Vol 70, No 4 (2024)

Articles

On nondegenerate orbits of 7-dimensional Lie algebras containing a 3-dimensional Abelian ideal

Atanov A.V., Loboda A.V.

Abstract

This paper is related to the problem of describing homogeneous real hypersurfaces of multidimensional complex spaces as orbits of the action of Lie groups and algebras in these spaces. We study realizations in the form of algebras of holomorphic vector fields in C4 of 7-dimensional Lie algebras containing only 3-dimensional Abelian ideals and subalgebras. Among 594 types of 7dimensional solvable indecomposable Lie algebras containing a 6-dimensional nilradical, there are five types of such algebras. The article describes all their realizations that admit nondegenerate in the sense of Levi 7-dimensional orbits. The presence of “simply homogeneous” orbits among the constructed hypersurfaces is shown.

Contemporary Mathematics. Fundamental Directions. 2024;70(4):517-532
pages 517-532 views 50

Correct solvability of problems for fractional-power operator equations

Baboshin S.D.

Abstract

In this paper, we consider the sum of linear fractional-power operators acting in a Banach space and satisfying weak positivity. We establish the correct solvability of the problem for the corresponding fractional-operator equation and we give the representation of the solution through the inverse operator with an exact estimate of its norm. The results are applied to problems without initial conditions for an equation with singular coefficients. We consider examples of such equations.

Contemporary Mathematics. Fundamental Directions. 2024;70(4):533-541
pages 533-541 views 50

Unimodality of the probability distribution of the extensive functional of samples of a random sequence

Virchenko Y.P., Tevolde A.M.

Abstract

We establish a criterion for the unimodality of the probability distribution of a functional that is represented by the sum of a set of independent identically distributed random nonnegative variables x~k with a random number of terms distributed according to Poisson. The general distribution of terms x~k is concentrated on the interval [0,1] and is such that Pr{x~k=0}0. Its absolutely continuous part is asymptotically close to a uniform distribution. We introduce the concept of smoothing functions and establish an explicit form of the distribution of any fixed number of terms uniformly distributed on [0,1].

Contemporary Mathematics. Fundamental Directions. 2024;70(4):542-560
pages 542-560 views 53

Integral inequalities for trigonometric polynomials in periodic Morrey spaces

Joseph D.J.

Abstract

In this paper, we present a detailed exposition of Bernstein’s inequality, inequalities of different metrics and different dimensions for trigonometric polynomials in periodic Morrey spaces.

Contemporary Mathematics. Fundamental Directions. 2024;70(4):561-574
pages 561-574 views 51

Interpolation of periodic functions and construction of biorthogonal systems using uniform shifts of the theta function

Zhadanova M.L.

Abstract

The problems of interpolation of periodic functions and construction of biorthogonal systems are considered. Uniform shifts of the third Jacobi theta function are used as a basis. Explicit formulas for the nodal function and the function generating the biorthogonal system are obtained. Exact values of the lower and upper Riesz constants are found.

Contemporary Mathematics. Fundamental Directions. 2024;70(4):575-585
pages 575-585 views 57

The problem of existence of feedback control for one nonlinear viscous fractional Voigt model

Zvyagin A.V., Kostenko E.I.

Abstract

In this paper, we study the feedback control problem for a mathematical model describing the motion of a nonlinear viscous fluid with infinite memory along the trajectories of the velocity field. The existence of an optimal control that gives a minimum to a given bounded and lower semicontinuous quality functional is proved. The proof uses the approximation-topological approach, the theory of regular Lagrangian flows, and the theory of topological degree for multivalued vector fields.

Contemporary Mathematics. Fundamental Directions. 2024;70(4):586-596
pages 586-596 views 55

Interpolation by Earl’s method in the space of functions of semiformal order

Kabanko M.V., Malyutin K.G.

Abstract

We consider the problem of simple free interpolation in the space of functions of finite order and normal type in a half-plane. We propose its solution by the method of shifting interpolation nodes. This solution is based on Earl’s method, who solved the problem of free interpolation in the space of analytic bounded functions in a unit circle.

Contemporary Mathematics. Fundamental Directions. 2024;70(4):597-609
pages 597-609 views 48

Ordered billiard games and topological properties of billiard books

Kibkalo V.A., Tuniyants D.A.

Abstract

We discuss the connection between the construction of an ordered billiard game introduced earlier by Dragovic and Radnovic and the class of billiard books proposed by Vedyushkina. In this paper, we propose a generalization of the concept of realization of a certain game using a billiard book and prove an analogue of the Dragovic-Radnovic theorem for such a realization. We present recent results by the authors, Tyurina, and Zav’ialov on topological properties of isoenergy manifolds of circular billiard books and topological invariants of specific series of elliptic billiard books.

Contemporary Mathematics. Fundamental Directions. 2024;70(4):610-625
pages 610-625 views 54

Inversion of a polynomial operator with the Maslov-Chebyshev symbol

Kostin A.V.

Abstract

The Maslov–Heaviside method is applied to the inversion of a polynomial operator by the Maslov–Chebyshev symbol introduced in the paper. The result is applied to the proof of a theorem on the Bessel operator in the Stepanov spaces Sp(Rn), \(1 n=1,2,. This significantly expands the scope of application of operator methods to the study of the correct solvability of equations with the Laplace operator, usually studied in Lp spaces.

Contemporary Mathematics. Fundamental Directions. 2024;70(4):626-635
pages 626-635 views 48

On the construction of the square root for some differential operators

Kostin V.A., Kostin D.V., Silaeva M.N.

Abstract

Using the Balakrishnan-Yosida approach to constructing fractional powers of linear operators in a Banach space by means of strongly continuous semigroups with densely defined generating operators, in this paper, a similar scheme is presented for constructing fractional powers of nondensely defined operators by means of semigroups with a summable singularity. It is found that the newly constructed semigroups also have a singularity at zero, and their sharp estimate is established, related to the order of the singularity of the original semigroup and the fractional power of the constructed operator, in particular, the square root. As an example, the obtained results are applied to semigroups with a singularity given in the paper [3] and in the doctoral dissertation of Yu. T. Silchenko, and a square root is also constructed for a nondensely defined operator.

Contemporary Mathematics. Fundamental Directions. 2024;70(4):636-642
pages 636-642 views 53

Dual Radon-Kipriyanov transformation. Basic properties

Lyakhov L.N., Kalitvin V.A., Lapshina M.G.

Abstract

The Radon–Kipriyanov transformation (Kγ) was introduced in 1998. In theoretical and applied studies, it is necessary to introduce its dual transformation, which is denoted by Kγ# in the paper. Theorems on the boundedness of the Kγ# transformation in the corresponding Schwartz subspace of the main functions are proved. A formula for representing the generalized convolution of Kγ#-transformations of functions belonging to the corresponding spaces of the main functions is obtained.

Contemporary Mathematics. Fundamental Directions. 2024;70(4):643-653
pages 643-653 views 47

Multiscale mathematical model of the spread of respiratory infection considering the immune response

Mozokhina A.S., Ryumina K.A.

Abstract

This work presents a multiscale mathematical model of the spread of respiratory viral infection in a tissue and in an organism, taking into account the influence of innate and adaptive immune responses based on systems of reaction-diffusion equations with nonlocal terms. The defining characteristics of such models, which have physiological significance, are the viral replication number, wave propagation speed, and total viral load. In this work, these characteristics are estimated and their dependence on immune response parameters is investigated.

Contemporary Mathematics. Fundamental Directions. 2024;70(4):654-668
pages 654-668 views 56

On the recovery of the solution of the initial-boundary value problem for the singular heat conduction equation

Polovinkina M.V.

Abstract

We present the results concerning the research of the problem of the best recovery of the solution of the initial-boundary value problem for the heat equation with the Bessel operator in the spatial variable from two approximately known temperature profiles.

Contemporary Mathematics. Fundamental Directions. 2024;70(4):669-678
pages 669-678 views 54

Linear inverse problems for integro-differential equations in Banach spaces with a bounded operator

Fedorov V.E., Godova A.D.

Abstract

In this paper, we study the questions of well-posedness of linear inverse problems for equations in Banach spaces with an integro-differential operator of the Riemann-Liouville type and a bounded operator at the unknown function. A criterion of well-posedness is found for a problem with a constant unknown parameter; in the case of a scalar convolution kernel in an integro-differential operator, this criterion is formulated as conditions for the characteristic function of the inverse problem not to vanish on the spectrum of a bounded operator. Sufficient well-posedness conditions are obtained for a linear inverse problem with a variable unknown parameter. Abstract results are used in studying a model inverse problem for a partial differential equation.

Contemporary Mathematics. Fundamental Directions. 2024;70(4):679-690
pages 679-690 views 57

On studying the spread model of the HIV/AIDS epidemic

Shashkin A.I., Polovinkina M.V., Polovinkin I.P.

Abstract

The aim of this work is to study sufficient conditions for the asymptotic stability of the stationary solution of the initial-boundary value problem for a system of nonlinear partial differential equations describing the growth and spread of the HIV/AIDS epidemic. The above-mentioned model takes into account not only the factors taken into account by classical models, but also includes migration processes.

Contemporary Mathematics. Fundamental Directions. 2024;70(4):691-701
pages 691-701 views 50