Bulletin of the Institute of Mathematics

Author: Abdukhakimov S. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences), Khasanov A. (Samarkand State University)

Abstract:
In this paper, we investigate the Schr\”odinger operators ${H}{\mu_1\mu_2}(K)$ associated with a system of two identical fermions on the three-dimensional lattice $\mathbb{Z}^3$. The fermions interact via first- and second-nearest-neighbor lattice sites along the lattice basis directions, with coupling constants $\mu_1\in\mathbb{R}$ and $\mu_2\in\mathbb{R}$, respectively, and $K$ denotes the quasi-momentum of the particle pair. We prove the existence of an invariant subspace for the operator ${H}{\mu_1\mu_2}(0)$ and describe the dependence of its restriction on the interaction parameters $\mu_1\in\mathbb{R}$ and $\mu_2\in\mathbb{R}$. For the corresponding reduced operator $H^{\theta}_{\mu_1\mu_2}(0)$, the plane of interaction parameters $(\mu_1,\mu_2)$ is decomposed into regions such that, within each connected component, the operator has a finite and constant number of eigenvalues. These eigenvalues are located either below the bottom or above the top of the essential spectrum.

Keywords: Two-fermion system; lattice Schr\”odinger operator; eigenvalue; essential spectrum; reduced subspaces.

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

Abstract:
In this paper we study the $p$-adic dynamical system generated by the M\”obius transformation $f(x)=\frac{x+a}{bx+c}$, where $a,b,c\in\mathbb{Q}p$, $b\neq 0$, $c\neq ab$, and $x\neq -\frac{c}{b}$. Our main objective is to analyze the geometry and dynamics of invariant spheres centered at the origin. We first obtain a complete characterization of all invariant spheres $S\alpha(0)$ for this transformation. The admissible radii are divided into four cases according to the relative sizes of $|c|_p$, $|ab|_p$, and $1$. For the first two cases, we show that the induced dynamics on the invariant sphere is contracting; consequently, each such sphere contains a unique fixed point, and every orbit converges to it. For the remaining cases, we investigate ergodicity with respect to the normalized Haar measure on the sphere. In Case~\textnormal{(III)}, we prove that the system is non-ergodic for $p\neq2$, while for $p=2$ we obtain a complete criterion for ergodicity. In Case~\textnormal{(IV)}, we prove non-ergodicity for $p>3$, establish a complete ergodicity criterion for $p=3$, and obtain an analogous criterion for $p=2$. The results provide a deeper understanding of the local structure of trajectories and reveal specific dynamical features that are not visible in the global phase space analysis.

Keywords: Rational dynamical systems; fixed point; invariant set; ergodic; $p$-adic number field; Haar measure.

Author: Ali M. (College of Education for Pure Sciences),(University of Karbala), Salman M. (College of Education for Pure Sciences),(University of Karbala)

Abstract:
In this article, a mathematical model with boundary conditions applied to the channel walls is developed to investigate the peristaltic transport of a pseudoplastic nanofluid in an asymmetrically rotating inclined channel filled with a porous medium and subjected to an inclined magnetic field. Following the development of a mathematical model that included equations for momentum, energy, and nanoparticle concentration. The governing nonlinear partial differential equations representing the flow model are compressed to a set of linear equations by assuming a long wavelength and low Reynolds number. Mathematica is used to obtain analytical expressions for the transformed equations. The influences of physical parameters, including the Hartmann number, Prandtl number, and rotation parameter, are examined graphically on the velocity profile, temperature profile, nanoparticle volume fraction, and pressure rise. Furthermore, the trapping and pumping characteristics of the flow are also discussed.

Keywords: Peristaltic transport; magnetohydrodynamics; rotation; pseudoplastic nanofluid; inclined magnetic field.

Author: Allakova Sh. (Termez State University)

Abstract:
This paper studies a new boundary value problem for a linear degenerate hyperbolic equation with a special singular coefficient. Posed in a characteristic triangle domain, the problem combines the following conditions: the function value is prescribed on the straight-line part of the boundary, a combined shift condition is imposed on parts of the boundary and interior characteristics, and an analogue of the Frankl condition is set on the degeneracy segment.
The main result establishes that the problem is uniquely solvable provided that a certain inequality involving the relevant parameters holds. In the course of the proof, a known integral formula is used to reduce the problem to a functional equation for an unknown function. This functional equation is solved by the method of iterations; the solution is constructed as an infinite series, and its uniform convergence is proved.
The paper also presents an example showing that a nontrivial solution of the corresponding homogeneous functional equation may be unbounded in a neighborhood of the boundary point, which justifies the need to specify precisely the class of solutions being sought.
The obtained results continue and generalize previous work on problems with displacement conditions, in particular by simultaneously incorporating an incomplete shift condition on one boundary characteristic, interior characteristics, and the Frankl condition.

Keywords: Singular coefficient; modified Cauchy problem; functional equation; iteration method; example; reconstruction of the solution.

Author: Arzikulov F. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences),(Andijon State University), Nabijonova F. (Fergana State University),(Fergana State Technical University)

Abstract:
In the present paper we study linear operators generated by elements of an infinite dimensional spin-factor and a general Baer JB-algebra of type I$_2$. We prove that every 2-local linear operator generated by elements of kind $ax$, $a(ax)$, $U_ax=2a(ax)-a^2x$ on a spin-factor and a Baer JB-algebra of type I$_2$ is linear. As a consequence, we obtain that every 2-local 1-automorphism on a spin-factor and a Baer JB-algebra of type I$_2$ is an automorphism.

Keywords: Jordan algebra; automorphism; 2-local automorphism; spin-factor; JB-algebra.

Author: Atajonov Kh. (National University of Uzbekistan)

Abstract:
We investigate local and (2)-local anti-derivations on a class of solvable Lie algebras whose nilradicals are filiform and whose ranks are zero or one. The main examples considered in the paper are the maximal solvable extensions (\mathcal W_n^+) and (\mathcal R_n^+) of the (n)-dimensional Witt and special filiform Lie algebras, respectively, as well as a metabelian filiform Lie algebra (\mathcal L).
For the algebras (\mathcal W_n^+) and (\mathcal R_n^+), we describe the spaces of anti-derivations and give explicit forms of all local anti-derivations. These descriptions show that the spaces of local anti-derivations properly contain the corresponding spaces of anti-derivations. At the same time, we prove that every (2)-local anti-derivation on (\mathcal W_n^+) and on (\mathcal R_n^+) is an anti-derivation.
For the metabelian filiform Lie algebra (\mathcal L), the situation is different: this algebra admits both local and (2)-local anti-derivations which are not anti-derivations. We also study the commutator structure of the obtained spaces. It is shown that
(\operatorname{LocDer}{-1}(\mathcal W_n^+)) and (\operatorname{LocDer}{-1}(\mathcal R_n^+)) are abelian Lie algebras, whereas (\operatorname{LocDer}_{-1}(\mathcal L)) is a non-abelian Lie algebra with respect to the commutator bracket.

Keywords: Lie algebras; anti-derivations; solvable Lie algebras; local anti-derivations; 2-local anti-derivations.

Author: Ayupov Sh. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences),(National University of Uzbekistan), Yusupov B. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences),(Urgench State University)

Abstract:
The aim of the present paper is to analyze local-type mappings on naturally graded quasi-filiform Leibniz algebras. Using the existing classification of the type I and type II cases, we determine the structure of all local derivations and local automorphisms associated with these algebras.
We further show that local derivations are stable under the commutator bracket, which endows their space with a finite-dimensional Lie algebra structure. Moreover, the family of local automorphisms is shown to carry the structure of a finite-dimensional Lie group.

Keywords: Derivation; local derivation; automorphism; local automorphism; Leibniz algebras; naturally graded Leibniz algebras.

Author: Boranbaev O. (Karakalpak State University named after Berdakh)

Abstract:
This work explores local and $2$-local $\tfrac{1}{2}$-derivations within the framework of the deformative Schr\”{o}dinger-Witt algebra, a prominent class of infinite-dimensional Lie algebras. We establish that every local and $2$-local $\tfrac{1}{2}$-derivation on this algebra necessarily coincides with a $\tfrac{1}{2}$-derivation.

Keywords: Infinite-dimensional Lie algebra; deformative Schr\”{o}dinger-Witt algebra; $\frac{1}{2}$-derivation; local $\frac{1}{2}$-derivation; 2-local $\frac{1}{2}$-derivation.

Author: Eshimbetov M. (Tashkent International University of Financial Management and Technologies),(Chirchik State Pedagogical University) Axmedova I. (Tashkent International University of Financial Management and Technologies), Rasulova G. (Tashkent International University of Financial Management and Technologies)

Abstract: For statistical mechanical models whose spin space is an arbitrary (possibly uncountable) set, translation-invariant Gibbs measures can be investigated through nonlinear Hammerstein integral operators. In the present paper, we consider a family of operators $H_k$, $k\in\mathbb N$, generated by a special class of degenerate kernels introduced in our previous studies. We prove that each operator $H_k$ possesses a unique positive fixed point. Furthermore, we establish that every such fixed point determines a translation-invariant Gibbs measure. As a consequence, the existence and uniqueness of the corresponding translation-invariant Gibbs measures are rigorously obtained.

Keywords: Hammerstein operator; Cayley tree; spin values; translational-invariant Gibbs measure; fixed point; degenerate kernel.

Author: Gaybullayev R. (National University of Uzbekistan), Solijanova G. (National University of Uzbekistan)

Abstract:
We investigate an infinite-dimensional Lie algebra which serves as a generalization of the positive Witt algebra. We prove that this algebra is pro-nilpotent. Furthermore, we give an explicit closed-form formula for indexing its basis elements, which allows for a clear and systematic listing of the entire basis.

Keywords: Infinite-dimensional Lie algebra; residually nilpotent Lie algebra; pro-nilpotent Lie algebra.

Author: Jalilov M. (Fergana State University)

Abstract:
In this paper, a nonlocal inverse problem of the Bitsadze–Samarskii type is studied for a fractional-order parabolic equation with the Caputo operator in a two-dimensional spatial domain. A spectral method is used to reduce the original problem to a spectral boundary value problem for a second-order ordinary differential equation with respect to a spatial variable. Under appropriate conditions on the given data, the existence and uniqueness theorems for the solution are established. The uniqueness proof is based on the completeness of the system of eigenfunctions associated with the corresponding spectral problem.

Keywords: Bitsadze-Samarskii type problem; fractional order equation; eigenvalues; eigen and associated functions; completeness; biorthogonality; Riesz basis.

Author: Khushvaktov N. (Tashkent International University of Financial Management and Technologies)

Abstract:
This paper is devoted to the study of nonlocal and backward problems for the fractional-order Barenblatt-Zheltov-Kochina equation involving the Caputo fractional derivative. In the present paper, we consider the Cauchy problem with a time-dependent source term $f(t)$. To analyze the non-local problem, we divide it into two auxiliary problems and use the solution of the corresponding Cauchy problem. We establish the existence and uniqueness theorems for the initial-boundary value problem, as well as, for the backward problem.

Keywords: Barenblatt-Zheltov-Kochina equation; Caputo fractional derivative; Fourier method; Mittag-Leffler functions.

Author: Poshakhodjaeva G. (Tashkent State University of Economics)

Abstract:
We study real-analytic critical circle maps $f$ with a unique cubic critical point $x_{cr}$ and golden mean rotation number $\rho_{\ast}.$ It is well known that the unique probability invariant measure $\mu_{f}$ of $f$ is singular with respect to Lebesgue measure $\ell$ on the unit circle $S^{1}=\mathbb{R}^{1}/ \mathbb{Z}^{1}\simeq [0,1).$ These two natural measures induced the measures $\widetilde{\mu}{f}$ and $\widetilde{\ell}$ on symbolic space $\Theta^{+}{\mathbb{A}}$ with alphabet $A={1,0,a}.$ Although these induced measures are not shift invariant in general, we prove that each of them admits a shift-invariant representative within its measure class. Moreover, the resulting shift-invariant measures are mutually singular, reflecting the singularity of the invariant measure with respect to the Lebesgue measure for critical circle maps. As a result, invariant, geometric, and Gibbs measures admit a unified symbolic description within the framework of thermodynamic formalism.

Keywords: Critical map; invariant measure; rotation number; symbolic space; thermodynamic formalism.

Author: Vasieva Kh. (National University of Uzbekistan)

Abstract:
This paper investigates linear differential games described by a system of linear neutral-type differential-difference equations under integral constraints on the players’ controls, concerning the possibility of completing pursuit in finite time. The problem formulation, basic definitions, and the objective of the pursuit problem are presented. New sufficient conditions for the solvability of the pursuit problem are established. The proposed approach is based on the ideas of the resolving function method, and by applying this method new sufficient conditions are derived for the completion of the differential pursuit game within a certain guaranteed time.

Keywords: Differential game; pursuit problem; integral constraint; resolving function; differential-difference equations of neutral type; terminal set; pursuer; evader; control.