Vol 51, No 4 (2018)

One of the models of discrete analog of the Rényi problem known as the “parking problem” has been considered. Let n and i be integers, n ≥ 0, and 0 ≤ i ≤ n–1. Open interval (i, i + 1), where i is a random variable taking values 0, 1, 2, …, and n–1 for all n ≥ 2 with equal probability, is placed on interval [0, n]. If n < 2, we say that the interval cannot be placed. After placing the first interval, two free intervals [0, i] and [i + 1, n] are formed, which are filled with intervals of unit length according to the same rule, independently of each other, etc. When the filling of [0, n] with unit intervals is completed, the distance between any two neighboring intervals does not exceed 1. Let X_n be the number of placed intervals. This paper analyzes the asymptotic behavior of moments of random variable X_n. Unlike the classical case, exact expressions for the first moments can be found.

This is the third paper in a series of reviews devoted to the scientific achievements of the Leningrad–St. Petersburg School of Probability and Statistics in 1947–2017. The paper deals with the studies on functionals of random processes, some problems of stochastic geometry, and problems associated with ordered systems of random variables. The first sections of the paper are devoted to the problems of calculating the distributions of various functionals of Brownian motion and consider the so-called invariance principles for Brownian local times and random walks. The second part is dedicated to limit theorems for weakly dependent random variables and local limit theorems for stochastic functionals. It provides information about the stratification method and the local invariance principle. The asymptotic behavior of the convex hulls of random samples of increasing size and limit theorems for random zonotopes are also considered. An important relation between Poisson point processes and stable distributions is explained. The final part presents extensive information on research related to ordered systems of random variables. The maxima of sequential sums, order statistics, and record values are analyzed in detail.

In 1964, A.N. Sharkovskii published an article in which he introduced a special ordering on the set of positive integers. This ordering has the property that if p ◃ q and a mapping of a closed bounded interval into itself has a point of period p, then it has a point of period q. The least number with respect to this ordering is 3. Thus, if a mapping has a point of period 3, then it has points of any periods. In 1975, the latter result was rediscovered by Li and Yorke, who published the paper “Period Three Implies Chaos.” In the present paper, an exact lower bound for the number of trajectories of a given period for a mapping of a closed bounded interval into itself having a point of period 3 is given. The key point of the reasoning consisted in solution of a combinatorial problem the answer to which is expressed in terms of Lucas numbers. As a consequence, an explicit formula for the number of necklaces of a special type is obtained. We also consider a piecewise linear unimodular mapping of [0, 1] into itself for which it is possible to find points of an arbitrary given period.

The monograph published by E.Ya. Remez in 1957 addressed numerical methods with Chebyshev approximations. Particularly, the problem of the best uniform approximation of a function that is convex on an interval with continuous piecewise linear functions with free nodes was considered. In 1975, A.M. Vershik, V.N. Malozemov, and A.B. Pevnyi developed a general approach for constructing the best piecewise polynomial approximations with free nodes. The notion of partition with equal deviations was introduced, and it was found that such partition exists and generates the best piecewise polynomial approximation. In addition, a numerical method for constructing a partition with equal deviations was proposed. This paper gives three examples to describe the general approach to solving the problem of the best piecewise linear approximation with free nodes. In the case of an arbitrary continuous function, its best piecewise linear approximation in general is not continuous. It is continuous when approximating strictly convex and strictly concave functions.

In parallel with the shadowing theory (which is now very well-developed), the theory of inverse shadowing has been advanced. The main difference between the two theories is that the shadowing property means that we can find an exact trajectory near an approximate one while inverse shadowing means that, given a family of approximate trajectories, we can find a member of this family that is close to any chosen exact trajectory. We generalize the property of inverse shadowing for group actions and prove the absence of this property for some linear actions of the Baumslag–Solitar group, which is often considered as a source of counterexamples.

The problem of rolling without sliding of a rotationally symmetric rigid body on a motionless sphere is considered. The rolling body is assumed to be subjected to forces whose resultant force is applied to the center of mass G of the body, directed to center O of the sphere, and depends only on the distance between G and O. In this case, the process of solving this problem is reduced to integrating the second-order linear differential equation with respect to the projection of the angular velocity of the body onto its axis of dynamic symmetry. Using the Kovacic algorithm, we search for Liouvillian solutions of the corresponding second-order linear differential equation. We prove that all solutions of this equation are Liouvillian in the case when the rolling rigid body is a nonhomogeneous dynamically symmetric ball. The paper is organized as follows. In the first paragraph, we briefly discuss the statement of the general problem of motion of a rotationally symmetric rigid body on a perfectly rough sphere. We prove that this problem is reduced to solving the second-order linear differential equation. In the second paragraph, we find Liouvillian solutions to this equation for the case when the rolling rigid body is a dynamically symmetric ball.