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The history of using machine learning algorithms to analyze statistical models is quite long. The development of computer technology has given these algorithms a new breath. Nowadays deep learning is mainstream and most popular area in machine learning. However, the authors believe that many researchers are trying to use deep learning methods beyond their applicability. This happens because of the widespread availability of software systems that implement deep learning algorithms, and the apparent simplicity of research. All this motivate the authors to compare deep learning algorithms and classical machine learning algorithms. The Large Hadron Collider experiment is chosen for this task, because the authors are familiar with this scientific field, and also because the experiment data is open source. The article compares various machine learning algorithms in relation to the problem of recognizing the decay reaction + + + at the Large Hadron Collider. The authors use open source implementations of machine learning algorithms. We compare algorithms with each other based on calculated metrics. As a result of the research, we can conclude that all the considered machine learning methods are quite comparable with each other (taking into account the selected metrics), while different methods have different areas of applicability.

In this paper, we propose an approach to solving the problem of recognizing skin lesions, namely melanoma, based on the analysis of dermoscopic images using deep learning methods. For this purpose, the architecture of a deep convolutional neural network was developed, which was applied to the processing of dermoscopic images of various skin lesions contained in the HAM10000 data set. The data under study were preprocessed to eliminate noise, contamination, and change the size and format of images. In addition, since the disease classes are unbalanced, a number of transformations were performed to balance them. The data obtained in this way were divided into two classes: Melanoma and Benign. Computer experiments using the built deep neural network based on the data obtained in this way have shown that the proposed approach provides 94% accuracy on the test sample, which exceeds similar results obtained by other algorithms.

This paper has a practical aim. For a long time, implementations of pseudorandom number generators in standard libraries of programming languages had poor quality. The situation started to improve only recently. Up to now, a large number of libraries and weakly supported mathematical packages use outdated algorithms for random number generation. Four modern sets of statistical tests that can be used for verifying random number generators are described. It is proposed to use command line utilities, which makes it possible to avoid low-level programming in such languages as C or C++. Only free open source systems are considered., Comment: in English, in Russian

This paper describes the created mathematical model that allows you to explore the dynamics of cavitation bubbles under the influence of a single negative pressure pulse. The time dependence and coordinates of the parameters of the carrier phase, the temperature and pressure of the vapor phase, the concentration and size of the bubbles are determined numerically. It is concluded that the model created gives a good agreement between the calculated and experimental data.

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.

Recently, the place of the main programming language for scientific and engineering computations has been little by little taken by Julia. Some users want to work completely within the Julia framework as they work within the Python framework. There are libraries for Julia that cover the majority of scientific and engineering computations demands. The aim of this paper is to combine the usage of the Julia framework for numerical computations and for symbolic computations in mathematical modeling problems. The main functional domains determining various variants of the application of computer algebra systems are described. In each of these domains, generic representatives of computer algebra systems in Julia are distinguished. The conclusion is that it is possible (and even convenient) to use computer algebra systems within the Julia framework., in English; in Russian

The most common physical formalisms are the Lagrangian formalism and the Hamiltonian formalism. From the superficial point of view, they are one and the same, but rewritten in other terms. However, it seems that the Hamiltonian formalism has a richer structure and is more convenient for studying the electromagnetic field, especially in the formalization of its geometrization. Unfortunately for field problems, there is a whole set of Hamiltonian formalisms. The authors study the applicability of different variants of the Hamiltonian formalism to the problems of electrodynamics. In this paper we consider the Hamiltonian formalism with constraints., Comment: in English; in Russian

Computer algebra systems are complex software systems that cover a wide range of scientific and practical problems. However, the absolute coverage cannot be achieved. Often, it is required to create a user extension for an existing computer algebra system. In this case, the extensibility of the system should be taken into account. In this paper, we consider a technology for extending the SymPy computer algebra system with a low-level module that implements a random number generator., Comment: in English; in Russian

A practical and simple stable method for calculating Fourier integrals is proposed, effective both at low and at high frequencies. An approach based on the fruitful idea of Levin, which allows the use of the collocation method to approximate the slowly oscillating part of the antiderivative of the desired integral, allows reducing the calculation of the integral of a rapidly oscillating function (with a linear phase) to solving a system of linear algebraic equations with a triangular or Hermitian matrix.

Different kinds of models are used to study various natural and technical phenomena. Usually, the researcher is limited to using a certain kind of model approach, not using others (or even not realizing the existence of other model approaches). The authors believe that a complete study of a certain phenomenon should cover several model approaches. The paper describes several model approaches which we used in the study of the random early detection algorithm for active queue management. Both the model approaches themselves and their implementation and the results obtained are described.

When the Maxwell equations are geometrized, the Maxwell Lagrangian is usually reduced to the Yang-Mills Lagrangian. In this case, the effective quadratic metric, usually corresponding to the Riemannian metric of our space, is considered. However, it is more reasonable to use Finsler approach to Maxwell's equations. In the paper the Finsler representation of the geometrized Maxwell equations is considered. The comparison with the Riemannian approach also is made., Comment: in English; in Russian

When modeling network protocols, the choice of a model approach and a software implementation tool is a problem. The specificity of this subject area is that for the description of protocols usually the discrete-event approach is used. However, the discrete model approach has several disadvantages. It is poorly scalable, not well suited for describing dynamic systems. As an alternative to the discrete approach, a continuous approach is usually considered. But when modeling discrete events, continuous description becomes unnecessarily complicated and heavy. Events take the form of some restrictions on the continuous system, which are often not explicitly included in the continuous model, but have the form of additional semantic descriptions. The authors propose to use a hybrid (continuous-discrete) approach when modeling such systems. In the framework of the hybrid approach, the discrete system is recorded in a continuous form, and the events take the form of discrete transitions inherent in the approach. In addition, if it is based on the description of events, a simulation model can be obtained on the basis of a hybrid approach. This paper demonstrates the use of a hybrid approach to describe systems with control by the example of the interaction of the TCP protocol and the RED algorithm. The simplicity of creating both computational and simulation models of the system is demonstrated. The Modelica language is used as the implementation language.

The development of physics in the XX-th century was closely linked to the development of the mathematical apparatus. The General Relativity demonstrated the power of the geometric approach. Unfortunately, the infiltration of this apparatus in other domains of physics is rather slow. For example, there were some attempts of integration of the geometric methods in electrodynamics, but until recently they remained only as a theoretical exercise. Interest to the geometric methods in electrodynamics is summoned by practical necessity. The following algorithm of designing of the electromagnetic device is possible. We construct the estimated trajectories of propagation of electromagnetic waves. Then we calculate the parameters of the medium along these trajectories. The inverse problem is also interesting. The paper considers the techniques of construction of optical devices based on the method of geometrization of Maxwell’s equations. The method is based on representation of material equations in the form of an effective space-time geometry. Thus we get a problem similar to that of some bimetric theory of gravity. That allows to use a well-developed apparatus of differential geometry. On this basis, we can examine the propagation of the electromagnetic field on the given parameters of the medium. It is also possible to find the parameters of the medium by a given law of propagation of electromagnetic fields.

In data transmission networks implemented as systems with control, the phenomenon of global synchronization can occur. Outwardly, this phenomenon manifests itself as a self-oscillating mode in the system. This mode negatively affects the characteristics of the entire system, such as throughput and transmission delays. Relevant is the problem of finding the areas of occurrence of self-oscillation. The authors investigated this problem for the system as a whole. Also, the problem of isolating the elements of the system responsible for the appearance of an autooscillatory regime is urgent. The complexity of this problem is caused by the essentially nonlinear character of the system and its elements. Often, the linearization method is used for the decomposition of the system. But with the linearization, the self-oscillatory regime disappears. There is a need to find a method of decomposition, non-destructive self-oscillating mode of the system. As such a method, the authors suggest using the method of harmonic linearization. This method is used in the control theory. However, we must admit that this mathematical apparatus is little known to researchers specializing in the study of networks. The authors tried to describe in as much detail the process of research using the method of harmonic linearization. The method is used to study the influence of the form of RED-type function on the occurrence of self-oscillation mode. Thus, this material is more methodical than exploratory one.

The Maxwell equations have a fairly simple form. However, finding solutions of Maxwell's equations is an extremely difficult task. Therefore, various simplifying approaches are often used in optics. One such simplifying approach is to use the approximation of geometric optics. The approximation of geometric optics is constructed with the assumption that the wavelengths are small (short-wavelength approximation). The basis of geometric optics is the eikonal equation. The eikonal equation can be obtained from the wave equation (Helmholtz equation). Thus, the eikonal equation relates the wave and geometric optics. In fact, the eikonal equation is a quasi-classical approximation (the Wentzel-Kramers-Brillouin method) of wave optics. This paper shows the application of geometric methods of electrodynamics to the calculation of optical devices, such as Maxwell and Luneburg lenses. The eikonal equation, which was transformed to the ODE system by the method of characteristics, is considered. The resulting system is written for the case of Maxwell and Luneburg lenses.

To construct realistic mathematical models from the first principles, the authors suggest using the stochastization method. In a number of works different approaches to stochastization of mathematical models were considered. In the end, the whole variety of approaches was reduced to two formalisms: combinatorial (state vectors) and operator (occupation numbers). In the article the authors briefly describe these formalisms with an emphasis on their practical application., Comment: in English; in Russian.arXiv admin note: text overlap with arXiv:1603.02205

This work is devoted to studying the possibilities of hybrid modeling of communication network algorithms in the Julia programming language. A simulation of the system, which consists of an incoming stream processed according to the Transmission Control Protocol (TCP) and a router processing traffic according to the Random Early Detection (RED) algorithm, is carried out. This model contains both continuous and discrete elements, for example, a discrete probability function of random packet dropping in the case of queue congestion.

It is known that modeling consists of a few stages. Analytic calculations are made first and afterwards, they are converted into numerical ones. It is a fairly complicated task, but it can easily be solved with the help of programming languages with the support of a numerical–analytical approach. In this case, the program can generate a numerical scheme for the solution based on the given analytical equation. The possibility of high-level work and differential equations’ modeling is obtained through the access to the additional packages. This article overviews one of the approaches, specifically the usage of the different integrated DSL packages in Julia language.

In the special theory of relativity (SR) it is usual to highlight so-called paradoxes. One of these paradoxes is the formal appearance of speed values grater then the light speed. In this paper we show that most of these paradoxes arise due to the incompleteness of relativistic calculus over velocities. Namely, operation over speeds form a group by composition. In this case, the extension to the field is usually carried out using non-relativistic operations.

In this paper, two approaches (combinatorial and operatorial) to the stochastization of the one-step processes are discussed for the closed and open version of the Lotka–Volterra model, respectively.

The transformation optics uses geometrized Maxwell's constitutive equations to solve the inverse problem of optics, namely to solve the problem of finding the parameters of the medium along the paths of the electromagnetic field propagation. The quadratic Riemannian geometry is usually used for the geometrization of Maxwell's constitutive equations, because of the usage of the general relativity approaches. However, the problem of the insufficiency of the Riemannian structure for describing the constitutive tensor of the Maxwell's equations arises. The authors analyze the structure of the constitutive tensor and correlate it with the structure of the metric tensor of Riemannian geometry. It was concluded that the use of the quadratic metric for the geometrization of Maxwell's equations is insufficient, since the number of components of the metric tensor is less than the number of components of the constitutive tensor. The possible solution to this problem may be a transition to Finslerian geometry, in particular, the use of the Berwald-Moor metric to establish the structural correspondence between the field tensors of the electromagnetic field., in Russian; in English

Nowadays the geometric approach in optics is often used to find out media parameters based on propagation paths of the rays because in this case it is a direct problem. However inverse problem in the framework of geometrical optics is usually not given attention. The aim of this work is to demonstrate the work of the proposed the algorithm in the framework of geometrical approach to optics for solving the problem of finding the propagation path of the electromagnetic radiation depending on environmental parameters. The methods of differential geometry are used for effective metrics construction for isotropic and anisotropic media. For effective metric space ray trajectories are obtained in the form of geodesic curves. The introduced algorithm is applied to well-known objects - Maxwell and Luneburg lenses. The similarity of results obtained by classical and geometric approach is demonstrated., Comment: in Russian; in English

This paper discusses stochastic numerical methods of Runge-Kutta type with weak and strong convergences for systems of stochastic differential equations in It\^o form. At the beginning we give a brief overview of the stochastic numerical methods and information from the theory of stochastic differential equations. Then we motivate the approach to the implementation of these methods using source code generation. We discuss the implementation details and the used programming languages and libraries

A series of theoretical and experimental works is known which investigated the magnetic properties of graphene structures. This is due, among other things, to the prospects of using graphene as a material for the needs of the future nanoelectronics and spintronics. In particular, it is known about the presence of ferromagnetic properties at temperatures up to 200 C and above in a single-layer graphene films that are free from impurities. Previously there was proposed a quantum field theoretical model describing the possible mechanism of ferromagnetism in graphene as a result of spontaneous breaking of spin symmetry of the surface density of valence electrons. The possible spatial configurations of the localized spin density were described. In this paper we investigate such spatially localized nonlinear spin configurations of the valence electron density on the graphene surface such as kinks, and their interactions, as well as quasibound metastable states of the interacting kinks and antikinks, that are breathers. The spectrum of such breathers is investigated. It is shown that under certain conditions, this spectrum has a discrete sector, which, in turn, allows us to speak about the possibility of coherent quantum generation of spin waves in graphene structures, which is important in terms of practical applications in nanoelectronics and spintronics.

When modeling such phenomena as population dynamics, controllable ows, etc., a problem arises of adapting the existing models to a phenomenon under study. For this purpose, we propose to derive new models from the rst principles by stochastization of one-step processes. Research can be represented as an iterative process that consists in obtaining a model and its further re nement. The number of such iterations can be extremely large. This work is aimed at software implementation (by means of computer algebra) of a method for stochastization of one-step processes. As a basis of the software implementation, we use the SymPy computer algebra system. Based on a developed algorithm, we derive stochastic di erential equations and their interaction schemes. The operation of the program is demonstrated on the Verhulst and Lotka-Volterra models., Comment: in English; in Russian

This paper considers three types of tensor computations. On their basis, we attempt to formulate criteria that must be satisfied by a computer algebra system dealing with tensors. We briefly overview the current state of tensor computations in different computer algebra systems. The tensor computations are illustrated with appropriate examples implemented in specific systems: Cadabra and Maxima., in Russian; in English

To synthesize Maxwell optics systems, the mathematical apparatus of tensor and vector analysis is generally employed. This mathematical apparatus implies executing a great number of simple stereotyped operations, which are adequately supported by computer algebra systems. In this paper, we distinguish between two stages of working with a mathematical model: model development and model usage. Each of these stages implies its own computer algebra system. As a model problem, we consider the problem of geometrization of Maxwell's equations. Two computer algebra systems---Cadabra and FORM---are selected for use at different stages of investigation., in Russian; in English

Presentation of the probability as an intrinsic property of the nature leads researchers to switch from deterministic to stochastic description of the phenomena. The procedure of stochastization of one-step process was formulated. It allows to write down the master equation based on the type of of the kinetic equations and assumptions about the nature of the process. The kinetics of the interaction has recently attracted attention because it often occurs in the physical, chemical, technical, biological, environmental, economic, and sociological systems. However, there are no general methods for the direct study of this equation. Leaving in the expansion terms up to the second order we can get the Fokker-Planck equation, and thus the Langevin equation. It should be clearly understood that these equations are approximate recording of the master equation. However, this does not eliminate the need for the study of the master equation. Moreover, the power series produced during the master equation decomposition may be divergent (for example, in spatial models). This makes it impossible to apply the classical perturbation theory. It is proposed to use quantum field perturbation theory for the statistical systems (the so-called Doi method). This work is a methodological material that describes the principles of master equation solution based on quantum field perturbation theory methods. The characteristic property of the work is that it is intelligible for non-specialists in quantum field theory. As an example the Verhulst model is used because of its simplicity and clarity (the first order equation is independent of the spatial variables, however, contains non-linearity). We show the full equivalence of the operator and combinatorial methods of obtaining and study of the one-step process master equation., in Russian; in English

Background. Spinors are more special objects than tensors. Therefore spinors possess more properties than the more generic objects such as tensors. The group of Lorentz two-spinors is the covering group of the Lorentz group. Purpose. Since the Lorentz group is the symmetry group of Maxwell equations, it is reasonable to use Lorentz two-spinors and not tensors when writing the Maxwell equations. Method. We write the Maxwell equations using Lorentz two-spinors. Also a convenient representation of Lorentz two-spinors in terms of the Riemann-Silberstein complex vectors is used. Results. In the spinor formalism (in the representation of the Lorentz spinors and Riemann-Silberstein vectors) we have constructed the Hamiltonian of Maxwellian optics. With the use of spinors, the Maxwell equations take a form similar to the Dirac equation. Conclusions. For Maxwell equations in the Dirac-like form we can expand research methods by means of quantum field theory. In this form, the connection between the Hamiltonians of geometric, beam and Maxwellian optics is clearly visible.

As a result of the application of a technique of multistep processes stochastic models construction the range of models, implemented as a self-consistent differential equations, was obtained. These are partial differential equations (master equation, the Fokker–Planck equation) and stochastic differential equations (Langevin equation). However, analytical methods do not always allow to research these equations adequately. It is proposed to use the combined analytical and numerical approach studying these equations. For this purpose the numerical part is realized within the framework of symbolic computation. It is recommended to apply stochastic Runge–Kutta methods for numerical study of stochastic differential equations in the form of the Langevin. Under this approach, a program complex on the basis of analytical calculations metasystem Sage is developed. For model verification logarithmic walks and Black–Scholes two-dimensional model are used. To illustrate the stochastic “predator–prey” type model is used. The utility of the combined numerical-analytical approach is demonstrated.

Background: Component-based modeling language Modelica (OpenModelica is open source implementation) is used for the numerical simulation of complex processes of different nature represented by ODE system. However, in OpenModelica standard library there is no routines for pseudo-random numbers generation, which makes it impossible to use for stochastic modeling processes. Purpose: The goal of this article is a brief overview of a number of algorithms for generation a sequence of uniformly distributed pseudo random numbers and quality assessment of the sequence given by them, as well as the ways to implement some of these algorithms in OpenModelica system. Methods: All the algorithms are implemented in C language, and the results of their work tested using open source package DieHarder. For those algorithms that do not use bit operations, we describe there realisation using OpwnModelica. The other algorithms can be called in OpenModelica as C functions Results: We have implemented and tested about nine algorithms. DieHarder testing revealed the highest quality pseudo-random number generators. Also we have reviewed libraries Noise and AdvancedNoise, who claim to be adding to the Modelica Standard Library. Conclusions: In OpenModelica system can be implemented generators of uniformly distributed pseudo-random numbers, which is the first step towards to make OpenModelica suitable for simulation of stochastic processes.

SymPy computer algebra library is used for automatic generation of ordinary and stochastic systems of differential equations from the schemes of kinetic interaction. Schemes of this type are used not only in chemical kinetics but also in biological, ecological and technical models. This paper describes the automatic generation algorithm with an emphasis on application details.

Self-oscillating modes in control systems of computer networks quite negatively affect the characteristics of these networks. The problem of finding the areas of self-oscillations is actual and important as the study of parameters of self-oscillations. Due to the significant nonlinearity of control characteristics, the study of the self-oscillating modes presents certain difficulties. This paper describes the technique of research of self-oscillating modes.

This paper describes Lorentz two-spinors and proposes using them in calculations with Dirac four-spinors and quaternions.

The problem of RED-module mathematical model results verification, based on GNS3 experimental stand, is discussed in this article. The experimental stand consists of virtual Cisco router, traffic generator D-ITG and traffic receiver. The process of construction of such stand is presented. Also, the interaction between experimental stand and a computer of investigation in order to obtain and analyze data from stand is revised. A stochastic model of the traffic management RED type module was built. Verification of the model was carried out on the NS-2 basis. However, we would like to conduct verification on a real router. As a result was the task of designing an experimental stand. It was decided to verify the clean RED algorithm based on Cisco router. For the construction of the stand software package GNS3 (Graphical Network Simulator) was chosen. Thus, the purpose of the study is to build on the GNS3 basis a virtual stand consisting of a Cisco router, a traffic generator and a receiver. A traffic generator D-ITG (Distributed Internet Traffic Generator) is used as., in Russian; in English