Vol 68, No 1 (2022): Science — Technology — Education — Mathematics — Medicine


Extension of Relative-Risk Power Estimator under Dependent Random Censored Data

Abdushukurov A.A.


In this paper, the considered problem consists in estimation of conditional survival function by right random censoring model in the presence of a covariate. We propose a new estimator of conditional survival function which is extension of relative-risk power estimator of independent censoring and study its large sample properties. We present result of asymptotic normality with the same limiting Gaussian process as for copula-graphic estimator.

Contemporary Mathematics. Fundamental Directions. 2022;68(1):1-13
pages 1-13 views

Smoothness of Solutions to the Damping Problem for Nonstationary Control System with Delay

Adkhamova A.S.


We consider the damping problem for a nonstationary control system described by a system of differential-difference equations of neutral type with smooth matrix coefficients and several delays. This problem is equivalent to the boundary-value problem for a system of second-order differential-difference equations, which has a unique generalized solution. It is proved that the smoothness of this solution can be violated on the considered interval and is preserved only on some subintervals. Sufficient conditions for the initial function are obtained to ensure the smoothness of the generalized solution over the entire interval.

Contemporary Mathematics. Fundamental Directions. 2022;68(1):14-24
pages 14-24 views

An Algebraic Condition for the Exponential Stability of an Upwind Difference Scheme for Hyperbolic Systems

Aloev R.D., Nematova D.E.


In the paper, we investigate the question of obtaining the algebraic condition for the exponential stability of the numerical solution of the upwind difference scheme for the mixed problem posed for onedimensional symmetric t-hyperbolic systems with constant coefficients and with dissipative boundary conditions. An a priori estimate for the numerical solution of the boundary-value difference problem is obtained. This estimate allows us to state the exponential stability of the numerical solution. A theorem on the exponential stability of the numerical solution of the boundary-value difference problem is proved. Easily verifiable algebraic conditions for the exponential stability of the numerical solution are given. The convergence of the numerical solution is proved.

Contemporary Mathematics. Fundamental Directions. 2022;68(1):25-40
pages 25-40 views

Polynomials on Regular Parabolic Manifolds

Atamuratov A.A.


In this work, we consider the regular parabolic manifold X and polynomials on it. We prove some properties of regular parabolic manifolds and describe polynomials on complements of Weierstrass algebroidal sets.

Contemporary Mathematics. Fundamental Directions. 2022;68(1):41-58
pages 41-58 views

Local and 2-Local Derivations of Locally Simple Lie Algebras

Ayupov S.A., Kudaybergenov K.K., Yusupov B.B.


In the present paper, we study local and 2-local derivations of the classical locally simple Lie algebras. Firstly, we prove that every local and 2-local derivations on classical locally simple Lie algebra is a derivation. Further, we show that every local derivation of Borel subalgebras of locally simple Lie algebras is a derivation.

Contemporary Mathematics. Fundamental Directions. 2022;68(1):59-69
pages 59-69 views

Systems of Matrix Differential Equations for Surfaces

Muminov K.K., Gafforov R.A.


Necessary and sufficient conditions for the equivalence of surfaces under the action of a special pseudo-orthogonal group are established.

Contemporary Mathematics. Fundamental Directions. 2022;68(1):70-79
pages 70-79 views

On Analytic Perturbations of Linear Equations in the Case of Incomplete Generalized Jordan Set

Rakhimov D.G., Akhmadzhanova D.


Based on the methods of the theory of bifurcations, the problem of perturbation of linear equations by small analytic terms is considered. In contrast to the work of Trenogin [7], the case of an incomplete generalized Jordan set of a linear Fredholm operator acting from one Banach space to another Banach space is studied. A technique is proposed that uses the regularization of the Fredholm operator by a specially constructed finite-dimensional operator.

Contemporary Mathematics. Fundamental Directions. 2022;68(1):80-94
pages 80-94 views

Local and 2-Local Derivations of Locally Simple Lie Algebras

Ayupov S.A., Kudaybergenov K.M., Yusupov B.T.


In this paper, we study a two-state Hard-Core (HC) model with activity λ>0{\lambda >0} on a Cayley tree of order k 2{k  \ge 2}. It is known that there are λcr{\lambda_{cr}}, λcr0{\lambda_{cr}^0} and λcr'{\lambda_{cr}^'} such that

  • for λλcr{\lambda \le \lambda_{cr}} this model has a unique Gibbs measure λ>0{\lambda >0}, which is translation invariant. The measure λ>0{\lambda >0}  is extreme for λ<λcr0{\lambda < \lambda_{cr}^0} and not extreme for λ>λcr'{\lambda > \lambda_{cr}^'};
  • for λ>λcr{\lambda > \lambda_{cr}} there exist exactly three 2-periodic Gibbs measures, one of which is μ*{\mu^*}, the other two are not translation-invariant and are always extreme.

The extremity of these periodic measures was proved using the maximality and minimality of the corresponding solutions of some equation, which ensures the consistency of these measures. In this paper, we give a brief overview of the known Gibbs measures for the HC-model and an alternative proof of the extremity of 2-periodic measures for k=2,3{k = 2 , 3}. Our proof is based on the tree reconstruction method.

Contemporary Mathematics. Fundamental Directions. 2022;68(1):95-109
pages 95-109 views

Applications of Quadratic Stochastic Operators to Nonlinear Consensus Problems

Saburov M., Saburov K.


Historically, the idea of reaching consensus through repeated averaging was introduced by De Groot for a structured time-invariant and synchronous environment. Since that time, the consensus, which is the most ubiquitous phenomenon of multi-agent systems, becomes popular in the various scientific fields such as biology, physics, control engineering and social science. In this paper, we give an overview of the recent development of applications of quadratic stochastic operators to nonlinear consensus problems. We also present some refinement and improvement of the previous results.

Contemporary Mathematics. Fundamental Directions. 2022;68(1):110-126
pages 110-126 views

Holomorphic Continuation of Functions Along a Fixed Direction (Survey)

Sadullaev A.S.


In this article, we give an overview of the most significant and important results on holomorphic extensions of functions along a fixed direction. We discuss the following geometric questions of multidimensional complex analysis: • holomorphic extension along a bundle of complex straight line, the Forelly theorem; • holomorphic continuation of functions with thin singularities along a fixed direction; • holomorphic continuation of functions along a family of analytic curves.

Contemporary Mathematics. Fundamental Directions. 2022;68(1):127-143
pages 127-143 views

Some Problems of Complex Analysis in Matrix Siegel Domains

Khudaibergenov G.K., Kurbanov B.T.


We give a review of recent results in multivariate complex analysis related to matrix Siegel domains.

Contemporary Mathematics. Fundamental Directions. 2022;68(1):144-156
pages 144-156 views

On the Inequalities for Moments of Branching Random Processes

Khusanbayev Y.M., Kudratov K.E.


We consider branching random processes with immigration starting from a random number of items. In this work, we provide estimates from above for the moments of such processes.

Contemporary Mathematics. Fundamental Directions. 2022;68(1):157-166
pages 157-166 views

Optimal Difference Formulas in the Sobolev Space

Shadimetov K.M., Mirzakabilov R.N.


Optimization of computational methods in functional spaces is one of the main problems of computational mathematics. In this paper, algebraic and functional assertions for the problem of difference formulas are discussed. For optimization of difference formulas, i.e., for construction of optimal difference formulas in functional spaces, an important role is played by the extremal function of the given difference formula. In this work, we explicitly find in Sobolev spaces the extremal function of the difference formula and compute the norm of the error functional of the difference formula. Furthermore, we prove existence and uniqueness of the optimal difference formula.

Contemporary Mathematics. Fundamental Directions. 2022;68(1):167-177
pages 167-177 views

On Boundedness of Fractional Hadamard Integration and Hadamard-Type Integration in LebesgueSpaces with Mixed Norm

Yakhshiboyev M.U.


In this paper, we consider the boundedness of integrals of fractional Hadamard integration and Hadamard-type integration (mixed and directional) in Lebesgue spaces with mixed norm. We prove Sobolev-type theorems of boundedness of one-dimensional and multidimensional Hadamard-type fractional integration in weighted Lebesgue spaces with mixed norm.

Contemporary Mathematics. Fundamental Directions. 2022;68(1):178-189
pages 178-189 views

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