# Mathematics

## 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.

## On Multiple Completeness of the Root Functions for a Class of the Pencils of Differential Operators

A polinomial pencil of ordinary differential operators of n-th order generated by a homogeneous differential expression with constant coefficients and by two-point boundary conditions of a special structure with lcondition sinzero only (1 ≤ l ≤ n−1) isconsidered in the space L 2 [0,1]. The case is studied, when the roots of the characteristic equation lieonaray coming from theorigin. Asufficient condition of m-fold completeness of the system of root functions for m ≤ n−l inthe space L 2 [0,1] isfound. Anaccuracy of obtained result is shown.

## Integrability of a Partial Case of the Lowner Equation

We give a quadrature solution to the partial case of the Lowner¨ equation for the upper half-plane.

## Asymptotic Properties of Polynomials pˆα,βn (x), Orthogonal on Any Sets in the Сase of Integers α and β

Asymptotic properties of polynomials *p*ˆ* ^{α,β}_{n }*(

*x*)

*,*orthogonal with weight (1−

*x*)

_{j}*(1+*

^{α}*x*)

_{j}*∆*

^{β}*t*on any finite set of

_{j }*N*points from segment [−1

*,*1] are investigated. Namely an asymptotic formula is proved in which asymptotic behaviour of these polynomials as

*n*tends to infinity together with

*N*is closely related to asymptotic behaviour of the Jacobi polynomials.

## Cohomology Rings of Semicubecal Sets

The aim of this paper is to define the structure of the ring over the graded cohomology group of a semicubecal set with coefficients in a ring with unity.