No 3 (2014)

In this paper we consider topological duplications of so-paracompact spaces and extensions of sc-mappings on duplicate. The notions of so-paracompact space and sc-mapping (which generalize the notions of paracompact space and continuous mapping respectively) were introduced by the authors recently and based on so-sets (so-set is the union of the open set and the nowhere dense set). The aim of the work is the investigation of the properties of duplicates of mentioned spaces and mappings. It is proved that Alexandroff duplicate of soparacompact space is so-paracompact. It is proved also that mentioned duplicate is almost paracompact space. It was established that natural extension of sc-mapping on duplicate of space is so-mapping which possess of supplementary properties. It is proved that the natural extension of quasi-continuous mapping is quasi-continuous.

The classic results of Gergen J. J., Dressel F. G. are generalized to the class of the functions with weak derivatives. We suppose that these derivatives could be estimated by the proper functions multiplied by the weighted functions which have singularities at isolated boundary points. The crucial point of the study is the iteration process used for the evaluations of the functions represented by the potential operators. As a result of such iterations we succeed in lowering the degree of kernel singularities of the potential operators. The above mentioned method is based on representation formula of I.N. Vekua for the functions whose weak derivatives are summable over domains. The analytic functions participating in these representations could be considered as generalized constants. We study the classes of those functions whose generalized constants have finite numbers of poles and zeros. We prove theorems on behavior of the above mentioned functions in neighborhood of their zeros. Besides we study these functions in the neighborhood of the points where they haven’t finite limits. The main result of the paper is the theorem on the number of zeros and poles of the functions under consideration. This result is the generalization of theorem from the paper of Gergen J. J., Dressel F. G.

It is assumed that the introduction of stochastic in mathematical model makes it more adequate. But there is virtually no methods of coordinated (depended on structure of the system) stochastic introduction into deterministic models. Authors have improved the method of stochastic models construction for the class of one-step processes and illustrated by models of population dynamics. Population dynamics was chosen for study because its deterministic models were sufficiently well explored that allows to compare the results with already known ones. To optimize the models creation as much as possible some routine operations should be automated. In this case, the process of drawing up the model equations can be algorithmized and implemented in the computer algebra system. Furthermore, on the basis of these results a set of programs for numerical experiment can be obtained. The computer algebra system Axiom is used for analytical calculations implementation. To perform the numerical experiment FORTRAN and Julia languages are used. The Runge- Kutta method for stochastic differential equations is used as numerical method. The program complex for creating stochastic one-step processes models is constructed. Its application is illustrated by the predator-prey population dynamic system. Computer algebra systems are very convenient for the purposes of rapid prototyping in mathematical models design and analysis.

The paper deals with the problem of the systems ‎‹M2/GI/1›‎ and ‎‹GI2/M/1› reliability characteristics sensitivity to the shape of their elements life and repair times distributions under restrictions on the availability of recovery. Partial differential equation for the time dependent and usual differential equations for the stationary micro-state probabilities of these systems are proposed. Explicit expressions for the micro-and macro-state stationary probabilities of these systems are given and they show their strong dependability on the shape of their elements life and repair times distributions. This dependence represents in terms of moment generation functions non-exponential distribution in the point of the exponential distribution parameters. Special software tool based on the MATLAB computer system has been developed for the numerical analysis of the system failure probability sensitivity to the shape of its elements life and recovery distributions and its comparison with the simplest Markov system. The numerical analysis shows that this dependence becomes negligible and vanishes for “fast” recovery (with recovery rate increasing). In particular, it has been shown that the failure probabilities of the systems ‎‹M2/GI/1›‎ and ‹GI2/M/1› with Gamma and Weibull-Gnedenko distributions instead of the general ones quickly converge to zero with increasing recovery rate and coincide with the simplest Markov system ‹M2/M/1› for special value of the particular value of the parameter c =1.0.

This article is devoted to the study of the maximization of the accumulated income for the model of the gas deposit on a finite horizon, a detailed analysis of the obtained results and their comparison with the results of the previously posted this same problem on an infinite horizon. So far the same tasks, based on a model with interacting wells, were solved at a constant price for gas. In reality, however, the price for the goods quite often has a nonlinear dependence and depends on the volume of purchases. Therefore, the statement of the problem is modified by the inclusion in its description of the procurement function. A major tool in the search for the solution to the maximization of income on a finite horizon is the Pontryagin’s maximum principle under the condition of its existence. There are two areas, separated from each other parametric dependence. On each of the selected areas with the use of the method of “phase diagram” the optimal solution is being found. The optimal solution of the problem of maximization on a finite horizon is explicitly described. Joint analysis of the obtained solutions in the problems of maximization on finite and infinite horizon revealed that under certain conditions a part of wells is used inefficiently. Several ways to solve this problem are recommended.

The procedure is described of application of the “principle of feedback on the quasi-accelerations” in the construction of auto-adjustment control vector for bringing the condition of mechanical and generalized systems without impact to a given manifold of phase state of systems in finite time, in full or partial uncertainty mass-inertial parameters of the system and disturbances acting on it. This process is called unstressed stabilization of system in a finite time. Varieties of the condition of systems are given by set of holonomic and nonholonomic soft links. A set of control vectors that provide a solution to this problem is obtained. Then from this set of vectors control vectors of minimal dimension and minimum Euclidean norm are allocated. The examples are shown of applying these results to solve problems of practical importance, such as process of control of unstressed docking of surface and underwater and spacecrafts objects, unstressed landing of landers to the moving platforms, and capture of fast moving objects, including “space debris”. In contrast to previous works of the author on the problems of control of mechanical systems, here, along with them, more general systems are also considered, including systems of other physical nature, such as the Helmholtz system, and a wide class of systems with variable masses, depending not only on generalized coordinates, but also on the generalized velocities. In addition, such systems may also include economic systems when considered as dynamic analogs of mechanical and generalized systems. It should be noted that in the above extension of the class of systems under study one have to reckon with the fact that the generalized matrix of quadratic form of the mass-inertial characteristics of the system may not be positive definite, unlike mechanical systems, but only to be nonsingular. This fact does not allow building a control without the use of elements of this matrix, it was possible in the case of mechanical systems. However, in the work a universal vestor could be built that does not depend on these elements, for any generalized systems, Multiplying that vector by this generalized matrix, the law of control of any given generalized system is determined. Thus, the results can be regarded as a significant contribution to the theory of auto-adjustment control of mechanical and generalized systems and their dynamic counterparts, when the goal of control is unstressed bringing of condition of system to given manifold, formed by program constraints, with incomplete information about the non-control forces and disturbances acting on system.

Common way to integrate the dynamical equations of projectile planar motion introduces two Cartesian coordinates x(t) and y(t) and attack angle ϑ(t), all depending on time t, and three coupled ordinary differential equations (ODE) each nominally of II-nd order. It leads to inevitable computational complexities and accuracy risks. The method proposed excludes the time variable and diminishes the number of functions to n = 2: the attack angle ϑ(b) and intercept a(b) of the tangent to the trajectory at the point with slope b = tan θ with the θ being the inclination angle. This approach based on Legendre transformation makes the integration more convenient and reliable in the studied case of quadratic in speed aerodynamic forces i.e. drag, lifting force, conservative and damping momenta and the wind affecting the flight. The effective dimensionality of new ODE system is diminished by 2 units and its transcendence is eliminated by simple substitution η = sin ϑ. Also the method enables to obtain easily and reliably the projectile trajectories in conditions of tail-, head-or side wind. Investigated are main ranges of aerodynamic parameters at which takes place different behavior of the attack angle ϑ vs slope b including quasi-stabilization and aperiodic auto-oscillations. In addition, it was revealed non-monotonous behavior of speed with two minima while projectile descending if launched at the angles θ0 close to 90 ∘ . The numerical method may implement into quality improvement of real combat or sporting projectiles such as arch arrow, lance, finned rocket etc.

In this paper the investigation of the laser wave parametric decay in the hot magnetized plasma is performed taking into account the relativistic electron mass. Strong external magnetic field affects essentially the efficiency of laser radiation energy input into plasma. The magnetic field of the wave modulates the external magnetic field which leads to the parametric acceleration of electrons in the crossing fields and to the amplification of the charge separation field. In this process up to 85% of the laser radiation energy transforms into the energy of plasma particles. The analysis of nonlinear dynamics of the extraordinary electromagnetic wave in the strong external magnetic field in the conditions of the parametric decay shows that the exponential increase in the amplitude of the secondary wave exited at half-frequency of the primary wave changes into a reverse process in which the energy returns to the primary wave and causes the large amplitude oscillations in plasma. Unlike the previous papers in this area the investigation considers the parametric decay in the plasma preliminary heated up to the relativistic temperature. The self-similar system of nonlinear equations in total derivatives which takes into account the relativistic heat electron mass is derived. The small perturbations of the heated plasma parameters are investigated using the dispersion equation which defines the phase and group velocities of the slow and fast extraordinary waves in the linear approximation. It is shown that unlike the cold plasma in the linear approximation the non-transparency band in the frequency region higher than upper-hybrid electron frequency disappears. Moreover, the asymptotes of the dispersion branches in the high frequency regions approach each other. In the final part of the paper the calculation of the parametric instability increment is performed. It reaches the maximum value when the exited wave frequency is equal exactly half the frequency of the laser pump wave. The analytical expression for the maximum increment is derived and its dependence on the electron thermal velocity is investigated.

The group-theoretical justification is presented for the original approach by Kirznits and Chechin which allows for the primary protons of ultra-high energy cosmic rays to overcome the energetic limit (about 50 EeV) of Greisen-Zatsepin-Kuzmin remaining in the scope of the usual ideas about the nature of the extra-galactic sources of the cosmic rays. But the experimental status of the GZK limit is at present not sufficiently definite due to the rareness of these events in this range of energies as well as due to the difficulty of their identification. Thus it seems reasonable to suggest one of the possible theoretical explanations of this limit without using any kind of so-called “new physics” (e.g., “cosmic strings” etc.). In this paper account is taken only for some natural extension of standard Lorentz kinematics as it formulated in the special relativity theory. It is shown that the explicit form of the factor deforming the Lorentz invariant in the energy-momentum space may be found on the grounds of the approximate transition from Lorentz symmetry to the conformal values of the Lorentz-factor of the order 10 10-10 11 . More technically, we replace the purely phenomenological approach of Kirznits and Chechin by more grounded regular expansion of special conformal transformation in terms of powers of inverse Lorenz-factor 1/y taking account only in the limiting case 1/y. In this way it occurs quite naturally that all the improved kinematics are capable to reproduce reasonably the observed data on the available sources of the extragalactic cosmic rays.

We have considered the potential fluid flow in porous medium taking into account variable diffusion coefficient in the tube of radius . The flow is supposed to be cylindrically-symmetric. The velocity has two components: ⃗ =(,0,). The Euler equation has in the right hand side the term which determines Darcy force: ⃗ = −⃗, where - inverse Darcy coefficient. Continuity equation has the term which describes transverse diffusion of flowing fluid. We have established that for Euler equations system the equality 2/ ≡ 2/ is fulfilled identically. It means that Euler equations system is compatible and integrable. For (,) we have obtained Bessel equation, for (,) - the equation with diffusion coefficient (). We have investigated the solution of the equation for (,) with three types of diffusion coefficient (). We have established that in all cases the equation has unstable solution.

Titanium dioxide films obtained by gel method are investigated. Optical properties of the fabricated films, such as thickness, refractive index and thermo-optic coefficient were studied by the methods of integrated optics which use waveguide propagation of radiation along the film. The parameters of the films fabricated by sol-gel and gel methods were compared. It was established that the pores in the films made by gel method contain smaller amounts of water and therefore have higher density. Refractive index of gel films was determined by the resonant angle of the waveguide excitation, calculated using the optical waveguide dispersion equations and amounted to 2.1-2.4. This value is higher than in the case of using sol-gel technology for fabrication of thin films (1.5-1.8). By the reflection and transmission spectra obtained using spectrophotometer, it was found that films produced under cetrain parameters of technological regime have anisotropic properties. It was established that the presence of anisotropy is due to the structure of the film in the form of a linear oligomer. The structure and morphology of the gel films was studied by electron microscopy. It is shown that the resulting films have a porous structure that allows their doping with substances allowing to create elements of integrated optics, such as lasers, amplifiers, etc.