Bulletin of the Institute of Mathematics

Author: Akhmedov O. (Kokand University Andijan branch), Muzaffarova D. (Andijan State University)

Abstract:
This paper rigorously establishes the coexistence of two distinct closed trajectories in
a three-dimensional quadratic dynamical system. The proof is constructed using the method of discrete numerical tracking (DN-tracking), which provides validated computations of the Poincaré map. Numerical evidence suggests the presence of two isolated periodic orbits with distinct basins of attraction. The DN-tracking procedure confirms that each cycle exists within a well-defined neighborhood in the phase space, thus providing a constructive proof of multistability in this system.

Keywords: Limit cycle; quadratic system; multistability; validated computation; Poincaré map; closed trajectory; coexistence of cycles; DN-tracking.

Author: Azamov S. (Tashkent State Transport University)

Abstract:
In this study, we investigate optimal quadrature formulas in a Hilbert space for functions involving exponential components. The main focus is on determining the norm of the error functional associated with such formulas. First, the extremal function corresponding to the error functional is constructed. Then, an explicit representation of the squared norm is derived. By minimizing this expression with respect to the coefficients, a system of linear equations is obtained and solved using Sobolev’s method. As a result, optimal coefficients of the quadrature formula are determined. Finally, an exact upper bound for the error is established.

Keywords: Optimal quadrature formulas; error functional norm; Hilbert space; Sobolev method; extremal function; numerical integration.

Author: Choriyeva I. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences)

Abstract:
In this work, we classify all symmetric Leibniz algebras corresponding to solvable Lie algebras whose nilradicals are naturally graded quasi-filiform. We focus on solvable Leibniz algebras with nonzero center and construct all symmetric Leibniz algebra structures whose underlying solvable Lie algebra is fixed. As a result, we obtain a complete classification of solvable symmetric Leibniz non-Lie algebras with quasi-filiform nilradicals.

Keywords: Symmetric Leibniz algebras; Lie algebras; solvable algebras; quasi-filiform algebras; automorphism; invariants.

Author: Dekhkonov F. (Namangan State University),(V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Science), Nematjonova M. (Namangan State University)

Abstract:
In this paper, we study the time-optimal control problem for a pseudo-parabolic equation with piecewise constant thermal conductivity. The generalized solution of the corresponding initial-boundary value problem is obtained. Using an additional integral condition, the control problem is reduced to a Volterra integral equation of the first kind. Estimates for the kernel of this integral equation are derived to ensure the existence and uniqueness of the solution. As a result, the admissibility of the control function is established, and the optimal estimate of the minimal time required to achieve a prescribed average temperature in the rod is obtained. Moreover, the minimal times corresponding to the pseudo-parabolic and purely parabolic cases are compared.

Keywords: Pseudo-parabolic equation; Volterra integral equation; admissible control; thin rod; weight function; minimal time.

Author: Dzhamalov S. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences), Shokirov A. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences)

Abstract:
This paper considers a coefficient inverse problem for the three-dimensional Tricomi equation in an unbounded parallelepiped. The problem consists of finding a triple of functions — the solution and two unknown coefficients — that satisfy the given equation, semi-nonlocal boundary conditions, and additional overdetermination conditions. A distinctive feature of the study is that the unknown coefficients are determined simultaneously with the solution.

The proposed method is based on reducing the coefficient inverse problem to a family of direct problems for loaded Tricomi integro-differential equations with semi-nonlocal boundary conditions in a bounded rectangular domain.

By applying the Fourier transform, the problem is reduced to a family of nonlinear integro-differential equations. To prove the existence and uniqueness of the solution, the methods of “$\varepsilon$-regularization” and successive approximations are employed. A priori estimates are established that ensure the convergence of the sequence of approximations.

Keywords: Three-dimensional Tricomi equation; coefficient-inverse problem with semi-nonlocal conditions; Fourier and $“ \varepsilon$-regularization$”$ methods; a priori estimates and sequences of approximations.

Author: Hasanov A. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences),(Ghent University), Ergashev T. (National Research University “TIIAME”),(V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences),(Ghent University), Okboev A. (National Research University “TIIAME”)

Abstract:
In this paper, Appell and Lauricella hypergeometric functions are investigated, and their known properties are applied to the solution of the Neumann problem for a three-dimensional degenerate elliptic equation. Fundamental solutions of the considered equation are expressed in terms of the Lauricella hypergeometric function of three variables, and an explicit solution of the Neumann problem in the first octant is obtained by means of the Appell hypergeometric function of the second kind. A decomposition formula for the Lauricella function is used to determine the order of singularity of the fundamental solutions. To justify the correctness of the obtained solution to the Neumann problem, an expansion formula for the Appell function is applied. The uniqueness of the solution of the posed problem is proved by the $abc$ -method (a variant of the energy method).

Keywords: Appell and Lauricella hypergeometric functions; three-dimensional degenerate elliptic equation; PDE-systems of hypergeometric type; fundamental solution; Neumann problem.

Author: Juraev B. (Andijan State University), Husanboeva K. (Namangan State University), Soyibboev U. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences),(Andijan State University)

Abstract:
We investigate linear pursuit-evasion game problems for two players, whose motions are described by linear differential equations of constant coefficients and whose control functions satisfy geometric constraints of exponential forms, in the Euclidean space $\mathbb{R}^n$. For solving the pursuit game, the $\Pi$-strategy (known as the parallel pursuit strategy) is constructed, and sufficient pursuit conditions are obtained. For solving the evasion game, a directional control function is devised, and sufficient evasion conditions are found. The set of meeting points of the players is explicitly defined by using the $\Pi$-strategy, a multi-valued mapping, and a support function. On the basis of the monotonicity of the set of meeting points of the players, necessary and sufficient conditions of the completion of pursuit in the “life-line” game are determined, and then, sufficient conditions of the possibility of evasion in this game are obtained.

Keywords: Differential game; exponential geometric constraint; parallel pursuit strategy; pursuer; evader; attainability domain of players; life-line.

Author: Karimov E. (Ghent University), Toshpulatov M. (Andijan State University)

Abstract:
In this article, we study the inverse source problem for a mixed fractional-order partial differential equation in a bounded domain. To analyze the problem, we employ the method of separation of variables. The solution is represented in terms of a bivariate Mittag-Leffler-type function. By utilizing known estimates for this function, we prove that the series representation of the solution is uniformly convergent. The novelty of this research lies in the complexity of the time-fractional derivative involved, which necessitates a careful analysis of the properties of bivariate Mittag-Leffler-type functions.

Keywords: Fractional-order mixed equation; Prabhakar derivative; wave-diffusion process; inverse source problem; bi-variate Mittag-Leffler-type function.

Author: Khashimov A. (Tashkent State Economic University), Kurbanov O. (Tashkent State Economic University)

Abstract: This article considers a boundary value problem for a third-order equation of the “pseudo-elliptic” type. Special energy estimates are established for the generalized solution of the equation. With the help of which you can build a solution to the boundary value problem in unlimited areas, in classes of functions growing at infinity, depending on the geometric characteristics of the boundaries of the area.

Keywords: Third-order equation; bounded domains; unbounded domains; boundary value problem; energy estimates; Saint-Venant’s principle; generalized solution; cutoff function.

Author: Mamadaliyev N. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences),(Kimyo International University in Tashkent), Toshbuvayev B. (Fergana State University), Juraev R. (National University of Uzbekistan)

Abstract:
This paper is devoted to the study of open-type soft mappings under the all-parameters neighbourhood convention. In this setting a soft open set is a neighbourhood of a soft point only when all parameter sections contain the underlying element, which leads to a genuinely multi–parameter behaviour. We introduce and analyse two central classes: soft almost-open mappings and soft pseudo-open mappings. Using finite, fully worked examples, we separate these notions from soft open mappings and from each other, showing that neither implication between them holds in general. Our main structural result is a parameterwise inheritance theorem: if a soft mapping $f_\psi$ is soft almost-open (respectively, soft pseudo-open), then every component map $f_\alpha$ is almost-open (respectively, pseudo-open) in the classical sense. We further prove that these soft open-type classes are stable under composition and under finite product constructions of the form $(\Pi^{n}f)_{\Pi^{n}\psi}$. Finally, we formulate problems for more general indexed products of soft topological spaces, indicating several directions for further research.

Keywords: Soft topology; soft mapping; soft open mapping; soft almost-open mapping; soft pseudo-open mapping.

Author: Muminov K. (National University of Uzbekistan), Juraboyev S. (Fergana State University), Gafforov R. (Fergana State University)

Abstract:
This work is dedicated to determining the necessary and sufficient conditions for the system of strongly regular paths in a finite set to be equivalent under the action of a special unitary group. Initially, in the case where the group $G$ is a special unitary group, the problem of $G$-equivalence for a pair of strongly regular paths is solved using $G$-invariant matrix functions and $G$-invariant bilinear differential forms. Using the obtained results, the generating system of the field of $G$-invariant differential rational functions is characterized. With the help of the generating system, the problem of $G$-equivalence for a system of a finite number of paths is solved.

Keywords: A system of regular paths; $G$-equivalent problem; $G$-invariant differential rational functions; special unitary group; a system of generating.

Author: Räıssouli M. (Moulay Ismail University), Chergui M. (CRMEF-RSK, EREAM Team, LaREAMI-Lab)

Abstract:
Recently, many researchers devoted their attention to study the extensions of the gamma and beta functions. In the present work, we focus on investigating some approximations for a class of Gauss hypergeometric functions by exploiting Gr\”{u}ss discrete inequality.

Keywords: Confluent hypergeometric function; Generalized Gauss hypergeometric function; Gr\”{u}ss type inequalities.

Author: Normatov Z. (Jilin University),(V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences)

Abstract:
We introduce the notion of mock-pre-Lie dialgebras, which encompasses anti-associative dialgebras as a special case. Furthermore, we provide new insights into the construction of anti-Leibniz algebras and the geometric structures associated with them.

Keywords: Anti-associative dialgebra; averaging operator; mock-pre-Lie algebra; affine transformation.

Author: Shadimetov Kh. (Tashkent State Transport University),(V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences), Shonazarov S. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences),(Tashkent International University)

Abstract:
A wide range of problems in numerical analysis-including integration, interpolation, and the finite difference approximation of ordinary differential equations-are fundamentally governed by the selection of nodes and the corresponding coefficients in quadrature, interpolation, and difference schemes. Moreover, any discrete problem admits a solution that resides in a finite-dimensional space. Among discrete approaches, the finite difference method is widely employed for the numerical solution of the Cauchy problem. The present work is devoted to the computation of the norm of the error functional associated with a finite difference formula. The minimization of the norm with respect to the coefficients, under fixed nodes, results in a system of linear algebraic equations whose solution yields the optimal coefficients of the finite difference formula. The system is shown to possess a unique solution, thereby guaranteeing both existence and uniqueness.

Keywords: Sobolev space; error functional; difference formula; extremal function; optimal coefficients.

Author: Turdibaev R. (New Uzbekistan University)

Abstract:
We investigate the invariant theory of pairs of ( n \times n ) complex matrices ( (X, Y) ) satisfying the anticommutation relation ( XY + YX = 0 ). The action of the general linear group $\mathrm{GL}_n(\mathbb{C})$ on these pairs by simultaneous conjugation defines a GIT-quotient, referred to as the invariant anti-commuting variety. For $n \leq 5$, we identify an optimal homogeneous system of parameters for the coordinate ring of this variety. These systems are constructed to minimize the sum or product of the degrees of the primary invariants.

Keywords: Trace rings; anti-commuting matrices; primary invariants; secondary invariants.