Mathematics

On Definability of Universal Graphic Automata by Their Input Symbol Semigroups

Authors: Farakhutdinov R. A., Molchanov V. A.

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

Authors: Mozhey N. P.

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

Authors: Ignatyev M. Y.

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

Quasi-Polynomials of Capelli. II

Authors: Antonov S. Y., Antonova A. V.

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

Authors: Lukashov А. L.

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).

Siegеl disks and basins of attraction for families of analytic functions

Authors: Gumenuk P. A.

Let be a hyperbolic domain, , let ∆ be a Stolz angle at with respect to the unit disk D, and W a domain containing the point λ0 . Consider an analytic family ; consisting of analytic functions in the domain U with the following expansion , λ ∈ W, for small z. Let be the maximal domain A ⊂ U, such that 0 ∈ A and f l (A) ⊂ A, or the set {0} if there exist no such domains. We prove, that if a sequence converges to λ0 and , then the sequence of the domains converges to S as to the kernel.

Isometry groups of Lobachevskian spaces, similarity transformation groups of Euclidean spaces and Lorentzian holonomy groups

Authors: Galaev A. S.

In the present paper transitively and simply transitively acting isometry groups of Lobachevskian spaces and transitively acting similarity transformation groups of Euclidean spaces are classified. A geometrical proof of the result of L. Berard Bergery and A. Ikemakhen about the classification of weaklyirreducible not irreducible subalgebras of so(1,n+1) is given. 

On Recovering Differential Operators on a Closed Set from Spectra

Authors: Yurko V. A.

The Sturm – Liouville differential operators on closed sets of the real line are considered. Properties of their spectral characteristics are obtained and the inverse problem of recovering the operators from their spectra is studied. An algorithm for the solution of the inverse problem is developed and the uniqueness of the solution is established. The statement and the study of inverse spectral problems essentially depend on the structure of the closed set.

Pieri Formulae and Specialisation of Super Jacobi Polynomials

Authors: Sergeev A. N., Zharinov E. D.

We give a new proof of the fact that the Euler supercharacters of the Lie superalgebra osp(2m + 1, 2n) can be obtained as a certain limit of the super Jacobi polynomials. The known proof was not direct one and it was mostly based on calculations. In this paper we propose more simple and more conceptional proof. The main idea is to use the Pieri formulae from the beginning. It turns out that the super Jacobi polynomials and their specialisations can be uniquely characterised by two properties.

An Inverse Spectral Problem for Sturm – Liouville Operators with Singular Potentials on Graphs with a Cycle

Authors: Vasilev S. V.

This paper is devoted to the solution of inverse spectral problems for Sturm – Liouville operators with singular potentials from class W2−1 on graphs with a cycle. We consider the lengths of the edges of investigated graphs as commensurable quantities. For the spectral characteristics, we take the spectra of specific boundary value problems and special signs, how it is done in the case of classical Sturm – Liouville operators on graphs with a cycle. From the spectra, we recover the characteristic functions using Hadamard’s theorem.

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