Mathematics

Approximation of Continuous 2 p-Periodic Piecewise Smooth Functions by Discrete Fourier Sums

Authors: Akniev G. G.

Let N be a natural number greater than 1. Select N uniformly distributed points tk = 2πk/N + u (0 6 k 6 N − 1), and denote by Ln,N(f) = Ln,N(f,x) (1 6 n 6 N/2) the trigonometric polynomial of order n possessing the least quadratic deviation from f with respect to the system {tk}N−1 k=0 . Select m + 1 points −π = a0 < a1 < ... < am−1 < am = π, where m > 2, and denote Ω = {ai}m i=0. Denote by Cr Ω a class of 2π-periodic continuous functions f, where f is r-times differentiable on each segment ∆i = [ai,ai+1] and f(r) is absolutely continuous on ∆i.

On Inverse Problem for Differential Operators with Deviating Argument

Authors: Yurko V. A.

Second-order functional differential operators with a constant delay are considered. Properties of their spectral characteristics are obtained, and a nonlinear inverse spectral problem is studied, which consists in constructin goperators from the irspectra. We establish the unique nessand develop a constructive procedure for solution of the inverse problem.

Hermite Interpolation on a Simplex

Authors: Khasyanov R. S.

In the paper, we solve the problem of polynomial interpolation and approximation functions of several variable sonann dimensional simplex in the uniform normus ingpoly nomials of the third degree.Wechoose interpolation conditions in terms of derivatives in the directions of the edges of a simplex. In the same terms we obtained estimates of the deviation of derivatives of polynomial from the corresponding derivatives of an interpolated function under the assumption that the interpolated function has continuous directional derivatives up to the fourth order inclusive.

Some Properties of 0/1-Simplices

Authors: Nevskii M. V., Ukhalov A. Y.

Let n ∈ N, and let Q n = [0,1] n . For a nondegenerate simplex S ⊂ R n , by σS we mean the homothetic copy of S with center of homothety in the center of gravity of S and ratio of homothety σ. Put ξ(S) = min{σ > 1 : Q n ⊂ σS}, ξ n = min{ξ(S) : S ⊂ Q n }.

Criterion for a Generalized Solution in the Class Lp for the Wave Equation to Be in the Class W1 p

Authors: Mokrousov I. S.

In this paper we consider the question of whether a generalized solution of the wave equation belongs to different function spaces. Consideration of classical solutions imposes substantial restrictions on the initial data of the problem. But if we proceed not from differential but from integral equations, then the class of solutions and the class of initial boundary value problems can be substantially expanded. To solve the boundary value problem for the wave equation obtained by the wave counting method, it is easy to obtain a sufficient condition for belonging to a particular class.

Non-reductive Homogeneous Spaces Not Admitting Normal Connections

Authors: Mozhey N. P.

The purpose of the work is the classification of three-dimensional non-reductive homogeneous spacesnot admitting normal connections, affine connections, their torsion tensors, curvature and holonomy algebras.The object of investigation arepointed-non-reductive spaces and connections on them. The basic notions, such as the isotropically-faithful pair, reductive space, afne connection, curvature tensor and torsion tensor, holonomy algebra and normal connection are defined.

An Asymptotic Relation for Conformal Radii of Two Nonoverlapping Domains

Authors: Zherdev A. V.

We consider a family of continuously varying closed Jordan curves given by a polar equation, such that the interiors of the curves form an increasing or decreasing chain of domains. Such chains can be described by the Löwner–Kufarev differential equation. We deduce an integral representation of a driving function in the equation.Using this representation we obtainan a symptotic formula, which establishes a connection between conformal radii of bounded and unbounded components of the complement of the Jordan curve when the bounded component is close to the unit disk.

Classification of Prolonged Bi-metric Structures on Distributions of Non-zero Curvature of Sub-Riemannian Manifolds

Authors: Galaev A. S.

The notion of the interior geometry of a sub-Riemannian manifold M is introduced, that is the aggregate of those manifold properties that depend only on the framing D ⊥ of the distribution D of the sub-Riemannian manifold as well as on the parallel transport of the vectors tangent to the distribution D along the curves tangent to this distribution.The maininvariantsof the interiorgeometry of a sub-Riemannianmanifold M are the following: the Schouten curvature tensor; the 1-form η defining the distribution D; the Lie derivative L ~ ξ g ofthemetrictensorg alongavectorfield

Special Examples of Superstable Semigroups and Their Application in the Inverse Problems Theory

Authors: Vu Nguyen Son Tung ..

Special examples of superstable (quasinilpotent) semigroups and their application in the theory of linear inverse problems for evolutionary equations are studied. The term “semigroup” means here the semigroup of bounded linear operators of class C 0 . The standard research scheme is used. The linear inverse problem with the final overdetermination in a Banach space for the evolution equation is considered. A special assumption is introduced, related to the superstability of the main evolutionary semigroup.

Polynomials Orthogonal with Respect to Sobolev Type Inner Product Generated by Charlier Polynomials

Authors: Sharapudinov I. I., Guseinov I. G.

The problem of constructing of the Sobol ev orthogonal polynomials sαr,n(x) generated by Char li er polynomialssαn(x) is considered. It is shown that the system of polynomials sαr,n(x) generated by Charlier polynomialsis complete in the space Wrl, consisted of the discrete functions, given on the grid Ω = {0, 1, . . .}. Wrlρisa H ilb ert space with the inner product hf, gi. An explicit formula in the form of sαr,k+r(x) =kPl=0brlx[l+r],where xm]= x(x −1) . . . (x − m + 1), is found.

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