Table of contents

We prove that for an arbitrary odd and every automorphism of the free Burnside group that stabilizes every maximal normal subgroup of infinite index is an inner automorphism. For the same values of  and , we establish that the subgroup of inner automorphisms of  is maximal among the subgroups in which the orders of the elements are bounded by .

We study the amenability property for the group of formal power series in one variable with coefficients in a commutative ring  with identity. We show that there exists an invariant mean on the space of uniformly continuous bounded functions on this group. This is equivalent to the fact that every continuous action of  on every compact space has an invariant probability measure.

We discuss the following problem: given an integer , a real number , and an arbitrary subset which is not contained in a multiplicative shift of a proper subfield of  and satisfies , where is the finite field of  elements, describe those positive integers and  for which we have a set-theoretic equality . In particular, we show that this equality holds for and .

We consider unbounded continuously invertible operators , on a Hilbert space such that the operator has finite rank. Assuming that and the semigroup , , is of class , we state criteria under which the semigroups , , are also of class . We give applications to the theory of mean-periodic functions. The investigation is based on functional models of non-selfadjoint operators and on the technique of matrix Muckenhoupt weights.

The Bohl index is associated with a one-parameter family of multi-valued maps of elliptic type , . It determines the asymptotic behaviour of solutions of the parabolic inclusion . Our main aim is to obtain lower bounds for the Bohl index. We study the nature of the dependence of solutions of the above inclusion on the initial value and the map . We prove that the Bohl index is stable with respect to perturbations that are small on the average.

We consider intersections of two real five-dimensional quadrics, which are referred to for brevity as real four-dimensional biquadrics. Their rigid isotopy classes were described long ago: there are 16 such classes. We prove that the rigid isotopy class of a non-singular real four-dimensional biquadric is uniquely determined by the topological type of its real part. To do this, we calculate the dimensions of the cohomology spaces of the real part of a four-dimensional biquadric.

We investigate optimal control problems for linear distributed systems which are not solved with respect to the time derivative and whose homogeneous part admits a degenerate strongly continuous solution semigroup. To this end, we first obtain theorems on the existence of a unique strong solution of the Cauchy problem. This enables us to formulate sufficient conditions for the solubility of the optimal control problems under consideration. In contrast to earlier papers on a similar topic, we substantially weaken the conditions on the quality functional with respect to the state function. The abstract results thus obtained are illustrated by an example of an optimal control problem for the linearized system of Navier-Stokes equations.

We obtain a result concerning the basis property in a weighted space on an interval  for a system of exponentials generated by the zeros of the Fourier transform of a function with singularities at the ends of the support interval . For an arbitrary we find a criterion for the basis property of the system in a weighted space on the interval and the systems of sines and cosines in a weighted space on the interval . The weight is everywhere a finite product of polynomial functions.