# No 2 (2015)

**Year:**2015**Articles:**15**URL:**https://journals.rudn.ru/miph/issue/view/505

## Articles

### Paracompactness of Extremally Disconnected Spaces

#### Abstract

In this paper we consider the ω-mappings deﬁned by semi-open sets, i.e. sets that are unions of open sets and subsets of their boundaries. This are quasicontinuous ω-mappings. (The mapping f : X → Y is called quasi-continuous if for any open set G ⊂ Y the set f-1(G) is a semi-open set). Characterization of paracompactness based on continuous ω-mappings is well known. Of interest is the question of to what extent it is possible to waive the requirement of continuity of ω-mappings in the characterization of paracompactness of topological spaces with those, or other additional properties. One of these properties is extreme disconnectness. The main goal of our work is to characterize extremely disconnected paracompact space by ω-mapping on the metric space, loosening the requirement of continuity. We have proved that extremely disconnected space X is paracompact if and only if for any open covering ω of X there exists a quasi-continuous ω-mapping on some metric space.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(2):7-10

7-10

### Index of Sobolev Problems Associated with Lie Group Action

#### Abstract

In relative elliptic theory or “Sobolev” problem as B. Yu. Sternin named it in his works one is required to construct a Fredholm elliptic theory and ﬁnd an index formula in the category of smooth pairs of manifolds (M,X), where X is a submanifold in M. From the point of view of (pseudo)diﬀerential equations the Sobolev problem deals with the comparison Du ≡ f(modX), where D is a pseudodiﬀerential operator, while the sign “ ≡” means that the left and right hand sides are equal modulo distributions supported on X. Obviously, if the dimension of the submanifold is greater than one, the comparison written above does not deﬁne a Fredholm operator, since its kernel is inﬁnite-dimensional. It turns out, that if we add to the comparison some operators B deﬁned on X, which are related by an algebraic condition (of coercitivity type) with operator D, then the obtained operator (D,B) is already Fredholm in appropriate Sobolev spaces. Remarkably, this condition can be formulated invariantly as an ellipticity condition of some operator, which is induced by the problem on the submanifold X. Hence, the ellipticity conditions of operators D and (D,B) together give us a Fredholm operator. This theorem and the corresponding index formula were proved by B.Yu. Sternin. Note that all operators appearing in this theory are pseudodiﬀerential. In particular, (D,B) is a pseudodiﬀerential operator, meanwhile, this enabled one to deﬁne its ellipticity. We have a quite diﬀerent situation, if the manifold M is endowed with an additional structure, for example, if it carries a Lie group action. In this case, (D,B) is in general no longer a pseudodiﬀerential operator and, hence, the question of its ellipticity, formally speaking, can not even be rised. However, in our work, under certain conditions, we can examine the resulting operator (D,B), deﬁne its symbol and prove its Fredholm property. Moreover, we give an index formula in this more general situation. This is the subject of this work.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(2):11-18

11-18

### On Solving Differential Kinematic Equations for Constrained Mechanical Systems

#### Abstract

This paper proposes a method for constructing the kinematic equations of the mechanical system, which imposed geometric constraints. The method is based on the consideration of kinematic constraints as particular integrals of the required system of diﬀerential equations. Runge-Kutta method is used for the numerical solution of nonlinear diﬀerential equations. The developed methods allow us to estimate the range of variation of the parameters during the numerical solution which determine conditions for stabilization with respect to constraint equations. The numerical results illustrate the dependence on the stabilization of the numerical solution is not only due to the asymptotic stability with respect to the constraint equations, but also through the use of diﬀerence schemes of higher order accuracy. To estimate the accuracy of performance of the constraint equations additional parameters are introduced that describe the change in purpose-built perturbation equations. It is shown that unstable solution, with respect to constraint equations, obtained by the Euler method can be stable by using Runge-Kutta method.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(2):19-27

19-27

### Probability Characteristic Computation Algorithm of ONUs Functioning in PON

#### Abstract

A problem in the mathematical teletraﬃc theory of passive optical networks (PON) with time division multiple access (TDMA) is to divide timing budget between optical network units to optimize the network performance. For the task we examine a segment of the passive optical network with wavelength division multiplexing (WDM) and time division multiple access technologies. An extend model of the segment is developed and an allocation problem of wavelength ﬁnite number between optical network units is resolved in the article. The optical network unit does not receive data from optical line terminal and send data to optical line terminal being in passive state. The algorithm for calculating of the probability of being each optical network unit in passive state is proposed. Finally, a problem for further research is formulated. It is the algorithm for calculation of call blocking probability in upstream traﬃc models for WDM-TDMA PON.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(2):28-32

28-32

### The Analysis of Queueing System with Two Input Flows and Stochastic Drop Mechanism

#### Abstract

The queueing system with two independent ﬂows of requests with diﬀerent types of priorities is considered. The incoming ﬂows are Poisson ﬂows with diﬀerent (non equal) rates. The service times of each type requests are independent and exponentially distributed. The priority requests at the end of its service can drop non-priority ones with probability q (renovation probability) or just leaves the system with probability p = 1 - q. For general case the two-dimensional Markov process is introduced and the system of equilibrium equations for steady-state probability distribution is presented. For special case, when drop probability q is equal to one, some probabilistic characteristics as the steady-state probability distribution of priority requests, the probability of idle period are obtained. Also the analytical expressions for some characteristics of non-priority requests, such as probability of being dropped (or serviced), waiting time distribution for non-priority requests (in terms of Laplace-Stieltjes transformation and generating function) and mean waiting time, are obtained.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(2):33-37

33-37

### Queuing System with Resource Allocation of the Random Volume

#### Abstract

The objective of this paper is the construction and analysis of the model of the wireless LTE network (Long Term Evolution) as a multiserver queuing system where losses are caused by a lack of resources required to service requests. Adopted by the service application takes a random amount of resources given to several types of distribution functions. Random vectors describing the requirements of applications to resources, processes do not depend on input ﬂow and service distribution are jointly independent and identically distributed. L independent Poisson ﬂows of requests enter the system, and there are N identical devices. Service times are distributed exponentially. The functioning of the system is described by the semi-Markov process, which takes into account the number of serviced requests, their types and amounts of resources they occupy. Explicit expressions for the stationary distribution of the semi-Markov process, and the theorem on product form solution are main results of the paper. Further studies suggest checking the hypothesis of invariance with respect to the form of the stationary distribution of the distribution of the service time and the development of numerical methods for the analysis of probability measures of the system.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(2):38-45

38-45

### The Problem of Data Placement in Distributed Systems

#### Abstract

Distributed system eﬀectiveness depends dramatically on the way it manages incoming tasks and data against limited computational resources that are at its disposal. Due to ever-inreasing amount of incoming data distributed systems are required to eﬃciently manage the way its storage and processing are being made. Nowadays the distributed system design is signiﬁcantly ﬂounced by the manner it leverages high load scenarios, provides data storage functionality and uses the underlying resources. An eﬀective distributed system’s resource management has to balance trade-oﬀs between single node resource consumption and the overall loss of data locality, that is inevitable due to data fragmentation. In this article we will formalize the problem of data placement by maximizing data storage locality in distributed data systems, which as it turns out is a NP-complete task. We will later describe a polynomial-time algorithm that is capable of providing us a solution that is within an additive constant from the optimal one.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(2):46-54

46-54

### Investigation of Solutions of a Quasipotential Equation for Complex Values of Energy with Piecewise Constant Potentials

#### Abstract

Solutions of the Schrödinger equation for complex values of the energy describe the quasistationary state. The energy spectrum of quasistationary states is quasidiscrete and consists of a series of broadened levels, width of which determines the lifetimes of the respective states. The introduction of quasistationary states only makes sense if the width of the respective quasidiscrete levels is small compared with the distances between levels. An investigation of the solutions of quasistationary states is carried out for the quasipotential equation with piecewise-constant potentials at various values of the parameter of the equation , included in the equation and the potential parameters. A comparative analysis of the solutions of the quasipotential equation for the diﬀerent values of with the solutions of the Schrödinger equation is performed. It was found that at → 0 the solutions of quasipotential equation tend to the solutions of the Schrödinger equation. With the increasing of the parameter the lifetime of quasilevels for the quasipotential equation increases as compared with the results obtained for the Schrödinger equation except the level that is close to the edge of the barrier. For comparison, the wave functions for the Schrödinger equation and the quasipotential equation for ﬁxed values of the potential parameters are shown.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(2):55-61

55-61

### Numerical-Analytical Investigation of the Model of Gradient Optical Waveguide for Obtaining Equidistant Spectrum of Waveguide Modes

#### Abstract

A model of the planar optical waveguide with linear, exponential and modiﬁed exponential proﬁles of the refractive index has been studied on the basis of numerical and analytical approaches to solving a parametric inverse Sturm-Liouville problem. The goal of the investigation is to determine the proﬁle parameters which provide proximity of the waveguide modes spectrum to an equidistant one. A software complex which we have developed in the MAPLE system is used for a numerical analysis. For solving a direct spectral problem with predetermined parameters of the model, a scheme has been proposed which uses an analytical representation of the general solution to the wave diﬀerential equation. The scheme is used for a supplementary accuracy control of the results, if a correct analytical general solution can be obtained by MAPLE tools. For the linear proﬁle model, a parameter domain has been deﬁned in which the Sturm-Liouville problem for description of the waveguide mode spectrum has three solutions. This domain borders on the domain where the Sturm-Liouville problem has two solutions only, and a bifurcation point is calculated over parameters. In a vicinity of this point we have calculated parameters that provide approximate equidistance of the waveguide mode spectrum. The results for exponential and modiﬁed exponential proﬁles have been recalculated in view of the calculated value of the parameter obtained for a linear proﬁle model. This parameter corresponds to the height of the waveguide layer. The characteristics of spectrum equidistance have been improved.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(2):62-68

62-68

### On a Probability Density Equation

#### Abstract

The stationary Schrödinger equation depending on spatial coordinates has been considered. The problem of obtaining a differential relationship for the wave function squared was posed. By extracting Schrödinger’s equation itself from this relationship a differential equation for a physically interpretable quantity, i.e. the probability density (wave function squared), has been formulated. As an example the one-dimensional case admitting a simple analytic solution was considered. The solution obtained is shown to be a solution squared of the corresponding nonlinear differential equation for the probability density. In the final section a more general non-stationary case was considered for the potential involving a time-dependent term, such potentials are found in the non-stationary perturbation theory. The constant in separating the variables remains real. Thus the procedure considered proves to be similar to that presented above for the stationary equation.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(2):69-72

69-72

### Description of Lepton and Baryon Phases in Skyrme-Faddeev Spinor Model

#### Abstract

8-spinor ﬁeld is suggested to unify Skyrme and Faddeev models describing baryons and leptons as topological solitons. In the Skyrme model the particles-solitons possess the topological charge of the degree type, which is interpreted as the baryon number. In the Faddeev model the particles are endowed with the topological invariant of the Hopf type, which is interpreted as the Lepton number. The special 8-spinor Brioschi identity is used to include leptons and baryons as two possible phases of the effective spinor ﬁeld model with Higgs potential depending on the jμjμ being added to the Lagrangian. In the present paper the generalization of the Mie electrodynamics within the scope of the eﬀective 8-spinor ﬁeld model is suggested. For this ﬁeld model the quadratic spinor quantities entering the Brioschi identity are constructed. Also we ﬁnd the symmetry groups, which generate S2- and S3-submanifolds in general S8 biquadratic spinor manifold. As a result we have the homotopy groups π3(S2) and π3(S3), which describes lepton and baryon phases. To unifying these phases, we ﬁnd common vacuum state which conserves only one component in two cases. Finally, we obtain the resulting 8-spinor model permitting uniﬁed description of baryon and lepton.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(2):73-77

73-77

### Anisotropy and Low-Frequency Dynamics of Charge Transport in Single-Domain Crystals of LiCu 2O 2 at Low-Temperatures and Sound Frequencies

#### Abstract

For the ﬁrst time, the single-domain (without twinning) crystals of LiCu 2O 2 were used to measure anisotropy of DC and AC conductivities along the principal crystal axes and low-frequency dynamics of charge transport in the temperature (4.2-295 K) and frequency (25-105 Hz) ranges. The temperature, frequency and ﬁeld properties of the DC and AC conductivities reﬂect the strong localization of charge carriers as result of a local lattice distortion due to the structural, compositional (extra oxygen content) defects and particularly electrical and magnetic polarization. Therefore the conductivity is by hopping transport between localized states near the Fermi level and its character (activated or variable-range hopping) depends on the temperature range and on relative direction to the crystal axes. The analysis of experimental data allows us to draw conclusions about the electronic energy structure near the Fermi level, about the anisotropic properties, low-frequency dynamics and mechanisms of charge transport.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(2):78-82

78-82

### The Interference of Two Oppositely Twisted Light Waves

#### Abstract

Recent researches in the ﬁeld of classical and quantum optics, have established the fact that the light can carry not only energy and linear momentum, but also angular momentum. To prove this, experiments were carried out on the creation and detection of twisted light. In the present work, the Maxwell theory of electromagnetism is generalized to the space with rotation. For this purpose the seven-dimensional model of space-time developed by the author is used. In this model the translational coordinates and time as well as the rotational coordinates are used. On the basis of generalized equations of electromagnetic ﬁeld, solutions describing the twisted light are obtained. Thus, the article proposes non-quantum approach to describe the twisted light, without using the concept of spin. We also consider the superposition of two diﬀerently twisted light waves. It is shown that the result this superposition is a wave with an unusual proﬁle of the amplitude that depends on the direction. The author suggests that such approach to describe the twisted waves can help to look diﬀerently at some of the issues of classical and quantum optics.

**Discrete and Continuous Models and Applied Computational Science**. 2015;(2):83-89

83-89

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**Discrete and Continuous Models and Applied Computational Science**. 2015;(2):90-91

90-91

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**Discrete and Continuous Models and Applied Computational Science**. 2015;(2):92-93

92-93