Vol 50, No 4 (2017)

The classical representation of record values in sequences of independent random variables with the standard exponential distribution E(1) as sums of exponentially distributed random summands plays an important role in the mathematical theory of records. A generalization of this representation is proposed. A new similar result that makes it possible to express the record values of sample ranges as sums of independent exponentially distributed random variables is obtained.

The problem of estimating the error of quasi-Monte Carlo methods by means of randomization is considered. The well known Koksma–Hlawka inequality enables one to estimate asymptotics for the error, but it is not useful in computational practice, since computation of the quantities occurring in it, the variation of the function and the discrepancy of the sequence, is an extremely timeconsuming and impractical process. For this reason, there were numerous attempts to solve the problem mentioned above by the probability theory methods. A common approach is to shift randomly the points of quasi-random sequence. There are known cases of the practical use of this approach, but theoretically it is scantily studied. In this paper, it is shown that the estimates obtained this way are upper estimates. A connection with the theory of cubature formulas with one random node is established. The case of Halton sequences is considered in detail. The van der Corput transformation of a sequence of natural numbers is studied, and the Halton points are constructed with its help. It is shown that the cubature formula with one free node corresponding to the Halton sequence is exact for some class of step functions. This class is explicitly described. The obtained results enable one to use these sequences more effectively for calculating integrals and finding extrema and can serve as a starting point for further theoretical studies in the field of quasi-random sequences.

In this paper, we consider, discuss, and update some recent results on simulation functions established by several authors. By using one lemma of Radenović et al. (Bull. Iran. Math. Soc., 2012, 38 (3), 625–645), we suggest much shorter and nicer proofs of some statements than the ones available in the literature.

We consider the problem of the explicit search for all solutions of a first-order nonstrict differential inequality. We use the formula of the general solution of the corresponding differential equation. Using an analog of the method of arbitrary constant variation or, in other words, a straightening diffeomorphism, we reduce the original inequality to the simplest form ẋ ≤ 0 or ẋ ≥ 0. Even if the equation is considered in the existence and uniqueness region, theoretical and practical problems arise. The first problem is related to the extension of solutions (i.e., to the interval of determination). The second problem is that the general solution may consist of several functions given on different intervals of the equation domain. As a result, the resulting inequality also may have a solution that is composed of different functions. The situation becomes more complicated when the equation has points of branching. In this case, the method of comparison of theorems cannot be used. In this paper, we describe a method for solving differential inequalities and estimating their solutions for this case as well. The result obtained in this study provides a unified approach to many theorems on differential inequalities available in the literature.

The paper presents the problem of triaxial stabilization of the angular position of a rigid body. The possibility of implementing a control system in which dissipative torque tends to zero over time and the restoring torque is the only remaining control torque is considered. The case of vanishing damping considered in this study is known as the most complicated one in the problem of stability analysis of mechanical systems with a nonstationary parameter at the vector of dissipative forces. The lemma of the estimate from below for the norm of the restoring torque in the neighborhood of the stabilized motion of a rigid body and two theorems on asymptotic stability of the stabilized motion of a body are proven. It is shown that the sufficient conditions of asymptotic stability found in the theorems are close to the necessary ones. The results of numerical simulation illustrating the conclusions obtained in this study are presented.

Nonstationary modes of direct and inverse passage through resonance of unbalanced rotor equipped with an imperfectly mounted autobalancing device (ABB) are investigated. Two mathematical models are considered. The first one is the passage of the critical region with constant angular acceleration, and the second one is the rotor motion under the influence of a constant external torque. The study of the first model revealed that the presence of ABB significant negatively influences on the process of direct passage through resonance, and the maximum amplitude of whirling in this case can exceed the maximum amplitude in the steady state case. At the same time, during inverse passage through resonance, the influence of ABB is positive, and the maximum amplitude of whirling is much less than in the case of the direct passage through resonance. The investigation of the influence of viscous damping in the ABB revealed that too small damping influences negatively on the transient process dynamics, delaying the time of the rotor resonant vibrations until the moment of their stop. The study of the processes of acceleration and rundown of the rotor according to the second model confirms the main conclusions obtained from the first model.

The theory of figures of equilibrium was extensively studied in the nineteenth century, when the reasons for which observed massive celestial bodies (such as the Sun, planets, and satellites) are almost ellipsoidal were discovered. The existence of exactly ellipsoidal figures was established. The gravitational potential of such figures can be represented by a Laplace series whose coefficients (the Stokes constants I_n) are determined by a certain integral operator. In the case of an ellipsoid of revolution with homothetic equidensites (surfaces of constant density), the general term of this series was found, and for some of the other mass distributions, the first few terms of the series were determined. In this paper, the general term of the series is found in the case where the equidensites are ellipsoids of revolution with oblateness increasing from the center to the surface. Simple estimates and asymptotics of the coefficients I_n are also found. It turns out that the asymptotics depends only on the mean density, the density on the surface of the outer ellipsoid, and the oblateness of the outer ellipsoid.