Mathematics

New Properties of Almost Nilpotent Variety of Exponent 2

Authors: Shulezhko O. V.

In the presented work we consider numerical characteristics of almost nilpotent variety of exponent 2, which was first constructing in article [1]. The main result of this paper is introduce the exact values of the multiplicities of the irreducible modules appearing in the expansion of the multilinear part of the variety. Meanwhile, we obtain as a consequence the formulas of codimension and colength of the variety of exponent 2.

The Gradient Methods for Solving the Cauchy Problem for a Nonlinear ODE System

Authors: Fominykh A. V.

The article considers the Cauchy problem for a nonlinear system of ODE. This problem is reduced to the variational problem of minimizing some functional on the whole space. For this functional necessary minimum conditions are presented. On the basis of these conditions the steepest descent method and the method of conjugate directions for the considered problem are described. Numerical examples of the implementation of these methods are presented. The Cauchy problem with the system which is not solved with respect to derivatives is additionally investigated.

Approximation of the Riemann–Liouville Integrals by Algebraic Polynomials on the Segment

Authors: Tyuleneva A. A.

The direct approximation theorem by algebraic polynomials is proved for Riemann–Liouville integrals of order r>0. As a corollary, we obtain asymptotic equalities for ε-entropy of the image of a Hölder type class under Riemann–Liouville integration operator.

Approximation of Functions by Fourier–Haar Sums in Weighted Variable Lebesgue and Sobolev Spaces

Authors: Magomed-Kasumov M. G.

It is considered weighted variable Lebesgue Lp(x)w and Sobolev Wp(⋅),w spaces with conditions on exponent p(x)≥1 and weight w(x) that provide Haar system to be a basis in Lp(x)w. In such spaces there were obtained estimates of Fourier–Haar sums convergence speed. Estimates are given in terms of modulus of continuity Ω(f,δ)p(⋅),w, based on mean shift (Steklov's function).

Approximation of Functions in Symmetrical and Connected Holder Spaces by Linear Means of Fourier–Vilenkin Series

Authors: Likhacheva T. V.

In this paper some summation methods are applied to Fourier-Vilenkin series in so called symmetric spaces. These methods use triangular matrix with sums in rows tending to zero and with some conditions on difference of coefficients. The triginometric counterpart of our results are due to M. L. Mittal, B. E. Rhoades, A. Guven, etc.

On Differential Operator in Compact Zero-dimensional Groups

Authors: Kruss Y. S.

We define strong derivative on zero-dimensional compact group and find conditions under which the differential operator does not depend from an orthonormal system that defines this derivative. For multidimensional case we find conditions under which the differential operator does not depend from method of conversion multidimensional group in one-dimensional group. We obtain a clear view of annihilators in a multidimensional compact zero-dimensional group.

Numerical Solution of Inverse Spectral Problems for Sturm–Liouville Operators with Discontinuous Potentials

Authors: Efremova L. S.

We consider Sturm–Liouville differential operator with potential having a finite number of simple discontinuities. This paper is devoted to the numerical solution of such inverse spectral problems. The main result of this work is a procedure that is able to recover both the points of discontinuities as well as the heights of the jumps. Following, using these results, we may apply a suitable numerical method (for example, the generalized Rundell–Sacks algorithm with a special form of the reference potential) to reconstruct the potential more precisely.

On an Approach to Approximate Solving of the Problem for the Best Approximation for Compact Body by a Ball of Fixed Radius

Authors: Dudov S. I., Osipcev M. A.

In this paper, we consider the problem of the best approximation of a compact body by a fixed radius ball with respect to an arbitrary norm in the Hausdorff metric. This problem is reduced to a linear programming problem in the case, when compact body and ball of the norm are polytops.

Asymptotic Values of Analytic Functions Connected with a Prime End of a Domain

Authors: Ganenkova E. G.

In 1954 M. Heins proved that for any analytic set A, containing the infinity, there exists an entire function with asymptotic set A. In the article we prove the following analog of Heins's theorem: for a multi-connected planar domain D with an isolated boundary fragment, an analytic set A, ∞∈A, and a prime end of D with impression p there exists an analytic in D function f such that A is the set of asymptotic values of f connected with p.

Synthesis in the Polynomial Kernel of Two Analytic Functionals

Authors: Volkovaya V. A.
Let ¼ be an entire function of minimal type and order ½ = 1 and let ¼(D) be the corresponding differential operator. Maximal ¼(D)-invariant subspace of the kernel of an analytic functional is called its C[¼]-kernel. C[¼]-kernel of a system of analytic functionals is called the intersection of theirC[¼]-kernels. The paper describes the conditions which allow synthesis ofC[¼]-kernels of two analytical functionals with respect to the root elements of the differential operator ¼(D). 

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