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# SCIENCE & EDUCATION

### Bauman Moscow State Technical University.   El № FS 77 - 48211.   ISSN 1994-0408

Mathematics and Mathematical Modeling

Analytical reservation of measuring systems of an aircraft
# 06, June 2013
DOI: 10.7463/0613.0571439
V.V. Dik, S.A. Krasnova, S.B. Tkachev
The problem of analytical reservation of measuring systems of an aircraft is solved with the use of state observers. The problem of tracking a program trajectory which implements landing maneuvers (approach and round-out) is solved by using estimates obtained by state observers. Aircraft movement in vertical plane was considered. It was assumed that the aircraft is exposed to limited unknown external disturbances. As a system of analytical reservation of board instruments, cascade system of state observers working on the sliding mode was proposed. This system allowed to estimate both some of the variables of the state vector and external disturbances acting on the aircraft. Simulation results were presented in the article.
Algorithm for calculating M-estimates of autoregressive field's parameters
# 07, July 2013
DOI: 10.7463/0713.0571094
УДК: 519.234.3
V.B. Goryainov
A process of a two-dimensional autoregression of order (1, 1) is considered in this article. Distribution of the innovation field of the autoregressive model is assumed to be unknown. An algorithm for calculating M-estimates of coefficients of the autoregressive field was constructed. The convergence of this algorithm was proved. The algorithm is an iterative version of the weighted least squares method. The weights are recalculated at each step. Each iteration represents the process of solving the system of linear equations. In contrast to the method of Newton (method of tangents) the algorithm converges from any point of the initial approximation.
Identification of random fields using methods based on signs of observation residuals
# 06, June 2013
DOI: 10.7463/0613.0571085
V.B. Goryainov
The process of two-dimensional autoregression of order (1, 1) was considered. Distribution of the innovation field of the autoregressive model is assumed to be unknown. Estimation of the autoregressive model parameters, based on the signs of observation residuals, was constructed. Consistency and asymptotic normality of these estimations was proved. Asymptotic relative efficiency of the estimation in relation to the least squares estimation was calculated. The conclusion was drawn on the advantage of the constructed estimation over the least squares estimation if the innovation field has double exponential distribution or Tukey distribution.
A new convection-diffusion global minimization method for solving inverse problems of chemical kinetics
# 04, April 2013
DOI: 10.7463/0413.0569246
V.V. Fedorov
A new deterministic global minimization method based on the numerical solution of the convection-diffusion boundary problem was developed. With the help of this method, the inverse problem of chemical kinetics was solved. Problems of identification of speed constants in chemical reactions are usually based on minimization of functions. In most cases it is necessary to deal with "stiff" systems of differential equations and, respectively, with many objective functions with ravine surface and local extremums, which makes application of existing multidimensional minimization methods complicated. Two main conceptions are included in this method: diffuse overcoming of local extremums and transforming multivariable minimization to one-dimensional minimization.
Robustness of estimates of spatial autoregression’s coefficients based on the sign tests
# 04, April 2013
DOI: 10.7463/0413.0569036
УДК: 519.12
V.B. Goryainov, E.R. Goryainova
In this article the process of two-dimensional autoregression of order (1,1) is considered. Distribution of the innovation field of the autoregressive model was assumed to be unknown. Definitions of the influence functional and the gross error sensitivity coefficient for autoregressive field parameter estimation were given. An explicit expression for the influence functional of sign estimation of the equation coefficients of the autoregressive field was obtained. It was shown that the sign estimation was robust. Sign estimation could be recommended as an alternative to least squares estimation with anomalously large errors when observing an autoregressive field.
Design of aircraft trajectories with non-monotonic change in energy
# 04, April 2013
DOI: 10.7463/0413.0554666
УДК: 517.977
A.N. Kanatnikov
A six-dimensional model, where the aircraft is treated as a material point, is used for planning trajectories of aircraft. In this case, the state variables are the coordinates of the aircraft in the trajectory coordinate system, and controls are longitudinal and transversal overloads and the roll angle of the transversal overload vector. In the context of this model, the author considers a terminal control problem in which it’s required to find such controls at which the aircraft is transferred from a specified initial point of the phase space to a specified destination point. Methods of solving this terminal control problem are well known if the flight time is given. Selection of flight time, if it is not known, is not an easy task, as this choice influences the shape of the flight trajectory. One of the methods to solve the terminal control problem with unknown time is the energy method based on replacement of the independent variable (time) by the normalized mechanical energy of the system. The energy method leads to flight trajectories with a monotonic variation of energy. It provides good solutions for landing and take-off of aircraft, but may not be applicable for complex maneuvers. The author considers methods of planning trajectories with non-monotonic variation of energy, which, nevertheless, are based on the energy method. These methods are based on special short-term maneuvers at the beginning and end of the trajectory (so-called transient maneuvers), and on selection of certain intermediate points through which the trajectory must pass.
On the conjugacy problem in subdirect products
# 04, April 2013
DOI: 10.7463/0413.0554654
УДК: 512.54.05
O.V. Kulikova
The conjugacy problem is one of the three fundamental algorithmic problems of the group theory, formulated by M. Dehn in 1912. The negative answer to this problem was obtained by P.S. Novikov in 1955: he created a finitely presented group with an unsolvable word equality problem; so he proved that in the class of finitely presented groups the conjugacy problem is also unsolvable. After that one began to study the conjugacy problem in particular classes of finitely presented groups and their subgroups. In 1971 C.F. Miller III constructed a subdirect product of free groups with the unsolvable conjugacy problem. This subdirect product is almost never finitely presented. In 2000 G. Baumslag et al. proved that there exist torsion-free word hyperbolic groups the direct product of which contains a finitely presented subgroup with the unsolvable conjugacy problem. For a direct product of free and surface groups M.R.Bridson and. Ch.F.Miller III proved in 2009 that any finitely presented subgroup of this direct product has a solvable conjugacy problem. The present paper is a sequel of the paper in which the conjugacy problem for subdirect products was studied with the use of pictures; and, in particular, it was proved that solvability of the conjugacy problem for subdirect products follows from atoricity and the small cancellation condition C(6). Pictures are geometric objects dual to van Kampen diagrams. In 1968 P. Schupp was the first to use van Kampen diagrams to solve the conjugacy problem in a group under the non-metric conditions C(p) and T(q). In this paper it is proved that under atoricity and the condition C(4)-T(4) or C(3)-T(6), the conjugacy problem for subdirect products is solvable. Also in this paper the sufficient condition for solvability of the conjugacy problem for subdirect products is expressed with small cancellation conditions only.
Movement stabilization along the given path for a two-wheeled robot with a differential drive
# 04, April 2013
DOI: 10.7463/0413.0547786
УДК: 517.93
G.A. Nefedov
The author considers the problem of movement stabilization along the given path for a two-wheeled robot with a differential drive. Wheeled robots belong to the class of non-holonomic systems, which, as it is generally known, cannot be stabilized in the vicinity of the equilibrium position of a continuously differentiable stationary state feedback. But the task of stabilizing the motion along the given curve is formulated only for part of the variables, so that the use of stationary feedback is possible. The robot model was transformed into a quasi-canonical form, all possible functions which allow one to obtain it were determined. The normal form for the system was built with the use of the distance to the target curve; its zero dynamics was investigated. On the basis of this form the problem of movement stabilization along the path was solved with the use of feedback linearization. The results of computer modeling are presented for particular cases: a circle and a line.
Mathematical model of a dynamically tuned gyroscope
# 04, April 2013
DOI: 10.7463/0413.0550951
V.N. Nikandrov
Mathematical model of a dynamically tuned gyroscope (DTG) includes an elastic and not ideal ball bearings of the gyro spin motor. The DTG is considered as a mechanical system with eight degrees of freedom. Selection of the number of degrees of freedom is based on the gyroscope design features. The angular and linear displacements of the gyro spin shaft, suspension ring and rotor became possible due to deformation of ball bearings. This model is necessary for modeling the DTG dynamics caused by natural vibration of bearings, for calculation of natural frequencies and for considering the influence of particular defects of ball bearings on the gyroscope drifts.
Modeling of dielectric properties of composite materials on the basis of asymptotic averaging
# 01, January 2013
DOI: 10.7463/0113.0531682
Yu.I. Dimitrienko, A.P. Sokolov, M.N. Markevich
In this paper, the authors consider calculation of effective dielectric parameters of composite materials with complex three-dimensional armoring. The method of asymptotic averaging of periodic structures is used for averaging of long-term quasi-static electric fields in composites with dielectric components. Series of local electrostatic tasks were formulated with the use of periodicity cells; electrostatic variational problems were also formulated. To solve spatial local problems for composite materials with three-dimensional complex armoring, the finite element numerical method was used. In this article, the authors proposed an algorithm for computation of effective tensor of dielectric permeability of three-dimensional armored composite materials. Test computations were carried out in order to obtain distribution of the local electric field, effective dielectric characteristics of three-dimensional orthogonal armored composite materials with different inclusion volume fractions.
Paleoclimate simulation with the use of a global climate model
# 01, January 2013
DOI: 10.7463/0113.0517953
V.P. Parhomenko
Influence of important climate parameters (atmosphere CO2 concentration, solar constant), configuration of continents, as well as distribution of ocean depths corresponding to the considered time periods, on climate was investigated. The author used a global climate model that includes a 3D model of the thermohaline circulation of the ocean, an energy and moisture balance model of the atmosphere and a sea ice evolution model. Numerical experiments corresponding to periods of 124 million years ago and 200 million years ago with drastically different parameters were carried out. It was established that a stationary climate system state is achieved after 1500-2000 years of development. General global and spatial climatic characteristics for the atmosphere, oceans and sea ice were obtained. Estimated global mean atmospheric temperature is agreed with the reconstructed observed data. Features of the ocean circulation were studied for the corresponding periods.
Construction of marginal probability density function of model parameters of cell population
# 12, December 2012
DOI: 10.7463/1212.0500575
The author considers a mathematical model describing dynamics of the population of stem cells in standard laboratory culture conditions without food restrictions. For this model, point estimates of parameters and expression for their joint probability density function are known. They are derived on the basis of the Bayesian approach and Jeffreys invariat theory with limited samples of experimental data on values ​​of the state vector. The author found marginal functions of probability density parameters of the mathematical model and interval estimates for these parameters.
Use of simultaneous approximations to study arithmetic nature of values of hypergeometric functions
# 12, December 2012
DOI: 10.7463/1212.0500464
P.L. Ivankov
To study the arithmetic nature of the values ​​of hypergeometric functions with irrational parameters, effective Padé approximants of first or second kind are commonly used. In this case, the approximant of second kind (joint approach) are simpler and often produce more general results. In this paper the author proposes an effective construction of simultaneous approximations for hypergeometric functions of general form and their derivatives (including the parameter). With this construction, the corresponding module is estimated below by a linear form. Some of the parameters of the functions are irrational.
The number of points moving along a metric graph: dependence on the permutation of edges
# 12, December 2012
DOI: 10.7463/0113.0515440
V.L. Chernyshev, A.A. Tolchennikov
This article deals with a discrete problem of points moving along a metric graph connected with studying the statistic of Gaussian packets in a three-dimensional network. This problem arises when considering the Cauchy problem for the non-stationary Schrödinger equation. For an arbitrary finite compact graph-tree, representation for the number of points occurring in the initial vertex was obtained. By the example of one particular graph, the number of points moving along this graph was obtained as a sum of solutions for slack linear inequalities. The authors found the highest term of difference of the number of points moving along the graphs which were obtained from various assemblies of edges; the highest term of symmetric difference of the number of moving points was also obtained.
Automorphisms of global semigroups of cyclic groups of prime order.
# 11, November 2012
DOI: 10.7463/1112.0495704
D.A. Stepanov
The global semigroup of some group is the set of all nonempty subsets of the given group with a natural operation induced by the operation of the group. In the paper we describe automorphisms of the global semigroup of a cyclic group of prime order. We show that for each prime with the exception of 3 and 5 all the automorphisms of the global semigroup are induced by the automorphisms of the given group.
Criteria for piecewise affine crumples and contractions.
# 11, November 2012
DOI: 10.7463/1112.0493524
N.S. Gusev
There are considered piecewise affine maps which are simplicial relative to a complete simplicial complex. Of such maps two kinds are studied: contractions (such that the complete preimage of a point is connected) and crumples (such that the image of a nonsingle-point straight-line segment is nonsingle-point broken line). Discrete criteria (with regard to properties of these maps at simplices of corresponding complexes) for such maps are proved.
Parametrical identification of a model of cooperating cellular populations on the basis of Bayesian approach
# 11, November 2012
DOI: 10.7463/1112.0490900
Algorithm, which allows one to obtain point estimations of a model’s parameters by using a limited sample of experimental data on the state vector’s values, was developed for a mathematical model that describes the dynamics of selective reproduction of clone-forming population of abnormal cells in the culture of human stem cells under standard laboratory conditions of cultivation without dietary restrictions. Bayesian approach was briefly described along with Jeffreys’ invariance theory. Probability density functions of specified mathematical model’s parameters were obtained with the use of these theories.
On isotropic solutions to Maxwell’s equations
# 11, November 2012
DOI: 10.7463/1112.0489647
V.N. Trishin
Null solutions of the source-free Maxwell equations in the Minkowski space-time are considered. It is well known that a principal null direction of the null electromagnetic field defines a shear-free geodesic null congruence. Conversely, null Maxwell fields can be obtained from the (analytic) shear-free congruence via the Robinson theorem. We present the explicit formula for the electromagnetic field derived from the Robinson and Kerr theorems. The solution depends on two analytic functions of two and three complex variables.
On piecewise linear solutions to the differential inclusion
# 10, October 2012
DOI: 10.7463/1112.0489571
A.V. Gorbunov
The problem of finding a solution to the differential inclusion in the form of piecewise linear function is examined. Sufficient conditions for the local existence of linear solution to the differential inclusion and procedure for the construction of piecewise linear solution to the differential inclusion were proposed. Illustrative example is considered. The obtained results can be used for mathematical simulation of systems with an incomplete description and optimization techniques.
Stabilization of affine systems with high transformability index to a quasicanonical form
# 09, September 2012
DOI: 10.7463/0912.0467824
S.B. Tkachev, A.A. Shevlyakov
For nonlinear dynamical systems with scalar control the equilibrium point stabilization problem is considered using a transformation of the system in a regular quasicanonical form.  Affine systems are investigated which can be transformed to quasicanonical form with  transformability index greater then two.   Zero dynamics of the system are not asymptotically stable.  The virtual outputs method of stabilizing  feedback design is propagated on the above mentioned systems.

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