# No 3 (2013)

**Year:**2013**Articles:**19**URL:**https://journals.rudn.ru/miph/issue/view/509

## Articles

### On Some Generalization of Paracompactness

#### Abstract

The generalization of paracompact spaces via so-sets, sets are unions of open and nowhere dense sets, was studied. The aim of this paper is to establish the relationship between so-paracompact spaces and other generalizations of paracompact spaces and clarify the conditions under which the so-paracompact space is compact. The problem is solved by methods of general topology. It is proved that the sequentially compact so-paracompact space is compact. It is proved that the so-paracompactness saved when multiplied by the compact. Previously, other authors introduced the concept of S-paracompact space, based on the semi-open sets. Class of so-paracompact spaces wider than the class S-paracompact spaces. This paper shows that there are so-paracompact spaces which are not S-paracompact.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):5-10

5-10

### Synthesis of 3D-dynamical Systems with Critical Points of Given Topological Structures

#### Abstract

The problem of synthesis of normal autonomous systems of ordinary differential equations which three-dimensional phase spaces have isolated equilibrium points with desired topolog- ical structure properties. To solve this problem a method based on the using special vector fields of comparison directions is proposed. While choosing these vector fields it should be taken into account that the local structure of an isolated equilibrium point is completely characterized by: a) a set of singular phase trajectories and surfaces that break up the neigh- borhood of the equilibrium point into elementary areas, and b) behavior of non-singular phase trajectories in these areas. Thus obtained vector fields allow, under certain conditions, to present the local topological structure properties of equilibrium point in an analytical form as algebraic expressions with respect to phase coordinates. These expressions are used to set up the equations equal in number to the number of dimensions of the phase space and which are the algebraic equations with respect to the right-hand sides of sought differential equations. The main purpose of the paper is to describe the general approach to the posed problem, so the solution is considered only in one particular case where all the elementary areas of the sought dynamical system equilibrium point are elementary areas of one of the possible types. Theoretical results of the article are illustrated by a concrete example. Presented in this paper is a partial generalization of the previously published results for solving inverse problems of the theory of dynamical systems on the plane.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):11-20

11-20

### Uniqueness of Solutions for One Class of Linear Equations of the First Kind with Two Variables

#### Abstract

This article is devoted to the study of the uniqueness of solutions of linear integral equations of the first kind with two independent variables in which the operator generated by the kernel, is not compact operator. The relevance of the problem is due to the needs in development of new approaches for the regularization and uniqueness of the solution of linear integral equations of the first kind with two independent variables. For approximate solutions of such tasks, stable to small variations of the initial data, we use the solutions derived by the method of regularization and belonging to the class of incorrectly formulated tasks. One of the classes of such ill-posed problems are integral equations of the first kind with two independent variables. The aim of the work is to prove the theorems of uniqueness for solving linear integral equations of the first kind with two independent variables. In the paper a theorem of the uniqueness of the solution of integral equations of the first kind with two independent variables is proved. To obtain the results formulated in the article the methods of functional analysis and method of nonnegative quadratic forms are used. The obtained results are new. The reliability of the result is set by prooves and illustrated by examples. The work has a theoretical character. The obtained theoretical results can be used in various fields of science and technology.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):21-29

21-29

### Uniqueness and Stability of Solutions for Certain Linear Equations of the First Kind with Two Variables

#### Abstract

The article is devoted to the study of uniqueness and stability of solutions of linear integral equations of the first kind with two independent variables. The relevance of the problem is due to the needs in development of new approaches for the regularization and uniqueness of the solution of linear integral equations of the first kind with two independent variables. Integral and operator equations of the first kind with two independent variables arise in theoretical and applied problems.Works of A.N. Tikhonov, M.M. Lavrentyev and B.K. Ivanov, in which a new concept of correctness of setting such targets is given, different from the classical, show tools for research of ill-posed problems, which stimulated the interest to the integral equations that are of great practical importance. At the present time the theory and applications of ill-posed problems have been rapidly developing. One of the classes of such ill-posed problems are integral equations of the first kind with two independent variables. As of approximate solutions of such problems, stable to small variations of the initial data, we use the solutions derived by the method of regularization. In this article we prove the theorem of uniqueness and obtain estimates of stability for such equations in families of sets of correctnesses. For the tasks solution the methods of functional analysis and method of nonnegative quadratic forms are used. The results of the work are new.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):30-35

30-35

### Necessary and Sufficient Conditions for the Potentiality of Nonlinear Differential-Difference Operator in Partial Derivative

#### Abstract

The purpose of the present paper is to investigate the potentiality of the differential difference operator with deviant arguments and to construct the functional, if the given operatoris a potential on a given set relatively to the some special bilinear form, i.e. the problem ofexistence of solutions of inverse problems of the calculus of variations for partial differentialdifference operators is investigated. Let ..,.. be normed linear spaces over the field of real numbers R. Take any operator .. : ..(..) > ..(..), where ..(..) . .., ..(..) . .. . A limit if it exists, is called the G.ateaux differential of .. at the point ... The operator ....(.., ·): .. > .. is called the G.ateaux derivative of .. at .. and will be denoted by ..... . Its domain of definition ..(..... ) consists of elements . . .. such that (.. + ...) . ..(..) for all .. sufficiently small. We obtain necessary and sufficient conditions for the adjusted partial differential difference op..,.. erator of ..class. The nonlinear differential operator of the second order and the nonlinear ..,.. differential operator of the second order with deviant arguments is consider as an example. Using the obtained conditions of potentiality corresponding functionals are constructed.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):36-41

36-41

### Nonlinear Waves Dynamics Modeling in Coaxial Geometrically And Physically Nonlinear Shell Containing Viscous Incompressible Fluid in between

#### Abstract

The present investigation is devoted to the analysis of non-linear deformation waves propagation in physically non-linear coaxial elastic cylinder covers, containing viscous incompressible liquid between them. Wave process in elastic cylinder cover without interactionwith liquid was investigated earlier on the basis of soliton theory. The presence of liquid demanded working out a new mathematical model and computer modeling of the processes,taking place in the system.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):42-51

42-51

### New Method for Constructing the Oscillator Functions of a Quantum System of Identical Particles in Symmetrized Coordinates

#### Abstract

The quantum model of a cluster, consisting of A identical particles, coupled by the internal pair interactions and aﬀected by the external ﬁeld of a target, is formulated in the new symmetrized coordinates. A new method and symbolic algorithm for generating (A − 1)-dimensional oscillator eigenfunctions, symmetric or antisymmetric with respect to permutations of A identical particles, is elaborated and implemented using the MAPLE computer algebra system. Examples of generating the symmetrized coordinate representation for composite systems of several identical particles in one-dimensional Euclidean space are given and their symmetry properties are analyzed. The systems composed from three to six particles in one dimensional Euclidean space were analyzed a correspondence between the representations of the symmetry groups D3 and Td for A = 3 and A = 4 and symmetric or antisymmetric oscillator functions was found. It is shown that the transformations of (A− 1)-dimensional oscillator functions from the symmetrized coordinates to the Jacobi coordinates, reducible to permutations of coordinates and (A − 1)-dimensional ﬁnite rotation, are implemented by means of the (A − 1)-dimensional oscillator Wigner functions. The examples of construction of the symmetric or antisymmetric oscillator functions in closed analytical form by means of mathematical induction and the algorithm are given. The approach is aimed at solving the problem of tunnelling the clusters, consisting of several identical particles, through repulsive potential barriers of a target.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):52-67

52-67

### Determining the Safety of Complex Technical Systems on the Basis of Measurements and Observations of its Parameters. Generalized Measure Security

#### Abstract

This article is based on a methodical approach to safety analysis report system based on measurements and observations of its parameters as a result of the impact of external and internal forces. During operation of a complex technical system physical, mechanical and technical properties of its structures (materials) and its components (elements, assemblies, modules) are changing. Change the physic-mechanical or technical properties can be both reversible and irreversible, and these changes give rise to incremental (wear) failure or in emergency cases to the instantaneous hald of components of the complex technical system. These failures characterize the reliability of the system, and the forces that lead to such consequences are internal. The impact of external forces on the system, especially targeted or erroneous from the operator (Manager of operational processes) in the amount of internal inﬂuence on safety of the work of analyzing complex technical systems. This paper presents the status of the dynamic system of the developed model of the vector and vector phase system parameters. Shows the choice of measured parameters to assess the State of the control system. Given the wording of the security of the system and the formalization of measuring security.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):68-71

68-71

### Methods of Observation of Dynamic Systems Functioning in Order to Ensure System Safety

#### Abstract

he article sets forth the methodological approach to the problem of formalization of the process of observation and measurement of parameters of dynamic systems. The necessity of selection of the measured parameters of the system for the assessment of the control is shown. The wording of the concept of security system is given and the formalization of the quantitative assessment of the safety of the operation and status changes of complex technical systems (STS) in time are proposed. Important process of ensuring the safety of the technical system, particularly for the special purposes, is the measurement and observation of the parameters. The process of the state and dynamics of the studied object (system) is described by the model in the form of dynamic equations in vector form. Dynamic management system and its measuring complex represent a linear stationary system, which is described by the model in a matrix form. When assessing the safety of technical systems diﬀerent algorithms of measurement of parameters may be used. In any case allowed operations are implemented by the operator, which for linear stationary dynamical system is also linear. The method of determining the optimal set of parameters observations characterizing the state of the system which is proposed in the article signiﬁcantly aﬀects the safety of the study of the dynamical system in case of internal disturbances. By varying these parameters, one can determine the time and the monitoring interval, when the current state of the dynamic system is determined with precision.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):72-75

72-75

### On the Models with Partial Distribution of Accuracy

#### Abstract

The majority of the models for describing any oscillatory processes have partial distributionof accuracy, i.e. the number of normal mode is higher, the model describes its evolution worse.Therefore the question about convergence of the normal waves series, taking the central place at classical approach, inevitably take out of applicability of model. At such approach this isa lack of models, one of many difficulty in the proof of series convergence and existence ofthe classical solution. In this article we discuss new approach to the description of such models which is simplerclassical: here the proof of convergence of series is replaced with research of uncertainty ofnormal waves amplitudes. The statement was illustrated with a concrete example of theelementary model with partial distribution of accuracy, i.e. problem about string osculations. In such problems there is some uncertainty in initial conditions. So usually the profile of initialvelocity, used for the description of blow by a hammer, we consider as step function or “hat”,but we can consider the whole class of suitable profiles, therefore the whole family of initial-boundary value problems. This uncertainty in initial values gives the chance to estimate an error for each mode separately. As one would expect, the error grows to infinity as numberof a mode tend to infinity. All solutions of considered family of problems are expanded innormal waves series and younger modes have close amplitudes. It allows to keep all classical statements about younger modes and to avoid a investigation of convergence of normal wavesseries, which is technically difficult and take out of applicability of model.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):76-80

76-80

### Solving Differential Equations of Motion for Constrained Mechanical Systems

#### Abstract

This paper presents an investigation of modeling and solving system of differential equations in the study of mechanical systems with holonomic constraints. A method is developed for constracting equation of motion for mechanical system with constraints. A technique is developed how to approximate the solution of the problem that is obtained from modeling of kinematic constraint equation which is stable. A perturbation analysis shows that velocity stabilization is the most efficient projection with regard to improvement of the numerical integration. How frequently the numerical solution of the ordinary differential equation should be stabilized is discussed. A procedure is indicated to get approximate solution when the systems of differential equations can’t be solved analytically. A new approach is applied for constructing and stabilyzing Runge-Kutta numerical methods. The Runge-Kutta numerical methods are reformulated in a new approach. Not only the technique of formulation but also the test developed for its stability is new.Finally an example is presented not only to demonstrate how the stability of the solution depends on the variation of the factor but also how to find an approximate solution of the problem using numerical integration.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):81-91

81-91

### Constructing Dynamic Equations of Constrained Mechanical Systems

#### Abstract

In this paper constructing equation of mechanical systems based on their kinetic energy, potential energy and dissipative force is discussed. Both the holonomic and non-holonomic constraints are considered. Equations of constraint forces resulting from ideal and non-ideal nature of the constraints are developed.It is shown that, the constraint force is a sum of two forces resulting from the ideal and non-ideal nature of the constraints. An explicit equation of the acceleration of the system is developed basing on the constraint forces from the nature of the constraints. For investigating the deviation of the system from the trajectory of the constraint equations, excess variables are included in the equations of the constraints. The stability of the system is based on determining the sign of constants emerging from developing the Lagrange’s equation of motion for the constraints. The determination of the sign of the constants is made based on Routh-Hurwitz Criterion for Stability. An example is used to demonstrate each of the equations developed in the paper and constructing state-space equation of the system.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):92-104

92-104

### Process Self-Adjusting Control of Non-Impact Bringing of the Condition of Mechanics Systems to Given Set

#### Abstract

The procedure for building auto-adjustment control vector to bring the state of the mechanical systems without impact in a given manifold for a finite period of time in the face ofuncertainty is described. Previously obtained the solution of the problem of bringing the phase state of the system ina given neighborhood of the manifold formed by the non-stationary holonomic program constraints. In this paper we extend this approach to the task of bringing non-impact phase of thesystem for a finite period of time in the manifold formed by the holonomic and nonholonomicprogram constraints. In this case, even the mechanical system can have besides stationary and non-stationary communications. Received a lot of control vectors that provide a solution to this problem of self-adapting control of feedback on the quasi-accelerations at discretepoints in time. Then this set of control vectors allocated dimension smaller than the numberof degrees of freedom of the system, including the minimum dimension vector. In cases where the vectors control more than the minimum, stand vectors with minimal Euclidean norm. The obtained results allow us to meet the challenges of an applied nature, such as processcontrol unstressed docking surface, swimming, aircraft and spacecraft as they move freely inspace, but also a process of unstressed landers landing on the moving platform, the nature of the movement are not fully known. To illustrate the effectiveness of the proposed method for solving such problems an exampleof a process control of non-impact bringing of position of the body in a predetermined orientation with haunting movement of the center of mass of the body on the basis of proportionalnavigation is proposed.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):105-112

105-112

### The Architecture of a Parallel-Pipeline Data Processing Complex for Heterogeneous Computing Environment

#### Abstract

A heterogeneous computing environment uses various types of computational units. An example of such environment is a GPU-cluster that contains general-purpose processors (central processing unit, CPU) and graphics processing units for special purposes (GPU). Today’s GPU is already far superior CPU performance and, despite the limitations imposed by developed under the concept of GPGPU-computing (general-purpose graphics processing units), parallel algorithms find their application in solving problems that require intensive computation. Organization of the so-called “GPU-cluster” may be an effective solution that have an acceptable “price/performance” ratio and, that most importantly, an ability to easily scale a computer system performance. There are several types of high-performance algorithms for concurrency that relevant for GPU-cluster too (including a task and data parallelism). In this paper produced an analysis of their applicability as a basis set of parallel-pipeline computations data processing. Investigated a variants of high-performance algorithms building, proposed previously developedsoftware adaptation scheme for a new conditions. Library of GPU-computing algorithms in the first place should have a thread-safe implementation (the code is thread-safe if it functions work correctly with multiple running parallel computing threads). An important and needs attention is the question of competing threads resource sharing. In order to assess theimpact of this factor on the effectiveness of applied problem, we performed an experiment,identifying GPU-cluster competing threads dealing bottlenecks. Have been estimated the effective threshold for increasing the number of processing threads that is expected to a further calculations accelerating.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):113-117

113-117

### Accounting for the Surface Energy in Spin Heisenberg’s Hamiltonians

#### Abstract

Using the quantum mechanical Bogolubov’s Hamiltonian hierarchy for localized electronicexcitations of the crystal system, we have obtained the Hamiltonian of the spin excitations ofthe Heisenberg model, taking into account the surface energy. This Hamiltonian is obtained in the zero approximation in the spin-spin interaction for a ferromagnetic crystal, in case ofrigid fixing of ions in the lattice sites. The corresponding expressions for the displacement ofions in the crystal lattice are also shown.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):118-128

118-128

### Topological Soliton Configurations in 8-Spinor Nonlinear Model

#### Abstract

We study the structure of the charged topological solitons in the lepton sector of the nonlinear 8-spinor model, at small distances the closed-string approximation being used. The mass, the spin and the magnetic moment of the soliton configuration with the unit leptonic number are estimated. The model is based on the well-known 8-spinor identity suggested by the Italian geometer Brioschi. Due to the identity the Dirac current appears to be time-like 4-vector that permits one to introduce the special form of the Higgs potential depending on the current squared. Within the framework of this model the natural classification of leptons and baryons can be realized via the Higgs mechanism. Concentrating on the lepton sector we study the simplest soliton configuration endowed with the unit Hopf index playing the role of the lepton number. Investigating the behavior of solutions at large and small distances we obtain the numerical estimate of physical characteristics of the topological soliton. The special symmetry group is used in our calculation, the combined rotations in ordinary and isotopic spaces being considered. The corresponding equivariant spinor fields involve phase functions linear with respect to azimuthal and toroidal angles. This property permits one to find explicit value of the topological invariant for the axially-symmetric configuration and to investigate the dependence of the physical characteristics on topology.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):129-136

129-136

### On the Movement of a Fluid with a Negative Pressure under the Action of its Own Gravitational Field

#### Abstract

We have considered in the nonrelativistic approach the movement of the three types of the ﬂuid with negative pressure (cosmic vacuum, quintessence, Chaplygin gas) under the action of the own gravitational ﬁeld. Introduction in cosmological models of a substance with negative pressure is one of alternative approaches to the explanation of existence of the accelerated expansion of the Universe. The space vacuum possesses not only a certain density of energy, but also a pressure. If density of space vacuum is positive, its pressure is negative. Connection between pressure and density, i.e. the equation of state, has an appearance for vacuum P + = 0. This equation of state is compatible with the deﬁnition of vacuum as energy form with everywhere and always constant density, irrespective of frame of reference. Study of properties of such ﬂuids represents certain scientiﬁc interest from the point of view of existence of usual hydrodynamic properties, in particular, existence of wave movements under the action of the own gravitational ﬁeld. The movement of the ﬂuids with constant negative pressure is considered in spherical coordinates when only radial component of velocity u(r,t) is considered. We have established that for ﬂuid type space vacuum with constant negative pressure the movement is possible only if the source function doesn’t depend on coordinates. In this case the velocity of the ﬂuid is linear function of the distance from the beginning of coordinates that reminds Hubble’s law in cosmology. For ideal ﬂuid with the equation of state of the type of quintessential we have established that movement of the ﬂuid under the action of the own gravitational ﬁeld for one-dimensional movement is possible in the case if its density exceeds some critical value, the movement of the ﬂuid takes place in some bounded region 0 ≤ x ≤ xmax and its velocity changes from some critical value ucr to u = 0. We also studied the movement of the medium with the equation of state of the Chaplygin gas under its own gravitational ﬁeld in one-dimensional case, and show that there are three diﬀerent ﬂow regimes.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):137-143

137-143

### Wave Processes Simulation in a Chain of Coupled Optical Microresonators Using a Virtual Radiofrequency Model

#### Abstract

Some results of wave processes simulation in chains of coupled optical microresonators bymeans of radiofrequency modeling are presented. Multisim 11 program was used to realize theoptical processes simulation using a radiofrequency model. Several virtual models consisting of chains of electrical circuits with inductive coupling was created and studied. The validityof radiofrequency simulation results in application to the optical frequency range is discussed.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):144-152

144-152

### Our authors

**Discrete and Continuous Models and Applied Computational Science**. 2013;(3):153-154

153-154