SCIENCE & EDUCATION

In this paper, the authors presented a mathematical model for calculating effective dielectric characteristics of complex structured composite materials influenced by an external alternating electric field. An asymptotic investigation of electro-dynamic differential equations with fast oscillating coefficients was carried out. A local electro-dynamic problem on the composite’s “periodicity cell” was formulated. An effective complex permittivity of composite materials with various volume fractions of the fine-dispersed ferroelectric ceramic inclusions was calculated by the finite element method and the biconjugate gradient stabilized method. The authors analyzed frequency dependence of dielectric characteristics and dielectric loss tangent of composite ceramics. The obtained results are good congruous to experimental data.

In this article a sufficient condition for exponential stability of a linear system with delay was proposed; a new estimating method of the degree of damping and magnitude of overregulation was developed on the basis of this condition. The exponential stability condition lies in solvability of an arbitrary non-linear matrix inequality for one matrix and two scalar unknown quantities. Explicit expressions for estimates of the lower bound of the degree of damping and the upper bound of overregulation in terms of solution to this matrix inequality were derived; a method for finding such solutions was also proposed. An example illustrating effectiveness of the proposed approach was considered.

In this paper the author proves that in groups with small cancellation conditions C(3)-T(6) an occurrence problem in a cyclic subgroup is solvable; in other words, for any elements (g,h) of such a group one can determine whether there exists an integer n≠ ±1:gⁿ=h. This result is the latest in a series of theorems on groups with small cancellation conditions C(p)-T(q), proved by the author: on resolvability of a root problem, on resolvability of a problem of conjugated appearance in a cyclic subgroup, on the norm forms of elements in the infinite order and on the characteristic property of elements in the finite order. The results were obtained with the use of the group diagram method. As a direction for further development, it is planned to solve a problem of power conjugation in the specified class of groups.

It is impossible to use the known in the theory of transcendental numbers Siegel's method directly for investigation of arithmetic properties of values of generalized hyper-geometric functions with irrational parameters because these functions do not belong to the class of E-functions. For that reason in such a situation one usually applies different variants of effective construction of linear approximating forms. In this paper we use one of such variants that makes it possible to consider the differentiated with respect to a parameter functions. The capabilities of this method are extended by a special choice of parameters of the functions under consideration.

The purpose of this work is to design a model of double-sided battles of numerous groups on the basis of discrete Markov processes. Mathematical relations which allow one to calculate probabilities of fighting strength preserved on each side during the battle and at the end of one. The present study addresses the question of influence of various factors on the battle outcome and main indicators of the mission. The proposed model may be used not only for development of new weapon models and defense equipment but also for estimation of their adequacy for execution of assigned missions.

One of the methods of qualitative analysis of a dynamical system is to estimate the position of its compact invariant sets closely associated with bounded trajectories of the system. As a solution to such a problem, one can use a localizing set, i.e. a set in the phase space which contains all in-variant compact sets of a system. In this paper the discrete-time dynamical Lozi system of second order was considered. This system was proposed as a piecewise linear analogue of the known discrete-time Chenon system which has a chaotic attractor for some parameter values. For positive invariant and negative invariant sets of the Lozi system a family of localizing sets was constructed and their intersections were determined. The results of this investigation were presented in figures.

In this paper a new approach was proposed to solve terminal control problems for affine systems. This approach is based on transformation of a system under consideration to a quasi-canonical form system. Moreover, it was assumed that all subsystems of the canonical form are two-dimensional. A sufficient condition for existence of a solution for the terminal control problem was proved. A numerical procedure was also proposed to construct a solution of the terminal control problem for affine systems which are equivalent to systems of the quasi-canonical form with two-dimensional subsystems of the canonical form. An example was given to illustrate the proposed approach.

This article deals with problems of global optimization, model updating and diagnostics of hydro-mechanical systems. It was assumed that criterion functions were continuous, Lipschitzian, multiextremal and not always differentiable. Two novel hybrid algorithms were proposed; these algorithms use the modern stochastic Particle Collision Algorithm based on an analogy with absorption and scattering processes for nuclear particles, for scanning a search space. Local search was implemented with the use of a linearization method with smoothing approximations of criteria for the first algorithm, and a convergent variant of the Nelder-Mead simplex method was used for the second algorithm. Some results on solving model problems on computational diagnostics of the coolant phase constitution in the reactor primary circuit were presented.

In order to obtain quantitative results in the theory of Diophantine approximations one uses functional linear approximating forms which have sufficiently high order of zero at z = 0. Such forms are constructed either by means of the Dirichlet principle or effectively. In this article, by means of effective construction of approximating linear forms, we obtain a low estimate of the modulus of a linear form in the values of an exponential function in different points of an imaginary quadratic field; numerators of these points are roots of unity. Precise estimates, with respect to the height, were obtained with computation of corresponding constants. The proposed construction could be used for obtaining analogous estimates in values of generalized hypergeometric functions.

This article deals with problems of global optimization and computational diagnostics of hydro-mechanical systems. It was assumed that criterion functions were continuous, Lipschitzian, multiextremal and incompletely differentiable. Novel hybrid algorithms were proposed; these algorithms use the modern stochastic Particle Collision Algorithm based on an analogy with absorption and scattering processes for nuclear particles, for scanning a search space. Local search was implemented with the use of two derivative-free methods: a space filling curve method and a convergent variant of the Nelder-Mead simplex method. Results of solving the model problem in identification of the coolant phase constitution anomalies in the reactor circuit were presented.

The Blasius benchmark problem dedicated to numerical simulation of a flow around a thin plate is considered in this paper. The laminar boundary layer was simulated using a modified numerical scheme of the vortex element method with tangent velocity components. This numerical scheme was implemented in POLARA software. Distributions of longitudinal and transversal velocity components over the boundary layer’s cross-section were obtained. Use of the modified scheme allowed to improve accuracy of determining the intensity of a vortex layer on the plate significantly at every step of the calculation. The obtained results appeared to be in good agreement with the analytic solution given by Blasius.

A problem of building a system of base functions suitable for solving motion equations of a wire by the Galerkin method was considered in this paper. Dependence of the peak-to-peak amplitude of oscillations on the number of base functions was studied by a test example. Two sets of wire parameters were considered; wire motion is qualitatively different for these sets. Use of one base function in each direction is reasonable for a taut wire and it leads to an error less than 4 %. For a slack wire usage of not less than six base functions is required for correct simulation of wire motion.

This paper deals with systems that are reduced to linear control systems by composition of a dynamic feedback transformation with the change of variables. Such systems form the widest class of systems for which control algorithms were developed. An invariant description of algebra of vector fields which define linearizing dynamic feedback was obtained by methods of infinite-dimensional differential geometry. This result can be used for checking dynamic linearizability of particular control systems and in theoretical studies for creating examples of dynamically linearizable systems with certain properties or describing the whole class of such systems with a given dimension.

Since the early nineties, the problem of controlling chaos attracts researches’ attention. The essence of the problem lies in synthesis of control which eliminates chaotic dynamics of a dynamic system. A normally closed system has one or several stable limit cycles. This article deals with the problem of controlling chaos in one of the Sprott systems and also describes the method of elimination of chaotic dynamics by stabilizing a cylindrical invariant sub-manifold in the phase space of the closed system which consists of system’s stable periodic solutions. The system under consideration belongs to the class of affine systems which are not equivalent to a regular system of the canonical form on any subset of the state space. The described method is applicable not only to the specific system in question but may be extended.

This article presents a simulation of kinetostatics of a 7-DOF surgical robotic manipulator by the example of sternotomy. Motion of the robot is performed along a predetermined path in the global coordinate system. The numerical simulation was conducted in MATLAB^®Robotics Toolbox + Simulink^® program package. The same motion was investigated in SolidWorks^® for calculating flexibility matrices with the use of the finite-element method. Flexibility matrices for the point of load application were found for 11 successive positions of the robot. Compliance matrices were obtained using ABAQUS CAE® software.

A general case of weakly investigated N-person matrix games was considered in this paper. The brute force Nash equilibrium method for mixed strategies, known in case of N = 2, was successfully generalized. Computational complexity of this algorithm is unacceptable; this algorithm is reduced to solving exponentially increasing (depending on the number of strategies) number of completed linear systems, where the number of variables also exponentially depends on N. The new playing equilibrium algorithm based on the methods of linear programming and duality theory was proposed in this work. The algorithm was developed with the use of a more simple auxiliary coalition-free game introduced by means of some problem of mathematical programming. An example of the numerical solution of a 3-person game is given in the article.

This paper presents results of the evaluation of conditional distribution of failures over the Poissonian flow by the binomial law. The determined assessment allows one to build new decision rules for testing the null hypothesis of correspondence of failure intensity for a component item’s type with a group of such items collected on the principle of uniformity of a manufacturing technology and similarity of application domains. The authors also give an example of using this estimation for testing the null hypothesis of the correspondence of the failure intensity for a component item’s type with the mean group failure intensity in case of an alternative hypothesis, assuming that the failure intensity of the type is larger than the mean group one.

This paper describes motion of a plane surface in an unbounded viscous medium with non-Newtonian properties under the action of a random force. It was shown that fluctuations in the velocity of such a surface are described by a stochastic integral equation, and are classified as non-Markov processes.  Statistical characteristics of changes in surface velocity, including a characteristic function of the first order and moments of the first and the second order were obtained. It was also shown that the effect of both dilettant and pseudo-plastic non-Newtonian fluids in the first approximation have similar statistical properties. It was determined that the variance of fluctuations of surface acceleration in case of a non-Newtonian medium was greater compared to the same parameter in a Newtonian fluid.

One method of the qualitative analysis of a dynamical system is to estimate the position of its compact invariant sets closely associated with bounded trajectories of the system. As a solution to such a problem, one can use a localizing set, i.e. a set in the phase space containing all invariant compact sets of the system. In this article two continuous two-dimensional dynamical systems describing behavior of some biological systems are explored. For each of these systems a family of localizing sets is constructed, and then the intersection of the family is calculated. For the first system the solution was obtained analytically and for the second one the numerical procedure of constructing localizing sets was proposed. The investigation results are shown in figures.

In this paper a point-to-point motion problem was considered by the example of brain biopsy. It was emphasized that this operation requires exceptional end-effector positioning accuracy to the target point. End-effector motion was considered as point-to-point Cartesian trajectory tracking. The desired trajectory in Cartesian space was transformed to joint coordinate trajectory in joint space through solving the inverse kinematic problem. Computed-torque control was selected as a control scheme. Desired joint positions, velocities and accelerations were input signals for the inverse dynamic problem with the control low to compute joint torques required for implementing the required end-effector motion. Current joint position, velocity and acceleration were computed through solving the direct dynamic problem. High point-to-point motion accuracy indicators prove adequacy of the available robotic manipulator model and suitability of the selected control scheme. The task of numerical simulation was solved with the use of MATLAB^® Robotics Toolbox + Simulink^®.