Vol 52, No 2 (2019)

The Seidel method for solving a system of linear algebraic equations and an estimate of the rate of its convergence are considered in this paper. It is proposed to construct an equivalent system for which the Seidel method also converges but yields a better rate of convergence. An equivalent system is constructed by a separate iterative process, where each step requires O(n) operations. The stability of this process is proved. Results of numerical experiments are presented that show an improvement in the estimate of the convergence rate.

An approach to the problem of rank-one approximation of positive matrices in the Chebyshev metric in logarithmic scale is developed in this work, based on the application of tropical optimization methods. The theory and methods of tropical optimization constitute one of the areas of tropical mathematics that deals with semirings and semifields with idempotent addition and their applications. Tropical optimization methods allow finding a complete solution to many problems of practical importance explicitly in a closed form. In this paper, the approximation problem under consideration is reduced to a multidimensional tropical optimization problem, which has a known solution in the general case. A new solution to the problem in the case when the matrix has no zero columns or rows is proposed and represented in a simpler form. On the basis of this result, a new complete solution of the problem of rank-one approximation of positive matrices is developed. To illustrate the results obtained, an example of the solution of the approximation problem for an arbitrary two-dimensional positive matrix is given in an explicit form.

One of the main tasks of mathematical diagnostics is the strict separation of two finite sets in a Euclidean space. Strict linear separation is widely known and reduced to the solution of a linear programming problem. We introduce the notion of strict polynomial separation and show that the strict polynomial separation of two sets can be also reduced to the solution of a linear programming problem. The objective function of the linear programming problem proposed in this paper has the following feature: its optimal value can be only zero or one, i.e., it is zero if the sets admit strict polynomial separation and one otherwise. Some illustrative examples of the strict separation of two sets on a plane with the use of fourth degree algebraic polynomials in two variables are given. The application efficiency of strict polynomial separation to binary data classification problems is analyzed.

This paper presents a method for deriving optimal lower and upper bounds for probabilities and conditional probabilities (given a σ-field) for various combinations of events. The optimality is understood as the possibility that inequalities become equalities for some sets of events. New generalizations of the Jordan formula and the Bonferroni inequalities are obtained. The corresponding conditional versions of these results are also considered.

This paper deals with the problem of uniaxial stabilization of the angular position of a rigid body exposed to a nonstationary perturbing torque. The perturbing torque is represented as a linear combination of homogeneous functions with variable coefficients. It is assumed that the order of homogeneity of perturbations does not exceed the order of homogeneity of the restoring torque, and the variable coefficients in the components of the disturbing torque have zero mean values. A theorem on sufficient conditions for the asymptotic stability of a programmed motion of the body is proven using the Lyapunov direct method. The determined conditions guaranteeing the solution to the problem of body uniaxial stabilization do not impose any restrictions on the amplitudes of oscillations of the disturbance torque coefficients. Results of numerical modeling are presented that confirm the conclusions obtained analytically.

This work considers the problem of bending a thick annular plate of variable thickness on point supports under the action of its own weight. The problem describes the stress-strain state of the primary mirrors of large optical telescopes when the mirror axis is directed to the zenith. The main feature of this problem is the transverse displacements of the plate reference surface, which coincides with the rear flat base of the plate where the supports are located, and the front surface from which the incident light is reflected. Errors of the wavefront of the reflected light, including the standard deviation and the magnitude of the wavefront, are associated with the transverse displacement. The problem is solved with the use of two nonclassical theories of plates, the Timoshenko-Reissner theory of plates and the Palii-Spiro theory of mean thickness shells. The case of the optimal location of the supports corresponds to the smallest values of the deviation of the wavefront magnitude and the standard deviation. Calculations according to the Timoshenko-Reissner and the Palii-Spiro theories gave the same optimal location of the supports. The Palii-Spiro theory, which takes into account the variation of the transverse displacement along the thickness of the plate, is preferable for calculating the distortion of the reflecting surface of the primary mirrors of optical telescopes.

The problem of rigid triple shear is solved in the framework of the endochronic theory of inelasticity with account for finite deformations. The numerical implementation of the algorithm for determining the orthogonal rotation tensor and vortex tensor is proposed. The strain tensor is constructed on their basis. Simultaneously, the strain tensor is calculated with a direct numerical method. The corresponding strain components obtained with both methods are compared and analyzed.