# No 4 (2014)

**Year:**2014**Articles:**12**URL:**https://journals.rudn.ru/miph/issue/view/519

## Articles

### Generalized Frames and Riesz Systems

#### Abstract

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.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(4):5-11

5-11

### On Radio Resource Allocation Scheme Model with Fixed Transmission Zone for Heterogeneous M2M Traffic in LTE Network

#### Abstract

Today human is surrounded by many technological devices (sensors, smartmeters, etc.) that become connected and will reshape the Internet as we know it today. These devices can transmit and receive data through wireless interfaces transmitting data independently and automatically, thereby generating M2M (Machine-to-Machine) traffic. Efficient service of M2M traffic remains a challenge for future mobile networks. Such massive connectivity offers novel attractive services and provides additional income for operators, but also raises significant challenges to manage large number of devices, typically transmitting only small data fragments. This is especially true for LTE (Long Term Evolution), which has been historically optimized for H2H users (Human-to-Human). Consequently, it is required to develop new methods for M2M traffic for each phase - from the connection establishment phase to the data transmission phase. The article proposes a scheme of dynamic radio resource allocation of LTE cell with fixed transmission zone for heterogeneous M2M traffic. The stationary probability distribution is obtained and the numerical analysis is performed.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(4):12-22

12-22

### Numerical Simulation of Supersonic Plane Gas Dynamics Problems on a Triangular Grid

#### Abstract

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.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(4):23-32

23-32

### Calculation Schemes for Solving Sturm- Liouville Problem by Finite-Element Method with Interpolating Hermite Polynomials

#### Abstract

Calculation schemes for solving Sturm-Liouville problem with first-, second-and third-type boundary conditions by finite-element method holding a continuity of derivatives of a required solution in its approximated solution are constructed. Recurrence relations for the calculation in analytical form of the interpolating Hermite polynomials with nodes of arbitrary multiplicity are derived. Using the interpolating Hermite polynomials, the basis piecewise-polynomial functions on finite-element grid with nonuniform step, approximating desired solution of the original problem are constructed and used for reduction to a generalized algebraic eigenvalue problem with banded stiffness and mass matrices. The stiffness and mass matrices are formed by sums of integrals containing the given coefficient and potential functions of the original self-adjoint second-order differential equation and the calculated interpolating Hermite polynomials and their derivatives on the finite element grid. The integrals are calculated using Gauss quadratures and in special cases, including the piecewise continuous polynomial coefficient and potential functions in analytical form. The efficiency and rate of convergence of the proposed calculation schemes and elaborated algorithms and programs implemented in Maple and Fortran is proved by benchmark calculations of exactly solvable Sturm-Liouville problems with continuous and piecewise continuous potential functions.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(4):33-49

33-49

### Vectorization and Parallelization of Algorithms for Selection and Reconstruction of J/ψ → e +e - Decaysin Real Time of the CBM Experiment

#### Abstract

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.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(4):50-67

50-67

### Gaze Tracking Acceleration using CUDA Technology

#### Abstract

Low-cost gaze tracking systems are in great demand due to their wide range of application. Commonly, extra devices are needed (for instance, head mounted cameras); however, in this investigation gaze tracking is performed in real-time based on the video stream from an infrared video camera. A comparative analysis of the existing analogues was executed and the main features of gaze tracking systems were highlighted and prioritized. These features are price, tracking accuracy, angle error, flexibility, and usability. A methodology was developed which allows to calculate a gaze direction vector according to the relative position of eye center and corneal reflection from an infrared diode. The centers of an eye and reflection are estimated using the vector field of image gradients and additional weighting. CUDA technology is used to accelerate the developed algorithms. The main advantage of the developed algorithm is the ability to detect and continuously track pupils’ centers, regardless of the head position, which significantly extends the scope of the gaze tracking system under consideration.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(4):68-84

68-84

### Program Constraints and Ensuring Stability of Movement of the Electromechanical Manipulator

#### Abstract

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.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(4):85-94

85-94

### Trajectory Tracking Control of Programmed Motion in Second Order Nonholonomic Systems

#### Abstract

The D’Alembert-Lagrange principle in general stands for all ideal holonomic and nonholonomic constraints of arbitrary order. But in practice the application of the principle is restricted to ideal holonomic and linear first order nonholonomic constraints. In recent years the direct application of this famous principle is made to model dynamic equation of acceleration level constrained systems. This paper uses the dynamic equation developed to establish a theoretical framework for trajectory tracking control of programmed motion with acceleration level constraints. The concept of dividing constraints based on their sources into natural and programmed constraints is employed. The trajectory tracking control is accomplished by two models called Reference Control Model constructed using both the programmed and natural constraints and a Dynamic Control Model developed by considering the natural constraints only. The Reference control model is used to plan the required trajectory based on a given acceleration or lower level programmed constraint. The Dynamic Control Model is utilized to control and stabilize the trajectory tracking process. Finally, to verify the effectiveness of the framework developed in the paper, a practical example is provided and simulation results are depicted.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(4):95-105

95-105

### Process Control of Unstressed Docking of Plurality of Moving Objects in an Ordered Time Points

#### Abstract

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.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(4):106-111

106-111

### Nonlinear Coherent Perfect Absorption

#### Abstract

We review some of the recent concepts and their realization exploiting the perfect destructive interference of light in micro and nano structures. One refers to optical structures where the effective absorption can be controlled and maximized to perfect absorption. The reported effects depend crucially on the coherent nature of the exciting radiation. Achieved with a single (two or more) incident plane wave (waves) the effect carries the name of critical coupling (coherent perfect absorption). Thus in a system supporting critical coupling (CC) or coherent perfect absorption (CPA) all the incident radiation can be absorbed leading to null scattering. In particular all the incident light energy can be channeled into a specified mode of a multimodal structure if such modes are supported by the system. We present a brief overview of CC and CPA in linear systems to recount their underlying concepts as time-reversed lasing and some of their futuristic applications. Next we review our work on the nonlinear extensions of CC and CPA where one or more of the layered media could be nonlinear with Kerr-type nonlinearity. The dispersive nonlinearity is shown to offer a practical handle over the process of perfect absorption by incident laser power. Further we show that the nonlinear periodic structure can support gap solitons which absorbs all the incident energy and do not scatter any light outside the hetero-guide.

**Discrete and Continuous Models and Applied Computational Science**. 2014;(4):112-133

112-133

### Information About Authors

**Discrete and Continuous Models and Applied Computational Science**. 2014;(4):134

134

### Guidelines for Authors

**Discrete and Continuous Models and Applied Computational Science**. 2014;(4):135-136

135-136