# No 4 (2016)

**Year:**2016**Articles:**10**URL:**https://journals.rudn.ru/miph/issue/view/886

## Full Issue

## Articles

### On Almost Global Half-Geodesic Parameterization

#### Abstract

The problem of existence of Global Half-Geodesic Surface Parameterization is considered. The problem is well known and it is yet unsolved in general case. It is known that for the twice-differentiable surfaces it has local solution. At the same time example of paraboloid of revolution proves that it is not possible in the general case to use local nets in order to construct the global halfgeodesic ones. In order to solve the problem the authors follow the way leading to the construction of isothermal parameterization for the surfaces with positive first quadratic form. To this end they deduce partial differential equation for the mappings giving necessary parameterization. In the contrast with the case of isothermal parameterization when the equation is Beltrami equation corresponding to the homogeneous elliptic system this equation is essentially non-linear one. Besides the new system admits degeneration at the points where the Jacobian of the solution is equal to zero or infinity. The speed of degeneration strongly affecting properties of the solutions is also unknown. In order to surpass these difficulties the authors change the challenge. Instead of the geodesics covering the whole surface they propose to find the geodesics covering the surface up to the set of Hausdorff null measure. Using the theory of -quasiconformal mappings they construct nonregular generalized solutions of non-linear Beltrami equation that nevertheless detect the necessary family of the geodesics. The constructed theory permits to study non-classical equilibrium forms of liquid drops.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(4):5-14

5-14

### The Application of Fluid Models to the Analysis of Peer-to-Peer Network

#### Abstract

The application of the fluid models to the analysis of the streams in information and communication networks is presented. The models, considered in the paper, take into account the specificities of widespread P2P networks (peer-to-peer), used for file-sharing, parallel computing, IP telephony, video streaming, etc. The review of the main types of P2P networks and their associated analytical models are presented in the paper. The fluid models, presented in the paper, describe network traffic in terms of the changes over time data stream rates between users and in terms of the number of network users. The first model is a system of ordinary differential equations and allows to analyze the average file download time. The second model is the extension of the first model and is represented in the form of partial differential equation. It takes into account a random amount of data requested by users. It can be used to analyze both the transient state and steady state during the download. This model is suitable to study the behavior of the system with a large number of users. In addition to the average download time the second model, taking into account the population in the network, allows to analyze such parameters of the network performance as the number of leechers and seeders in the network.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(4):15-25

15-25

### Multiservice Queuing System with Elastic and Streaming Flows and Markovian Arrival Process for Modelling LTE Cell with M2M Traffic

#### Abstract

Internet of Things (IoT) is thought to become the third wave of the Internet and to bring important changes into both technological and business aspects of telecommunications. However, for this to happen, an infrastructure should be developed in order to provide network access and management functions to millions, if not billions of IoT-enabled devices. LTE networks could play the key role in the IoT communications landscape, provided that their capabilities are enhanced to efficiently support IoT devices and provide massive machine-to-machine connections without hampering human-to-human communications. Our paper addresses resource allocation in an LTE cell with both human-to-human and machine-to-machine connections. The cell is modeled as a multiservice queuing system with streaming and elastic jobs flows. Resources for machine-to-machine connections are allocated in batches of fixed size; requests for them arrive according to a Markovian arrival process. We obtain the stationary probability distribution of the system and formulas for request blocking probabilities.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(4):26-36

26-36

### Dynamic Non-Linear Model of Distribution and Changes of Linguistic Information in the Indo-European Model Language Communit

#### Abstract

The paper considers the nonlinear dynamic mathematical model describing the distribution and variation of linguistic information in the Indo-European linguistic community. When constructing a mathematical model of linguistic information propagation and changes in the linguistic community as a priori information data from independent studies both linguistics and other scientific fields, such as history, archeology and genetics were used. Within the framework of this model the spread of linguistic information in a model Indo-European language community, including at the initial stage of its formation was numerically studied. The preliminary results of theoretical analysis and computer simulation are given. It was found that the mathematical model of the distribution and modification of linguistic information shows both regular and typical chaotic behavior. As one of quantitative characteristics of considered nonlinear process of distribution of the linguistic information it is offered to consider number of arising cycles as number of the arisen modern languages, in the given language community. Results of computer modeling show, that from two main hypotheses of formation of the Proto-Indo-Europeans - Anatolian and Kurgan, the latter better matches temporary estimates obtained by us.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(4):37-48

37-48

### About One Method of Differentiation of a Flat Discrete Planar Curve in Image Processing

#### Abstract

The problem of receiving points with high curvature (singular points) of contours for identification of the shape of objects on images is solved. Analysis of existing methods of numerical differentiation in the given aspect is held. The new method of differentiation of the flat discretely defined curves, which are dots (pixels) of circuits, based on variations of Arch Height method is considered. Features of such method of differentiation are shown using various formulas of calculation of a derivative. Dependency aspects of the accuracy of the derivative on the chord length are analyzed. It is shown, that with an increase in its length differentiation accuracy degrades, and the result tends to the module of curvature of a curve at the given point. Comparison of the developed method with other known methods is made. The analysis of area of applicability and variability of parameters of differentiation is made. The accuracy aspects of calculation of derivatives for various parameters of differentiation are investigated. Examples of differentiation of various curves, both set analytically, and the functions-contours received from real images are considered. It is shown, that the offered method allows to get rid of the ambiguity in position of points of a contour with high curvature and consequently to raise quality of recognition of the shape of objects. Possible scopes of the given method in various areas of science and technics are stated.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(4):49-55

49-55

### Algorithms for Solving the Boundary-Value Problems for Atomic Trimers in Collinear Configuration using the Kantorovich Method

#### Abstract

The model of atomic trimers with molecular pair interactions for collinear configuration is formulated as a 2D boundary-value problem (BVP) in the Jacobi and polar coordinates. The latter is reduced to a 1D BVP for a system of second-order ordinary differential equations (ODEs) by means of the Kantorovich method using the expansion of the desired solutions over a set of angular basis functions, parametrically dependent on the (hyper)radial variable. The algorithms for solving the 1D parametric BVP by means of the finite element method (FEM) and calculating the asymptotes of the parametric angular functions and effective potentials of the system of ODEs at large values of the parameter are presented. The efficiency of the algorithms is confirmed by comparing the calculated asymptotic solutions and effective potentials with those of the parametric eigenvalue problem obtained by applying the FEM at large values of the parameter. The applicability of the algorithms is demonstrated by calculating the asymptotic expansions of the parametric BVP solution, effective potentials and sets of binding energies for the beryllium trimer in the collinear configuration.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(4):56-76

56-76

### A Geometric Approach to the Lagrangian and Hamiltonian Formalism of Electrodynamics

#### Abstract

In solving field problems, in particular problems of electrodynamics, we commonly use the Lagrangian and Hamiltonian formalisms. Hamiltonian formalism of field theory has the advantage over the Lagrangian, which inherently contains a gauge condition. While the gauge condition is introduced ad hoc from some external reasons in the Lagrangian formalism. However, the use of the Hamiltonian formalism in the field theory is difficult due to the non-regularity of the field Lagrangian. We must use such variant of the Lagrangian and the Hamiltonian formalism, which would allow us to work with the field models, in particular, to solve the problem of electrodynamics. We suggest using the modern differential geometry and the algebraic topology, in particular the theory of fiber bundles, as a mathematical apparatus. This apparatus leads to greater clarity in the understanding of mathematical structures, associated with physical and technical models. Using the fiber bundles theory allows us to deepen and expand both the Lagrangian and the Hamiltonian formalism. We can detect a wide range of these formalisms. We can select the most appropriate formalism. Actually just using the fiber bundles formalism we can adequately solve the problems of the field theory, in particular the problems of electrodynamics.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(4):77-83

77-83

### Spherically Symmetric Solution of the Weyl-Dirac Theory of Gravitation and its Consequences

#### Abstract

The Poincar´e and Poincar´e-Weyl gauge theories of gravitation with Lagrangians quadratic on curvature and torsion in post-Riemannian spaces with the Dirac scalar field is discussed in a historical aspect. The various hypotheses concerning the models of a dark matter with the help of a scalar field are considered. The new conformal Weyl-Dirac theory of gravitation is proposed, which is a gravitational theory in Cartan-Weyl spacetime with the Dirac scalar field representing the dark matter model. A static spherically symmetric solution of the field equations in vacuum for a central compact mass is obtained as the metrics conformal to the Yilmaz-Rosen metrics. On the base of this solution one considers a radial movement of an interplanetary spacecraft starting from the Earth. Using the Newton approximation one obtains that the asymptotic line-of-sight velocity in this case depends on the parameters of the solution, and therefore one can obtain, on basis of the observable data, the values of these parameters and then the value of a rest mass of the Dirac scalar field.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(4):84-92

84-92

### Information about the authors

**Discrete and Continuous Models and Applied Computational Science**. 2016;(4):93-94

93-94

### Guidelines for Authors

**Discrete and Continuous Models and Applied Computational Science**. 2016;(4):95

95