Table of contents

A complete description is given of the dual of the Bergman space on a plane bounded convex domain.

We study subspaces invariant under differentiation in spaces of analytic functions in convex domains of the complex plane. For arbitrary convex domains we get a criterion for analytic continuation of functions from arbitrary non-trivial closed invariant subspaces admitting spectral synthesis.

We find weak asymptotics of approximation characteristics related to the problem of recovering (reconstructing) the derivative from the function values at a given number of points, Stechkin's problem for the derivation operator, and the problem of describing asymptotics of diameters for non-isotropic Nikol'skii and Besov classes.

We give a formula for factorizing the full twist in the braid group in terms of four factorizations of the full twist in. This formula is used to construct a symplectic 4-manifold  and two regularly homotopic generic coverings branched along cuspidal Hurwitz curves (without negative nodes) having different braid monodromy factorization types. The class of fundamental groups of complements of affine plane Hurwitz curves is described in terms of generators and defining relations.

An undirected graph is said to be edge-regular with parameters if it has  vertices, each vertex has degree , and each edge belongs to  triangles. We put . Brouwer, Cohen, and Neumaier proved that every connected edge-regular graph with (equivalently, with ) is strongly regular. In this paper we construct an example of an edge-regular, not strongly regular graph on 36 vertices with . This shows that the estimate above is sharp. We prove that every connected edge-regular graph with (equivalently, either satisfies , or has parameters or , or is strongly regular.

We prove that by deforming the multiplication in a prime commutative alternative algebra using a C-operation we obtain a prime non-commutative alternative algebra. Under certain restrictions on non-commutative algebras this relation between algebras is reversible. Isotopes are special cases of deformations. We introduce and study a linear space generated by the Bruck C-operations. We prove that the Bruck space is generated by operations of rank 1 and 2 and that "general" Bruck operations of rank 2 are independent in the following sense: a sum of n operations of rank 2 cannot be written as a linear combination of (n–1) operations of rank 2 and an arbitrary operation of rank 1. We describe infinite series of non-isomorphic prime non-commutative algebras of bounded degree that are deformations of a concrete prime commutative algebra.

We prove the theorem of Markov on the existence of an algorithmically non-recognizable combinatorial n-dimensional manifold for every n≥4. We construct for the first time a concrete manifold which is algorithmically non-recognizable. A strengthened form of Markov's theorem is proved using the combinatorial methods of regular neighbourhoods and handle theory. The proofs coincide for all n≥4. We use Borisov's group [8] with insoluble word problem. It has two generators and twelve relations. The use of this group forms the base for proving the strengthened form of Markov's theorem. (The author is indebted to S.I. Adian for this idea.)