Vol 39, No 2 (2018)

We obtain a new class of solutions to the inverse problems of the logarithmic potential in the form of a logarithmic function of a ratio of polynomials of the same degree. We give examples of finite solvability of the inverse problems.

We consider a periodic problem of conjugation on the real axis for a linear differential elliptic equation of second order with constant coefficients. There is known a representation of general solution of the equation in terms of analytic functions depending on affine connected variables. We introduce auxiliary analytic functions enabling us to reduce the problem to a system of two periodic Riemann boundary value problems. That problem was solved first by L. I. Chibrikova. We use her results for solving of our problem. We obtain its explicit solution and conditions of solvability, evaluate the index and defect numbers.

We study separating function sets. We find some necessary and sufficient conditions for C_p(X) or C_p² (X) to have a point-separating subspace that is a metric space with certain nice properties. One of the corollaries to our discussion is that for a zero-dimensional X, C_p(X) has a discrete point-separating space if and only if C_p² (X) does.

In this paper, we prove some important results of 2-absorbing ideals in a commutative ring with 1 ≠ 0. The conceptof n-weakly prime ideal in a commutative ring with 1 ≠ 0 is introduced and prove number of results concerning to n-weakly prime ideals in commutative rings.

We consider linear normed spaces of measurable functions dominated by positive measurable function powered by real positive parameter. Also, we consider its dual and predual, and we propose a method for constructing a limit spaces of these functional spaces taken by power parameter. We prove that these limit spaces are (LF)-spaces and also prove that the limit spaces presume the relation of duality, i.e., the limit space of predual spaces is predual for the limit space of dominated functions, and the limit space of duals is dual for it. Also, the limit space of predual spaces is embedded into the limit space of dual spaces.

We considered 6-fold linear equation in the class of analytic functions outside the equilateral triangle and vanishing at infinity. We proposed a method of equivalent regularization, using the theory of boundary value problem of Carleman. Also, the paper is dedicated to applications to the problem of moments for entire functions of exponential type. In particular, we construct a system of functions biorthogonal whit piecewise-quasipolynomial weight to some power system on the three rays. The indicator of such functions is a piecewise-trigonometric function, with a period of 2π/3.

In this paper it is given a full description of normal subgroups of index eight and ten for the group representation of a Cayley tree and counted all of them.

We consider a class of anisotropic elliptic differential equations of second order with divergent form and variable exponents. The corresponding elliptic operators are pseudo-monotone and coercive. We obtain solvability conditions for the Dirichlet problem in unbounded domains Ω ⊂ ℝ_n, n ≥ 2. The proof of existence of solutions is free of restrictions on growth of data for |x| → ∞.

We consider the space obtained from spaces Triebel–Lizorkin with help interpolation by the real method. Embedding theorems for these spaces are studied. It is shown that the space of traces belong to the same class of spaces.

We deduce a systemofODEs describing behavior of critical points and poles of a smooth one-parametric family of rational functions uniformizing a given family of ramified coverings of the Riemann sphere.

The problem of the existence of twice periodic solutions of a nonlinear elliptic system of second-order equations on a plane whose main periods are given is studied. Conditions for the existence of solutions with prescribed poles, as well as with prescribed zeros and poles, are obtained.

For a complete Cartan foliation (M,F) we introduce two algebraic invariants g₀(M,F) and g₁(M,F) which we call structure Lie algebras. If the transverse Cartan geometry of (M,F) is effective then g₀(M,F) = g₁(M,F). Weprove that if g₀(M,F) is zero then in the category of Cartan foliations the group of all basic automorphisms of the foliation (M,F) admits a unique structure of a finite-dimensional Lie group. In particular, we obtain sufficient conditions for this group to be discrete. We give some exact (i.e. best possible) estimates of the dimension of this group depending on the transverse geometry and topology of leaves. We construct several examples of groups of all basic automorphisms of complete Cartan foliations.

In the present paper we obtain new metric invariants of metric spaces. These invariants can be used for classification of the finite metric spaces, their recognition and in the research of Tammes’ problem for sphere.