Self-oscillating modes in computer networks control systems quite negatively affect the characteristics of these networks. The problem of finding the areas of self-oscillations is actual and important as the study of parameters of self-oscillations. Due to the significant nonlinearity of control characteristics, the study of the oscillatory modes presents certain difficulties. This paper describes the technique of research of self-oscillating modes on the basis of the control theory. This material is rather methodical than exploratory one. © Springer International Publishing AG 2018.

SymPy computer algebra library is used for automatic generation of ordinary and stochastic systems of differential equations from the schemes of kinetic interaction. Schemes of this type are used not only in chemical kinetics but also in biological, ecological and technical models. This paper describes the automatic generation algorithm with an emphasis on application details.

Nowadays the geometric approach in optics is often used to find out media parameters based on propagation paths of the rays because in this case it is a direct problem. However inverse problem in the framework of geometrized optics is usually not given attention. The aim of this work is to demonstrate the work of the proposed the algorithm in the framework of geometrized approach to optics for solving the problem of finding the propagation path of the electromagnetic radiation depending on environmental parameters. The methods of differential geometry are used for effective metrics construction for isotropic and anisotropic media. For effective metric space ray trajectories are obtained in the form of geodesic curves. The introduced algorithm is applied to well-known objects - Maxwell and Luneburg lenses. The similarity of results obtained by classical and geometric approach is demonstrated.

Spinors are more special objects than tensor. Therefore possess more properties than the more generic objects such as tensors. Thus, the group of Lorentz two-spinors is the covering group of the Lorentz group. Since the Lorentz group is a symmetry group of Maxwell's equations, it is assumed to reasonable to use when writing the Maxwell equations Lorentz two-spinors and not tensors. We describe in detail the representation of the Maxwell's equations in the form of Lorentz two-spinors. This representation of Maxwell's equations can be of considerable theoretical interest. © Published under licence by IOP Publishing Ltd.

The paper considers the technics of construction of optical devices based on the method of geometrization of Maxwell's equations. The method is based on representation of material equations in the form of an effective space-time geometry. Thus we get a problem similar to that of some bimetric theory of gravity. That allows to use a well-developed apparatus of differential geometry. On this basis, we can examine the propagation of the electromagnetic field on the given parameters of the medium. It is also possible to find the parameters of the medium by a given law of propagation of electromagnetic fields.

For the study and verification of our mathematical model of telecommunication systems a discrete simulation model and a continuous analytical model were developed. However, for various reasons, these implementations are not entirely satisfactory. It is necessary to develop a more adequate simulation model, possibly using a different modeling paradigm. In order to modeling of the TCP source it is proposed to use a hybrid (continuous-discrete) approach. For computer implementation of the model the physical modeling language Modelica is used. The hybrid approach allows us to take into account the transitions between different states in the continuous model of the TCP protocol. The considered approach allowed to obtain a simple simulation model of TCP source. This model has great potential for expansion. It is possible to implement different types of TCP.

In solving field problems, for example problems of electrodynamics, we commonly use the Lagrangian and Hamiltonian formalisms. Hamiltonian formalism of field theory has the advantage over the Lagrangian, which inherently contains a gauge condition. While the gauge condition is introduced ad hoc from some external reasons in the Lagrangian formalism. However, the use of the Hamiltonian formalism in the field theory is difficult due to the non-regularity of the field Lagrangian. We must use such variant of the Lagrangian and the Hamiltonian formalism, which would allow us to work with the field models, in particular, to solve the problem of electrodynamics. We suggest to use the modern differential geometry and the algebraic topology, in particular the theory of fiber bundles, as a mathematical apparatus. This apparatus leads to greater clarity in the understanding of mathematical structures, associated with physical and technical models. The usage the fiber bundles theory allows us to deepen and expand both the Lagrangian and the Hamiltonian formalism. We can detect a wide range of these formalisms. Also we can select the most appropriate formalism. Actually just using the fiber bundles formalism we can adequately solve the problems of the field theory, in particular the problems of electrodynamics.

The Hamiltonian formalism is extremely elegant and convenient to mechanics problems. However, its application to the classical field theories is a difficult task. In fact, you can set one to one correspondence between the Lagrangian and Hamiltonian in the case of hyperregular Lagrangian. It is impossible to do the same in field theories. In the case of irregular Lagrangian the Dirac-Bergman Hamiltonian formalism with constraints is usually used, and this leads to a number of certain difficulties. The paper proposes a reformulation of the problem to the case of a field without sources. This allows to use a instantaneous (symplectic) Hamiltonian formalism.

In this work the symbolic algorithm for the calculations of transition probabilities for hydrogen-like atoms in terms of quantum mechanics with non-negative probability distribution function is proposed. The problem was solved in terms of eigenvalues of the finite-approximated Ritz matrices. All the necessary functions, including wave functions, Sturmian functions and their Fourier-transforms, Clebsh-Gordan coefficients etc. were united in one single framework. The program is written using Maple. Results were compared with the data provided by NIST Atomic Spectra Database.

This paper studies program implementation problem of pseudo-random number generators in OpenModelica. We give an overview of generators of pseudo-random uniform distributed numbers. They are used as a basis for construction of generators of normal and Poisson distributions. The last step is the creation of Wiener and Poisson stochastic processes generators. We also describe the algorithm to call external C-functions from programs written in Modelica. This allows us to use random number generators implemented in the C language.