SCIENCE & EDUCATION

The new method is proposed to solve a terminal problem for regular systems of a quasicanonical form with two-dimensional zero dynamics and scalar control. The example of terminal problem solving by means of the method proposed is given. The controllability sufficient condition for regular systems of a quasicanonical form with scalar control and two-dimensional zero dynamics is proven. The example is represented to illustrate the condition received.

 The authors consider a generalization of the well-known combinatorial structure of rook polynomials for boards of arbitrary dimension. The authors obtained basic formulas and examples of their application in practice. An algorithm of computing rook polynomials was developed.

In the paper the presentation of push-down automation as oriented multigraphs is described. The language of  push-down automata, which is represented by its multigraph, is defined and equivalence of this definition with standard one is proved. In terms of this representation some properties of push-down automata (such as regularity  of  push-down stack strings set) are considered. 

For multi-input nonlinear dynamic systems the problem of state feedback design stabilizing an equilibrium point is solved using the method of virtual outputs. Affine systems are considered for which smooth function (a system output) defining transformation of the system to a normal form with a vectorial relative level of an output (2, …, 2) is known. If zero dynamics of the system isn't asymptotically stable, that is the nonlinear system isn't minimum-phase, for the specified class of systems necessary and sufficient conditions of existence of such new outputs having the relative level (2, …, 2) for which the corresponding normal form has asymptotically stable zero dynamics are proved. The received results generalize the results received earlier for affine systems with scalar input.

The three-dimensional polynomial dynamical system dx/dt = y+z, dy/dt = -x+αy, dz/dt = x²-z with complex behavior is considered. In the particular case α = 0,5 this system reduces to one of systems with the chaotic behavior, which was found by J.C. Sprott. For the specified system the problem of localization of invariant compact sets, i.e. a problem of construction of such set in phase space of the system which contains all invariant compact sets of this system is solving. In article  the family of localizing sets for invariant compact sets is received using the functional method of localization proposed by A.P.~Krishchenko. The intersection of this family is found by numerical optimization methods.

The Cauchy problem for a parabolic partial differential equation with biharmonic operator and additive perturbation is considered in this note. Such equations are used in different  domains of physics, chemistry, biology, and computer sciences.  The solution of the considered problem is represented by Feynman formulae, i.e. by limits of iterated integrals of elementary functions when multiplicity if integrals tends to infinity.  The main part of these formulae is proved  with the help of Chernoff's theorem; some formulae are obtained on the base of the Yosida approximations.  Different types of Feynman formulae are presented in this work: Lagrangin and Hamiltonian. Lagrangian Feynman formulae are suitable for computer modeling of the considered dynamics. Hamiltonian Feynman formulae are related to some phase space Feynman path integrals; such integrals are important objects in quantum physics.

In the article the author describes the Smoothed Particle Hydrodynamics (SPH) method theory and its parallel implementation for simulation of incompressible fluids. Implementation of Predictive-Corrective Incompressible SPH (PCISPH) method is described in detail. Parallel techniques are introduced to specify and to process initial and boundary conditions in fluid simulation. The author presents results for two- and three-dimensional cases of collapse of column of fluid such as comparison of simulation with real experiments and with results obtained by other authors, performance comparison of CPU and GPU implementations.

Two-dimensional phase unwrapping problem was considered. Phase unwrapping is a key step of generating digital elevation model with the use of topographic radar interferometry. Difficulty of phase unwrapping problem is caused by unknown location of discontinuities on the interferogram and large interferogram sizes. On the base of Bayesian approach the mathematical model of the phase slopes on radar topographic interferograms was developed. Phase unwrapping may be considered as an optimization problem. For this statement of the problem a mathematical method based on the developed mathematical model was proposed. The computational experiment confirmed practical value of the obtained results.

In this paper we consider two etimations of the unknown parameter in the Lehmann-Cox power-law model. These estimates are obtained by minimization of two different functionals: Kolmogorov-Smirnov and Savage ones. The type of these functionals was obtained by using non-parametric statistics proposed by authors in their previous articles. We demonstrated the advantage of one statistics over the other by use of the methods of statistical modeling when the Lehmann-Cox power-law model is valid. It was proposed using a correction coefficient calculated with Monte Carlo method to eliminate bias of estimates for small sample sizes.

Investigation of arithmetic nature of values of generalized hypergeometric functions usually starts with applying a linear approximating form. Such a form must have a high order of zero at the coordinate basic origin and can be constructed by means of Dirichlet principle. The results obtained in this way are considerably general but the capabilities of such a method are restricted when precise quantitative assessment is required. Additional difficulties arise for functions with irrational parameters. In some cases the approximating form can be constructed effectively. Such a construction makes it possible to obtain more precise low estimates of linear forms for functions with rational parameters and to consider a case of irrational parameters. In this paper a new construction of Pade approximation for hypergeometric functions and their derivatives (also with respect to parameter) is proposed. This construction is applied for investigation of arithmetic properties of such functions.

 The authors consider an aircraft with four propellers (four-propeller rotorcraft). The propellers are attached to two bars rigidly fastened in the middle. The propellers on different bars are rotated in opposite directions. Changes of tractive forces of the propellers allows to control motion of the rotorcraft. The mathematical model of such an aircraft represents a dynamical system with a 12-dimensional state and 4-dimensional control. Dynamic feedback linearizing this system is constructed in the article. The feedback is used to track given reference trajectories with stability at the stages of take-off and landing. Acceptability of the found control is verified. Results of numerical simulation demonstrate efficiency of the proposed approach.

 In this paper the authors define sufficient conditions for existence of optimal thickness of a flat isotropic wall with a heat insulator and an interlayer in the form of a heat-activated padding which operates on the principle of feedback providing a minimum steady hottest-spot temperature of the wall. The unprotected side of the wall is cooled by the medium with a constant temperature and the heat transfer coefficient, and the side of the wall whith the heat insulator is affected by a heat flow in pulse-periodic mode.

The paper considers the possibility of organizing the spatial patterns of plankton's population densities defined solely by biological factors, under the condition of homogeneous environment. To simulate self-organization of two plankton's species populations a mathematical model of the "predator-prey" type with regard to the effect of limited seeking by the predator was studied. The simulation results showed the fundamental possibility of spatial structures organization through limited self-movement of plankton.

 The article presents a review of investigation of kinetic processes such as diffusion and thermal conduction in media with microstructure, near to the particles of micron and nanometer size, as well as microfilaments. The flow features of kinetic processes on small spatial and temporal scales are described. It was shown that these processes belong to the class of non-Markov processes and require the use of integral transformations for their description.

The author introduces a family of Boolean functions which can be used as local transition functions in generalized cellular automates for usage as part of stream cipher. The functions from this family are balanced. Функции из этого семейства являются равновесными, их нелинейность близка к максимальной. Each of these functions is a complete system and allows to prove the lower bound estimate of the length of the output sequence of the generalized cellular automaton. These properties are important for cryptographic applications of generalized cellular automatons.

One of the problems of teaching modern sections of natural sciences and mathematics is the absence of illustrative examples of application of the general theory due to complexity of analytical description of events. At the Physics Department of BMSTU the special course “Non-linear transport processes” has been reading for several years as a part of profession “Applied physics”. This article deals with analytical solution of one specific problem of the non-linear boundary layer theory in the non-Newtonian liquid that confirms the general theory.

This article deals with problems of creating generalized cellular automatons with specified lower bounds of the period. This property is very important for constructing pseudorandom sequence generators and stream ciphers based on generalized cellular automatons.

This article deals with geotechnical systems, such as engineering constructions with foundation in particular geological medium, for instance, even-frozen or thawing rock mantle. The calculation and forecast problems of temperature field extension into even-frozen ground under building’s foundations in cryolithozone were studied. The mathematical formulation of the stationary thermal conductivity problem with different boundary conditions was given. Predicted results of ground’s thermal condition in different spatial sections were included in the article.

Novel hybrid multiobjective optimization algorithms are presented for solving problems of computational diagnostics. A vectorial variant of the linearization method is implemented. Global solutions for individual criteria are determined by use of hybrid algorithms that combine the Metropolis algorithm scanning the space of variables and deterministic methods for local search. The vector optimization algorithms generate a set of non-dominated solutions to approximate a Pareto-optimal front. Simulation results for one of the benchmarks and corresponding valuations of the algorithm computational efficiency are received. The proposed hybrid algorithms can be applicable for computational diagnostics systems, problems of intellectual models teaching, complex dynamic systems control, and other intellectual technologies.

In the present paper a new method of investigation and description of linear dynamics was considered. This method was based on representations of corresponding evolution semigroups (or, what is the same, representations of solutions of the corresponding equations) with Feynman formulas, i.e. with limits of finite multiple integrals when n tends to infinity. Sometimes one succeeds to get Feynman formulas containing only integrals of elementary functions. Such Feynman formulas allow to calculate solutions of evolution equations directly, to approximate transition probabilities of stochastic processes, to model stochastic and quantum dynamics numerically. The limits in Feynman formulas agree with some functional integrals with respect to probability measures or Feynman type pseudomeasures. Nowadays, functional integrals (or path integrals) play one of the central roles in mathematical apparatus of theoretical physics; they are important objects of quantum field theory, especially in the theory of gauge fields. To solve a variety of problems it is worth to apply Hamiltonian formalism of quantum mechanics and to deal with (Hamiltonian) Feynman path integrals in phase space. There are many different approaches to define such integrals mathematically rigorously. And different classes of integrable functions arise in the frame of each approach. In this paper the approach of Smolyanov and his coauthors was used. This approach allows to connect Feynman path integrals in pahse space with Hamiltonian Feynman formulas for evolution semigroups. This method was actively used last decade to describe different types of dynamics in domains of Euclidean spaces and Riemannian manifolds, in infinite dimensional linear and non-linear spaces, to investigate p-adic analogues of equations of mathematical physics. The present work is expository, it brings together some of the results of recent articles of the author (joint with Boettcher, Grothaus, Schilling, Smolyanov), in which the method of Feynman formulas was subsequently developed to investigate Feller semigroups and the connection of such formulas with Feynman path integrals in phase space was studied. In this paper Feynman formulas for Feller semigroups and semigroups generated with different quantizations of a quadratic Hamilton function were obtained; a construction of Feynman path integral in phase space was introduced; Feynman path integrals in phase space for Feller semigroups and semigroups generated with different quantizations of a quadratic Hamilton function were presented.