## Vol 27, No 4 (2019)

**Year:**2019**Articles:**7**URL:**http://journals.rudn.ru/miph/issue/view/1285**DOI:**https://doi.org/10.22363/2658-4670-2019-27-4

###### Abstract

The article is an overview. We carry out the comparison of actual machine learning libraries that can be used the neural networks development. The first part of the article gives a brief description of TensorFlow, PyTorch, Theano, Keras, SciKit Learn libraries, SciPy library stack. An overview of the scope of these libraries and the main technical characteristics, such as performance, supported programming languages, the current state of development is given. In the second part of the article, a comparison of five libraries is carried out on the example of a multilayer perceptron, which is applied to the problem of handwritten digits recognizing. This problem is well known and well suited for testing different types of neural networks. The study time is compared depending on the number of epochs and the accuracy of the classifier. The results of the comparison are presented in the form of graphs of training time and accuracy depending on the number of epochs and in tabular form.

**Discrete and Continuous Models and Applied Computational Science**. 2019;27(4):305-315

###### Abstract

Robinson-Schensted-Knuth (RSK) correspondence occurs in different contexts of algebra and combinatorics. Recently, this topic has been actively investigated by many researchers. At the same time, many investigations require conducting the computer experiments involving very large Young tableaux. The article is devoted to such experiments. RSK algorithm establishes a bijection between sequences of elements of linearly ordered set and the pairs of Young tableaux of the same shape called insertion tableau and recording tableau . In this paper we study the dynamics of tableau and the dynamics of different concrete values in tableau during the iterations of RSK algorithm. Particularly, we examine the paths within tableaux called bumping routes along which the elements of an input sequence pass. The results of computer experiments with Young tableaux of sizes up to 108 were presented. These experiments were made using the software package for dealing with 2D and 3D Young diagrams and tableaux.

**Discrete and Continuous Models and Applied Computational Science**. 2019;27(4):316-324

###### Abstract

A new analytical and numerical solution of the electrodynamic waveguide problem for leaky modes of a planar dielectric symmetric waveguide is proposed. The conditions of leaky modes, corresponding to the Gamow-Siegert model, were used as asymptotic boundary conditions. The resulting initial-boundary problem allows the separation of variables. The emerging problem of the eigen-modes of open three-layer waveguides is formulated as the Sturm-Liouville problem with the corresponding boundary and asymptotic conditions. In the case of guided and radiation modes, the Sturm-Liouville problem is self-adjoint and the corresponding eigenvalues are real quantities for dielectric media. The search for eigenvalues and eigenfunctions corresponding to the leaky modes involves a number of difficulties: the problem for leaky modes is not self-adjoint, so the eigenvalues are complex quantities. The problem of finding eigenvalues and eigenfunctions is associated with finding the complex roots of the nonlinear dispersion equation. To solve this problem, we used the method of minimizing the zero order. An analysis of the calculated distributions of the electric field strength of the first three leaky modes is given, showing the possibilities and advantages of our approach to the study of leaky modes.

**Discrete and Continuous Models and Applied Computational Science**. 2019;27(4):325-342

###### Abstract

Pair-copula constructions have proven to be a useful tool in statistical modeling, particularly in the field of finance. The copula-based approach can be used to choose a model that describes the dependence structure and marginal behaviour of the data in efficient way, but is usually applied to pairs of securities. In contrast, vine copulas provide greater flexibility and permit the modeling of complex dependency patterns using the rich variety of bivariate copulas which may be arranged and analysed in a tree structure. However, the number of possible configurations of a vine copula grows exponentially as the number of variables increases, making model selection a major challenge in development. So, to learn the best possible model, one has to identify the best possible structure, which necessitates identifying the connections between the variables and selecting between the multiple bivariate copulas for each pair in the structure. This paper features the use of regular vine copulas in analysis of the co-dependencies of four major Russian Stock Market securities such as Gazprom, Sberbank, Rosneft and FGC UES, represented by the RTS index. For these stocks the D-vine structures of bivariate copulas were constructed, which models are described by Gumbel, Student, BB1and BB7 copulas, and estimates of their parameters were obtained. Computer simulations showed a high accuracy of the approximation of the explored data by D-vine structure of bivariate copulas and the effectiveness of our approach in general.

**Discrete and Continuous Models and Applied Computational Science**. 2019;27(4):343-354

###### Abstract

The aim of this work was to study the spectrum of LF and HF oscillations generated by plasma bunches created and confined in the volume of a microwave cavity immersed in the magnetic field of a mirror trap. The registration of electrostatic oscillations in the plasma was carried out using two flat electrodes mounted diametrically opposite in the central part of the cavity close to its wall. This diagnostic showed the presence of low-frequency oscillations with frequencies of 130 kHz and 450 kHz. The oscillation spectrum in the microwave range was recorded at the minimum of the magnetic trap using a real-time spectrometer and a loosely coupled loop antenna. The registration of the spectra in the 40 MHz band revealed a regular change in the frequency of the fundamental oscillation mode of the cavity and the presence of two harmonics of the synchrotron radiation of the plasma bunch at frequencies of 2.25 GHz and 4.52 GHz, respectively. According to the obtained data, the parameters of the formed bunch (density, shape, volume, energy spectra of plasma components) can be restored.

**Discrete and Continuous Models and Applied Computational Science**. 2019;27(4):355-364

###### Abstract

For the first time, the theoretical model of the spin-electron structure of a singlelayer graphene film was proposed by Wallace. The literature also describes ferromagnetism generated by none of the three common causes: impurities, defects, boundaries. We believe that the source of ferromagnetism is the spontaneous breaking of spin symmetry in a graphene film. The classical field model describing spontaneously broken symmetry is necessarily non-linear. Among non-linear models, the simplest is the well-known 4 model. We believe that, as a first approximation, we can describe with its help all the characteristics of spin waves that interest us, their spectra, and the domain structure of ferromagnetism in graphene. The model admits kink and anti-kink exact solutions and a quasiparticle breather, which we modeled numerically. We use the kink-anti-kink interaction energy obtained numerically to solve the Schrödinger equation, which simulates the quantum dynamics of breathers, which underlies the description of spin waves. The solution of the Schrödinger equation by the Ritz method leads to a generalized problem of eigenvalues and eigenvectors, the solution of which is mainly devoted to this work.

**Discrete and Continuous Models and Applied Computational Science**. 2019;27(4):365-377

###### Abstract

We have developed an efficient computational scheme for integration of the classical Hamilton equations describing the ion dynamics confined in the radio-frequency field of the Paul trap. It has permitted a quantitative treatment of cold atom-ion resonant collisions in hybrid atom-ion traps with taking into account unremovable ion micromotion caused by the radio-frequency fields (V.S. Melezhik et. al., Phys. Rev. A100, 063406 (2019)). The important element of the hybrid atom-ion systems is the electromagnetic Paul trap confining the charged ion. The oscillating motion of the confined ion is defined by two frequencies of the Paul trap. It is the frequency of the order of 100 kHz due to the constant electric field and the radio-frequency of about 1-2 MHz defined by the alternating electromagnetic field of the ion trap. The necessity to accurately treat the ion motion in the combined field with two time scales defined by these two very different frequencies has demanded to develop the stable computational scheme for integration of the classical Hamilton equations for the ion motion. Moreover, the scheme must be stable on rather long time-interval of the ion collision with the cold atom ∼ 10 × 2/ defined by the atomic trap frequency ∼ 10 kHz and in the moment of the atom-ion collision when the Hamilton equations are strongly coupled. The developed numerical method takes into account all these features of the problem and makes it possible to integrate the system of coupled quantum-semiclassical equations with the necessary accuracy and quantitatively describes the processes of atomic-ion collisions in hybrid traps, including resonance effects.

**Discrete and Continuous Models and Applied Computational Science**. 2019;27(4):378-385