Bulletin of the Institute of Mathematics

Author: Boboyarova N.(Urgench Ranch Technological University), Islamova M. (Urgench State University)

Abstract:
The present research investigates a two-dimensional discrete dynamical system in which each variable evolves according to the product of its previous value and a linear function of the other variable. One variable increases or decreases depending on the value of the second variable multiplied by a constant parameter, while the second variable changes in a similar way according to the first variable and another parameter. This system represents mutual interaction between two quantities and can display a variety of dynamic behaviors such as equilibrium, stability changes, and chaotic motion depending on the parameter values. The model describes the time evolution of two interacting variables and serves as a prototype for analyzing nonlinear interactions in discrete time. From a theoretical perspective, the study focuses on the existence and stability of fixed points, bifurcation scenarios, and the emergence of complex behaviors such as chaos. The system also demonstrates connections to well-known universal properties in dynamical systems, including the Feigenbaum constant and Sharkovskii’s ordering.

Keywords: Stability; Schwarzian derivative; bifurcation diagram; filled Julia set; Feigenbaum number; unimodal maps; chaos; cobweb diagram.

Author: Bozorkulov A. (Fergana State University)

Abstract:
In this paper, we consider three-state Hard-Core (HC) model on a Cayley tree under the condition “wand” \; with activity parameter $\lambda>0$. In this case we study alternative Gibbs measures. Specifically, the new conditions for the existence of alternative Gibbs measures (non-translation-invariant) are obtained on a Cayley tree of arbitrary order. Moreover, the dynamics of some solutions corresponding to Gibbs measures has been stud.

Keywords: Cayley tree; configuration, hard-core model; Gibbs measure; translation-invariant measure; alternative Gibbs measure; iteration dynamics.

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), Tulakova Z.(Fergana State Technical University)

Abstract:
This work is a continuation of the research published in [1]. Hypergeometric functions are divided into complete and confluent functions. This paper presents new integral representations of the Euler type for some hypergeometric Gauss functions of three variables. The main results were obtained using the properties of the gamma and beta functions. New integral representations were derived for $E_{68}$ — $E_{395}$ functions from the list of confluent hypergeometric functions of three variables. Thus, all derived integrals can be considered as generalized representations of the Euler type for classical hypergeometric Gauss functions of one and two variables.

Keywords: Srivastava-Karlsson hypergeometric functions; confluent hypergeometric function of three variables; integral representation; improper integral representations; multiple integral representations.

Author: Jabborov Kh.(V.I.Romanovskiy Institute of Mathematics Uzbekistan Academy of Sciences),(University of World Economy and Diplomacy),(Jizzakh Polytechnic Institute)

Abstract:
This article presents a novel and effective approach to finding an approximate solution for the Fredholm integral equation of the second kind using optimal quadrature formulas. This method, distinguished by its high accuracy and simplicity, involves constructing optimal quadrature formulas in the sense of Sard and providing error estimates in the Hilbert space of differentiable functions. The form of the squared norm of the error functional for the quadrature formula in the Hilbert space $L_2^{ (2) } (0, 2\pi) $ has been determined. To minimize this error, a system of linear equations with respect to the formula’s coefficients has been derived, and a unique solution has been found. An explicit expression for these optimal coefficients has been obtained.

Keywords: Hilbert space; singular integral; error functional; optimal quadrature formula; singular integral equations; Hilbert kernel.

Author: Karimov Sh.(Fergana State University), Mamajonova R.(Fergana State University),(Presidential School in Fergana)

Abstract:
In this paper, the Cauchy problem for a fourth-order time-fractional differential equation posed in the half-plane is investigated. The order of the problem is reduced twice, transforming it into an initial value problem for an auxiliary equation whose solution is already known. Subsequently, by using the solution of the auxiliary problem, an explicit analytical solution to the original problem is obtained.

Keywords: Cauchy problem; beam vibration equation; Schrödinger equation; Mittag–Leffler function; Wright function; Caputo fractional derivative; Riemann–Liouville fractional integral.

Author: Karimov U.(National university of Uzbekistan)

Abstract:
In this paper, we study the derivations acting on the algebra $S(M)$, which consists of all measurable operators affiliated with a finite real W-algebra $M$. It is shown that if $M$ is a finite real W-algebra endowed with a faithful normal semi-finite trace $\tau$ and equipped with the locally measurable topology $t$,
then every $t$-continuous derivation $D : S(M) \to S(M)$ is inner.
An analogous conclusion also holds for derivations on the algebra $S(M,\tau)$ of $\tau$-measurable operators endowed with the measure topology $t_\tau$.

Keywords: Derivation; real W*-algebra; *-algebra of all measurable operators.

Author: Mutalliev N.(Namangan State Technical University)

Abstract:
This study examines the two-state Hard-Core model defined on a Cayley tree. Focusing on the case where the underlying group contains a normal subgroup of index four, we derive sufficient conditions guaranteeing the uniqueness of the weakly periodic Gibbs measure for all values of the activity parameter $\lambda>0$. In addition, we demonstrate the existence of new weakly periodic (non-periodic) Gibbs measures that are distinct from those previously reported in the literature. The obtained results provide further insight into the structural properties and classification of Gibbs measures associated with the Hard-Core model on Cayley trees.

Keywords: Cayley tree; admissible configuration; HC model; Gibbs measure; periodic Gibbs measure; weakly periodic Gibbs measure.

Author: Rafiqov A.(Fergana State University), Tursunova E.(Fergana State University)

Abstract:
The Cauchy problem for a differential equation involving the regularized Prabhakar fractional derivatives has been targeted for unique solvability. By reducing the considered problem to a second-kind Volterra integral equation and solving it via the successive iteration method, an explicit solution was obtained. This solution is represented via bivariate and trivariate Mittag-Leffler type functions. Based on certain properties of bivariate and trivariate Mittag-Leffler type functions, the obtained result can be applied to the study of direct and inverse problems for partial differential equations involving two regularized Prabhakar fractional derivatives in the time variable.

Keywords: The Cauchy problem; regularized Prabhakar fractional derivative; bivariate Mittag-Leffler type function; Trivariate Mittag-Leffler type function; Volterra integral equation.

Author: Rakhmonov Z.(National University of Uzbekistan), Yarmetova D.(National University of Uzbekistan)

Abstract: This article investigates the life span of solutions to a nonlinear diffusion-reaction equation with nonlinear boundary conditions in the half-space $R_{+}^{N} $. The problem features a doubly nonlinear diffusion operator, an internal source term, and a nonlinear Neumann boundary condition of the form $\left| \nabla u^m \right|^{p-2} \frac{\partial u^m}{\partial x_1} = u^q $ at $ x_1=0$. The focus is on determining conditions under which solutions exist globally in time or blow up in finite time. Using energy methods, comparison principles, and suitable sub- and super-solution constructions, the article derives upper and lower bounds for the life span of solutions in terms of the parameters $m,p,q,\beta$ and the initial data. The results highlight the subtle interplay between nonlinear diffusion, internal reaction, and boundary flux that determines the emergence of a finite-time blow-up or the global existence of solutions.

Keywords: Nonlinear diffusion; comparison principle; self-similar solution; Neumann problem; nonlinear boundary flux; life-span; blow-up; global solution.

Author: Rasulov T.(Bukhara State University), Jurakulova F.(Bukhara State University)

Abstract:
This study is devoted to the analysis of a (3\times3) operator matrix (\mathcal B_\mu) depending on a spectral parameter (\mu>0), interpreted as the interaction strength.
The model originates from lattice quantum systems describing the coupling of two identical bosons with an additional particle of a different nature, in which the particle number is not conserved. The operator acts in the Hilbert space given by the direct sum
of the zero, one-particle, and two-particle sectors of the bosonic Fock space. Within this framework, we formulate a Faddeev-type operator equation and find the corresponding Fredholm determinant (\Omega_\mu(\cdot)). It is established that the discrete part of the spectrum of the operator matrix (\mathcal B_\mu) is completely characterized by the zeros of the function (\Omega_\mu(\cdot)).

Keywords: Block operator matrix; bosonic Fock space; Hamiltonian; annihilation and creation operators; the Faddeev type operator equation; the Fredholm determinant; eigenvalues; eigenvector; essential spectrum; discrete spectrum.

Author: Samijonova N.(Namangan State University)

Abstract:
In this paper, we study weakly periodic $p$-adic quasi Gibbs measures for the $q$-state Potts model on the Cayley tree of order three and investigate the conditions under which a phase transition occurs. Specifically, we show that a phase transition exists if $|q(q-1)|_p = 1$ with $\sqrt{1 – q} \in \mathbb{Q}_p$, or if $|q|_p < 1$. Moreover, if $|q – 1|_p < 1$, we establish the existence of phase and quasi phase transitions under certain additional assumptions.

Keywords: $p$-adic numbers; Potts model; $p$-adic quasi Gibbs measure; weakly periodic Gibbs measure; phase transition, quasi phase transitions.

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

Abstract:
This paper investigates the construction of an optimal quadrature formula in a Hilbert space of real-valued periodic functions. The upper bound of the absolute error of the proposed quadrature formula is characterized by evaluating the norm of the associated error functional. To this end, the extremal function corresponding to the quadrature formula is utilized to derive the error functional norm. An explicit analytical representation of the error functional norm is obtained, and the methodology for determining the coefficients of the quadrature formula in the prescribed function space is presented. The primary objective of this study is to establish the structure of the error functional norm based on the derived coefficients.

Keywords: Optimal numerical integration formula; Hilbert functional space; error functional analysis; Fourier analytical methods; extremal functions; Sobolev-type spaces; optimal weight coefficients.

Author: Shermukhammedov B.(Samarkand State University)

Abstract:
In the paper, an approximate solution of second-order nonlinear ordinary differential equations of the Lane–Emden type is obtained using the piecewise constant argument method. Several differential equations that model different physical problems with given initial conditions are solved numerically to illustrate the efficiency and accuracy of the piecewise constant argument method. The obtained results are compared with other numerical results and exact solutions.

Keywords: Approximate solution; initial value problems; Lane-Emden equation; non-linear second order ordinary differential equation; Piecewise constant argument.