Table of contents

For a number of non-matrix varieties of associative algebras finite-dimensional (so-called model) algebras generating these varieties are calculated in this paper. In essence, the multiplicity and representability indices of such varieties are estimated. Simultaneously, an estimate of the multiplicity for a certain class of varieties of arbitrary complexity is given, which continues the previous studies of the author. The connection with quasipolynomials is also considered.

A version of Bogolyubov's first theorem is established for infinite-dimensional parabolic inclusions. Sufficient conditions for the asymptotic stability of the trivial solution of a parabolic inclusion with non-stationary homogeneous principal part are stated.

In the first part of the paper the following conjecture stated by Dal'bo and Starkov is proved: the geodesic flow on a surface of constant negative curvature has a non-compact non-trivial minimal set if and only if the Fuchsian group  is infinitely generated or contains a parabolic element. In the second part interesting examples of horocycle flows are constructed: 1) a flow whose restriction to the non-wandering set has no minimal subsets, and 2) a flow without minimal sets. In addition, an example of an infinitely generated discrete subgroup of with all orbits discrete and dense in is constructed.

The existence of isothermic coordinates is proved for special classes of irregular surfaces. The smoothness properties of isothermic representations are investigated.

A new method is proposed for constructing Hamilton-minimal and minimal Lagrangian immersions and embeddings of manifolds in  and in . In particular, using this method it is possible to construct embeddings of manifolds such as the -dimensional generalized Klein bottle , , , , and others.

A semilinear parabolic equation is considered in the union of two bounded thin cylindrical domains and adjoining along their bases, where  is a domain in , . The unknown functions are related by means of an interface condition on the common base . This problem can serve as a reaction-diffusion model describing the behaviour of a system of two components interacting at the boundary. The intensity of the reaction is assumed to depend on  and the thickness of the domains, and to be of order . Under investigation are the limiting properties of the evolution semigroup , generated by the original problem as (that is, as the domain becomes ever thinner). These properties are shown to depend essentially on the exponent . Depending on whether  is equal to, greater than, or smaller than 1, the original system can have three distinct systems of equations on  as its asymptotic limit. The continuity properties of the global attractor of the semigroup as are established under natural assumptions.

A Kac-Moody algebra is said to be hyperbolic if it corresponds to a generalized Cartan matrix of hyperbolic type. Root subsystems of root systems of algebras of this kind are studied. The main result of the paper is the classification of the maximum-rank regular hyperbolic subalgebras of hyperbolic Kac-Moody algebras.

Let be an entire function of exponential type in  with indicator function ; let , , be a subsequence of zeros of the entire function of exponential type ; let be a complex number sequence and assume that

A simple construction of a sequence of entire functions of exponential type  transforming  into a subsequence  of zeros of an entire function of exponential type such that is put forward (an approximation theorem). This result is applied to stability problems of zero sequences and non-uniqueness sequences for spaces of entire functions of exponential type with constraints on the indicators and to the problem of the stability of the completeness property of exponential systems in the space of germs of analytic functions on a compact convex set.