Table of contents

We use the method of similar operators to study the spectral properties of Dirac operators, and obtain results on the asymptotic behaviour of the spectra of Dirac operators and the convergence of spectral expansions.

We consider a two-dimensional periodic self-adjoint second-order differential operator on the plane with a small localized perturbation. The perturbation is given by an arbitrary (not necessarily symmetric) operator. It is localized in the sense that it acts on a pair of weighted Sobolev spaces and sends functions of sufficiently rapid growth to functions of sufficiently rapid decay. By studying the spectrum of the perturbed operator, we establish that the essential spectrum is stable, the residual spectrum is absent, and the set of isolated eigenvalues is discrete. We obtain necessary and sufficient conditions for the existence of new eigenvalues arising from the ends of lacunae in the essential spectrum. In the case when such eigenvalues exist, we construct the first terms of asymptotic expansions of these eigenvalues and the corresponding eigenfunctions.

We construct a series of counterexamples showing the unimprovability of the hypotheses of certain results related to the extension problem in the theory of convolution equations.

We consider a two-dimensional lattice of coupled van der Pol oscillators obtained under a standard spatial discretization of the non-linear wave equation , , , on the unit square with the zero Dirichlet or Neumann boundary conditions. We shall prove that the corresponding system of ordinary differential equations has attractors admitting no analogues in the original boundary-value problem. These attractors are stable invariant tori of various dimensions. We also show that the number of these tori grows unboundedly as the number of equations in the lattice is increased.

We consider real algebraic varieties that are intersections of three real quadrics. For brevity they are referred to as real triquadrics. We construct triquadrics that are -varieties and calculate the cohomology groups of the real parts of such triquadrics with coefficients in the field of two elements using relations between triquadrics and plane curves.

We consider non-singular intersections of three real five-dimensional quadrics. For brevity they are referred to as real three-dimensional triquadrics. We prove the existence of real three-dimensional -triquadrics with components, where is any integer in the range .

In the problem of the best uniform approximation of a continuous real-valued function in a finite-dimensional Chebyshev subspace , where is a compactum, one studies the positivity of the uniform strong uniqueness constant . Here stands for the strong uniqueness constant of an element of best approximation of , that is, the largest constant such that the strong uniqueness inequality holds for any . We obtain a characterization of the subsets for which there is a neighbourhood  of  satisfying the condition . The pioneering results of N. G. Chebotarev were published in 1943 and concerned the sharpness of the minimum in minimax problems and the strong uniqueness of algebraic polynomials of best approximation. They seem to have been neglected by the specialists, and we discuss them in detail.

We use proof-nets to study the algorithmic complexity of the derivability problem for some fragments of the Lambek calculus. We prove the NP-completeness of this problem for the unidirectional fragment and the product-free fragment, and also for versions of these fragments that admit empty antecedents.