Mathematics

CMS Operators Type B(1, 1) and Lie Superalgebra osp(3, 2)

Authors: Movsisyan G. S., Sergeev A. N.

The main purpose of this article is to study the realation between the representations theory of Lie superalgebras osp(3, 2) and the Calogero –Moser – Sutherland (CMS) B(1, 1) type differential operator. The differential operator depends polynomially on three parameters. The corresponding polynomial eigenfunctions also depend on three parameters; but in the general case, the coefficients of these eigenfunctions have a rational dependence on the parameters.

Multipoint Differential Operators: „Splitting“ of the Multiple in Main Eigenvalues

Authors: Mitrokhin S. I.

We study the boundary value problem for the differential operator of the eighth order with a summable potential. The boundary conditions of the boundary value problem are multipoint. We derived the integral equation for solutions of differential equation which define the studied differential operator. The asymptotic formulas and estimates for the solutions of the corresponding differential equation for large values of the spectral parameter are obtained.

Dini – Lipschitz Test on the Generalized Haar Systems

Authors: Shcherbakov V. I.

Generalized Haar systems, which are generated (generally speaking, unbounded) by a sequence {pn} ∞n=1 and which is defined on the modification segment [0, 1]∗ , thai is on a segment [0, 1], where {pn} — rational points are calculated two times and which is a geometrical representation of zero-dimensional compact Abelians group are considering in this work.

Bernstein Polynomials for a Standard Module Function on the Symmetric Interval

Authors: Tikhonov I. V., Sherstyukov V. B., Petrosova M. A.

Bernstein polynomials are studied on a symmetric interval. Basic relations connected with Bernstein polynomials for a standard module function are received. By the Templ’s formula we establish recurrence relations from which the Popoviciu’s expansion is derived. Suitable formulas for the first and second derivatives are found. As a result an explicit algebraic form for Bernstein polynomials is obtained. We also notice some corollaries. 

On a Quotient Topology in Topological Semigroups and Groups

Authors: Sultanov S. R.

The paper discusses the definution of the topological factor semigrop using the open congruence relations on this topological semigroup. Based on this approach, a description of all open gomomorfic images of topological semigroup is obtained. Similarly, this approach is used to describe all open homomorphic images of a topological group. 

Three-dimensional Homogeneous Spaces, Not Admitting Invariant Connections

Authors: Mozhey N. P.

The purpose of the work is the classification of three-dimensional isotropy-faithful homogeneous spaces, not admitting invariant connections. The local classification of homogeneous spaces is equivalent to the description of effective pairs of Lie algebras. If there exists at least one invariant connection then the space is isotropy-faithful, but the isotropy-faithfulness is not sufficient for the space in order to have invariant connections.

Resolvent Approach to Fourier Method in a Mixed Problem for Non-homogeneous Wave Equation

Authors: Kornev V. V., Khromov A. P.

Fourier method of obtaining classic solution is being justified in a mixed problem for non-homogeneous wave equation with a complex potential and fixed boundary conditions under minimal conditions on initial data. The proof is based on resolvent approach which does not need any information on eigen and associated functions of the corresponding spectral problem. 

Spectral Analysis of a Class of Difference Operators with Growing Potential

Authors: Garkavenko G. V., Uskova N. B.

The similar operator method is used for the spectral analysis of the closed difference operator of the form (A x)(n) = x(n + 1) + x(n − 1) − 2x(n) + a(n)x(n), n ∈ Z under consideration in the Hilbert space l2(Z) of bilateral sequences of complex numbers, with a growing potential a : Z → C. The asymptotic estimates of eigenvalue, eigenvectors, spectral estimation of equiconvergence applications for the test operator and the operator of multiplication by a sequence a : Z → C.

The Fourier Series of the Meixner Polynomials Orthogonal with Respect to the Sobolev-type Inner Product

Authors: Gadzhimirzaev R. M.

In this paper we consider the system of discrete functions {ϕr,k(x)} ∞ k=0 , which is orthonormal with respect to the Sobolev-type inner product hf, gi = Xr−1 ν=0 ∆ ν f(−r)∆ν g(−r) + X t∈Ωr ∆ r f(t)∆r g(t)µ(t), where µ(t) = q t (1−q), 0 < q < 1. It is shown that the shifted classical Meixner polynomials © M−r k (x + r) ª∞ k=r together with functions n (x+r) [k] k! or−1 k=0 form a complete orthogonal system in the space l2,µ(Ωr) with respect to the Sobolev-type inner product.

Graphs with Contours in Multiresolution Analysis on Vilenkin Groups

Authors: Berdnikov G. S.

The aim of this article is to study the problem of constructing mutiresolution analysis on Vilenkin group. Previous papers by S. F. Lukomskii, Iu. S. Kruss and the author present an algorithm for constructing scaling functions ϕ with compact support, Fourier transform of which also has compact support. The description of such algorithm is tightly connected with directed graphs of special structure, which are constructed with the help of so-called N-valid trees. One of the special properties of these graphs is the absence of directed cycles — contours.

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