Mathematics

Affine System of Walsh Type. Completeness and Minimality

Authors: Al-Jourany K. H., Mironov V. A., Terekhin P. A.

The question on completeness and minimality of Walsh affine systems is considered. On the basis of functional-analytical structure of multishift in Hilbert space, which being the generalized analogue of the operator of simple one-side shift and closely connected with Cuntz algebra representations, we give definition of Walsh affine system. Various criteria and tests of completeness of affine systems of functions are established. A biorthogonal conjugate system is found and its completeness is proved. 

On the Distribution of the Numbers with Binary Expansions of a Special Type in Arithmetic Progressions

Authors: Naumenko A. P.

Let 2 be the primitive root mod p. Let N0 be a set of natural numbers whose binary expansions contain even numbers of 1. Numbers from N0 are uniformly distributed in arithmetic progressions with the difference p.

Operator Integration with an Involution Having a Power Singularity

Authors: Kornev V. V., Khromov A. P.

Spectral properties of the integral operator with an involution of special type in the upper limit are studied and an equiconvergence theorem for its generalized eigenfunction expansions is obtained.

Spectral Sequences of Fibre Tolerance Spaces

Authors: Klyaeva I. A.

The paper presents the theoretical base for the construction of spectral sequences of tolerant exfoliations. Namely, the authors give a number of important qualities of singular cubes in tolerant exfoliations. The fundamental base group operation on the group of fiber homology of tolerant exfoliation theorem is proved. According to the general theory of spectral sequences the first and the second terms of spectral sequence of tolerant exfoliations are got.

The Theorem on Equiconvergence for the Integral Operator on Simplest Graph with Cycle

Authors: Burlutskaya M. S.

The paper deals with integral operators on the simplest geometric two-edge graph containing the cycle. The class of integral operators with range of values satisfying continuity condition into internal node of graph is described. The equiconvergence of expansions in eigen and adjoint functions and trigonometric Fourier series is established.

Generalized Regular Systems of Differential Equations

Authors: Aldibekov T. M.

From the set of systems of differential equations we consider the class of systems with solutions asymptotic defined by generalized exponents. The generalization some known signs correct theorems are obtained.

On Numerical Approximation of Differential Polynomials

Authors: Fryantsev A. V.

A numerical approximation formula was devised for differential operators of a special form. An absolute approximation error value was indicated. It was shown that the mentioned approximation is accurate for polynomials.

On Solvability of Certain Classes of Irregular the Second Order Variation Problems

Authors: Pokornyi Y. V., Bakhtina Z. I., Ischenko A. S.

In the work the irregular model of the Stiltjes string on segment [0, ℓ] with boundary conditions u(0) = u(ℓ) = 0 is discussed. The solvability conditions of the mentioned problem are described.

Research of Stability for Extremal Rotation Surfaces

Authors: Medvedeva N. M.

In this work we obtain the first and second variations of area type functional for rotation surfaces formulas. We proof the feature of stability and instability in the terms of the local coordinates and special integrals. We consider some examples by application our results for research if stability for rotation surfaces.

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