Table of contents

The International Scientific and Practical Conference "Mathematical Modeling, Programming and Applied Mathematics" was held at the Yaroslav-the-Wise Novgorod State University on June 27-28, 2019. It was attended by prominent scientists from Moscow and St. Petersburg, Petrozavodsk and Kazan, Tver and Voronezh, as well as from foreign countries: China, Egypt and Ukraine.

The scholars presented their latest scientific achievements, shared the best practices and discussed mathematical modeling in applied mathematics, programming, physics. They analyzed and evaluated innovative solutions in the conducted research in terms of their relevance, effectiveness and prospects for further development.

This journal presents the articles of the conference participants. The collection of works fully represents the current state in the field of mathematical modeling and, in our opinion, will be very interesting to scientists around the world – both novice researchers and professionals with significant experience.

Editor:

Stefan I. Eminov

The Institute of Electronic and Information Systems of Novgorod State University - Federal State Budgetary Educational Institution of Higher Education "Yaroslav-the-Wise Novgorod State University" (NovSU)

E-mail: conf-ieis@novsu.ru

All papers published in this volume of Journal of Physics: Conference Series have been peer reviewed through processes administered by the proceedings Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing.

In our paper we discuss the k-ary Euclid Algorithm for counting the great common divisor (GCD) of two or more integers and suggest some improvements. This gives us a possibility to parallelize and speed up the calculating of GCD, which has a variety of applications in the Number Theory, Modular Arithmetic and the Cryptography Algorithms such as RSA, ElGamal encryption system and others.

Now new technologies within the Industry 4.0 develop. Convergence of various applications of applied mathematics for a research of a synergetic coevolution the sociotechnical landscapes (STL) is intensified. STL is presented in the form of a taxonomical ontologic matrix which elements are symbols of the Cartesian work of sets social the practician and digital technologies. The train of indicator signs which is unambiguously characterizing levels of tension of STL is applied to classification of STL. The method of coding of indicator signs for calculation of value of the qualifier based on calculation of a binary code on function of accessory of each sign which carrier is the natural logarithm of a relative deviation of nominal rate of sign from registered is described. The reversibility of formation of the qualifier allows carrying out decrease in dimension of feature space without information losses. Results of researches STL of Kursk region are given in work (the train was presented by 20 indicators). The conclusion about prospects of application of the considered applied method of the mathematical analysis of the solution of classification tasks is drawn.

The article presents the derivation of the integral equation for a cylindrical perfectly conducting symmetric vibrator located vertically at a certain height from the interface of the two media. An example of numerical calculation is considered.

In the article the output of integral equations symmetric cylindrical dipole antenna of superconductive material. The antenna is located parallel to the interface between the two media. An example of calculating the current distribution over the antenna depending on the parameters of the medium is given.

We are considering the use of a two-stage Residue number system (RNS) in data transmission systems with a special set of modules {2^l₁, 2^l₂, 2^l₃,..., 2^l_p − 1}. We propose the forward converter with such type of RNS and demonstrate its advantage in hardware costs and delay. This article discusses the FPGA simulation of a forward conversion circuit to a two-stage RNS using adders and the standard modulo function and their comparison in terms of delay and hardware costs.

We analyze aggregated traffic dynamics obtained from strongly linked network communities. Our results based on two empirical data traces from university campus networks indicate that neglecting the statistical links between traffic patterns generated by individual network nodes leads to the drastic underestimation of both waiting and sojourn times. We also show that similar effects can be observed in simulated traffic patterns obtained by agent based modeling. Moreover, we suggest several indices that could be used to quantify the links between nodes and show their relation with the queuing system performance indicators.

We introduce the generalized shift operator related to the Bessel differential operator and the Hankel integral transform. It generalizes the operator introduced by J Delsarte and studied by J. Delsarte, B.M. Levitan and Ya.I. Zhitomirskii. The considered operator is not the generalized shift operator of the Levitan's type. The basic properties of the shift operator are proved. The visualization of shift action is given.

The paper presents a developed software tool that includes software systems Prolog, JavaScript, Python, as well as database management systems PostgreSQL and MySQL.With the help of the created integrated development environment, experimental versions of specialized expert systems in the field of education and health care were developed. In the field of education, a prototype of an expert system focused on the support of the discipline "development and analysis of requirements for the creation of a software product" was created. This prototype of the expert system should provide an opportunity for students to build various software systems based on the formulated requirements and rules from the knowledge base of the expert system, taking into account the specifics of the relevant subject areas. In the field of health care, a prototype of an expert system focused on supporting the processes of vaccination of patients has been created. The knowledge base of the expert system includes clinical characteristics of patients, as well as a set of rules governing the decision - making processes for vaccines.

This paper considers the target locating against the clutter background in radars with quasi-continuous mode of transmission and reception on common aerial of signals with pseudorandom amplitude- and phase-shift keying. Pseudorandom amplitude-phase-shift keyed signals with irregular bi-level amplitude spectrum are proposed for use. The high-level spectrum components of returns from the moving and stationary targets can be partially overlapped. It is proposed to perform the frequency rejection of high-level components of the stationary targets before the correlation signal processing. As a result, significant part of the energy of interfering reflections is cut out and just a small part of the energy of the useful signal is lost. The signal-to-clutter ratio increases. The synthesis method of pseudorandom amplitude-phase-shift keyed signal with irregular bi-level amplitude spectrum is considered. The spectrum of the synthesized signals is analyzed. Increase of the signal-to-clutter ratio was obtained in the processing of signals with frequency interference rejection. The result of detection modeling of a moving target against the background of reflections from a stationary object is presented.

The article discusses an approach to speeding up the search complex errors in software – reverse debugging. In this approach, debugging is divided into two phase. The first phase is to save the behaviour to a log file. The second phase is the reproduction of behaviour and the search for errors. During the second phase, both past and future program states can be examined. The article describes methods that support this approach and general principles for such methods and their differences. A classification of reverse debugging methods is proposed by debugging scope and replay scope. Examples of modern reverse debugging tools and their place in the proposed classification are given.

We investigate the k-error linear complexity over ${\mathbb{F}}$_p of binary sequences of length 2p with optimal three-level autocorrelation. These balanced sequences are constructed from cyclotomic classes of order four using a method presented by Ding et al.

We estimate the k-error linear complexity of some interleaved sequences with period 4N over the finite field of order N (N an odd prime). Furthermore, we obtain the exact value of the k-error linear complexity for small value of k of the interleaved sequences obtained from the cyclotomic sequences. In particular, we study the interleaved sequences obtained from Legendre sequences and Hall's sextic residue sequences, respectively. Our results show that these sequences are quite stable.

In this paper, we study the linear complexity of series of binary sequences with optimal autocorrelation magnitude of length 4N. These sequences are obtained from almost-perfect binary sequences and binary sequences with optimal autocorrelation of length N. The construction of these sequences were presented E.I. Krengel and P.V. Ivanov. We show that considered sequences have the high linear complexity. Also we derive the 1-error linear complexity of these sequences.

The trace representation of sequences is a powerful tool for the analysis and for the design of pseudorandom sequences. Z. Dai et al. (2011) reduce the problem of determining trace representation of series of binary eth power residue sequences to that of determining the values of generating polynomials of cyclotomic classes. We derive the values of generating polynomials of cyclotomic classes of order 4, 6, 8 and consequently solve three problems pointed by Z. Dai et al. In fact, we study the discrete Fourier transform of cyclotomic sequences of order 4, 6, 8.

The solution of navigation problems the ice situation problems in the Arctic region need to study the acoustic parameters and local reflective properties of ice in real conditions. Information about the local sea ice acoustic properties essentially helps to solve the problems of statistical forecasting of reflective and scattering properties of ice cover. The successful solution of these problems mostly depends on the metrological parameters of the probing signals emitted by the primary piezoelectric transducers. This means, that to increase the resolution of the control and measuring equipment it is better to radiate and to receive short acoustic pulses. The article presents the results of analysis the pulse mode of operation of cylindrical piezoelectric transducers designed to measure the velocity of sound in sea ice. To reduce the length of acoustic pulses, an inductive-resistive load is connected to the electrical input of the radiator. The length and the amplitude of the emitted acoustic pulse were estimated using equivalent circuits of piezoelectric transducers and a spectral method based on the Fourier transforms. The optimal parameters of the electrical circuit, ensuring the minimum length of the acoustic pulses are calculated. The technique of possible measurements is described. The results are presented in the most general form, making possible to use them within different frequency ranges.

Streaming networks have a wide range of options for practical application: information flows in information and communications networks, transport traffic, electrical power systems. The tasks of large dimension, including those which are associated with finding the frequency-optimum traffic, should be solved by the decomposition method. The paper considers the algorithm of the "pinwheel" decomposition of the installation and switching space and synthesis of the streaming network with optimal adjustment placement of different-sized radio elements on the surface of the construct.

This article presents an analysis of trends in the development of the information technology market. The development of the content management systems (CMS) is traced from its origin. The tools designed to create such systems are considered. First of all, these are programming languages and technology companies working in this field. It is no secret that the development of the Internet is primarily associated with the development of the sources of information, that is, with the websites. Currently, there constantly appear much easier ways of website creation and the procedure of information uploading is getting significantly simplified year by year. A brief analysis of programming languages for creating websites is given. The possibilities and tools for creating CMS augmented with a number of frameworks to speed up the development exponentially and having other advantages as well are dwelt upon. The functional capabilities of CRM-systems both in general and for the development of the whole business, especially the online segment are presented. The article substantiates the necessity for the implementation of CRM system and describes the main features of the system.

In this paper, we consider the synthesis of a compact (SPICE) Schottky diode model on GaN. Synthesis of a compact (SPICE) Schottky diode model is implemented on the basis of a modified model of a conventional diode on a p-n junction by extracting the parameters contained in it from experimental data. To create the structures of the Schottky diodes, epitaxial structures of gallium nitride grown by the MOCVD method on a sapphire substrate were used. Ohmic contacts of diode structures were formed on ion-doped epitaxial layers. The result of the simulation of diode structures is the extraction from experimental data of parameters of the compact model of a Schottky diode on GaN.

Metal optics is actively used in IR optics, which makes it necessary to clean it from dust and other operational contaminants. This increases the life of these notably expensive products. Usually their mechanical wiping is excluded, since it leads to the appearance of scratches and other defects on the surface, markedly increasing the scattering of radiation. Methods for the selection of solvents have been analyzed and experiments have been carried out that make it possible to accept efficient their use in the purification of metal optics from contaminations. Contaminations have different physicochemical properties and are present, as a rule, together on the surface of metal optics (taking into account the corrosion properties of the latter). Solvent compositions based on azeotrope mixes with freon-114B2, having a solubility parameter (δ) corresponding or close to the solubility parameter (δ) of the main (by weight) contamination or a mixture of contaminants which are present on the optical surface, were used.

We consider a problem of simulating strictly stationary random series. A modification of an autoregressive–moving-average model of orders 1, 1 is proposed. Proposed algorithm allows to simulate stationary random series with uniform distribution. Properties of resulting random series are examined.

The article provides information on the temperature dependence of the main characteristics of an uncooled photodetector based on the InGaAsSb compound, for the spectral range 1.7 ... 2.3 μm. Techniques have been developed, data on integral sensitivity, spectral, noise and threshold characteristics of PHD in the temperature range have been studied and presented (223 - 323 Ƙ). Analytical expressions for the temperature dependence of the characteristics are obtained. Noise model PHD is proposed. It was found that the increase in noise with increasing temperature is associated with a change in the differential resistance of the photodiode, and the frequency dependence was determined by the characteristics of the preamplifier. The volt sensitivity of the PHD to the ABB radiation of 800 K reached 40·10³ V/W, and the detecting ability D*(λ_max,1000.1)=9.4·10¹⁰ W^-1·cm·hz^1/2 at room temperature.

According to the theory of relatively p-bounded operators, we study the Heat Conduction model involving two temperatures for isotropic material, which describes, the rate of change of internal energy due to the movement of the heat flux form one medium to its complement with general Wentzell boundary conditions. In particular, we consider spectrum of one-dimensional Laplace operator on the segment [0,1] with general Wentzell boundary conditions. We examine the relative spectrum in one-dimensional Heat Conduction equation involving two temperatures, and construct the resolving group in the Cauchy-Wentzell problem with general Wentzell boundary conditions. In the paper, these problems are solved under the assumption that the initial space is a contraction of the space L²(0,1).

This paper contains the general solution for potential A_μ of electromagnetic field with given sources. It consists of two terms having clear physical sense and includes gauge additive.

In this article a feedforward error backpropagation artificial neural network is investigated and the analysis of its illogical behaviour is presented. The problem of illogical behavior arises in various models of artificial neural networks. In the presented work a classifying artificial neural network (CANN) is considered and several learning algorithms were implemented and compared. CANN was designed for automatic differentiaition of cyanobacterial strains during environmental monitoring and some of trained networks demonstrated illogical behavior in further testing. Several original techniques were elaborated for estimation of the quality and accuracy of classification in addition to the traditional ones. Novel visualization methods were suggested for classification and generalization results representation.

The article discusses the features of the formation of spectrozonal images corresponding to narrow registration areas by optical and digital means. The processes of the corresponding optical and signal transformations are analyzed. The results of computer simulation are discussed with a view to a qualitative assessment of the degree of consistency between optical spectrozonal visualization and the digital spectrozonal visualization carried out by the differential method. Examples of images of test and real objects obtained by a differential method from the original spectral images are given.

As part of a computer experiment with the use of the finite element method, a method was developed for analyzing the morphology of the non-stationary temperature field of a heat pipe which has a localized structure defect in its case. It is shown that within the model of a steady start of HP (τ = 0 ÷ 120 s.) and maximum temperatures of PHFS T_PHFS = 350 ÷ 400 K, the presence of a defect causes the variability of the shape of isotherms over time and the nature of the temperature distribution along the HP axis. In this case, the distance from the center of the HP to the front of the isothermal line uniquely determines changes in the morphology of the temperature field.

Energy field plays an important role in commercial world and makes a vital part of humanity. The paper considers the prediction problem concerning spent electrical energy in the Republic of Tatarstan, Russia. In fact, the task of electrical load prediction was set for each hour of the specified period. We solve the problem using Machine learning methods. Four models are considered. These are Linear Regression, Decision Tree, Random Forest and Gradient Tree Boosting. Linear Regression method shows good results, the average relative error is 3.98%. Decision Tree and Random Forest show the worst result, the average relative error is 10.44%. Gradient Tree Boosting show the best result, the average relative error made 2.17%. At the same time, Linear Regression model is much faster than other ones and more useful in industry. In the paper we show that using several techniques we can improve results for Linear Regression, such that it will be close to another advanced algorithms. The average relative error that is less than 5% is considered as a high enough result. The solution of the problem with a small error allows us to prevent the accidents related to electric overload. We assume that load depends on date, time and temperature. Then input variable models were obtained from these data. It is common approach to use that parameters as input data but let us note that the results obtained by other researchers are not suitable for this area, since each area has individual climatic, geological and social features.

The article discusses the additive-multiplicative fixed-pattern noise model of the infrared photodetector and a new way of its compensation, invariant to a change in exposure time. Examples of images showing the effectiveness of a new method for compensating fixed-pattern noise when the exposure time of a photodetector is changed are given.

The paper is devoted to the development of the energy harvesting system of microwave range. The block diagram of the system was developed and stands for measuring antenna patterns and study of the transmission voltage dependence from distance were made. To perform the study were selected six types of antennas: dipole, wire dipole, stripline, horn, parabolic and stripline antenna array. The study revealed that the parabolic antenna has the advantage in distance of energy transfer out of the six considered antennas. The achieved value of the potential on the receiving device is about 23 mV at the distance of 9 m. The obtained results allow realizing the energy harvesting system in a small industrial premise or room.

Method of V.M. Alekseev was used to obtain the clarifications of lower estimates for the lower singular exponent and of upper estimates for the upper singular exponent of a linear system of differential equations. In contrast to the classical results of the freezing method, which assume that the derivatives of the elements of the matrix of the coefficients of the system are small, the new estimates admit that these derivatives are unbounded.

The article describes the method of measuring blood flow velocity by a non-invasive method of magnetizing blood with a magnetic pulse and detecting a magnetized substance using a magnetoelectric magnetic field sensor. The influence of the magnetic field on the parameters of blood and blood flow is considered. The methods of measuring blood flow velocity are considered, the distinguishing features of each method are given. The use of a magnetoelectric composite structure of bidomain lithium niobate\nickel\metglas is proposed as a sensitive element for detecting blood flow velocity.

Electronic structure calculation of the stishovite as close packing SiO₂ modification was performed with the use of the linearized augmented plane wave method in framework of Wien2k code. Band diagram, total and partial density of states were calculated. X-ray emission Si K and Si L_2,3 spectra characterizing valence band were obtained. By "core hole" approximation XANES Si L_2,3 was calculated. X-ray stishovite spectra reveals good agreement with the experimental data. The explanations are given for the observed density of states and emission spectra fine structure features.

In the conditions of transformation and modernization of society, the use of mathematical modeling is of great importance. This is due to the fact that at the present stage a distinctive feature is the introduction of science and technology, complex technical systems in all areas of activity, one of which is the development of diagnostic systems, in particular, the construction of a mathematical model of tomography scanner. The article develops a mathematical model of a two-contour tomography scanner, describes the fields created by a single-point, two-point, single-circuit source, shows the construction of the Green function for the parameter k = 1. The advantage of a two-contour tomography scanner is the presence of a sphere among the equipotential surfaces created by this device, which is convenient for analytical calculations. The final results are accurate, not numerical. This plays a significant role in medical diagnostic studies, where the patient's health depends on the accuracy of the diagnosis.

The article is devoted to the problem of numerical evaluation of the proximity of a thematic text to the most rational (reference) language version of the description of the piece of knowledge it represents. This problem is relevant for the implementation of targeted selection of textual information without losing the useful semantic component. Examples of practical applications here can be the selection of articles for publication in scientific journals, as well as the development of training courses and educational portals. In the proposed solution, the basis for assessing the proximity of a text to a semantic pattern (i.e. sense standard) is the division of the words of each phrase into classes according to the value of the TF-IDF measure relative to the texts of the corpus pre-formed by an expert. The analyzed texts considered in the paper are the abstracts of scientific articles along with their titles. At the same time, the semantic images of the texts closest to the standard determine the words with the highest TF-IDF values, which, being neighbors in a linear series, are most likely related by meaning and form key combinations. The proposed numerical estimate of the proximity to the standard makes it possible to rank articles according to the significance of the described fragments of knowledge with respect to a given subject area, as well as to the non-redundancy of the description itself.

We consider nonhomogeneous pseudorandom sequences of nonmaximal length formed by a shift register with linear feedback (Fibonacci generators), and with internal half-adders (Galois generators). As a basis, we consider characteristic polynomial raised to the power of n of a form ${\rm{\varphi }}(\rm{x})={{\rm{\varphi }}}_{0}^{m}({x}){{\rm{\varphi }}}_{1}(\rm{x})$, where φ₀ (x) and φ₁ (x) are primitive polynomials respectively raised to the power of m₁ and m₁, m₀ · m + m₁ = n. We discovered periodic polynomial structures. Examples demonstrate a diversity of generated practical sequences, which are organized and ordered from elements of direct and inverse M-sequences. We investigated probabilistic properties of the formed sequences.

The article presents a set of calibration techniques for thermal imaging systems calibration. The proposed algorithms allow high-precision calibration of the IR photodetector with finding and subsequent correction of the entire spectrum of defective pixels: by noise, amplitude and sensitivity. Part of the work is devoted to the development of methods for correcting the influence of ambient temperature on the optical system of a thermal imaging channel, which helps to prevent the effect of super saturation of the IR image pixels.

The article discusses the magnetoelectric effect in the thickness-shear mode region in a layered structure of a piezoelectric bimorph and a magnetostrictive material. Numerical estimates according to the proposed theoretical model of the magnetoelectric effect in the structure of the langatate bimorph and yttrium-iron garnet show that the value of ME voltage coefficient is much higher than for the bilayer of langatate and yttrium-iron garnet.

The paper considers the change in the spectrum of ferromagnetic resonance in ferrite-piezoelectric layered structures in an external electric field. The induced magnetic anisotropy through the magnetoelastic interaction is generally non-uniform in thickness due to the presence of bending deformations. As a result, the applied electric field leads to a shift and broadening of the magnetic resonance line, which are determined by the composition and geometric dimensions of the structure layers. The paper presents a theoretical modeling of the magnetic resonance line broadening when additional buffer layers are inserted between the ferrite and piezoelectric layers to reduce the effect of a high-permittivity piezoelectric on the microwave field structure in the ferrite. The simulation results are considered on the example of layered structures of the composition of iron-yttrium garnet - lead titanate-zirconate. It is shown that at certain thicknesses of the magnetic and piezoelectric layers, the broadening of the magnetic resonance line as a function of the thickness of the buffer layer has a maximum. The results can be used in frequency-selective microwave devices with electrically tunable range.

In the article, we considered recommender models based on matrix factorization demonstrate excellent performance in collaborative filtering. The standard Matrix Factorization approach in MLlib deals with clear ratings. To work with implicit data, we used the trainImplicit method. To simulate the processing of real-time data streams, we used the Spark Streaming library, which is responsible for receiving data from the input source and converting the raw data into a discretized stream discretized stream (DStream) consisting of Spark RDD. The rank parameter determines the number of hidden features in the low rank approximation matrices. As a rule, the greater the number of factors, the better, but for a large number of users or elements, it will directly affect the memory usage of the computing system and the amount of data required for training. Therefore, in our problem it was a compromise solution.

In the article, we consider a discerning authorization algorithm that eliminates the contradictions in the assignment of access rights to security principals in complex infrastructures, which are using concepts such as group and inheritance. It also can use for the identification of security risks and it ensures that the security risk assessments produce consistent, valid and comparable results.

The work is devoted to the study of rotational features of methane molecules in the methane solid state in different temperature ranges. We formulated earlier, the concept of the description of collective rotational motion of the CH₄ molecules with the use of a topon quasi-particle. This approach received a new experimental confirmation in papers by Yu. A. Dmitriev and N. P. Benetis on studying by electron paramagnetic resonance (EPR) quantum mechanical rotation of CH₃ radicals matrix isolated in solid methane and in a work by P. Pyykko using nuclear magnetic resonance (NMR) technique to test rotation of host molecules in methane at temperatures of orientational ordering (below the α−β transition). Comparison with the behavior of the CH₃ radical in other orientation-ordered matrices, CO₂, N₂, CO, N₂O suggests that the observed change in the barrier at such low temperatures is associated with the reorientation of the methane matrix molecules, which, provided that their orientation ordering is preserved, is possible only as a result of collective tunnel rotation.

Microwave filters made according to the technology of low-temperature co-fired ceramics (LTCC) are considered. The main features of LTCC technology are described. The parameters of the domestic glass-ceramic material are given. The stages of designing an integrated microwave filter, all elements of which are in the volume of ceramics, are considered. In the course of this work, a model of a LTCC microwave filter with a nominal frequency of 300 MHz was developed taking into account the parameters of domestic materials. According to the modeling outcomes, conclusions were made.

Transformation of modulated by amplitude and phase signals in the spin-torque nanooscillator is investigated. Frequency dependences of transmission ratios of amplitudes and oscillation phases at the modulating current are received. Existence of detuning of a locking frequency of rather natural frequency of the spin transfer nano-oscillator and frequency dependence of a threshold parameter near the natural frequency leads to emergence of cross transmission ratios of variations of amplitude in a variation of a phase and variations of a phase in amplitude variation Thus, we showed, that the network of the mutually coupled oscillators can be used as the modulator/demodulator of a phase and frequency at strong nonisochronous.

We have proposed to use the nanoparticle (NP) melting temperature T_m to distinguish between the coalescence and sintering processes on the nanoscale. According to our molecular dynamics results, obtained on Au NPs, the coalescence of nanoproplets may be interpreted as a hydrodynamic phenomenon on the nanoscale, and the characteristic coalescence time τ is a linear function of the initial particle radius r₀. In turn, the sintering of two crystalline NPs (T < T_m) relates to a grain boundary formation as a result of an alignment of the crystallographic orientations of the sintering NPs, and in this case a dependence of τ on r₀ has not been revealed.

A condition of the mechanical stability of non-volatile nanoparticles in a chemically inert medium is derived from the non-negativity of the second variation of the Helmholtz free energy of the 'small object - gaseous medium' system. Two cases have been analyzed: (i) a non-volatile nanoparticle with the surface tension depending on its radius; (ii) a limiting case of bigger objects when the surface tension corresponds to its constant macroscopic value. It has been shown that the external pressure should increase stability of nanoparticles. However, the effect under consideration should be noticeable at high pressures only.

Indefinite integrals arising in the projection methods applied for analysis of wave scattering by penetrable bodies of revolution in spherical coordinates are considered. Explicit expressions for the integrals of products of associated Legendre functions of different degrees and the same orders are derived. Recurrent relations for the integrals of Legendre functions of the same degrees and the same orders, as well as expressions for the indicated integrals via integrals of products of functions of the same degree but of the order smaller by unity, are obtained. All the integrals are expressed via the associated Legendre functions themselves providing a convenient algorithm for using in the MATLAB software.

An automaton is a synchronizing if it has an input word that transfers it from any state to a particular state. There are two versions of synchronization in partial deterministic automata: careful synchronization and exact synchronization. In this paper we focus on the exact version; we survey the complexity of testing exact synchronization and describe a SAT solver based algorithm for calculating the length of the shortest exact synchronizing word.

The operation of the mirror descent algorithm in a weakly inhomogeneous random environment is considered. A weakly inhomogeneous random environment describes a data stream each item of which can be processed by two alterative methods, and probabilities of successive processing of both methods may vary during the control process. Algorithms for one-by-one data processing and batch data processing are given. Using numerical modeling, the behavior of the presented algorithms in a weakly inhomogeneous environment is studied and the obtained results are presented.

In this article, the problem of detection of lightning discharge time and location within the area of the Republic of Kazakhstan is considered. The mathematical model offered in this article is based on a method of timepoints recording by mutually spaced detectors of the network receiving and identifying radio signals, which are generated by cloud to ground lightning discharges (the TOA method − time-of-arrival). According to this method the system of the equations of model is constructed by comparison of distance from a lightning discharge point to stations with the same distance passed by a radio signal taking into account the moment of its records at stations. For solution finding the system of the equations is reducing to continuously differentiable functional with respect to geographical coordinates and the time moment of lightning discharge. The conditions which performing the reached minimum of functional with a practical accuracy provides determination of coordinates and time moment of lightning discharge are considered. The test calculations with estimated accuracy depending on an error of the time moments of lightning discharge detected at the stations are performed.

The theory of the integral equation for azimuthal currents in the axisymmetric problem of diffraction on a disk is constructed. The study is based on the selection of the main positive operator. Existence and uniqueness theorems are obtained. An orthonormal basis satisfying the physical condition by Meixner on the edge is constructed. Calculations are made and good convergence of the reduction method is shown.

The convergence rate of the quasi – Monte Carlo method of search for extremum is examined. It is shown that, if the objective function is nonsingular, then the number of its evaluations required to obtain the desired accuracy ε in the solution can be a slowly (namely, logarithmically) growing function as ε approaches zero.

A program that implements a Markov homogeneous monotonous random search algorithm of an extremum with normal distributions is presented. This program allows to solve a fairly wide class of problems of finding the global extremum of an objective function with a high accuracy.

A program that implements a Markov inhomogeneous monotonous random search algorithm of an extremum with normal distributions is presented. This program allows to solve a fairly wide class of problems of finding the global extremum of an objective function with a high accuracy.

A program that implements a Markov inhomogeneous monotonous random search algorithm of an extremum is presented. This program allows to solve a fairly wide class of problems of finding the global extremum of an objective function with a high accuracy.

A program that implements a Markov homogeneous monotonous random search algorithm of an extremum with Ingber's distribution is presented. This program allows to solve a fairly wide class of problems of finding the global extremum of an objective function with a high accuracy.

A three-parameter probability distribution is presented, modeled on the basis of the negative binomial distribution. This distribution corresponds structurally to the author's distribution of the hyperbolic cosine type by the form of the characteristic function. The difference lies in the use of standard trigonometric cosine and sine for the characteristic function instead of the corresponding hyperbolic functions. The moment-forming polynomials of two arguments are found, derived from the recurrent differential relation to calculate the moments of distribution. A recurrent algebraic formula to produce the integer coefficients of these polynomials is introduced. The set of the coefficients depending on three arguments is ordered and geometrically interpreted as a numerical prism. Numerical prism sections are numerical triangles and numerical sequences. Among the sections of the prism there are well-known and new numerical sets, for example, Stirling numerical triangle, Bessel numerical triangle with alternating signs of elements, alternating tangential numbers, etc. A connection with the numerical prism derived from the hyperbolic cosine type distribution is indicated.

The paper presents a method for modeling probability distributions based on the base normal distribution. The transformation of the base distribution includes truncating the distribution, splitting it into three components in accordance with two specified quantiles, multiplying the density function in each of the intervals by a linear factor. The distribution obtained in this way has a complex nonstandard configuration, including mixtures of the normal distribution and the Rayleigh distribution. A computer program has been developed that makes it possible to model the distribution according to its parameters and implements the criterion for the agreement of sample data with the distribution of the specified type with automatic adjustment of the parameters. The construction algorithm allows for use of other base distributions.

The calculation of the total electron energy E_total and the enthalpy of formation ${{\rm{\Delta }}}_{f}{H}_{298}^{0}$ of the compounds GeO₂, C₂H₆, NH₂CHO, C₆H₆, C₅H₅N and C₆H₄O₂ has been performed using methods B1LYP, B1PW91, B3LYP, BHandH, BHandHLYP, BLYP, BP, CAMY-B3LYP, HTBS, KMLYP, LCY-BLYP, LCY-BP86, LCY-PBE, LDA, M06, M06-2X, M06-HF, M06L, mPBE, mPW, MPW1K, MPW1PW, O3LYP, OLYP, OPBE, OPBE0, PBE, PBE0, PBEsol, PW91, revPBE, revTPSS, RPBE, TPSS, TPSSH, X3LYP and HF in the slater basis QZ4P, aug-TZ2P and TZ2P, as well as DZ, and the errors of calculation of E_total and Δ ${{\rm{\Delta }}}_{f}{H}_{298}^{0}$ at the variation limit were determined. The QZ4P basis set and the settings of the ADF program were configured so that these errors can be attributed to the structure of DFT functionals. The dependence of the E_total and ${{\rm{\Delta }}}_{f}{H}_{298}^{0}$ based on type (size) of the basis set was studied. It has been demonstrated that the functional M06 has the smallest error independent from the incomplete basis set, and the optimal combinations are DFT/TZ2P, where DFT = OPBE, X3LYP, B3LYP, revTPSS, M06L, M06, TPSS, CAMY-B3LYP.

This article describes the formulation of a mathematical model for optimizing the structure of a distributed information-measuring system based on a multi-level topology. The paradigm of cloud, fog and boundary computing is used, which allows building a modern network infrastructure within the framework of the industrial IoT concept. The most typical example of the use of this paradigm may be intelligent power supply networks and distributed systems for monitoring the transportation of gas or oil products. To solve the problem, it is proposed to use an approach based on a genetic algorithm.

The article presents the simulated media player for faster switching between available bit rates with the media stream. Also the investigation includes the results of simulated behavior of the media player while it is receiving media streams in different networks. It is important to identify the real bit rates that are used by the media player for playing media streams at each quantum of time. These results allow better determination of the process of media streams' encoding bit rates which fit best for playing media without delays among the maximum number of users and streams.

A brief review of the methods of calculation of the basic types of binary phase diagrams is done in the framework of the generalized lattice model. The construction of binary phase diagrams with unlimited and limited solubility of the components in the solid state, the phase diagrams of the eutectic types is considered.

The formed sets of ergodic stochastic matrices are focused on solving the problem of classifying probabilistic automaton models by certain criteria/parameters of similarities or differences in the structures of ergodic stochastic matrices using the methods of applied multivariate mathematical statistics. The algorithms developed allow forming a variety of the sets of stochastic matrices through changing the transition and random entry function in a probabilistic automaton. Transition function of the autonomous probabilistic automaton allows using the probabilistic automaton operation algorithm proposed to form the sets of ergodic stochastic matrices that differ in their potencies and structures, based on implementing state set permutations with repetitions and changing the random input variable probability distributions. Defining various autonomous deterministic automaton output functions, we can use the algorithm developed to form the sets of ergodic stochastic matrices that differ in their potencies, with a specified limiting vector, based on implementing the permutations of a set of output letters with repetitions. We are also presenting the evaluations of the potencies of the sets of ergodic stochastic matrices with rational elements, represented by autonomous probabilistic automata under the given constraints.

Kink solutions of the Landau-Khalatnikov equation have been calculated in the case when elastic deformation by twisting is applied to a semi-infinite rod and a finite one. As a crystal without an inversion center was considered, Lifshitz invariants were taken into account in the thermodynamic potential. The specific features of kink propagation at varied boundary conditions, phase velocities and modulus of the torsion moment have been studied.

This paper discusses the solution to the problem of constructing latent complex indexes of a change in a system's quality for several observations in the absence of training. The algorithm for constructing complex indexes is implemented with the definition of non-random variables of the principal component characterizing the structure of the system under discussion. The algorithm uses a new approach to choose the principal component number, determine the weights of the considered variables and subsystems, and to determine the information content of the complex index based on the selected signal-to-noise ratio parameter. The algorithm was used to obtain complex indexes of quality of life for Russia's constituent entities for 2007-2016.

Characteristics of the quality of many systems in the economics, politics, society is in the form of a composite index, take the form of a linear composite index that linearly aggregates over several dimensions using a weight vector. The construction of the complex indexes of a system can be considered as the task of extracting the useful signal from the background noise. The signal in this case is the weights of the linear convolution of the indicators, which should reflect the structure of the system being evaluated. Principal component analysis determines the structure of the principal components for successive observations differently. The reason for this may be the presence of fatal errors in the data used. A modification of the principal component method that takes into account the presence of errors in the data used determines the structure of the system unambiguously. If the nature of the aggregation method of variables and the choice of weights adequately reflect the system quality model, then to study the reliability of the composite index, it is necessary to investigate the stability of ratings over time. The consequence of stability is on average a slight change (increment) in the rating of objects for different measurements. This increment can be a posteriori estimated by a set of observations on the proposed dispersion criterion. The results of assessing the stability of different integral characteristics by this criterion are given. Complex indexes, calculated by the author's method, show good stability.

This paper presents the mathematical model of moving target detection against the clutter background of the radar with quasi-continuous mode of transmission and reception of signals with pseudorandom law of amplitude-phase-shift keying. The modeling results of the optimal pair of "signal - processing method" research, which provides the maximum range of target detection under noise conditions, are presented.

A non-probabilistic interpretation of the distribution functions is obtained. The cases of instantaneous and retarded interactions are considered. The derivation is based on the use smoothing operator that does not change the symmetry of the equation of motion with respect to time inversion.