Vol 67, No 4 (2021): Science — Technology — Education — Mathematics — Medicine


Existence and Uniqueness Theorems for the Pfaff Equation with Continuous Coefficients

Abduganiev A.A., Azamov A.A., Begaliev A.O.


In this paper, the Pfaff equations with continuous coefficients are considered. Analogs of Peano’s existence theorem and Kamke’s theorem on the uniqueness of the solution to the Cauchy problem are established, and a method for the approximate solution of the Cauchy problem for the Pfaff equation is proposed.

Contemporary Mathematics. Fundamental Directions. 2021;67(4):609-619
pages 609-619 views

α-Subharmonic Functions

Abdullaev B.I., Imomkulov S.A., Sharipov R.A.


In this paper, we study the class of α-subharmonic functions. A number of important properties of α-subharmonic functions are proved, and an equivalent, more convenient definition of α-subharmonicity is given. The geometric structure of removable singularities for some classes of α-subharmonic functions is also described.

Contemporary Mathematics. Fundamental Directions. 2021;67(4):620-633
pages 620-633 views

Generalized Localization and Summability Almost Everywhere of Multiple Fourier Series and Integrals

Ashurov R.R.


It is well known that Luzin’s conjecture has a positive solution for one-dimensional trigonometric Fourier series, but in the multidimensional case it has not yet found its confirmation for spherical partial sums of multiple Fourier series. Historically, progress in solving Luzin’s hypothesis has been achieved by considering simpler problems. In this paper, we consider three of these problems for spherical partial sums: the principle of generalized localization, summability almost everywhere, and convergence almost everywhere of multiple Fourier series of smooth functions. A brief overview of the work in these areas is given and unsolved problems are mentioned and new problems are formulated. Moreover, at the end of the work, a new result on the convergence of spherical sums for functions from Sobolev classes is proved.

Contemporary Mathematics. Fundamental Directions. 2021;67(4):634-653
pages 634-653 views

Statistical Ergodic Theorem in Symmetric Spaces for Infinite Measures

Veksler A.S., Chilin V.I.


Let (Ω,μ) be a measurable space with σ -finite continuous measure, μ(Ω)=∞. A linear operator T:L1(Ω)+L(Ω)→L1(Ω)+L(Ω) is called the Dunford-Schwartz operator if ||T(f)||1<||f||1 (respectively, ||T(f)||<||f||) for all f∈L1(Ω) (respectively, f∈L(Ω)).  {Tt}t>0 is a strongly continuous in L1(Ω) semigroup of Dunford-Schwartz operators, then each operator At(f)=1t0tTs(f)dsL1(Ω){{{A_t(f)} ={\frac{1}{t}} {\int_0^t} {T_s(f)} ds \in L_1(\Omega)}} has a unique extension to the Dunford-Schwartz operator, which is also denoted by At, t>0. It is proved that in the completely symmetric space of measurable functions on (Ω,μ) the means At converge strongly as t→+∞ for each strongly continuous in L1(Ω) semigroup {Tt}t>0 of Dunford-Schwartz operators if and only if the norm ||.||E(Ω) is order continuous.

Contemporary Mathematics. Fundamental Directions. 2021;67(4):654-667
pages 654-667 views

Weierstrass Polynomials in Estimates of Oscillatory Integrals

Ikromov I.A., Sadullaev A.S.


In this paper, estimates are obtained for the Fourier transform of smooth charges (measures) concentrated on some nonconvex hypersurfaces. The summability of the maximal Randall function is proved for a wide class of nonconvex hypersurfaces. In addition, in the three-dimensional case, estimates are obtained depending on the Varchenko height. The accuracy of the obtained estimates is proved. The proof of the estimate for oscillatory integrals is based on the Weierstrass preparatory theorem.

Contemporary Mathematics. Fundamental Directions. 2021;67(4):668-692
pages 668-692 views

Functor of Idempotent Probability Measures with Compact Support and Open Mappings

Ishmetov A.Y.


In this paper, we show that the functor of idempotent probability measures with compact support acting in the category of Tikhonov spaces and their continuous mappings is normal. It is found that this functor is monodic. Further, it is proved that the functor of idempotent probability measures with compact support preserves the openness of continuous mappings of Tikhonov spaces.

Contemporary Mathematics. Fundamental Directions. 2021;67(4):693-706
pages 693-706 views

Separable Algorithmic Representations of Classical Systems and Their Applications

Kasymov N.K., Dadazhanov R.N., Ibragimov F.N.


The main results of the theory of separable algorithmic representations of classical algebraic systems are presented. The most important classes of such systems and their representations in the lower classes of the arithmetic hierarchy - positive and negative - are described. Special attention is paid to the algorithmic, structural and topological properties of separable representations of groups, rings and bodies, as well as to effective analogs of the Maltsev theorem on embedding rings in bodies. The possibilities of using the studied concepts in the framework of theoretical informatics are considered.

Contemporary Mathematics. Fundamental Directions. 2021;67(4):707-754
pages 707-754 views

Injectivity and Nuclearity Properties for Real C*-Algebras

Rakhimov A.A., Nurillaev M.E., Boltaev K.K.


In this paper, we study injective and nuclear real W*- and C*-algebras. The connection of these concepts with similar concepts of enveloping W*- and C*-algebras is considered. The equivalence of the concepts of injectivity and nuclearity for real C*-algebras is shown. As a consequence, nuclear real factors of types II1, II, III1, III0 and IIIλ (0< λ<1) are completely described.

Contemporary Mathematics. Fundamental Directions. 2021;67(4):755-765
pages 755-765 views

Fokas Method for the Heat Equation on Metric Graphs

Sobirov Z.A., Eshimbetov M.R.


The paper presents a method for constructing solutions to initial-boundary value problems for the heat equation on simple metric graphs such as a star-shaped graph, a tree, and a triangle with three converging edges. The solutions to the problems are constructed by the so-called Fokas method, which is a generalization of the Fourier transform method. In this case, the problem is reduced to a system of algebraic equations for the Fourier transform of the unknown values of the solution at the vertices of the graph.

Contemporary Mathematics. Fundamental Directions. 2021;67(4):766-782
pages 766-782 views

Volterra-type Quadratic Stochastic Operators with a Homogeneous Tournament

Tadzhieva M.A., Eshmamatova D.B., Ganikhodzhaev R.N.


As is known [1], each quadratic stochastic operator of Volterra type acting on a finitedimensional simplex defines a certain tournament, the properties of which make it possible to study the asymptotic behavior of the trajectories of this Volterra operator. In this paper, we introduce the concept of a homogeneous tournament and study the dynamic properties of Volterra operators corresponding to homogeneous tournaments in the simplex S4.

Contemporary Mathematics. Fundamental Directions. 2021;67(4):783-794
pages 783-794 views

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