaxis in Sobolev weighted spaces are established.

In this paper it is shown that the nilpotency class of sandwich subalgebras in Lie algebras of Cartan type and in Melikyan algebras over a field of characteristic is equal to , where is the sum of the heights of the variables. The only exception is the Zassenhaus algebra (and also the Hamiltonian and contact algebras for ).

We study codimension schemes in that are scheme-theoretic intersections of three hypersurfaces. The results of Peskine, Szpiro and Rao about invariant smooth three-generated schemes are generalized to Cohen-Macaulay schemes. We also give criteria for stability and splittability of the associated vector bundles.

The main result is that any hyperbolic group that is almost cyclic is SQ-universal.

The Dirac operator with elliptic finite-gap potential

is considered. An Ansatz for the Krichever curves associated with elliptic (in ) finite-gap solutions of the 'decomposed' non-linear Schrödinger equation

and of the modified () equation

is presented. Examples of two- and three-sheeted coverings associated with the one- and two-gap Dirac potential are discussed.