Vol 52, No 3 (2019)

We consider a generalization of the well-known problem to randomly fill a long segment by unit intervals. On the segment [0, x], x ≥ 1, we place an open unit interval according to the F_x distribution law, which is the distribution of the left-hand endpoint of the unit interval, concentrated on the segment [0, x – 1]. Let the first allocated interval (t, t + 1) divide the segment [0, x] into two parts [0, t] and [t + 1, x] and they are filled independently of each other according to the following rules. On the segment [0, t] a point t₁ is selected randomly according to the law F_t and the interval (t₁, t₁ + 1) is placed. A point t₂ is selected randomly in the segment [t + 1, x] such that u = t₂ – t – 1 is a random variable distributed according to the law F_{x – t – 1}, and we place the interval (t₂, t₂ + 1). In the same way, the newly formed segments are then filled. If x < 1, then the filling process is considered to be complete and the unit interval is not placed on the segment [0, x]. At the end of the filling process, unit intervals are located on the segment [0, x] such that the distances between adjacent intervals are less than one. In this article, we consider distribution laws F_x with distribution densities such that their graphs are centrally symmetric with respect to the point (x – 1/2, 1/x – 1). In particular, this class of distributions includes the uniform distribution on the segment [0, x – 1] (the corresponding filling problem was previously investigated by other authors). Let N_x be the total amount of single units placed on the segment [0, x]. Our concern is the properties of the distribution of this random variable. We obtain an asymptotic description of the behavior of central moments and prove the asymptotic normality of the random variable N_x. In addition, we establish that the distributions of the random variables N_x are the same for all the distribution laws of the specified class.

Two classes of time-periodic systems of ordinary differential equations with a small parameter ε ≥ 0, those with “fast” and “slow” time, are studied. The corresponding conservative unperturbed systems \({{\dot {x}}_{i}}\) = \( - {{\gamma }_{i}}{{y}_{i}}{{\varepsilon }^{\nu }}\), \({{\dot {y}}_{i}}\) = γ_i(\(x_{i}^{3}\) – \({{\eta }_{i}}{{x}_{i}}\))ε^ν (i = \(\overline {1,n} \), ν = 0, 1) have 1 to 3ⁿ singular points. The following results are obtained in explicit form: (1) conditions on perturbations independent of the parameter under which the initial systems have a certain number of invariant surfaces of dimension n + 1 homeomorphic to the torus for all sufficiently small parameter values; (2) formulas for these surfaces and their asymptotic expansions; (3) a description of families of systems with six invariant surfaces.

The Seidel method for solving systems of linear algebraic equations x = Ax + f  is considered in this paper. This study is a continuation of a previous author’s work proposing an algorithm to estimate the rate of convergence of the Seidel method. A more exhaustive proof of the correctness of this algorithm is given here. The estimate obtained by the algorithm is somewhat better than the estimate in the monograph Computational Methods of Linear Algebra by D.K. Faddeev and V.N. Faddeeva; however, a separate iterative process is needed to derive it. It is shown that this iterative process has at least a linear rate of convergence and its single step requires O(n) operations. The rate of convergence can be estimated as \(\left| {\mu ({{A}_{{k + 1}}}) - \mu {\text{*}}} \right|\) < \(C\left| {\mu ({{A}_{k}}) - \mu {\text{*}}} \right|\), where C = 1 – \(\frac{{{{m}^{5}}}}{{12}}\), m is the modulus-least element of the matrix A, μ* is the limit value of the iterative process (the best estimate of the rate of convergence of the Seidel method), and μ(A_k) and μ(A_k + 1) are the estimates obtained at the kth and (k + 1)th steps of the iterative process, respectively.

In 1964, A. N. Sharkovskii published a paper in which he introduced an ordering relation on the set of positive integers. His ordering had the property that if a continuous self-map of an interval has a periodic point of some period p, then it also has periodic points of any period larger than p in this ordering. The least number in this ordering is 3. Thus, if a continuous self-map of an interval has a point of period 3, then it has points of any period. In 1975, this result was rediscovered by Lie and Yorke, who published it in their paper “Period three implies chaos.” Their work has led to the international recognition of Sharkovskii’s theorem. Since then, numerous papers on properties of self-maps of an interval have appeared. In 1994, even a conference named “Thirty Years after Sharkovskii’s Theorem: New Perspectives” was held. One of the research directions is estimating the number of periodic trajectories which a map satisfying the conditions of Sharkovskii’s theorem must have. In 1985, Bau-Sen Du published a paper in which he obtained an exact lower bound for the number of periodic trajectories of given period. In the present paper, a new, significantly shorter and more natural, proof of this result is given.

A multidimensional optimization problem is formulated and solved in terms of tropical mathematics that is concerned with the theory and applications of semi-rings with idempotent addition. The problem, whose objective function is defined by a matrix, is proposed to be solved via idempotent algebra and tropical optimization tools. A strict lower bound is first derived for the objective function, used for solving the problem, to allow the evaluation of its minimum value. The objective function and its minimum value are then combined into an equation whose complete solution is obtained in the form of all eigenvectors of the matrix. A practical application of the problem is considered using the example of an explicit solution for the optimal scheduling of a project that consists of a set of activities defined by constraints on their start and end times. The optimality criterion for scheduling is defined to minimize the maximum, over all activities, of the working cycle time, which is described as the time interval between the start and the end of the activity. The analytical result extends and supplements the existing algorithmic numerical solutions to optimal scheduling problems. As an illustrative example, the solution of a problem to schedule a project consisting of three activities is presented to illustrate the result.

The effect of the ellipticity of a race of an automatic ball balancer (ABB) on the dynamics of a statically unbalanced symmetrically fixed rotor is discussed in this paper. The analysis of stationary and non-stationary movement modes of the system showed that the shape of ABB's race deviating from an ideal circle can lead to loss of the functionality of the ABB.

Geometric properties of spaces of Keplerian orbits are of interest for celestial mechanics problems related to the search for groups of celestial bodies with close orbits. Those groups include asteroid families and meteor streams. Studying these groups provides important information on the evolution of the Solar System, as well as on the characteristics of objects within a family and their parent bodies. The local properties of a distance function between orbits are of primary importance for the problems of the search for families of related celestial bodies, because orbits of family members cluster together in a small region of the orbit space. Several metrics on the set of Keplerian orbits \(\mathbb{H}\) and its quotient sets are considered in this paper. For each of these metrics we solve the question: is there a normed vector space that is locally isometric to the orbit metric space? In two of the considered cases, the answer turns out to be positive: the quotient space of \(\mathbb{H}\) by the equivalence relation neglecting the magnitude of the pericenter argument of the orbit can be isometrically embedded into \({{\mathbb{R}}^{4}}\). The embedding into \({{\mathbb{R}}^{3}}\) also exists for the quotient space by the pair of elements: the longitude of ascending node and the pericenter argument. It is shown in this paper that for other metrics, the answer to the stated question is negative. The possibility of an isometric embedding of the orbit space or its part into Euclidean space is useful in application to the aforementioned problems of celestial mechanics. The isometric map helps to define the mean of the orbit family in a natural way: the arithmetic mean of images corresponds to the orbit with the minimum square deviation of distances from the orbits of the family.