Table of contents

An expression is found for the Gaussian torsion of a surface in 4-dimensional Euclidean space defined implicitly by a system of two equations. An application of this formula for the intersection of two quadrics is considered as an example.

Wagner's celebrated theorem states that a finite affine plane whose collineation group is transitive on lines is a translation plane. The notion of an orthogonal decomposition (OD) of a classically semisimple associative algebra introduced by the author allows one to draw an analogy between finite affine planes of order and ODs of the matrix algebra into a sum of subalgebras conjugate to the diagonal subalgebra. These ODs are called WP-decompositions and are equivalent to the well-known ODs of simple Lie algebras of type into a sum of Cartan subalgebras. In this paper we give a detailed and improved proof of the analogue of Wagner's theorem for WP-decompositions of the matrix algebra of odd non-square order an outline of which was earlier published in a short note in "Russian Math. Surveys" in 1994. In addition, in the framework of the theory of ODs of associative algebras, based on the method of idempotent bases, we obtain an elementary proof of the well-known Kostrikin-Tiep theorem on irreducible ODs of Lie algebras of type in the case where is a prime-power.

The large-time asymptotic behaviour is studied for a system of non-linear evolution dissipative equations

where is the Fourier transform. Under the assumptions that the initial data , are sufficiently small, where

is a Sobolev weighted space, and that the total mass vector is non-zero it is proved that the leading term in the large-time asymptotic expansion of solutions in the critical case is a self-similar solution defined uniquely by the total mass vector of the initial data.

The behaviour of solutions of non-linear elliptic systems is studied in the neighbourhood of a singular point, finite or infinite. It is shown that if the order of the singularity lies in a certain interval depending on the modulus of ellipticity of the system, then it is equal to the order of the singularity of some singular solution for the poly-Laplacian operator. Sharp two-sided energy estimates in the neighbourhood of the singular point are obtained for such solutions. Global solutions are considered, for which lower energy bounds are derived from upper estimates. Counterexamples constructed for second-order equations and systems demonstrate that the interval of regular behaviour of the order of the singularity is precisely described.

The asymptotic behaviour of the circular parameters of the polynomials orthogonal on the unit circle with respect to Geronimus measures is analysed. It is shown that only when the harmonic measures of the arcs making up the support of the orthogonality measure are rational do the corresponding parameters form a pseudoperiodic sequence starting from some index (that is, after a suitable rotation of the circle and the corresponding modification of the orthogonality measures they form a periodic sequence). In addition it is demonstrated that if the harmonic measures of these arcs are linearly independent over the field of rational numbers, then the sets of limit points of the sequences of absolute values of the circular parameters and of their ratios are a closed interval on the real line and a continuum in the complex plane, respectively.

The study of the birational geometry of Fano fibrations whose fibres are Fano double hypersurfaces of index 1 is continued. Birational rigidity is proved for the majority of families of this type, which do not satisfy the condition of sufficient twistedness over the base (in particular, this means that there exist no other structures of a fibration into rationally connected varieties) and the groups of birational self-maps are computed. The principal components of the method of maximal singularities are considerably improved, chiefly the techniques of counting multiplicities for fibrations into Fano varieties over the line.