# No 1 (2013)

**Year:**2013**Articles:**29**URL:**https://journals.rudn.ru/miph/issue/view/507

### A Study of the Integrability of the Derivatives of the Solutions of the Conjugate (nonlinear) Beltrami Equation

#### Abstract

In this article we study degenerate elliptic equations. Using integral representations for the function posessing generalised derivatives we prove theorems of existence of solutions of these equations in the class of quasiconformal mappings in the mean, and in the more general case. We prove higher integrability of the solutions of these equations in the case of the boundary degeneration.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):5-18

5-18

### Optimal Recovery of Functions from Inaccurate Data on the Radon Transform for Classes Deﬁned by the Degree of the Laplacian

#### Abstract

We consider the problem of optimal recovery of function in Schwartz space from inaccurate data (in the mean square metric) on it’s Radon transform. We present explicit expressions for the error of optimal recovery and a set of optimal methods. As a consequence we prove one inequality for functions in Schwartz space.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):19-25

19-25

### Suﬃcient Conditions of Existence of Unique Periodic Solutions for Second Order One-Dimension Equations

#### Abstract

In this work suﬃcient conditions of existence of unique periodic solutions for second order one-dimension equations are derived. The goal of the paper is not only in getting such results, but also in demonstration of new approach, which may be applied to more general class of diﬀerential equations, i.e. it may be considered not only second order, but also grater order diﬀerential equations.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):26-36

26-36

### An Analytical Model of Load Distribution Schemes in LTE Heterogeneous Networks

#### Abstract

The resource allocation problem in Orthogonal Frequency Division Multiple Access relay-enhanced heterogeneous cellular networks is studied. An analytical model of the downlink channel in discrete time is suggested. We derive and analyze various resource allocation algorithms. In order to evaluate the role of diﬀerent resource allocation schemes we obtain blocking probabilities and other performance metrics of interest.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):37-44

37-44

### Mathematical Framework for the Performance Evaluation of an All-Optical Packet Switch with FDL´s Utilization

#### Abstract

Nowadays all-optical network is the most perspective. Traﬃc transmits in optical networks by commutation technology, one of which is packets commutation. WDM and full wavelength conversion service for eﬀective using of optical ﬁber bandwidth. A vital problem in optical networks is collision initiation, when ever two or more packets are switched on the same output wavelength, at the same time. FDLs are used for solving this problem. They allow to delay packets for deﬁnite time, preventing packets dropping and reducing packet blocking when all output wavelengths are engaged. In this paper we propose a mathematical framework for the performance evaluation of an all-optical packet switch, including ﬁber delay lines and wavelength reservation. WDM and full wavelength conversion are also used. We present the formulas for the steady-state blocking probabilities and the PBP calculation in one destination ﬁber. Numerical analysis of these characteristics is also provided.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):45-51

45-51

52-57

### Spectral Approaches of Gaussian Conditional Simulations in History Matching Problems

#### Abstract

In the paper we consider optimization problem arised in history matching of reservoir model. In such type of problems the unknown parameter often is a distributed ﬁeld of the physical quantity such as permeability and porosity ﬁelds. The way the parameter ﬁeld is parametrized greatly inﬂuences eﬃciency of the complete optimization approach. We propose an eﬃcient technique of a spectral-domain parameterization based on the Cholesky decomposition of the covariance matrix in Fourier domain. The approach signiﬁcantly reduce the number of simulation runs. A comparative analysis of history matching of the proposed algorithm and standard spectral method is performed using PUNQ-S3 test model.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):58-66

58-66

### The Equations of Population Dynamics in the Form of Stochastic Diﬀerential Equations

#### Abstract

In this paper we consider the equation of Fokkera-Planck for models of population dynamics and the method to obtain stochastic diﬀerential equation written as Langevin equations. And we received the Fokker-Planck equation for the model “predator-prey” and its variants and for models of symbiosis and competition.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):67-76

67-76

### Speciﬁc Implementations of Symplectic Numerical Methods

#### Abstract

The paper illustrates the use of the tensor notation for writing symplectic numerical schemes. Symplectic conditions are given for the partitioned Runge–Kutta and Runge–Kutta–Nystr̈om methods. The speciﬁc implementations of symplectic numerical methods are reviewed.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):77-89

77-89

### High-Order Vector Nodal Finite Elements with Harmonic, Irrotational and Solenoidal Basis Functions

#### Abstract

In the present paper a concept of vector nodal ﬁnite element has been introduced, algorithms of construction of the vector nodal basis functions with high approximate properties from special functional spaces are presented. Examples of high-order interpolation of harmonic, irrotational vector ﬁelds by the developed ﬁnite elements illustrate their approximate advantage in comparison with the standard Lagrange elements.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):90-98

90-98

### Numerical Investigation of Generalized Nested Factorization Preconditioner for Reservoir Simulation Problems

#### Abstract

In this paper we present some results on application of Generalized Nested Factorization (GNF) preconditioner for real-life reservoir simulation problems. The test problems considered in the paper share some important features with reservoir models arise in real-life practice, e.g., a presence of non-local cell connections and unstructured computational grids. Numerical results are compared with the ones obtained using standard ILUT preconditioning techniques. Spectral properties of preconditioned matrices are analyzed. As a result very good performance of GNF preconditioner is observed for real-life reservoir simulation problems.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):99-110

99-110

### Numerical Simulation of Freak Waves in the Ocean

#### Abstract

Modern approaches to studying of freak waves at ocean by means of computing experiments are consi-dered. The description of computing experimental installation is given, used numerical methods are described, and the review of the received results is given too.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):111-119

111-119

### Analysis of Electrical Turbulence Characteristics in Thunderstorm Clouds

#### Abstract

It has been performed the numerical analysis of structure functions for an electric turbulence in thunderstorm clouds by usage of experimental data on altitude proﬁle of the electric ﬁeld vertical component. Numerical calculations of the structure functions were performed and inertial intervals of electric turbulence were detected in the small scales range and the middle scale one. Scaling exponents, Herst index, the curtosis were determined. The structure functions behaviour has explained by the presence of intermittency and coherent structures which inﬂuence on scaling exponents magnitudes. It has been shown that for data considered the generalized scale invariability (GSI) of electric turbulence is observing and GSI scaling exponent has been calculated. Results obtained are of the great interest for following investigations of intense atmospheric vortices charged subsystems contribution to the hydrodynamical helicity generation and vortices dynamics including tropical cyclones formation.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):120-128

120-128

### Theoretical Analysis of Gaussian Beam Diﬀraction on the Phase Step Structure

#### Abstract

The paper analyzes the interaction of a Gaussian beam with a phase step structure (PSS). The shape of the spatial spectrum at the output of such a scheme is examined. The most detailed description was done for the case when the size of the PSS phase diﬀerence is 180 degrees. The eﬀects that occur at the displacement of SPS in the plane perpendicular to the direction of incidence of the laser beam were considered. Discussions for the possible practical application of the scheme were carried out.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):129-140

129-140

### Estimation of Losses in Optical Film Waveguides with Rough Surfaces and Absorption

#### Abstract

The method of small indignations with use of the optics-geometrical approach solves a problem of light propagation in a ﬁlm waveguide with rough surfaces and absorption. Experimental researches speciﬁed waveguide structures are carried out. Results of these researches substantially do not correspond to conclusions of the developed theory. As a result of the analysis of the revealed discrepancies and additional calculations more exact model of scattering process in ﬁlm waveguides with rough surfaces is oﬀered.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):141-147

141-147

### Analysis of Planar Graded-Index Optical Waveguides with Strong Asymmetry of a Refractive Index Proﬁle by the Beam Propagation Method

#### Abstract

In the paper application of the beam propagation method (BPM) to investigation of wave processes in regular and nonregular graded-index waveguide structures with strongly asymmetric refractive index proﬁle is analyzed. On purpose to match the BPM requirement to smoothness of the refractive index proﬁle, the proﬁle with strong asymmetry, at the problem solution, is approximately exchanged by a symmetrical one. For exponential and Gaussian proﬁles of the refractive index the BPM calculations are compared to the strict and WKB solution respectively. Calculation of a wave pattern of a mode ﬁeld coupling into the substrate at the tapered edge of a waveguide with strongly asymmetric proﬁle is demonstrated.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):148-154

148-154

### Propagation of Nonlinear Electromagnetic Wave Across Strong Magnetic Field in Plasma

#### Abstract

The propagation of the large amplitude electromagnetic wave in a plasma across the external magnetic ﬁeld is considered. In this interaction electrons are accelerated by the longitudinal electric ﬁeld up to relativistic energies. The system develops into the strongly nonlinear regime with increasing the wave amplitude. It is shown that in the strong magnetic ﬁeld near plasma or cyclotron resonance the envelope soliton solution exists. This solution is qualitatively similar to the Langmuir soliton in a plasma without magnetic ﬁeld.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):155-163

155-163

### On the Relativistic Theory of Motion of the Charged Particle at the Resonant Conditions

#### Abstract

General relativistic theory of motion of the charged particles in the electromagnetic waves at the resonant conditions in the drift-eikonal approximation is presented. The motion of the particle in the vicinity of cyclotron, parametric and half-cyclotron resonances is considered.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):164-180

164-180

### The Anisotropy Properties of a Background Radiation in the Fractal Cosmological Model

#### Abstract

We consider the anisotropy properties of a background radiation in the fractal cosmological model. The space of this model includes self-similar domains. The metric tensors of any two domains are connected by the discrete scaling transformation. Photons of the background radiation cross the domain and their energies change. Any observer receives these photons from diﬀerent domains and detects spots with diﬀerent brightness. The power spectrum of the brightness anisotropy of the background radiation in the fractal cosmological model is calculated. It is shown this spectrum is closed to the observed angular power spectrum of the SDSS-quasar distribution on the celestial sphere. Only qualitatively it conforms to the angular power spectrum of CMB (WMAP-7).

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):181-188

181-188

### Fractal Properties of the Universe

#### Abstract

The large-scale structure of the Universe is revealed to be characterized by a range of power-laws. The power-laws are evidences of fractality because they may be interpreted through a conception of the Universe as an assembly of self-similar space–time domains. We accept the hypothesis that the matter of the Universe is described by the scalar charged meson ﬁeld possessing the rotary symmetry. On basis of the hypothesis, the fractal cosmological model with scale invariant Lagrange’s ﬁeld equation and Einstein’s equation permitting physical explanation of these properties is constructed. The ﬁeld energy densities (which are constant) and the space–time metrics of diﬀerent domains diﬀer in constant factors only. Therefore, the space–time domains are geometrically similar and evolve similarly. Fractal properties of initial cosmological density perturbations remain and lead to presence of the fractal properties of the Universe’s large-scale structure which formed from them. The nonsingular, compacted, pulsating and doubly-connected cosmological model as a partial solution for the homogeneous, isotropic and ﬂat case is constructed. A background radiation power spectrum has been computed. The spectrum is shown to be close to the observable angular power spectrum of the SDSS-quasar distribution on the celestial sphere.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):189-201

189-201

### Tesla Energy Space for Mie–Schwinger Continuous Electron

#### Abstract

Tesla reading of electricity through invisible energy ether between visible bodies corresponds to the found radial solution for a continuous source. The Mie–Schwinger distributed electron extends over the very structure of its Coulomb radial ﬁeld. The electric charge is not a basic concept of Maxwell-Tesla electrodynamics but is the ﬁeld energy distribution under the uniﬁed, non-dual approach to matter-energy in the nonempty world space. Electric self-energy of such a uniﬁed nonlocal carrier is ﬁnite despite the latter ﬁlls the inﬁnite Universe with Tesla material ether everywhere (without empty space regions). Maxwell’s equations can describe both local balances of electric self-energy currents and nonlocal Tesla resonances within the global world overlap of moving continuous carriers of energy. Material Tesla space for overlapping electric energy sources in Maxwell’s equations calls for radial mass-energy sources in the Einstein equation.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):202-211

202-211

### Determining the Trajectory of a Gyroscopically Stabilized Projectile in Air by Using Dual-Projective Variables

#### Abstract

Early developed method of integrating the ballistic equations in dual-projective variables was applied for the gyroscopically stabilized projectile under action of relatively weak lift force L = γV 2 ≈ 0.1mg and the force of air drag R = αV 2 ≫ mg, both their coeﬃcients α() and γ() being strongly dependent on angle of attack . Obtained are both exact and approximate expressions for the resolventa function f(b) = a bb′′(b) , with the a(b) and b = tg being an intercept and slope of trajectory respectively.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):212-223

212-223

224-228

### Fundamental Principles of Theoretical Physics and Concepts of Quasiaverages, Quantum Protectorate and Emergence

#### Abstract

In the present paper we discuss the interrelation of the advanced interdisciplinary concepts of modern physics such as symmetry breaking, quantum protectorate, emergence and the Bogoliubov’s concept of quasiaverages in the context of modern theoretical physics, and, in particular, quantum and statistical physics. The main aim of this analysis was to demonstrate the connection and interrelation of these conceptual advances of the many-particle physics and to try to show explicitly that those concepts, though diﬀerent in details, have certain common features. Some problems in the ﬁeld of statistical physics of complex materials and systems e.g. foundation of the microscopic theory of magnetism and superconductivity were pointed in relation to these ideas. The main suggestion is that the emphasis of symmetry breaking concept is on the symmetry itself, whereas the method of quasiaverages emphasizes the degeneracy of a system. The concept of quantum protectorate reveals essential diﬀerence in the behavior of the complex many-body systems at the low-energy and high-energy scales. Thus the notion of quantum protectorate might provide distinctive signatures and good criteria for a hierarchy of energy scales and the appropriate emergent behavior.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):229-244

229-244

### Degenerate 4-Dimensional Matrices with Semi-Group Structure and Polarization Optics

#### Abstract

In polarization optics, an important role play Mueller matrices — real four-dimensional matrices which describe the eﬀect of action of optical elements on the polarization state of the light, described by 4-dimensional Stokes vectors. An important issue is to classify possible classes of the Mueller matrices. In particular, of special interest are degenerate Mueller matrices with vanishing determinants. With the use of a special technique of parameterizing arbitrary 4-dimensional matrices in Dirac basis, a classiﬁcation of degenerate 4-dimensional real matrices of rank 1, 2, 3. is elaborated. To separate possible classes of degenerate matrices we impose linear restrictions on 16 parameters of 4 × 4 matrices which are compatible with the group multiplication law.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):245-259

245-259

### End of the General Relativity

#### Abstract

The Einstein–Cartan theory as a natural generalization of the General Relativity is proposed. The cosmological term is connected with string additions for de Sitter world having Planck density before the Big Bang. Is proposed that this world was having the Lemaitre atom form with mass of modern Methagalaxy mass and the diameter 10 −13 cm. We propose that the Big Ban is the result of the transforming of the topological energetical string modes into oscillations one. The hypothesis of Fridmon particles as particles of dark matter with masses near one billion GeV and corresponding to dualized weak interaction symmetry group is proposed.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):260-273

260-273

### On Nonstationary Solutions to Yang–Mills Equations

#### Abstract

We study Yang–Mills ﬁelds with SU(2) symmetry generated by classical ﬁeld sources. It is shown that in this case the Yang–Mills equations can be regarded as a reasonable nonlinear generalization of the equations of Maxwell’s electrodynamics. We seek new classes of solutions to the examined Yang–Mills equations and ﬁnd their nontrivial solutions in the case of nonstationary spherically symmetric sources and a wide class of their non-Abelian wave solutions.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):274-283

274-283

284-287

### Educational Trajectories for Students in Mathematical Specialties

#### Abstract

The paper discusses the role of mathematics in the development of scientiﬁc research, the possibility of crafting an individualized educational trajectory for students in the higher education as a mean of creating of the mathematical competence.

**Discrete and Continuous Models and Applied Computational Science**. 2013;(1):288-289

288-289