## Vol 25, No 2 (2017)

**Year:**2017**Articles:**12**URL:**http://journals.rudn.ru/miph/issue/view/948**DOI:**https://doi.org/10.22363/2312-9735-2017-25-2

Mathematics

On Application of M.N. Lagutinski Method to Integration of Differential Equations in Symbolic Form. Part 1

###### Abstract

The method of M.N. Lagutinski (1871-1915) allows to find rational integrals and Darboux polynomials for given differential ring and thus can be used for integration of ordinary differential equations in symbolic form. A realization of Lagutinski method was made under free opensource mathematics software system Sage and will be presented in this article with application for symbolic integration of 1st order differential equations. In the first part of the article basic concepts of the Lagutinski method is briefly stated for polynomials rings. Then this method is applied to search of algebraic integrated curves for given ordinary differential equations of the form d + d with , Q[, ]. It is shown how the Lagutinski method allows to look for curves of the given order or to prove that there are not such curves. In particular questions about the optimization of computations and integration in micronomials are considered. The theory and its realization in Sage are tested on numerous examples from standard for Russia text-book by A.F. Filippov. Some recommendations for optimization of the Lagutinski method usage are made in the conclusion of the article.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(2):103-112

On Application of M.N. Lagutinski Method to Integration of Differential Equations in Symbolic Form. Part 2

###### Abstract

The method of M.N. Lagutinski (1871-1915) allows to find rational integrals and Darboux polynomials for given differential ring and thus can be used for integration of ordinary differential equations in symbolic form. A realization of Lagutinski method was made under free opensource mathematics software system Sage and will be presented in this article with application for symbolic integration of 1st order differential equations. The second part is devoted to integration of given differential equation d + d with , Q[, ] in quadratures. According to the theorem of M. Singer the problem of integration in quadratures is equivalent to the finding of integrating factor of the form = exp d + d where , Q[, ]. The function can be found as a root of Darboux polynomial for some auxiliary differentiation of the ring Q[, , ]. By Lagutinski method we can find all Darboux polynomials for given differentiation of polynomial ring if degrees of required polynomials are less than given boundary and thus we can find integration factor of the form stated above. The theory and its realization in Sage are tested on numerous examples from standard for Russia text-book by A. F. Filippov.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(2):113-122

Mathematical Teletraffic Theory and Telecommunication Networks

Analysis of Model for Multichannel Peer-to-Peer TV Network with View-Upload Decoupling Scheme

###### Abstract

In recent years, video streaming systems such as P2PTV successfully use P2P-based networks to allow users watching numerous streaming TV channels. Various designs of an overlaid network were proposed to improve the quality of services of P2PTV. In this paper, we explore View-Upload Decoupling-scheme (VUD), which strictly decouples data to what peer uploads and what it personally views. It’s based on the split of downloaded user data streams into two types: the stream of the chosen TV channel, and the stream (one or more) of the other TV channel, exclusively to deliver it to other users. Such peers form the distribution swarm which is assigned the streams of the channels with low popularity, so that would guarantee the stability of multichannel systems. The mathematical model of VUD scheme is proposed, which considers two classes of users - homogeneous type (all users have the same upload rate) and heterogeneous systems (there are two types of users - with low and high upload rate). We develop the method for calculation the probability of universal streaming - one of the key performance indicators in streaming TV - when all users receive the requested video data with guaranteed quality, defined in the service level agreement (SLA). We propose a method for calculating the probability of universal streaming for a single channel. Statistically significant results for a small network in comparison to VUD and ISO schemes are presented.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(2):123-132

The Construction and Analysis of Models of the Input Switch in a Network with Optical Switching

###### Abstract

Currently, there are two generally recognized principles of switching of information signals in high-speed networks: networks with wave routing, and networks with the principle of optical packet switching. In networks with wave routing it is not required to produce opto-electrical and electro-optical conversions and to create a buffer, but with this switching principle the working range of wavelengths is not efficiently used. In networks with optical packet switching the traffic is transmitted in packets, which consist of a header and an information part of a consistent size. In this case, using of the frequency range is the most complete, but there is a need of optical-electronic conversions. In an effort to combine the advantages of two optical switching technologies, a new combined switching principle was proposed, called optical switching bursts. In this technology there are not buffering and electronic processing in intermediate nodes, there is a reservation of the channel for a limited time. For the effective implementation of such a network connection, we must calculate its probability characteristics. To assess probabilistic characteristics of the network the methods of theory of mass service are widely used. The input switch is one of the key devices on the network. The article describes the input switch of the network with the optical switching of bursts, calculates the probable characteristics of the network using analytical and simulation models. Examples of the calculation of the probability of blocking of packets flowing in the input switch are presented.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(2):133-140

Mathematical Modeling

Surface Electromagnetic Waves at the Interface of Two Anisotropic Media

###### Abstract

The article discusses the existence of surface electromagnetic wave at the interface of a structure formed by two identical anisotropic media, each of which is rotated in opposite directions at an angle relative to the desired direction of propagation of electromagnetic wave. Earlier, in the pioneering papers on the subject Diakonov M.I. and Averkiev N.S. (1988, 1990) considered only uniaxial anisotropic media. This article presents calculations for the general case of biaxial media. In a particular case, the obtained results describe a case of uniaxial media. The paper does not use any approximation, except, perhaps, the concept of plane waves. Exact analytical expressions were obtained, which relate the values of the phase velocity of the surface wave with an angle of rotation of the axes of symmetry of anisotropic media relative to the direction of the wave vector of the surface wave. In addition, the transverse distributions of the fields of such a wave were found, and these distributions uniquely characterizes this wave as surface wave.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(2):141-148

Elements of Gestalteconomy. Part 2

###### Abstract

In this article proceeds with consideration of a problem of a global trend of development of world economy and its management. The productive role of a constructive labour, energy and knowledge and their influence on world economy is underlined. Connection and difference between the thermodynamic system which condition is characterized by entropy, and the economic system characterized, including, by the negentropy is established. The interrelation between the negentropy and the information and knowledge is considered. To maintain and continue its development, mankind must go from unlimited use of limited natural resources to the use of an unlimited resource - our knowledge and skills. It is stated that the development of the exchange of information and knowledge will significantly reduce the cost of limited natural resources. And adding of the knowledge used into the integral cost of production ensures long-term conservation of all natural resources, instead of maximizing short-term income of the minority. When implementing such a paradigm the humanity could go from the method of operation, based on the strategies of conflict in conditions of shortage, to the management and operation under conditions of excess.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(2):149-160

Theoretical Mechanics

General Integral for a Class of NonSteady Atmospheric Flights and Applications to Trajectory Analysis

###### Abstract

A complete analytical integration of the aircraft kinematic and dynamic equations of motion is presented. Different applications of defined integrals to trajectory analysis are considered. The dynamic equations are obtained under the assumptions, that acceleration due to aerodynamic lift, the difference between the accelerations due to propulsive thrust and aerodynamic drag are not changed, the aircraft body rate about the velocity axis is zero and the sideslip angle is zero. The general integral of these equations consists of six independent first integrals of motion and describes a class of non-steady flight trajectories in a maneuver plane. It will be shown that the dynamic equations can be derived and completely integrated in a closed-form for more general assumptions. The problem of computing thrust for a given trajectory has been considered. The trajectory is defined by constraint equation. Constraints stabilization equations, which have asymptotically stable trivial solution, are constructed. Explicitness can make the integrals applicable to modeling the trajectories of spacecraft, re-entry vehicles and missiles, and to the design of on-board targeting and guidance. An illustrative example is presented.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(2):161-169

Control over the Process of Unstressed Capture of Unpredictably Moving Target by the Robotic Arm

###### Abstract

An algorithm is constructed for controlling a non-impact capture of unpredictably moving target by the robotic arm. Capture is performed in a finite time. The solution is obtained without the use of the information on the non-control forces, including the disturbing forces and forces of inertia. The object is achieved in four stages. First, the principal vector of the forces is obtained, which provides movement of the center of mass of the robot body in the mode of persecution on the basis of proportional navigation in the pursuit of the object. Second, the principal moment of the forces about the center of mass of the body is obtained, which provides bringing of the one of the principal central axes of inertia of the moving coordinate system associated with the robot’s body, in a position coinciding with the line of sight. Third, an additional driving force is determined, which provides the unstressed bringing of the attachment point of the first link of the arm with a robot body at a distance of “manipulator arm” from the target on the line of sight to provide capture. Fourth, the expression of forces and moments is constructed for the management of the translational and rotational motion of the links of the manipulator relative to each other, allowing bumpless capture of the pursued object. The self-adjusting method is proposed to automatically select the optimal values of the control. It is carried out by the “principle of feedback on the quasi-acceleration” at discrete points in time. This principle was first proposed by I.A. Mukhametzyanov in an article published in the Bulletin of Peoples’ Friendship University series “Mathematics. Information Sciences. Physics”, No 3, 2013.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(2):170-181

Physics

Coherent Perfect Absorption Mediated Enhancement and Optical Bistability in Phase Conjugation

###### Abstract

Coherent perfect absorption has been one of the important research directions in optics in recent years because of its ability to absorb all the incident light. It has been extended to nonlinear regime to show multistability and gap solitons in nonlinear periodic structures. We study yet another nonlinear effect, namely, phase conjugation in a Kerr nonlinear composite slab when the counter propagating pump waves are completely absorbed by means of coherent perfect absorption. The theory is developed under the undepleted pump approximation, when the pump waves can be decoupled from the signal and the phase conjugated waves. Dynamical phase matching is also incorporated. The coupling constant and the phase conjugated reflectivity are shown to undergo a substantial increase. They also exhibit multivalued response. Both downward and upward switching are shown to be possible. The effect can be used for efficient switching of the phase conjugated reflectivity in photonic circuits and can find several application in photonic logic gates.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(2):182-191

Rotating Cosmological Bianchi Type VIII Models with Anisotropic Fluid, Scalar Field and Radiation

###### Abstract

Within the general theory of relativity the Bianchi type VIII cosmological models with rotation and expansion have been built. The matter includes 3 components: perfect anisotropic fluid, imitating the rotating dark energy, clear radiation and scalar field. Different types of scalar field potential have been observed: a square one, Higgs’s potential and a power of four pontential. Evolution of the potential function is given in the way similar to inflation, at the same time the equations of state are not postulated initially. When solving the Einstein’s equation we obtain evolution of density and pressure of the liquid, also it has been found that when the potential is square, the fluid’s equation of state becomes vacuum-like and the fluid becomes asymptotically isotropic. The analysis of absence of closed time-like curves has been done, so the model has been proved to be casual. The order of present angular velocity value, calculated within the cosmological model, has been found to be quite satisfactory. The found solutions may be used for effects taking place nowadays and also during the inflationary stage.

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(2):192-198

Articles

Information about the authors

###### Abstract

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(2):199

Guidelines for Authors

###### Abstract

**Discrete and Continuous Models and Applied Computational Science**. 2017;25(2):200-201