Vol 25, No 2 (2017)

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.

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.

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.

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.

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.