Bulletin of the Institute of Mathematics

Author: Ibragimov G.(V.I.Romanovskiy Institute of Mathematics), (Tashkent State University of Economics), Tursunaliev T.(V.I.Romanovskiy Institute of Mathematics), (Tashkent International University of Financial Management and Technologies)

Abstract:
In this paper, we analyze an evasion differential game involving two pursuers and a single evader. The dynamics of variables x1 and x2 are described by linear differential equations. The control functions of players are subjected to geometric constraints. The control sets of pursuers is the unit ball, and that of evader is the ball of radius σ,σ > 1. If xi(t)= 0 for all i ∈ {1,2} and t ≥ 0, then we say that evasion is possible. We construct an evasion strategy for the evader which ensures the possibility of evasion from any initial positions of objects. In addition, we introduce the concept of approach times and establish that the number of approach times does not exceed 3. 

Keywords: Differential game; evasion; control; evasion strategy; maneuver; evader; two pursuers.

Author: Isakov B. (Andijan State University)

Abstract:
We consider the three-state Ising-Potts model on the Cayley tree of order three. We study the translation-invariant splitting Gibbs measures for this model. Moreover, we show the existence of a phase transition.

Keywords: Cayley tree; Ising-Potts model; Gibbs measure; translation-invariant Gibbs measure.

Author: Khajiev I.(National University of Uzbekistan), (Turin Polytechnic University in Tashkent), Shobdarov E.(National University of Uzbekistan)

Abstract: 
The initial boundary value problem for a differential equation composed of first-order and parabolic type with changing direction of time differential operators is investigated. This problem is an ill-posed problem in the sense of J. Hadamard. Therefore, according to the theory of A. Tikhonov, it is assumed that the solution of the problem exists. Uniqueness and conditional stability in the set of correctness are proved. As a result, a regularized solution of the problem is constructed.

Keywords: Composite differential equation; parabolic equation with changing direction of time; ill-posed problem; uniqueness; conditional stability; set of correctness; regularized solution.

Author: Kim D. (National University of Uzbekistan), (Tashkent Branch of the G.V. Plekhanov Russian University of Economics)

Abstract: We study AW-algebras, which are an algebraic generalization of W-algebras. Uniformly hyperfinite finite AW-algebras are considered, and an analogue of Dixmier’s result on the approximation of uniformly hyperfinite finite AW-algebras by matrix algebras are obtained.

Keywords: Quasitrace; AW-algebra; C-algebra; UHF-algebras; dτ-metric; projection; partial isometry; quasinorm.

Author: Makhammadaliev M. (Namangan state university)

Abstract: 
In this paper, we consider a two-state Hard Core (HC) model with two competing interactions (nearest-neighbor, one-level next-nearest-neighbour) on the Cayley tree. We show that at some values of the parameters the model exhibits a phase transition. We also prove that for the model under some conditions there is no two-periodic Gibbs measures.

Keywords: Cayley tree; Gibbs measure; HC model; competing interactions.

Author: Pirmatov Sh. (Tashkent State Technical University)

Abstract: 
The problem of expansion in eigenfunctions of the Schr¨ odinger operator with singular potential is investigated in an arbitrary N-dimensional domain. It is proved that if partial Riesz means converge at some point, then the mean value of a function of order α > 1/2 at the same point is equal to this limit.

Keywords: Eigenfunctions; eigenvalues; spectrum; Fourier series; convergence.

Author: Shakarova M. (V. I. Romanovskiy Institute of Mathematics)

Abstract: 
An inverse problem of determining the right-hand side of the abstract subdiffusion equation with a fractional Caputo derivative is considered in a Hilbert space H. For the forward problem, the non-local in-time condition u(0) = u(T) is taken instead of the Cauchy condition. The right-hand side of the equation has the form g(t)f with a given function g(t) and an unknown element f ∈ H. As an over-determination condition, we adopt the integral condition Tu(t)dt = ψ. Here ψ ∈ H is a given element. For a sign-preserving function g(t), the existence 0 and uniqueness of a solution are proved. If the function g(t) changes sign, then necessary and sufficient conditions for the existence of a solution are found. All the presented results are new for diffusion equations as well.

Keywords: Subdiffusion equation; non-local condition; inverse problem; the Caputo derivatives; the Fourier method.

Author: Sharipov O.(National university of Uzbekistan), (V.I.Romanovskiy Institute of Mathematics), Muxtorov I.(V.I.Romanovskiy Institute of Mathematics)

Abstract:
We consider a law of large numbers for random variables with values in Lp[0,1]. Our main results are law of large numbers for triangular arrays of weakly dependent random variables with values in Lp[0,1], 1 ≤ p < ∞. We assume that random variables satisfy some modified mixing conditions. 

Keywords: Law of large numbers; mixing condition; triangular arrays.

Author: Turmetov B.(Akhmet Yassawi University), Abdullayev O.(Alfraganus University), V.I.Romanovskiy Institute of Mathematics), Sobirjonov A. (Alfraganus University)

Abstract: 
A class of inverse problems is considered for a fractional-order loaded diffusion equation with involution perturbation using four different boundary conditions. Proved theorems on the existence and uniqueness of solutions to these problems are presented. Solutions are obtained in the form of series expansion using a set of appropriate orthogonal basis for each problem. The convergence of the obtained solutions is also proved.

Keywords: Inverse problems; fractional-order loaded diffusion equation; involution perturbation; existence and uniqueness of solution; series expansion.

Author: Allakov I. (Termez State University), Erdonov B.(Termez State University)

Abstract:  Let $X$ be a sufficiently large real number, $b_{1},b_{2},b_{3}$ be natural with the condition $1\le {{b}{1}},{{b}{2}},{{b}{3}}\le X$, $ a{ij}, (i=1,2,3;\,\,\, j=\overline{1,5})$ integers, $p_{1},…,p_{5}$ prime numbers. We define: $ B = \max\limits_{i=1,2,3;\,\, j=\overline{1,5}} 3|a_{ij}|,\,\,\, \vec{b} = (b_{1},b_{2},b_{3}),\,\, K=36\sqrt{3}B^{5}|\vec{b}|, \,\,\,\, E_{3,5}(X)=card{(b_{1},b_{2},b_{3}) |1\le {{b}{i}}\le X,\,\,b{i}\neq a_{i1}p_{1}+\cdots+a_{i5}p_{5},\,\,i=1,2,3}$. In this work, the solvability of the system $b_{i}=a_{i1}p_{1}+\cdots+a_{i5}p_{5},\,\,(i=1,2,3)$, in prime numbers $p_{1},\cdots,p_{5}$ is studied and a power estimate for the exceptional set $E_{3,5}(X)$ is obtained for the first time, and a lower estimate for the $R(\vec{b})$ number of solutions of the system under consideration in prime numbers, namely, it is proved that if $\varepsilon \,\, (0<\varepsilon<1)$ is a sufficiently small real number, then there exists a number $A$ such that for $X>B^{A}$ the estimate $E_{3,5}(X)<X^{3-\varepsilon}$ is valid and for $R(\vec{b})$ for a given $\vec{b}=(b_{1}, b_{2},b_{3}),\,\, 1\le {{b}{1}},{{b}{2}},{{b}{3}}\le X$ the estimate $R(\vec{b})\gg K^{2-\varepsilon}(\ln K)^{-5}$, for all $(b{1},b_{2},b_{3})$ except for at most $X^{3-\varepsilon}$ triples.

Keywords: Estimate; positive solvability; congruent solvability; Euler’s constant; effective constant; fixed number; prime number; system of linear equations; power estimate.

Author: Eshimbetov M. (Tashkent International University of Financial Management and Technology), (Kimyo International University in Tashkent), Eshimbetov J.(Tashkent University of Human Scienses), Abduraimov Y.(Termez State University)

Abstract: This research work is devoted to study the solutions of the max-plus linear equation system. Similarly in classical linear algebra, we got an analogue of Cramer’s method for solving the max-plus linear equation system in max-plus algebra. Then, we provided examples of finding the roots of a max-plus system and plotted the graphs of these max-plus equations in the Cartesian coordinate system.

Keywords: Max-plus algebra; idempotent semi-ring; tropical determinant; max-plus linear equation system.

Author: Kadirkulov B. (Alfraganus University), (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences), Uzaqbaeva D. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences)

Abstract:
In this paper, the spectral properties of nonlocal Samarskii–Ionkin type problems for a fourth-order ordinary differential equation are investigated. The eigenvalues and corresponding eigenfunctions are found, and their completeness is studied. The spectral properties of the adjoint problem are also examined. Furthermore, the Riesz basis property of the systems of root functions for these problems is proved.

Keywords: Samarskii–Ionkin type problem; non-self-adjoint problem; root functions; completeness; Riesz basis.

Author: Mamadaliev N.(National University of Uzbekistan), Ibaydullaev T. (Andijan State University)

Abstract:
This work is devoted to the study of the game problem of control of trajectory bundles described by a system of differential-difference equations of neutral type under geometric and integral constraints on the players’ controls. To solve this problem, sufficient conditions are obtained for the possibility of transferring a trajectory bundle in some finite time.

Keywords: Differential games; the pursuit problem; differential-difference equations of neutral type; terminal set; pursuer; evader; controls.

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

Abstract: 
In this paper, multipoint Bitsadze-Samarskii problem for an equation of elliptic type $u_{tt}(t)-Au(t)=F(t)$ is studied, i.e., the boundary conditions are replaced by non-local conditions: $ u(0)=\sum\limits_{j=1}^m \alpha_j u(\xi_j)+\varphi;
$ $||u(t)||$ is bounded, if $ t\to \infty$, here $F(t)\in C([0,\infty),H)$ is a given function, $\varphi\in H$ is known element, $\alpha_j,\,\,j=1,2,..m;$ are real constants, $\xi_j,\,\,j=1,2,..m$ are fixed points of interval $(0,\infty)$ and $0<\xi_1<\xi_2< \cdots < \xi_m.$ For the parameters $\alpha_j,\,\,j=1,2,..m$ and the function $F(t)$, uniqueness criteria and sufficient conditions were found to ensure the existence of a solution to the problem. The conditions for the existence and uniqueness of the solution were derived by the Fourier method.

Keywords: Bitsatse-Samarskii problem; Fourier method; abstract operator; complete system; non-local problems; elliptic type equation.

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 paper investigates a nonlocal problem for a mixed elliptic-hyperbolic-type equation in an unbounded domain. The uniqueness of the solution of the problem is proved using the extremum principle. The existence of a solution is proved using the method of integral equations. The problem studied is equivalently transformed into solving a singular integral equation. By applying the Carleman-Vekua method, an explicit solution is obtained.

Keywords: Mixed-type equation; singular coefficient; half-strip; extremum principle; integral equation; uniqueness of the solution; existence of the solution; Bessel function.

Author: Zaynabiddinov I. (Andijan State Pedagogical Institute)

Abstract:
This paper investigates linear optimal pursuit-evasion games in a Hilbert space. We prove the existence and uniqueness of mild solutions for linear differential equations. Subsequently, we analyze a linear differential game subject to integral constraints on the players’ control parameters. The optimal strategy for the pursuer is formulated to minimize the time required to bring the state of the system to the origin within the space. In contrast, the optimal strategy for the evader is designed to maximize the time required to prevent the system from reaching the origin. We construct the players’ strategies using the Gramian operators, which serve as the primary tool for achieving optimal pursuit time.

Keywords: Differential game; Hilbert space; pursuer; evader; control parameter; optimal strategy; Gramian operator.

Author: Artikbaev A.(Tashkent State Transport University), Sultanov B.(Urgench State University), Akhmedov I.(Urgench State University)

Abstract:  This paper explores the invariance of geometric characteristics of surfaces under a specific nonlinear transformation in Galilean space. It is shown that the Gaussian curvature of a surface remains unchanged under such a transformation, indicating that the intrinsic geometry of the surface is preserved. Furthermore, it is demonstrated that the total curvature and the surface area of a domain are also invariant under the nonlinear transformation. These results provide a deeper understanding of how certain geometric properties remain constant during nonlinear deformations within the framework of Galilean geometry.

Keywords: Nonlinear transformation; invariant; surface; Gaussian curvature; complete curvature; domain surface.

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

Abstract: The present paper is devoted to the study of a boundary value problem for an inhomogeneous mixed equation, which for $0<t\le \beta$ is a subdiffusion equation with a fractional derivative in the sense of Caputo, and for $-\alpha\le t<0$ is an equation of hyperbolic type with a classical derivative. The Laplace operator is involved in the space-variable of mixed equation. A feature of the problem under consideration is that neither initial nor final conditions are involved. Instead, three gluing conditions are specified on the line of type-changing of the equation. As a result, the usual well-posedness conditions for the numbers $\alpha$ and $\beta$ are removed from the boundary of the domain under consideration. The existence and uniqueness of a solution to the problem are proved using the Fourier method. However, a disadvantage of the problem under consideration is that if the right-hand side of the equation does not depend on the time variable $t$, then, as shown in this paper, the dependence of the problem under consideration on $t$ is lost.

Keywords: Mixed equation; Caputo fractional derivative; uniqueness and existence of a solution; boundary value problem; gluing condition; Fourier method.

Author: Dzhalilov A.(Turin Polytechnic University in Tashkent), Imomaliev J. (Turin Polytechnic University in Tashkent), ( Tashkent International University of Financial Management and Technologies)

Abstract:
{In this paper, we study nonlinear moving average (MA(1)) processes defined via Möbius transformations. These processes belong to the class of stationary random processes, and their covariance functions and joint distributions are evaluated. Additionally, certain limit theorems for MA(1) processes are proved.

Keywords: Moving average process; Möbius transformation; strictly stationary random process; covariance function.

Author: Fayziev Y.(National University of Uzbekistan),
Jumaeva Sh. (V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences),
Abdullaeva F. (V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences),

Abstract:
This paper studies the inverse problem of identifying an unknown source term in a Langevin-type fractional differential equation involving the composition of fractional derivatives of orders $\alpha$ and $\beta$ (with $ \alpha, \beta \in(0, 1)$) in a Hilbert space. We establish the existence and uniqueness of the solution pair ${u(t), f}$ to the inverse problem.

Keywords: Inverse problem; Langevin-type differential equation; Caputo fractional derivative; Mittag-Leffler function.

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: 
Hypergeometric functions are devided into the complete and confluent functions.
Srivastava and Karlsson have defined, in 1985, 205 complete triple series. At present, integral representations for these functions are known. Recently, all possible 395 hypergeometric series of three variables have been published in the scientific literature, all of which are analogous to the double confluent series of Horn and Humbert (20 functions in total) and are confluent forms of the known complete hypergeometric functions of three variables. This paper firstly presents some integral representations of the Euler type for new 28 confluent hypergeometric functions of three variables. The main results were obtained using the properties of the gamma and beta functions. Thus, all derived integrals can be considered as generalized representations of the Euler type for classical hypergeometric functions of one and two variables.

Keywords: Complete and confluent hypergeometric functions; Euler type integral representations; Horn and Humbert hypergeometric functions of two variables; confluent hypergeometric functions of three variables, beta function.

Author: Hoitmetov U.(Urgench State University), Musaeva F.(Urgench State University)

Abstract:
In this work, we prove effectiveness of the inverse scattering method for solving the initial value problem associated with the Hirota equation, a nonlinear model with time-dependent parameters and an additional term, within the class of rapidly decreasing functions. The main attention in this paper is paid to the study of the time evolution of the spectral properties of the Dirac operator when the potential is defined as a solution of the Cauchy problem for the nonstationary Hirota equation including variable coefficients and an additional term from the class of rapidly decreasing functions. In addition, a detailed step-by-step algorithmic procedure is developed to solve this problem, the practical realization of which is demonstrated on a concrete example.

Keywords: Dirac operator; Jost solutions; Gelfand-Levitan-Marchenko integral equation; integrable nonlinear evolution equations; soliton solutions.

Author: Kadirkulov B.(Alfraganus University), (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences), Otarova J.(Karakalpak State University),
Uzaqbaeva D.(V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences)
Abstract:
In this work, for a degenerate fourth-order mixed-type equation in a rectangular domain, nonlocal problems are solved with boundary conditions connecting the values of the sought solution or normal derivatives on the lower and upper bases of the given rectangle, which belong to different types of the studied equation. Using the spectral analysis method, a criterion for the uniqueness of the solution to the problem, was established, and the solution is constructed as the sum of a Fourier series. The spectral properties of the Samarskii-Ionkin type problem for an ordinary differential equation of the fourth order were also studied, the eigenvalues and corresponding eigenfunctions were found, their completeness and basis property were proven, and the adjoint problem was investigated.
Keywords: Degenerate equation; mixed-type equation; Samarskii-Ionkin type problem; spectral method; completeness; Riss basis; existence; uniqueness.

Author: Karimov K.(Fergana State University), (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences), Olimova D. (Fergana State University),(Fergana Military Academic Lyceum of the Ministry of Defense “Temurbek School”)

Abstract:
In this paper, we study a non-local boundary value problem in a semi-infinite parallelepiped for a three-dimensional equation of mixed type with two singular coefficients. The proof of the uniqueness of the solution and its construction are carried out by the method of spectral analysis. The solution to the problem is constructed as a double Fourier series by the sum of trigonometric and Bessel functions. When substantiating the uniform convergence of the constructed series, the problem of small denominators arises. In this regard, an estimate of the separation from zero of the small denominator with the corresponding asymptotes is found. The obtained estimate made it possible to prove the convergence of the obtained series and its derivatives up to the second order inclusive, as well as the existence theorem in the class of regular solutions.

Keywords: Semi-infinite parallelepiped; mixed type equation; non-local problem; singular coefficient; spectral method.

Author: Abraev B. (Chirchik State Pedagogical University)

Abstract:
We study a three-state SOS (solid-on-solid) model on a Cayley tree and reduce the characterization of Gibbs measures to the analysis of a system of non-linear equations. Each solution to this system corresponds to a distinct Gibbs measure. We establish conditions on the model parameters that guarantee the existence of multiple Gibbs measures. Furthermore, we review several known classes of Gibbs measures, including translation-invariant, periodic, and non-periodic types, and compare them with the newly obtained measures. Our analysis demonstrates that the Gibbs measures constructed in this work are fundamentally different from the previously known ones, thereby establishing the novelty of these solutions.

Keywords: SOS model; Cayley tree; Gibbs measures; phase transitions.

Author: Rabil A.(National Aviation Academy of Azerbaijan), Vugar K. (Ministry of Science and Education Republic of Azerbaijan Institute of Mathematics and Mechanics), Narmin A. (Baku State University)

Abstract:
This paper describes an interpolation method for obtaining a priori estimates of strong solutions of quasilinear elliptic equations and systems with unbounded singularities in the right-hand side provided that there is the first a priori estimate in the space of summable functions. The theorem of solvability of boundary value problems for nonlinear elliptic equations and systems is proved. This approach was developed by S.I. Pokhozhaev.

Keywords: Quasilinear elliptic equation; Sobolev space; interpolation method; a priory estimate.

Author: Hasanov I. (Bukhara State University)

Abstract:
This paper investigates the Cauchy problem for a partial differential equation with Gerasimov–Caputo fractional derivatives. Both the direct problem and the inverse problem of determining the time-dependent coefficient at the highest-order derivative term are studied. For the direct problem, conditions for the unique solvability are established, and an explicit representation of the solution is constructed using the fundamental solution. The inverse problem is equivalently reduced to a nonlinear Volterra integral equation of the second kind. To prove the existence and uniqueness of the solution to this integral equation, the Banach fixed point theorem is employed. The results are presented for the general case with an arbitrary number of fractional derivatives and a time-dependent potential term $
q(t)$.

Keywords: Gerasimov – Caputo fractional derivatives; Laplace transform; Fourier transform; Mittag-Leffler functions; $H$-Function.

Author:Haydarov T.(Jizzakh Polytechnic Institute)

Abstract:
The paper considers a parameterized model of a control system under conditions of uncertainty of the initial state and external influences. Applying the minimax principle to this system, the optimal control problem with a non-smooth terminal functional of the maximum type is studied. The theorems of the existence of a solution to this problem and the necessary and sufficient conditions for optimality are obtained.

Keywords: Control system; structural parameter; uncertainty conditions; minimax problem; non-smooth optimization problem; existence of a solution; optimality conditions.

Author: Hayotov A.(V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences), (Central Asian University), (Bukhara State University),
Ravshanova M.(V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences)

Abstract:
This paper explores the construction and properties of optimal quadrature formulas for exponential functions in the sense of Sard. The accuracy and efficiency of quadrature formulas, methods for determining their optimal coefficients, and the principles of constructing optimal formulas using the (\varphi)-function method are analyzed.
The research results indicate that when the nodes and coefficients of optimal quadrature formulas are correctly chosen, computational accuracy improves significantly. If the nodes are evenly distributed within the given interval, the optimal quadrature formula reduces to the Euler-Maclaurin type formula, enhancing the precision of the calculations.
The obtained results can be applied in integral computations, particularly in physics and engineering problems. This study is expected to contribute to the future application of optimal quadrature formulas to a broader class of functions and the further improvement of numerical integration techniques.

Keywords: Optimal quadrature formula; Sobolev space; optimal approximation; the (\varphi)-function method; Euler-Maclaurin type formula.

Author: Husanov E. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences), (Tashkent State Technical University named after Islam Karimov)

Abstract:
In the article, the Cauchy problem for a fractional Benney-Luke type equation with a fractional Caputo derivative is studied, where the order of the fractional derivative is $1<\rho<2$. The existence and uniqueness of the solution of the Cauchy problem is proved. Moreover, the existence and uniqueness of the solution of the inverse problem of finding the right-hand side of the equation are shown.

Keywords: Benney-Luke type equation; fractional order derivative; forward and inverse problem; Caputo fraction derevative.

Author: Islamov E. (Fergana State University)

Abstract:
In this study, the problem of approximate computation of the inverse Erdélyi–Kober operator using splines is investigated. A cubic spline interpolation method on a non-uniform grid (with unequal step sizes) is proposed for approximating the inverse Erdélyi–Kober operator. The convergence rate of the method is derived, and its stability is analyzed. The method is validated using numerical examples, and the results are presented through graphs and tables.

Keywords: Erdélyi–Kober operator; inverse Erdélyi–Kober operator; non-uniform grid; cubic spline interpolation; convergence rate; stability.

Author: Karimov Sh.(Fergana State University), Yulbarsov Kh.( Fergana State Technical University)

Abstract: This paper investigates the Cauchy problem for a third-order hyperbolic equation with the Bessel operator. By applying the Erdélyi–Kober transmutation operator to the given equation and the initial conditions, the problem is reduced to an auxiliary equation without singular coefficients. The solution of the auxiliary problem is obtained in an explicit form, and then the solution of the original Cauchy problem is derived by applying the inverse transmutation. The obtained results contribute to the further development of the theory of singular differential equations and can be applied to the analysis of a wide class of such equations.

Keywords: Cauchy problem; third-order hyperbolic equation; Bessel operator; transmutation operator.

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

Abstract:
We consider the critical Galton-Watson branching process starting with one particle and possibly with the infinite variance the number of descendants of one particle. Limit theorem for such processes proved.

Keywords: Critical Galton-Watson branching process; generation function; limit theorem.

Author: Nuraliyeva N. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences), (Karshi State University)

Abstract:
In this paper, we study a time-fractional subdiffusion equation with a nonlinear nonlocal initial condition involving the unknown solution at the final time. The considered problem is formulated using the Caputo fractional derivative of order $0 < \alpha < 1$, along with homogeneous Dirichlet boundary conditions. The nonlocal initial condition is of the form $u(x,0) = g(x, u(x,T)) $, where $g$ is a nonlinear function satisfying a Lipschitz condition. The main challenge arises from the implicit dependence on the unknown final state. Using an explicit representation of the solution in terms of the Green function and applying the Banach fixed point theorem, we establish the existence and uniqueness of a regular solution. We also provide uniform estimates for the Green function and analyze the influence of the Lipschitz constant on solvability

Keywords: Fractional subdiffusion equation; Caputo fractional derivative; nonlocal boundary condition; Green’s function; Lipschitz condition; Banach fixed point theorem; existence of solution.

Author: Rakhimov A. (Karshi State University)

Abstract:
In this paper, a mixed problem for a poroelasticity system described by non-homogeneous equations with three elasticity parameters in a reversible hydrodynamic approximation is considered. Using the Fourier method, a solution to the mixed problem with non-homogeneous boundary conditions is found, and formulas for solving this problem are obtained. It is proved that the solution is unique. The obtained solution formulas can be used in numerical methods for solving problems related to wave dynamics in poroelasticity.

Keywords: Porous medium; mixed problem; poroelasticity; porosity; initial-boundary value problem; Fourier method.

Author: Samatov B. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences), Umaraliyeva N. (Namangan State University)

Abstract: In this paper, a discrete pursuit game with Langenhop constraints on the players’ controls is considered. The relationship between geometric, Langenhop, and summary constraints on control classes is investigated. The main tool for solving the problem is the parallel approach strategy ($\Pi$-strategy). New sufficient conditions for the solvability of the discrete pursuit problem are obtained. Based on the conditions of the theorems, an animation model is developed, which is used to verify the obtained theoretical results.

Keywords: Discrete game; pursuer; evader; Langenhop constraint; pursuit strategy; guaranteed capture step.

Author: Sobirov Z. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences), (National University of Uzbekistan),
Turemuratova A.(National University of Uzbekistan), (Branch of the Russian Economic University named after G. V. Plekhanov in Tashkent)

Abstract:
In this paper, we explore both direct and inverse problems related to the space-time fractional diffusion equation on metric graphs. Using the method of separation of variables, we derive an explicit solution in the form of a Fourier series. The convergence of the resulting Fourier series is shown. The uniqueness of a solution to the considered problem is demonstrated using a priori estimates.

Keywords: Metric graph; fractional derivatives and integral; space-time fractional diffusion equation; inverse source problem.

Author: Gulkhayo S. (National University of Uzbekistan)

Abstract:
In this paper, we construct solvable Poisson algebras whose nilradical has a Lie part isomorphic to a nilpotent Lie algebra with characteristic sequence $(n_1, n_2, 1)$ denoted by $\mathfrak n_2$. We prove that if the Lie part is isomorphic to the maximal solvable extension of $\mathfrak{n}_2$, then it is trivial. Furthermore, we show that if the Lie part is isomorphic to the semidirect sum of $\mathfrak{n}_2$ and a two-dimensional torus, then it possesses a nontrivial associative commutative part.

Keywords: Lie algebra; Poisson algebra; nilpotent; solvable.

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

Abstract:
In this work, we present the main parts of the verification of the Cauchy problem for an equation with singular coefficients and the Prabhakar fractional derivative. The solution to the given problem is shown to be unique using the generalized Hankel transform. Using the Fourier method, we find the general solution of the given equation. The unknown coefficients are found using the Hankel transformation. We have also presented important statements the bivariate Mittag-Leffler function $E_{2} \left(x,{\rm \; }y\right)$ and the Bessel function.

Keywords: Cauchy problem; Prabhakar fractional derivative; Hankel transform; bivariate Mittag-Leffler function; Bessel function.

Author: Allakov I.(Termez State University), Imamov O. (Termez State University)

Abstract:
The work studies the problem of representing the natural number $n$ as the sum of the squares of four prime numbers from an arithmetic progression. The number of natural numbers that cannot be represented in the specified form has been estimated, i.e. the exceptional set of the problem, is estimated.

Keywords: Diophantine equation; congruent solution; positive solution; exceptional zero; Dirichlet $L$-function; Legendre symbol; minor arc; major arc; singular series; singular integral.

Author: Hasanov A. (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences),(Ghent University). Yuldashova H (V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences)

Abstract:
In this paper, by decomposing the confluent hypergeometric function $E_{1} $ into eight parts, we demonstrate how some useful and generalized relations between the hypergeometric functions of Srivastava $F^{\left(3\right)} $ and $E_{1} $ can be obtained. It is shown that other main results are specified in order to derive certain relations between the functions $F_{3} $, ${{\Phi }{1}}$, ${{\Phi }{2}}$, ${{\Phi }{3}}$, ${{\Xi }{1}}$, ${{\Xi }{2}}$, ${}{p}{{F}{q}}$ and $F{l:m;n}^{p:q;k}$. Some other interesting functional relations involving the exponential function, hyperbolic functions, and modified Bessel functions are also considered.

Keywords: Confluent hypergeometric function; generalized hypergeometric series; functional identities; modified Bessel functions; exponential function.

Author: Khakimov R.(V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences), Ergashboeva L. (Namangan State University)

Abstract:
We study the fertile three-state Hard-Core (HC) model with activity parameter $\lambda>0$ in the case of a graph of type “Wand” on a Cayley tree. In this case on a Cayley tree of order six translational invariance conditions for alternative Gibbs measures are found. Furthermore, we determine exact critical values of $\lambda>0$, for which alternative non-translation-invariant Gibbs measures arise and are non-unique.

Keywords: Cayley tree; Hard-Core model; Gibbs measure; Phase transition.

Author: Kuldoshev K.(National University of Uzbekistan)

Abstract:
In this paper, the concept of the $P_m$-capacity introduced in the class of $m$-subharmonic functions is generalized by defining the weighted $P(m,\psi,\delta)$-capacity. Several fundamental properties of this capacity are established, including monotonicity, boundedness, limit transition for sequences, and countable subadditivity. Moreover, the relationship between the $P(m,\psi,\delta)$-capacity and the corresponding external $C(m,\psi,\delta)$-capacity is investigated, and comparison inequalities between them are derived. The obtained results extend previously known properties of the unweighted $P_m$-capacity to the weighted case and contribute to the development of potential theory in the class of $m$-subharmonic functions in complex spaces.

Keywords: $m-$subharmonic function; $\mathcal{P}_m-$capacity; weighted $\mathcal{P}_m-$capacity; $m-$polar set.

Author: Kurbonov O. (Tashkent State Economic University)

Abstract:
This paper presents a study of a nonlinear boundary value problem for a nonlinear third-order equation with multiple characteristics in a curvilinear domain. The unique solvability of this problem is established using the energy method. To prove the existence of a solution to this problem, an auxiliary problem was considered, for which the Green’s function was constructed. By solving an auxiliary problem, the original problem was reduced to a system of Hammerstein integral equations. The solvability of a nonlinear system was proven by the contraction mapping method.

Keywords: Nonlinearity; uniqueness; existence; system of Hammerstein equations; the contraction mapping method.

Author: Mamadaliev N.(National University of Uzbekistan),(V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences), Mustapokulov Kh. (National University of Uzbekistan)

Abstract:
In this paper, we study differential games described by a system of linear differential-difference equations of neutral type under integral restrictions on the players’ controls. An analogue of the third pursuit method has been developed and applied to solve the pursuit problem described by a system of neutral-type differential-difference equations. New sufficient conditions are proposed that ensure the end of the game in a certain guaranteed time.

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

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

Abstract:
This article investigates the unique solvability of a time and boundary-nonlocal inverse boundary value problem for a fractional subdiffusion equation with an inner point condition. To analyze the solvability of the inverse problem, we first reduce the original problem to an auxiliary system with trivial data and establish its equivalence, under certain conditions, with the original formulation. By applying the Banach fixed-point theorem, we then demonstrate the existence and uniqueness of a solution to the auxiliary system. Finally, leveraging the established equivalence, we prove an existence and uniqueness theorem for the classical solution of the inverse coefficient problem for a sufficiently small time interval.

Keywords: Subdiffusion equation; inverse problem; Mittag-Leffler function; classical solution; existence; uniqueness.

Author: Muydinov I. (Fergana State Technical University)

Abstract:
This article solves the Goursat problem for a higher-order hyperbolic equation using the Riemann function and obtains the solution in an explicit form. The primary focus is directed toward deriving the exact (open-form) analytical solution based on the Riemann function.

Keywords: Higher-order hyperbolic equation; Goursat problem; Riemann function; Green’s formula; coupled equations.

Author: Rasulov T. (Bukhara State University), Ismoilova D.(Bukhara State University)

Abstract: In this paper, we consider a third-order bounded, self-adjoint operator matrix $\mathcal{A}$ that acts on a fermionic Fock space. First, we introduce a third-order discrete parameter operator matrix to study its spectral properties. Next, we construct the first Schur complement $S_{1}^{(\mathrm{s})}(\lambda)$ corresponding to this operator matrix and use it to define the first Schur complement corresponding to the operator matrix $\mathcal{A}$. We establish connection between the eigenvalues of third-order matrix and its corresponding first Schur complement. We prove that the function $N_{(-\infty;0)}(S_{1}(\lambda))$ related with the eigenvalues of the first Schur complement of the operator matrix $\mathcal{A}$ is monotonically increasing and continuous with respect to the spectral parameter $\lambda$.

Keywords: Fermionic Fock space; operator matrix; spectral relation; Schur complement; spectral parameter; negative eigenvalue; resolvent operator.

Author: Rozikov U.(V.I.Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences), Juraev I.(Namangan State University)

Abstract:
In this paper, the dynamics of quadratic Volterra operators defined on the four-dimensional simplex of idempotent measures are investigated. The authors determine all fixed points of quadratic absolute stochastic Volterra operators (QASVO) and classify them as attracting, repelling, or hyperbolic. Moreover, the limit points of trajectories for all initial conditions are completely described. The study employs methods from idempotent algebra, max-plus analysis, and tropical geometry. The obtained results provide explicit descriptions of the operator dynamics in the 4D case and open possibilities for generalization to higher-dimensional simplices.

Keywords: Quadratic operator; Volterra operator; simplex; idempotent measure; fixed point; trajectory; dynamical system; attracting; repelling; chaos.

Author: Abdullayev J.(Samarkand State University), Ergashova Sh.(Samarkand State University)  

Abstract: 
In this paper, the problem of finding a generalized solution of an integro-differential equation of the hyperbolic type is studied in an infinite field on the spatial variable x. The existence of a single solution of a given equation when the integral limit is known has
been proved by the method of sequential approximations.

Keywords: Schr ̈odinger operator; Hamiltonian; invariant subspace; eigenvalue; eigenfunction; quasimomentum; boson.

Author: Allakov I.(Termez State University)0, Siti Hasana Sapar.(Universiti Putra Malaysia), Fatanah binti Deraman (Institute of Engineering Mathematics Universiti Malaysia Perlis),Shahrina I. (Universiti Sains Islam Malaysia)

Abstract:  Non homogeneous Beatty sequences play important rules in Wythoff games and invariant games such as on how to beat your Wytoff games opponent on three fronts and give properties into a decision of the procedure relying only on a few algebraic tests. This paper discusses on the cardinality of character sums and their estimation with respect to non homogeneous Beatty sequences βα = ⌊αn + β : n = 1, 2, 3…⌋ where β in real numbers and α greater than zero is irrational. In order to estimate the cardinality, the discrepancy is used to measure the number of uniform distribution for Beatty sequences. Pigeonhole principle is discussed on the estimation of the fractional part of Beatty sequences involve. Meanwhile, Cauchy inequalities is applied to expand the double character sums. Then, the cardinality of double character sums is obtained by applying the extension properties of additive and multiplicative character sums. The result obtained is depend on the existing of identity of additive and multiplicative character sums and the uniformly distribution modulo 1. The result of the estimation in this study over composite moduli is more general compared to previous studies, which only cover prime moduli.

Keywords:

Author: Aripov M.(National University of Uzbekistan),Atabaev O. (Andijan State University)

Abstract: In this paper, we investigate a doubly degenerate parabolic problem with source and absorption terms within a bounded domain, subject to homogeneous Dirichlet boundary conditions and a nonnegative initial condition. We discuss the global existence and nonexistence of solutions to the problem. Precise estimates for the blow-up rates for non-global solutions are also established. Furthermore, there is an additional emphasis on the persistent existence of solutions without extinction.

Keywords: Doubly degenerate parabolic equation; boundedness; global existence; blow-up; non-extinction; finite time extinction.

Author: Muzaffarova M.  (Bukhara State University)

Abstract: The article provides brief information about dynamic systems. Stationary points of the continuous-time Volterra quadratic stochastic operator (dynamic system) were found and stability was studied. It is shown that the analytical solution of the dynamic system cannot be found. Numerical solutions of the dynamic system at different initial values were found using the MathCAD mathematical package, graphs and phase space were drawn, and relevant conclusions were written. The Runge-Kutta method was used to solve the problem using the MathCAD mathematical package.

Keywords: Quadratic stochastic operator; dynamic system; numerical solution; analytical solution; fixed

Author: Olimov U. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  We consider an infinite-dimensional non-linear operator related to a hard core (HC) model with a countable set N of spin values. It is known that finding the fixed points of an infinite- dimensional operator is generally impossible. But we have fully analyzed the fixed points of an infinite-dimensional operator by applying a new technique of reducing an infinite-dimensional operator to a two-dimensional operator. The set of parameters is divided into subsets Ai,j , where the index i means the number of fixed points on the line y = x, j means the number of fixed points outside of y = x. The number of fixed points can be up to seven, and the explicit form of each fixed point was found.

Keywords: Fixed point; invariant set; infinite-dimensional operator.

Author: Azamov S.(Tashkent State Transport University)

Abstract:  In this paper, we consider the construction of optimal quadrature formulas in a specific Hilbert space of exact functions of a first-degree polynomial multiplied by an exponential. From the beginning, a maximizing element is found, i.e. extremal function of the error functional of the quadrature formula under consideration. Then the squared norm of the error functional is calculated. By minimizing this norm by coefficients, a system of equations is obtained for finding the optimal coefficients of quadrature formulas. The existence and uniqueness of the resulting system is proven. By solving this special system using the Sobolev method, the optimal coefficients of the quadrature formulas are found.

Keywords: Hilbert space; extremal function; generalized functional; operator; optimal quadrature formula.

Author: Djabbarov O. (Karshi State University)

Abstract:  Paper discusses the global solvability of the Cauchy problem for a double nonlinear parabolic equation with a damping term and the asymptotic behavior of the solution. The global solvability of the Fujita type Cauchy problem is proved by the method of comparison and self-similar analysis of solutions.

Keywords: Finite speed; perturbation; global solutions; solution estimate; critical case; asymptotics; numerical analysis.

Author: Dzhamalov S.(V.I.Romanovskiy Institute of Mathematics), Sipatdinova B.(V.I.Romanovskiy Institute of Mathematics) 

Abstract: In this paper, we study the correctness of a linear inverse problem with a nonlocal boundary condition of periodic type for a three-dimensional equation of mixed type of the second kind and second order in an unbounded parallelepiped. For this problem, using the methods of “ε -regularization”, a priori estimates, and a sequence of approximations using the Fourier transform, the existence and uniqueness theorems for a generalized solution of a linear inverse problem with a nonlocal boundary condition of periodic type in a certain class of integrable functions are proved.

Keywords: Equations of mixed type of the second kind of the second order; linear inverse problem with a nonlocal boundary condition of periodic type; correctness of the problem; methods of “ ε-regularization”; a priori estimate; sequence of approximation; Fourier transform.

Author: Zikirov O.(National University of Uzbekistan), Sagdullayeva M.(National University of Uzbekistan)

Abstract: We consider a non-local initial boundary value problem with an integral condition for a third-order equation with a heat operator in the main part. We prove the unique solvability of the problem under study using an integral representation of the solution to the first boundary value problem for the heat equation. The problem is equivalently reduced to the Volterra integral equation of the second kind.

Keywords: Boundary value problem; nonlocal condition; integral condition; heat equation; Green’s function; integral equation.

Author: Kalandarov T. (V.I.Romanovskiy Institute of Mathematics), (Karakalpak State University)

Abstract: This paper is devoted to the study of 2-local inner automorphisms of Arens algebras Lω(M, τ ) associated with the von Neumann algebra M with a faithful normal semi-finite trace τ . It is shown that every surjective 2-local automorphism Φ of the Arens algebras L ω(M, τ ) is an automorphism. And if M is a von Neumann algebra of type I, then Φ is an inner automorphism, that is, there exists a unitary element u ∈ Lω(M, τ ) such that Φ(x) = uxu∗ for all x ∈ Lω(M, τ ).

Keywords: Arens algebra; automorphism; 2-local automorphism.

Author: Rakhmonov F.(National University of Uzbekistan), Tankeeva A.(Aktobe regional university)

Abstract:  In the present paper, the unique classical solvability and the construction of the solution of mixed problems for a nonlinear integro-differential equation involving the square of the Barenblatt-Zheltov-Kochina operator have been studied. The method of Fourier series based on the separation of variables is used. A countable system of nonlinear integral equations is obtained. Sufficient coefficient conditions for the unique regular solvability of the mixed problem are established. The method of successive approximations combined with the method of compressing mapping are applied.

Keywords: Mixed problem; nonlinear integro-differential equation; fourth order equation; square of the Barenblatt-Zheltov-Kochina operator; regular solvability.

Author: Ruziev M.(V.I.Romanovskiy Institute of Mathematics), Mirsaburova G.(Termez State University), Mamatmuminov D.(Termez State University)

Abstract: In this paper we study a boundary value problem with an analogue of the Bitsadze–Samarskii condition for degenerate hyperbolic equations with singular coefficients. To prove the unique solvability of the problem under study, a functional equation is solved using the iteration method.

Keywords: In this paper we study a boundary value problem with an analogue of the Bitsadze–Samarskii condition for degenerate hyperbolic equations with singular coefficients. To prove the unique solvability of the problem under study, a functional equation is solved using the iteration method.

Author: Usmonov D. (Fergana State University)

Abstract:  In this work, in a rectangular domain, we study an initial boundary problem for a degenerate fourth-order differential equation containing an integrodifferential operator with a Bessel function in the kernel. Applying the method of separation variables to the considered problem a spectral problem for an ordinary differential equation has been obtained. The existence of eigenvalues and the system of eigenfunctions of the spectral problem was proved. A theorem is proved for expanding a given function into a uniformly convergent series with respect to the system of eigenfunctions. The solution of the considered problem is written as the sum of the Fourier series with respect to the system of eigenfunctions of spectral problem. An estimate for the solution of the problem is obtained, from which follows its continuous dependence on the given functions.

Keywords: Degenerate equation; initial-boundary problem; Bessel function; integrodifferential operator; spectral method; Green’s function; integral equation.

Author: Holboyev A. (Tashkent State Pedagogical University)

Abstract: This research is devoted to pursuit-evasion game on the edges of the dodecahedron. It was studied how changing the minimum number of pursuers which catch an evader when an edge is excluded and it was proved that this number is equal to two.

Keywords: Pursuit problem; evasion problem; strategy; dodecahedron; geometric graph.

Author: Khadjiev Dj. (V.I.Romanovskiy Institute of Mathematics), (National University of Uzbekistan)

Abstract: Let B = {gij (x), Lij (x), i, j = 1, 2, . . . , n} be the set of all coefficients of the first and second fundamental forms of a hypersurface x in a Euclidean space Rn+1. Using computations of invariants from B for some hypersurfaces, it is proved that B is a minimal complete system of SM(n + 1)-invariants of a regular hypersurface in Rn+1, where SM(n + 1) is the group of all Euclidean motions of Rn+1.

Keywords: Bonnet system; invariants of a hypersurface; group of Euclidean motions.

Author:  Ergashova Sh.  (National University of Uzbekistan)

Abstract:  This paper studies the geometry of the Liouville foliation generated by completely integrable Hamiltonian systems in six-dimensional Euclidean space. It is shown that the regular leaves of the Liouville foliation are three-dimensional sub-manifolds whose Gaussian curvature is equal to zero.

Keywords: Completely integrable Hamiltonian system; Liouville foliation; regular leaf of a singular foliation; Gauss curvature.

Author: Sobirov Z. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  We considered two inverse source problems for a time-fractional pseudo-subdiffusion equation involving the Hilfer fractional derivative on a metric graph. Using the method of separation of variables we find the exact solution of the investigated problems in the form of the Fourier series. Results on the existence and uniqueness of solutions to these problems are presented.

Keywords: Inverse source problem; Hilfer fractional derivative; pseudo-subdiffusion equation; metric graph.

Author: Gaziev K. (Fergana State University)

Abstract: In this work, we study boundary value problems for a parabolic-hyperbolic equation in a finite domain. This problem differs from other problems in that non-local conditions are formulated on the the parabolic boundary of the domain. The uniqueness of the solution has been proved by the extreme principle, and the existence of the the solution is reduced to the system of Volterra integral equations.

Keywords: Volterra integral equation; Cauchy problem; extremum principle; parabolic-hyperbolic equation.

Author: Dzhamalov S.(V.I.Romanovskiy Institute of Mathematics), Khalkhadjayev B.(V.I.Romanovskiy Institute of Mathematics), Yusupov Sh. (V.I.Romanovskiy Institute of Mathematics)

Abstract: In this article using the methods of Faedo Galerkin, «ε-regularization» and a priori estimation, the uniqueness and existence of one solution to a semi-nonlocal boundary value problem for the fourth-order mixed type equation in Sobolev space is proved.

Keywords: Mixed-type fourth order equations of the second kind; semi-nonlocal boundary value problem; method of Faedo-Galerkin; a priori estimates and «ε-regularization»; unique solvability of a regular generalized solution.

Author:  Durdiev U. (Bukhara State University), (V.I.Romanovskiy Institute of Mathematics)

Abstract:  This study is devoted to the inverse problem for determining of kernel, which represents the memory of the medium in the integro-differential equation of forced vibrations of a beam. First, the initial boundary value problem (direct problem) is considered. Using the Fourier method, this problem is reduced to equivalent integral equations. Then, using the technique of estimating these functions and the generalized Gronwall inequality, we obtain a priori estimates of the solution in terms of the unknown coefficient, which will be used to study the inverse problem. The inverse problem is reduced to an equivalent integral equation of Volterra type. To show the existence of a unique solution to this equation, the method of contraction mappings in the space of continuous functions with exponential weight norm is used. Results are obtained on the global solvability of the inverse problem under consideration and an estimate of the conditional stability of the solution.

Keywords: Initial-boundary value problem; integro-differential equation; beam vibration equation; Gronwall inequality; inverse problem; global solvability.

Author: Ibragimov M.(Karakalpak State University)

Abstract: In this paper we establish a morphism between lattices of geometric tripotents in a dual space of strongly facially symmetric spaces. More precisely, we show that this morphism is a quantum-logical isomorphism and study its properties.

Keywords: Facially symmetric spaces; quantum logic; morphism; quantum-logical isomorphism.

Author: Karimov K.(Fergana State University), Shokirov A.(Fergana State University)

Abstract:  In this paper we study the Gellerstedt problem for a three-dimensional mixed-type equation with singular coefficient in a mixed region, the elliptic part of which is a half-cylinder and the hyperbolic part is two triangular straight prisms. In the investigated problem, the method of separation of variables is used and non-trivial partial solutions (eigenfunctions) in the hyperbolicity and ellipticity regions of the equation are found. With the help of these functions, the solutions of the problem in the regions of hyperbolicity and ellipticity of the equation are constructed in the form of a double series. The asymptotic estimates of the Bessel functions of the real and imaginary arguments, as well as the properties of the Gauss hypergeometric functions were used to justify the uniform convergence of the constructed series. On their basis we obtained estimates for each term of the series, which allowed us to prove the convergence of the obtained series and its derivatives up to the second order inclusive, as well as the existence theorem in the class of regular solutions.

Keywords: Singular coefficient; Gellerstedt problem; Bessel function; Gauss hypergeometric function; Tricomi problem; Tricomi-Neumann problem.

Author: Oripov Sh. (Fergana State University)

Abstract:  In this article, the Cauchy problem has been formulated and studied for the fourth-order hyperbolic equation with a singular coefficient. In contrast to traditional methods, to solve this problem, the fractional order Erdelyi – Kober operator was used and an explicit formula for the desired solution was found.

Keywords: Fourth-order equation; Cauchy problem; Bessel operator; transmutation operator; Erdelyi – Kober operator.

Author: Rakhimov B. (Jizzakh Polytechnic Institute)

Abstract:  In this paper we consider a model of dynamic system under conditions of indeterminacy – differential inclusion with control parameters. For the model, the problem of controllability of an ensemble of trajectories to a mobile terminal set has been investigated. The necessary end sufficient conditions of controllability are
obtained.

Keywords: Differential inclusion; ensemble of trajectories; terminal set; problem of controllability; conditions of controllability.

Author:  Tulakova Z.(Fergana branch Tashkent University of Information Technologies)

Abstract: Known that in studying boundary value problems for degenerate elliptic equations, their fundamental solutions play an important role. These fundamental solutions are expressed in terms of hypergeometric functions of one or more variables depending on the number of singular coefficients of the elliptic equation. In this
work, using the well-known fundamental solutions of a three-dimensional elliptic equation with three singular coefficients, three mixed problems in the first octant of a ball are studied. The uniqueness of the solution to the problems posed is proved by the method of energy integrals, and the existence – by the Green’s function method. The sought solutions are explicitly represented through the triple Lauricella hypergeometric function, moreover, in forms convenient for further research.

Keywords: Lauricella hypergeometric function; fundamental solution; three-dimensional singular elliptic equations; Green’s function; mixed problem.

Author: Khasanov A.(Samarkand State University), Normururodov Kh.(Samarkand State University)

Abstract:  In this paper, the inverse spectral problem method is used to integrate a nonlinear equation (mKdV-sG) with an additional term in the class of periodic infinite-gap functions. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of six times continuously differentiabl periodic infinite-gap functions is proved. It is shown that the sum of a uniformly convergent functional series constructed by solving the system of Dubrovin equations and the first trace formula satisfies the mKdV-sG equations with an additional term.

Keywords: Modified Korteweg-de Vries-sine Gordon (mKdV-sG) equation; Dirac operator; spectral data; Dubrovin’s system of equations; trace formulas.

Author: Khodjibayev V.(Namangan Engineering-Construction Institute), (V.I.Romanovskiy Institute of Mathematics), Atakhujayev A.(Tashkent branch of the National Research nuclear university MEPhI)

Abstract:  The paper considers a homogeneous random process with independent increments and zero drift. Two-sided inequalities for the ruin probability are established.

Keywords: Stochastic processes with independent increments; ruin probability; excess over boundary.

Author: Abdurakhimova Sh.(Namangan State University)

Abstract:  In this paper, we explore a class of quadratic stochastic operators that act as evolution operators on a population comprising two species. These operators are characterized by three parameters. We demonstrate that, under certain conditions on these parameters, some operators within this class can be topologically conjugate to others. By leveraging these conjugacy relations, we have streamlined the theorems established in our work [1].

Keywords: Dynamical systems; fixed point; periodic point; evolution operator; topological conjugacy.

Author: Aralova K. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  In the present paper, we study dynamical systems generated by stochastic operators which are superpositions of extremal Volterra and non-Volterra quadratic stochastic operators defined on the two-dimensional simplex. It is described the set of all periodic and the set of all fixed points of such operators. Further for such operator we showed that for an initial point their trajectory either converges to a periodic trajectory or diverges.

Keywords: Quadratic stochastic operator; Volterra operator; non-Volterra operator; trajectory; fixed point; periodic point.

Author:  Dekhkonov F. (Namangan State University)

Abstract: This paper considers a boundary control problem for a fourth-order parabolic equation in a bounded one-dimensional domain. The considered problem by separating variables is reduced to the Volterra integral equation of the first kind. The existence of the control function was proved by the Laplace transform method and the estimate of the minimum time to reach the given average temperature in the rod was found.

Keywords: Initial-boundary value problem; fourth order parabolic equation; admissible control; minimal time; integral equation; Laplace transform.

Author: Dzhalilov A.(Turin Polytechnic University in Tashkent), Abdusalomov X. (National University of Uzbekistan)

Abstract: In the present paper we prove limit theorems for nonlinear moving-average (MA) models. Also, for piecewise linear moving-average (MA(pl)) models, we investigate the dependence the autocovariance and find the limiting values on the parameter of generating a piecewise
linear map.

Keywords: Moving average process; piecewise–linear map; strictly stationary; autocovariance function.

Author: Elmuradova H.(Bukhara State University)

Abstract:  In this paper, we study a nonlinear inverse problem for a one-dimensional pseudoparabolic integro-differential equation with overdetermination conditions. First, we studied direct problem. Using the Fourier method, this direct problem is reduced to equivalent integral equation. Then, using the estimate of the Mittag-Leffler function and generalized singular Gronwall inequalities, we obtain an estimate for the solution of the direct problem via the norm of unknown functions. Inverse problem reduces to an equivalent system of integral equations. To solve this system, we use the principle of constraint mapping. The results of local existence and global uniqueness of the solution to the problem were obtained.

Keywords: Pseudoparabolic equation; Fourier method; inverse problem; integral equation; Banach fixed point theorem.

Author: Nuriddinov O.(Andijan State University)

Abstract:  In the present paper we investigate local derivations on finite dimensional Jordan algebras. We give a description of local derivations on all Jordan algebras of dimension less than or equal to four.

Keywords: Algebra; Jordan algebra; basis of an algebra; derivation; local derivation; dimension of an algebra.

Author: Saparbayev R. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  This work is devoted to the study of a non-local problem for the telegraph equation that is the initial conditions are replaced by non-local conditions: Here, A is a non-negative selfadjoint operator, acting in a Hilbert space H; D is the Caputo
fractional derivative. Conditions are found for the functions φ1, φ2 and the right side of the equation that guarantee both the existence and uniqueness of the solution of the non-local problem. The influence of the constant β on the existence of a solution to the problem is investigated. We also found stability estimates which are important for the applications. It should be noted that the non-local problem for the fractional order telegraph equation is studied for the first time.

Keywords: Non-local problem; the Caputo derivative; Fourier method; fractional telegraph equation.

Author: Toshpulatov M.(Andijan State University) 

Abstract: In this paper, we consider a unique solvability of an initial-boundary problem for mixed equation with space-variable coefficient involving the regularized Prabhakar fractional derivative.We have presented important statements on the bi-variate Mittag-Leffler function. Namely, Euler-type integral representations and certain estimations for the bi-variate Mittag- Leffler type function E2(x, y). Since the key tool of investigation is the method of separation of variables, Most of our evaluations are linked with the proof of uniform convergence of infinite series. We impose certain conditions on given unctions in order to provide uniform convergence of infinite series corresponding to the solution of formulated problem. Certain properties of Legendre polynomials, bi-variate Mittag-Leffler type function have been used to prove the convergence of infinite series.

Keywords: Fractional order mixed equation; Prabhakar derivative; Legendre polynomials; bi-variate Mittag- Leffler type function.

Author: Yuldashova H. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  In this article, we aim to study Mittag-Leffler-type function 2F ̄1 (x), which correspond, to the familiar Gaussian hypergeometric functions 2F1 (a, b; c1, x). Among the various properties and characteristics of this Mittag-Leffler-type function, which we investigate in this article, include its relations with other extensions and generalizations of the classical Mittag-Leffler function, its convergence regions and even and odd parts, its special cases and Euler-type integral representations, its one-dimensional Laplace transforms and connections with the Riemann- Liouville operators of fractional calculus, and ordinary differential equation associated with the function 2F ̄1 (x).

Keywords: Extended Mittag-Leffler type function; hypergeometric function; special (or higher transcendental) function; Lauricella function; integral representation; system of partial differential equations; Laplace transforms; Riemann-Liouville fractional integral; Riemann-Liouville fractional derivative; Appell and Kamp ́ede F ́eriet functions; Srivastava-Daoust hypergeometric function.

Author: Kabulov A.(National University of Uzbekistan), Babadzhanov A.(National University of Uzbekistan), Saymanov I.(National University of Uzbekistan)

Abstract: This paper proposes a consideration of various pattern recognition models. In this case, it is proposed to consider models in the form of two operators: a recognition operator and a decision rule. Algebraic operations are introduced on recognition operators and, based on the use of these operators, a family of recognition algorithms is created. An upper bound is constructed for the model to guarantee the completeness of the expansion.

Keywords: Recognition; disjoint classes; model; calculations of estimates; operator; algorithm.

Author:  Karimov Sh.(Fergana State University), Bogdan A.(Fergana State University) 

Abstract: In this work, we investigated an analog of the characteristic problem for one non-classical fourth-order partial differential equation with singular coefficients. To solve the problem, the method of transmutation operators was used. The two-dimensional generalized Erdelyi-Kober operator of fractional order was used as a transmutation operator. The solution to the problem is constructed in explicit form.

Keywords: Transmutation operator; generalized Erd ́elyi-Kober operator; Bessel operator; Bessel-Clifford function; normalized Bessel function; multidimensional Laplace operator.

Author:  Nasirova D.  (Tashkent State Technical University)

Abstract:  This work is devoted to a formulation and an investigation of a boundary value problem with Gellerstedt conditions on parallel characteristics for a loaded parabolic-hyperbolic type equation of the second kind. Using by the extremum principle and the method of energy integrals, was proved the uniqueness of solution of the formulated problem, and the existence of a solution to the problem – by the method integral equations.

Keywords: Equation of the second kind; loaded equation; Gellerstedt condition; extremum principle; method of energy integrals; Fredholm equation of the second kind.

Author: . Ruziev M.(V.I.Romanovskiy Institute of Mathematics), Yuldasheva N.(V.I.Romanovskiy Institute of Mathematics)

Abstract: For a fractional order diffusion equation and a degenerate hyperbolic type equation with singular coefficients, a nonlocal problem is studied in an infinite domain, the boundary condition of which contains a linear combination of generalized fractional integro-differentiation operators. The uniqueness of the solution to the problem is proved by the energy integral method. The existence of a solution to the problem is obtained as a solution of fractional derivative equations of different orders.

Keywords: Degenerate hyperbolic equation; singular coefficient; boundary value problem; generalized fractional integro-differentiation operator; Wright function; fractional differential equation.

Author: Urinov A.(Fergana State University), Oripov D. (Fergana State University)

Abstract:  In this paper, for a degenerate partial differential equation of high even order containing the Gerasimov-Caputo fractional differentiation operator of order γ, where γ ∈ (p − 1, , p], p ∈ N, , p ≤ 2, an initial boundary value problem is formulated and studied in a rectangle. By applying the Fourier method, based on the separation of variables, to the problem under study, a spectral problem for an ordinary differential equation is obtained. Green’s function of the latter problem is constructed equivalently reduced to the Fredholm integral equation of the second kind with a symmetric kernel, implying the existence of eigenvalues and a system of eigenfunctions of the spectral problem. A theorem for the expansion of a given function into a uniformly convergent series in
a system of eigenfunctions is proved. Using the derived integral equation and Mercer’s theorem, the uniform convergence of some bilinear series depending on the obtained eigenfunctions is proved. The order of the Fourier coefficients is established. The cases γ ∈ (0, , 1), γ = 1, γ ∈ (1, 2), and γ = 2 are considered separately. For each of these cases, the solution to the problem under study is expressed as the sum of a Fourier series over the system of eigenfunctions of the spectral problem. An estimate for the solution is obtained, which implies its continuous dependence on the given functions.

Keywords: Degenerate equation; initial boundary value problem; method of separation of variables; spectral problem; method of Green functions; integral equation; Fourier series.

Author: Khaydarov I. (Fergana State University)

Abstract:  The present research is devoted to studying the unique solvability of a problem with an integral condition for a parabolic-hyperbolic equation with two lines of type change. The uniqueness of the solution has been proved by applying the extremum principle. The existence of the solution to the problem has been proved using the theory of integral equations.

Keywords: Integral condition; mixed-type equation; parabolic-hyperbolic equation.

Author: Khamrayev A.(Karshi State University), Norov A.(Karshi State University)

Abstract:  In this paper, we study dynamical systems defined by a cubic operator. For a cubic non-Volterra operator on the two-dimensional simplex, all fixed points were found and the behavior of the trajectories of this operator was completely studied.

Keywords: Cubic operator; two-dimensional simplex; theory of dynamical systems; fixed point.

Author: Khujamiyorov I.(Samarkand State University)

Abstract: We consider the three-particle discrete Schr ̈odinger operator Hμ,γ(K), K ∈ T
4, associated with the Hamiltonian of a system of three particles (two are fermions with mass 1 and one – arbitrary with mass m = 1/γ < 1), interacting using paired repulsive contact potentials μ > 0 on the four-dimensional lattice Z 4 . It has been proven that there are critical values of the mass ratios γ = γ1 and γ = γ2 such that if γ ∈ (0, γ1), then the
operator Hμ,γ(0) has no eigenvalues, if γ ∈ (γ1, γ2), then it has a single eigenvalue, if γ > γ2, then it has four eigenvalues, taking into account multiplicity, lying to the right of the essential spectrum for all sufficiently large values of the interaction energy μ.

Keywords: Lattice; Schr ̈odinger operator; Hamiltonian; contact potential; fermion; eigenvalue; quasi- momentum; invariant subspace; operator Faddeev.

Author: Artykbaev A.(Tashkent State Transport University), Toshmatova M.(Tashkent State Transport University) 

Abstract:  This article is dedicated to the study of the geometric properties of osculating circles in a set of spatial curves. Osculating circles formed by spatial curves are considered, and the surface generated by these circles is introduced and analyzed. An equation defining this surface is provided, and it is applied to optimize the geometric design of railway tracks. Particular attention is given to transition curves and the optimal choice of the radius of the osculating circle in curved sections of railway tracks, ensuring smooth and safe transitions from straight to curved parts. Our methods and results have practical significance for the geometric modeling and design of railway tracks.

Keywords: Osculating circle; curvature; tangent circle; surface of osculating circles.

Author: Arzikulov F.(V.I.Romanovskiy Institute of Mathematics), Alijonova K. (Andijan State University)

Abstract:  In the present paper, we investigate derivations and local derivations on low-dimensional Jordan algebras. We give a description of derivations and local derivations on four-dimensional 2-step nilpotent noncommutative Jordan algebras.

Keywords: Algebra; Jordan algebra; basis of an algebra; derivation; local derivation; dimension of an algebra.

Author: Dusanova U.  (V.I.Romanovskiy Institute of Mathematics)

Abstract: The work is devoted to the study of initial-boundary value and inverse problems for a mixed- type equation with the Caputo operator and to the indication of conditions for the existence and uniqueness of a solution. The solution is constructed as the sum of an orthonormal series over the eigenfunctions of the elliptic part of the equation. For the forward problem, the condition u(x, −α) = φ(x) is taken instead of the Cauchy condition. The right-hand side of the equation has the form f(x, t). The inverse problem of determining the right-hand side of the equation is also considered. When solving the inverse problem, it is assumed that the unknown function depends only on x and an additional condition is taken in the form u(x, β) = ψ(x).

Keywords: Mixed type equation; the Caputo derivatives; forward and inverse problems; Fourier method.

Author: Gaybullaev R.(National University of Uzbekistan), Urazmatov G.(National University of Uzbekistan)

Abstract: The present paper is devoted to the study of nilpotent n-Lie algebras with maximal rank and to proving the uniqueness of maximal solvable n-Lie algebras with a hyponilpotent ideal of maximal rank. Furthermore, a 16-dimensional maximal solvable 3-Lie algebra with a given maximal hyponilpotent ideal is constructed using the maximal torus.

Keywords: n-Lie algebra; nilpotent ideal; hyponilpotent ideal; solvable n-Lie algebra; derivation; maximal rank; maximal torus.

Author:Kazakova M.(V.I.Romanovskiy Institute of Mathematics), Ruzimurodova D.(V.I.Romanovskiy Institute of Mathematics)

Abstract:  It is shown that trajectories of a linear system different from a equilibrium state those, do not lie in the hyperplane, are in general a state. It is proved that the curvature and torsion coefficients of such trajectories keep their sign when the order of the system is equal to 2 and 3.

Keywords: Linear system; trajectory; regular lines; curvature; torsion; regularity.

Author: Qudaybergenov A.(National University of Uzbekistan)

Abstract:  The problem of finding the temperature on the outer boundary of a straight circular cylindrical domain under known conditions on the inner boundary is considered. The existence and uniqueness of the solution to this problem have been proved.

Keywords: Elliptic equations; heat conduction equation; Cauchy problem; existence; uniqueness.

Author: Hitesh K.(Maharaja Agarsen University)

Abstract:  The main purpose of this paper is to study the results of 2-absorbing ideals and weakly 2-absorbing ideals in a commutative Γ-semiring, which is a generalization of prime ideals of a commutative Γ-semiring. We prove a characterizations theorem of 2-absorbing ideals in a commutative Γ-semiring. Finally, we give the relationship between the 2-absorbing (resp., weakly 2-absorbing) ideals of R and the 2-absorbing (resp., weakly 2-absorbing) ideals of R/J.

Keywords: k-ideals; Q-ideal; 2-absorbing ideals; weakly 2-absorbing ideals.

Author: Nuraliyeva N.  (V.I.Romanovskiy Institute of Mathematics)

Abstract:  The non-local problem, Dρt u(t) + Au(t) = f(t) (0 < ρ < 1, 0 < t < T), αu(0) + βu(T) +γu(η)dη = φ, (α, β and γ are constants), in an arbitrary separable Hilbert space H with
the strongly positive selfadjoint operator A, is considered. The operator Dρt is the Caputo derivative. Existence and uniqueness theorems are established for the solutions of the problem. Criteria ensuring the uniqueness of the solution are identified. The influence of the constants α, β and γ on the existence and uniqueness of a solution to problems is investigated.

Keywords: Fourier method; subdiffusion equation, three-parameter nonlocal problems; Caputo derivatives; integral condition.

Author: Ruzieva D. (National University of Uzbekistan)

Abstract:  We consider random fields with values in some Banach spaces and satisfying some dependence conditions. Namely we assume that the random field can be represented as a functional of another random field of independent identically distributed random variables. For such random fields we prove strong law of large numbers.

Keywords: A strong law of large numbers; a random field; Banach space; a moment inequality.

Author: Yusupbaeva Kh.(V.I.Romanovskiy Institute of Mathematics), Khakimov O. (V.I.Romanovskiy Institute of Mathematics)

Abstract: The present research focuses on analyzing the dynamics of the Ising mapping over the field of 2-adic numbers. First, we examine the set of all “bad points” of the mapping. And we will find the necessary and sufficient conditions for the finiteness of this set. Next, we detail the set of all 2-periodic points of the Ising mapping, which are crucial in our investigations. We show that the positive trajectory of any initial point either converges or forms a two-period cycle. This demonstrates that each 2-periodic point of the Ising mapping exhibits an attractive characteristic.

Keywords: p-adic numbers; Ising mapping; positive trajectory; dynamical system.

Author: Allakov I., Imamov O.

Abstract:  The work proves a lower estimate for the quantity of a natural number represented as a sum of five squared prime numbers from an arithmetic progression, more precisely, it proves a lower estimate for a number solving an equation of prime numbers from an arithmetic progression pj ≡ tj (mod d), (j = 1, . . . , 5). Here a1, …, a5, N−are integers that satisfy the conditions a1 · · · a5 ̸= 0, (a1, …, a5) = 1 and 1 < d < N δ 420 – any integer. The obtained result complements the corresponding results of Hua Loo-Keng,
Ming-Chit Liu and Kai-Man Tsang, Tian Ze Wang.

Keywords: Diophantine equation; congruent solution; positive solution; exceptional zero; Dirichlet L−function; Legendre symbol; minor arc; major arc; Hua Loo-Keng method; ternary Goldbach problem.

Author: Ibragimov G.(V.I.Romanovskiy Institute of Mathematics), Gulomov S. (Andijan State University), Qushakov X.(Andijan State University)

Abstract:  In the present paper, we study a differential game of evasion described by infinite systems of differential equations in the Hilbert space l2. This game involves many pursuers and one evader, where the control functions of players are subject to integral constraints. We say that evasion is possible in the game if the states of all systems do not coincide with the origin of the space l2. The main goal of the study is to develop a strategy for the evader to avoid capture by pursuers. We derive a sufficient condition for evasion from any initial state and construct an evasion strategy for the evader.

Keywords: Differential game; infinite system; integral constraint; evasion problem; Hilbert space.

Author: Oripov Sh. (Fergana State University)

Abstract:  The article investigates the Cauchy problem for a fourth-order pseudo-hyperbolic equation with singular coefficients. The problem is solved using the Riemann’s method. The Erd ́elyi-Kober transmutation operator is applied to construct the Riemann’s function. The Riemann’s function for this equation is derived and expressed in terms of hypergeometric functions. Based on the Riemann’s function, an explicit formula for the solution to the Cauchy problem is found.

Keywords: Cauchy problem; pseudo-hyperbolic equation; singular coefficient; transmutation operator; Riemann’s function.

Author: Sharipov A.(National University of Uzbekistan), Topvoldiyev F. (Fergana State University)

Abstract: This article studies the problem of restoration isometric surface by sections, by their conditional curvature. It is known that the concept of isometry by sections differs from isometry, and isometric surfaces may not be isometric by sections and vice versa. Therefore, solving the problem of restoration, surface isometric by sections requires a special approach. The article introduces the concept of convergence of polyhedron to a surface by sections and, based on this approach, solves the restoring problem.

Keywords: Convex polyhedron; convex surface; convergence by polyhedrons to the surface; convergence by sections; conditional curvature.

Author: Ergashev T. (National Research University “TIIIMSKh”)

Abstract: Thanks to great successes in the study of the theory of the hypergeometric Gaussian function (with single variable), the corresponding theory for functions of two variables has developed significantly, for which have been obtained integral representations, reduction formulas, transformations and systems of partial differential equations, corresponding to 14 complete and 20 confluent hypergeometric second order functions from Horn’s list. Currently, 600 (of which 205 are complete and 395 are confluent) hypergeometric functions of three second-order variables are known. In the works of A. Hasanov and M. Ruzhansky, for the 205 complete hypergeometric functions of three variables the systems of partial differential equations were compiled and integral representations were established. In addition, in the origin linearly independent solutions (if such solutions exist) of some systems of differential equations have been found. This work is devoted to the formulation of systems of partial differential equations satisfied by 395 confluent hypergeometric functions of three variables.

Keywords: Gauss hypergeometric function; Appel functions; Horn functions; Humbert functions; confluent functions; hypergeometric functions in three variables; systems of partial differential equations of hypergeometric type.

Author: Aktamov F. (Chirchik State Pedagogical University)

Abstract: In the present paper in a partially ordered linear space we consider a base that generates a convex Hausdorff topology in it. Further using elements of this base we obtain a metric and a norm on the given space. We show the achieved space is a space with an order unit. The way obtaining a partially ordered topological linear space gives the possibility to investigate the spaces with an order unit with a new point of view.

Keywords: Metric; norm; order unit; A-subspace.

Author: Arzikulov F.(V.I.Romanovskiy Institute of Mathematics), Khakimov U.(Andijan State University)

Abstract: In the present paper, we introduce and study counterparts of Rickart involutive algebras, i.e., almost inner Rickart algebras. We prove that a nilpotent associative algebra, which has no nilpotent elements with a nonzero square root, is an almost inner Rickart algebra. A nilpotent associative algebra, which has no nilpotent elements with a square root b such that b^3 ̸= 0, is not an almost inner Rickart algebra if there exists a nonzero element a such that a^2 ̸= 0. As a main result of the paper, we describe a finite-dimensional almost inner Rickart algebra A over a field F, isomorphic to Fn+ ̇ N , n = 1, 2, with a nilradical N . Also, we classify finite-dimensional almost inner Rickart algebras over the real or complex numbers with a nonzero nilradical N .

Keywords: Associative algebra; nilradical; nilpotent associative algebra; nilpotent element; idempotent

Author: Kuchkarova S. (National University of Uzbekistan)

Abstract:  The present research is studied differential games described by an infinite system of 2-block differential equations in the Hilbert space l2. We find the guaranteed pursuit time for the differential game.

Keywords: Pursuer; evader; geometric constraint; guaranteed pursuit time.

Author: Rajabov S. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  The present article deals with dynamics of a non-Volterra quadratic stochastic operator defined on the two-dimensional simplex. We showed that it has at least four fixed points and we found the types of all fixed points. Also we proved that for any values the parameters the trajectory of an arbitrary initial point approaches to a fixed point.

Keywords: quadratic stochastic operator; Volterra operator; non-Volterra operator; regularity.

Author: Rahmonov A. (V.I.Romanovskiy Institute of Mathematics)

Abstract: In this paper, we investigate backward and inverse problems for time-fractional integro-differential diffusion equation. We prove a local and global in time existence and uniqueness theorems for the inverse problem of memory reconstruction for abstract fractional integro-differential equation in Banach space.

Keywords: Integro-differential diffusion equation; Gerasimov-Caputo derivative; backward problem; Banach space; Mittag-Leffler function.

Author: Zaitov A.(V.I.Romanovskiy Institute of Mathematics), Eshtemirova Sh. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  In the category of Tychonoff spaces and their continuous maps, we offer a new type of extension of the functor of idempotent probability measure acting in the category of compact Hausdorff spaces and their continuous maps. Note that the obtained extension is an expected one. In the paper (in the T ych sense) normality properties with slight modifications are established.

Keywords: Tychonoff space; idempotent measure; normal functor, k-covering map.

Author: Dzhamalov S.(V.I.Romanovskiy Institute of Mathematics), Khudoykulov Sh.(V.I.Romanovskiy Institute of Mathematics)

Abstract:  In this paper, the correctness of a linear two-point inverse problem for a three-dimensional wave equation has been studied. Using the methods of a priori estimates, Galerkin’s, sequences of approximations, and contracting mappings, the authors prove the unique solvability of a generalized solution of a linear two-point
inverse problem for a three-dimensional wave equation.

Keywords: Three-dimensional wave equation; linear two-point inverse problem; unique solvability; methods of a priori estimates, Galerkin’s, sequences of approximations, contracting mappings.

Author: Ibragimov M.(Karakalpak State University), Arziev A. (V.I.Romanovskiy Institute of Mathematics)

Abstract: In this paper, based on the generality of the physical meanings of GL-projectors on pre-symmetric space and geometric Peirce projectors on facially symmetric space, we define rigidly collinearity of geometric tripotents in the dual space of the facially symmetric space. As a result of studying the properties of geometric tripotents, we have determined the conditions under which rigidly collinearity coincides with collinearity. Additionally, the relationship between geometric concepts belonging to certain classes of geometric tripotents has been revealed.

Keywords: Facially symmetric spaces; regularity; collinearity and rigidly collinearity of geometric tripotents.

Author: Mirsaburov   M.(Termez State University), Mamatmuminov D. (Termez State University)

Abstract:  In the present paper, the theorems on the existence and uniqueness of the solution to the problem with Frankl-type conditions on the characteristics and degeneration line, as well as general gluing conditions for a mixed-type equation with singular coefficients, are proved. To achieve this, nonstandard singular integral equations are solved. The general gluing condition is considered on the degeneration line.

Keywords: Boundary value problem; singular coefficient; Frankl-type condition; mixed type equation; singular integral equation; regularization; Wiener-Hopf integral equation; Fredholm integral equation.

Author: Oripov D. (Fergana State University)

Abstract: In this paper, an initial-boundary value problem has been formulated and studied for a degenerate partial differential equation of high even order in a rectangle. By applying the Fourier method, a spectral problem was obtained for an ordinary differential equation. Using Green’s function method, this latter problem was equivalently reduced to a Fredholm integral equation of the second kind with a symmetric kernel, implying the existence of eigenvalues and eigenfunction systems of the spectral problem. Using the obtained integral equation and Mercer’s theorem, the uniform convergence of certain bilinear series dependent on the found eigenfunctions has been proved. The solution of the studied problem is expressed as a Fourier series sum over the eigenfunction system of the spectral problem. The uniqueness of the solution to the problem was established using the method of energy integrals. An estimate for the solution of the problem was obtained.

Keywords: Degenerate equation; initial-boundary value problem; method of separation of variables; spectral problem; method of Green functions; integral equation; Fourier series.

Author: Abdukodirov A.(Fergana State University), Tulkinboev T.(Fergana State University)

Abstract:  Although many works have been dedicated to the study of boundary value problems for sixth-order partial differential equations, the issues of complete classification of linear sixth-order partial differential equations and determination of their canonical forms remain unexplored. This work addresses the issues of classification and canonical form transformation of sixth-order linear partial differential equations with multiple and complex characteristics, where the coefficients of the equation are constant. If a sixth-order partial differential equation has multiple and complex characteristics, it is easy to see that it consists of nine cases. Therefore, in this article, the issues of classification and determination of the canonical forms of the equations are investigated in nine cases, respectively. The article first provides the general formula for the coefficients of the new sixth-order equation, obtained after the transformation of variables, and formulates three lemmas that play an important role in finding the canonical form of the equation. Additionally, a theorem is proven regarding the canonical forms of sixth-order partial differential equations with multiple and complex characteristics, and the corresponding canonical forms of sixth-order equations are determined.

Keywords: Classification of differential equations; canonical form of differential equations; multiple characteristics; equations of characteristics.

Author: Akhmedov O. (Fergana State University)

Abstract:  This paper investigates p-adic quasi-Gibbs measures for the four-state SOS model on a Cayley tree of order two. We find the conditions that ensure the existence of at least three translation-invariant (TI) and at least two $G^(2)_2$-periodic p-adic quasi-Gibbs measures. By analyzing the boundedness of obtained p-adic quasi-Gibbs measures, the existence of a phase transition is established.

Keywords: Cayley tree; configuration; p-adic quasi-Gibbs measure; p-adic SOS model; periodic p-adic quasi-Gibbs measure; p-adic numbers.

Author: Eshbekov R. (Samarkand State University)

Abstract:  In this paper, the inverse spectral problem method is used to integrate a Hirota-type equation with finite density and additional terms in the class of periodic infinite-gap functions. The evolution of the spectral data of the periodic Dirac operator is deduced, and the coefficient of the Dirac operator is a solution to a Hirota-type equation with finite density and additional terms. A simple algorithm for deriving the Dubrovin system of differential equations is proposed. The solvability of the Cauchy problem for a Dubrovin infinite system of differential equations in the class of six times continuously differentiable periodic infinite-gap functions is proven. It is proven that there is a global solution to the Cauchy problem for a Hirota-type equation with finite density and additional terms for sufficiently smooth initial conditions.

Keywords: Nonlinear Hirota-type equation with finite density and additional terms; Dirac operator; spectral data; Dubrovin system of differential equations; trace formulas.

Author: Karimov E.(V.I.Romanovskiy Institute of Mathematics), Mirzaeva M. (Fergana State University)

Abstract: In the present investigation, we targeted an inverse problem of finding an unknown time moment for the mixed wave-diffusion equation involving the Riemann-Liuoville fractional derivative. We have used a solution of the first boundary problem for the time-fractional diffusion-wave equation. The explicit form of the solution was obtained using the Green function, which was constructed using the Wright-type function. Certain properties of this function were utilized effectively to prove the solvability of the proposed inverse problem.

Keywords: Inverse problem; Riemann-Liouville fractional derivative; Diffusion-wave equation; Wright-type function.

Author: Hasanov A.(V.I.Romanovskiy Institute of Mathematics), Nematjonov F.(Fergana State University)

Abstract: The article discusses the study of Mittag-Leffler-type functions of two variables, denoted as $D_1 (x, y), …., D_5 (x, y)$ modeled after the Appell hypergeometric functions. Horn’s method was employed to identify the convergence region of $D_2 (x, y)$. Additionally, integral representations for the generalized hypergeometric function $D_2 (x, y)$, including Euler-type integrals, were derived. Using this integral representation, an upper estimate has been established. One- and two-dimensional Laplace transforms of the functions were established, and a related system of partial differential equations was formulated based on $D_2(x, y)$.

Keywords: Bivariate Mittag-Leffler type function; Integral representation; Integral transforms; Appell hypergeometric function.

Author: Khusainov D.(Taras Shevchenko National University of Kyiv), Shatyrko A.(Taras Shevchenko National University of Kyiv), Mustafaeva R.(Karakalpak State University)

Abstract: A linear differential-difference equation of neutral type is considered in the article. Conditions for the asymptotic stability of the zero solution are obtained. The research was performed using the second Lyapunov method with the functional of the Lyapunov-Krasovskii integral type.

Keywords: Differential-difference equations of neutral type; degree of instability; second Lyapunov method; Lyapunov-Krasovskii functional.

Author: Nurbayev A.(Gulistan State University)

Abstract: The first and second fundamental forms of the surface in the four-dimensional Galilean space are defined in a specially selected curvilinear coordinate system. It is shown that the full curvature of the surface can not be expressed only by the coefficients of the first fundamental form. The part of the total curvature of the surface that does not depend on the coefficients of the first fundamental form is called its defect. In the four-dimensional Galilean space, from the set of surfaces projected onto the hyperplane with one value, the surfaces with a total curvature defect equal to zero are divided into classes. The geometric properties of the class of separated surfaces are shown and their characteristic features are defined.

Keywords: Fundamental forms; curvilinear coordinate; Galilean space; special hyperplane; vector product; curvature indicatrix; the total curvatures; mean curvatures; principle curvatures.

Author: Rakhmatova N. (V.I.Romanovskiy Institute of Mathematics)

Abstract: A unique solvability of a direct problem for a time-fractional wave equation with the Caputo and Bessel operators is investigated. Using the spectral expansion method, we present explicit solutions to formulated problems involving the Mittag-Leffler, and first and second kind Bessel functions.

Keywords: Time-fractional wave equation; Caputo fractional derivative; Bessel operator; Fourier series; Mittag-Leffler function.

Author: Turdiev Kh. (Fergana State University)

Abstract: This article presents two nonlocal boundary value problems for an equation consisting of a combination of two telegraph equations in a mixed pentagonal domain and investigates its unique solution. Initially, the solutions of the Goursat and Cauchy problems in rectangular and triangular domains are written down. Then, using the transmitting and nonlocal conditions, the second-kind Volterra integral equation was derived concerning the unknown function. Using the theory of integral equations, the uniqueness of the solution to the stated nonlocal problems has been proven.

Keywords: Telegraph equations; nonlocal problems; mixed domain; integral equation; Cauchy problem, Goursat problem, Prabhakar fractional order derivative; Mittag-Leffler function.

Author: Allakov I.(Termez State University), Muzrapova N.(Termez State University)

Abstract: The work examines the solvability of equation $b = a_1p_1 + a_2p_2 + a_3p_2^3^
in prime numbers $p_1, p_2, p_3$. Here $a_1, a_2, a_3$, b are integers. If the coefficients $a_1, a_2, a_3$ satisfy some additional conditions, then the considered equation is solvable in prime numbers, and a lower estimate of the solution for $b$, $X/2 < b ≤ X has been proved.

Keywords: Dirichlet character; prime numbers; basic intervals; additional intervals; estimation; equation; solvability; nontrivial zeros; exceptional zero.

Author: Akhmedova D.(Andijan State University), Muminov U. (Fergana State University)

Abstract: This paper provides a complete description of the dynamics of trajectories, as well as, using bigraphs, the location of fixed points of Lotka-Volterra operators in a four-dimensional simplex, and a set of limit points is found. And the dynamics of these points is explained. Also, under the influence of the Lotka-Volterra operator, the identification of signature signs, the division of the simplex into polyhedra corresponding to the signature signs, and the route, the route of the trajectory through the polyhedra, are determined.

Keywords: Complete oriented bigraphs; signature and corresponding partitions of the simplex into polyhedra; route of the trajectory along polyhedra.

Author: Ibragimov G.(V.I.Romanovskiy Institute of Mathematics), Kurbanov A.(V.I.Romanovskiy Institute of Mathematics), Tilavov A.(V.I.Romanovskiy Institute of Mathematics) 

Abstract: This article considers a differential game of evasion involving two pursuers $x_1, x_2$ and one evader y, all moving with simple motions. The control sets of the pursuers are circles of radius 1, with centers at the origin, while the control set of the evader is a sector of a circle with radius $σ > 1.$ We will say that evasion is possible in the game if $x_i(t) ̸= y(t)$ for all $t ≥ 0$ and $i = 1, 2.$ A new strategy is constructed for the evader. This strategy guarantees the possibility of evasion for arbitrary initial positions of the players.

Keywords: Differential game; evasion; closing time; strategy; control.

Author: Kadirkulov B.(V.I.Romanovskiy Institute of Mathematics), Begimqulov F. (Tashkent Perfect University)

Abstract: In this research, a nonlocal Bitsadze-Samarskii type problem is investigated for a fractional analogue of the Laplace equation in a vertical half-strip $Ω = {(x, y) : 0 < x < 1, y > 0}.$ This problem relates the value of the unknown function on the right boundary to the value of the function at an interior point of the domain. The uniqueness and the existence of a solution to the problem is established by the the spectral method. In this paper we also studied the spectral properties of the Bitsadze-Samarskii type problem for an ordinary differential equation of the second order, found the eigenvalues, as well as the corresponding eigenfunctions, proved their completeness and basis property, and also investigated the adjoint problem.

Keywords: Bitsadze-Samarskii problem; fractional order equation; derivative in the sense of Caputo; sequential derivative; fractional analogue of the Laplace equation; completeness; Riesz basis.

Author: Abdukahorova Z. (Namangan State University)

Abstract:  In this paper we consider a p-adic Ising model on the Cayley tree of order two. We show the existence of $H_A$-weakly periodic generalized p-adic Gibbs measures for the p-adic Ising model on the Cayley tree of order two.

Keywords: Caylee tree; p-adic numbers; p-adic Ising model; $H_A$ weakly-periodic Gibbs measure.

Author: Akbarov A.(Andijan State University)

Abstract:  In the present paper, the simple motion pursuit-evasion problems with constraints given in the form of hyperbolic functions on the rates of controls are studied. For solving the pursuit problem, the strategy of parallel approach (for brevity, Π-strategy) is suggested and sufficient conditions guranteeing the pursuit are determined. Here, an attainability domain of evader and its important properties are presented. For solving the evasion problem, escape conditions are found.

Keywords: Differential game; Pursuit; Evasion; Gr ̈onwall’s inequality; Pursuer; Evader; Strategy; Attainability domain.

Author: Baratov B. (Karshi State University)

Abstract: In this article, we consider a class of separable cubic stochastic operators defined on a finite-dimensional simplex. Namely, cubic stochastic operators, which have the form as a product of three linear operators defined on a simplex. For one separable cubic stochastic operator defined on a two-dimensional simplex, it is shown that the vertices of the simplex are fixed points. It is proved that for an arbitrary initial point the trajectory of such a separable cubic stochastic operator is convergent.

Keywords: Cubic stochastic operator; separable cubic stochastic operator; trajectory.

Author: Boltaev N. (Tashkent State Transport University)

Abstract:  In the present paper, optimal quadrature formula for approximate evaluation of Fourier coefficients is constructed for functions of the space. At the same time, explicit formulas for the optimal quadrature formulas are obtained. The obtained formula is exact for the trigonometric functions sin ωx, cos ωx and for polynomials of degree m − 3.

Keywords: Hilbert space; extremal function; generalized function; optimal quadrature formula.

Author: Dekhkonov F.

Abstract: In this paper, we consider a problem of boundary control of the heat transfer process. The considered problem is reduced to a system of Volterra integral equations of the first type by the Fourier method. The existence of a solution of this system of integral equations is proved by means of the Laplace transform.

Keywords: Heat exchange process; system of integral equations; initial boundary value problem; Laplace transform.

Author: Kurbanov Kh.(Academy of the Armed Forces of the Republic of Uzbekistan)

Abstract: In this paper, for a given group (G, X, α) of topological transformations, the group (OS(G, X), OS(X), OS(α)) topological transformations . The group OS(G, X) is endowed with a topology such that the induced topology from OS(G, X) to the group G coincides with the original topology on G. It is shown below that each mapping taken from the group OS(G, X) is open.

Keywords: group of topological transformations; semi-additive functional; open display

Author:  Samatov B. (V.I.Romanovskiy Institute of Mathematics), Uralova S. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  In this article, we study pursuit-evasion problems for one class of linear differential games with Langenhop type constraints. To solve the pursuit problem, we propose a strategy that allows players to approach each other in parallel. For the problem of evasion, the lower bound of the approach function of players is found.

Keywords: Differential game; pursuit; evasion; Langenhop constraint; pursuer; evader; strategy.

Author:  Formanov Sh. (National University of Uzbekistan), Sirojitdinov A. (National University of Uzbekistan)

Abstract:  In this paper we present another proof of Essen’s theorem on the form of the main term in the asymptotic expansion in the central limit theorem for the distributions of the sum of independent identically distributed random variables. This proof uses an auxiliary assertion (lemma) about the estimates of the exponential function in the complex domain.

Keywords: Truncated random variables; distribution function; Essen theorem.

Author: Kahhorov A.(Academic Lyceum of TSTU named after I.A. Karimova), Khusanov Dj. (Jizzakh State University)

Abstract: The paper substantiates the use of Lyapunov-Krasovsky functionals with a diagonal component in the study of the stability of non-autonomous systems with nonlinear separability and variable delay.

Keywords: Stability; method of Lyapunov-Krasovsky functionals; nonlinear systems with delay; non-autonomous equations with variable delay.

Author: Makhammadaliev M.(Namangan State University), Kholmirzaev J.(Namangan State University) 

Abstract: In this work the existence and uniqueness conditions of weakly periodic Gibbs measures for a HC model with a countable set of spin values for a normal divisor of index four on a Cayley tree of order two and three are found.

Keywords: Cayley tree; configuration; Hard-Core model; Gibbs measure; weakly periodic Gibbs measure.

Author: Matchanova A. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  This work is devoted to the unique solvability of a local problem for the parabolic-hyperbolic type equation of the third order involving Caputo derivatives. Considering problem involves third boundary condition on the parabolic domain and discontinuous gluing condition on the line y = 0. The existence of the solution is proved using the theory of integral equations of the Volterra type.

Keywords: Caputo fractional operator; parabolic-hyperbolic type; third-order equation; unique solvability.

Author:  Mirsaburova D. 

Abstract: For the Gellerstedt equation with singular coefficient, the uniqueness of the solution of the problem with Bitsadze-Samarsky condition on boundary and parallel internal characteristic of mixed field and Frankl condition on degeneracy line is proved using the extremum principle and the existence of the solution of the problem is proved using the theory of integral equation.

Keywords: Gellerstedt equation; Bitsadze-Samarsky condition on parallel characteristics; Frankl condition; extremum principle; non-standard singular integral equation; Wiener-Hopf integral equation; Fourier integral; index.

Author:  Mustapokulov Kh.(National University of Uzbekistan), Mamadaliev N.(National University of Uzbekistan), Abdualimova G. (Andijan State University)

Abstract:  In this article, the issue of strong and weak invariance of invariant multivalued reflection with respect to delayed oscillatory systems is considered. Sufficient conditions for the given multivalued reflection to be strongly and weakly invariant are given.

Keywords: Integral constraint; geometric constraint; multivalued display; control; weak invariance; strong invariance; concentrated parameters.

Author:  Safarov J. (V.I.Romanovskiy Institute of Mathematics)

Abstract: In this paper, an inverse problem for an inhomogeneous integro-differential equation of hyperbolic type is studied. The right issue has been studied before. Then, using the value given at x = 0 of the generalized solution of the direct problem, the equation of the integral kernel was obtained, that is, the problem was brought to the system of integral equations of the Volterra type, and the principle of reduced reflection was applied. It was proved that the unique solution of the inverse problem exists.

Keywords: Hyperbolic equation; integro-differential equation; delta function; integral kernel; inverse problem.

Author: Shadimetov Kh.(Tashkent State Transport University),
Davlatova F.(V.I.Romanovskiy Institute of Mathematics)

Abstract: In this paper, the squared norm of the error functional was calculated using the Riss element in the Sobolev quotient space and using a discrete analog of the differential operator an analytic representation of the coefficients that give a minimum to the norm of the functional error.

Keywords: Quadrature formulas; extremal function; error functional; Sobolev space.

Author:  Urinov A.(Fergana State University), Usmonov D.(Fergana State University)

Abstract:  In this paper, we study the Cauchy problem for an inhomogeneous ordinary differential equation containing a fractional differential operator in the sense of Caputo with a Bessel function in the kernel. The considered problem is equivalently reduced to a Volterra integral equation of the second kind. The solution of the integral equation is found by the method of successive approximations. It has been proved that the obtained
solution really satisfies the conditions of the problem. An estimate for the solution is obtained. When deriving a formula for solution to the problem, a new special function was derived, which in a particular case follows the Mittag-Leffler function. The properties of the introduced function are studied, in particular, differentiation formulas for it are written out.

Keywords: Bessel function; integro–differential operator; ordinary differential equation; the Cauchy problem.

Author: Usmonov D.A. (Fergana State University)

Abstract: You shouldn’t use formulas and citations in the abstract. In this article, for an equation of mixed type of the fourth order, degenerating inside and on the border of the domain, in a rectangular domain, a non-local initial boundary value problem has been formulated and investigated. Applying the method of separation variables to the considered problem a spectral problem for an ordinary differential equation has been obtained. The existence of eigenvalues and the system of eigenfunctions of the spectral problem ware proved. A theorem is proved for expanding a given function into a uniformly convergent series with respect to the system of eigenfunctions. The solution of the considered problem is written as the sum of the Fourier series with respect to the system of eigenfunctions of spectral problem. An estimate for solution of the problem is obtained, from which follows its continuous dependence on the given functions.

Keywords: degenerate equations of mixed type; boundary problem; spectral problem, Green’s function, integral equation, Fourier series, separation of variables method.

Author: Aliyev A. (V.I.Romanovskiy Institute of Mathematics), Begmatov A. (National University of Uzbekistan)

Abstract: In this work, we study asymptotical behavior of renormalizations of circle homeomorphisms with two break points not lying on the same orbit and with irrational rotation number. We show that the renormalizations of such homeomorphisms are approximated by M ̈obius transformations in $C^2$-norm.

Keywords: Rotation number; break point; dynamic partitioning; renormalization.

Author: Jafarov S. (Faculty of EducationMus Alparslan University), (Institute of Mathematics and Mechanics, Azerbaijan)

Abstract: We investigate the approximation of the functions by means of the trigonometric Fourier series in subspace of weighted generalized grand Lebesgue spaces.

Keywords: Trigonometric approximation; weighted grand Lebesgue spaces; Muckenhoupt weight; weighted L(α, p), ω) class; modulus of continuity.

Author: Khujakulov J. (V.I.Romanovskiy Institute of Mathematics), (Chirchik state pedagogical University)

Abstract:  This work is devoted to study an initial boundary value problem for a time-fractional equation involving the regularized Prabhakar fractional derivative on a star graph. Using the method of separation of variables we find exact solution of the investigated problems in the form of Fourier series which attended the bi-variate Mittag-Leffler type function $E_2(x, y)$.

Keywords: Prabhakar derivative; bi-variate Mittag-Leffler function; star metric graph; initial boundary value problem.

Author:  Mamadaliyev B.(Fergana State University)

Abstract: In the article, two-dimensional surfaces in a five-dimensional pseudo-Euclidean space of index two are considered. Geometry on two-dimensional planes of this space can be of three types, Euclidean, Minkowski, and Galilean. Therefore, two-dimensional surfaces are also divided into three types according to the geometry on the tangent plane. A special class of two-dimensional surfaces given by a vector equation is considered. For the chosen class of a two-dimensional surface, an analog of the normal space called by the authors the dual space, is constructed. Using the dual space, the geometry of a two-dimensional surface is studied, reduced to a Euclidean or pseudo-Euclidean surface of a three-dimensional space. Conditions are revealed and theorems are proved on the existence of a surface that does not lie in a four-dimensional hyperplane and has tangent planes with one internal geometry.

Keywords: Manifold; pseudo-Riemannian manifold; multidimensional geometry; dual space; pseudo-Riemannian space; sphere of real; imaginary; and zero radius; isotropic cone.

Author: Qudayberganov A.(National University of Uzbekistan), Sharipova S.(National University of Uzbekistan) 

Abstract:  In this paper, we prove the uniqueness of the solution of the Cauchy problem for an elliptic equation with discontinuous coefficients in the class of generalized functions.

Keywords: Elliptic equations; the Cauchy problem; uniqueness; auxiliary equation.

Author: Qushaqov Kh.(Andijan State University)

Abstract: In the present paper, we are interested in finding a time to complete an evasion differential game given by an infinite system of ternary differential equations. The game is considered in Hilbert space $l_2$. The control parameters of pursuer and evader are subject to geometric constraints. Our purpose is to construct strategies for the evader to avoid being captured in the game. The game is completed when the state of the system is brought to the origin of Hilbert space $l_2$. We find equations for time to guaranteed evasion for the evader.

Keywords: Differential game; evader; control; strategy; infinite system of differential equations; geometric constraint.

Author: Taifour S. (Oran – Maurice AUDIN), Zoubir H. (Oran – Maurice AUDIN), Medjati R. (Oran – Maurice AUDIN)

Abstract: We study translation surfaces in the 3-dimensional Lorentz Heisenberg space which are of coordinate finite type with respect to the first fundamental form, i.e. their position vector r satisfies the relation $∆r = (∆r1, ∆r2, ∆r3) = (λ1r1, λ2r2, λ3r3), λi ∈ R, i = 1, 2, 3. We show that the minimal translation surface is the only translation surface which possesses the above property.

Keywords: Lorentz-Heisenberg space; Laplace operator; translation surface; mean curvature.

Author: Zhabborov N. (Belorussian-Uzbek joint intersectoral institute of applied technical qualifications in Tashkent), Husenov B.(Bukhara State University)

Abstract: In this paper, we prove an analogue of Fatou’s theorem on radial for A(z)–analytic functions. The boundary uniqueness theorem for bounded A(z)–analytic functions is also proved.

Keywords: Beltrami equation; radial limits; set of positive measure; A(z)–lemniscate.

Author: Abduolimova G. (Andijan State University)

Abstract: In this paper, the main attention is paid to the study of the game problem of control of pencils of trajectories, which is described by a system of differential-difference equations of neutral type under geometric and integral restrictions on the players’ controls. Analogues of the first method of pursuit and the method of pursuit in direction are developed and applied for solving game problems of controlling bundles of trajectories. In the process of studying this problem, new sufficient conditions were obtained for the solvability of the pursuit problem.

Keywords: differential game; pursuit task; differential-difference equations of neutral type; terminal set; bundle of trajectories; pursuer; evader; control.

Author:  Ibragimov M.(Karakalpak State University)

Abstract: In this paper the geometrical properties of the lattice $L_ω$ of geometrical tripotents which are not larger than a fixed geometrical tripotent $ω$ of the dual space $Z∗$ of the neutral strongly facially symmetric space $Z$ are investigated. More precisely, it is shown that if the symmetric faces corresponding to geometrical tripotens from $L_ω$ are strongly splitting, then $L_ω$ is a Boolean algebra.

Keywords: Facially symmetric spaces; strongly split faces; geometrical tripotent.

Author:  Irgashev B.(Namangan Engineering-Construction Institute)

Abstract: In the article, in a rectangular domain, the Fourier method is used to study the unique solvability of an initial-boundary value problem for an equation of high even order with a fractional derivative, which has degeneration in two variables.

Keywords: Equation; high order; boundary value problem; fractional derivative; eigenvalue; eigenfunction; Kilbas-Saigo function; series; convergence; existence; uniqueness

Author:  Ismoilov A. (Fergana State University)

Abstract: This article considers the Goursat problem for the nonhomogeneous generalized Euler-Poisson-Darboux equation in the characteristic triangle. The solution of the problem is found using the Riemann function. It is proved that it satisfies the equation and the stated conditions are satisfied.

Keywords: Goursat problem; nonhomogeneous generalized Euler-Poisson-Darboux equation; Riemann method; Riemann-Hadamard function.

Author:Ruziev M.(V.I.Romanovskiy Institute of Mathematics), Rakhimova G.(Fergana State University) 

Abstract: In this paper we study a boundary value problem for a differential equation with partial fractional derivative in an unbounded domain. We prove the unique solvability of the considered problem.

Keywords: Boundary value-problem; operator; fractional derivative; differential equation.

Author: Turakulov Kh.  (V.I.Romanovskiy Institute of Mathematics)

Abstract:  In this paper, it is investigated a linear inverse problem with a periodic boundary condition for the three-dimensional Tricomi equation in an unbounded parallelepiped domain. In the considered domain, the uniqueness and existence of the generalized solution of this problem is studied using the methods of “ε-regularization” , a priori estimates, successive approximation and Fourier transform.

Keywords: Three-dimensional Tricomi equations; linear inverse problem with periodic boundary condition; problem correctness; methods “ε -regularization” ; a priori estimates; sequence of approximation; Fourier transforms.

Author: Tursunov F.(Samarkand State University), Shodiev D. (Samarkand State University)

Abstract:  The article studies the problem of continuation of the solution and the stability estimate of the Cauchy problem for the biharmonic equation in a domain G by its known values on the smooth part S of the boundary ∂G . The considered problem belongs to the problems of mathematical physics, in which there is no continuous dependence of solutions on the initial condition. It is assumed that the solution of the problem exists and also is continuously differentiable in a closed domain with exactly given Cauchy condition. For this case, an explicit formula for the continuation of the solution is established. Stability estimates for the solution of the Cauchy problem in the classical sense are obtained.

Keywords: Cauchy problem; ill-posed problems; biharmonic equation; Carleman function; regularization; continuation formulas; stability estimate.

Author: Urinboyev F.(Namangan State University)

Abstract: In the present article, properties of inner derivations and local inner derivations of finite-dimensional Jordan algebras are studied. The main goal of the scientific work is to construct algorithms for describing in matrix form inner derivations and local inner derivations of finite-dimensional Jordan algebras. In addition, we apply
the constructed algorithms to two-dimensional and three-dimensional Jordan algebras. It is proved that every local inner derivation of two-dimensional and three-dimensional Jordan algebras is an inner derivation.

Keywords: Jordan algebras; derivation; inner derivation; local inner derivation.

Author: Urinov A.(Fergana State University), Abduqodirov A.(Fergana State University) 

Abstract: In this paper, necessary and sufficient conditions have been found for fifth-order partial differential equations with non-multiple characteristics. A lemma on the invariance of the number and multiplicity of real and complex roots of an algebraic equation of the fifth degree under a transformation of variables that admits an inverse transformation has been formulated and proved. Cononic types of fifth-order differential equations with two independent variables and non-multiple characteristics are given.

Keywords: Canonical form of partial differential equations; characteristic; characteristic equation; hyperbolic operator; elliptic operator.

Author: Fayazov K. (Turin Polytechnic University in Tashkent) , Khudayberganov Y. (National University of Uzbekistan)

Abstract: This paper is devoted to the study of the initial-boundary value problem for the mixed type fourth-order partial differential equation with two degenerate lines. The problem under consideration is ill-posed in the sense of J. Hadamard. According to the idea of A.N. Tikhonov on approximate solution of ill-posed problems we proof the theorems of the uniqueness and conditionally stability.

Keywords: boundary problem; mixed type equation with two degenerate lines; ill-posed problem; a priori estimate; estimate of conditional stability; uniqueness; set of correctness.

Author: Khasanov A.(V.I.Romanovskiy Institute of Mathematics), Komilov A. (Fergana State University)

Abstract: In this paper studied the properties of the Campe de Feriet function of two arguments with third-order. A system of hypergeometric partial differential equations of the third order, which is satisfied by the function, is defined. It is indicated that the resulting system of hypergeometric type has nine linearly independent solutions at the origin.

Keywords: Multivariable hypergeometric functions; system of equation of hypergeometric type; linearly independent solutions.

Author: Aralova K. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  In this article, we study the dynamics of an operator consisting of a superposition of Volterra and non-Volterra quadratic stochastic operators defined in a two-dimensional simplex. It is shown that the dynamics of a non-linear operator that is a superposition of non-ergodic and regular quadratic stochastic operators is regular.

Keywords: quadratic stochastic operator; Volterra operator; non-Volterra operator; trajectory.

Author:  Dushatov N.  (Tashkent State Technical University)

Abstract: The article considers the problems of estimation and investigation of the asymptotic properties of risk ratio estimates in the case when the data are censored and strictly moved.

Keywords: Relative risk; censored data; strong mixing; cumulative hazard.

Author: Khamdamov I. (National University of Uzbekistan)

Abstract:  This study is devoted to clarifying the contribution of the extreme terms of the variational series to the behavior of the sum of independent random variables. In particular, the asymptotic properties of the truncated sum with respect to the extreme terms in the sum are established for the case when the initial distribution belongs to the domain of attraction of the stable law. It is proved that the limiting distribution for a truncated sum is a two-parameter mixture of infinitely divisible laws, the spectral measure of each of which is a narrowing of the spectral measure of the limiting stable distribution to an asymmetric interval, the ends of which are the parameters of the mixture. In this case, the density of the limiting joint distribution of the maximum and minimum terms in the truncated sum serves as a weight function.

Keywords: Truncated sum; infinitely divisible distribution; stable distribution; spectral measure; ordered statistics.

Author: Kurbanbaev T. (Karakalpak State University), (V.I.Romanovskiy Institute of Mathematics)

Abstract:  The present paper is devoted to almost inner derivations of solvable Leibniz algebras. We prove that any almost inner derivations on a solvable Leibniz algebra whose nilradical is naturally graded filiform algebras are inner.

Keywords: Leibniz algebra; solvable Leibniz algebra; derivation; inner derivation; almost inner derivation.

Author: Rozikov U.(V.I.Romanovskiy Institute of Mathematics), Jumayev J.(Karshi State University)

Abstract:  In this paper we find all fixed points and invariant sets for a two-parameter quadratic operator mapping a two-dimensional simplex to itself, defined by a non-stochastic cubic matrix. Moreover, under certain conditions on the parameters, the limit points of the trajectory of an arbitrary point of the simplex result from operator.

Keywords: simplex; quadratic operator; fixed point; invariant set; limit point.

Author: Topvoldiyev F. (Fergana State University)

Abstract:  It is known that one of the main problems of modern differential geometry is the problem of restoring a surface according to a given geometric characteristic. In this article, a new invariant for polyhedras in isometric on sections, the concept of a conditional full angle of a polygonal angle is defined. Using it, a conditional full angle around the vertex of the cone with special edges was found.

Keywords: isometry on sections; invariant; conditional full angle; cone with special rib.

Author: Zaitov A.(Tashkent Architecture and Civil Engineering University),(V.I.Romanovskiy Institute of Mathematics), Karimov S. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  For a Tychonoff space $X$, we consider the space $Pτ (X)$ of probability $τ$ -smooth measures on $X$, and exp $X$ of all closed nonempty subsets of $X$. In the paper we establish that for every positive integer $n$ and $m$-bounded space $X$ the spaces $Pτ, n(X)$ and expn $X$ are also $m$-bounded spaces.

Keywords: $τ$ -smooth measure; hyperspace; $m$-bounded spaces.

Author:  Zhavliev S.(National University of Uzbekistan), Kasymov N.(National University of Uzbekistan)

Abstract:  It has been established that for uniformly computably separable equivalence, the finite approximability of any algebra represented over it is equivalent to the immune of the characteristic transversal of this equivalence. It is shown that there is such an infinite computably separable equivalence, a characteristic transversal, which is not immune, over which only finite approximable algebras are represented. Close questions were also considered.

Keywords: Enumerated algebras; morphism; representability of a universal algebra over equivalence and η-algebra, characteristic transversal of equivalence and enumerating; uniformly computably separable enumerating; finite approximability.

Author:  Mirsaburov M.(Termez State University), Ergasheva S.(Termez State University) 

Abstract: In this article, for equation, considered in a mixed domain, uniqueness, and existence theorems for the solution of the problem with an analog of the Frankl condition and missing Tricomi condition on one boundary characteristic are proved.

Keywords: mixed type equation; singular coefficient; missing Tricomi condition; analogue of Frankl condition; non-standard Tricomi singular integral equation with a shift and non-Fredholm operator; Wiener-Hopf equation; index.

Author:  Umirzakova K.(Namangan State University)

Abstract: In this article, one of the NN models with observable states. It is known that there are four such types of models: in one of them, the exact number of translation-invariant Gibbs measures found in the village of Cayley is of order k = 4, k = 5.

Keywords: Cayley tree; configuration; NS model; Gibbs measure; fertile graph, translation-invariant Gibbs measures.

Author: Shermatova Kh. (Fergana State University)

Abstract:  In this paper, we study a boundary value problem for a third-order parabolic-hyperbolic equation of the form in a pentagonal domain when the characteristic of the operator a is greater than one. The unique solvability of the considered problem is proved using the method of constructing a solution.

Keywords: Differential equation; method of construct solution; boundary value problem; parabolic-hyperbolic type; unique solvability; pentagonal domain.

Author: Erisbaev S.(Karakalpak State University)

Abstract:  In partially informative regression model Koks of random censorship from the right the semiparametric estimator for conditional distribution function is constructed and the property of uniform strong consistency of the estimator is studied.

Keywords: Cox regression model; random censoring; kernel estimator; regression parameter; covariate.

Author: Azizov M.(V.I.Romanovskiy Institute of Mathematics)

Abstract:  The present paper is devoted to the description of small dimensional Leibniz dialgebras. More precisely, small dimensional Leibniz dialgebras, constructed by null-filiform Leibniz algebras are classified.

Keywords: Leibniz algebra; solvable Leibniz algebra; derivation; inner derivation; almost inner derivation.

Author: Dustov S. (Navoi State Pedagogical Institute)

Abstract:  A family of the generalized Friedrichs models with the perturbation of rank one is considered. We obtain an absolutely convergent expansion for eigenvalue at, the coupling constant threshold. The expansion is dependent to a large extent on whether the upper bound of the essential spectrum is a threshold resonance or a threshold eigenvalue.

Keywords: Generalized Friedrichs models; coupling constant threshold; Hamiltonian; dispersion relation; threshold resonance; threshold eigenvalue.

Author: Ibragimov M. (Andijan State University)

Abstract:  In the present paper we consider genetic algebras corresponding to quadratic automorphisms defined on the two-dimensional simplex. Main goal is to describe the set of all idempotent elements and one-dimensional subalgebras of such genetic algebras. It is showed correspondence between the one-dimensional subalgebras and the set which contains all idempotent elements and absolute nilpotent elements.

Keywords: Non-associative algebra; genetic algebra; quadratic stochastic operator; quadratic automorphism.

Author: Ismoilov Sh.(National University of Uzbekistan), Kholmurodova G.(Tashkent State Transport University) 

Abstract: Total curvature is defined by first and second fundamental forms in Euclidean and isotropic spaces. In this article, theorems are proved about the connection of two-dimensional surfaces satisfying Cauchy-Riemann conditions in Euclidean and isotropic spaces.

Keywords: Isotropic space; Monge-Ampere equation; movement; conditions Cauchy-Riemann; full curvature; average curvature; Gauss’s theorem; fundamental forms.

Author: Jumayev J.(V.I.Romanovskiy Institute of Mathematics), Atoyev D. (Bukhara State University) 

Abstract:  In this paper, the inverse problem of determination kernel from integro-differential heat equation with the nonlocal initial boundary and additional conditional is studied. The solvability of the direct problem is proved using Fourier’s method and Banach’s principle. To investigate the inverse problem, an auxiliary equivalent to the original problem is obtained. Then, using the Fourier method, the problem is reduced to a system of closed integral equations equivalent to the unknown functions. The theorem about the existence and uniqueness of the solution of this integral equation using the principle of contraction mapping is proved.

Keywords: integro-differential equation; non-local initial-boundary problem; inverse problem; integral equation; Banach principle.

Author: Nazarov Z. A. (V.I.Romanovskiy Institute of Mathematics)

Abstract: We examine the population growth system called Q-processes. This is defined by the Galton- Watson Branching system conditioned on non-extinction of its trajectory in the remote future. In this paper we observe the total progeny up to time n in the Q-process. By analogy with branching systems, this variable is of great interest in studying the deep properties of the Q-process. We find that the sum total progeny as a random variable approximates the standard normal distribution function under the third moment assumption for the initial Galton-Watson system offspring law.

Keywords: Branching system; Q-process; Markov chain; generating function; transition probabilities; invariant distribution; extinction time; total progeny; positive recurrent; central limit theorem; law of large numbers.

Author: Samatov B.(V.I.Romanovskiy Institute of Mathematics), Uralova S.(V.I.Romanovskiy Institute of Mathematics) 

Abstract:  This work is focused on the pursuit problem between two controlled objects(a pursuer and an evader). The pursuit problem is formulated by linear differential equations. The objects apply controls with the Langenhop constraints of integral type. A strategy of parallel convergence (Π-strategy) is constructed for the pursuer that provide completion of the pursuit in a finite time.

Keywords: Differential games; linear game; pursuit problem; Langenhop constraint; pursuer; evader; Π-strategy.

Author: Sharipov O.(National University of Uzbekistan), Kobilov U.(National University of Uzbekistan)

Abstract:  In this note we give sufficient conditions of almost sure convergence of the series of mixing random variables. We assume that random variables are from domain of attraction of stable laws.

Keywords: Random series; mixing conditions; almost sure convergence.

Author: Sheraliyeva S. (V.I.Romanovskiy Institute of Mathematics)

Abstract: In this work we consider one-dimensional extension of solvable Leibniz algebras whose nilradical is null-filiform and naturally graded filiform algebras.

Keywords: Leibniz algebra; solvability; nilpotency; filiform Leibniz algebras; cental extension.

Author: Sobirov Z.(V.I.Romanovskiy Institute of Mathematics), Saparbayev R.(V.I.Romanovskiy Institute of Mathematics) 

Abstract:  This work is devoted to the Cauchy problem for a time-fractional differential equation on the metric graph that consists of three incoming bonds, a series of 3-bridges that connect the consequent vertices, and three outgoing bonds. The problem is reduced to the number of IBVPs in finite, semi-infinite intervals and the Cauchy problem on the line. We found an exact solution to the problem in the form of integral representation via given data. The uniqueness of the solution is proved using a-prior estimate for the solution.

Keywords: Fractional derivatives; Fractional PDE; subdiffusive equation; time-fractional heat equation; integral representation of the solution.

Author: Abdukodirov A. (Fergana State University)

Abstract:  In this work, we present canonical forms of differential equations of the fifth order with two independent variables with multiple characteristics.

Keywords: canonical form; partial differential equations; characteristics; characteristic equation.

Author:  Allambergenov A.(Karakalpak State University)

Abstract: This paper is devoted to the study of 2-local derivations on Okubo algebra. It is proved that every 2-local derivation of the Okubo algebra is a derivation.

Keywords: Okubo algebras; derivation; 2-local derivation.

Author:  Kadirkulov B.(Tashkent State Institute of Oriental Studies), Ergashev O.(Tashkent Institute of Engineers irrigation and agricultural mechanization)

Abstract: The paper investigates the spectral questions of a non-self-adjoint problem of the Bitsadze-Samarskii type. Eigenvalues corresponding to eigenfunctions are found, and conditions for the existence of associated functions are found. Spectral questions of the adjoint problem are also studied. Next, we prove the completeness and also the Riesz basis property of the systems of root functions of these problems.

Keywords: Bitsadze-Samarskii type problem; non-self-adjoint problem; eigenfunctions and associated functions; completeness; Riesz basis.

Author: Rahmonov Z.(Institute of Mathematics named after. A. Juraeva), Allakov I.(Termez State University),  Abrayev B. (Termez State University)

Abstract: An asymptotic formula is obtained for the number of representations of a sufficiently large natural N in the form $b_1p_1 + b_2p_2 + b_3p_3 = N$ with the conditions where $b_i$ — natural numbers, $b_1, b_2, b_3, N$ are pairwise coprime, $B_i$ — arbitrary fixed positive numbers.

Keywords: ternary Goldbach problem; almost equal terms; short exponential sum with primes; small neighborhood of centers of major arcs.

Author: Usmonov D. (Fergana State University)

Abstract:  In this work, in a rectangular domain, we study an initial boundary value problem for a degenerate fourth-order differential equation containing an integral-differential operator with a Bessel function in the kernel. Applying the method of separation variables to the considered problem a spectral problem for an ordinary differential equation has been obtained. The existence of eigenvalues and the system of eigenfunctions of the spectral problem were proved. A theorem is proved for expanding a given function into a uniformly convergent series with respect to the system of eigenfunctions. The solution of the considered problem is written as the sum of the Fourier series with respect to the system of eigenfunctions of spectral problem. An estimate for the solution of the problem is obtained, from which follows its continuous dependence on the given functions.

Keywords: degenerate equation; initial-boundary value problem; Bessel function; integral–differential operator; spectral method; Green’s function; integral equation.

Author: Shamsiddinov N.(Academic Lyceum of TSTU), Anorov O. (Tashkent State Transport University)

Abstract:  In this paper, we construct quadratic stochastic operators generated by the finite regular partition of a countable set of states and prove that such operators are regular transformations.

Keywords: absolutely continuous measure; Puasson distribution; finite regular partition; regular transformation.

Author:

Abstract:  As is known, self-similar solutions of a degenerate partial differential equation are found by reducing it to some ordinary differential equation with regular points, the solution of the latter being expressed in terms of the well-studied generalized hypergeometric function. In this work, linearly independent solutions of fourth and fifth-order ordinary differential equations with a singular point at the origin are constructed. The application of these results makes it possible to express in explicit forms the self-similar solutions of several degenerating PDE up to seventh-order inclusive.

Keywords: Gauss hypergeometric function; generalized hypergeometric function; degenerating PDE of higher-order; ordinary differential equation of higher order; linearly independent solutions.

Author: Hayotov A.(V.I.Romanovskiy Institute of Mathematics), (National University of Uzbekistan), (Bukhara State University)
Doniyorov N. (V.I.Romanovskiy Institute of Mathematics)

Abstract: In this article, using S. L. Sobolev’s method, an optimal interpolation formula in the sense of Sard in a certain Hilbert space is constructed, and basis functions for finite element methods are constructed using the coefficients of this optimal interpolation formula. Applying these basis functions, the boundary value problem for the second-order ordinary differential equation is approximately solved and the error of the approximate solution is studied.

Keywords: Basis functions; ordinary differential equation; boundary value problem; finite element; interpolation; interpolation formula; bilinear form.

Author:  Juraev B. (Andijan State University)

Abstract: In this paper consider a simple pursuit-evasion problem, where the pursuer’s controls are subject to an integral constraint, and the evader’s controls are subject to a Gr ̈onwall-type constraint. To solve the pursuit problem, the parallel pursuit strategy (Π-strategy) is used, which allows the best approach of the players. To solve the problem of evasion, a special control is used that allows the evader to evade pursuit. For the pursuit problem, the attainability domain of the players is studied. New sufficient conditions are obtained for solving the pursuit-escape problems.

Keywords: Differential game; Gr ̈onwall constraint; integral constraint; pursuer; evader; strategy; attainability domain.

Author: Khakimov O. (V.I.Romanovskiy Institute of Mathematics)

Abstract: The research is devoted to studying the dynamics of Riesz-type stochastic operators (in short, RSO) defined on a finite-dimensional simplex. A complete description of the dynamics of nonlinear stochastic operators, even in the quadratic one, has not yet been obtained. Recently, we introduced a new class of nonlinear stochastic operators called Riesz-type stochastic operators (see [3]). The Cesaro operator is a special and more interesting case of positive RSOs. The dynamics of the Cesaro operators were fully described in [3]. This work is a continuation of the research in [3]. In this paper, we will consider positive RSOs. We prove that the positive RSOs are regular, and the dynamics of such kinds of operators are very close to the dynamics of the Cesaro operator.

Keywords: Dynamical systems; fixed point; nonlinear operator; stochastic operator.

Author: Kurbanbaev T.(Karakalpak State University),(V.I.Romanovskiy Institute of Mathematics), Iskenderov A. (Nukus State Pedagogical Institute)

Abstract:  The present study is devoted to the study of almost inner derivations of solvable Leibniz algebras whose nilradical is a quasi-filiform Leibniz algebra of maximum length. Inner and almost inner derivations of n-dimensional quasi-filiform non-Lie Leibniz algebras of maximum length are considered.

Keywords: Leibniz algebra; solvable Leibniz algebra; derivation; inner derivation; almost inner derivation.

Author: Zaitov A.(Tashkent Architecture and Civil Engineering University), (V.I.Romanovskiy Institute of Mathematics), Eshimbetov M. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  For a compact Hausdorff space X, we consider a space $I_B(X)$ of all idempotent probability measures on $X$, which define as set-funtions on the $σ$-algebra of Borel sets in $X$, and a space $I_C (X)$ of all normed max-plus linear functionals on the set of all continuous functions on $X$, equipped with idempotent operations. The main result declares that the spaces $I_B(X)$ and $I_C (X)$ are homeomorphic.

Keywords: idempotent measure; max-plus-linear functional; Borel sets; continuous functions.

Author: Zunnunov R. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  In the articel, a displacement problem for the generalized Tricomi equation with a spectral parameter in an unbounded domain, the elliptic part of which is the upper half-plane, is studied. The uniqueness of the solution to the considered problem was proved by the method of energy integrals. To prove the existence of the solution to the problem, the methods of Green’s functions and integral equations were used.

Keywords: Green’s function method; integral equation method; nonlocal problem; unbounded domain; generalized Tricomi equation with spectral parameter.

Author: Ibaydullaev T.(Andijan State University)

Abstract:  This article are devoted to study pursuit and evasion problems with special integral constraints on player controls. Such constraints were first set by A.A. Azamov. From a physical point of view, these conditions on the players’ controls mean that the resources given to them are continuously (sometimes periodically) renewable. Such conditions are given, if they are done, then it will be a win for pursuer, otherwise, a win of evader.

Keywords: Differential games; pursuit problem; evasion problem; Π-strategy; A.A.Azamov constraints.

Author: Ismoilov A. (Fergana State University)

Abstract:  The article discusses the Cauchy-Goursat problem for the generalized inhomogeneous Euler-Poisson-Darboux equation in the characteristic triangle. The formula for the solution to the problem was found by the Riemann method. It was proved that the found solution does indeed satisfy the equation and boundary conditions.

Keywords: Cauchy-Goursat problem; inhomogeneous generalized Euler-Poisson-Darboux equation; Riemann method; Riemann-Hadamard function.

Author:  Mirsaburov M.(Termez State University), Makhmudov A. (Termez State University)

Abstract:  For the Gellerstedt equation with a singular coefficient, uniqueness and existence theorems for a solution to the problem with local and nonlocal conditions on parts of the boundary characteristic are proved.

Keywords: partition of the boundary characteristic into two parts; Bitsadze-Samarsky condition on parallel characteristics; Tricomi integral equation with a non-Carleman shift in the “non-singular” part of the kernel; kernel with a first-order singularity at an isolated singular point; Wiener-Hopf equation; index.

Author: Murzambetova M. (Nukus State Pedagogical Institute)

Abstract:  The present research is devoted to studying forward and inverse problem for a mixed-type equation involving the Caputo fractional derivative. The conditions for the existence and uniqueness of the solution were obtained by the Fourier method.

Keywords: Mixed-type equation; Caputo fractional derivative; existence and uniquenass of the solution; forward and inverse problem; conjugation condition; Fourier method.

Author: Okboev A. (Fergana State University)

Abstract: The present paper a higher-order Cauchy-type problem is formulated and studied for a degenerating hyperbolic equation of the second kind in the characteristic triangle. The solution to the problem is constructed and the uniqueness of the solution is proved. First, the general solution of the given degenerate hyperbolic equation is constructed. Then the uniqueness of the solution to the problem is proved by the estimation of the unknown function and its high-order derivatives with respect to the second argument. In proof of the theorem the method of mathematical induction, the Gaussian hypergeometric function, Euler’s gamma function, Pochhammer’s symbol are used.

Keywords: degenerate equation of hyperbolic type; Cauchy problem; uniqueness of solution.

Author:  Sobirov Sh. (Urgench State University)

Abstract: In this article, the problem of integration of the system of nonlinear integro-differential equations with additional term in the class of rapidly decreasing functions is considered. The method of inverse problems of the scattering theory was used to integrate this system of nonlinear equations.

Keywords: Jost solutions; modified Korteweg-de Vries equation; evolution of scattering data; the system of Dirac equations; Gelfand-Levitan-Marchenko integral equation.

Author: Urinov A.(Fergana State University), Abduqodirov A.(Fergana State University)

Abstract: In the article, the canonical forms of $n^th$-order partial differential equations of two variables are studied.

Keywords: canonical form; partial differential equations; characteristics; characteristic equation.

Author: Azamov A.(V.I.Romanovskiy Institute of Mathematics), Samatov B.(V.I.Romanovskiy Institute of Mathematics), Turgunboeva M.(Namangan State University)

Abstract:  This paper considers the l-catch problem in a nonlinear differential game involving two opposing objects, the pursuer and the evader. The movements of each object occur in space Rn in the field of dynamic influences of a different nature. To solve the l-catch problem, a strategy is generalized that allows for the best l-approximation with the evader in simple cases. To obtain sufficient conditions for solvability, the well-known Gr ̈onwall lemma is used. The results obtained are verified using illustrative examples.

Keywords: Differential game; l-catch; pursuer; evader; geometric constraint; strategy; guaranteed time.

Author: Diyorov A. (The Samarkand branch of TUIT)

Abstract: In this short note, we study a dynamical system generated by a three-parametric quadratic operator mapping 4-dimensional simplex to itself. This is an evolution operator of the frequencies of gametes in a three-locus system. We find the set of all (a continuum set) fixed points and show that each fixed point is non-hyperbolic. For any initial point (taken from the 4-dimensional simplex) we find an invariant set containing the initial point and a unique fixed point of the operator, such that the trajectory of initial point converges to this fixed point.

Keywords: Loci; gamete; dynamical system; fixed point; trajectory; limit point.

Author: Kholikova F. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  In the present paper, we consider Volterra cubic stochastic operators depending on three parameters and study their trajectory behaviours. We describe the invariant sets and find all fixed points on the two-dimensional simplex. We give a complete description of the set of limit points and we show that such operators have the ergodic property.

Keywords: Cubic stochastic operator; Volterra operator; fixed point.

Author: Mizomov I.(V.I.Romanovskiy Institute of Mathematics)

Abstract:  For an elliptic curve E, a point τ ∈ E, and a pair of coprime numbers 1 ≤ k ≤ n, Feigin and Odesski defined a family of algebras Qn,k(E, τ ) which generalize Sklyanin algebras. The Calabi-Yau property of Qn,k was first established by Smith and Stafford for the case when n = 3, and later proved by Bocklandt, Schedler, and Wemyss for n = 4. It has been conjectured that Qn,k possess the Calabi-Yau property for all pairs (n, k). In the present paper, we focus on the algebras Q5,1(E, τ ), examining specific values of τ . In this case, we show that these algebras indeed exhibit the Calabi-Yau property.

Keywords: Superpotential; Koszul Calabi-Yau algebra; Sklyanin algebra; free algebra; Koszul dual algebra; noncommutative partial derivative.

Author: Mustafoyeva Z.(V.I.Romanovskiy Institute of Mathematics)

Abstract:  The present research is devoted to studying the Potts model with a countable set of spin values and with competing interactions of radius r = 2 on the Cayley tree of order k = 3.

Keywords: Potts model; Cayley tree; Hamiltonian; configuration; basic condition;
countable set of spin values.

Author: Turdiev Kh. 

Abstract:  In this work a unique solvability of a boundary problem for diffusion-wave equation involving the regularized Prabhakar fractional derivative has been proved. Using a method of separation of variables, spectral problem with a spectral parameter in boundary condition, and solution of the Cauchy problem for differential equation involving the regularized Prabhakar derivative, the main result of the investigation is proved.

Keywords: Diffusion-wave equation; regularized Prabhakar fractional order derivative; spectral problem with non-classical boundary condition; Mittag-Leffler function.

Author: Ashurov R.(V.I.Romanovskiy Institute of Mathematics), Nuraliyeva N.(V.I.Romanovskiy Institute of Mathematics)

Abstract:  In this paper, a non-local time problem for a hyperbolic equation is studied, i.e. the initial conditions are replaced by non-local conditions: u(x, ξ) = αu(x, 0) + φ(x), ut(x, ξ) = βut(x, 0) + ψ(x), where ξ ∈ (0, T] is an arbitrary fixed number and α and β are arbitrary real numbers. For the parameters, α and β criteria of uniqueness and sufficient conditions ensuring the existence of the solution of the problem are found. The conditions of existence and uniqueness of the solution were derived from the Fourier’s method.

Keywords: Hyperbolic equation; time-non-local problem; uniqueness and existence of the solution; Fourier method.

Author: Dzhamalov S.(V.I.Romanovskiy Institute of Mathematics), Khudoykulov Sh. (V.I.Romanovskiy Institute of Mathematics)

Abstract: In this paper, we study the well-posedness of one linear two-point inverse problem for the three-dimensional heat equation. Using the methods of a priori estimates, Galerkin, a sequence of approximations and contracting mappings, we prove the unique solvability of a generalized solution of a linear two-point inverse problem for three-dimensional heat equation.

Keywords: Three-dimensional heat equation; linear two-point inverse problem; generalized solution; a priori estimate; method of Galerkin; successive approximations; method of contracting mappings.

Author: Dosanova U.(V.I.Romanovskiy Institute of Mathematics), Nuraliyeva N.(V.I.Romanovskiy Institute of Mathematics)

Abstract: The present research is devoted to the studying of the uniqueness and existence of the solution to the non-local problem for the parabolic-hyperbolic type equation involving the Caputo fractional derivative. The conditions for the existence and uniqueness of the solution were obtained by the Fourier method, while the stability was proved by the estimation and Cauchy-Bunyakovsky inequalities.

Keywords: Non-local problem; Caputo fractional derivative; Fourier method; parabolic-hyperbolic type equation; Cauchy-Bunyakovsky inequality.

Author: Dushatov N.  (Tashkent State Technical University)

Abstract: In this paper, we consider estimation of mean residual life function from incomplete dependent right censored data. The Archimedean copula function was used to construct the estimate. The consistency property of the estimate is proved.

Keywords: Incomplete-censored from right data; mean residual life time; archimed-copula functions; hazard rate function.

Author: Kurbanov B. (Karakalpak State University)

Abstract: The work is devoted to the problem of holomorphic continuation from the boundary of a domain in a multidimensional complex space. It asserts the possibility of a holomorphic continuation of the function f from the boundary ∂D of the domain D ⊂ C n under the condition that the integrals of f over the boundaries of analytic disks lying on ∂D. In this article, using the properties of the Poisson integral, we prove a boundary version of Morera’s theorem in the Siegel matrix domain of the second kind.

Keywords: Morera’s theorem; Poisson kernel; holomorphic function; Siegel domain.

Author: Mustapokulov Kh.(National University of Uzbekistan), Mamadaliev N. (National University of Uzbekistan), (V.I.Romanovskiy Institute of Mathematics)

Abstract: In this paper, we consider the issues of strong and weak invariance of a given constant set-valued mapping with respect to a system with distributed parameters. The system is described by the heat equation, in the right part of which the impulse control is in the additive form.

Keywords: Invariance; control; weak invariance; strong invariance; lumped parameters.

Author: Khalkhuzhaev A.(V.I.Romanovskiy Institute of Mathematics), Boymurodov J. (Navoi State Pedagogical Institute)

Abstract:  We consider the three-particle Schr ̈odinger operator Hμ,λ,γ(K), K ∈ T
3, associated with a system of three particles (two of them are bosons with mass 1 and one is arbitrary with mass m = 1/γ < 1), interacting with the help of paired contact potentials μ > 0 and λ > 0 on the three-dimensional lattice Z^3. It is proved that there exists a critical value of the mass ratio γ = γ_1 such that the operator Hμ,λ,γ(π), π = (π, π, π) has: for γ ∈ (0, γ1) at least one eigenvalue, for γ ∈ (γ1, +∞) there are at least four eigenvalues lying to the left of the essential spectrum for sufficiently large μ > 0 and fixed λ > 0.

Keywords: Schr ̈odinger operator; lattice; Hamiltonian; zero-range potential; boson; eigenvalue; quasimomentum; invariant subspace; Faddeev operator.

Author: Kurudirek A.(Stirling Education, KRG-IRAQ)

Abstract: 
Galileo’s geometry, taxicab and geometry are little used endpoints. It is difficult to explain to secondary school students spherical and hyperbolic geometry. Students, future mathematicians, need to study non-Euclidean geometry and methods of his training. In this article I share my experience in exploring the vast horizons of non-Euclidean geometry

Keywords: Galilean; taxicab; limited (finite) point; Lobachevsky; non-Euclidean geometry.

Author: Samatov B. T.(Namangan State University), Jurayev B. I. (Andijan State University) 

Abstract: 
In this paper, we study a differential pursuit game when the pursuer’s control is chosen from class of measurable functions satisfying inequalities Gronwall type, and for the control functions of the evader a geometric type constraint is imposed.

Keywords: Differential game; Gr ̈onwall constraint; geometrical constraint; pursuer; evader; strategy; pursuit; guaranteed time.

Author: Azizov M. S. (Fergana State University)

Abstract: 
In the present work an initial-boundary value problem for a higher even order partial differential equation with a Bessel operator and the existence, uniqueness and stability of the solution of the considered problem have been investigated.

Keywords: even order partial differential equation; Bessel operator; initial-boundary value problem; spectral method; Green’s function; integral equation; existence; uniqueness and stability of the solution

Author: Ganikhodjaev N. N. (V.I.Romanovskiy Institute of Mathematics), Dusmurodova G. Kh.(Chirchik Pedagogical Institute)

Abstract: 
In this paper, we consider four- and five-dimensional algebras created by quadratic stochastic operators and establish the conditions under which the corresponding algebra is associative.

Keywords: quadratic stochastic operator; associative; unit element; (n−1)-dimensional simplex; regularity; partial reversible; extreme Volterra operator.

Author: Davlatov Sh. O. (Karshi Engineering and Economic Institute)

Abstract: 
This article discusses the finite element method for symmetric t-hyperbolic systems. In approximation of a mixed problem for symmetric t-hyperbolic systems, the finite element method is used to discretize in space, and the time discretization is done using finite difference. An implicit difference scheme of a mixed problem for symmetric t-hyperbolic systems was constructed. Created an algorithm for a numerical solution of a mixed problem for symmetric t-hyperbolic systems by finite elements on an irregular grid. Based on this algorithm, a program has been created for a numerical solution of a mixed problem for symmetric t-hyperbolic systems by the method of finite elements on an irregular grid. The numerical calculation of the model problem is given.

Keywords: finite element method; algorithm; mixed problem; hyperbolic system; basic
functions; implicit difference scheme.

Author: Jumaev D. I.(Tashkent Architectural and Construction Institute),
Beshimova D. R. (Bukhara State University)

Abstract: In the paper it is shown that each topological transformation group on a Tychonoff (in particular, a compact Hausdorff) space generates a topological transformation group on the hyperspace. Further, it is proved that a continuous map between hyperspaces is equivariant if the map between the original Tikhonov spaces, which induces it, is equivariant. Hence it follows that a continuous map between hyperspaces is an equivalence if the map between given Tychonoff spaces, which induces it, is an equivalence.

Keywords: A group of topological transformations; hyperspace; equivariant map.

Author: Imomov A. A.(Karshi State University), Nazarov Z. A.(V.I.Romanovskiy Institute of Mathematics) 

Abstract: 
In the paper we consider a stochastic model which called Markov Q- processes that forms a continuous-time Markov population system. Markov Q-processes are defined as stochastic Markov branching processes with trajectories continuing in the remote future. Estimation of the structural parameter of the Markov Q-process is the main goal of this paper. To estimate this parameter, an unbiased estimator of the Lotka-Nagaev type is proposed. An asymptotic expansion of the variance of this estimator is found.

Keywords: Markov branching systems; Markov Q-processes; transition probabilities; generating function; structural parameter; unbiased estimator.

Author: Ismoilov A. I. (Fergana State University)

Abstract: 
This paper discusses the Darbu problem for the Euler-Poisson- Darbu equation in a characteristic triangle. The formula for solving the problem by the Riemann method has been found. The function defined by the found formula has been proved to satisfy the equation and boundary conditions indeed.

Keywords: Euler-Poisson-Darboux equation, inhomogeneous equation, Darboux problem, Riemann-Hadamard function.

Author: Makhammadaliev M. T. (Namangan State University)

Abstract: In this work the existence and non-existence conditions of
translation-invariant and $G^2_k$ – periodic Gibbs measures for a
HC model with a countable set of spin values on a Cayley tree
of order two are found.

Keywords: Cayley tree; configuration; Hard-Core model; Gibbs measure; translation-invariant Gibbs measure; periodic Gibbs measure.

Author: Rasulov M. S.(V.I.Romanovskiy Institute of Mathematics), Norov A. K.(V.I.Romanovskiy Institute of Mathematics) 

Abstract: 
In this paper studied the problem for a competition-diffusion system with two different free boundaries. To solve the problem, a priori estimates of the H ̈older norms are established. On the basis of a priori estimates, the existence and uniqueness of the
solution is proved.

Keywords: Free boundaries; system of quasilinear parabolic equations; a priori estimates; existence and uniqueness of solutions.

Author: Rasulov S. I.(Tashkent State Technical University), Mustapokulov Kh. Y.(Tashkent State Technical University), Kuralov B. A.(Tashkent State Technical University) 

Abstract: 
The article discusses estimates of the Monte Carlo method for Fredholm determinants. It is shown that the problem of calculating eigenvalues and eigenfunctions is one of the
difficult branches of computational mathematics, and also that it plays an exceptional role in the science of technology and is of great theoretical and applied importance. A new method for calculating the characteristic numbers of the Schrodinger equation for a harmonic oscillator is presented. Monte Carlo estimates are constructed for calculating the coefficients of the Fredholm determinants. It is proved that the mathematical expectation of a random variable is equal to the coefficient of the polynomial.

Keywords:Eigenvalue; eigenfunction; determinant; Monte Carlo; kernel; polynomial; expectation; variance; estimate.

Author: Rakhmonov F. D. (National University of Uzbekistan)

Abstract: 
In a two-dimensional spatial domain, we consider a Benney–Luke type partial integro- differential equation of an even high order with mixed conditions. We study the unique solvability of a nonlinear inverse problem for the case when the kernel of the integral term in a given equation is degenerate. The solution of this partial differential equation is studied in the class of regular functions. The Fourier series method is used in combination it with the degenerate kernel method. The inverse problem is reduced to solving countable systems of nonlinear functional-integral equations. In proving the existence and uniqueness of the Fourier coefficients of the unknown functions, the successive approximation method is used in combination it with the contraction mapping method. The Cauchy-Schwarz inequalities and the Bessel inequality are applied in proving the absolute and uniform convergence of the obtained Fourier series

Keywords: Benney–Luke type equation; integro-differential equation; degenerate
kernel; inverse problem; existence and uniqueness of the solution.

Author: Turakulov Kh. Sh. (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
In this paper, using the “ε -regularization” methods, a priori estimates and the Fourier transform, the unique solvability of a generalized solution of one semi-periodic boundary value problem for the three-dimensional Tricomi equation in an unbounded parallelepiped is studied.

Keywords: the Tricomi equation, semi-periodic boundary value problem, Fourier transform, ” ε -regularization”methods and a priori estimates.

Author: Urinov A.K. (Fergana State University)

Abstract: 
In the present work Abel’s integral equation type integral equation (Bessel’s function in the kernel) was considered. Firstly, it was found the formulas of the solutions of this equation then it was shown that these solutions really satisfy considered equation. After that using the solution of the considered integral equation (as in Abel’s integral equation), the concepts of generalizing Riemann-Liouville fractional order integrals and derivatives have been introduced and some properties about superposition and extremum principles for them have been formulated. At the end of the article, the introduced fractional integrals and derivatives of some functions were calculated.

Keywords: Abel’s integral equation; fractional order integral; fractional order derivative; integral-differential operators of Riemann-Liouville.

Author: Yuldasheva A.V. (Branch of Moscow State University named after M.V. Lomonosov in Tashkent)

Abstract:  In this paper, we prove a unique solvability of the Cauchy problem for nonlinear peridynamic equation.

Keywords:nonlinear integro-differential equation; peridynamics; Cauchy problem; contraction mapping.

Author: Durdiev D. K.(V.I.Romanovskiy Institute of Mathematics), Shishkina E.L (Belgorod State National Research University), Rahmonov A. A.(V.I.Romanovskiy Institute of Mathematics)

Abstract:
This article is devoted to obtaining an explicit solution n-dimensional wave equation with fractional derivative Gerasimov-Caputo in an infinite domain with a non-zero initial condition and a condition for vanishing at infinite infinity. It is shown that this equation can be obtained but from the classical homogeneous hyperbolic integro-differential equation with memory with Mittag kernel Leffler. Based on Laplace and Fourier transforms using the properties of the Fox function and the convolution theorem, an explicit solution to the problem under consideration is obtained.

Keywords: fractional wave equation; Gerasimov-Caputo fractional derivative; Laplace transform; Fourier transform; explicit solution.

Author: Mizomov I.E. (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
The article considers the central extension of the three-dimensional Sklyanin algebra, two Koszul Artin algebras, Shelter and Ore extension through the differential differential of the polynomial algebra $C[x_0, x_1, x_2]$. Our main result is that these algebras represent Calabi-Yau. In doing so, we use a new characteristic of Koszul Calabi-Yau algebras.

Keywords: Koszul Calabi-Yau algebra; central extension; Ore extension.

Author: Mustapokulov Kh. Y. (Tashkent State Technical University),
Bekchonov Sh. E. (Tashkent State Technical University)

Abstract: 
In this work, we study the issue of strong and weak invariance of sets with respect to systems with distributed parameters described by the heat equation. Geometric constraints on control are considered. Necessary conditions are obtained for a set to be strongly and weakly invariant with respect to a given system.

Keywords: Invariance; control; weak invariance; strong invariance; lumped parameters.

Author: Normatov Z. (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
In this article, a method is given for finding the defining relations between the generators of the coordinate ring of the Calogero-Moser space.

Keywords: Calogero-Moser space, coordinate ring; matrix invariants.

Author: Xudayarov S. S. (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
This article constructs several examples of quadratic stochastic processes (QSP) of type (σ|D), where σ is a type of stochastic cubic matrices, and D means a specific multiplication between cubic matrices.

Keywords: quadratic stochastic process; cubic matrix; time; Kolmogorov-Chapman equation

Author: Abdullayev R. Z. (Tashkent University of Information Technologies), Madaminov B. A. (Tashkent Chemical-Technological Institute)

Abstract: 
In this paper, the isometries of F-spaces of integrable functions with logarithm have been studied. In particular, using passports of Boolean algebra, a necessary and sufficient condition of isometry F-spaces of integrable functions of logarithm with
respect to strictly positive σ-finite measures is proved. In this work,external, internal and generalized log algebras are considered separately.

Keywords: Functional spaces; F-spaces; log-algebras; boolean algebras; complete boolean algebras; homogeneous Boolean algebras; passport of boolean algebra; strictly positive; σ- finite measures; internal log-algebras; external log-algebras; generalized log-algebras; isomorphisms; isometries.

Author: Zuparov T. M.(Tashkent State University of Uzbek Language and Literature named after Alisher Navoi), Khalkhadjayev B. B. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  The work is devoted to the study of the Abelian sum with
random coefficients $S_n (η) = \sum_{j=0}^n η^j ξ_j$, where η− is a random
variable having non-negative finite moments of any order, independent of the sequence ${ξj , j = 0, 1, 2, …}$. It is proved that the terms of a random Abelian sum $ζ_j = η^j ξ_j , j = 0, 1, …, n,$ satisfy the condition of association of the k th order $(A(k))$ for any $k ∈ N$ if the sequence ${ξj}$ has all moments up to $2k$ -th order and satisfies the independence condition.

Keywords: Abelian sum; association; finite moments; structure of the dependence.

Author: Ibragimov M. M. (Karakalpak State University)

Abstract: 
In this paper, the geometric properties of unit balls of weakly and strongly facially symmetric spaces are studied. It is shown that in a finite-dimensional real neutral strongly facially symmetric space, any symmetric face of a unit ball is contained in a set of fixed points of linear isometry corresponding to a symmetric face that lies in thr given face. The results related to infinite-dimensional complex and finite – dimensional real strongly facially symmetric spaces are analyzed.

Keywords: Weakly and strongly facially symmetric spaces.

Author: Rakhimov D. G.(Branch of the Russian University of Oil and Gas named after I.M. Gubkin in Tashkent), Akhmadjanova D. D. (National University of Uzbekistan)

Abstract: 
In this paper, we prove a Fredholm property of E. Schmidt’s eigenvalues of the model problem of the theory of electromagnetic oscillations in resonators without losses.

Keywords: E. Schmidt’s eigenvalues; Fredholm eigenvalues; linear operators; Hilbert spaces.

Author: Ruziev M. Kh.(V.I.Romanovskiy Institute of Mathematics), Yuldasheva N. T. (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
In this paper we have studied a nonlocal boundary value problem for Gellerstedt equation with singular coefficient in an unbounded domain. With the help of the method of integral equations and the principle of extremum we have proved the unique solvability of the considered problem.

Keywords: Principle of extremum; uniqueness of a solution; existence of a solution; index of equation; singular coefficient; integral equations.

Author: Safarov J.Sh. (V.I.Romanovskiy Institute of Mathematics), (Tashkent University of Information Technologies)

Abstract: In this paper, the problem of finding a generalized solution of an integro-differential equation of the hyperbolic type is studied in an infinite field on the spatial variable x. The existence of a single solution of a given equation when the integral limit is known has been proved by the method of sequential approximations.

Keywords: Hyperbolic equation; integral-differential equation; delta function; Gronwall’s inequality.

Author: Fozilov D. Sh. (Samarkand branch of Tashkent University of Information Technologies)

Abstract:  In this paper, we consider the problem of continuation of a
bianalytic function into a domain in terms of its values and the
values of the derivative on a part of the boundary. In addition,
for bianalytic functions, the question of the transformation of a
Cauchy-type integral according to V.V. Pokazeev into a Cauchy
integral formula is considered.

Keywords: bianalytic function; Sokhotskiy-Plemelh formula;
Carleman’s formula; continuation formula; regularization family.

Author: Khatamov N. M.(V.I.Romanovskiy Institute of Mathematics),
Ibragimov I. I. (Namangan State University)

Abstract:  The DNA molecule is considered as a configuration of the Ising
model on the paths of the Cayley tree. For this model, new Gibbs measures on the Cayley tree of the second order are studied. It is shown that there exists a critical temperature Tc such that for T > T_c there is a single translation-invariant Gibbs measure, for T = T_c− two Gibbs measures, one of them is not translation- invariant and, for T < T_c− three Gibbs measures, two of which are not translation-invariant.

Keywords: Cayley tree, Ising model, Gibbs measure.

Author: Sharipov A. S.(National University of Uzbekistan), Abdishukurova G. M. (National University of Uzbekistan)

Abstract:  In this paper, we investigated the isometry group $I_{soF}(M)$ of
an n− dimensional foliated manifold with F− compact open topology. This topology depends on the foliation F and coincides with compact open topology when F is an n− dimensional foliation. If co-dimension of F is equal to n, convergence in this topology coincides with point-wise convergence. Some properties of isometry group IsoF (M) of foliated manifolds are proved.

Keywords: Manifold, foliation, foliated manifold; isometry of foliated manifolds, topological group, F− compact open topology.

Author: Eshkabilov Y. Kh.(Karshi State University), Baratov B. S.(Karshi State University)

Abstract:  In this paper, we studied one of the classes of cubic stochastic
operators, called separable. Separable cubic stochastic operators
$W_(A,B,C)$ were generated by three quadratic matrices A, B and
C. A types of the fixed points and dynamics of one separable
cubic stochastic operator with two parameters were studied on
the two-dimensional simplex.

Keywords: Keywords:Cubic stochastic operator; simplex; separable cubic
stochastic operator; trajectory.

Author: Yuldashev T. K.(National University of Uzbekistan), Egamnazarova M. G.(National University of Uzbekistan), Rasulova S. Kh. (Jizzakh State Pedagogical Institute)

Abstract: 
The problems of the unique classical solvability and the construction of a solution of a multidimensional mixed problem for a homogeneous fifth order partial integral-differential equations with a degenerate kernel and time-reflective argument are studied. The multidimensional Fourier series method, based on the separation of many variables, is used. A system of countable systems of algebraic equations is derived. Iteration process of solving the problem is constructed. Sufficient coefficient conditions for the unique classical solvability of the mixed problem have been established.

Keywords: Mixed problem; integral-differential equation; time-reflective argument; degenerate kernel; Fourier series; classical solvability.

Author: Abbasova M. O. (Namangan State University)

Abstract: In 2012, Wang obtained a whole class of recursive formulas for hypergeometric Appel functions of two variables. Inspired by Wang’s research, in this paper we prove recursive formulas for hypergeometric Lauricella functions of three variables.

Keywords: Appell hypergeometric functions; Lauricella hypergeometric functions in three variables; recursion formulas.

Author:Abdushukurov A. A.(Moscow State University named after M.V.Lomonosov, Tashkent Branch), V.I.Romanovskiy Institute of Mathematics), Erisbaev S. A. (Nukus State Pedagogical Institute)

Abstract: In this article, we consider the competing risks when random censoring occurs at intervals other than observations. We present some useful representations of the Fisher Information Matrix and finally establish performance measures for parameter estimation.

Keywords: random censoring; nonobservation intervals; Competing Risks Model; Fisher information; regularity conditions.

Author: Abdushukurov F. A. (V.I.Romanovskiy Institute of Mathematics, Chupronov A. N.(Almetyevsk State Oil Institute)

Abstract:  Consider the probability P(n,N) of the event: when placing 2n distinguishable particles across N different cells, provided that each cell contains an even number of particles and probability P'(n,N) of an event: when placing 2n indistinguishable particles in N different cells, each cell contains an even number of particles. For various types of convergence $N,n→ ∞$ we study the asymptotic properties P(n, N) and P'(n, N)

Keywords: allocation scheme; Gaussian random variable; Poisson limit theorem; Berry-Essen inequality; limit theorem; local limit theorem.

Author: Abduganiyev A. A.(V.I.Romanovskiy Institute of Mathematics), Bakhramov J. A.  (V.I.Romanovskiy Institute of Mathematics)

Abstract: The speed problem for a controlled heat conduction equation in a rod is studied. By restricting the initial three terms to the Fourier expansion, the problem is reduced to a three-dimensional linear control system and an optimal control is found for it.

Keywords: Heat equation; time optimality; Fourier expansion; finite dimensional reduction; feedback control.

Author: Alimov Z.S. (Fergana State University)

Abstract:  The basis of this article is the unique solvability boundary value problem for a degenerate wave equation in prismatic region. Variable separation method By solving the Cauchy problem for the telegraph equation, as well as using the properties of Legendre polynomials and the Bessel function, the unique solvability of the problem under study was proved.

Keywords: Wave equation; Legendre polynomials; telegraph equation; Fourier-Legendre series.

Author: Boltaev A. K.(V.I.Romanovskiy Institute of Mathematics), Davronov J. R.(V.I.Romanovskiy Institute of Mathematics)

Abstract:  In this work, a discrete analogue of the differential operator $d^4/dx^4 + 1$ is constructed, which is important in finding analytical expressions for the optimal interpolation, quadrature and difference formulas in the Hilbert space $L^(2,0)_2(0, 1)$.
Instead, some important properties of the constructed discrete analogue are proven.

Keywords: Hilbert space; generalized function; operator; error functional, discrete analogue.

Author: Hasanov A. H.(V.I.Romanovskiy Institute of Mathematics),
Ruzhansky M. V. (Ghent University)

Abstract: 
This paper defines a system of partial differential equations of hypergeometric type and their linearly independent solutions for 205 hypergeometric functions of three second-order variables.

Keywords: Hypergeometric functions of Gauss type; system of differential equations; linearly independent solutions.

Author: Imomnazarov Kh. Kh.( Institute of Computational Mathematics and Mathematical Geophysics,), Khujayev L. Kh.(Tashkent University of Information Technologies Karshi branch), Yangiboev Z. Sh.( Karshi State University)

Abstract:  This work is devoted to the study of solvability inverse problem with an unknown nonstationary coefficient for the lowest term of the system of poroelasticity equations and the uniqueness of its solution. The problem is considered in a rectangular area. The conditions for the direct initial boundary value problem and the conditions for redefining the integral are necessary for finding an unknown factor. When proving solvability, parameter continuation methods are used, fixed point, cutoff and regularization.

Keywords: Inverse problem; systems of equations; Darcy coefficient; partial density; regularization; porous medium.

Author: Juraboyev S. S. (Fergana State University)

Abstract:  This article is devoted to solving the problem of describing generators of the differential field of invariant differential rational functions with respect to the action of the group of real representations of symplectic transformations of n-dimensional quaternion space properties and a certain relationship between them.

Keywords: A group of real representations; invariant polynomial; invariant rational function; differential ring; differential field.

Author: Khusanbayev Y. M.(V.I.Romanovskiy Institute of Mathematics), Toshkulov Kh. A.(National University of Uzbekistan) 

Abstract: 
This paper examines Galton’s critical branching processes Watson, starting with a large number of particles in the case where the number is subsequently The size of one particle has infinite dispersion. The asymptotic behavior of finite-dimensional distributions of such processes is found. A recurrence relation has been found, which makes it possible to determine the Laplace transforms of finite-dimensional distributions of a limiting process.

Keywords: Branching process; generating function; weak convergence.

Author: Turdiev Kh. Kh.(Bukhara State University), Suyarov T. R. (Bukhara State University)

Abstract: 
In this article, the reduction of a linear system of equations for an incompressible viscoelastic polymer fluid at rest. The system of linear equations is reduced to canonical form, changes in Fourier transforms with respect to the variable x. In cononic form, a direct problem is posed for a system of incompressible viscoelastic polymer fluid at rest. Replacement tasks -are defined by a closed system of integral equations of the second kind Volterra type. Integral equations of Volterra type with continuous kernel and free term have a continuous solution in a closed region. Thus, the theorem is proved existence and uniqueness of solutions to the problems posed.

Keywords: Hyperbolic system; incompressible viscoelastic polymer fluid; integral equation; integro-differential equations; rheological model of Vinogradov-Pokrovsky.

Author: Abdullaev O. Kh. (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
This paper is devoted to the unique solvability of inverse
problems for a parabolic-hyperbolic equation with fractional
derivative and non-linear load terms. Investigated problems are
equivalently reduced to the Fredholm type non-linear integral
equation. A unique solvability of the obtained non-linear integral
equation has been proved using the method of successive
approximations.

Keywords:  Inverse problem; parabolic-hyperbolic equation
involving Caputo derivatives; non-linear load; non-linear integral
equation; a method of successive approximations; unique
solvability.

Author: Dadajanov R. N. (National University of Uzbekistan)

Abstract: 
An example of a non-computable negative model is constructed,
which is the only constant-generated model of a suitable
computably enumerable set of existential sentences.

Keywords: data model, negative representation, standard
enrichment, existential specification.

Author: Djabbarov N.(Joint Belarusian-Uzbek Intersectoral Institute of Applied Technical Qualifications in the city of Tashkent), Djalilov Sh. (Samarkand Institute of Economics and Service)  

Abstract:  Suppose is a preserving orientation homeomorphism with break at the points and irrational rotation number of bounded type i.e. the elements its continued fraction decomposition are bounded. Let us log has a bounded variation on the circle.
We show that the orbits and of break points diverges.

Keywords: circle homeomorphism; rotation number; break
point; dynamical partition; orbit.

Author: Ikromov I. A.(V.I.Romanovskiy Institute of Mathematics), Safarov A. R. (V.I.Romanovskiy Institute of Mathematics)

Abstract: We consider the problem on uniform estimates for oscillatory
integrals with the smooth phase functions having singularities
of type . The estimate is sharp and analogy to estimates of
the work by V.N.Karpushkin.

Keywords: Phase; deformation; singularity.

Author: Kudaybergenov K. K. (V.I.Romanovskiy Institute of Mathematics)

Abstract: The article is devoted to the history of solution of Ayupov’s problem which was posed by academician Shavkat Abdullaevich Ayupov in the late of twentieth century to
describe derivations of algebras of measurable operators affiliated with von Neumann
algebras. In the early twentieth century, the concept of the classical derivative was
generalized by a number of mathematicians. We shall consider relation between
these generalizations and Ayupov’s problem in the case of abelain algebras. Also
we shall consider a relation with the Wickstead problem posed 50 years ago. The
main results obtained by a number of mathematicians in the process of solving the
Ayupov’s problem and also new questions that have arisen during investigation of
these problems are also considered.

Keywords: Ayupov’s problem; von Neumann algebra; measurable operator; derivation;
inner derivation.

Author: Urinov A. K.(Fergana State University), Mirsaburova U. M. (Termez State University)

Abstract:  In the article, the existence and uniqueness of the solution
of local and nonlocal conditional problems in the boundary
characteristic parts for an equation of hyperbolic type with
singular coefficient degenerating inside the domain has been
proved.

Keywords: hyperbolic type equation with singular coefficient;
fragmentation of boundary characteristics; Bitsadze-Samarsky
condition in boundary characteristics; Non-standard singular
integral equation of Tricomi.

Author: Ergasheva D. A. (National Research University “TIIIMSH Tashkent)

Abstract: In this work, for products of four, six, eight, etc. 2n Pochhammer
symbols, expansion formulas are found in the form of finite
sums of products of two Pochhammer symbols with different
indices. When proving the finite expansions, the properties
of the Pochhammer symbol and the identities established for
hypergeometric functions of several variables are used.

Keywords: Pochhammer symbol; Gaussian hypergeometric
function; hypergeometric function of Kampe de Feriet ; a
high-order hypergeometric function of several variables; finite
expansions for products of Pochhammer symbols.

Author: Abdushukurov A. A. (Mosсow State University named after M.V.Lomonosov, Tashkent Branch), (V.I.Romanovskiy Institute of Mathematics),
Sayfulloyeva G. S. (Navoi State Pedagogical Institute),

Abstract:  The article discusses the estimation of the distribution function using empirical Katz statistics, which is obtained is calculated by replacing the number of terms with independent ones Poisson random variables. We prove Gaussian approximation result for modified versions of the vector-valued estimator.

Keywords: competing risks model; characterization of proportional hazards; empirical estimators.

Author: Aripov M. M.(National University of Uzbekistan), Bobokandov M. M. (National University of Uzbekistan)

Abstract:  In this article, the Cauchy problem for a doubly nonlinear parabolic equation of nondivergent form was studied with the source. It has been proven that in a limited time there is dits a blow-up event to solve the Cauchy problem.

Keywords: quasi-linear parabolic equations; self-similar solution; self-similar equation; global solution; asymptotic behavior.

Author: Aripov M. M.(National University of Uzbekistan), Matyakubov A. S.(National University of Uzbekistan), Khasanov J. O.(Urgench state university)

Abstract: In this paper, we study the conditions for the existence of global solutions to the Cauchy problem in a system of cross-diffusion equations of non-divergent parabolic type with variable density and source, and study the qualitative properties
properties of the asymptotic solution and numerical results were obtained. received.

Keywords: cross-diffusion; system of nonlinear equations; non-divergence; global solution; self-similar solution.

Author: Xadjiyev D.(National University of Uzbekistan), Ayupov Sh. A.(V.I.Romanovskiy Institute of Mathematics), Beshimov G. R.(National University of Uzbekistan)

Abstract:  Let $R$ be the field of real numbers. Let us denote by $E_2$ the two-dimensional Euclidean space over the field $R$. Let $H$ be the set having the smallest
three elements. The definition of the $H$-parametric figure in $E_2$ is given and the definition of the movement of the $H$-parametric figures in $E_2$. A complete system of G-invariants H-parametric figure in $E_2$ for groups $G =O(2,R) and $G=SO(2, R)$ of transformations of the space E_2. Complete received system of G-invariants of motions of an H-parametric figure in $E_2$ for Galilean groups $G = Gal1(2, 1)$ and $Gal1 +(2, 1)$
space transformations.

Keywords: Euclidean geometry; invariant; figure.

Author: Kilichov O. Sh. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  The article studies a boundary value problem for a fourth-order partial differential equation. The article considers a local problem when, in the initial and boundary conditions, In this condition, a high derivative with respect to t is specified, exceeding order of the equation. The Fourier variable separation method is used. The existence of a unique solution has been proven this task.

Keywords: Boundary value problem; Fourier method; existence of the solution; uniqueness of the solution.

Author: Gafforov R. A. (Fergana State University)

Abstract: The criterion of equivalence of regular surfaces with respect to
the action of the special pseudo-orthogonal group in established

Keywords: pseudo-orthogonal group,special pseudo-orthogonal
group,transposed matrix.

Author: Irgashev B. Y. (Namangan Engineering-Construction Institute), (V.I.Romanovskiy Institute of Mathematics)

Abstract:  In the article, self-similar solutions of a high-order equation with
multiple characteristics are constructed. With the help of these
solutions, a solution of a boundary value problem in an infinite
domain for a fourth-order equation is found.

Keywords: High-order equation; self-similar solution; boundary
value problem; Fourier transform; Green’s function.

Author: Mirsaburov M.(Termez State University), Ergasheva S. B.(Termez State University)

Abstract:  In this article, in an unbounded domain, for one class of mixed- type equations with a singular coefficient, we study the well- posedness of the problem when one of the internal characteristics is freed from the Gellerstedt condition and this missing local condition is replaced by an analogue of the Frankl condition on the degeneration line. The uniqueness of the solution of the formulated problem is proved using the extremum principle, and the existence of a solution to the problem is proved by the method of integral equations.

Keywords: Mixed-type equations with a singular coefficient;
unbounded mixed domain; missing Gellerstedt condition;
analogue of the Frankl condition.

Author: Omirov B. A.(V.I.Romanovskiy Institute of Mathematics), Rakhimov I. S.(V.I.Romanovskiy Institute of Mathematics)

Abstract:  This article is devoted to the exposition of standing of the
algebraic school of Sh.A. Ayupov and a short presentation of
its main results, as well as the influence of the this algebraic
school on specialists in the field of non-associative algebras in
other countries. In addition, some recognitions of this scientific
school in Uzbekistan as well as abroad are presented.

Keywords: Leibniz algebra; superalgebra of Leibniz; Zinbiel
algebra; nipotent algebras; differentiation.

Author: Fayziev Yu. E. (National University of Uzbekistan)

Abstract:  The control problem for fractional order equations is studied.
The issue of how to choose a source function for the distribution
as a previously given function of the average temperature in the
time interval [0, T] in a rectangular area is proved.

Keywords: Control problem; fractional equations; source
function.

Author: Khodjibaev V. R.(Namangan Engineering-Construction Institute), (V.I.Romanovskiy Institute of Mathematics), Atakhujaev A. A. (Tashkent branch of the National Research Nuclear University “MEPhI”)

Abstract:  In this paper, we find asymptotic expansion in the powers of
$e^{−b}$ for the distribution of excess over boundary $b → ∞$ under
Cramer condition on the distribution of homogeneous stochastic
processes with independent increments.

Keywords: Stochastic processes with independent increments;
excess over boundary; asymptotic expansion.

Author: Khurramov N. Kh.(Termez State Pedagogical Institute)

Abstract:  In the article, for the equation in a mixed domain, we prove the theorems of existence and uniqueness of the solution of the problem with local and nonlocal
conditions on parts of the boundary characteristic.

Keywords: mixed type equation; partition of the boundary characteristic into two parts; Bisadze-Samarskiy condition on parallel characteristics; Tricomi integral equation with non- Carleman shift in the “nonsingular”part of the kernel; kernel with singularity of the first order at an isolated singular point; Wiener–Hopf equation; index.

Author: Shadimetov Kh. M. (Tashkent State Transport University), Nuraliyev F. A.(V.I.Romanovskiy Institute of Mathematics), Ulikov Sh. Sh.(V.I.Romanovskiy Institute of Mathematics)

Abstract:  In this paper, we find an extremal function of the error
functional of a quadrature formula in the Sobolev quotient space.
The square of the norm of the error functional is calculated,
minimizing the norms, a system of the Wiener-Hopf type is
obtained to find k [β], the existence and uniqueness of the
solution of the resulting system are proved.

Keywords: Quadrature formulas; extremal function; error
functional; Sobolev space.

Author: Ashurov R. R.(V.I.Romanovskiy Institute of Mathematics), Fayziev Y. E.(National University of Uzbekistan), Sulaymonov I. A. (National University of Uzbekistan)

Abstract:  A Cauchy type problem for the subdiffusion equation is considered with Hadamard’s fractional derivative and an arbitrary self-adjoint positive operator. The uniqueness and existence theorems are proved by the Fourier method. Next, The inverse problem of determining the source function is studied. The value of the decision at some point in time $\tau$ is taken as the overdetermination condition

Keywords: Fractional equation; Hadamard derivatives; forward and inverse problems; determination of the source function; the Fourier method.

Author: Azamatov A. Sh. (Urgench state university)

Abstract:  In this paper, an algorithm is derived for finding a solution to the Cauchy problem for a system of the Caup-Boussinesq type with a loaded term using the inverse scattering problem method. Defined evolution of scattering data of a quadratic beam of the Sturm-Liouville equations.

Keywords: Kaup equation, loaded Kaup-Boussinesq type system; quadratic pencil of Sturm-Liouville operators; inverse scattering method, soliton solution.

Author: Hoitmetov U. A. (Urgench state university)

Abstract:  The work is devoted to the integration of the sine-Gordon equation
with variable, time-dependent coefficients and additional members in the fast -decreasing class functions using the inverse scattering problem method. Found
evolution of scattering data for the Dirac operator, whose potential is a solution to this problem. Several examples are given to illustrate application of the obtained results. Also considered integrability of the loaded sine-Gordon equation with self-consistent source.

Keywords: Sine-Gordon equation; Jost solutions; inverse scattering problem; Gelfand-Levitan-Marchenko integral equation; evolution of the scattering data.

Author:  Imomov A.(Karshi State University), Meyliev A.(Karshi State University), Murtazaev M. (V.I.Romanovskiy Institute of Mathematics)

Abstract: Consider a critical homogeneously continuous Markovian a branching system with an average value of branching intensity, equal to one. Our basic assumption is that
The generating function of the branching intensity is correctly changing function in which the character slowly changing factor is concentrated at infinity with explicit form
remainder member. We will establish an estimate for the rate of convergence in the theorem Yaglomov type on approximation of the distribution of normalized population size $Z(t)/t$ to the standard exponential lawas $t → ∞$ on positive trajectories of the branching process. In contrast to previous results, we established the estimate is predominantly based on the condition of finite moment the intensities of the branching law are of order $1 + ν$ for all $ν ∈ (0, 1)$.

Keywords: Markov branching processes; slowly varying functions with remainder; q-matrix; transition probabilities; generating functions; Yaglom theorem; convergence rate.

Author: Karimov R. S. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  This paper considers the extremal function and norm of the error functional of the difference formula. In this work, the extremal function of the error functional is first found. Then an expression for the norm of the functional is obtained errors of the difference formula.

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

Author: Karimov E. T. (Fergana State University), (V.I.Romanovskiy Institute of Mathematics), Turdiyev X. N. (Fergana State University)

Abstract:  In this article, we will prove the unique solvability of the direct and inverse problems with non-classical boundary conditions. expressions for the subdiffusion equation with the generalized fractional derivative of Hilfer order. Using the method of separation of variables and based on the spectral problem with a spectral parameter in the boundary condition, as well as using the solution of the Cauchy problem for a fractional order differential equation with the generalized Hilfer operator, the main scientific result of this article has been proven.

Keywords: Sub-diffusion equation; generalized Hilfer operator; spectral problem with non-classical boundary condition; Mittag-Leffler function.

Author:  Rakhmonov U. S.  (Tashkent State Technical University)

Abstract:  This work gives a description of automorphisms of the matrix ball $\[B_{m,n}^{\left( 3 \right)}\]$ associated with classical domains third type, some properties are also given matrix ball of the third type.

Keywords: Classical domain; matrix ball; automorphism of the matrix ball; Shilov’s boundary.

Author:  Utebaev D.(Karakalpak State University), Nurullaev Zh. A. (National University of Uzbekistan)

Abstract:  In this work, multiparameter difference schemes of the finite element method of high order accuracy for the equation Sobolev type of fourth order. In particular, boundary value problems for the equation of waves from a compressible stratified rotating fluid are considered. A high order of accuracy of the scheme is achieved through special discretization of time and spatial variables. The presence of parameters in the scheme allows you to regularize schemes with the purpose of optimizing the algorithm implementation and accuracy. Using the method of energy inequalities, an a priori estimate in a certain weak norm is obtained. Based on this estimate and the Bramble-Hilbert lemma, we prove the convergence constructed schemes in classes of generalized solutions. Algorithm proposed implementation of the difference scheme.

Keywords: Sobolev type equation; difference schemes; finite difference method; finite element method; stability; convergence; accuracy.

Author: Allakov I.(Termez State University), Muzropova N. S. (Termez State University)

Abstract: A new estimate is obtained from below for the numbers presentations of a natural number, in the form of the sum of the five squares of prime numbers.

Keywords: Dirichlet character; prime numbers; basic intervals; additional intervals; estimation; equation; solvability; nontrivial zeros; exceptional zero.

Author: Allambergenov A. Kh.(Nukus State Pedagogical Institute)

Abstract:  This paper is devoted to the study of local derivations on Okubo
algebras.

Keywords: Okubo algebras; derivation; local derivation.

Author: Ashurov R. R.(V.I.Romanovskiy Institute of Mathematics), Fayziev Yu. E. (National University of Uzbekistan), Khushvaktov N. Kh. (National University of Uzbekistan)

Abstract:  It is considered direct problems for a fractional equation in partial derivatives of the Barenblatt-Zheltov-Kochina type. Given conditions that are guarantee the existence and uniqueness solution of the Cauchy problem.

Keywords: Cauchy problem; existence and uniqueness of the solution.

Author: Azamov A. A.(V.I.Romanovskiy Institute of Mathematics), Abdullaev A. X.(V.I.Romanovskiy Institute of Mathematics), Tilavov A. M. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  The article is devoted to the filling a logical gap in the derivation of an inequality that gives an error in the approximate solution of the initial value problem using the Taylor expansion method.

Keywords: Initial value problem; approximate solution; the Taylor formula; accuracy of estimate; stringency of a proof; inequality.

Author: Azizov M.S.  (Fergana State University)

Abstract:  In this article, for a partial differential equation of high even order with a
Bessel operator in a rectangle, an initial boundary value problem has been
formulated. Applying the method of separation of variables to the considered
problem, a spectral problem was obtained for an ordinary differential equation
of high even order. The solution of the considered problem study was found
as the sum of a Fourier series with respect to the system of eigen-functions of
the spectral problem. The uniform convergence of this series, as well as the
series obtained from it by term-by-term differentiation, have been proved. The
uniqueness of the solution of the problem has been proved by the method of
energy of integrals. An estimate for solution of the problem was obtained, from
which follows its continuous dependence on the given functions.

Keywords: initial-boundary value problem; spectral problem; the existence;
uniqueness and stability of the solution; the method of separation of variables.

Author:    Eshkobilova D. T. (Termez State University)

Abstract:  In the present paper for a given topological transformation group (G, X, α) a topological transformation group (I(G, X), I(X), I(α)) is built. The group I(G, X) is equipped such a topology, the induced topology from it to the group G coincides with the initial topology on G. Further, it is shown that the functor of idempotent probability measures preserves a property of maps to be equivariant (in particular, to be
equivalent).

Keywords: Topological transformations group; idempotent
probability measure; equivariant map.

Author: Kadirkulov B. J.(Tashkent State Institute of Oriental Studies), Jalilov M. A.(Fergana State University)

Abstract:  In this article, it is the fourth that breaks down in a rectangular area nonlocal posed problem for an order mixed-type equation studied. The conditions for the existence and uniqueness of the solution to the problem are determined, and the continuous dependence of the found solution on the given ones is shown

Keywords: mixed-type equation; nonlocal boundary value problem; the existence and uniqueness of a solution; operator of fractional integro-differentiation; operator Kaputo; the Kilbas-Saigo function.

Author: Karimov N. R. (National University of Uzbekistan)

Abstract: It is proved that every negative representable data structure has an enrichment, which is the only model built from constants for a suitable computably enumerable set of existential sentences. A counterexample is given for sets of universal sentences.

Keywords: computability; negative representability; standard numbering; negative and positive diagram; universal and existential definability.

Author: Karimov Sh. T.(Fergana State University), Oripov Sh. A.(Fergana State University), Khujakhonov Z. Z. (Fergana Polytechnic Institute)

Abstract: In this paper we develop a method for investigating the Cauchy problem for a degenerate differential equation of high even order. Applying the generalized Erd ́elyi – Kober operator, the formulated problem reduces to the problem for an equation without degeneracy. Necessary and sufficient conditions for reducing the order of the equation are proved to be twofold. Two examples demonstrating the application of the developed
method are given.

Keywords: Fractional integrals and derivatives; generalized Erdelyi-Kober operator; Bessel differential operator; degenerate differential equations.

Author:  Kurbanov O. T. (Tashkent State Economic University)

Abstract: The regular solvability is investigated of the problem for a third-
order equation with multiple characteristics.

Keywords: A equation with multiple characteristics; a boundary
value problem; integral condition; method of integral equations.

Author: Muminov U. B(Samarkand State University), Hudayorov U. O. (Samarkand Institute of Architecture and Construction)

Abstract:  In this paper, a defocused nonlinear Schrodinger equation with source and loaded terms using the inverse spectral problem method is integrated in a class of infinitely zoned periodic functions. When the coefficients of the periodic Dirac operator consist of a solution of the loaded term and the source defocused nonlinear Schrodinger equation, the evolution of its spectral data is given.

Keywords: Nonlinear Schrodinger equation; Dirac operator; spectral data; inverse spectral problem; Dubrovin’s system of equations; trace formulas.

Author: Sharipov O.Sh.(National University of Uzbekistan),(V.I.Romanovskiy Institute of Mathematics), Muhtorov I.G. (National University of Uzbekistan)

Abstract: In this paper moment inequality for the sum of weakly dependent random variables with values in type $p$ Banach spaces is proved.

Keywords: Moment inequality; type p Banach space; weakly
dependent random variables.

Author: Sirojitdinov A. A. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  In this paper, we obtain an estimate for the rate of convergence in the central limit theorem without any moment conditions on terms of random variables. The results obtained are generalized and refined by one of the results of $W$. Feller.

Keywords: Truncated random variables; distribution function; Berry-Essen theorem.

Author: Yuldasheva N. T. (V.I.Romanovskiy Institute of Mathematics)

Abstract: For a mixed-type equation with a partial fractional Riemann- Liouville derivative, a non-local boundary value problem with a generalized fractional integro-differentiation operator in the boundary condition is studied. The uniqueness of the solution of the problem is proved using the extremum principle for a nonlocal parabolic equation and for operators of fractional differentiation in the sense of Riemann–Liouville. To prove the existence of a solution, the problem is reduced to solving a differential equation of fractional order.

Keywords: Boundary value problem; fractional order differential equation; Gauss hypergeometric function; principle of extremum; uniqueness of a solution; existence of a solution; singular coefficient.

Author: Boltaev Kh. Kh.(Tashkent State Pedagogical University)

Abstract:  This article shows that any hyperfinite factor has an involutive *- antiautomorphism. It has been proven that a real subfactor is irreducible if and only if its enveloping factor is irreducible. Using the constructed examples in the complex case, as well as an involutive *-antiautomorphism of the W*-algebra; similar examples are constructed for real factors.

Keywords: real W*-algebra; real subfactor; index of subfactor; graphs.

Author: Dadakhodjaev R. A.(V.I.Romanovskiy Institute of Mathematics), Zokirov F. M(Tashkent State Pedagogical University)

Abstract:  In this note we will describe ring isomorphisms
between algebras of measurable operators associated with
real W∗-algebras of type $II_1$.

Keywords: Real W*-algebra; algebra of measurable operators; ring isomorphisms.

Author: Ergashova Sh. R. (National University of Uzbekistan)

Abstract: In this article, we will prove that the product of discretely generated spaces is not discretely generated. In addition, we have constructed a topological space as a counterexample that is sequential but not discretely generated.

Keywords: discretely generated space; Frechet-Urysohn space; sequential space.

Author: Hayotov A. R.(V.I.Romanovskiy Institute of Mathematics), Khayriev U. N.(V.I.Romanovskiy Institute of Mathematics), (Bukhara State University)

Abstract:  In this work in the Hilbert space periodic complex-valued functions, an optimal quadrature formula is constructed with exponential. The optimality of the direct formula is shown polygons in periodic space functions for ω = 0 and m ≥ 2. In addition, numerical results show that the order of approximation of the optimal quadrature formula constructed in space is equal to 2.

Keywords: Optimal quadrature formula; periodic function; Hilbert space; extremal function; error functional.

Author:  Rajabov S. M. (V.I.Romanovskiy Institute of Mathematics)

Abstract: The article studies the dynamics of a non-Volterian quadratic operator in a two-dimensional simplex. For such an operator sets of periodic and fixed points were found, and sets of limiting points of trajectories are also described.

Keywords: quadratic stochastic operator; Volterra operator; non-Volterra operator; trajectory.

Author: Abdushukurov A.A. (Branch of Moscow State University in Tashkent) ,Sayfulloyeva G.S. (Navoi State Pedagogical Institute)

Abstract: This article considers the modified Kac estimates of integral
intensity function in the general random censorship model and
its approximation with Gaussian processes.

Keywords: Kac estimates; Gaussian process; integral hazard function; empirical estimates.

Author: Allakov I.(Termez State University), Abraev B. Kh.(Termez State University)

Abstract: The paper studies the question of simultaneous representation of a pair of natural numbers by the sum of four primes. A new power-law estimate is proved for the number of pairs that are unrepresentable in this form, i.e. for an exceptional set of the
problem under consideration.

Keywords: equation; system of linear equations; prime numbers; solvability criterion; estimation; power estimation; Dirichlet series; Dirichlet character; exceptional zero.

Author: Ahmadov I.A. (National University of Uzbekistan)

Abstract:  In this paper, strong solvability, and Volterness of a boundary problem with local condition for an equation of mixed parabolic- hyperbolic type with a Gerasimov-Caputo fractional order differential operator has been considered.

Keywords: Strong solvability; local boundary conditions; equation of parabolic-hyperbolic type of fractional order in a sense of Gerasimov-Caputo; Wright and Green’s functions; conjugate problem.

Author: Boltaev A. K.  (V.I.Romanovskiy Institute of Mathematics)

Abstract:  In the present paper the norm of the error functional of the
optimal quadrature formulas was calculated in the Hilbert space when m is odd, and the existence and uniqueness of the obtained system solution for the optimal coefficients was proved.

Keywords: extremal function; error functional; Hilbert space; quadrature formulas.

Author:  Djamalov S. Z.(V.I.Romanovskiy Institute of Mathematics), Turakulov Kh. Sh. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  In this paper, it is investigated a linear inverse problem with a
nonlocal boundary condition for the three-dimensional Tricomi
equation in unbounded prismatic domain. In the considered
domain, the uniqueness and existence of the generalized solution
of this problem have been studied by using the methods of “ε
-regularization” , a priori estimates, sequence of approximation
and Fourier transform.

Keywords: three-dimensional Tricomi equations; linear inverse
problem with nonlocal boundary condition; problem correctness;
methods “ε -regularization” ; a priori estimates; sequence of
approximation; Fourier transforms.

Author: Zarifzoda S. K. (Tajik National University)

Abstract: The paper is devoted to the investigation of a class of the operator-differential equations, constructed by spacial differential operator. The technique of integral transformation is a main tool. In three cases, depending of the roots of the corresponding characteristic equation, the explicit solutions of the investigated equation in terms of an elementary functions are found.

Keywords: Special differential operator; integral transformation;
operator-differential equation; characteristic equation.

Author:  Ibragimov G. I.(Universiti Putra Malaysia), Qushaqov H. Sh.(Andijan State University), Muhammadjonov A. A. (Andijan State University)

Abstract: In this paper, we consider an infinite system of 2-systems of differential equations with nonnegative coefficients. The system consists of 2-systems corresponding to 2 × 2 Jordan blocks. We investigate the system of differential equations in Hilbert space $l_2$. We prove the existence and uniqueness of the solution of system of differential equations in a specified space of functions.

Keywords: Differential equations; infinite system; existence and uniqueness of solution; Hilbert space.

Author: Mamajonov S.M.(V.I.Romanovskiy Institute of Mathematics)

Abstract: In this paper, for an non-homogeneous fourth-order equation
with lower terms with constant coefficients, one boundary value
problem in a rectangular domain is considered. The uniqueness
of the solution to stated problem is proved by the method
of energy integrals. The solution is written in terms of the
constructed Green’s function. In substantiating the uniform
convergence, the “small denominator” is established to be
nonzero.

Keywords: Differential equation; fourth order; boundary value
problem; multiple characteristics; uniqueness; existence; Green’s
function.

Author:  Mamatov A. R. (Samarkand State University)

Abstract:  An algorithm for solving the problem of transferring the
trajectory of a dynamical system controlled by two players whose
integers are opposite from a given point to a terminal set is
proposed.

Keywords: the game; maximin problem; dual problem; support; algorithm

Author: Nurmukhamedova N. S. (National University of Uzbekistan)

Abstract:  In this paper we consider an inhomogeneous competing risks
model. For the likelihood ratio statistics, proved the theorem
on the local asymptotic normality in the competing risk model
corresponding to inhomogeneous and randomly right-censored
observations.

Keywords: Competing risks model; random censoring; local
asymptotic normality; likelihood ratio statistics.

Author: Khalkhuzhaev A.M. (V.I.Romanovskiy Institute of Mathematics), Mahmudov H.Sh. (Samarkand State University)

Abstract: We consider the Hamiltonian $hˆμ, μ > 0$, of a system of two quantum particles (fermions) interacting at the nearest neighbor sites of the $d ≥ 1 $- dimensional lattice $Z$. Found convergent expansions of the eigenvalues the corresponding two-particle Sch ̈odinger operator $hμ(k)$, are obtained, $k ∈ T$ is the two-particle total quasimomentum, in the left neighborhood of the essential spectrum for small , where $μ$ are threshold values of the coupling constant.

Keywords: Two-particle discrete Schr ̈odinger operator; system of two quantum particles; eigenvalue; essential spectrum; decomposition.

Author:  Khodjamuratova I. A.(National University of Uzbekistan)

Abstract: It is established that a translationally complete finitely
generated algebra is representable over any negative equivalence.
It is shown that this is not true for positive equivalences. The
questions of representability of semigroups over equivalences are
also considered.

Keywords: Universal algebra, equivalence, numbered algebra,
negativity, positivity, translation complete algebra, semigroup.

Author: Sharipov O. Sh.(National University of Uzbekistan), Kobilov U. Kh.(National University of Uzbekistan) 

Abstract:  Limit theorems for the maximum of weakly dependent random
variables are proved.

Keywords: Maximum; weakly dependent random variables; limit
theorems.

Author: Ergashev T. G.(National Research University TIIIMSH), Vokhobov F. F.(Kokand State Pedagogical Institute), Makhmudov B. B. (Kokand State Pedagogical Institute)

Abstract:  Well-known Horn’s List, which consisted of 34 hypergeometric functions of two variables of the order two, is divided into two parts: the first part of the list consists of complete hypergeometric functions, and the second part is formed from all possible confluent hypergeometric functions, which are the limit forms of the complete functions included in the first part of the list. Published in 1985, Srivastava and Karlsson’s List of 205 complete hypergeometric functions of three variables of the order two is analogous to the first part of Horn’s List, which included 14 complete hypergeometric functions of two variables. In this paper, confluent hypergeometric functions of three variables of the order two are defined as limit forms of complete functions from the Srivastav-Karlsson List.

Keywords: Horn’s List; a complete and confluent hypergeometric
series of the order two; Srivastava-Karlsson’s List; the confluent
hypergeometric series of three variables.

Author:  Erisbaev S. A. (Nukus State Pedagogical Institute)

Abstract: In the model of hybrid form of competing risks, when random variables censored from the right by I and II type of censoring we give methods of calculating of Fisher information function.

Keywords: Fisher information; Cramer-Rao inequality; competing risks
model; type I and II censoring.

Author: Eshmamatova D. B.(Tashkent State Transport University), Tadjieva M. A.(National University of Uzbekistan), Yusupov F. A.(Tashkent State Transport University)

Abstract:  In the paper, we consider the asymptotic behavior of trajectories of Lotka-Volterra mappins, with transitive tournaments operating in a two-dimensional simplex. Fixed points are found for compositional mapping , and the characters of these points
are studied

Keywords: Composition operator; simplex; transitive tournament; fixed point; spectrum.

Author: Mirzaeva M. M. (National University of Uzbekistan)

Abstract: 
This work presents a solution to the boundary value problem of an integro-differential equation with the operator Caputo-Fabrizio obtained by the Green’s function method. Here we consider integro-differential equations, which are reduced to differential equations order logo. The resulting solution was used to solve the Dirichlet problem for the partial differential equation wave type. Note that the result can be used in the study of inverse problems for partial differential equations.

Keywords: Caputo-Fabrizio integral-differential operator; Green’s function; integral-differential equation; wave-type equation; differential equation with variable coefficient; Gilbert’s theorem.

Author:  Sharipov A. S.(Tashkent Branch of the National Research Nuclear University MEPHI),  Topvoldiyev F. F. (Fergana State University)

Abstract: In this article, invariants of surfaces isometric along sections are found. In particular, the invariance area of ​​the cylindrical image, when displayed preserving isometry along sections.

Keywords: Isometry on sections, isometry; spherical image; cylindrical image; completely additive function.

Author: Allakov I.(Termez State University), Ismoilov M. (Termez State University)

Abstract: 
In the work, a general formula for the dependence on other parameters is derived, which is constant involved in the remainder in the asymptotic formula, proved by L.Dirichlet for the number of points with integer coordinates in a circle with radius √N. Using this dependence formula, the numerical value of this constant is determined. Several corollaries of independent interest were derived from the general theorem. It is known that in many problems of number theory various estimates or asymptotic formulas are used. But they usually involve the symbols “o” or “O”. There are separate tasks that need precise values of the constants. In such problems it is impossible to use from estimates or asymptotic formulas that involve the symbols “o” or “O”. In such cases, one can use results of the type obtained in the present work.

Keywords: arithmetic function, multiplicative function, primes, composite numbers, canonical expansion, number of divisors, sum of divisors, points with integer coordinates, asymptotic formula, symbols “o” and “O”, estimate.

Author: Abdushukurov A. A.(Branch of Moscow State University named after M.V. Lomonosov in Tashkent),() Kakadjanova L. R. (Branch of Moscow State University named after M.V. Lomonosov in Tashkent)

Abstract: 
For empirical characteristic process of independence limit Gaussian process is established. For testing of zero hypothesis some statistics are presented.

Keywords: empirical characteristic process; metric entropy; law of iterated logarithms.

Author: Kadirkulov B. J.(Tashkent State Institute of Oriental Studies), Kayumova G. A.(Karshi Engineering Economic Institute)

Abstract: 
In this article, non-local problem is investigated for the second-order mixed-type equation with the Hilfer operator in rectangular domain. The conditions for the existence and uniqueness of the solution of the problem are defined and it is shown that the solution depends continuously on the given functions.

Keywords:mixed type equation; non-local boundary-value problem; the existence and a uniqueness of a solution; operator of fractional integro-differentiation; Hilfer operator; Mittag-Leffler function.

Author:  Rajabov E. O.(National University of Uzbekistan),(Tashkent State Technical University), Sharipov Kh. F.(National University of Uzbekistan)

Abstract: 
In this paper it is studied geometry of foliation generated by orbits conformal vector fields.

Keywords: Conform vector field; orbit; foliation; invariant function.

Author: Urinov A. K.(Fergana State University), Usmonov D. A. (Fergana State University)

Abstract: 
In the work it was found forms of the general solutions of a second kind equation and it was showed that satisfying these solutions to the equation. Using representations of the general solutions, it was formulated and investigated initial problems for considered equation.

Keywords: hyperbolic equation; degenerated equation; second kind equation; general solution; Cauchy problem.

Author: Khamdamov I. M. (National University of Uzbekistan)

Abstract: 
The role of the maximum term in the formation of the probability that the truncated sum reaches the level xn, when xn increases in a certain way, is investigated in the article. Here, the tail of the distribution of the original random variable is regularly varying function.

Keywords: truncated sum; extremal terms; variation series; probabilities of large deviations.

Author: Khalilov K S. (Fergana State University)

Abstract: 
In the work, it has been formulated and investigated non-classical problems with integral conditions for a third order parabolic-hyperbolic equation in a mixed domain. It is proved the uniqueness and existence of the solution of the considered problems.

Keywords: parabolic hyperbolic equation; integral condition; uniqueness of the solution; existence of the solution.

Author: Absalamov A. Т.(Samarkand State University), Rozikov U. A.(V.I.Romanovskiy Institute of Mathematics)

Abstract: 
In this paper, we study dynamical systems generated by the gonosomal evolutionary operator of a bisexual population. Explicit types of all (infinite set) fixed points of the operator are found. It is shown that each fixed point has its own values ​​less than or equal to 1. In addition, it has been proven that each trajectory converges to a fixed point, i.e. operator is regular. There is an infinite number of invariant sets, each of which contains a single fixed point. Thus, there is a one-to-one correspondence between such invariant sets and the set of fixed points. Any trajectory starting at some point of the invariant set converges to the corresponding fixed point.

Keywords: Dynamical systems; fixed point; invariant set, limit point.

Author: Durdiev D. Q.(V.I.Romanovskiy Institute of Mathematics), Boltaev A. A.(Bukhara State University)

Abstract: 
For a two-dimensional system of integro-differential viscoelasticity equations in an isotropic medium, the direct problem of determining the stress vector and particle velocity is studied. The problem is reduced to an equivalent system of integral equations of the second kind of Volterra type with respect to Fourier transform with respect to one of the spatial variables of the solution to the direct problem. Next, the method of successive approximations in the class of continuous functions is applied to this system. Thus, the existence and uniqueness of a solution to the problem posed is proved.

Keywords: hyperbolic system; direct problem; integral equation; fixed point theorem.

Author: Jalilov A. A.(V.I.Romanovskiy Institute of Mathematics)

Abstract: 
Erman Mapping and Sturm Sequences Let $f$ be a piecewise linear homeomorphism of a circle that preserves orientation with two break points $a^(0), c^(0)$ and an irrational rotation number $ρf$. Let us denote by $σf (xb):= {Df−(xb)}/{Df+(xb)}, $x_b = a^(0), c(0)$, the jump coefficients $f$ at two discontinuity points and $ σf := σf (a^(0))·σf (c^(0))$ is its general jump coefficient. Such mappings were first studied by M. Erman. Invariant measures, renormalizations, and rigidity properties play an important role in studying the behavior of the sequence $Df^n(x), n ≥ 1.$ We show that the encoding sequence associated with $t_n := Df^n(x) mod1, ∈ S ^1, n ≥ 1$, is Sturmovskaya.

Keywords: Mellin transform; Lerch function; Logarithm function; Contour Integral.

Author: Reynolds  M. R. (York University)

Abstract: 
Finding the Mellin transform for applying the contour integral method in the study of integrals introduced by Anatoly Prudnikov and representing them through Lerch functions is the main result of this article. A table of definite integrals that were not included in the known literature and can be added to new similar literature is given.

Keywords: Mellin transform; Lerch function; Logarithm function; Contour Integral.

Author:Javliyev S. K.(National University of Uzbekistan), Xodjamuratova I. A.(National University of Uzbekistan)

Abstract: 
The existence of a natural subclass of the class of subdirectly indecomposable algebras is established, any Hausdorff enumerations of which are negative.

Keywords:Subdirectly indecomposability; computable and effective topologies; topological enumerated algebras; positivity; negativity; effective separability.

Author: Djalilov A. A.(Turin Polytechnic University in Tashkent), Abdukhakimov S. Kh.(National University of Uzbekistan)

Abstract: 
In the theory of Feigenbaum’s universality the Feigenbaum’s and its unstable separatrice passing through. The unstable separatrice is the one-parameter family of unimodal maps of the interval [−1, 1]. in the present paper we study the sequence
$x_n = g(x, t) + ξ_n, n ≥ 1 $ of small stochastic perturbations of maps g(x, t) from unstable separatrice Γ (u)(g). It is proved the theorem on behaviour of variance of linear part of random values x_n.

Keywords: Feigenbaum map, unstable separatrice, stochastic perturbations, unimodal map.

Author: Kurbanov Kh. Kh.(Academy of the Armed Forces of the Republic of Uzbekistan), Zaitov A. A.(Tashkent Institute of Architecture and Construction)

Abstract: In the paper it is given a description of the space of normed, max-plus-homogenous, max-plus-semiadditive functionals. It is shown some natural Z- and σ-Z-sets, and Fσ- and Gδ-sets of the space of normed, max-plus-homogenous, max-plus-semiadditive functionals.

Keywords: Normed, max-plus-homogenous, max-plus- semiadditive functional; Z-set; Fσ-set; Gδ-set.

Author: Kilichov O. Sh.  (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
In this paper a boundary value problem for a fourth order partial differential equation is investigated. The existence of unique solution of this problem is proved.

Keywords: boundary value problem; method of Fourier; the existence of the solution; the uniqueness of the solution.

Author: Kulturayev D. J.(Karshi State University)

Abstract:  The discrete spectrum of self-adjoint operators is studied in the Friedreich model. Sufficient conditions are given the existence of an infinite number of eigenvalues in the Friedreich model. The infinity of negative eigenvalues of one discrete Schrodinger operator.

Keywords: Friedreich model; spectrum; essential spectrum; discrete spectrum, non-degenerate kernel

Author: Mukhamedov A. K.(National University of Uzbekistan)

Abstract: In the paper almost sure convergence of the empirical survival function estimates of stationary random variables of linearly positive quadrant dependent sequences is proved.

Keywords: empirical survival function, strong convergence, linearly positive quadrant dependence.

Author: Sagdullayeva M. M. (National University of Uzbekistan)

Abstract: 
In this paper, we prove a unique solvability of a initial–boundary value problem with integral condition for the third–order partial differential equation with heat operator in main part.

Keywords: A boundary value problem; nonlocal condition; nonlocal problem; parabolic equation; Green’s function; integral equations.

Author: Srajdinov I. F. (Almalyk branch of Tashkent State Technical University)

Abstract: 
In this article, a problem with initial and boundary condition for a composite system if formulated. Solution of this problem is represented by series and the convergence of these series are proved.

Keywords: Initial-boundary value problem; system of equations of component type; Fourier series; initial condition; elliptic and hyperbolic type equations.

Author: Khamdamov I. M. (National University of Uzbekistan)

Abstract: 
This work is devoted to the study of the properties of convex hulls generated by independent observations of a random vector with a homogeneous Poisson distribution in a cone on the plane.

Keywords: Convex hull; Poisson point process; random vector.

Author: Hayotov A. R.(V.I.Romanovskiy Institute of Mathematics), Xayriyev U. N.(V.I.Romanovskiy Institute of Mathematics), Maxkamova D.(National University of Uzbekistan)

Abstract:  In the present paper, in the Sobolev space $L ̃ ^(2)_2 (0, 1]$ of periodic,
complex-valued functions, we construct an optimal quadrature formula of the form (2). We obtain an approximation formula on the basis of the optimal quadrature formula. Here, this formula is used to numerical calculation of the Fourier transform of some functions. The examples show that the order of approximation of the obtained approximation formula is 2.

Keywords: Optimal quadrature formula; periodic function; Sobolev space; extremal function; discrete argument function.

Author: Choriyeva S. T. (Termez State University), Bobomurodov U. E.(Termez State University)

Abstract: 
In this paper, the existence and uniqueness of the solution of a nonlocal boundary problem with general conjugation condition with the value of the boundary characteristic of the solution sought with the values of n special lines lying within the field connected with Bitsadze-Samarskiy condition for a hyperbolic equation degenerating inside the domain with singular coefficients are proved.

Keywords: hyperbolic equation with a singular coefficient degenerating inside the domain; general connection condition; n special lines lying within the field; Bitsadze-Samarskiy condition.

Author: Ergashev T. G.(Tashkent Institute of Irrigation and Agricultural Mechanization Engineers), Ergashev O. T. (Tashkent Institute of Irrigation and Agricultural Mechanization Engineers)

Abstract: 
Fundamental solutions of the generalized bi-axially symmetric elliptic equation are expressed in terms of the well-known Appell hypergeometric function in two variables, the properties of which are required to study boundary value problems for the above equation. In this paper, by using some properties of the Appell hypergeometric function $F_2$, we prove limiting theorems and derive integral equations concerning a denseness of the double- and simple- layer potentials. We apply the results of the constructed potential theory to the study of the generalized Holmgren problem for the two-dimensional elliptic equation with two singular coefficients in the domain bounded in the first quarter of the plane.

Keywords: Appell hypergeometric function in two variables; generalized bi-axially symmetric elliptic equation; potential theory; generalized Holmgren problem.

Author: Eshimbetov M. R. (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
For some extensions of idempotent τ -smooth measures it is proved that they are idempotent τ -smooth measures. Further, for each idempotent τ -smooth measure any Luzin measurable function induces an idempotent τ -smooth measure.

Keywords: Luzin measurable function; idempotent τ -smooth measure.

Author: Aripov M. M.(National University of Uzbekistan), Muqimov A. Sh.(National University of Uzbekistan)

Abstract: In this work we study the asymptotics (for t → ∞) solutions to a system of semilinear heat conduction problems with absorption at a critical parameter. The asymptotic behavior was established using the standard equation method. The proofs were carried out using the method comparison of solutions and the maximum principle.For numerical calculations, we used the long-time asymptotic behavior of the solution as an initial approximation. Numerical experiments and visualizations were carried out for one-dimensional and two-dimensional cases.

Keywords: heat conduction problem; semileinear system; critical value of parameter; absorption; maximum principle; numerical computation; visualization.

Author: Khudoyberdiyev A. Kh. (National University of Uzbekistan), Shermatova Z. Kh.(V.I.Romanovskiy Institute of Mathematics)

Abstract: In this article we look at some properties perfect Leibniz algebras. We distribute some results obtained for perfect Lie algebras, in the case of Leibniz algebras.

Keywords: Leibniz algebra; ideal; nilradical; radical; center; derivation.

Author: Rozikov U. A.(V.I.Romanovskiy Institute of Mathematics), Hamidov Sh. Sh. (Bukhara State University)

Abstract: 
We study p-adic dynamical systems generated by function a $\frac{a}{x^2+1}$ to the p-adic field Cp of complex numbers. This function has three fixed points. We find some regions of the parameters a and p where the fixed point is attracting (or indifferent, or repulsive). We construct the corresponding Siegel disks of these fixed (and some 2-periodic) points and find a sufficiently small set containing the set of limit points.

Keywords: Rational dynamical systems; fixed point; Siegel disk; complex p-adic field.

Author: Akhmadaliyev G. N.(Tashkent State Transport University)

Abstract: Using S.L. Sobolev we will construct a spline interpolation, minimizing the seminorm in K_2(P_2) , where K_2(P_2) is space of functions φ such that φ’ absolutely continuous but, φ” belongs to L_2 (0, 1) and $\int_0^1(φ”(x) − ω^2 φ(x))^2dx < ∞$. Explicit formulas for the coefficients of the interpolation spline are obtained. The resulting interpolation spline is accurate for the hyperbolic functions sinhx and coshx. Finally, several numerical examples present the qualities of the resulting spline.

Keywords: Interpolation splines; Hilbert space; seminorm minimizing property; S.L. Sobolev’s method; discrete argument function; discrete analogue of a differential operator.

Author: Tukhtabayev A. M. (Namangan State University), Bunazarov Kh. K. (Namangan Institute of Engineering-Construction)

Abstract: 
In this paper, we study the existence of a prime number p such
that a fixed integer has a square root in the field Q_p. Moreover,
we show that there exists a polynomial such that it has roots in
R and in all Q_p, at the same time, it has not any root in Q.

Keywords: p-adic number; congruence; quadratic residue; Legendre symbol.

Author: Artiqbayev A.(Tashkent State Transport University), Mamadaliyev B. M.(Fergana State University)

Abstract: 
In this paper, the subspaces geometries and manifolds of a five-dimensional pseudo-Euclidean space with two index are investigated. Geometry in the sphere with real
radius is defined. The existing geometries of three and four dimension subspaces are shown, as well as the measure of manifold with maximum dimension is proved.

Keywords: Manifold; subspace; five-dimensional pseudo-Euclidean space; pseudo-isotropic spaces; hyperbolic spaces; hyperplane; norm; distance; degenerate metric;
isotropic vectors; sphere of real; imaginary; zero radius; isotropic cone.

Author: Boltayev A. K. (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
In the present paper, in the$W_2^{(3,0)}(0,1)$ Hilbert space the first part of the problem of construction of optimal interpolation formulas is solved, i.e. norm of the error functional of a interpolation formula in the $W_2^{(3,0)}(0,1)$ space is calculated.

Keywords: extremal function; error functional; interpolation; Laplace transform.

Author:  Djamalov S. Z. (V.I.Romanovskiy Institute of Mathematics), Ashurov R. R.(V.I.Romanovskiy Institute of Mathematics), Turakulov H. Sh. (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
In this paper, it is studied the unique solvability and smoothness of the generalized solution of nonlocal boundary value problem of periodic type for the three-dimensional Tricomi equation in an unbounded prismatic domain by the methods of “ε –
regularization” and a priori estimates are defined by using the Fourier transform.

Keywords: Tricomi equation; nonlocal boundary value problem of periodic type; Fourier transform, “ε -regularization”method; a priori estimates.

Author: Imanbaev N. S.( South Kazakhstan State Pedagogical University), Kurmish E.(South Kazakhstan State Pedagogical University)

Abstract: 
In this paper, we consider the question on computation of eigenvalues and eigenfunctions of a third-order composite type equation in a rectangular region
D of the space W^3_2 (0, 1) satisfying the following boundary conditions where D = {x, y : 0 < x < 1, 0 < y < 1}. All eigenvalues and eigenfunctions of the considered spectral problem are found, looking for a solution of the problem by the Fourier method. Characteristic determinant of the spectral problems is an entire analytic function, which coincides with the exponential type quasi-polynomial with commensurable exponents, a conjugate indicator diagram of the function, which is a regular hexagon on the complex plane and the adjoint operator is constructed.

Keywords: composite type equation; regular, periodic boundary value conditions; eigenvalyes; eigenfunctions; adjoint operator; characteristic determinant; zeros of entire functions.

Author: Madirimov M.(Tashkent State Pedagogical University), Zaitov A. A.(Tashkent Institute of Architecture-Construction)

Abstract: 
In the paper it is shown that each group of topological transformations on a Tychonoff (in particular, a compact Hausdorff) space generates a group of topological transformations on the space of probability measures. Further, it is proved that a continuous map between spaces of probability measures is equivariant if the map between the original Tikhonov spaces, which induces it, is equivariant. Hence it follows that a continuous map between spaces of probability measures is an equivalence if the map between given Tychonoff spaces, which induces it, is an equivalence.

Keywords: A group of topological transformations; probability measure; equivariant map.

Author: Muratova Kh. A. (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
This article devoted to the description of solvable Leibniz superalgebras with four dimensional nilradical. In this paper we describe the derivation of four-dimensional nilpotent Leibniz superalgebras with nilindex 4 and classify solvable Leibniz superalgebras with such nilradicals.

Keywords: nilindex; nilradical; derivation; Leibniz superalgebras; solvable Leibniz superalgebras.

Author: Safarov J. Sh. (V.I.Romanovskiy Institute of Mathematics), (Tashkent University of Information Technologies)

Abstract: 
In this paper, we prove a unique solvability of a boundary value problem with integral form conjugation condition on the line of type changing for a mixed parabolic-hyperbolic type equation.

Keywords: inverse problem; delta function; integral equation; kernel of the integral.

Author: Tadjieva M. A. (National University of Uzbekistan), Eshmamatova D. B. (Tashkent State Transport University), Eshniyozov A. I.(Gulistan State University), Ganikhodjayev R. N.(National University of Uzbekistan)

Abstract: 
In this paper, all homogeneous tournaments with 5 vertices were described and the spectrum of the Jacobian of the Lotka-Volterra mappings corresponding to these homogeneous tournaments at fixed points were studied.

Keywords: Lotka-Volterra quadratic mapping; simplex; graph; tournament; homogeneous tournament; fixed point; spectrum.

Author: Khoitmetov U. A.(V.I.Romanov Institute of Mathematics, Khorezm Branch)

Abstract: 
This paper is devoted to the integration of the loaded modified Korteweg-de Vries equation by the inverse scattering method.

Keywords: Loaded modified Korteweg-de Vries equation; Jost solutions; scattering data; Gelfand-Levitan-Marchenko integral equation.

Author: Shadimetov Kh. M.(V.I.Romanovskiy Institute of Mathematics), Nuraliyev F. A.(Tashkent State Transport University), Esanov Sh. J.(Termez State University)

Abstract: 
In the present paper the optimal, lattice cubature formulas with derivative are constructed in the space of two variable differentiable periodic functions. Here analytic expressions of optimal coefficcients are found.

Keywords: Optimal qubature formula; Hilbert space; the error functional; discrete argument function.

Author: Artykbaev A.(Tashkent State Transport University), Ismoilov Sh. Sh.(National University of Uzbekistan) 

Abstract: 
This paper studies some properties of convex surfaces in the isotropic space $R^2_3$. The definition of $R^2_3$ is given, the first and second quadratic forms of the surface and found analogue of the sphere of Euclidean space. Sphere of isotropic space is a paraboloid of revolution. It has been proven that the intersection of a sphere of isotropic space is always an ellipse. The definition of the dual image of a plane relative to a sphere of isotropic space is given. Using the dual image, a generalized surface mapping is defined. Some properties of the dual surface have been proven.

Keywords: Paraboloid; plane; isotropic space; dual mapping; dual surfaces.

Author: Karimov E. T. (Fergana State University), Sobirov Z. A.(University of Geological Sciences), Khujakulov J. R.(V.I.Romanovskiy Institute of Mathematics) 

Abstract: 
We study a local problem for a time-fractional differential equation involving the fractional derivative Hilfer on the stellar metric graph. Using the method of separation of variables, we find an explicit solution to the problem under study in the form of a Fourier series.

Keywords: Hilfer operator; metric graph; method of separation of variables; Mittag-Leffler function.

Author: Lakaev S. N.(Samarkand State University), Abdukhakimov S. Kh.(Samarkand State University) 

Abstract: 
We construct a two-particle discrete Schre type operator dinger $H_μ(k)=H_0(k) + μV, k∈T^2$, associated with a system of two fermions on a two-dimensional cubic lattice
$Z^2$ interacting through a short-range potential, where the unperturbed part is $H_0(k), k∈T^2$ type operator convolution with the dispersion relation $Eˆk(·)$ defined on the torus $T^2$ and having a degenerate minimum at $0∈ T^2$. We prove the existence of eigenvalues ​​lying below the essential spectrum for the operator $H_μ (k)$ for $k = 0$ and all $μ > 0$.

Keywords: system of two fermions; discrete Schr ̈odinger operator; Hamiltonian; dispersion relation; degenerate minimum; bound state.

Author: Ochilov Z. Kh.(Samarkand State University)

Abstract: The paper considers a new class of problems of integral geometry of the Voltaire type with a weight function of a special form. A uniqueness theorem and inversion formulas are derived, stability estimates in Sobolev spaces are obtained, thereby showing weak incorrectness assigned task. A formulation and proof of the existence theorem for a solution to the problem of restoring a function from a family of parabolas with a weight function is given special type in the strip.

Keywords: integral geometry problems; ill-posed problems; existence and uniqueness theorems for solutions.

Author: Rakhimov I. D.(Urgench state university)

Abstract: 
The article shows that the solution of a loaded nonlinear. The Schrödinger equation can be found by the inverse scattering problem method.

Keywords: inverse scattering method; loaded nonlinear Schrodinger equation; nonlinear equations; Zakharov- Shabat system; Cauchy problem.

Author: Samatov B. T.(Namangan State University), Khorilov M. A.(Namangan State University), Akbarov A. Kh.(Andijan State University) 

Abstract: 
In this paper we study differential games pursuit of escape at non-stationary integral
restrictions on player controls. To solve the problem In favor of the pursuing player, a parallel approach strategy is proposed and solvability conditions are found pursuit tasks. To solve the problem in favor of the evader, conditions for escaping are given and a function limiting from the bottom for the distance of the players is found.

Keywords: Differential game, integral constraint, pursuer, evader, strategy, pursuit, evasion.

Author: Tagaymurotov A. O.(Chirchik State Pedagogical Institute)

Abstract: 
The correspondence between the spaces of probability measures and idempotent probability measures is one of the pressing issues in category theory. The paper provides a description of the space of idempotent probability measures by the pole of the space of probability measures. Then the Minkowski functional interpreted as a metric on the space of probability measures.

Keywords: Minkowski functional, probability measure, polar.

Author: Azizov M. S.(Fergana State University)

Abstract: In this article an inverse problem for the fourth-order partial differential equation with a singular coefficient, has been considered in a rectangle. Theorems on the uniqueness and existence of the solution to the problem have been proved.

Keywords: rectangular domain; unknown right-hand side; fourth order partial differential equation; singular coefficient; inverse problem; spectral method.

Author: Gaybullayey R. K. (National University of Uzbekistan)

Abstract:  In this paper, it is given a classification of maximal solvable 4-Lie algebras with a given maximal filiform hypo-nilpotent ideal.

Keywords: 4-Lie algebras; nilpotent n-Lie algebras; hypo-nilpotent ideals; solvable n- Lie algebras; derivation; the characteristic sequence.

Author: Ibragimov G. I.(University Putra Malaysia), Kuchkarova S. A.(National University of Uzbekistan)

Abstract:
We consider an infinite system of 2-block of differential equations. The questions of existence and uniqueness of the solution of the infinite system are discussed in the Hilbert space $l_2$. Under some conditions, we prove the existence and uniqueness of the solution of the infinite system of 2-block of differential equations.

Keywords: Hilbert space; differential equations; infinite system; existence and uniqueness of the solution.

Author:  Irgashev B. Yu. (Namangan Engineering-Construction Institute)

Abstract: 
In this article, two methods are used to construct particular solutions with a singularity for a higher-order equation with a fractional derivative in the sense of Hilfer. Further, these solutions are used to obtain a solution to the problem with initial conditions.

Keywords: Higher order equation; fractional derivative in the sense of Hilfer; self-similar solution; Wright function; Fourier transform; existence; uniqueness.

Author: Ismoilov A. I.(Fergana State University)

Abstract: 
In the work a Cauchy-Goursat problem for nonhomogeneous Euler-Poisson-Darboux equation has been formulated and the solution of this problem has been found by the method of Riemann. And it was checked this solution indeed satisfies the equation and the conditions of the problem.

Keywords: Euler-Poisson-Darboux equation; Cauchy-Goursat problem; Riemann’s method; Riemann-Hadamard function.

Author: Kuldoshev Kh. M., Azamov S. S., Makhmudov M.M.

Abstract: 
In this paper the discrete analogue of the differential operator $ d^{4}/dx^{4}-\sigma^{2} d^{2}/dx^{2}$ is constructed and some of its properties are studied.

Keywords: Hilbert Space; extremal function; generalized function; operator; optimal quadrature formula.

Author: Mamanazarov A. O.(Fergana State University)

Abstract: 
In this paper, problems with an integral condition have been formulated and studied for a parabolic-hyperbolic equation with non-characteristic line of changing type .

Keywords: parabolic-hyperbolic equation; integral condition; method of integral equations.

Author: Mukhamedov A. K. , Kobilov U. Kh. 

Abstract:  In the paper almost sure convergence of kernel estimates of density function of stationary pairwise positive quadrant dependent sequences is proved.

Keywords: density estimation, random variables of pairwise positive quadrant dependent random variables.

Author: Islomov B. I.(National University of Uzbekistan), Dusanova U. Kh. (Karshi Branch of the Tashkent Institute of Irrigation and Agricultural Mechanization Engineers)

Abstract:
This work is devoted to proving the uniqueness and existence of a solution to a nonlocal problem with an integral gluing condition for a loaded equation parabolic-hyperbolic type, including the operator Caputo fractional order. The uniqueness of the problem posed is proved by the method of energy integrals, and its existence – by the method of integral equations. 

Keywords: Loaded equation; parabolic-hyperbolic type; fractional order operator in the sense of Caputo; integral gluing condition; uniqueness and existence of solution; Volterra type integral equation.

Author: Xakimov  O.  N.(V.I.Romanovskiy Institute of Mathematics),  Abdullaeva  G.  Sh.(V.I.Romanovskiy Institute of Mathematics) 

Abstract: In this article, the dynamics of the Ising-Potts map on the field of 2-adic numbers is studied. All fixed points of the Ising-Potts map are found.

Keywords: p-adic numbers; Ising model; Gibbs measure.

Author: Tishaboyev  J.  K.(National University of Uzbekistan), Tirkasheva  G.  D. (National University of Uzbekistan)

Abstract: 
In this work, some geometric properties of the A− lemniscate, the uniqueness theorem and the conserved domain theorem were proved, and the invariance condition in the Carathéodory metric for A(z)-analytic functions.

Keywords: A(z) – analytic functions; A-lemniscate; Caratedore metric; uniqueness theorem; open mapping theorem.

Author:Turdiev Kh.Kh. (Bukhara State University)

Abstract: 
In this paper, for the reduced canonical two-dimensional systems of Maxwell’s integro-differential equations direct and inverse problems of determining the electromagnetic voltage field and the diagonal memory matrix are posed. Problems are replaced by a closed system of integral equations of the second kind of Volterra type relative but the Fourier image in variables x_1 of the solution to the direct problem and the unknowns of the inverse problem. Further to this system the method of compressive mappings in space is used continuous functions with weight norm. Thus, global existence and uniqueness theorems for solutions to the problems posed are proven.

Keywords: Hyperbolic system; two-dimensional system of Maxwell’s equations; integral equation; integro-differential equations; contraction mapping principle.

Author: Boltayev A. K.(V.I.Romanovskiy Institute of Mathematics), Davronov J. R.(V.I.Romanovskiy Institute of Mathematics)

Abstract:  We consider a quadrature formula of the form $int_0^1φ(x)dx ∼= \sum^N_{β=0}C[β]φ[β]$. Then we estimate the error of this quadrature formula. We find the extremal function of the error functional. We will also use a discrete analog $D_1[β]$ of the differential operator $d^2/dx^2 − 1$. We find the analytical expressions of the optimal coefficients of the quadrature formula and, finally, we calculate the square of the norm of the error functional of the optimal quadrature formula.

Keywords: Sobolev space; the error functional; the extremal function; optimal quadrature formula.

Author: Abdushukurov A. A.(Branch of Moscow State University named after M.V. Lomonosov in Tashkent), (V.I.Romanovskiy Institute of Mathematics),
Erisbayev S. A. (Nukus State Pedagogical Institute)

Abstract: 
In competing risks model of incomplete data we present useful representations of Fisher information function in applications and establish Cramer-Rao lower bound for variance of unbiased estimator.

Keywords: Fisher information; Cramer–Rao inequality; Competing Risks Model; Proportional Hazards Model.

Author: Allakov I.(Termez State University), Safarov A. Sh.(Termez State University)

Abstract: 
The work studies the question of representing numbers as the sum of two primes from an arithmetic progression, that is, the binary Goldbach problem, when primes are taken from an arithmetic progression. New estimates are proved number of even natural numbers that are (possibly) not representable as a sum of two primes from an arithmetic progression and for a number representing a given natural number, as a sum of two primes from an arithmetic progression.

Keywords: The Dirichlet character; Dirichlet L-function; exceptional set; representation numbers; exceptional zero; exceptional character; main member; remaining member;

Author: Aripov M. M.(National University of Uzbekistan), Nigmanova D. B.(National University of Uzbekistan), Raimbekov J. R. (Inha University in Tashkent)

Abstract: 
The paper considered the critical exponent of the non-divergent form of polytropic filtration equation with nonlinear boundary conditions. The global critical exponent and the critical Fujita exponent were obtained using various upper and lower self-similar solutions.

Keywords: Non-Newtonian equation of non-divergent polytropic filtration; Non-linear boundary flow; Exacerbation mode; Critical exponent.

Author:  Zarifzoda  S. K. (Tajik National University)

Abstract: 
The paper is devoted to the investigation of the Riemann problem for a second order differential equation with four singular points. In three cases, in depending of the roots of the corresponding characteristic equation, there are found the explicit solutions of this problem by means of an elementary functions. In the first case, it is proved that the obtained function satisfies all the conditions enumerated by Riemann and indeed will be the Riemann P-function.

Keywords: The Riemann-Hilbert problem; the Riemann equation; special differential operator; the class of Fuchsian equations.

Author: Karimjonov I. A.(Andijan State University),(V.I.Romanovskiy Institute of Mathematics), Khabibullayev A. R.(Andijan State University), Khakimova N. R.(Andijan State University)

Abstract: 
In this paper, we give the description of Rota-Baxter operators of weight 0 and 1, Reynold operators, Nijenhuis operators and average operators on 3-dimensional nilpotent algebras over complex numbers.

Keywords: Nilpotent algebras; Rota-Baxter operators; Reynold operators; Nijehuis operators; average operators.

Author: Kudratov Kh. E. (National University of Uzbekistan)

Abstract: 
In this paper, the critical Galton-Watson branching process is considered in the case when the number of descendants of particles has a geometric distribution. An estimate is obtained for the rate of convergence in the theorem of A.M. Yaglom.

Keywords: Galton-Watson process; generating function;
exponential distribution; fractional linear generating function.

Author: Mirsaburov M.(Termez State University), Allakova Sh. I. (Termez State University)

Abstract: 
Problem in an unbounded domain for the Gellerstedt equation with singular coefficients with the Bitsadze-Samarskiy condition on parallel characteristics In this paper theorems on the existence and uniqueness of the problem with Bitsadze-Samarskiy condition in boundary and internal parallel characteristics and Tricomi condition not fully given in boundary characteristics in unbounded mixed domain consisting of characteristic triangle in hyperbolic part and y ≥ 0 half plane in elliptic part for a class of mixed type second order linear equation with a singular coefficient are proved. The extremum principle was used to prove the uniqueness of the solution of the problem, and the theories of singular integral equations, Wiener-Hopf integral equations and Fredholm integral equations were used to prove the existence of the solution. This issue is being studied for the first time in an unbounded domain.

Keywords: Mixed type equations with a singular coefficient; infinite mixed domain; comitted Tricomi condition; Bitsadze-Samarsky condition; non-standard singular integral equation; Wiener-Hopf equation; index.

Author: Turmetov B. Kh.(International Kazakh-Turkish University), Kadirqulov B. J.(Tashkent State Institute of Oriental Studies) 

Abstract: 
In this article non-local problem for a second-order mixed- type equation involved Caputo fractional differential operator was studied in the rectangular domain. The conditions for the existence and uniqueness of the solution of the problem are defined and the connection between given functions and continuous unique solution is shown.

Keywords: nonlocal equation; mixed type equation; degeneration equation; nonlocal boundary value problem; operator of fractional integro-differentiation; Kilbas-Saigo
function; Fourier series.

Author: Aliyev A. F. (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
In this article, we consider $x ̄_n= T( ̄x_n−1) +σ_n ξ_n, x_0 =x ∈ S^1$ model for circle homeomorphisms $T∈C^2(S_1) and prove the central limit theorem for this model, where $ξ_n$ is a sequence of random variables with strong mixing and $σ_n$ parameter.

Keywords: Circle homeomorphism; central limit theorem; strong mixing random variables; circle dynamics.

Author:  Ishmetov A. Ya.(Tashkent Institute of Architecture and Civil Engineering), Kurbanov Kh. Kh.(Academy of the Armed Forces of the Republic of Uzbekistan)

Abstract: 
The paper shows that the functor of semi-additive functionals preserves homotopy. Natural transformations have been introduced, called unit and multiplication. It is proved that the functor of semi-additive functionals together with these unity and multiplication forms a monad.

Keywords: Functor of semiadditive functionals; monad; homotopy.

Author: Khalkhuzhaev A. M.(V.I.Romanovskiy Institute of Mathematics),
Pardabaev M. A (Samarkand State University)

Abstract: 
In this paper we consider the family $h:=\delta\delta-\mu\nu, \mu\inR$ of discrete Schrödinger-type operators in a d-dimensional lattice $Z^d,$ where $\delta$ is the discrete Laplacian and $ \nu$ contact potential. We will prove that there is a coupling constant threshold $μo ≥ 0$ such that for any $μ \in [0, μ_o]$ the discrete spectrum $hμ$ is empty, and for any $μ>μo$ the discrete spectrum $hμ$ is a singleton set ${e(μ)}$. In addition, we we study the asymptotics of $e(μ)$ for $μ & μ_o.$ The asymptotics is strongly depends on the lattice dimension d.

Keywords: Discrete bilaplacian, essential spectrum; discrete spectrum; eigenvalues; asymptotic; expansion

Author: Nazarov Kh. A.(Fergana State University)

Abstract: 
The article is devoted to the description of -automorphisms of the real factor B(H_r). It has been proven that on a real W- in the algebra B(H_r) every *-automorphism is internal.

Keywords: real factor; *-automorphism; inner *-automorphism.

Author: Rakhimov  A.  A.(National University of Uzbekistan),    Karimov  U.  Sh.(National University of Uzbekistan)

Abstract: 
Using the result of M. Brechard, it is shown that any local differentiation of a real factor is a differentiation. In addition, it is proven that the restriction of local derivations on the Hermitian part of a real W *-algebra is also a derivation.

Keywords: real W*-algebra; real factor; local derivation.

Author: Utebaev D.(Karakalpak State University),  Utepbergenova G.Kh.(Karakalpak State University),  Tleuov K.O.  (Nukus branch of Tashkent University of Information Technologies)

Abstract: 
Difference schemes of the finite element method of increased order of accuracy have been proposed and investigated for the nonstationary Aller-Lykov moisture transfer equation. A high order of accuracy is achieved in account for special discretization of time and space variables. The stability and convergence of the constructed numerical algorithms have been proven, and corresponding a priori estimates in various norms, which are used later to obtain estimates of the accuracy of the scheme under weak assumptions on the smoothness of solutions to the original differential problem.

Keywords: Hallaire-Luikov equation; finite element method; difference schemes; stability; a prior estimates; convergence; accuracy.

Author: Yuldayev T.K. (National University of Uzbekistan), Fayziev A.K. (Tashkent State Technical University)

Abstract: 
This article investigates a nonlocal boundary value problem for a system of ordinary differential equations with impulse actions and maxima. Boundary value problem is specified using the integral condition. Used the method of successive approximations combined with the compression mapping method. Proven existence and the uniqueness of the solution to the boundary value problem. The continuous dependence of solutions on the right side of the boundary value is shown conditions.

Keywords: Impulsive differential equations; nonlocal boundary condition; successive approximations; existence and uniqueness of solution; continuous dependence of solution.

Author: Nuritdinov J. T. (National University of Uzbekistan)

Abstract: 
In this work, the sufficiency and necessity conditions for the non-emptiness of the Minkowski difference of the given triangles in the Euclidean plane are obtained. Methods for finding the Minkowski difference of some groups of triangles by vectors corresponding to their sides are also shown. At the end of the article a theorem on Minkowski difference of triangles is presented.

Keywords: Minkowski difference, Minkovsky sum; convex set; in-circles and circumcircles.

Author: Abdullayev A. A.(Tashkent Institute of Irrigation and Agricultural Mechanization Engineers), Islomov Kh.(Termez State University) 

Abstract: 
In this paper, we study a boundary value problem with a conormal condition for an equation of elliptic type of the second kind. Using the properties of generalized solutions, a modified Dirichlet problem is studied, and its solutions are found in a form convenient for further research. The uniqueness of the solution to Problem E is proved by the method of energy integrals. The existence of a solution to the problem E under study is equivalently reduced to a singular integral equation, and the unique solvability of the singular integral equation is investigated by the Carleman – Vekua regularization method.

Keywords: generalized solution of the class R_2 ; problem with co-normal condition; degenerate equation of the second kind; integral equation; method of energy integrals; Green’s function.

Author: Babaev  M. M. (Military Academic Lyceum “School of Temurbeks” of the State Security Service)

Abstract: 
In this paper, we study a mixed problem for a fractional-order equation in Sobolev classes with pseudodifferential operators associated with Laplace operators with nonlocal boundary conditions and a lagging argument in time. The solution to the initial boundary value problem is constructed as the sum of a series in the system of eigenfunctions of the multidimensional spectral problem. The eigenvalues of the multidimensional spectral problem are found and the corresponding system of eigenfunctions is constructed. It is shown that this system of eigenfunctions is complete and forms a Riesz basis in Sobolev subspaces. Based on the completeness of the system of eigenfunctions, a uniqueness theorem for the solution of the problem is proved. The existence of a regular solution to the stated initial-boundary value problem is proved in Sobolev subspaces.

Keywords: partial differential equation with lagging argument; fractional time derivative; initial-boundary value problem; pseudodifferential operator; spectral problem; Riesz basis.

Author: Kuryazov D. M.  (National University of Uzbekistan)

Abstract: 
Stability of digital signature algorithm GOST R 34.10 94 is based on the complexity of calculating discrete logarithms in a finite field. This algorithm is modified on elliptic curves, as a secret parameters of signature selected coordinates of elliptic curves and thereby stability was increased. Since 2001 this variant with increased stability has been accepted as the standard of Russian Federation.In the proposed work a new digital signature algorithm on the basis of the complexities of calculating the discrete logarithm problem in a finite field and factoring sufficiently large odd number was developed.

Keywords: cryptosystem, cryptographic protocol, digital signature, blind signature, factorization problem, discrete logarithm problem

Author: Kushmurodov A. A.(V.I.Romanovskiy Institute of Mathematics), Sharipov O. Sh.(National University of Uzbekistan)

Abstract: 
In this paper, law of large numbers of the Marcinkiewicz-Zygmund type is proved for a mixing sequence of identically distributed random variables with values in separable type p Banach spaces.

Keywords: type p Banach space; identically distributed random variables; Marcinkiewicz-Zygmund law of large numbers.

Author: Madgoziev  G.  T.(Academic Lyceum at TUIT)

Abstract: 
In [8] the investigation of a model with the interaction radius 2 is reduced to a HC-model with activity $λ > 0$. For $λ > λ_{cr} ≈2.6639$ we show that the translation-invariant Gibbs measure of this model in not extreme.

Keywords: Configuration; Cayley tree; HC-model; Gibbs measure

Author: Rakhmonov  F.  D. (National University of Uzbekistan)

Abstract: 
In a two-dimensional domain, a Benney-Luke type partial differential equation of an even higher order with conditions in integral form is considered. The unique solvability of the nonlinear inverse problem is studied. The unique solvability of the nonlinear inverse problem is studied. The solution of this partial differential equation is studied in the class of regular functions. The method of Fourier series of separation of variables is used. The inverse problem is reduced to solving systems of two nonlinear integral equations. In proving the existence and uniqueness of the Fourier coefficients of the unknown function, the successive approximation method is used in combination with the contraction mapping method. The Cauchy-Schwarz inequalities and the Bessel inequality are used to prove the absolute and uniform convergence of the obtained Fourier series.

Keywords: Benney-Luke type differential equation; integral conditions; inverse
problem; nonlinear functional equation; nonlocal boundary value problem.

Author: Fayazov K. S.(Turin Polytechnic University in Tashkent), Khajiev I. O.(National University of Uzbekistan)

Abstract: 
In this paper, we investigate an initial-boundary value problem for the inhomogeneous equation of mixed type in the multidimensional case. A priori estimates were obtained with the methods of logarithmic convexity and the energy integral for the solutions of the problems under consideration. Theorems of uniqueness and conditional stability on the well-posedness sets were proved.

Keywords: Mixed type equation; ill-posed problem; a priori estimate; a set of correctness; uniqueness; conditional stability.

Author: Allen I.K.(University of Wisconsin-Madison), Duggal D.(The Georgia Institute of Technology), Nasir S.(Vassar College), Karimov E.T. (V.I.Romanovskiy Institute of Mathematic)

Abstract:
In this article, a boundary value problem for a wave equation with fractional order derivatives in the sense of Riemann-Liouville and Atangana-Baleanu is considered. Using the Fourier method, the variables are separated and the equation in time
the belt is solved using the Laplace transform. The solution to the problem is expressed in the form of an infinite series.

Keywords: Riemann-Liuoville derivative; Atangana-Baleanu derivative; fractional wave equation; Laplace transformation; Mittag-Leffler function.

Author: Alimov Sh.A.(National University of Uzbekistan),Yuldasheva A.V.(M.V. Lomonosov Moscow State University Tashkent Branch)

Abstract:
The article proves the unique solvability of the Cauchy problem for an equation of Boussinesq type.

Keywords: Boussinesq-type equation; Cauchy problem; integral equation.

Author: Babajanov B.A.(Urgench State University),Ruzmetov M.M(Urgench State University), Babajanov A.B.(Urgench State University)

Abstract:
In this work, the inverse spectral problem method is applied to the integration of a Toda chain type equation with integral source.

Keywords: Toda-type chain; discrete Sturm-Liouville operator; inverse scattering method; self-consistent integral type source; soliton solution.

Author: Muydinjanov D.R. (Kokand State Pedagogical Institute)

Abstract:
Fundamental solutions to the multidimensional Helmholtz equation with three singular coefficients were constructed in 2019 and they are expressed through a confluent hypergeometric function of four variables. In this paper, we study the Dirichlet problem for the three-dimensional Helmholtz equation with three singular coefficients
and the only solution to the problem posed is obtained in explicit form.

Keywords: Helmholz equation; Dirichlet problem; confluent hypergeometric function.

Author: Ruzimuradova D.H.(National University of Uzbekistan), Azamova N.A.(IT Akademiya (IT Park))

Abstract:
The topological properties of the ω-limit set of dynamical systems are studied. An analytical system with ω- limit set consisting of four lines in the space R^3 and this property was proved.

Keywords: Analytical system; vector field; limit set; Hamiltonian function; non-connectivity.

Author: Abdushukurov F.A.(Kazan Ferreral University),Chuprunov A.N.(Kazan Ferreral University)

Abctract:
In this paper we obtain asymmetric and symmetric estimates of the rate of convergence of centered and normalized Poisson random variables to the standard Gaussian random variable.

Key words: Berry-Essen inequality, central limit theorem;
Gaussian random variable; Poisson random variables.

Author: Azamov S.S.(Tashkent Institute of Railway Transport Engineers)

Abstract:  In the paper a discrete analogue of the differential operator $d^{6n}/{dx^{6n}}+d^{6n−2}/{dx^{6n−2}}+d^{6n−4}/{dx^{6n−4}}$ is constructed. It is used in the construction of the optimal quadrature and interpolation formulas.

Keywords: Hilbert Space; extremal function; generalized function; operator; optimal quadrature formula.

Author: Kadirkulov B.J.(Tashkent State Institute of Oriental Studies), Jalilov M.A.(Fergana State University)

Abstract: 
In this article, the Hilfer operator on a rectangular field for the involved fourth-order mixed type equation existence and uniqueness of the solution of the nonlocal problem
shown. The solution of the problem is continuous to the given one the connection has been proven.

Keywords: Mixed type equation; boundary problem; the existence and a uniqueness of a solution; fractional differential operator; Mittag-Leffler function; Riesz basis; Fourier series; operator Hilfera.

Author: Mamadaliev U.X. (National University of Uzbekistan)

Abstract: 
In this paper, we describe solvable Leibniz algebras with a nilradical that has a characteristic sequence equal to (m, n). In addition, it is proved that the solvable Leibniz algebra is complete when the dimension of the complementary subspace of a nilradical has a maximum value.

Keywords: Leibniz algebra; derivation; nilpotent algebra; nilradical; solvable Leibniz algebra.

Author: Mamanazarov A.O.(Fergana State University)

Abstract: 
In this paper, two variants of Tricomi problem were investigated for a parabolic-hyperbolic equation with singular coefficients. The uniqueness and existence of the solution of the problems were proved.

Keywords: parabolic-hyperbolic type equation; Tricomi problem; method of integral equations.

Author: Nurmuhamedova N.S. (National University of Uzbekistan)

Abstract: 
In this paper, we investigate the property of local asymptotic normality for likelihood ratio statistics in competing risk model under combined hybrid random censoring.

Keywords: Competing risks model; local asymptotic normality; random censoring.

Author: Oqboev A.B. (Fergana State University)

Abstract:  In this article, the Tricomi problem for a parabolic-hyperbolic type equation in a mixed domain has been investigated. The solution of the problem in the hyperbolic sub-domain is found as a solution to the Cauchy problem, and in a parabolic subdomain as a solution to the first boundary value problem.

Keywords: parabolic-hyperbolic type equation, mixed domain, Tricomi problem, Cauchy problem, first boundary value problem.

Author: Fayziev A.A.(Tashkent State Agrarian University), Turgunov T. T.(Tashkent State Agrarian University)

Abstract: 
Using the methods of statistical econometric analysis of time series, the article studies the statistical regularities of the series of dynamics of the average cotton yield in the Republic of Uzbekistan on the basis of materials from the Central Statistical Bureau for 2003-2018. In addition, a linear regression model was compiled, estimates of unknown parameters were given, numerical characteristics of the regression equations were
calculated, hypotheses about the linearity of the trend and the normality of the average cotton yield estimation were tested, and an interval estimate was construct.

Keywords: discrete, dynamic, trend, yield, regression, hypothesis, auto-correlation,
mean, dispersion, coefficient of variation, standard deviation, normal distribution, criterion, level significance, interval estimates.

Author: Fayazova Z.K.(Tashkent State Technical University)

Abstract:
In this paper, we investigate boundary control problem for heat exchange process with Neuman condition in bound of region with integral condition. Considering problem is transformed by Fourier separable method to first kind Volterra integral equation. By investigating this integral equation we proof the existence of the solution initial problem.

  Keywords: equation of heat exchange process, boundary control, integral equation, parameter of control

Author: Fayazov K.S.(Turin Polytechnic University in Tashkent), Xajiyev I.O. (National University of Uzbekistan)

Abstract: 
In this paper we consider the boundary value problem for a second-order partial differential equation with one degeneration line. An apriori estimate is obtained by the method of energy integral. Theorems on the uniqueness of a solution and its conditional stability on the set of correctness are proved. Using the methods of quasi-inversion and Tikhonov regularization, sequences of functions are constructed that tend in the norm of the corresponding space to the exact solution of the desired problem on the set of correctness.

Keywords: equation with one line of degeneracy, ill-posed boundary value problem, a priori estimate, set of correctness, regularization, quasi-inverse.

Author: Ergashev T.G. (V.I.Romanovskiy Institute of Mathematics), Safarbaeva N.M. (Tashkent Institute of Irrigation and Agricultural Mechanization Engineers)

Abstract:
Fundamental solutions for the multidimensional Helmholtz equation with one singular coefficient in the half-space were found recently. In this paper, Holmgren problem for the above-mentioned elliptic equation in a finite simply connected domain is studied. Using the properties of one of the fundamental solutions, the Green’s function was constructed, with the help of which the solution of the problem in the finite region bounded by the multidimensional hemisphere was found explicitly.

Keywords: Multidimensional elliptic equations with one singular coefficient; Holmgren problem; fundamental solution; Green’s function method.

Author: Babajanova A.K.(Urgench State University)

Abstract: 
In this work, equations for the time evolution of scattering data associated with the Ablowitz-Ladik matrix spectral problem are derived. The resulting equations make it possible to integrate the Ablowitz-matrix system Ladika with a self-consistent source by the inverse scattering problem method.

Keywords: Discrete integrable systems; inverse scattering problem; self-consistent source; matrix Ablowitz-Ladik spectral problem.

Author: Prenov B.B. (Nukus State Pedagogical Institute)

Abstract: 
This article discusses some systems of non-algebraic equations. Their consideration is based on the results of the article by A.M. Kytmanov, E.K. Myshkina. Residue integrals are found over cycles associated with the systems. Their connection with power sums of the roots of the system is established.

Keywords: non-algebraic systems of equations; residue integrals; power sums of roots.

Author: Rozikov U.A.(V.I.Romanovskiy Institute of Mathematics), Shoyimardonov S.K.(V.I.Romanovskiy Institute of Mathematics)

Abstract: 
In this article we will look at a model for forecasting COVID-19 pandemic in Uzbekistan. This model was successfully used to predict the development of the pandemic situation in China and South Korea (see [2]). This simple model requires both government intervention and public response, and in this article we will highlight some categories of these indicators for Uzbekistan and give a forecast for the development of the pandemic situation in Uzbekistan until
May 10, 2020.

Keywords: Сoronavirus, pandemic situation, prediction.

Author: Seyrov Sh.J. (National University of Uzbekistan), Ganikhodzhayev R.N.(National University of Uzbekistan)

Abstract: 
In this paper, we show a graphical analysis method for some two-dimensional dynamical systems. Using graphical analysis method, we study some properties of the Julia and Mandelbrot sets for given mappings.

Keywords: Method of graphical analysis; Julia set; Mandelbrot set.

Author: Abdurasulov K.K. (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
In this paper we present an elementary proof of the isomorphism
between algebra of inner derivations n-Lie algebra of Jacobians
and infinite-dimensional algebra $W_{n−1}$.

Keywords: n-Lie algebra; Lie algebra; inner derivations; isomorphic.

Author: Boltayev A.K. (V.I.Romanovskiy Institute of Mathematics) Akmedov D.M. (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
In the present paper in the $W^(4,0)_2(0, 1)$ Hilbert space the first part of the problem of construction of optimal quadrature formulas is solved, i.e. in the $W^(4,0)_2 (0, 1)$ space the extremal function of a quadrature formula is found. Using the extremal function the norm of the error functional of the quadrature formula is calculated.

Keywords: extremal function; error functional; definite integral; Fourier transform.

Author: Boltayev N.D. (Tashkent Institute of Railway Transport Engineers)

Abstract: 
In the present work in the space $K_2(P^4)$ the optimal quadrature formula is constructed for evaluation of Fourier integrals.

Keywords: Hilbert Space; extremal function; generalized function; optimal quadrature formula.

Author: Islomov B.I.(National University of Uzbekistan), Ubaydullayev U.Sh. (National University of Uzbekistan)

Abstract: 
In this paper, we study a unique solvability of a boundary value problem with second kind borders conditions for the parabolic hyperbolic type equation with the Caputo operator in the rectangular domain. In the study of formulated boundary value problem, three gluing conditions were used, while some part of a borders is free from the boundary condition. The solution of the problem is constructed in the form of a series and the convergence of series in the class of regular solutions, of
considering equation are proved.

Keywords: Mixed type equation; regular solution; Caputo operator; rectangular domain; Fourier series; gluing conditions; boundary conditions.

Author: Ismoilov A.S. (Samarkand State University)

Abstract: 
In this paper, we consider the problem of reconstructing a function from a family of parabolas in the upper half-plane with a weight function having a singularity. The uniqueness theorem and the existence of a solution to the problem are proved. It is shown that the solution of the problem posed is weakly ill-posed, that is, stability estimates are obtained in spaces of finite smoothness.

Keywords: Weakly ill-posed problems, Fourier transform, uniqueness theorems and existence, weight function, finite function.

Author: Karimov K.T.(Fergana State University)

Abstract: 
The first boundary value problem for a three-dimensional equation of mixed type with three singular coefficients in a semi- infinite parallelepiped in the class of regular solutions is studied. The study of the problem is carried out using the method of
separation of Fourier variables and spectral analysis. For the problem posed, using the Fourier method, two one-dimensional spectral problems are obtained. Based on the completeness property of systems of eigenfunctions of these problems, the uniqueness theorem is proved. The solution of the problem under study is constructed in the form of the sum of a double Fourier-Bessel series.

Keywords: Dirichlet problem; equations of mixed type; spectral method; Bessel function; singular coefficient.

Author: Kakadjonava L.R. (National University of Uzbekistan)

Abstract:  For a sequential integral process of independence the uniform variants of the strong law of large numbers and the central limit theorems by indexed classes are established.

Keywords: empirical processes, metric entropy, Glivenko- Cantelli’s and Donsker’s theorems.

Author: Muxamedov A.K.(National University of Uzbekistan), Kovilov U.X. (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
In the paper we prove an asymptotic normality of two sample Wilcoxon-Mann-Whitney U-statistic of stationary strongly positive orthant dependent sequences.

Keywords: U-statistics, Wilcoxon-Mann-Whitney statistic, central limit theorem, strongly positive orthant dependent random variables.

Author: Safarov J. Sh. (Tashkent University of Information Technologies)

Abstract: 
In this paper, a theorem of existence and uniqueness of local solutions of the inverse problem of integro-differential equation in the area bounded by x.

Keywords: integro-differential equation; inverse problem; kernel of integral; Banach theorem.

Author: Usmonov B.Z. (National University of Uzbekistan)

Abstract: This paper is devoted to the investigation of non-local problem
for the equation with third-order elliptic-hyperbolic operator, by reducing to the inverse problem for a mixed type second order differential equation with unknown right-hand sides. Using the new extremum principle for a third-order equations, the uniqueness of the problem has been proved. General representation and the existence of solution of the investigated problem is proved by the method of inverse problem.

Key words:Third order equation; non-local problem; regular solution; representation of a general solution; extremum principle; Fredholm equation.

Author: Aralova K.A.(V.I.Romanovskiy Institute of Mathematics), Jamilov U.U. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  The article studies the dynamics of operators consisting of superpos-positions of non-Volterian quadratic stochastic operations rators in two-dimensional and three-dimensional simplexes. For such operators, sets of periodic and fixed-
nal points, and also describes the sets of limit points trajectories.

Keywords: Volterra quadratic stochastic operator; trajectory; fixed point; periodic point; limit point.

Author: Durdiev D.K.(Bukhara State University), Nuriddinov Zh.Z.(Bukhara State University)

Abstract: The problem of determining the multidimensional kernel of the titanium integral is considered. Convolution pas with respect to the time variable depending on t and the (n−1)-dimensional spatial variable x_0 = (x_1, …, x_n−1) in the n-dimensional heat equation with a variable thermal conductivity coefficient. The direct problem is the Cauchy problem for this equation. The integral term in the main equation is in the form of convolution of the kernel and solution of the direct problem. As additional information, the solution of the direct problem on the hyperplane x_n = 0 is given. The problem is reduced to an auxiliary problem, which is more convenient for further research. Next, the auxiliary problem is replaced by an equivalent system of integral equations of the type Volterra regarding unknown functions. Applying the method
contraction mappings in the class of Hölder functions, a theorem on the local existence and uniqueness of the solution is proved inverse problem.

Keywords: Heat equation; memory kernel; H ̈older space; convolution integral; auxiliary problem; contraction mapping.

Author: Eshkabilov Yu.Kh.(Karshi State University), Nodirov Sh.D.(Karshi State University)

Abstract: 
The article considers a quartic operator with positive coefficients on R^2. Positive fixed points of the quartic operator are studied. Theorems on the number of positive fixed points of the quartic operator are proved.

Keywords: Fixed point; quartic operator; Ferrari’s method; Descartes rule; translation-invariant Gibbs measure; Cayley tree; model; hamiltonian; potential.

Author: Imomkulov A.N. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  In this article, we will prove that if there is absolutely nilpotent element of an arbitrary finite-dimensional algebra, then there are absolutely nilpotent elements of evolutionary algebra, which is its approximation and their number is exactly found.

Keywords: Algebra; evolution algebra; absolute nilpotent elements; approximation.

Author: Ishmetov A.Ya.(Tashkent Institute of Architecture and Civil Engineering)

Abstract: In this work it is shown that the kappametrizability Tikhonov space X implies kappa-metrizability space I_β(X) of idempotent probability measures
with compact carrier. Built series max-plus-convex subfunctors of the functor I_β. Further, it is established that the functor I_β preserves the openness of continuous mappings of Tikhonov spaces.

Keywords: Idempotent measure; open map; kappa-metric.

Author: Mamatov M.Sh.(National University of Uzbekistan), Nuritdinov J. T. (National University of Uzbekistan)

Abstract: 
This article studies the properties of the geometric difference and the Minkowski sum over the sets that are used in the study of differential games. Some theorems and their proof are presented. At the end of the article, we consider a differential game described by linear equations.

Keywords: Minkowski difference; Minkowski sum; multiplication of a set by number; strongly convex sets; pursuit; escape; terminal set.

Author: Mustapokulov Kh.Y. (Tashkent State Technical University)

Abstract: 
This article addresses the issue of invariance of a given multivalued mapping relative to a system with distributed parameters. The system is described equation of thermal conductivity, within the boundaries of which pulse control is located in additive form.

Keywords: Invariance; control; weak invariance; strong invariance; lumped parameters.

Author: Rahmonov E.S.(National University of Uzbekistan), Beshimov G.R.(National University of Uzbekistan)

Abstract: 
In this article, an explicit form of the elements of the orthogonal group and special orthogonal group in two dimensions bilineometric space over the field of rational numbers with the form x_1 y_1−2x_2 y_2. Complete systems of invariants of these groups are obtained for systems of m points in this space.

Keywords: Invariant of an m-tuple; m-point invariant.

Author: Mohd .D.S. (Jazan University)

Abstract: 
The purpose of the study is to study generalized Yamabe solitons in trans-Sasaki varieties. In addition, we discuss that the Yamabe soliton on trans-Sasaki manifolds contracts under some specific conditions.

Keywords: Generalized Yamabe soliton; Trans-Sasakian manifold.

Author: Sheraliyev Sh.N. (Lomonosov Moscow State University, Tashkent Branch)

Abstract: 
A quasilinear integro-differential equation of peridynamics with nonlinearity in the form of the Urysohn integral operator is considered. The existence and uniqueness of a continuous periodic solution is proved.

Keywords: The quasi-linear periodic integro-differential equation; Uryson’s integral operator; Volterra integral equation of the second kind.

Author: Umrzaqov N.M.( Andijan State University)

Abstract: 
This article proves that every additive local inner derivation of a nilpotent algebra of upper triangular matrices is an inner derivation. And also, it is proved that any additive local internal differentiation of the nilpotent algebra of upper triangular infinite-dimensional matrices is an internal differentiation.

Keywords:Derivation; inner derivation; local inner derivation; associative algebra of matrices.

Author: Usmonov J.B. (V.I.Romanovskiy Institute of Mathematics), Kodirova M.A. (Namangan State University)

Abstract: 
The article finds fixed points of a dynamic system defined by a one-dimensional piecewise continuous function with parameters a and b, and shows that 2 periodic points are not exists, and conditions for the existence of 3 periodic points are set in accordance with the parameters.

Keywords: Evolution operator, Volterra QSO, fixed point, periodic points.

Author: Aripov M.M.(National University of Uzbekistan), Djabborov O.R.( Karshi State University)

Abstract: 
In this paper, using the solution to the Hamilton-Yakobi equation, we study the estimation and asymptotics of solutions to a parabolic equation with double nonlinearity and damping. An estimate of the weak solution and the asymptotic behavior of regular, unbounded, and finite solutions to the stationary equation are obtained. A condition of spatial localization of the solution to the Cauchy problem is found.

Keywords: Parabolic equation; asymptotic; double non-linearity; qualitative properties of parabolic type equations; localization of solutions.

Author: Kasimov Sh. G.(National University of Uzbekistan), Madrahimov U. S.(National University of Uzbekistan), Koshanov A. P.( Karakalpak State University)

Abstract: 
In this paper, we study a problem with initial conditions for a more general equation of beam oscillations, one end of which is patched, and the other floating, in the multidimensional case. The solution of the initial boundary-value problem is constructed as the sum of a series in the system of eigenfunctions of a multidimensional spectral problem. The obtained spectral problem eigenvalues and the
corresponding system of eigenfunctions is constructed. It is shown that
this system eigenfunctions is complete and forms a Riesz basis in
Sobolev space. Based on completeness system of eigenfunctions, the
uniqueness theorem for solving the problem is proved. In Sobolev classes
the existence of a regular solution to the stated initial-boundary value
problem is proved.

Keywords: beam equation; initial-boundary value problem; spectral method; eigenvalues; eigenfunctions; completeness; Riesz basis; uniqueness; existence; series.

Author: Irashev B. Yu. ( Namangan Engineering-Construction Institute)

Abstract:  In this article, self-similar solutions for degenerate equations of higher order are constructed. The conditions for the linear independence of these solutions are found.

Keywords: Partial differential equation derivatives; high order; degeneration; self–similar solution; hypergeometric function.

Author: Ruziev M.X.(V.I.Romanovskiy Institute of Mathematics)

Abstract: In this paper, we study a boundary value problem with conditions on the inner characteristic and on some parts of the degeneration line for mixed type equation with singular coefficients in unbounded domain. We prove the uniqueness of solution of the mentioned problem with the help of the extremum principle. The proof of the existence solution is based on the theory of singular integral equations and Wiener-Hopf,
Fredholm integral equations.

Keywords: Mixed type equation; a boundary value problem; method of integral equations; index equation.

Author: Umarova G.B. (Kokand State Pedagogical Institute)

Abstract: 
In this paper, in infinite three-dimensional domains, the analogue of the Tricomi problem (problem T) is formulated and studied for a parabolic-hyperbolic equation with two type changing planes. The main research method of the problem T is the Fourier transform. Based on the Fourier transform, the problem and equations is reduced to a planar analogue of the Tricomi problem (problem T_λ) with a spectral parameter, both
in the equation and in the boundary conditions.

Keywords: Equation with two planes of change of type; Fourier transform; regular solution; extremum principle; asymptotic behavior of a solution; Volterra equation.

Author: Hayotov A.R.(V.I.Romanovskiy Institute of Mathematics), Rasulov R.G.(V.I.Romanovskiy Institute of Mathematics)

Abstract: 
In the paper we consider an extension problem of the Euler-Maclaurin quadrature formula in the space W^(5,4)_2 by constructing an optimal quadrature formula.

Keywords: Optimal quadrature formula; Hilbert space; the error functional; Sobolev’s method; discrete argument function.

Author: Xaboev A.G. (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
In this paper, the game between a group of n pursuers and one evader moving along 1-skeleton of the given three-dimensional Archimedean solids is considered. Under the condition that the maximal speed of all players equal to 1, a minimum number of pursuers that win (i.e, catch the Evader) in the game is found.

Keywords: pursuit-evasion game; approach problem; evasion problem; positional strategy; counter strategy; polyhedron; graph; one-dimensional graph.

Author: Xurramov N.X. (Termez State University)

Abstract:  In the article, for the equation $(signy)|y|^m u_{xx} + u_{yy} −(m/2{y})u_y = 0$, in a mixed domain, we prove existence and uniqueness theorems for a solution of the problem with the Gellerstedt i condition on part of the boundary characteristic
and the condition on an internal characteristic parallel to it.

Keywords: mixed type equation with singular coefficient; boundary equation; nonstandard singular integral equation of Tricomi, Wiener-Hopf integral equation; index.

Author: Shaimkulov B.A.(Karshi State University), Bozorov J.T. (Karshi State University)

Abstract: In this article, necessary and sufficient conditions for the holomorphic continuation into a unit polydisc of a function defined on the polydisc skeleton or on its part are given.

Keywords: Polydisc; Coshy integral formula; golomorph continuation.

Author: Babaev S.S.(V.I.Romanovskiy Institute of Mathematics), Davronov J.R. (Bukhara State University), Mamatova N.H.(Bukhara State University)

Abstract: 
Consider the interpolation formula φ(x) ∼= Pφ(x) =N β=0 Cβ ·φ(xβ). Then we will estimate the error of this interpolation formula. Let us find the extremal function of the error functional. Also, we build a discrete analogue D(hβ) of the differential operator $d^2/
d^{x^2} −σ^2$. Finally we we find the explicit form of the optimal coefficients of the interpolation formula.

Keywords: Hilbert space; the error functional; the extremal function; optimal interpolation formula.

Author: Iskanadjiev I.M. (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
In this work we study some approximative properties of the lower Pontryagin operator for nonlinear differential games with unfixed ending time. Based on these properties, simplified formulas for constructing the lower operator. In particular, An exact formula has been proposed for calculating this operator when the structure of the game has a simpler form.

Keywords: Pontryagin’s lower operator; differential game; pursuer; evader; control; strategy; approximation.

Author: Rasulov T.H. (Bukhara State University), Bahronov B.I.(Bukhara State University)

Abstract: 
This article considers the restricted and self-adjoint Friedrichs model A(μ1, μ2), μ1, μ2 > 0 and discusses the one-dimensional case with a two-dimensional perturbation. Typically, such models are associated with a system of two quantum particles in a one-dimensional lattice. The numerical range of the operator A(μ1, μ2) with respect to the parameters μ_1 and μ_2 is studied. Let us find the critical value of the parameter μ_α, α = 1, 2 guaranteeing the coincidence of the spectrum and numerical range of values ​​of the operator A(μ1, μ2).

Keywords: Friedrichs model; perturbation; quantum particles; non-local interaction operator; numerical range; spectral inclusion; spectrum; eigenvalue.

Author: Alikulov E.K. (Tashkent University of Information Technologies)

Abstract: 
In this paper, we formulate and study a boundary value problem with local boundary conditions on characteristics parallel to the plane (problem AG1) for a loaded parabolic-hyperbolic equation of the third order in an infinite three-dimensional domain. Based on the Fourier transform, problem and the equation are reduced to a planar analogue of the Gellerstedt problem (problem AG1λ) with a spectral parameter, both in the equation and in the boundary conditions. The uniqueness of the solution of the problem AG1 and AG1λ is proved using the new extremum principle for loaded equations of mixed type of the third order. Using the general representation of the solution, we prove the existence of the solution of problem AG1 and AG1λ by the method of integral equations. In addition, we study the asymptotic behavior of the solution of problem AG1 at larger values of the spectral parameter.

Keywords: third-order equation; loaded equation; Gellerstedt problem; Fourier transform; regular solution; extremum principle; solution estimate.

Author: Azizov M.S.(Fergana State University)

Abstract:  
In the present work with spectral method it was proved the uniqueness and existence of the solution of the mixed problem for a fourth order equation with singular coefficients in a rectangular domain.

Keywords: The fourth order differential equation, singular coefficient, boundary-value problem, spectral method, the uniqueness of the solution, the existence of the solution.

Author: Fayazov K.S. (Turin Polytechnic University in Tashkent), Rahmatov H.Ch.(Turin Polytechnic University in Tashkent)

Abstract:
In this paper we investigate the initial-boundary value problem for a fourth-order pseudo-differential equation. This problem, in general, is ill-posed in the sense of J. Hadamard, namely, there is no continuous dependence of the solution on the initial
data of the problem. Based on the idea of A. N. Tikhonov, an apriori estimate is obtained, and theorems on the uniqueness and conditional stability of a solution on the set of correctness are proved.

  Keywords: pseudo-differential equation, ill-posed boundary value problem, set of correctness, conditional stability.

Author: Jurayev Sh. Yu. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  In this note, the Galton – Watson branching critical
process Z_n, Z_0 = 1, n = 1, 2, 3, …. is considered. The rate of convergence of the conditional distribution law to the exponential distribution is estimated.

Keywords: Galton-Watson critical branching processes; generating function;conditional distribution; probability of process degeneration; convergence rate.

Author: Karimov K.T. (Fergana State University)

Abstract: In a cylindrical domain for a three-dimensional elliptic equation with a singular coefficient, some boundary value problems are investigated. The study of the tasks is carried out using the method of spectral analysis. Based on the completeness of the systems of eigenfunctions of the posed problems, a uniqueness theorem is proved. The solutions to the problems under study are constructed as the sum of the Fourier-Bessel series. Under certain conditions with respect to the parameters and given functions, the uniform convergence of the constructed series is proved.

Keywords: Dirichlet problem; Dirichlet-Neumann problem; Keldysh problem; singular coefficient; spectral method; cylindrical domain.

Author: Kasimov X.N.( National University of Uzbekistan), Javliyev S.K.( National University of Uzbekistan)

Abstract: 
The computability of natural positive and negative linearly ordered unary algebras proved and built example a negative unar whose semigroup of computable automorphisms is not is a group.

Keywords:Automorphism; unary algebra; semigroup;factor-algebra.

Author: Okboev A.B. (Fergana State University)

Abstract:  In the work a Cauchy-Goursat problem was studied for a hyperbolic type equation. A class of regular solutions was introduced and an explicit solution of the problem which belongs to this class was found. It was shown that taken solution really
satisfies all conditions of the problem.

Keywords: Equation of hyperbolic type; second kind degenerated equation; Cauchy-Goursat problem; regular solution.

Author: Khakimov R.M.(V.I.Romanov Institute of Mathematics, Namangan Branch), Umirzakova K.O. (Namangan State University)

Abstract: 
We consider fertile HC models with three states on the Cayley tree of order k ≥ 2. It is known that there are four types of such models. In this paper for one of these models the uniqueness conditions of the periodic Gibbs measure is found.

Keywords: Cayley tree; configuration; HC-model; Gibbs measure; translation-invariant measures; periodic measures.

Author: Khoturaev X.F. (Tashkent Institute of Irrigation and Agricultural Mechanization Engineers)

Abstract: 
In the paper some Z-sets of the space of idempotent probability
measures are allocated. For a metrizable infinite compact X and
its closed sets A1 ⊆ A2 ⊆ . . . such that. Ai is dense in X and $∪^{∞}_{i=1}
A_{i6}= X$ it is proved that couples (I(X), $∪^{∞}_{i=1} I(A_{i})$ and (Q, B(Q)) are homeomorphic. It is shown that subsets SI (Bd Q), SI (S) and I(Bd Q) of the space of idempotent probability measures contain skeletons for compacta.

Keywords: idempotent measure, compact, Z-set, skeleton.

Author: Khurramov N.X. (Tezmez State University)

Abstract: 
In the article, for the equation $(signy)|y|^m u_{xx} + u_{yy} −(m/{2y})u_y = 0$, in a mixed domain, we prove the theoremsof uniqueness and existence of the solution of the problem with local and nonlocal conditions on parts of the boundary characteristics and with disconnecting conditions of gluing on the degeneration line.

Keywords: partition of the boundary characteristic into two parts, Bisadze-Samarskiy condition on parallel characteristics, Tricomi integral equation with non-Carleman shift in the “nonsingular”part of the kernel, kernel with singularity of the first order at an isolated singular point, Wiener-Hopf equation, index.

Author: Khusanov Dj. X.( Jizzakh Polytechnic Institute), Berdiyorov F. Sh.(Jizzakh Polytechnic Institute), Buranov J. I. (Academic Lyceum named after I. Karimov of Tashkent State Polytechnic University)

Abstract: 
The intensive development of science and technology, the creation of new information technologies determine the need to develop new approaches to the study of the stability problem by controlled systems modeled by differential equations. A significant place in solving this problem is occupied by the development of qualitative research methods, one of which is the direct Lyapunov method and its development – the method of comparison with the Lyapunov vector function. The comparative method is a universal method for analyzing various properties of systems, regardless of their complexity, nature and structure, effective in the dynamics of systems and control theory. The article considers the problem of applying the comparative method with the Lyapunov vector function to the problem of stability with respect to some variables.

Keywords: differential equations; sustainability; direct Lyapunov method; Lyapunov vector function, comparative method.

Author:Bozarov B.I.( V.I.Romanovskiy Institute of Mathematics)

Abstract: 
This work is devoted to the construction of optimal quadra- formulas with weight function sin x in the Sobolev space L^(m)_2 [0, π]. Here the coefficients of optimal quadrature formulas are found and the general form of the norm is obtained error functional of these formulas.

Keywords: Optimal quadrature formulas; error functional; optimal coefficients; Sobolev space; the weight function

Author: Eshimbetov M.R. (National University of Uzbekistan )

Abstract: 
The initial boundary value problem for the heat equation is studied conductivity on a metric graph in the form of a ladder. Using the generalized Fokas method, we obtained an exact solution to the problem in the form of an integral representation for the given data.

Keywords: Heat equation; metric graph; Fokas method; unified transformation; Fourier transformation; initial-boundary value problem.

Author: Khudoyberdiyev A. Kh.(National University of Uzbekistan ), Shermatova Z. Kh. (V.I.Romanovskiy Institute of Mathematics) 

Abstract: 
We generalize some results obtained for the maximal torus of Lie algebras. to the case of Leibniz algebras. In this paper, we explicitly define the maximal torus of quasi-filiform Leibniz algebras using the matrix method.

Keywords: Leibniz algebra; nilpotent ideal; nilradical; derivation; maximal torus.

Author: Abdurasulov K.K.(V.I.Romanovskiy Institute of Mathematics), Solijanova G.O.(National University of Uzbekistan)

Abstract: 
In this paper we present the descriptions of maximal pro-solvable Lie algebras with maximal positive graded ideals of length 3/2.

Keywords: pro-solvable Lie algebra, pro-nilpotent Lie algebra, derivation, potentially nil-independent derivation.

Author: Dzhamalov S. Z.(Russian State University of Oil and Gas named after I.M. Gubkin in Tashkent), Ruziyev U. Sh.(Russian State University of Oil and Gas named after I.M. Gubkin in Tashkent), Abdullayev O. K.(Russian State University of Oil and Gas named after I.M. Gubkin in Tashkent),  Turaqulov Kh. K.(Kokand State Pedagogical University) 

Abstract: 
In this paper, under certain conditions, on the coefficients of a model loaded mixed type equation of the second kind of the second order in a rectangle, the unique solvability of the solution of one seminonlocal boundary value problem in Sobolev space has been proved.

Keywords: Mixed type loaded equation; seminonlocal boundary value problem; ε – regularization method; the uniqueness and existence of the solution.

Author: Nuraliyev  F. A.(V.I.Romanovskiy Institute of Mathematics), Akhmedov D. M.(V.I.Romanovskiy Institute of Mathematics )

Abstract: 
In this paper in the S.L.Sobolev space L^(m)_2 (0, 1) the extremal function of Hermite type interpolation formulas is found. For optimal coefficients of these formulas the system of linear equations is obtained.

Keywords: extremal function; error functional; interpolation formula.

Author: Rahimova G.G. (National University of Uzbekistan)

Abstract: 
The conditions for the convergence of an empirical characteristic process with random sample size to the corresponding Gaussian process(invariance principle) are studied and an estimate of the convergence rate in this invariance principle is obtained.

Keywords: empirical distribution function, empirical characteristic function, empirical process, invariance principle, rate of convergence.

Author: Khasanov T. G.( Urgen State University), Normuradov Kh. N.( Samarkand State University)

Abstract: 
In this paper, the method of the inverse spectral problem is applied to the integration of the loaded Korteweg-de Vries equation with a free term independent of the spatial variable in the class of periodic functions.

Keywords: In this paper, the method of the inverse spectral problem is applied to the integration of the loaded Korteweg-de Vries equation with a free term independent of the spatial variable in the class of periodic functions.

Author: Khusainova B. B. (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
This article provides another proof of Lindeberg’s theorem based on the analytical method of characteristic functions and using the method of truncation of random variables.

Keywords: Infinitely divisible distribution, distribution function, random variable, condition of uniform infinite smallness, weak convergence, central limit theorem, class of limit distributions, characteristic function.

Author: Sharipov  S. O.(V.I.Romanovskiy Institute of Mathematics), Toshqulov Kh. A.(National University of Uzbekistan)

Abstract: This paper deals with central limit theorem for nearly critical branching processes with dependent immigration.

Keywords: Branching process, immigration, φ-mixing

Author: Sharipov O. Sh.(National University of Uzbekistan), Ruziyeva D. S. (National University of Uzbekistan)

Abstract: 
In the paper the central limit theorem for c_0 space-valued random fields satisfying some dependence condition is proved

Keywords: Random field, central limit theorem, c_0 space.

Author: Karimov U.Sh.(National University of Uzbekistan)

Abstract: 
The article is devoted to local differentiations on real W * -algebras. It is proved that the restriction of local derivations to the Abelian real W*- subalgebra is a differentiation.

Keywords: real W*-algebra; derivation; local derivation.

Author: Zunnunov A.O. (National University of Uzbekistan)

Abstract: 
The article deals with the pursuit problem with simple motions in the sense of l-capture on a plane in a circle. A structure is proposed for constructing piecewise constant persecution directorates which will ensure the completion of the game in a finite time. Established a score above the time of the game to complete the pursuit.

Keywords: Pursuit; pursuer; evader; pursuit control; evasion control.

Author: Abdullayev J. Sh. (National University of Uzbekistan)

Abstract: 
The aim of this work is to find optimal estimates for the Bergman kernels for the classical domains R_I (m, k), R_II (m), R_III (m) and R_IV (n) through the Bergman kernels of balls in the spaces C^{mk }, C ^{m(m+1)/2} , C^{m(m−1)/2} and C^n, respectively.

Keywords: Classical domains, Bergman’s kernel, homogeneous domain; symmetric domain; orthonormal system..

Author: Juraev R. M. (National University of Uzbekistan)

Abstract: 
In this paper the introduction of a uniform structure in the space of G-permutation degree is presented using statement. Such a structure we denoted by $SP^n_G U$. In addition it is presented that the weight of the space ($S P^n_G X, S P^n_G U$) does not exceed the weight of the space (X, U). In the work [2] is studied some cardinal properties of space of the permutation degree, in particular: it is shown that the weight of a topological space X is equal to the weight of a space $SP^n_G X$. In [3] is studied topological properties of space of the permutation degree, in particular: it is proven that for a topological space X to be zero-dimensional it is necessary and sufficient that $SP^n_G X$ is zero dimensional.

Keywords: A space of the G-permutation degree; uniform space; totally bounded; uniformly metrizable.

Author: Kushmurodov A. A. (V.I.Romanovskiy Institute of Mathematics)

Abstract:  In this paper we prove strong law of large numbers for m- dependent random fields with values in separable type p Banach space.

Keywords: m − dependent random fields, Banach space, strong law of large numbers.

Author: Madrakhimov U. S. (National University of Uzbekistan)

Abstract: 
In this paper, we study the Cauchy problem in Sobolyev’s spaces for a fractional-order equation with a pseudo-differential operator coefficient in a multidimensional case. In this case, the pseudo-differential operator is considered to be related to the oscillation equation of a plate with one end fixed and the other free. Using the completeness of the system of eigenfunctions, the theorem on the uniqueness of the solution of the problem is proved. The regular solution of the Cauchy problem under consideration has been proved to exist in the Sobolyev space.

Keywords: pseudo-differential operator, Cauchy problem, spectral problem, eigenvalue, eigenfunction, uniqueness, existence, series.

Author: Mamanazarov A. O. ( Fergana State University)

Abstract:  In this paper, boundary-value problems have been formulated and investigated for a fractional order mixed parabolic equation in unbounded domains. The uniqueness and existence of the solution of the considered problems was proved.

Keywords: fractional order mixed parabolic equation; a boundary value problem; method of energy integrals; method of integral equations.

Author: Seidullayev A. K.(Karakalpak State University)

Abstract: 
In this paper, we consider the problem of integral geometry on a family of broken lines with a piecewise constant weight function. An exact inversion formula is obtained, on the basis of which the uniqueness theorem is proved and stability estimates for its solution are obtained. If the right-hand side of the equation is given approximately, we will consider the Tikhonov regularization based on the obtained inversion formula.

Keywords: Radon transform; the Tikhonov regularization; weighted function.

Author: Tugenov Z. T.(V.I.Romanovskiy Institute of Mathematics)

Abstract: 
In this paper, the concept of a periodic real measure on [0; 1) is introduced.This periodicity depends on the partition of the set [0; 1). For one specific partition of [0, 1) the general form of periodic measures is found. For an arbitrary partition of the set [0; 1), a sufficient condition for the non-existence of periodic measures is found.

Keywords: p-adic periodic measures; real periodic measures; potential field.

Author: Kholturayev Kh. F. (Tashkent Institute of Irrigation and Agricultural Mechanization Engineers)

Abstract: 
For idempotent probability measures max-plus variant of Fubini theorem is established. Further, it is proved metrizability of a given compact if the space of idempotent probability measures given on the compact, is hereditary normal.

Keywords: Idempotent measure; compact Hausdorff space; metrizability.

Author:  Khomidov M. K. (National University of Uzbekistan)

Abstract: 
In this paper, we study Gibbs distributions corresponding to the infinite Toda chain (see [1-2]).

Keywords: Toda chain; Gibbs distributions; relatively compact.

Author: Khusainova B. B. (V.I.Romanovskiy Institute of Mathematics)

Abstract: 
This article provides another proof of Lindeberg’s theorem based on the analytical method of characteristic functions and using the method of truncation of random variables

Keywords: Infinitely divisible distribution, distribution function, random variable, condition of uniform infinite smallness, weak convergence, central limit theorem, class of limit distributions, characteristic function.

Author: Tukhtasinov M.T. (National University of Uzbekistan), Abduolimova G.M.(Andijan State University), Hayitqulov B.Kh.(National University of Uzbekistan)

Abstract:
The paper deals with the third boundary value problem of parabolic type. The distribution of heat in the body under consideration is controlled by a function that is at
the boundary of the body. The question of the possibility of transferring the initial body position to the desired state in case of conflict is being resolved.

Keywords: Parabolic type equation; a boundary value problem; boundary function; control problem.

Author: Abdullayev A.(Inha University in Tashkent), Rozikov U.A. (V.I.Romanovskiy Institute of Mathematics)

Abstract:
In this article, we give a brief overview of mathematical methods in the humanities and social sciences. Note that mathematical methods are used in the study of complex natural systems and for understanding calculus and statistics. When mathematically modeling a social system, a specific method may be needed that may not be mentioned in this article. Consequently, it is impossible to give a complete list of mathematical methods in the humanities and social sciences. Here we list some mathematical methods in voting systems, social networks, statistical and multi-agent systems.

Keywords: voting system; social network; multi-agent system.

Author: Karimov E.T. (V.I.Romanovskiy Institute of Mathematics)

Abstract:
In this article, we study the unique solvability one boundary value problem with an integral conjugation condition in a mixed domain consisting of a characteristic triangle and a rectangle for a mixed type equation with the Hilfer derivative. Using methods of the theory of differential equations of integer and fractional orders, integral equations, proven the corresponding theorem.

Keywords: Mixed type equation; boundary value problem; integral conjugation condition; Hilfer operator, method of integral equations.

Author: Ruziev M.Kh.(V.I.Romanovskiy Institute of Mathematics)

Abstract:
The paper studies a boundary value problem with displacement for
mixed type equations. The uniqueness of the solution to the problem is proven using the extremum principle. When proving the existence of a solution to the problem, the theory of singular integral equations, Wiener-Hopf equations and Fredholm integral equations is used.

Keywords: Mixed type equation; boundary value problem; extremum principle; method of singular integral equations.

Author: Bekiev A.B. (Karakalpak State University)

Abstract:
In this paper we prove the existence and a uniqueness of a solution of the boundary problem for the fourth order mixed type equation in a rectangular domain. Continuous
dependence of the solution on the given data has been shown.

Keywords: Mixed type equation; boundary problem; the existence and a uniqueness of a solution.

Author: Mamanazarov A.O. (Fergana State University)

Abstract:
In the paper, a problem with shifting condition for a parabolic-hyperbolic equation was studied and a uniqueness and the existence of the solution of the considered problem was proved. A uniqueness of the solution of the problem was proved by the method of integral energy and the existence part was proved by
the method of integral equations.

Keywords: Parabolic-hyperbolic equation; problem with shifts; second kind Fredholm’s integral equation; Abel’s integral equation, method of energy integrals.

Author: Rozikov U.A.(V.I.Romanovskiy Institute of Mathematics), Khakimov R.M.(Namangan State University)

Abstract:
In this paper we study the extremality of translation-invariant Gibbs measure for the HC-model on a Cayley tree. It is known that for this model the translation invariant measure is unique. We give a new proof of this statement and found regions of the extremality of this measure on the Cayley tree of order k.

Keywords: Cayley tree; admissible configuration; HC-model; Gibbs measure; translation-invariant measure; extremality of measure.

Author: Ergashev T.G. (V.I.Romanovskiy Institute of Mathematics)

Abstract:
In this paper, we prove a unique solvability of a boundary value problem with integral form conjugation condition on the line of type changing for a mixed parabolic-hyperbolic type equation.

Keywords: Multidimensional elliptic equations with one singular coefficient; Holmgren’s problem; fundamental solution; Green’s function method.

Author: Jalilov M.A.(Fergana State University)

Abstract:
In this paper, for equations of mixed type of the fourth order with the Caputo fractional derivative in a rectangular domain, we study the Samarsky-Ionkina. Using the method of separation of variables, a theorem on the uniqueness and existence of a regular solution to this problem is proved.

Keywords: mixed type equation; fractional differential operator; Mittag-Leffler function; Samarskii-Ionkin type problem; completeness, Riesz basis; Fourier series.

Author: Murzambetova M.B. (Nukus State Pedagogical Institute)

Abstract:
In this article, we study one boundary value problem for a partial differential equation fourth order order with the lowest term in the rectangular region. To solve the problem, an a priori estimate was obtained, from which it follows that the solution to the problem is unique. To prove the existence of a solution to the problem, the method of separation of variables is used. The solvability of the problem is reduced to the Fredholm integral equation of the second kind with respect to the desired function, which is solved by the method of successive approximations.Sufficient conditions are found to ensure absolute and uniform convergence of the series representing the solution to the problem and the series obtained from it by differentiating it four times with respect to x and once with respect to t.

Keywords: Boundary value problem; Fredholm integral equation; a priori estimate; regular solvability; resolvent.

Author: Kholikov D.K. (National University of Uzbekistan)

Abstract:
The work investigates the solvability of a nonlocal problem with an integral condition for a loaded third-order pseudoparabolic equation. The existence and uniqueness of a classical solution to the problem under consideration is proved by the Riemann method.

Keywords: Loaded equation, Riemann function, non-local condition, pseudoparabolic equation.

Author: Irgashev B.Yu.(Namangan Engineering-Construction Institute)

Abstract:
In this paper, we are studying the Dirichlet-Neumann problem for higher order mixed type equation. The conditions for a uniqueness and the existence of the solution are found.

Keywords: Mixed type equation; a boundary value problem; uniqueness; existence; eigenvalue; eigenfunction; infinite series; small denominators; convergence.

Author: Rarovik.R.I(Vitus Bering State University)

Abstract:
In this paper, using the harmonic balance method, we study the forced oscillations of the fractional Duffing oscillator. The amplitude-frequency and phase-frequency characteristics are obtained. The relationship between the fractional parameters in the model equation with the quality factor of the oscillatory system is shown.

Keywords: fractional Duffing oscillator; harmonic balance method; amplitude-frequency characteristics; phase-frequency characteristics.

Author: Sirojitdinov A.A.(V.I.Romanovskiy Institute of Mathematics)

Abstract:
In this article we consider sequences of independent random vectors with values in S-dimensional Euclidean space. Some asymptotic estimates for the distributions of sums of independent random vectors are obtained.

Keywords: Random vectors; quadratic form; characteristic function; covariance matrix.

Author: Khashimov A.R. (Tashkent Financial Institute)

Abstract:
The article deals with a boundary value problem for a non-stationary third order equation of composite type, in which the value of the desired functions and their derivatives are specified as a linear combination on the boundary of the region. The
correctness of the boundary value problem is proved. The proof used the method of integrals of energy, the theory of potentials. In addition, the asymptotic properties of the fundamental solutions of the equation are studied.

Keywords: Third order composite type equation; energ integrals; method of potentials.

Author: Kurbaev T.K. (V.I.Romanovskiy Institute of Mathematics)

Abstract:
In this article, Leibniz algebras are constructed using algebras. Lie hebra sl2 and infinite-dimensional sl2-modules.

Keywords: Lie algebra; Leibniz algebra; simple sl2-module.

Author: Karimov E.T.(V.I.Romanovskiy Institute of Mathematics),
Toshtemirov B.H.(Ferghana Politechnical Institute)

Abstract:
In this work, the unique solvability of a Tricomi-type problem with an integral conjugation condition for a mixed-type equation with a fractional Hyper-Bessel operator in a mixed domain consisting of characteristic sky triangle and rectangle. The obtained scientific result was proven by the methods of integral energy and integral equations.

Keywords: Mixed type equation; a boundary value problem; integral conjugation condition; Hyper-Bessel operator; method of integral equations.

Author: Artikov M.(Termez State University), Mirsaburova U.M(Termez State University)

Abstract:
In the article, the equation (signy)|y|^m u_{xx} + u_{yy} − (m/{2y})u_y =0, is considered in a mixed domain, the existence and the uniqueness of the solution of the problem with insufficient Tricomi condition on the boundary characteristic and Frankl type condition on the internal characteristic for the mixed-type equation are proved.

Keywords: mixed type equation with singular coefficient; boundary equation; Tricomi-Frankl type condition; nonstandard singular integral equation of Tricomi, Wiener-Hopf integral equation; index.

Author: Zikirov O.S.(National University of Uzbekistan) Zhumagul G.O.(Kazakh National Pedagogical University)

Abstract:
In the paper we study of the correctness of the problem Goursat–Dirichlet problem for a hyperbolic equation of the third order in the characteristic triangle. The effects of the influence of younger members on the correctness of the problem under consideration.
In violation sufficient conditions for the coefficients of the lower derivatives in equations, some examples of what the problem can be incorrect.

Keywords: Goursat-Dirichlet problem; third order hyperbolic equation; a uniqueness of a solution.

Author: Norjigitov A.F.(V.I.Romanovskiy Institute of Mathematics), Sharipov O.Sh. (National University of Uzbekistan)

Abstract:
In this paper, the law of large numbers for weakly dependent random variables with values in D[0, 1] is proved.

Keywords: Weakly dependent random variables; law of large numbers; D[0, 1] space.

Author: Okboev A.B. (Fergana State University)

Abstract:
In the work the problems with shift condition for second kind degenerated equation of hyperbolic type were investigated for various values of coefficient of the term which involves derivative with respect to y.

Keywords: Equation of hyperbolic type; second kind degenerated equation; boundary value problem; problem with shift condition; general solution.

Author: Khomidova S.M.(Ferghana State University), Nazarov H.A.(Ferghana State University), Karimov U.Sh.(National University of Uzbekistan)

Abstract:
In this paper we prove that every differentiation of simple real C*-algebras with identity is also internal.

Keywords: real C*-algebra; derivation; inner derivation.

Author: Abdullaev B.I.(Urgench State University ), Sharipov R.A. (Urgench State University)

Abstract:
In this paper using Green function we prove equality of the local and global α−polar sets.

Keywords: α−subharmonic function; α−polar set; α−regular domain; Green function; strongly m−subharmonic function.

Author: Allakov I.(Termez State University), Safarov A.Sh. (Termez State University)

Abstract:
Let X it is enough big real number and let M denote the set natural numbers not exceeding X which cannot be written as a sum prime and fixed degree a prime number from arithmetical progression with difference d. Let Ed(X) = cardM.. In persisting
work is received new numerical sedate estimation for set Ed(X) and estimation from below for number presentation n /∈ M in specified type. Proved estimations is revision and generalization for arithmetical progression earlier got result by V.A.Plaksin.

Keywords: The Dirichlet character; Dirichlet L− function; exceptional set; representation numbers; exceptional zero; exceptional nature; main member; remaining member.

Author: Juraev T.F. (Tashkent State Pedagogical University)

Abstract:
In this paper the Milutin, Dugundji and Mihael spaces are considered and their geometrical, topological properties under acting of some functories of T ych-Tychonov space and continius maps to itself are studied.

Keywords: Spaces, compact Dugundji; absolute extensors at dimension n; linear operators; operators exactly and operators extensions.

Author: Imanbaev N.S.(South Kazakhstan State Pedagogical University),
Ospanov M.N. (Eurasian National University)

Abstract:
In this paper, we study the question about distribution of eigenvalues of the third-order composite type equations with regular, more precisely, with periodic boundary value conditions. After, applying the Fourier method, the original problem splits into two problems on eigenvalues of third-order ordinary differential operators with periodic boundary value conditions in L_2(0, 1).

Keywords: Composite differential equations; Fourier method; entire functions, conjugate indicator diagram; adjoint operator.

Author: Parovik R.I. (Kamchatka State University)

Abstract:
In the work, the Cauchy problem for the Airy fractional oscillator is investigated. The solution to the Cauchy problem is presented in terms of the three-parameter Airy function. Analogs of the Airy functions of the first and second genera are found.

Keywords: Airy functions, Cauchy problem, fractional oscillator, Gerasimov-Caputo operator.

Author: Sobirov Z.A.(National University of Uzbekistan),
Rakhimov K.U. (National University of Uzbekistan)

Abstract:
In this article, we studied the Cauchy problem for the Airy equation with a fractional derivative in a star-shaped graph. An a priori estimate was obtained and a uniqueness theorem was proved. It was studied the properties of the potentials. Found the exact view of the problem’s solution using the method of potentials.

Keywords: Airy equation; Cauchy problem; fractional-order differential equation; method of potentials; Korteweg-de Vries equation.

Author: Hasanov A. (V.I.Romanovskiy Institute of Mathematics), Ergashev T.G.(V.I.Romanovskiy Institute of Mathematics)

Abstract:
At present, for all hypergeometric functions in one and two variables, analytical continuation formulas are known. In this paper, we find analytic continuation formulas for Lauricella hypergeometric functions in three variables.

Keywords: Lauricella hypergeometric functions; analytical continuation formulas; hypergeometric functions in three variables.

Author: Sharakhmetov Sh.(V.I.Romanovskiy Institute of Mathematics)

Abstract:
In this paper, an equality connecting joint distribution with marginal distributions is proved. Some applications are given to: the proof of the basic properties of joint distributions and the structure of the dependence of binomial random variables; calculating functionals and finding their extreme values; building joint distributions with given marginal distributions. Also proposed a new statistical estimation for the unknown distribution.

Keywords: joint distribution; binomial random variable; extreme values; functionals; statistical estimation.

Author: Baratov I.I.(National University of Uzbekistan), Dekhkonov F.N.(National University of Uzbekistan), Komilov N.M.(Andijan State University)

Abstract:
This article examines the inverse problem associated with the heat equation. The regularization of the problem using the Lattes-Lyons quasi-inversion method is shown. An estimate has been found for the deviation uδ from the exact solution u.

Keywords: Inverse problem; boundary conditions; Laplace operator.

Author:Akhmedov S.A.(Andijan State University)), Ablazova K.S. (Andijan State University)

Abstract:
In this article the control charts for defining linear dependence or independence of two measurable quantities submitting to two-dimensional normal distribution are given. These charts may be used to control technological processes.

Keywords: two-dimensional normal distribution; hypothesis; control chart; quantile; correlation function.

Author: Djalilov A.A.(Turin Polytechnic University in Tashkent), Abdukhakimov S.Kh.( National University of Uzbekistan)

Abstract:
It is well known, that the Feigenbaum unstable separatrice is the family of unimodal maps of the interval [−1, 1] with one critical point. It is established imlicit formulas for periodic orbits of this family, and the relations between orbits of different maps of this family.

Keywords: periodic point; fixed point; critical point; separatrice Feigenbaum.

Author: Juraev Sh.Yu. (V.I.Romanovskiy Institute of Mathematics)

Abstract: The Galton Watson branching random process {Zn : n ≤1, Z0 = 1} generated by a generating function of the formF(x) = (a + bx)/(c + dx). Exact formulas are found for the function generating the number of particles of the n− generation Zn Fn(x) = ExZn , |x| ≥ 1 and probability distributions P(Zn =k), k = 0, 1, 2, . . ..

Keywords: Galton-Watson branching processes; linear fractional generating function; probability of process degeneration; critical process.

Author: Irgashev B.Yu. (Namangan engineering-construction Institute,)

Abstract:
In this paper, we study a boundary-value problem for a degenerate high-order equation. The conditions for the uniqueness and the existence of the solution are found.

Keywords: Partial differential equation; high order; degeneration; boundary value problem; uniqueness; the existence; eigenvalue; eigenfunction; integral equation; series; convergence.

Author: Kushmurodov A.A. (V.I.Romanovskiy Institute of Mathematics)

Abstract:
In this paper we prove strong law of large numbers for m-dependent random variables with values in separable Banach space type p.

Keywords: m-dependent random variables; separable Banach space; strong law of large numbers.

Author: Mamurov B.J. (Bukhara State University)

Abstract:
In the present article the evolutionary equations for finite-dimensional homogeneous cubic stochastic processes are studied.

Keywords: Finite-dimensional simplex; initial state; transition functions; homogeneity; cubic stochastic processes.

Author: Nishonova Sh.T. (Fergana State University)

Abstract:
In the article problems with the first and second kind integral conditions were investigated for an elliptic-parabolic equation in a mixed domain.

Keywords: elliptic-parabolic equation; nonlocal problem; integral condition; unique solvability.

Author: Parovik R.I(Kamchatka State University)

Abstract:
Using the harmonic balance method, the bistable behavior of the fractional Duffing oscillator is studied, the region of transition from one stable mode to another (jump effect) is found, the condition for the existence of a bistable mode is determined,
and resonance amplitude-frequency characteristic curves are constructed.

Keywords: jump effect; fractional Duffing oscillator; amplitude-frequency characteristic.

Author: Khusanbaev Y.M. ,(V.I.Romanovskiy Institute of Mathematics) Kudratov Kh. E. (National University of Uzbekistan)

Abstract:
In this paper an above bound is presented for the deviation of branching random processes with immigration from the conditional expectation.

Keywords: Branching random process with immigration;
Gal’ton-Vatson process; momets

Author: Ruziev .M.Kh.(V.I.Romanovskiy Institute of Mathematics)

Abstract: In this paper, we prove a unique solvability of a boundary value problem for a mixed type equation with singular coefficient.

Keywords: Mixed type equation; a boundary value problem; method of integral equations; functional equation; iteration method.

Author: Fayazov K.S. (Turin Polytechnic University in Tashkent), Khudayberganov
( National University of Uzbekistan)

Abstract:
second-order partial derivatives with two degenerate lines. This paper is devoted to the study of the initial-boundary value problem for a system of second-order partial differential equations of mixed type with two degenerate lines. Considered
problem is ill-posed in the sense of H’Adamard. Following to the idea of A.N. Tikhonov of approximate solution ill- posed problems we proof the theorems of the uniqueness and conditionally stability.

Keywords: boundary problem, system of equations of mixed type with two degenerate lines, ill-posed problem, a priori estimate, estimate of conditional stability, uniqueness, set of correctness.

Author: Sharipov O.Sh. (National University of Uzbekistan), Ruzieva D.S. (National University of Uzbekistan)

Abstract:
In the paper the central limit theorem for Hilbert space-valued random fields satisfying some dependence condition is proved.

Keywords: Random field; central limit theorem; Hilbert space.

Author: Hasanov A.,(V.I.Romanovskiy Institute of Mathematics), Ruzhansky M.( Ghent University)

Abstract:
It is known, there are 205 hypergeometric functions in three variables of second order, regions of convergence of which have been given in the literature. In this paper, using the properties of the beta-function, Euler-type integral representations are constructed and proved for the indicated hypergeometric functions.

Keywords: Euler-type integral representations; hypergeometric
function.

Author: Dzhamalov S.Z. (V.I.Romanovskiy Institute of Mathematics)

Abstract:
In the present work a uniqueness, existence and smoothness of
the solution of nonlocal boundary value problem for the second
order mixed-type equation of the second kind are proved in
certain Sobolev spaces.

Author: Urinov A.K. (Fergana State Univerisity), Ismoilov A.I. (Fergana State Univerisity)

Abstract:
In this paper we study the Cauchy-Goursat problem
for the generalized Euler-Poisson-Darboux equation in the
characteristic triangle. By the Riemann method, a formula
is found for solving the problem under study. It is proved
that under certain conditions on given functions, this formula
actually gives a solution of the Cauchy-Goursat problem.

Author: Khakimov R.M. (Namangan State University), Ayubjonova M.S.(Namangan State University).

Abstract: In this paper at any values of the parameters of the four state
ferromagnetic Potts model it is proved that all periodic Gibbs
measures are translation-invariant.

Author: Khakimov R.M. (Namangan State University), Tojiboev B.Z.(Namangan State University).

Abstract:In this paper we find conditions under which the extreme Gibbs
measures are not unique.

Author: Khudoyberdiyev A.Kh. (National University of Uzbekistan), (V.I.Romanovskiy Institute of Mathematics), Sattarov A.M. (National University of Uzbekistan)

Abstract:
It is known that an arbitrary nilpotent Leibniz algebra of nilindex has a nonnilpotent Leibniz differential differentiation of order [s/2]+1. This paper shows the existence of a nilpotent Leibniz algebra, for which every Leibniz differentiation of order [s/2]+1
nilpotent.

Author: Ruziev M.Kh. (V.I.Romanovskiy Institute of Mathematics)

Abstract:
The paper studies a boundary value problem for a mixed type equation. The uniqueness of the solution to the problem is proved using the extremum principle, and the existence of the solution method of integral equations.

Author: Amanov Dj. (V.I.Romanovskiy Institute of Mathematics), Kilichov O. (Bukhara Engineering and Technology Institute)

Abstract:
In the present paper, we study a boundary value problem for
the fourth order equation of mixed type in a rectangular domain
and prove the existence of the unique solution of this problem. In
the theory of boundary value problems for mixed type equations
usually two conjugation conditions are in use, in general. In this
case, for the solvability of boundary value problem governed
by mixed type equation containing a hyperbolic equation in a
rectangular domain a certain condition (on the sizes of the sides
of the rectangle) appears. However, in this paper, we give three
conjugation conditions so that such condition does not appear.

Author: Dzhamalov S.Z. (V.I.Romanovskiy Institute of Mathematics)

Abstract:
In the present work, the correctness of an inverse problem for
the second order mixed type multidimensional equation of the
first kind with semi-periodic conditions has been investigated.
For this problem, the theorems on existence and uniqueness of
the solution are proved in a certain class by “ε-regularization”,
a priori estimations and of successive approximations methods.

Author: Rakhimova G.G. (National University of Uzbekistan)

Abstract:
We consider sequential confidence estimation of functionals of
an unknown distribution function. Conditions of asymptotic
consistency of fixed-width confidence intervals and asymptotic
efficiency of stopping times are obtained.

Author: Sagdullaeva M.M. (Tashkent University of Information Technologies)

Abstract:
In the paper we study non-local boundary-value problem for the
third-order equation with heat operator in the main part. We
prove the theorem of the existence and uniqueness of regular
solution for considered problem.

Author: David J.(Florida State University), Nolte A.(Tufts University), Sherman J.(University of Minnesota)

Abstract:
The paper proves the unique solvability of the boundary value problem for a three dimensional wave equation with the Bessel operator and the fractional differentiation operator in the sense Caputo. Under certain conditions for specified functions the existence, uniqueness and continuous dependence of the solution to the problem under study is proven.

Author: Ashurov R.R. (V.I.Romanovskiy Institute of Mathematics), Dzhamalov S.Z.(V.I.Romanovskiy Institute of Mathematics)

Abstract:
In the present work the problems of correctness of an inverse problem for the second order mixed type multidimensional equation of the second kind with nonlocal conditions has been investigated. For this problem the theorems on existence and uniqueness of the solution are proved in a certain class by “ε-regularization a priori estimations and of successive approximations methods.

Author: Sabitov K.B.( Sterlitamak branch of the Bashkir state university),
Sidorov S.N.( Sterlitamak branch of the Institute of Strategic Studies of the Republic of Bashkortostan)

Abstract:
In this paper, inverse problems are posed and studied to determine the factors of the right-hand sides of a mixed parabolic-hyperbolic type with a degenerate parabolic part,
depending on time. On the basis of the theory of integral equations, the corresponding uniqueness theorems and the existence of solutions of inverse problems were proved, and explicit formulas for the solution were obtained.

Author: Ergashev T.G. (V.I.Romanovskiy Institute of Mathematics),
Hasanov A.(V.I.Romanovskiy Institute of Mathematics)

Abstract: When studying boundary value problems for some differential equations of athematical physics, arose Students of applied mathematics often have to study systems of partial differential equations that are satisfied by hypergeometric functions of several variables. In this communication it is argued that the solutions to two systems of partial differential equations are hypergeometric functions of two variables of the third and fourth order.

Author: Mirsaburov M.(Termiz State University), Ruziev M.Kh.(V.I.Romanovskiy Institute of Mathematics), Amonov B.B.(Termiz State University)

Abstract:
In this paper, we find the fundamental solutions of a degenerate elliptic equation with singular coefficients, study the properties of these solutions, and give an integral representation of the solution of the equation.

Author: Balkizov Zh.A.( Institute of applied mathematics and automation of the Kabardino-Balkarian scientific center of the Russian Academy of Sciences)

Abstract: A boundary-value problem with an offset for a model in-homogeneous third-order parabolic-hyperbolic type equation is studied, in case when a linear combination (with variable coefficients) of the derivatives of the desired function on the characteristics and on the line of type changing is specified as one of the inner-boundary conditions. The necessary and sufficient conditions for the existence and uniqueness of a regular solution of the problem are found. In some particular cases, the
representation of the solution of the problem being studied, is written out explicitly.

Author: Berdyshev A.S.(Abai Kazakh National Pedagogical University),
Hasanov A.(V.I.Romanovskiy Institute of Mathematics)

Abstract:
In this work, eigenvalues and some properties of eigenfunctions of a local boundary problem for the third order degenerating composity type equation have been studied.

Author: Karachik V.V.(South Ural State university),
Turmetov B.X. (International Kazakh-Turkish University)

Abstract:
In this paper we study new classes of well-posed boundary-value problems for the biharmonic equation. The considered problems are Bitsadze-Samarskii type nonlocal boundary value problems. The investigated problems are solved by reducing them to the Neumann and Dirichlet type problems. In this paper, theorems on existence and uniqueness of the solution are proved, and exact conditions for solvability of the problems are found. In addition, integral representations of the solution are obtained.

Author: Karimov Sh.T.(Fergana State University)

Abstract: multidimensional generalized Erd ́elyi-Kober operator with differential operators of the higher order, in particular, with powers of the Bessel differential operator. The applications of the proved properties of the Erd ́elyi-Kober operator to the solution of the analogue of the Cauchy problem for a multidimensional polycaloric equation with the Bessel operator acting in all spatial variables are shown. An integral formula for solving the formulated problem is constructed.

Author: Urinov A.K.,(Fergana State University), Okboev A.B.(Fergana State University)

Abstract: In the article a problem of A. M. Nakhushev type for the second kind degenerated hyperbolic equation with spectral parameters has been formulated and the unique solvability of the considered problem has been investigated.

Author: Khashimov A.R. (Tashkent Financial Institute)

Abstract:
In the paper, the second boundary value problems with nonlocal conditions for non stationary of the equation of the third order of the composite type is considered. By methods of the integrals energy and potentials we have found a unique regular solution
of the considered problem.

Author: Imanbaev N.S.( South Kazakhstan State Pedagogical University),
Kanguzhin B.Ye.(Kazakh National University)

Abstract: This article considers the eigenvalue problem for the Cauchy-Riemann operator with homogeneous boundary conditions of the Dirichlet type. It is proved that the spectral problem under consideration does not have eigenvalues, that is, for any complex λ, it has only a zero solution.

Author: Irgashev B.Yu.(Namangan Engineering and Construction Institute)

Abstract:
For an equation of higher order, auto-model solutions were found. For particular cases (n = 1, n = 2), it was shown that a fundamental solution can be expressed through these solutions.

Author: Kadirkulov B.J.(Tashkent State University of Oriental Studies),
Turmetov B.Kh.(International Kazakh-Turkish University)

Abstract:
In this paper, we propose a new method for constructing a solution of the integro-differential equations of fractional order with Hadamard type operators. The particular solutions of the homogeneous and of the inhomogeneous equation will be constructed. Note that this method is based on construction of normalized systems functions with respect to the integro-differential operator of fractional order.

Author: Shermatova Kh.M. (Fergana State University)

Abstract: In the present work some boundary-value problems for the third order parabolic- hyperbolic equation with two lines of type changing are formulated and one of these problems has been studied.

Author: Abdullayev A.A. (Tashkent Institute of Irrigation and Agricultural Mechanization Engineers)

Abstract: In this work, we prove the uniqueness of the solution to the a problem of the type of the Poincare – Tricomi problem for an equation of elliptic-hyperbolic type of the second kind.

Author: Abdullaev O.Kh. (National University of Uzbekistan), Matchanova A.A. (National University of Uzbekistan)

Abstract: This paper is devoted to the unique solvability of a nonlocal problem for a loaded third-order parabolic-hyperbolic equation with Caputo and Riemann-Liouville operators.

Author: Tashpulatov S.M.(Institute of Nuclear Physics of Uzbekistan Academy of Sciences)

Abstract:
This paper considers the energy operator of five electron systems in the Hubbard model and investigates the structure of the essential spectrum and the discrete spectrum of the system in the fifth doublet state.

Author: Apakov Yu.P.( Namangan Engineering and Construction Institute),
Juraev A.Kh(Namangan Engineering and Construction Institute)

Abstract:
In the article, the third boundary problem for the third order equation with multiple characteristics. The uniqueness of the solution studied by the method of energy integrals and the existence has been proved using Fourier’s method.

Author: Dalabaev U.(The University of World Economy and Diplomacy)

Abstract:
In the present work solutions of applied problems by means of a method of moved nods are observed. Problems are observed: flow in a flat pipe, heat extending to a plate, a magnetohydrodynamic flow of Kuetta, an unsteady flow of a viscous fluid between parallel walls and a flow in ellipsoidal pipe.

Author: Karimov K.T. (Fergana State University)

Abstract:
In this paper, a nonlocal problem with an integral condition of the first kind for a three-dimensional elliptic equation with two singular coefficients has been investigated by the method of spectral analysis.

Author: Mamanazarov A.O. (Fergana State University)

Abstract:
In the article a nonlocal problem for a parabolic-hyperbolic type equation was formulated and a uniqueness and the existence of the solution of the considered problem was proved. The uniqueness of the solution of the problem has been proved by
the method of energy integrals and the existence by the method of integral equations.

Author: Khudayarov B.A.( Tashkent Institute of Irrigation and Mechanization Engineers), Turaev F.J. (Tashkent Institute of Irrigation and Mechanization Engineers),
Komilova Kh. (Tashkent Institute of Irrigation and Mechanization Engineers)

Abstract:
The article presents the results of a study of vibration process in pipelines conveying fluid or gas. When studying vibrations of gas and fluid conveying pipelines, a model is used in the form of a cylindrical shell and a two-parameter model of the Pasternak viscoelastic foundation. The hereditary Boltzmann-Volterra theory of viscoelasticity is used to describe viscoelastic properties. The effects of the parameters of the Pasternak
foundations, the singularity in the heredity kernels and geometric parameters of the pipeline on vibrations of structures with viscoelastic properties are numerically investigated.

Author: Shikhiev R.M.(Karakalpak State University), Baltabaeva R.B.(Karakalpak State University)

Abstract:
In this work, one property of the mathematical model of salt-dust movement in the two-component nonlinear environment considering convectivity has been studied.

Author: Chorieva S.T. (Termez State University)

Abstract:
In this work, for the Gellerstedt equation with singular coefficient, we study the problem with the Bitsadze-Samarskiy and Frankl conditions. A uniqueness of the solution for the
formulated problem is proved.

Author: Churikov V.A.(Tomsk Polytechnic University)

Abstract:
In this paper, d-operator of fractional differentiation and integration of variable order has been studied.