Mathematics

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). 

Foliation on Distribution with Finslerian Metric

Authors: Bukusheva A. V.
A distribution D with a admissible Finsler metric is defined on a smooth manifold X. Let F be a foliation on X. On the distribution of D as on a smooth manifold foliation F corresponds to the foliation TF. Using this foliation and connection over distribution we define analog exterior derivative. Exterior differential forms is applied to a special form. 

Classical solution by the Fourier method of mixed problems with minimum requirements on the initial data

Authors: Khromov A. P., Burlutskaya M. S.

The article gives a new short proof the V. A. Chernyatin theorem about the classical solution of the Fourier method of the mixed problem for the wave equation with fixed ends with minimum requirements on the initial data. Next, a similar problem for the simplest functional differential equation of the first order with involution in the case of the fixed end is considered, and also obtained definitive results. These results are due to a significant use of ideas A. N. Krylova to accelerate the convergence of series, like Fourier series.

Everywhere divergence of Lagrange processes on the unit circle

Authors: Tyshkevich S. V., Shatalina A. V.

We study the convergence of Lagrange interpolation processes in the closed unit disk. Choosing a matrix with a certain distribution of interpolation nodes allowed to construct the set, completely covering the unit circle, and the function for which the process diverges everywhere on this set.

The new approach to solving the Riemann boundary value problem with infinite index

Authors: Salimov R. B., Karabasheva E. N.

This research considers Riemann–Hilbert boundary value problem with infinite index where edge condition of problem is established by the real axis. To solve this problem the approach based on the removal of the infinite discontinuity of the argument of boundary condition coefficient is used. The approach is analogous to the one which, in the context of the finite index of the problem in researches by F. D. Gakhov, helps to remove a discontinuity of initial genre of boundary condition coefficient with specially created functions, different from the ones in this research.

On Poisson Customary Polynomial Identities

Authors: Ratseev S. M.
We study Poisson customary and Poisson extended customary polynomials. We show that the sequence of codimensions {rn(V )}n¸1 of every extended customary space of variety V of Poisson algebras over an arbitrary field is either bounded by a polynomial or at least exponential. Furthermore, if this sequence is bounded by polynomial then there is a polynomial R(x) with rational coefficients such that rn(V ) = R(n) for all sufficiently large n. We present lower and upper bounds for the polynomials R(x) of an arbitrary fixed degree.

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