No 4 (2014)

In this paper a generalization of frame systems is made. First description of systems of this type was made by T. P. Lukashenko. In 1997 he introduced a class of generalized similar to orthogonal systems, and in 2006 proposed an idea to expand of frame-based systems on the generalized space. This question is considered in this paper. Firstly, the paper gives the description of well-studied, as for now, discrete and integral frames, as well as describes the main practical applications of such frame systems. The paper considers generalized systems, similar to orthogonal, introduced by T. P. Lukashenko, and these systems are extended to generalized frames. Given examples indicate that an input class is more inclusive than previously considered discrete and integral frames, and more general than the generalized orthogonal system (examples are Fourier transformation and the Hilbert transformation). The concept of generalized Riesz system is introduced and the relationship between frames and Riesz systems in a generalized way is studied. Two theorems are proved in the work to establish close links between the introduced generalized frames and generalized Riesz systems. The theorem give the necessary and sufficient criteria for the system to be a generalized frame. Parseval’s identity analog is deduced for generalized frame systems.

This study focuses on modification of the Davydov’s method (large particles) in case of the triangular grid. Numerical approach to the solution of two-dimensional equations of non-viscous perfect gas flow (flat case) using triangular grids is developed. The only class of triangular cells is used in this method, instead of the two classes of cells of differential grid (fractional cell directly beside the body and regular cells in other cases) that are used in classical method of large particles, which simplifies the logic of computations. Vector notation is used to write equations of the method instead of matrix notation in the case of a regular grid. Due to the usage of triangular mesh formulas of all three stages of the method are considerably changed, while the ideology of the method remains the same: splitting of the initial equations on physical factors. Triangular mesh, except for the undoubted advantages associated with the construction of a body of complex shape, introduces additional complexity in the numerical calculations: the generation of grid itself (triangulation); neighboring triangles are not necessarily have adjacent indexes; calculation time increases for the movable body due to rebuild of the grid; additional memory for storing the geometry of the computational domain. Also in this paper the comparison of numerical solutions of the perfect nonviscous gas flows on an irregular grid using different methods is carried out. The comparison of numerical results obtained by the method of large particles in the case of a triangular mesh and for the case of a regular grid is carried out. The comparison of numerical results with the approximate analytic ones is carried out.

The measurements of J/ψ decays is one of the key goals of the CBM experiment. The technique of J/ψ registration in its dielectron channel has been developed, which includes a chain of methods of trajectories and momentum reconstruction of charged particles with STS, electron/positron identification with RICH, TRD and TOF, as well as construction of the J/ψ-candidates and their characteristics using the KFParticle package. Taking into account that selection and reconstruction decays of J/ψ → e +e - are planned to be carried out in real time of the CBM experiment, the used methods and algorithms should be not only effective but also fast. In this paper the time-consuming estimation of the existing algorithms based on their acceleration via code vectorization by means of SIMD instructions and parallelization between the processor cores that are implemented using OpenMP, OpenCL software environments and TBB library has been carried out. This analysis allowed to establish weak points in this chain, which are under investigation, as well as offer a fast and efficient parallel algorithm for the identification of the charged particles with TRD based on the ωn k criterion.

For the theoretical study of the dynamics of manipulation robots, define design parameters and control laws, you must have a current mechanical models that accurately describe the properties of real robots. The choice of the computational model in each case is determined by the kinematic scheme of the manipulator, mechanical properties (inertial, elastic, dissipative, and the like) parts and assemblies, type and characteristics of the drives, as well as the required accuracy of the calculation. The objective of the control is to ensure the motion of the mechanical system under some requirements that make up its program. Program motion of the system can be performed by the application to the system of control of forces, the system settings change in the process, building of special control devices (controllers) or a combination of these. The original objectives of the control theory are inverse problems of classical dynamics. From the mathematical point of view, calculation model manipulation robot is a system of differential equations. This model may include equations describing the phenomena non-mechanical nature, for example, electrical processes in the circuits of the motors of the actuators. In this article the author examines the issues of ensuring conditions of the asymptotic stability software movement mechanical and electromechanical systems with holonomic and nonholonomic constraints. For example, the three-tier model controllable electromechanical manipulator conditions of the asymptotic stability of a given movement. The described approaches to ensuring the asymptotic stability of electromechanical systems can be used in the study of stability of motion proprietary mechanical systems, mechanics of controlled motion in the solution of management tasks manipulators, transport and space systems.

An algorithm is constructed for controlling a plurality of moving objects, pursuing unpredictably moving body in space, with the aim of unstressed docking with it in an ordered time points. The pursuing objects move on the principle of proportional navigation. To solve the problem, we use the equation of the relative motion, in which there are random forces, both active and inertia. These unknown disturbances we consider continuous and bounded. We introduce a control force, which is the sum of continuous and piecewise-constant functions. The piecewise-constant function is of a variable sign, large enough to offset the presence of perturbations. As a result, the process becomes a “quasi-similar” with the process of the unstressed docking of pursuing objects and the goal in ideal conditions. The self-adjusting method is proposed to automatically select the optimal values of the control. It is carried out by the “principle of feedback on the quasi-acceleration” at discrete points in time. This principle was first proposed by I.A.Mukhametzyanov in an article published in the Bulletin of Peoples’ Friendship University, Series “Mathematics. Information Sciences. Physics” No 3 for 2013. The control system of the pursuing objects automatically selects the piecewise-constant control force for unstressed docking, depending on the parameters of their convergence. It uses information about the distance between the centers of mass of the persecuted and persecuting objects. The control system calculates the second time derivative of this distance. Docking of pursuing objects and the goal shall be held alternately at predetermined intervals of time. Solution of the problem is obtained in the cases of pursuing objects of permanent and variable masses. In the second case, when the motion of controlled objects is carried out by reactive forces, the value of masses spent in the process of the control is estimated. In contrast to previous studies by the authors, such an assessment is made of fuel consumption not only for continuous, but also for the piecewise-constant control.