# Mathematics

## Recovering singular differential pencils with a turning point

Second-order pencils of differential equations on the half-line with turning points are considered. We establish properties of the spectrum and study the inverse spectral problem of recovering coefficients of the pencil from the spectral data.

## An extension of the ordering to the set of probability measures

A general method for extension of the ordering to the set of the probability measured. It based on the Galois connection between all such extensions and subsets of isotone mappings of the given ordered set in the real numbers. The canonical extension is defined as extension determined by the set of all isotone mappings. For canonical extension, an effective description is given and the maximal measures in convex polyhedra are found. Some applications of considered methods for decision making problems are indicated.

## On regularity of self-adjoint boundary conditions

In this paper we expound the favourable decision of Kamke's (Камке) hypothesis that self-adjoint boundary conditions are regular and we also establish an analogue of Jordan-Dirichlet theorem on uniform convergence of trigonometric Fourier series for the case of the expansions in eigen functions of self-adjoint integral operators from the certain class.

## On optimal choise of interpolation spline on triangular net

In this paper we find a Hermite Spline on atriangle for the approximation error of its derivatives with respect to a side of this triangle are inversely proportional to length of this side.

## The Principle of Localization at the Class of Functions Integrable in the Riemann for the Processes of Lagrange –Sturm – Liouville

Let us say that the principle of localization holds at the class of functions F at point x0 ∈ [0, π] for the Lagrange –Sturm – Liouville interpolation process L SL n (f, x) if limn→∞ L SL n (f, x0) − L SL n (g, x0) = 0 follows from the fact that the condition f(x) = g(x) is met for any two functions f and g belonging to F in some neighborhood Oδ(x0), δ > 0.

## On Definability of Universal Graphic Automata by Their Input Symbol Semigroups

Universal graphic automaton Atm(G, G′ ) is the universally attracting object in the category of automata, for which the set of states is equipped with the structure of a graph G and the set of output symbols is equipped with the structure of a graph G′ preserved by transition and output functions of the automata. The input symbol semigroup of the automaton is S(G, G′ ) = End G×Hom(G, G′ ). It can be considered as a derivative algebraic system of the mathematical object Atm(G, G′ ) which contains useful information about the initial automaton.

## On the Geometry of Three-dimensional Pseudo-Riemannian Homogeneous Spaces. I

The problem of establishing links between the curvature and the topological structure of a manifold is one of the important problems of geometry. In general, the purpose of the research of manifolds of various types is rather complicated. Therefore, it is natural to consider this problem for a narrower class of pseudo-Riemannian manifolds, for example, for the class of homogeneous pseudo-Riemannian manifolds.

## Asymptotics of Solutions of Some Integral Equations Connected with Differential Systems with a Singularity

Our studies concern some aspects of scattering theory of the singular differential systems y ′ − x −1Ay − q(x)y = ρBy, x > 0 with n × n matrices A, B, q(x), x ∈ (0, ∞), where A, B are constant and ρ is a spectral parameter. We concentrate on investigation of certain Volterra integral equations with respect to tensor-valued functions. The solutions of these integral equations play a central role in construction of the so-called Weyl-type solutions for the original differential system.

## Quasi-Polynomials of Capelli. II

This paper observes the continuation of the study of a certain kind of polynomials of type Capelli (Capelli quasi-polynomials) belonging to the free associative algebra F{X S Y } considered over an arbitrary field F and generated by two disjoint countable sets X and Y . It is proved that if char F = 0 then among the Capelli quasi-polynomials of degree 4k − 1 there are those that are neither consequences of the standard polynomial S − 2k nor identities of the matrix algebra Mk(F).

## Rational interpolation processes on several intervals

It is considered the Lagrange interpolation processes such that rational functions with fixed denominators play the role of polynomials vanishing at interpolation nodes. An estimate for Lebesgue constants is obtained for the case of rational functions deviated least from zero on a given system of intervals with maximally possible number of deviation points, and when the matrix of fixed poles is contained in a compact set outside of the system of intervals. V. N. Rusak and G. Min found earlier particular case (for the case of one interval).