Vol 99, No 5-6 (2016)

The C*-simplicity of n-periodic products is proved for a large class of groups. In particular, the n-periodic products of any finite or cyclic groups (including the free Burnside groups) are C*-simple. Continuum-many nonisomorphic 3-generated nonsimple C*-simple groups are constructed in each of which the identity xⁿ = 1 holds, where n ≥ 1003 is any odd number. The problem of the existence of C*-simple groups without free subgroups of rank 2 was posed by de la Harpe in 2007.

For weighted symmetric (or rearrangement-invariant) spaces with nontrivial Boyd indices and weights from suitable Muckenhoupt classes, the basis property of the Haar system in these spaces and two versions of the direct theorem on the approximation by polynomials in the Haar system are established.

Generalized hypergeometric differential equations of arbitrary order are considered. Necessary and sufficient conditions for the algebraic independence of solutions of collections of such equations, as well as of their values at algebraic points, are obtained.

In the paper, first-order complex sequences with finite maximal angular density are studied. A criterion for such a sequence to be a part of a regularly distributed set with a given angular density is obtained. Using this criterion, we present complete solutions of fundamental principle problems and basis for an invariant subspace of analytic functions in a bounded convex domain.

It is shown that, between the values of the activity a = 1 and a < 1, there is a gap, which can be overcome by using additional energy. This energy is defined on the spinodal a = 1 (μ = 0) on the P–Z diagram and gives, in the parastatistical distribution, an additional term of Bose condensate type, which is also preserved for μ < 0. This term is the right-hand side of the Fermi–Dirac distribution. In this paper, it is also shown how to find the “liquid–amorphous body” binodal.

Several characteristics of the solutions of a differential system are defined and studied from a unified standpoint, namely, they are arranged in a certain order and unite all known and some new Lyapunov characteristics describing various oscillation and wandering properties. For second-order equations, all of these characteristics coincide with each other, and for autonomous systems, the set of values of each of these characteristics contains all absolute values of the imaginary parts of eigenvalues of the operator of the system.

We study the behavior of the set of time-periodic solutions of the three-dimensional system of Navier–Stokes equations in a bounded domain as the frequency of the oscillations of the right-hand side tends to infinity. It is established that the set of periodic solutions tends to the solution set of the homogenized stationary equation.

Asymptotic solutions of a nonlinear magnetohydrodynamic system rapidly varying near moving surfaces are described. It is shown that the motion of jump surfaces is determined from a free boundary problem, while the main part of the asymptotics satisfies a system of equations on the moving surface. In the “nondegenerate” case, this system turns out to be linear, while, under the additional condition that the normal component of the magnetic field vanishes, it becomes nonlinear. In the latter case, the small magnetic field instantaneously increases to a value of order 1.

For the generalized Lauricella hypergeometric function F_D^(N), Jacobi-type differential relations are obtained and their proof is given. A new system of partial differential equations for the function F_D^(N) is derived. Relations between associated Lauricella functions are presented. These results possess a wide range of applications, including the theory of Riemann–Hilbert boundary-value problem.

In the present paper, a 2mth-order quasilinear divergence equation is considered under the condition that its coefficients satisfy the Carathéodory condition and the standard conditions of growth and coercivity in the Sobolev space W^m,p(Ω), Ω ⊂ Rⁿ, p > 1. It is proved that an arbitrary generalized (in the sense of distributions) solution u ∈ W₀^m,p (Ω) of this equation is bounded if m ≥ 2, n = mp, and the right-hand side of this equation belongs to the Orlicz–Zygmund space L(log L)ⁿ⁻¹(Ω).

The paper deals with the question of the divergence of Fourier series in function spaces wider than L = L[−π, π], but narrower than L^p = L^p[−π, π] for all p ∈ (0, 1). It is proved that the recent results of Filippov on the generalization to the space ϕ(L) of Kolmogorov’s theorem on the convergence of Fourier series in L^p, p ∈ (0, 1), cannot be improved.

The notion of Hadamard decomposition of a semisimple associative finite-dimensional complex algebra generalizes the notion of classicalHadamard matrix, which corresponds to the case of commutative algebras. The algebras admitting Hadamard decompositions are called Hadamard algebras. A relation for the values of an irreducible character of a Hadamard algebra on the products of involutions forming an orthogonal basis of the algebra is obtained. This relation is then applied to describe the Hadamard decompositions in an algebra of dimension 8.

The first boundary-value problem in the half-strip for a parabolic-type equation with Bessel operator and Riemann–Liouville derivative is studied. In the case of the zero initial condition, the representation of the solution in terms of the Fox H-function is obtained. The uniqueness of the solution for a class of functions vanishing at infinity is proved. It is shown that when the equation under consideration coincides with the Fourier equation, the obtained representation of the solution becomes the known representation of the solution of the corresponding problem.