# Mathematics

## Approximating Properties of the Powers of the Differentiation Operator Resolvent

The families of operators are constructed and their approximating properties are investigated in the problems of approximating the derivative of function sand the smooth solutions of integralequations.

## The Mixed Problem for the Differential Equation with Involution and Potential of the Special Kind

For the solution of some mixed problem with involution and real symmetrical potential, explicit analytical formula has been found with the use of the Fourier method. Techniques allowing to avoid term-byterm differentiation of the functional series and impose the minimum conditions for initial problem data, are used.

## On a Approximate Solution of the Problem of Aspherical Convex Compact Set

We examine a finite-dimensional problem of minimizing the ratio radius of the ball given a compact convex set (in an arbitrary norm) to the radius of the inscribed sphere through the choice of a common center of these balls. The article offers an approach to building the numerical method of its solution.

## The Inverse Problem of Spectral Analysis for the Matrix Sturm – Liouville Equation

The inverse spectral problem is investigated for the matrix Sturm - Liouville equation on a finite interval. The article provides properties of spectral characteristics, a constructive procedure for the solution of the inverse problem along with necessary and sufficient conditions for its solvability has been obtained.

## Uniqueness of the Solution of the Inverse Problem for Differential Operators on Arbitrary Compact Graphs

An inverse spectral problem is studied for Sturm – Liouville operators on arbitrary compact graphs with standard matching conditions in internal vertices. A uniqueness theorem of recovering operator’s coefficients from spectra is proved.

## Finite Closed 3(4)-Loops of Extended Hyperbolic Plane

This article considers finite closed *n*-loops of the extended hyperbolic plane *H*^{2}. The paper deals with topological and metric properties of the finite closed 3(4)-loops. Pasha statement analogues have been also obtained. We proved the existence of two types 4-loops and convexity of the plain 4-loop.

## Approximate Solution of an Optimal Control Problem with Linear Nonhomogeneous Control System in Hilbert Space

For an optimal control problem with a linear differential equation in Hilbert space and quadratic criteria necessary and sufficient conditions of control functions optimality and approximate formulas of the expansions of these functions in eigenfunctions of the control system operator have been obtained.

## On Analogue of Jordan – Dirichlet Theorem about the Convergence of the Expansions in Eigenfunctions of a Certain Class of Differential-Difference Operators

An analogue of Jordan – Dirichlet theorem is established of convergence of the expansions in eigen functions of the operator Ly = αy′(x) − y′(1 − x) with the boundary condition U(y) = ay(0) + by(1) − (y,ϕ) = 0.

## Symmetry Axes of Planar Polynomial Differential Systems

The notion of *N*-type axis of symmetry is introduced. It is proved that the vector field defined by system of the differential equations with *n*order polynomials in a right hand, cannot have even number of axes of symmetry *N*-type at *n *=2*m,m *∈N. For *n *=2*,*3 full research of the given system on *N*-symmetry is carried out.

## Аbout Some Boundary Problems in the Semispace for a Class of Pseudo-Differential Equations with Degeneracy

Boundary problems in the halfspace for one class of the pseudodifferential equations are considered. The coercetive a priori estimations and theorems of the existence of solutions for these problems are established.