Nurzhan Temirgaliyev

• Kazakh University of the Humanities and Law

In the article is studied, according to our opinion, the problematic moment "An experiment was performed,the outcome is known. Did the event occur?" and the associated calculations of the frequencies of the occurrence of thisevent in the context of determining the probability. Namely, theoretical and practical detailed discussions of these topicsar...

The approximate calculation of integrals is a common part of research in various fields of mathematics, primarily in the theory of approximations and computational mathematics.In its turn, within the theory of numerical integration, distinctions take place depending on the features of the integrand, the scale of which is “Quadrature (cubature) form...

In this paper, we propose a program for studying Radon transforms in accordance with the computational (numerical) diameter (C(N)D) scheme by applying the uniform distribution theory. The main result is that Radon transforms are qualified as optimal among all possible linear functionals that are used to extract numerical information for generating...

We study approximate calculation of integrals of products of two functions by the method of tensor products of functionals. Under the assumption that one of them belongs to the Sobolev class with the dominant mixed derivative, and the other does with fast oscillation, we obtain unimprovable (in the sense of order) estimates of approximation error....

We discuss a C(N)D-statement, consisting of the known and elaborating in decades C(N)D-1 statement that can be and should be interpreted as quantitative statement of approximation theory and computational mathematics, which, in common with new prolongations of both C(N)D-2 and C(N)D-3, is suggested as a natural theoretical and computational scheme...

We discuss two questions. First, we consider the existence of close to optimal quadrature formulas with a “bad” L²-discrepancy of their grids, and the second is the question of how much explicit quadrature formulas are preferable to sorting algorithms. Also, in the model case, we obtain the solution to the question of approximative possibilities of...

In this article was proved, that any linear equation $Lu=f$ in the case of any linear versions of the Galerkin method, has at least as many unsolved right-hand sides in the form of linear combinations $f=L\psi_1 +...+L\psi_N +L\psi_{N+1} +...+L\psi_{5N} +...+L\psi_{T}$, as there are finite-dimensional linear subspaces with dimensionality as much th...

For a given finite set of linear functionals we construct functions vanishing on them and give order estimates of their derivatives. We also give their different applications.

As noted in circumstantial monograph “Geometry of Numbers” by P. M. Gruber and C. G. Lekkerkerker, lately there is no considerable contribution to classical theory of calculation and estimates of critical determinants. In this paper, in a certain sense, we fill a gap in a retardation occurred in developing of this theme. We propose a new theoretica...

We prove the embedding theorems of the Sobolev–Morrey spaces into the space of uniformly continuous functions so extending the classical Sobolev Theorems.

In this paper the spectral analysis of all possible linear congruent sequences with a maximum period is conducted and the best random number generators are selected among them.

Based on algebraic theory of number, we determine discrete Fourier transforms with further concrete definitions. At that, the sets of specification of discrete function are interconnected with various optimization problems, quasi-Monte Carlo method including.

The computational (numerical) diameter is used to completely solve the problem of approximate differentiation of a function given inexact information in the form of an arbitrary finite set of trigonometric Fourier coefficients.

In Ul’yanov classes, we compare computational aggregates constructed by the method of tensor products of functionals by means of trigonometric Fourier series.

We show that replacement of the Lebesgue norm in definitions of functional classes by the Morrey semi-norm provides unimprovable embedding theorems, which complement the classical results on this subject.

We study three concretizations of the notion of computer (computing) diameter, namely, the discretization of solutions to the Klein-Gordon equation, numerical differentiation, and function recovery.

This paper examines the relationship between the degree of uniformity of distribution of grids, including Smolyak grids, with the intention of choosing weights to obtain efficient quadrature formulas.

We find sharp lower bounds for the accuracy in the approximation to solutions of the wave equation by computational aggregates constructed on the basis of numerical information obtained from all linear functionals (whose total number is given) applied to two initial conditions in Sobolev classes. We compute the sharp order of approximation by all l...

In this paper with the help of Smolyak quadrature formulas we calculate exact orders of errors of the numerical integration of trigonometric Fourier coefficients of functions from generalized classes of Korobov and Sobolev types. We apply the obtained results to the recovery of functions from their values at a finite number of points in terms of th...

A study was conducted to demonstrate tensor products of functionals and their application. The study aimed at describing the idea underlying Smolyak's method and to extend it to the general case of orthonormal systems. Each function expandable in a Fourier-Lebesgue series in terms of such a system was associated everywhere with a unique set of Four...

1 Eurasian National University named after L.N.Gumilev, Astana, Kazakhstan 2 M. O. Auezov South Kazakhstan State University, Shymkent, Kazakhstan

Based on the theory of divisors, an effective theoretical algorithm designed previously by the authors for constructing good quadrature formulas with a Korobov grid (i.e., an algorithm for finding optimal coefficients) is used to develop a computer search method that produces tables of optimal coefficients giving more accurate integration error est...

For the Klein-Gordon equation with initial conditions in Nikol'skii classes, we ob- tain sharp orders of discretization errors with respect to precise information. In particular, these orders imply qualitatively and quantitatively that raising the smoothness of one of the initial conditions relative to the other does not improve the discretization...

The informativeness of all the linear functionals in the recovery of functions in the classes is investigated. The optimal recovery orders of functions in are found. These are completely determined by embedding theorems, similarly to the case of function classes with smoothness described in terms of numerical parameters. Bibliography: 28 titles.

The author exhibits an application of ideal theory to the exact integration of trigonometric polynomials with arbitrarily prescribed coefficient spectrum and to the approximate integration of functions of certain multidimensional classes E, W, and H.

An algorithm is proposed for the numerical integration of an arbitrary function represent-able as a sum of an absolutely converging multiple trigonometric Fourier series. The resulting quadrature formulas have identical weights, and the nodes form a Korobov grid that is completely defined by two positive integers, of which one is the number of node...

V rabote polucheno neravenstvo E. A. Storozhenko v sluchae modulei nepreryvnosti, predlozhennykh Dittsianom i Totikom, i dany ee primeneniya k teorii vlozhenii klassov funktsii.

Sharp estimates (in the power scale) are obtained for the discretization error in the solutions to Poisson’s equation whose right-hand side belongs to a Korobov class. Compared to the well-known Korobov estimate, the order is almost doubled and has an ultimate value in the power scale.

In this paper, we study the informativeness of linear functionals in reconstruction problems and obtain exact orders of the informativeness of linear functionals in the Besov and Sobolev classes W and SW.

For function classes with dominant mixed derivative and bounded mixed difference in the metric ofL q (1<q≤2), quadrature formulas are constructed so that the following properties are achieved simultaneously: the grid is simple, the algorithm is efficient and close to the optimal algorithm for constructing the grid, and the order of the error on the...

The present article offers a method of constructing quadrature formulas based on the theory of divisors of the field of algebraic numbers.