Table of contents

The following two classical problems are considered: the existence and the estimate of a solution of an equation defined by a map in the neighbourhood of a point ; necessary conditions for an extremum at of a smooth function under equality-type constraints defined in terms of a non-linear map . If the range of the first derivative of at is not closed, then one cannot use classical methods of analysis based on inverse function theorems and Lagrange's principle. The results on these problems obtained in this paper are of interest in the case when the range of the first derivative of at is non-closed; these are a further development of classical results extending them to abnormal problems with non-closed range.

Necessary and sufficient conditions for the representation of the index of elliptic operators on manifolds with edges in the form of the sum of homotopy invariants of symbols on the smooth stratum and on the edge are found. An index formula is obtained for elliptic operators on manifolds with edges under symmetry conditions with respect to the edge covariables.

Fix several non-zero elements of an algebraic field with linearly independent logarithms. Consider the set of elements of the field whose logarithms can be expressed in terms of the logarithms of the fixed numbers using rational coefficients. The corresponding vectors of coefficients make up a lattice with the standard integral lattice as a finite-index sublattice. An improved upper bound for this index is given in terms of the extended logarithmic heights of the quantities involved. On the way an estimate for the coefficients in integer linear relations between the logarithms of algebraic numbers is obtained.

A non-local problem (with respect to time) for the heat equation is considered for , : find a function such that

An explicit formula for the solution is found. The question of its applicability is discussed. A description of well-posedness classes is presented. The main conjecture is as follows: as , the solution grows no more rapidly than with .

The subgroups of discrete Euclidean groups generated by reflections are classified.

Spaces of local deformations of classical Lie algebras with a homogeneous root system over a field K of characteristic 2 are studied. By a classical Lie algebra over a field K we mean the Lie algebra of a simple algebraic Lie group or its quotient algebra by the centre. The description of deformations of Lie algebras is interesting in connection with the classification of the simple Lie algebras.