# No 2 (2016)

## Articles

### Probability Characteristic Analysis and Computation Algorithm of One ONU Upstream in WDM-TDMA PON

#### Abstract

Nowadays, the telecommunication industry undergoes fundamental changes, associated with transition from voice to data-link systems. It‘s due to telecommunication networkrevolution, rapid growth of user count, increasing number of provide service and quality of service. Access network evaluation is being conducted in both directions, such as high bitrate access development for providing high quality of service and decrease length of cooper wiring in local line networks. Data traﬃc dominates in the networks and requires the creation of networks with high bandwidth based on packet switching. Therefore, it‘s paid special attention to the networks, which are based on optical and optoelectronic components. Passive optical network is an all optical network based on passive optical components only, which exclude the conversion of electrical signal into optical form and vice verso. Traﬃc transmission in the networks may be implemented using time division multiple access (TDMA) and wavelength division multiplexing (WDM) technologies. In the present paper, we propose a fragment of the multiservice passive optical network with upstream traﬃc carrying considering the functioning process of optical network units (ONU) and the principle of wavelength dynamic distribution. These results are used in the blocking probability analysis of the model.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(2):5-12

5-12

### Local-Cubic Spline for Approximate Solution of Boundary Value Problems

#### Abstract

We have constructed an explicit local-cubic spline for the approximation of the smooth functions and have studied the behavior of the approximation. To solve numerically boundary value problems, a spline-scheme based on the properties of the local-cubic spline and the standard cubic spline collocation is proposed. The scheme is implemented by sequentially solving two tridiagonal systems, which allow to use the three-point sweep method and diﬀer from each other only by matrix of the right-hand side of the equation. It indicates that this algorithm is eﬃcient. The number of operations depends linearly on the number of grid nodes. It is proved that the constructed spline possesses the same approximation properties as the local-cubic spline. Thus, in this paper we actually considered the approximation of the solutions of the boundary value problems. The proposed scheme also allows to ﬁnd the ﬁrst and second derivatives of the solution of the boundary value problem on the uniform grid nodes of the fourth-order accuracy with respect to the step-size of the grid. The numerical experiments conﬁrm the theoretical order of convergence. Due to good approximation properties and the simplicity of the algorithm implementation, the proposed method can be applied to solve numerically the boundary value problems for the second order ordinary diﬀerential equations, which often occur in mathematics, physics, and in the ﬁeld of natural and engineering sciences.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(2):13-23

13-23

### Symbolic-Numeric Solution the Shroedinger Equation for Rotating Solid Body by Diagonalisation Method

#### Abstract

The eigenvalues and wave functions of the rotational quantum top Hamiltonian with a diﬀerent three moment of inertia by the diagonalisation method in the basis function system that realized the all four irreducible representation of the discrete D2 group are obtained. For the low rotational moment value J = 1,2,3,4 the analytical formulae are calculated. But in the case of any rotational moment values the systems of equation are obtained that with the mean of the modern computer program packages allow very easy to calculate the spectrum and eigenfunctions of asymmetric quantum top. As example, for the rotational moment value J=50 by the help of Maple system eigenvalues are performed and its dependence versus of the parameter asymmetry are presented.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(2):24-36

24-36

### On a Method of Two-Dimensional Smoothing

#### Abstract

Regression analysis has the task of ﬁnding a functional relationship between the observed values the studied process. The raw data is the realization of a random variable, it is therefore considered dependent on the expectation. This problem can be solved by “smoothing” the raw data. Smoothing is the process of removing the noise and insigniﬁcant fragments while preserving the most important properties of the data structure. It is similar to ﬁnding the expectation of data. Data smoothing usually attained by parametric and nonparametric regression. The nonparametric regression requires a prior knowledge of the regression equation form. However, most of the investigated data cannot be parameterized simply. From this point of view, nonparametric and semiparametric regression represents the best approach to smoothing data. The aim of the research is development and implementation of the fast smoothing algorithm of two-dimensional data. To achieve this aim previous works in this area have been analyzed and its own approach has been developed,improving the previous ones. As a result, this paper presents the algorithm that quickly and with minimal memory consumption cleanses the data from the “noise” and “insigniﬁcant” parts. To conﬁrm the “eﬃciency” of the algorithm the comparisons with other generally accepted approaches were carried out on simulated and real data with other generally accepted approaches. The results of these comparisons are also shown in the paper.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(2):37-43

37-43

### Kuryshkin-Wodkiewicz Model of Quantum Measurements for Atoms and Ions with One Valence Electron

#### Abstract

The structural form of the Kuryshkin-Wodkiewicz model of quantum measurement was developed in detail for quantum Kepler problem. For more complex objects such quantumstructure is unknown. At the same time, a standard (non-structural) model of quantum measurement proposed by Holevo-Helstrom is suitable for any quantum object. The aim of thiswork is to spread the structural model of quantum measurement to a broader class of quantum objects - a model of quantum measurements of optical spectra of atoms and ions with one valence electron. In this work the Kuryshkin-Wodkiewicz model with implementation of Weyl-Kuryshkin quantization rule is applied to the extended quantum Kepler problem of quantum systems with one valence electron, such as alkali metal atoms. The proof of the consistency of the model is based on two Kato theorems about compact perturbations of operators. In the proof process the explicit form of the discrete spectrum of the valence electron for various spectral series was achieved with dependence on the serial parameters of the disturbance spectrum of an isolated object in the process of quantum measurement.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(2):44-52

44-52

### Stability of Non-Linear Vibrations of Doubly Curved Shallow Shells

#### Abstract

Large amplitude (geometrically non-linear) vibrations of doubly curved shallow shells with rectangular boundary, simply supported at the four edges and subjected to harmonicexcitation normal to the surface in the spectral neighbourhood of the fundamental mode are subject of investigation in this paper. The ﬁrst part of the study was presented by the authors in [M. Amabili et al. Nonlinear Vibrations of Doubly Curved Shallow Shells. Herald of Kazan Technological University, 2015, 18(6), 158-163, in Russian]. Two diﬀerent non-linear strain-displacement relationships, from the Donnell’s and Novozhilov’s shell theories, are used to calculate the elastic strain energy. In-plane inertia and geometricimperfections are taken into account. The solution is obtained by Lagrangian approach. The non-linear equations of motion are studied by using (i) a code based on arclengthcontinuation method that allows bifurcation analysis and (ii) direct time integration. Numerical results are compared to those available in the literature and convergence of the solution is shown. Interaction of modes having integer ratio between their natural frequencies, giving rise to internal resonances, is discussed. Shell stability under dynamic load is also investigated by using continuation method, bifurcation diagram from direct time integration and calculation of the Lyapunov exponents and Lyapunov dimension. Interesting phenomena such as (i) snap-through instability, (ii) subharmonic response, (iii) period doubling bifurcations and (iv) chaotic behavior have been observed.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(2):53-63

53-63

### Symmetric Encryption on the Base of Splitting Method

#### Abstract

This article shows a method of secured transmitting of information by using splitting encryption algorithm which replaces each character in plaintext by k-integer in ciphertext.Splitting algorithm is a generalization of the secured transmission procedure with secret key that. This study shows how to use a set of cryptographic keys which are generated using genetic algorithm and pseudorandom number generators, to solve some of serious problems in the modern cryptography.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(2):64-72

64-72

### Properties of Titanium Dioxide Films with Metallic Nanoparticles

#### Abstract

The physicochemical properties of titanium dioxide thin ﬁlms prepared by the gel technology, doped with gold nanoparticles, were investigated. The diﬀerences betweentechnologies for the synthesis of titanium dioxide were compared. It is experimentally shown that the developed gel technology allows to get almost 100% phase of nanostructured anatase that was conﬁrmed by high-resolution microscopy and X-ray results. The topography and morphology of the ﬁlms samples were investigated. The photoactivity of the synthesized ﬁlms was studied by EPR spectroscopy. It is shown that the photoactivity of the ﬁlms is increased by the UV irradiation. Titanium dioxide was modiﬁed by nanoparticles of gold with various concentrations. Has been investigated the depending of the ratio of the solution components in the manufacture of gel ﬁlms, as well as of the annealing temperature of their formation on transmission spectra. It is shown that the absorption spectra depend signiﬁcantly on the parameters of the technology. A study of the absorption spectra of titanium dioxide ﬁlms containing gold nanoparticles showed signiﬁcant changes in the spectra, exactly, there is an additional absorption peaks of varying intensity and the observed shift in the passband region. These changes are caused, presumably, by changes of the ﬁlm structure, and the aggregation of gold nanoparticles. Studies have shown the prospects of the gel method for the synthesis of titanium dioxide and its modiﬁcation of nanoparticles.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(2):73-86

73-86

### Remarks to the Standard Theory of K0, ¯K0 Meson Oscillations. -Strangeness and -Violation in Weak Interactions in System of K0, ¯K0 Mesons

#### Abstract

Usually it is supposed that K0, ¯K0 meson oscillations are realized through KS, KL meson states. It is necessary to remark that KS, KL meson states are produced at CP violation in the weak interactions, besides these states are nonorthogonal states. Since KS, KL meson states are nonorthogonal states they cannot generate K0, ¯K0 meson oscillations. For this aim can be used only orthogonal states. In reality at strangeness - S violation K0, ¯K0 mesons are transformed into superpositions of orthogonal K10, K20 meson states. Then through these K10, K20 meson states there are realized oscillations of K0, ¯K0 mesons. Further K10, K20 states at CP violation are transformed into superpositions of KS, KL meson states and then arise interference of KS, KL meson states but not oscillations. This picture is well in agreement with experiments. So we come to conclusion: K0, ¯K0 meson oscillations are realized through K10, K20 mesons, but not through KS, KL.

**Discrete and Continuous Models and Applied Computational Science**. 2016;(2):87-94

87-94

### Information about the authors

**Discrete and Continuous Models and Applied Computational Science**. 2016;(2):95

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### Guidelines for Authors

**Discrete and Continuous Models and Applied Computational Science**. 2016;(2):96-97

96-97