Vol 49, No 2 (2016)

This work is the first in a series of papers devoted to classifying of two-dimensional homogeneous cubic systems based on partitioning into classes of linear equivalence. Principles have been developed that are capable of constructively distinguishing the structure of a simplest system in each class and a canonical set that defines the admissible values that can be assumed by its coefficients. The polynomial vector in the right-hand part of this system identified with a 2 × 4 matrix is called the canonical form (CF) and the system itself is called the normal cubic form. One of the main objectives of this series of papers is to maximally simplify the reduction of a system with a homogeneous cubic polynomial in the unperturbed part to the various structures of a generalized normal form (GNF). Generalized normal form refers to a system in which the perturbed part has the simplest form in some sense. The constructive implementation of the normalization process depends on the ability to explicitly specify the conditions of compatibility and possible solutions of the so-called bonding system, which is understood to be a countable set of linear algebraic equations that specify the normalizing transformations of the perturbed system. The above principles are based on the idea of the maximum possible simplification of the bonding system. This will allow one to first reduce the initial perturbed system by an invertible linear substitution of variables to a system with some CF in the unperturbed part, then reduce the resulting system, which is optimal for normalization, by almost identical substitutions to various structures of the GNF. In this paper, the following tasks are carried out: (1) the general problem is set, close problems are formulated, and the available results are described; (2) a bonding system is derived that is capable of determining the equivalence of any two perturbed systems with the same homogeneous cubic part, the possibilities of its simplification are discussed, the GNF is defined, and the method of resonant equations is given allowing one to constructively obtain all its structures; (3) special forms of recording homogeneous cubic systems in the presence of a common homogeneous factor in their right-hand parts with a degree of 1–3 are introduced, and the linear equivalence of these systems, as well as of systems without a common factor is studied, and key linear invariants are offered.

A problem concerning the dimensions of the intersections of a subspace in the direct sum of a finite number of the finite-dimensional vector spaces with the pairwise sums of direct summands, provided that the subspace intersection with these direct summands is zero has been discussed. The problem is naturally divided into two ones, namely, the existence and representability of the corresponding matroid. Necessary and sufficient conditions of the existence of a matroid with the specified ranks of some subsets of the base set have been given. Using these conditions, necessary conditions of the existence of a matroid with a base set composed of a finite series of pairwise disjoint sets of the full rank and the given ranks of their pairwise unions have been presented. A simple graphical representation of the latter conditions has also been considered. These conditions are also necessary for the subspace to exist. At the end of the paper, a conjecture that these conditions are sufficient has also been stated.

The bilinear Chapman–Kolmogorov equation determines the dynamical behavior of Markov processes. The task to solve it directly (i.e., without linearizations) was posed by Bernstein in 1932 and was partially solved by Sarmanov in 1961 (solutions are represented by bilinear series). In 2007–2010, the author found several special solutions (represented both by Sarmanov-type series and by integrals) under the assumption that the state space of the Markov process is one-dimensional. In the presented paper, three special solutions have been found (in the integral form) for the multidimensional- state Markov process. Results have been illustrated using five examples, including an example that shows that the original equation has solutions without a probabilistic interpretation.

An algorithm for the numerical solution of a system of two parabolic equations of a special form has been presented, and its stability has been proved. Estimates of the numerical solutions obtained have been made. Conditions guaranteeing the applicability of the sweep method are determined, and the time step at which the iterative method has the inner convergence has been estimated. An example of the application of the algorithm in soil science has been presented.

The Foster–Stuart test is often used in the analysis of time series to determine trends. The test is based on calculating the lower and upper time series records x₁, …, x_n. Unlike other tests of randomness, tests based on records are not invariant under a reversal of the direction of the time variable. To construct invariant round-trip tests, it is necessary to count the records in both forward and backward time variables. Thus far, the construction of this test has been impossible because the variances of test statistics were not known when the null hypothesis was true. Variances for Foster–Stuart round-trip test statistics D and S have been found in the present paper. The test was designed to identify positive and negative trends in means and variances of x₁, …, x_n time series. Dispersions of S and D test statistics have been found under the H₀ randomness null hypothesis. Asymptotic approximations were obtained for dispersions. Thus, it has turned out to be possible to construct a full-fledged invariant test for two-sided alternatives. The Foster–Stuart round-trip test application example has been reviewed.

Buckling analysis of a thin cylindrical shell stiffened by rings with T-shaped cross section under the action of uniform internal pressure in the shell is performed. An annular plate stiffened over the outer edge by a circular beam is used as the ring model. The classical ring model, which is a beam with a T-shaped cross section, is inappropriate in this problem, since in the case of the loss of stability, buckling deformations are localized on the ring surface. The beam model does not allow one to find the critical pressure that corresponds to such a loss of stability. In the first approximation, the problem of the loss of stability of the annular plate connected with the shell is reduced to solving the boundary value problem for finding eigenvalues of the annular plate bending equation. Approximate formulas for determining critical pressure are obtained under the assumption that the plate width is much smaller than its inner radius. The results found using the Rayleigh method and the shooting method differ slightly from each other. It has been demonstrated that the critical pressure for rings with rectangular cross section is higher than that for rings with a T-shaped cross section.

In their previous papers, the authors have considered the possibility of applying the theory of motion for nonholonomic systems with high-order constraints to solving one of the main problems of the control theory. This is a problem of transporting a mechanical system with a finite number of degrees of freedom from a given phase state to another given phase state during a fixed time. It was shown that, when solving such a problem using the Pontryagin maximum principle with minimization of the integral of the control force squared, a nonholonomic high-order constraint is realized continuously during the motion of the system. However, in this case, one can also apply a generalized Gauss principle, which is commonly used in the motion of nonholonomic systems with high-order constraints. It is essential that the latter principle makes it possible to find the control as a polynomial, while the use of the Pontryagin maximum principle yields the control containing harmonics with natural frequencies of the system. The latter fact determines increasing the amplitude of oscillation of the system if the time of motion is long. Besides this, a generalized Gauss principle allows us to formulate and solve extended boundary problems in which along with the conditions for generalized coordinates and velocities at the beginning and at the end of motion, the values of any-order derivatives of the coordinates are introduced at the same time instants. This makes it possible to find the control without jumps at the beginning and at the end of motion. The theory presented has been demonstrated when solving the problem of the control of horizontal motion of a trolley with pendulums. A similar problem can be considered as a model, since when the parameters are chosen correspondingly it becomes equivalent to the problem of suppression of oscillations of a given elastic body some cross-section of which should move by a given distance in a fixed time. The equivalence of these problems significantly widens the range of possible applications of the problem of a trolley with pendulums. The previous solution of the problem has been reduced to the selection of a horizontal force that is a solution to the formulated problem. In the present paper, it is offered to seek an acceleration of a trolley with which it moves by a given distance in a fixed time, as a time function but not a force applied to the trolley, while the velocities and accelerations are equal to zero at the beginning and end of motion. In this new problem, the rotation angles of pendulums are the principal coordinates. This makes it possible to find a sought acceleration of a trolley on the basis of a generalized Gauss principle according to the technique developed before. Knowing the motion of a trolley and pendulums it is easy to determine the required control force. The results of numerical calculations are presented.