Vol 49, No 3 (2016)

In this work, we obtain new characterizations of certain probability distributions by relations with different ordered random variables. Such variables include order statistics, sequential maxima, and records. We consider relations that include not only upper, but also lower record values. The presented ordered objects are based on sequences of independent random variables with a common continuous distribution function. We also investigate equalities in the distribution of sequential maxima exposed by various random shifts. These shifts (one-sided or two-sided) have exponential distributions. Certain theorems and their corollaries present corresponding characterizations of distributions by relations of such a type. In addition, we consider exponentially shifted order statistics such that simple relations among them also characterize certain probability distributions. All of the presented results yield a set of characterizations of various distributions. For particular cases, we present the relations that characterize families of classical exponential and logistic distributions.

Abstract Dynkin algebras are studied. Such algebras form a useful instrument for discussing probabilities in a rather natural context. Abstractness means the absence of a set-theoretic structure of elements in such algebras. A large useful class of abstract algebras, separable Dynkin algebras, is introduced, and the simplest example of a nonseparable algebra is given. Separability allows us to define appropriate variants of Boolean versions of the intersection and union operations on elements. In general, such operations are defined only partially. Some properties of separable algebras are proved and used to obtain the standard intersection and union properties, including associativity and distributivity, in the case where the corresponding operations are applicable. The established facts make it possible to define Boolean subalgebras in a separable Dynkin algebra and check the coincidence of the introduced version of the definition with the usual one. Finally, the main result about the structure of separable Dynkin algebras is formulated and proved: such algebras are represented as set-theoretic unions of maximal Boolean subalgebras. After preliminary preparation, the proof reduces to the application of Zorn’s lemma by the standard scheme.

The rapidly growing field of parallel computing systems promotes the study of parallel algorithms, with the Monte Carlo method and asynchronous iterations being among the most valuable types. These algorithms have a number of advantages. There is no need for a global time in a parallel system (no need for synchronization), and all computational resources are efficiently loaded (the minimum processor idle time). The method of partial synchronization of iterations for systems of equations was proposed by the authors earlier. In this article, this method is generalized to include the case of nonlinear equations of the form x = F(x), where x is an unknown column vector of length n, and F is an operator from ℝⁿ into ℝⁿ. We consider operators that do not satisfy conditions that are sufficient for the convergence of asynchronous iterations, with simple iterations still converging. In this case, one can specify such an incidence of the operator and such properties of the parallel system that asynchronous iterations fail to converge. Partial synchronization is one of the effective ways to solve this problem. An algorithm is proposed that guarantees the convergence of asynchronous iterations and the Monte Carlo method for the above class of operators. The rate of convergence of the algorithm is estimated. The results can prove useful for solving high-dimensional problems on multiprocessor computational systems.

Three series of number-theoretic problems with explicitly defined parameters concerning systems of Diophantine dis-equations with solutions from a given domain are considered. Constraints on these parameters under which any problem of each series is NP-complete are proved. It is proved that for any m and m′ (m < m′) the consistency problem on the segment [m, m′] for a system of linear Diophantine dis-equations, each of which contains exactly three variables (even if the coefficients at these variables belong to {–1, 1}), is NP-complete. This problem admits a simple geometric interpretation of NP-completeness for the determination of whether there exists an integer-valued point inside a multidimensional cube that is not covered by given hyperplanes that cut off equal segments on three arbitrary axes and are parallel to all other axes. If in a system of linear Diophantine dis-equations each dis-equation contains exactly two variables, the problem remains NP-complete under the condition that the following inequality holds: m′–m > 2. It is also proved that if the solution to a system of linear Diophantine dis-equations, each containing exactly three variables, is sought in the domain given by a system of polynomial inequalities that contain an n-dimensional cube and are contained in an n-dimensional parallelepiped symmetric with respect to the point of origin, its consistency problem is NP-complete.

Let (j₁,..., j_n) be a permutation of the n-tuple (1, ..., n). A system of differential equations \(\dot x = {f_i}\left( {{x_{{j_i}}}} \right),i = 1, \ldots ,n\) in which each function f_i is continuous on ℝ is considered. This system is said to have the property of generation of solutions with a small period if, for any number M > 0, there exists a number ω₀ = ω₀(M) > 0 such that if 0 < ω ≤ ω₀ and h_i(t, x₁, ..., x_n) are continuous functions on ℝ × ℝⁿ ω-periodic in t that satisfy the inequalities |h_i| ≤ M the system \(\dot x = {f_i}\left( {{x_{{j_i}}}} \right),i = 1, \ldots ,n\) has an ω-periodic solution. It is shown that a system has the property of generation of solutions with a small period if and only if f_i(ℝ) = ℝ for i = 1,..., n. It is also shown that the smallness condition on the period is essential.

The iteration algorithm is used to solve systems of linear algebraic equations by the Monte-Carlo method. Each next iteration is simulated as a random vector such that its expectation coincides with the Seidel approximation of the iteration process. We deduce a system of linear equations such that mutual correlations of components of the limit vector and correlations of two iterations satisfy them. We prove that limit dispersions of the random vector of solutions of the system exist and are finite.

A 3D problem of the deformation of an elastic orthotropic spherical layer that is subjected to normal pressure applied to its outer and inner surfaces is analyzed. Asymptotic first-order approximation solutions are obtained for a slightly orthotropic layer for which the elastic moduli in the meridional and circumferential directions have similar values. The solutions that are obtained are used for analyzing the scleral shell under intraocular pressure; however, they can also be used for solving the inverse problem of analyzing the stress–strain state of a human eye during intravitreal injections. The influence that the meridional and circumferential elastic moduli have on the magnitudes of changes in the relative layer thickness and in the length of the anteroposterior eye axis due to elevated intraocular pressure is studied.

A Laplace series of spherical harmonics Y_n(θ, λ) is the most common representation of the gravitational potential for a compact body T in outer space in spherical coordinates r, θ, λ. The Chebyshev norm estimate (the maximum modulus of the function on the sphere) is known for bodies of an irregular structure:〈Y_n〉 ≤ Cn^–5/2, C = const, n ≥ 1. In this paper, an explicit expression of Y_n(θ, λ) for several model bodies is obtained. In all cases (except for one), the estimate 〈Y_n〉 holds under the exact exponent 5/2. In one case, where the body T touches the sphere that envelops it,〈Y_n〉 decreases much faster, viz.,〈Y_n〉 ≤ Cn^–5/2pⁿ, C = const, n ≥ 1. The quantity p < 1 equals the distance from the origin of coordinates to the edge of the surface T expressed in enveloping sphere radii. In the general case, the exactness of the exponent 5/2 is confirmed by examples of bodies that more or less resemble real celestial bodies [16, Fig. 6].