Vol 25, No 4 (2017)

Original results of a research of an efficient computing method - group method of data han-dling are presented. Key shortcomings on each significant procedure of a classical algorithm arerevealed and systematized, and also ways of their elimination, including author’s modificationsare presented. In particular, the use of dispersion and an assessment of dispersion (Fischer’scriterion) is proposed as an assessment of accuracy of the received result, additional “internal”criterion for evaluation of adequacy of model in various tests during the fixing of input dataand changing of characteristics of an algorithm, and determining the optimal complexity of themodel. To solve the convergence problem of the classical algorithm, it was proposed to usethe methods of dispersion, factor and correlation analysis to eliminate non-informative features,modify the criterion for stopping the algorithm. The use of regularizing functionals is suggestedto solve the problem of multicollinearity of input characteristics and increase the stability of theobtained model, etc. A complex of computer modeling programs was developed, realizing an ef-ficient modified algorithm of GMDH with the considered modifications and also methods of adispersion analysis, correlation analysis, component analysis, elements of the regression analy-sis and others. The conducted researches and the received practical results can become a basisfor development with use of Machine Learning and Data Science technologies of the automaticsystem of computer modeling, the intellectual analysis and the data processing.

In the paper we study integral properties of convolutions of functions with kernels generalizingthe classical Bessel-Macdonald kernels (), ∈ , 0 < < . The local behavior of Bessel-Macdonald kernels in the neighborhood of the origin are characterized by the singularity ofpower type ||-. The kernels of generalized Bessel-Riesz potentials may have non-powersingularities in the neighborhood of the origin. Their behavior at the infinity is restricted onlyby the integrability condition, so that the kernels with compact support are included too. In thepaper the general criteria for the embedding of potentials into rearrangement invariant spacesare concretized in the case when the basic space coincides with the weighted Lorentz space.We obtain the explicit descriptions for the optimal rearrangement invariant space for such anembedding.

The multidimensional models of the population dynamics are considered in the paper. Thesemodels are the generalizations of the Lotka-Volterra model in case of interaction of the finitenumber of populations. The deterministic description of the models is given by the systemsof the ordinary nonlinear differential equations presented in the paper in the form of themultidimensional vector differential equations. The qualitative properties of the specified modelsare sufficiently well studied by means of Lyapunov methods. However, the probabilistic factorsinfluencing on the behavior of models are not taken into account at the deterministic descriptionof models. The new approaches to the modeling and stability analysis are of theoretical andapplied interest in the nondeterministic case.In this paper, the methods for design of multidimensional nondeterministic models ofinteraction of populations are considered. The first method is connected with the transitionfrom the vector nonlinear ordinary differential equation to the corresponding vector differentialinclusions, fuzzy and stochastic differential equations. On the basis of the reduction principle,which makes it possible to reduce the problem of the stability of solutions of a differentialinclusion to the problem of stability of solutions of other types of equations, stability conditionsfor the constructed models are obtained. The second method is connected with the technique ofdesign of the self-consistent stochastic models. The scheme of interaction is received on the basisof this technique. This scheme includes a symbolical record of possible interactions between thesystem elements. The structure of the multidimensional stochastic Lotka-Volterra models isdescribed, and the transition to the corresponding Fokker-Planck vector equations is carriedout by means of the system state operators and the system state change operator. The rules forthe transition to the multidimensional stochastic differential equation in the Langevin form areformulated. The execution of the numerical experiment with the application of the developedprogram complex for solving the systems of the stochastic differential equations is possible forthe models which are the concretizations of the studied general models. The described approachto the modeling of the stochastic systems can be applied in the problems of comparing of thequalitative properties of the models in deterministic and stochastic cases. The obtained resultsare aimed at the development of the methods for the analysis of nondeterministic nonlinearmodels.

Within the general theory of relativity the Bianchi type VIII cosmological models with rotationand expansion have been built. The first case includes a field of radiation, the second one -a perfect fluid with dust-like equation of state. Perfect anisotropic fluid imitates the rotatingdark energy. Static and dynamic cosmological modes have been observed, at the same timethe equations of state are partly postulated in the first case for the anisotropic fluid and inthe second case - for the perfect isotropic fluid, that imitates baryon matter. The analysisof absence of closed time-like curves has been done, so the models have been proved to becasual when the metric parameters satisfy the found conditions. Also the conditions, when theanisotropic fluid’s equation of state becomes vacuum-like, the energy of the fluid dominatesand it becomes asymptotically isotropic, have been cleared out. Specialities of the oscillatingmode have been observed. The order of present angular velocity value, calculated within thecosmological models, has been found to be quite satisfactory when expanding from the Plank scale to the present size of observed part of the Universe. The found solutions may be used foreffects taking place nowadays and also during the inflationary stage.

In this paper of the nonlinear laser wave propagation in the hot plasma along the strong exter-nal magnetic field under the electron cyclotron resonance conditions is investigated. The strongnonlinearity of such a process is caused by the relativistic electron movement and resonancewave ponderomotive force acting on the electrons. The system of equations for the enveloperight hand polarized laser pulse is derived using the hydrodynamics and Maxwell’s equations.The numerical integration of this system for the cold plasma case discovered the soliton solu-tions. This kind of solutions take a form of the envelope solitons containing inside them the plasma oscillations. The analytical expression for the energy density integral in a cold plasma isderived. It follows from the numerical results that for a hot plasma under cyclotron resonanceconditions the soliton solution becomes unstable. In this case the energy density conservationbreaks down, but electron momentum density conserves. It is concluded that the nonlinear sat-uration of the field amplitude is due to the plasma charge separation under electromagneticradiation pressure. In this case the discrete set of the envelope soliton carrier frequency is de-termined by the ratio of the frequency of the nonlinear longitudinal electron oscillations to theLangmuir frequency of plasma. For the low density plasma the discrete frequency spectrumobtained by the numerical integration transforms to the continuous one.

The task of integrating information contained in databases of various companies that providecargo carriers is considered in the article. It is assumed that these companies are independentand each of them maintains its own database about its employees, customers, number and pa-rameters of carries, services and orders. The article simulates three such independent databases(information sources) and describes a scheme for constructing an integrated system based onthem, in particular, for the convenience of customers when searching for carriers with the neces-sary parameters (dimensions and load-carrying capacity). To solve this problem, it is proposedto use an approach combining the advantages of mediator and data repository technologies. Itis assumed that all sources have similar conceptual schemes, but they have their own specificcharacteristics. In particular, there may be different names of tables and attributes in differ-ent sources and different distribution of attributes throw tables. Certain attributes may alsobe absent at some sources. We construct correspondence tables and a mediator that translateuser queries to the sources. To identify the necessary sources, a small auxiliary repository ismaintained that contains information about the parameters of the carriers that each source has.The user of the integrated system enters data on the weight and dimensions of his cargo, then,using the information of the repository, the system determines the appropriate trucking compa-nies to which the user request is broadcast. After combining all the answers that came from thesources, the client receives full information with a list of suitable companies, prices and otherconditions for the execution of the order.