# Mathematics

## On Classic Solution of the Problem for a Homogeneous Wave Equation with Fixed End-Points and Zero Initial Velocity

The paper gives necessary and sufficient conditions of classic solution for a homogeneous wave equation with a summable potential, fixed end-point, and zero initial velocity. With the use of Fourier method and Krylov method of improving series rate convergence an analogue of d’Alembert formula is derived in the form of exponentially convergent series. The paper essentially supports and extends the results of our work carried out in 2016. The suggested new method, based on the use of divergent (in Euler’s sense) series, is very economical in using well-known mathematical facts.

## Value Regions in Classes of Conformal Mappings

The survey is devoted to most recent results in the value region problem over different classes of holomorphic univalent functions represented by solutions to the Loewner differential equations both in the radial and chordal versions. It is important also to present classical and modern solution methods and to compare their efficiency. More details are concerned with optimization methods and the Pontryagin maximum principle, in particular.

## Analytic Embedding of Geometries of Constant Curvature on a Pseudosphere

In mathematical studies, the geometries of maximum mobility are important. Examples of such geometries are Euclidean, pseudo-Euclidean, Lobachevsky, symplectic and so on. There is no complete classification of such geometries. They are distinguished as the geometries of the max- imum mobility in general, for example, the geometries from the Thurston list, and the geometries of the local maximum mobility. V. A. Kyrov developed a method for classifying the geometries of local maximum mobility, called the method of embedding.

## The Solvability of a System of Nonlinear Integral Equations of Hammerstein Type on the Whole Line

In recent years, the interest has grown in nonlinear integral equations of convolution type in connection with their application in various fields of mathematical physics, inparticular, inthep-adic theory of an open-closed string, kinetic theory of gases, in the theory of radiation transfer in spectral lines. The paper is devoted to the questions of construction of nontrivial solutions and the study of their asymptotic behavior for one system of nonlinear integral equations of convolution type with a symmetric kernel on the whole axis.

## Harmonic Analysis of Operator Semigroups Slowly Varying at Infinity

The article focuses on studying of strongly continuous bounded operator semigroups. In the space of uniformly continuous functions with values inacomplex Banach space weconsider the subspace of integrally vanishing at infinity functions. This subspace includes the subspace of vanishing at infinity functions, but it is wider. We study the properties of the subspace under consideration.

## Multiple Completeness of the Root Functions of the Pencils of Differential Operators with Constant Coefficients and Splitting Boundary Conditions

In the space of square summable functions on the main segment [0,1], the class of polynomial pencils of ordinary differential operators of the n-th order is considered. The coefficients of the differential expression are assumed to be constants. The boundary conditions are assumed to be splitting and two-point at the ends 0 and 1 (l of boundary conditions is taken only at the point 0, and the remaining n − l is taken at the point 1). The differential expression and the boundary forms are assumed to be homogeneous, that is, they contain only main parts.

## Martingale Inequalities in Symmetric Spaces with Semimultiplicative Weight

Let (Ω,Σ,P) be a complete probability space, F = {F n } ∞ n=0 be an increasing sequence of σ- algebras such that ∪ ∞ n=0 F n generates Σ. If f = {f n } ∞ n=0 is a martingale with respect to F and E n is the conditional expectation with respect to F n , then one can introduce a maximal function M(f) = sup n>0 |f n | and a square function S(f) =?∞P i=0|f i − f i−1 | 2 ¶ 1/2 , f −1 = 0. In the case of uniformly integrable martingales there exists g ∈ L 1 (Ω) such that E n g = f n and we consider a sharp maximal function f ♯ = sup n>0 E n |g − f n−1 |.

## The Il’in Spectral Method for Determination of the Properties of the Basis Property and the Uniform Convergence of Biorthogonal Expansions on a Finite Interval

The paper discusses the basics of the spectral method of V. A. Il’in on an example of a simple second order differential operator on a segment of the number line. The ﬁrst theorem of Il’in on the unconditional basis property is stated. Its detailed proof is given. A chain of generalizations of this theorem is traced. A recently established a theorem on the unconditional basis property for the differential operators with general integral boundary conditions is formulated.

## Necessary and Sufficient Condition for an Orthogonal Scaling Function on Vilenkin Groups

There are several approaches to the problem of construction of an orthogonal MRA on Vilenkin groups, but all of them are reduced to the search of the so-called scaling function. In 2005 Yu. Farkov used the so-called “blocked sets” in order to ﬁnd all possible band-limited scaling functions with compact support for each set of certain parameters and his conditions are necessary and sufﬁcient. S. F. Lukomskii, Iu. S. Kruss and G. S.

## Nonlocal Boundary-Value Problems in the Cylindrical Domain for the Multidimensional Laplace Equation

Correct statements of boundary value problems on the plane for elliptic equations by the method of analytic function theory of a complex variable. Investigating similar questions, when the number of independent variablesis greater than two, problems of a fundamental nature arise. Avery attractive and convenient method of singular integral equations loses its validity due to the absence of any complete theory of multidimensional singular integral equations. The author has previously studied local boundary value problems in a cylindrical domain for multidimensional elliptic equations.