SCIENCE & EDUCATION

 Step backward problem for generalized cellular automaton was proved to be NP-hard in this article. This problem can be described as follows: to obtain the initial states of defined automaton when its states after the first step are given. The existence of polynomial solution algorithm for this problem was also proved for the case when local coupling function of generalized cellular automaton was binary. Obtained results have theoretical significance and can be applied to justify cryptosecurity of encryption algorithms based on generalized cellular automaton.

 For model systems with parallel structure established formula for the main characteristics of quality and reliability of the system, which then gives a formula for numerical calculation of optimal critical level ofbeginning repair in the system, as well as established  approximate analytic formula for the basic characteristics of an optimal rule of repair.

We simulated the dynamics of temperature field soil base of the building, installed on a ground floor in a continuous permafrost in areas with seasonal freezing and thawing of the soil. The model is based on boundary-value problem for the heat equation with phase transitions. The equation describes the variation with time of the temperature field in a rectangular area of ground under the building and take into account the thermal effect of heated buildings on the ground and seasonal changes in climatic conditions (air temperature). The problem was solved numerically using method of control volume, the solution is implemented in the environment of Compaq Visual Fortran. The simulation results allow to analyze the possibility of dangerous cryogenic processes, resulting in a building in an emergency situation (heave, settlement, etc.).

Position stabilization problem for non-linear dynamic system with scalar control was considered based on differnetial-geometrical approach. Basic theoretical fundamentals about controlled dynamic system transformation into quasi-cannonical form, and controlled dynamic system with an output transformation into normal form. Attention was paid to the case when zero dynamics of the system wasn’t asymptotically stable, that is non-linear system wasn’t minimal-phase. Usage of the method of virtual outputs for stabilization of these systems isn’t always possible because of the difficulties raised from the search of virtual controls with required properties. It was shown that for building of virtual outputs with the required properties linearization of the subsystem defining zero dynamics could be used. For the dynamic system describing inverted pendulum placed on the truck the state feedback control synthesis problem was solved with the method of virtual outputs and linearization of the part of variables; this solution allowed to stabilize both upper equilibrium position of pendulum and given position of the truck.  

Mathematical model of selective replication dynamics of clone forming population of abnormal cells in the culture of human stem cells in the laboratory (in vitro), considering the influence of density factor on cells multiplication. Model is a system of second order ordinary differential equations with polynomial special right member. All stationary points of this model with non-negative coordiantes with at least one zero coordinate were found; stability analysis for these points was carried out with different values of model parameters. It was shown that system couldn’t have more than two points with both positive coordinates. The results of numerical simulation were included in the article.

Class of parabolic second order equations, produced by different quantization types of one classical system’s quadratic Hamiltonian, was considered. Solution of Cauchy-Dirichlet problem for considered class of equations on the interval was represented as Hamiltonian Feynman’s formula, that is as a limit of finite-multiplicity elementary integrals when multiplicity approached infinity. New formula for direct computation of the solution to formulated problem and computer simulation of corresponding dynamics was obtained in this work. Connection between differential operators corresponding to different quantization types of quadratic Hamiltonian and connection of obtained Hamiltonian Feynman’s formula with Feynman’s path integrals in phase space were also discussed in the article.

Bogley W.A and Pride S.J. have researched aspherical relative copresentations. Relative copresentation were considered in atoral aspect in this article. Sufficient condition for atoricity was obtained, just like Bogley and Pride had obtained sufficient condition for asphericity. Properties of commutation elements in the group, represented as oriented atoral relative copresentation were investigated. Some properties of conjugate and commutation elements in the group represented as oriented relative copresentation were obtained. There were no connected reduced circular images over this copresentation .

Running waves with matrix coefficients were introduced. Permutation properties were defined for these coefficients. Wave intaeraction operator was built; its derivatives were found. Operator was applied to the solution of Cauchy-Kovalevskaya’s first order system. This system was defined as a sequence of numeric matrices.

Cauchy-Neyman problem for parabolic equation on half-line with variable coefficients depending on coordinate was considered.Solution of this problem was presented as a limit of elementary multiple integrals, containing coefficients of equation and initial conditions, when multiple approached infinity. Such formulas were called “Feynman’s formulas”. Similar presentations of evolution equations’ solutions could be used for direct computation and computer simulation of researched dynamics. Among other things, limits of finite-multiplicity integrals in Feynman’s formulas agreed with some functional integrals in some probability measures on set of trajectories in the areas where equations were considered. By this means, Feynman’s formulas allowed to approximate functional integrals and, consequently, transition probabilities (which usually weren’t expressed in terms of elementary functions) of corresponding random processes. Method for obtaining Feynman’s formulas for evolution equations has been proposed in the works of Smolyanov O. G. and his co-authors in 1999 – 2003 years. This method was based on the application of Chernoff’s theorem and allowed to obtain Feynman’s and Feynman-Katz’s formulas for extensive class of evolution equations on different geometric structures.

Asymptotic normality of M-estimations with not necessarily convex regret function for 2D-autoregression of (1,1) order was recognized. Stability of these estimations when filling observation data with rough upwards excursions was recognized with computer simulation.

Numeric computation of gas-dynamic parameters when flow blown from the crack in the plate interacted with main supersonic gas flow moving parallel to the plate with sublimated surface was considered in this article. Numeric simulation of gas-dynamic parameters, structure and main dependences of supersonic flow past a jet, blown from the crack in he plate on superficial mass transfer was carried out. Influence of plate sublimation on gas flow parameters was numeric investigated.

Implementation of input-output mapping as equations of state was considered. Methods of differential geometry were used in the research. Early developed implementation algorithm included the test of first distribution integrals existence. Reliable test of existence based on Frobenius condition in differential forms language was proposed. To this end, module decomposition was built, containing differentials of equations’ variables. Method of codistribution basis building was created based on this decomposition. The results of this work could be used for the test of distribution integratability in the automatic control problems. It’s recommended to use symbolic mathematic software when using the results of this work.

A problem of finding rank of non-square functional matrix, whose elements are smooth functions, is considered. Such matrices arise in course of analysis of regularity and involutivity of vector distributions. To find rank of these matrices, it is convenient to use computer algebra systems, which can perform symbolic computations. Alternative to checking all possible minors is the Gauss algorithm of transforming matrix to row-echelon form. When considering regularity and involutivity, it is often of interest to know the rank of a matrix in the neighborhood of a point. Usual Gauss algorithm cannot answer this question, since it is possible for basis minor to become zero at chosen point.  A modified algorithm is presented. This algorithm allows to say if there exists a neighborhood of constant rank containing a chosen point. Examples are presented.

The description of the statistical behavior of Gaussian packets on a metric graph is considered. Semiclassical asymptotics of solutions of the Cauchy problem for the Schrodinger equation with initial data concentrated in the neighborhood of one point on the edge, generates a classical dynamical system on a graph. In a situation where all times for packets to pass over edges are linearly independent over the rational numbers, a description of the behavior of such systems is related to the number-theoretic problem of counting the number of lattice points in an expanding polyhedron. In this paper we show that for the final compact graph packets almost always are distributed evenly. The formula for the leading coefficient of the asymptotic behavior of the number of packets with an increasing time is obtained. The situation is also discussed where the times of passage over the edges are not linearly independent over the rationals.

Stabilization algorithm of the wheeled vehicle on the planned path is studied by means of numerical modeling. Synthesis of stabilization algorithm is based on theory of normal form of the affine system. Smoothing algorithm of the first derivative of curvature in the case of setting the path by parametric cubic spline is suggested. Methods of finding distance from the wheeled vehicle to the planned path are considered. Efficiency of the method is shown, high sensitivity of control to curvature of the path is established.

A new method of describing linear dynamics is considered.  This method is based on representations of corresponding evolution semigroups (or, what is the same, representations of solutions of corresponding evolution equations) by Feynman formulae, i.e. by limits of n-fold iterated integrals, when tends to infinity. Green functions of many initial-boundary value problems are not known explicitly, whereas   it is possible to obtain Feynman formulae containing only elementary functions as integrands for some of these problems. Such Feynman formulae, that allow to calculate solutions of evolution equations directly, to approximate  transition probabilities of stochastic processes, are useful  for computer modeling of quantum and stochastic dynamics. The notion “Feynman formula” (in this context) and the method to obtain such formulae were introduced in works of Smolyanov and his coauthors in the late  nineties. In the last decade it has been actively applied to describe different types of dynamics in domains of Euclidean spaces and Riemannian manifolds, in infinite dimensional linear and non-linear spaces. In the present note the multiplicative perturbations of generators of strongly continuous semigroups on a Banach space of some continuous functions are considered.  A Feynman formula   is obtained for the semigroups with   perturbed generators.  Therefore, a new formula is given for the description and the investigation of the perturbed dynamics. Also some particular examples, when the obtained Feynman formula contains only elementary functions as integrands, are considered in this note.

The paper describes KOKON software package developed at the Mathematical modeling Department (FN-12) of Bauman Moscow State Technical University. The software allows to construct visual three-dimensional models of spacecrafts and spacestations. Realistic 3D Stereo visualization is also included. The process of spacecraft motion and, in particular, angular rotation of the spacecraft under the influence of attitude control can be simulated and visualized. The developed software also contains tools for visual design of three-dimensional models.

 The author considers the problem of a Lagrange's system equilibrium point asymptotic stabilization with delay in the control loop. For linear Lagrangian systems of a general type the author found dynamic feedback providing the solution of a task in view. The offered approach is based on the use of change of variables excluding delay from the control loop, keeping the plant equations of motion without changes. The considered change of variables represents the modified transformation known as an Artstein reduction. The energy method is applied to solving the stabilization problem for the system in the new variables. Efficiency of the offered control method is shown trough numerical simulations. The results received in this note can be used to design control algorithms for mechanical systems under conditions of limited speed of reception, processing and transfer of an operating signal. 

 The problem under consideration is a generalized concept of a principal ideal ring and does not suggest that the ring does not have divisors of zero. The problem of classification of principal ideal rings in the new sense and their connection with principal ideal rings in the usual sense are investigated. It is proved that each principal ideal ring (maybe with divisors of zero) without nilpotent elements can be decomposed into a direct product of finitely many principal ideal rings without divisors of zero. For principal ideal rings with nilpotent elements the classification problem is not studied.

Complicated mathematical models are usually used for processing measurements of the Earth magnetic field carried out onboard an Earth artificial satellite. It is desirable to verify the measurement data by humble tools before such processing. If a few onboard magnetometers made their measurements at the same instants, then one ought to check a simple geometrical compatibility of the measurement data. One can estimate constant shifts in the data and transition matrices between proper coordinate systems for each pair of magnetometers in case the check proved to be successful. Below, we described the method for checking the compatibility of the measurement data produced by two magnetometers. We illustrate the method by checking the compatibility of the magnetic measurements produced by the equipment Mirage onboard Foton M-2.