Vol 65, No 1 (2019): Contemporary Problems in Mathematics and Physics

Articles
Interpretation of Geometry on Manifolds as a Geometry in a Space with Projective Metric
Artikbaev A., Saitova S.S.
Abstract
In this paper, we give essential concepts of geometry of three-dimensional spaces in vector formulation in an aﬃne-vector space An.
Contemporary Mathematics. Fundamental Directions. 2019;65(1):1-10
On Formulation of Modiﬁed Problems for the Euler-Darboux Equation with Parameters Equalto 1/2 in Absolute Value
Dolgopolov M.V., Rodionova I.N.
Abstract
We consider the Euler-Darboux equation with parameters equal to 1/2 in absolute value. Since the Cauchy problem in the classical formulation in ill-posed for such values of parameters, we proposeformulations and solutions of modiﬁed Cauchy-type problems with the following values of parameters: a)α = β = 1 , b) α = - 1 , β = - 1 , c) α = β = - 1 . In the case а), the modiﬁed Cauchy problem is solved2 2 2 2by the Riemann method. We use the obtained result to formulate the analog of the problem Δ1 in the ﬁrst quadrant with shifted boundary-value conditions on axes and nonstandard conjunction conditions on thesingularity line of the coeﬃcients of the equation y = x. The ﬁrst condition is gluing normal derivatives of the solution and the second one contains limiting values of combination of the solution and its normal derivatives. The problem is reduced to a uniquely solvable system of integral equations.
Contemporary Mathematics. Fundamental Directions. 2019;65(1):11-20
Covariant Functors and Shapes in the Category of Compacts
Zhuraev T.F., Tursunova Z.O., Zhuvonov K.R.
Abstract
In this paper, we consider covariant functors F : Comp → Comp acting in category of shape-preserving compact sets [2], inﬁnite compact sets, and shape equivalence [9]. Also we study action of compact functors and shape properties of the compact space X consisting of connected components ОX of the compact X as well as shape identity ShX = ShY of inﬁnite compacts X and Y for the space P (X) of probability measures and its subspaces.
Contemporary Mathematics. Fundamental Directions. 2019;65(1):21-32
Application of A-analytic Functions to the Investigation of the Cauchy Problem for a Stationary Poroelasticity System
Imomnazarov K.K., Jabborov N.M.
Abstract
In a reversible hydrodynamic approximation, a closed system of second-order dynamic equations with respect to the displacement vector of an elastic porous body and pore pressure has been obtained. The Cauchy problem for the obtained system of poroelasticity equations in the stationary case is considered. The Carleman formula for the Cauchy problem under consideration has been constructed.
Contemporary Mathematics. Fundamental Directions. 2019;65(1):33-43
A Fuzzy MLP Approach for Identiﬁcation of Nonlinear Systems
Marakhimov A.R., Khudaybergenov K.K.
Abstract
In case of decision making problems, identiﬁcation of non-linear systems is an important issue. Identiﬁcation of non-linear systems using a multilayer perceptron (MLP) trained with back propagation becomes much complex with an increase in number of input data, number of layers, number of nodes, and number of iterations in computation. In this paper, an attempt has been made to use fuzzy MLP and its learning algorithm for identiﬁcation of non-linear system. The fuzzy MLP and its training algorithm which allows to accelerate a process of training, which exceeds in comparing with classical MLP is proposed. Results show a sharp reduction in search for optimal parameters of a neuro fuzzy model as compared to the classical MLP. A training performance comparison has been carried out between MLP and the proposed fuzzy-MLP model. The time and space complexities of the algorithms have been analyzed. It is observed, that number of epochs has sharply reduced and performance increased compared with classical MLP.
Contemporary Mathematics. Fundamental Directions. 2019;65(1):44-53
Geometry of Orbits of Vector Fields and Singular Foliations
Narmanov A.Y.
Abstract
The subject of this paper is the geometry of orbits of a family of smooth vector ﬁelds deﬁned on a smooth manifold and singular foliations generated by the orbits. As is well known, the geometry of orbits of vector ﬁelds is one of the main subjects of investigation in geometry and control theory. Here we propose some author’s results on this problem. Throughout this paper, the smoothness means C∞-smoothness.
Contemporary Mathematics. Fundamental Directions. 2019;65(1):54-71
Reductional Method in Perturbation Theory of Generalized Spectral E. Schmidt Problem
Rakhimov D.G.
Abstract
In this a paper perturbations of multiple eigenvalues of E. Schmidt spectral problems is considered. At the usage of the reductional method suggested in the articles [10, 11] the investigation of the multiple E. Schmidt perturbation eigenvalues is reduced to the investigation of perturbation of simple ones. At the end, as application of the obtained results the problem about the boundary perturbation for the system of two Sturm-Liouville problems with E. Schmidt spectral parameter is considered.
Contemporary Mathematics. Fundamental Directions. 2019;65(1):72-82
Continuation of Analytic and Pluriharmonic Functions in the Given Direction by the ChirkaMethod: a Survey
Abstract
In this paper, we provide a survey of results on analytic and plurisubharmonic continuations of functions that have this set of singularities along a ﬁxed direction. We show the advantages of using the pluripotential theory and the Jacobi-Hartogs series for description of the singular set of such functions.
Contemporary Mathematics. Fundamental Directions. 2019;65(1):83-94
Carleman’s Formula for Solutions of the Generalized Cauchy-Riemann System in Multidimensional Spatial Domain
Sattorov E.N., Ermamatova F.E.
Abstract
In this paper, we consider the restoration problem for solutions of the generalized Cauchy- Riemann system in a multidimensional spatial domain using their values on a piece of the boundary of the domain, i. e., the Cauchy problem. We construct an approximate solution of this problem based on the Carleman matrix method.
Contemporary Mathematics. Fundamental Directions. 2019;65(1):95-108
Spectra of the Energy Operator of Three-Electron Systems in the Impurity Hubbard Model.Second Doublet State
Tashpulatov S.M.
Abstract
We consider the three-electron systems in the impurity Hubbard model and investigated the spectra of the system in the second doublet state in the ν-dimensional lattice Zν .
Contemporary Mathematics. Fundamental Directions. 2019;65(1):109-123
ε-Positional Strategies in the Theory of Diﬀerential Pursuit Games and the Invariance of a Constant Multivalued Mapping in the Heat Conductivity Problem
Tukhtasinov M., Mustapokulov K.Y.
Abstract
In this paper, we consider two problems. In the ﬁrst problem, we prove that if the assumption from the paper [1] and one additional condition on the parameters of the game hold, then the pursuit can be ﬁnished in any neighborhood of the terminal set. To complete the game, an ε-positional pursuit strategy is constructed.In the second problem, we study the invariance of a given multivalued mapping with respect to the system with distributed parameters. The system is described by the heat conductivity equation containing additive control terms on the right-hand side.
Contemporary Mathematics. Fundamental Directions. 2019;65(1):124-136
The Cyclical Compactness in Banach C∞(Q)-Modules
Chilin V.I., Karimov J.A.
Abstract
In this paper, we study the class of laterally complete commutative unital regular algebras A over arbitrary ﬁelds. We introduce a notion of passport Γ(X) for a faithful regular laterally complete A- modules X, which consist of uniquely deﬁned partition of unity in the Boolean algebra of all idempotents in A and of the set of pairwise diﬀerent cardinal numbers. We prove that A-modules X and Y are isomorphic if and only if Γ(X)= Γ(Y ). Further we study Banach A-modules in the case A = C∞(Q) or A = C∞(Q)+ i · C∞(Q). We establish the equivalence of all norms in a ﬁnite-dimensional (respectively, σ-ﬁnite-dimensional) A-module and prove an A-version of Riesz Theorem, which gives the criterion of a ﬁnite-dimensionality (respectively, σ-ﬁnite-dimensionality) of a Banach A-module.
Contemporary Mathematics. Fundamental Directions. 2019;65(1):137-155