Bulletin of the Institute of Mathematics

Author: Abdukhakimov S. (Jizzakh branch of the National University of Uzbekistan), Abdukhakimova M. (Jizzakh branch of the National University of Uzbekistan)

Abstract:
In this work, we study critical circle homeomorphisms of class $C^{3}(S^{1})$ with a unique cubic critical point and an irrational rotation number given by a periodic continued fraction. Using the renormalization group formalism, we analyze the infinite orbit of the critical point and the associated first return maps, and construct the corresponding sequence of dynamical partitions. The main result establishes explicit scaling relations for the iterates of the critical map near the critical point. These relations are expressed in terms of universal normalizing maps and products of renormalization scaling factors. The obtained formulas provide a rigorous description of the self-similar geometric structure of orbit segments and determine the asymptotic behavior of first return maps along the renormalization levels.

Keywords: Circle homeomorphisms; critical point; rotation number; renormalization; dynamical partitions; orbit trajectory.

Author: Babaev S. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences),(Tashkent International University)

Abstract:
This study looks at how to numerically approximate two-dimensional (2D) right-sided Riemann-Liouville fractional integrals. Inspired by the challenges of modelling anomalous diffusion and memory effects in complex media, we develop a numerical method using optimal cubature formulas. We extend the optimal coefficients from one-dimensional quadrature formulas to two dimensions by exploiting the tensor-product structure of the integral operator. We present the mathematical formulation, computational algorithm, and numerical experiments that demonstrate the efficiency and accuracy of the proposed method for functions of varying regularity.

Keywords: An optimal quadrature formula; Cubature formula; Riemann-Liouville fractional integral; Hilbert space; Optimal coefficients.

Author: Bakhromov S. (National University of Uzbekistan), Qobilov S. (Tashkent University of Information Technologies), Makhmudjanov S. (Tashkent University of Information Technologies)

Abstract:
Various mathematical models are being developed in the scientific-methodological study of the problems arising in the industry, and these models are of great importance. Therefore, the use of spline functions, which are considered effective and important in the development of modern science and technology, is an urgent issue. One of the remarkable properties of the spline function is that it has been proven to have a number of advantages over the existing classical interpolation multipliers. In this work, we consider the error estimation and application of the local interpolation cubic spline function in $W^{1} [a_{1},b_{1} ]$, which is constructed on the basis of a linear combination of parabolas with 2 common points. All error properties of local interpolation cubic spline functions are analyzed based on digital processing of geophysical signal obtained on the basis of several experimental data. The local interpolation error of the cubic spline functions in the derived area was evaluated and verified using graphs and numerical analysis based on specific examples.

Keywords: Spline function; signal processing; geophysical signal.

Author: Durdiev D. (Bukhara Branch of the Institute of Mathematics at the Academy of Sciences of the Republic of Uzbekistan),(Bukhara State University), Jumaev J. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences), Abdullaev B. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences)

Abstract:
We study an inverse problem for determining the kernel of an integro-differential pseudoparabolic equation. The direct problem is formulated with initial and nonclassical boundary conditions. First, we analyze the corresponding spectral problem, where the completeness of the eigenfunctions, the construction of a biorthogonal system in the space $L_2(0,1)$, and the properties of the eigenfunctions are established. By applying the Fourier method, the direct problem is reduced to a Volterra-type integral equation of the second kind. The existence and uniqueness of its solution are proved using Gronwall’s inequality and the properties of functional series. Using the solution of the direct problem together with an additional condition, we derive an integral equation for the unknown kernel function. Finally, the existence and uniqueness of the solution to the inverse problem are proved using the fixed-point theorem.

Keywords: Integro-differential pseudoparabolic equation; nonclassical boundary conditions; Fourier method; spectral problem; inverse problem; Banach fixed-piont theorem; existence; uniqueness.

Author: Karimov E. (Ghent University), Khasanov Sh. (Fergana State University)

Abstract:
In this paper, we introduce novel trivariate and quadri-variate Mittag-Leffler-type functions and provide a systematic investigation of their fundamental analytical properties. In particular, we establish their regions of convergence, derive suitable integral representations, and obtain explicit estimates. Several notable special cases are also discussed. Furthermore, the integral representations and upper estimates of the six-parameter Mittag-Leffler function, as well as the bivariate Mittag-Leffler-type function, are examined in detail. Finally, we present information on the application of the proposed functions to the study of a specific partial differential equation.

Keywords: Bivariate Mittag-Leffler-type function; six-parametric Mittag-Leffler function; trivariate Mittag-Leffler-type function; quadri-variate Mittag-Leffler-type function; integral representation; $\psi$-Prabhakar integral-differential operator.

Author: Khomidov M. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences),(International Islamic Academy of Uzbekistan)

Abstract:
This paper studies a nonlinear AR(1) process generated by a fractional-linear map under the influence of Bernoulli strict white noise. Conditions for stationarity are investigated, and a functional equation characterizing the stationary distribution is derived. It is shown that this equation admits a unique solution, implying the existence of a stationary distribution for the process. The results contribute to the theory of nonlinear time series and stochastic processes.

Keywords: White noise; AR process; nonlinear AR-process; stationary distributions; fractional-linear maps.

Author: Kucharov R. (National University of Uzbekistan), Tuxtamurodova T. (National University of Uzbekistan)

Abstract:
In this paper, the spectral properties of Fredholm-type partial integral operators defined in a Hilbert space are investigated. The unperturbed operator is given in the form of a multiplication operator, while the perturbation is treated as a diagonal operator defined with respect to an orthonormal system. The integral structure of the operator allows the application of a fiber-wise decomposition based on the direct integral framework, which makes it possible to describe the spectrum of the global operator in terms of the spectra of parameter-dependent fiber operators.

The spectral structure of partial integral operators and their sums is studied, with particular attention to the existence of the essential and discrete spectra. It is shown that, under diagonal non-compact perturbations, no discrete spectrum arises, and the spectrum is mainly composed of spectral bands generated by the eigenvalues of the fiber operators together with their accumulation points. Within the framework of a Friedrichs-type model, explicit bounds for the essential spectrum are obtained using the minimax principle. The presented examples and graphical illustrations clearly confirm the theoretical results.

Keywords: Essential spectrum; discrete spectrum; spectral lines; spectral boundaries; fiber operators; direct integral; non-compact perturbations; Kato theory; Weyl criterion; Friedrichs model.

Author: Madrakhimov T. (Urgench State University), Vaisova N. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences)

Abstract: This research is devoted to the investigation of $\frac{1}{2}$-derivations and local $\frac{1}{2}$-derivations within the class of $n$-dimensional naturally graded quasi-filiform Leibniz algebras of type II. Specifically, we determine the general matrix representations for both categories of derivations for the algebras under consideration. Our analysis demonstrates that the matrix structure for $\frac{1}{2}$-derivations does not align with the matrix form of local $\frac{1}{2}$-derivations. As a result, it is proven that these naturally graded quasi-filiform Leibniz algebras of type II possess local $\frac{1}{2}$-derivations that fail to be standard $\frac{1}{2}$-derivations[cite: 1].

Keywords: Leibniz algebras; naturally graded quasi-filiform Leibniz algebras; $\frac{1}{2}$-derivations; local $\frac{1}{2}$-derivations.

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

Abstract:
It is shown that whenever $R\subset B(H)$ is a unital real C*-algebra such that each maximal abelian self-adjoint subalgebra (MASA) of is a real W*-algebra, the algebra $R$ itself must also be a real von Neumann algebra. As a consequence, if $B_c=B+iB$ is a MASA with $B\subset R$, then $B$ is a MASA in $R$. It is also given conditions under which a real AW*- or C*-algebra is monotone complete. Specifically, any real AW*-algebra that has a separating family of completely additive states is a real von Neumann algebra, and a unital real C*-algebra whose MASAs are monotone complete is a real AW*-algebra.

Keywords: Real $C^{*}$-algebras; Real $AW^{*}$-algebras; von Neumann algebras; Monotone closed subalgebras; Completely additive states; MASA.

Author: Ruziev M. (V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences), Kazakbaeva K. (V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences)

Abstract:
This work is devoted to the study of the solvability of a boundary value problem for a degenerate elliptic type equation with a singular coefficient in a vertical strip. The uniqueness of the solution is established by applying the extremum principle. The problem under consideration is reduced to a singular integral equation. Based on the theory of singular and Fredholm’s integral equations, the existence of a solution is established.

Keywords: Singular coefficient; vertical strip; Bessel function; Hankel transform; Fourier method; extremum principle; integral equation; uniqueness of solution; existence of solution.

Author: Sharipova S. (National university of Uzbekistan)

Abstract:
The present research is devoted to studying a solvability of the Cauchy problem for biharmonic equation. First, we solve the Cauchy problem, which is represented in series form. And the existence and uniqueness of the solution of the given problem are proved.

Keywords: Biharmonic equation; Cauchy problem; ill-posed problem; existence; uniqueness.

Author: Solijanova G. (National University of Uzbekistan)

Abstract:
This paper is devoted to the study of an open problem posed by Nicoletta Cantarini and Victor G. Kac concerning the existence of infinite-dimensional simple $n$-Lie superalgebras over an algebraically closed field of characteristic zero for $n > 2$, excluding the algebras of Wronskians and Jacobians. In particular, we investigate whether there exist infinite-dimensional simple $n$-Lie superalgebras that are not $n$-Lie algebras and whose even part is isomorphic to $W^n$ for $n=2$ and $n=3$. We construct simple Lie superalgebra whose even part is isomorphic to $W^2$.

Keywords: Lie algebra; $n$-Lie algebra; infinite-dimensional Lie (super)algebras; simple $n$-Lie (super)algebra.