https://bphm.knu.ua/index.php/bphm/issue/feedBulletin of Taras Shevchenko National University of Kyiv. Physics and Mathematics2023-08-03T16:56:27+03:00Irina Rozora // Ірина Розораrozora.iryna@gmail.comOpen Journal Systems<p>The journal contains results of new research in various fields of mathematics, computer science, mechanics, and physics. For researchers, teachers, postgraduates, engineers and students. </p>https://bphm.knu.ua/index.php/bphm/article/view/371Calculation of the magnetic susceptibility of a degenerate ideal gas of electrons in the case of anisotropic effective mass2023-08-03T16:29:26+03:00V. O. Andreyevvolodymyr.andreev@knu.ua<p><em>Dependence of the diamagnetic susceptibility of a degenerate ideal gas of electrons on the orientation of the magnetic field strength vector relative to the crystal axes, which is due to the anisotropy of the effective mass of electrons is considered. The presence of a diamagnetic susceptibility of a degenerate ideal gas of electrons is one of the fundamental results of the application of quantum mechanics and statistical physics to the description of magnetic phenomena. For an ideal gas of electrons in vacuum, the magnetic susceptibility has diamagnetic and paramagnetic components, the origin of which is due to different mechanisms. At low values of the magnetic field strength and high temperatures, their values are proportional to each other. Since the total value of the susceptibility is measured, it is impossible to separate them. Moreover, when measuring the magnetic susceptibility of a crystal, the contribution of conduction electrons cannot be separated from the contributions of the electrons of atoms, unless some features of the behavior of the magnetic susceptibility of conduction electrons are known. In this article, the calculation of the angular dependence of the effective mass of electrons due to the anisotropy is carried out. The angular dependence of the diamagnetic susceptibility is calculated for the practically important case when the distances between the nearest magnetic levels are small compared to the temperature. It is shown that the angular dependence of the diamagnetic susceptibility coincides with the angular dependence of the cyclotron mass squared, which can be determined from independent cyclotron resonance experiments. It is proposed to use the results obtained to separate the diamagnetic susceptibility of conduction electrons from susceptibilities due to other mechanisms.</em></p> <p><em><strong>Pages of the article in the issue</strong></em>: 108 - 111</p> <p><em><strong>Language of the article</strong>: Ukrainian</em></p>2023-07-13T00:00:00+03:00Copyright (c) 2023 V. O. Andreyevhttps://bphm.knu.ua/index.php/bphm/article/view/377Сumulants and parameters сalculation of distribution function by Monte Carlo and from the equation for it2023-08-03T16:35:25+03:00V. O. Andreyevvolodymyr.andreev@knu.uaYa. V. Choliycharlie@mail.univ.kiev.uaM. V. Makaretsmmv@univ.kiev.ua<p><em>Calculations results of the moments, central moments, cumulants and parameters of the distribution function of a beam of ions implanted in a solid body were analyzed. To analyze the differences between the results of modeling this process by the Monte Carlo method, which is widely used for practically important targets, and the results of the solution of the corresponding integrodifferential equations that describe ions distribution analytically, in the simple case of axial symmetry of the ion beam, when all moments of odd order along the transverse Cartesian coordinate is considered equal to zero due to the symmetry of the problem. It is shown that the same moments obtained by the Monte Carlo method is not exactly equal to zero, but slowly decrease with an increase in the number of ions, as predicted by statistics, and then remain approximately constant. Increasing the number of ions for Monte Carlo simulation reduces the statistical component of this error, but does not affect on the component arising from the application of a simplified model of ion-atom collisions.</em></p> <p><em><strong>Pages of the article in the issue</strong></em>: 112 - 115</p> <p><em><strong>Language of the article</strong>: Ukrainian</em></p>2023-07-13T00:00:00+03:00Copyright (c) 2023 V. O. Andreyev, Ya. V. Choliy, M. V. Makaretshttps://bphm.knu.ua/index.php/bphm/article/view/378Comparative analysis of some anti-missile defense/air defense systems using fuzzy set theory2023-08-03T16:56:27+03:00S. I. Kolesnykua.serhii.kolesnyk@gmail.comM. V. Maksyutamaksyuta.nik@gmail.comV. V. Obukhovskyobukhovsky.v@gmail.com<p><em>The article is devoted to the problem of using air defense / missile defense systems to protect critical and military infrastructure during hostilities. Attention is drawn to the fact that these systems can be attacked and damaged, which can lead to significant losses of critical infrastructure. To prevent (or reduce) these losses, this paper proposes to use information that can be obtained on the basis of a comparative analysis of the performance characteristics of the used air defense / missile defense systems. In particular, on the example of five such systems, it is demonstrated how this comparative analysis can be carried out using the theory of fuzzy sets (for this, the work considered inclusion operations, dominance relations, calculated the so-called linear and quadratic fuzzy indices, and also constructed various sets of α-levels).</em></p> <p><em><strong>Pages of the article in the issue</strong></em>: 116 - 122</p> <p><em><strong>Language of the article</strong>: Ukrainian</em></p>2023-07-13T00:00:00+03:00Copyright (c) 2023 S. I. Kolesnyk, M. V. Maksyuta, V. V. Obukhovskyhttps://bphm.knu.ua/index.php/bphm/article/view/356Number-theoretic functions for Gaussian integers2023-03-12T21:14:24+02:00Nikita Arskyinikita.arskiy.25@gmail.com<p><em>The classical number-theoretic functions – a number of divisors τ(n), sum of the divisors σ(n) and product of the divisors π(n) of a positive integer n – were generalized to the ring Z[i] of Gaussian integers. For the evaluation of the corresponding functions τ*(α), σ*<sub>m</sub>(α) and π*(α), obtained were the explicit formulae that use the canonical representation of α. A number of properties of these functions were studied, in particular, estimates from above for the functions τ*(α) and σ*<sub>m</sub>(α) and the properties connected with divisibility of their values by certain numbers. Researched are also sums of products of powers of the divisors for α∈Z[i].</em></p> <p><em><strong>Pages of the article in the issue</strong></em>: 11 - 20</p> <p><em><strong>Language of the article</strong>: English</em></p>2023-07-13T00:00:00+03:00Copyright (c) 2023 Nikita Arskyihttps://bphm.knu.ua/index.php/bphm/article/view/365Derivations of infinite-dimensional Lie superalgebras2023-07-15T13:20:11+03:00D. I. Bezushchakbezushchak@gmail.com.uaO. O. Bezushchakbezushchak@knu.ua<p><em>We study infinite-dimensional analogs of classical Lie superalgebras over an algebraically closed </em><em>field F of zero characteristic. Let I be an infinite set. For an algebra M_∞ (I) of infinite I × I matrices </em><em>over a ground field F having finitely many nonzero entries, we consider the related Lie superalgebra </em><em>gl_∞ (I1, I2) and its commutator sl_∞ (I1, I2) for a disjoint union of nonempty subsets I1 and I2 of the </em><em>set I; and we describe derivations of the Lie superalgebra sl_∞ (I1, I2).</em></p> <p><em><strong>Pages of the article in the issue</strong></em>: 21 - 25</p> <p><em><strong>Language of the article</strong>: English</em></p>2023-07-13T00:00:00+03:00Copyright (c) 2023 The Author(s)https://bphm.knu.ua/index.php/bphm/article/view/366Addendum to the paper “Corepresentations of Munn matrix algebras”2023-07-15T13:41:27+03:00V. M. Bondarenkovitalij.bond@gmail.com<p><em>This paper is an addendum to the paper, published in Bulletin of Taras Shevchenko National Uni</em><em>versity of Kyiv (Series: Physics & Mathematics), 2022, No. 3, pp. 42-44.</em></p> <p><em>In the original version does not given a clear definition of the corepresentation of an algebra with </em><em>identity what led to some formal inaccuracies (regarding notations, writing of defining relations, etc.). </em><em>All the results of the original paper are essentially correct, but an error was made in the last part of </em><em>the proof of Theorem 1, which does not affect the correctness of the idea of proof, but requires extended </em><em>considerations.</em></p> <p><em>In this addendum the author provides detailed information on everything related to statements </em><em>about the generating elements and defining relations of the Munn algebras over algebras with identity.</em></p> <p><em><strong>Pages of the article in the issue</strong></em>: 26 - 29</p> <p><em><strong>Language of the article</strong>: English</em></p>2023-07-13T00:00:00+03:00Copyright (c) 2023 The Author(s)https://bphm.knu.ua/index.php/bphm/article/view/367Extinction and persistence in stochastic predator population density-dependent predator-prey model with jumps2023-07-25T10:19:52+03:00O. D. Borysenkoborysenko13101953@gmail.comO. V. Borysenkooborisenko1373@gmail.com<p><em>The non-autonomous stochastic density dependent predator-prey model with Holling-type II functional response disturbed by white noise, centered and non-centered Poisson noises is investigated. Corresponding system of stochastic differential equations has a unique, positive, global (no explosions in a finite time) solution. Sufficient conditions are obtained for extinction, non-persistence in the mean, weak and strong persistence in the mean of a predator and prey population densities in the considered stochastic predator-prey model.</em></p> <p><em><strong>Pages of the article in the issue</strong></em>: 30 - 36</p> <p><em><strong>Language of the article</strong>: English</em></p>2023-07-13T00:00:00+03:00Copyright (c) 2023 The Author(s)https://bphm.knu.ua/index.php/bphm/article/view/358Cross-validation for local-linear regression by observations from mixture2023-06-12T13:21:25+03:00Daniel Horbunovdanielhorbunov@knu.uaRostyslav Maiborodarostmaiboroda@gmail.com<p><em>We consider a generalization of local-linear regression for estimation of compnents' regression functions by observations from mixture with varying concentrations. A cross-validation technique is developed for the bahdwidth selection. Performance of the obtained estimator is compared with the modified Nadaraya-Watson estimator performance by simulations.</em></p> <p><em><strong>Pages of the article in the issue</strong></em>: 37 - 43</p> <p><em><strong>Language of the article</strong>: Ukrainian</em></p>2023-07-13T00:00:00+03:00Copyright (c) 2023 Daniel Horbunov, Rostyslav Maiborodahttps://bphm.knu.ua/index.php/bphm/article/view/368Mathematical modeling of the dynamics of an insurance company based on macro indicators2023-07-30T19:12:22+03:00V. P. Zubchenkovolodymyr.zubchenko@knu.uaP. V. Aleksandrovaaleksandrova.polina@knu.ua<p><em>In the paper we investigate the dynamics of a life insurance company depending on changes in macro indicators. For the sensitivity study, profitability testing is performed for the average life insurance contract. Profitability testing is based on an estimate of the expected present value of future cash flows. Various approaches to the choice of discount rates are investigated. The choice of strategy for using the rates of the zero-coupon yield curve built on the UAH domestic government bonds is substantiated. The dependence of the zero-coupon yield curve on the change in the key rate is shown. The dynamics of the model at different levels of the interest rates, namely guaranteed investment income, average profitability of investment activity, key rate, is considered. The scenario of the predicted decrease in the key rate at the stage of post-war reconstruction of Ukraine is considered. The profitability of the insurance company was tested based on the macro parameters of the specified scenarios. The expediency of using the chosen approach for researching the dynamics of a life insurance company in the conditions of the current high volatility of the Ukrainian economy is shown.</em></p> <p><em><strong>Pages of the article in the issue</strong></em>: 44 - 47</p> <p><em><strong>Language of the article</strong>: Ukrainian</em></p>2023-07-13T00:00:00+03:00Copyright (c) 2023 V. P. Zubchenko, P. V. Aleksandrovahttps://bphm.knu.ua/index.php/bphm/article/view/369Detection of hidden periodicities in models with discrete time and long range dependent random noise2023-07-31T10:08:12+03:00A. V. Ivanovalexntuu@gmail.comI. V. Orlovskyiorlovskyi@matan.kpi.ua<p><em>Trigonometric regression models take a special place among various models of nonlinear regression analysis and signal processing theory. The problem of estimating the parameters of such models is called the problem of detecting hidden periodicities, and it has many applications in natural and technical sciences.</em></p> <p><em>The paper is devoted to the study of the problem of detecting hidden periodicities in the case when we observe only one harmonic oscillation with discrete time, where random noise is a local functional of Gaussian random sequence with singular spectrum. In particular, the random sequence in the model can be strongly dependent.</em></p> <p><em>For estimation of unknown parameters the periodogram estimator is chosen. Sufficient conditions of the consistency of the amplitude and angular frequency periodogram estimator of the model described above are obtained in the paper.</em></p> <p><em>The proof of Lemmas 1 and 2 gave an important asymptotic properties of the random noise functional related to the periodogram estimator which necessary for the proof of the main results. Series expansion of random noise in terms of Hermite polynomials and the Diagram formula are main tools that were used to obtain this lemmas.</em></p> <p><em><strong>Pages of the article in the issue</strong></em>: 48 - 58</p> <p><em><strong>Language of the article</strong>: Ukrainian</em></p>2023-07-13T00:00:00+03:00Copyright (c) 2023 A. V. Ivanov, I. V. Orlovskyihttps://bphm.knu.ua/index.php/bphm/article/view/370On interaction of liquid drops located in radiation field of the acoustic wave2023-07-31T11:08:11+03:00O. P. Zhukopzhuk1939@gmail.comY. A. Zhuky.zhuk@i.uaT. V. Klimchuky.zhuk@i.ua<p><em>Propagation of the plane acoustic wave along the center line of two liquid spherical drops placed into a space filled with another liquid is under investigation. An approach is elaborated to characterize the interaction between the liquid drops caused by the acoustic radiation forces that are the time-constant components of hydrodynamic forces acting upon the drops located in the outer liquid. Investigation of the acoustic radiation forces influencing the drops in the acoustic field is performed in the frame of two-step procedure. The first step comprises solution of the linear problem of incident wave diffraction on the drops while the second one is calculation of the hydrodynamic forces acting upon each spherical drop followed by time averaging of forces determined. The analytical formula for the acoustic radiation force calculation is derived for the case under consideration.</em></p> <p><em><strong>Pages of the article in the issue</strong></em>: 61 - 64</p> <p><em><strong>Language of the article</strong>: Ukrainian</em></p>2023-07-13T00:00:00+03:00Copyright (c) 2023 O. P. Zhuk, Y. A. Zhuk, T. V. Klimchukhttps://bphm.knu.ua/index.php/bphm/article/view/372Comparative dynamics of liquid outflow from containers of certain geometric shapes depending on the change in the axis of rotation2023-07-31T14:50:41+03:00V. O. Bilyivaleri_y@ukr.netO. G. Bilyibelyi.oleksandr@gmail.com<p><em>In metallurgical enterprises, in some plants and factories, in technological processes, pouring of various types of liquids (cast iron, steel, gold, silver, chocolate, etc.) from large containers of various geometric shapes into smaller standard forms by the method of turning the container relative to a fixed point or a fixed axis and pouring over the edge It is important that the flow of liquid is uniform, which in reality is not true even at a constant angular speed of rotation. This paper examines the process of pouring from containers of conical and parabolic shapes, which are the most widely used in practice. In the development of the authors' works on this topic, another mechanism of rotation, which is often found, is considered, namely, rotation relative to the diameter of the upper part of the boiler. We analyze the obtained results of the study of the dynamics of changes in fluid consumption depending on the angle of inclination in comparison with the results of the previous works of the authors and provide conclusions and recommendations. Taking into account these theoretical studies, the operation of rotary mechanisms in each specific case can be adjusted by developing an algorithm of sequential actions that ensure the uniformity of the pouring process.</em></p> <p><em><strong>Pages of the article in the issue</strong></em>: 65 - 68</p> <p><em><strong>Language of the article</strong>: Ukrainian</em></p>2023-07-13T00:00:00+03:00Copyright (c) 2023 V. O. Bilyi, O. G. Bilyihttps://bphm.knu.ua/index.php/bphm/article/view/373Review of approaches for paraphrase identification2023-07-31T15:44:32+03:00V. N. Vrublevskyivitalii.vrublevskyi@gmail.comA. A. Marchenkorozenkrans17@gmail.com<p><em>The article is devoted to a review of approaches to solving the problem of identifying paraphrases. This problem's relevance and use in tasks such as plagiarism detection, text simplification, and information search are described. Several classes of solutions were considered. The first approach is based on manual rules - it uses manually selected features based on the fundamental properties of paraphrases. The second approach is based on lexical similarity and various databases and ontologies. Machine learning-based approaches are also presented in this paper and describe different architectures that can be used to identify paraphrases. The last approach considered is based on deep learning and modern models of transformers.</em></p> <p><em><strong>Pages of the article in the issue</strong></em>: 71 - 78</p> <p><em><strong>Language of the article</strong>: Ukrainian</em></p>2023-07-13T00:00:00+03:00Copyright (c) 2023 V. N. Vrublevskyi, A. A. Marchenkohttps://bphm.knu.ua/index.php/bphm/article/view/374Exact pattern matching. Current achievements and research2023-07-31T16:15:17+03:00Anton Zuievant0n3uev@gmail.com<p><em>The problem of exact pattern matching is an essential programming problem. Different algorithms that solve this problem are core elements of search engines, version control systems, text editors, DNA analyzers, and many others. For simplification reasons articles usually denote pattern as P or p and pattern length as M or m. Similarly, the text is usually denoted as T or t and its length - N or n. Alphabet is denoted Σ and its length - |Σ|. Based on these notations the problem of pattern matching can be written as follows:</em></p> <p><em>Find all positions/ amount of i, such that P[0...m] = T[i...i + m], or: Find all positions i in text for which substring starting at position i of the text of length m is equal to the pattern.</em></p> <p><em>The main parameters of this problem are pattern length and alphabet size. The length of the text usually doesn’t matter because, for any long enough text of a specific structure, the run time of the algorithm per character will be close to constant. Besides that, the specifics of the input data and text may also impact the performances of the algorithms. All of that makes the problem both very nuanced and interesting to investigate. This problem features a lot of different existing solutions developed over the course of the last 5 decades. The main part of the work provides short descriptions and analyses of a set of algorithms that are still relevant in the field. Besides that, some remarks are made on the topic of their theoretical regions of efficiency and how they depend on the specifics of the input. The results of the practical experimentation on the variety of randomly generated test data are provided. The conclusion provides some analysis of the received results and algorithms’ class efficiency based on the input as well as a visual representation of the received results in a form of a table representing the most efficient algorithm for each pair of pattern length and alphabet size.</em></p> <p><em><strong>Pages of the article in the issue</strong></em>: 79 - 88</p> <p><em><strong>Language of the article</strong>: Ukrainian</em></p>2023-07-13T00:00:00+03:00Copyright (c) 2023 Anton Zuievhttps://bphm.knu.ua/index.php/bphm/article/view/375Development of a Web Application Testing Automation Platform2023-07-31T17:38:08+03:00D. O. Lipskyilipsky@knu.ua<p><em>In the modern fast-paced world of web development, automated testing of web applications has become an integral part of the software development process. This article provides an overview of a platform for automating web application testing, which simplifies the process of creating, executing, and analyzing automated tests. The main components of the platform, their functional capabilities, and advantages are discussed.</em></p> <p><em>One of the key components of this platform is a module for interacting with the web application, which provides a convenient way to control the browser and perform actions on web pages, such as navigation, form filling, clicking on elements, and more. This enables automated verification of various aspects of web application functionality and correctness.</em></p> <p><em>Another important component is the reporting module, which ensures the collection and storage of test </em><em>results. It allows for the creation of detailed reports on executed tests, identified errors, and other useful </em><em>information. This module helps understand the state of the software, facilitates the detection of issues, and </em><em>simplifies error reproduction.</em></p> <p><em>The platform also includes a module for creating test data, which can be useful for generating random data, loading test datasets from databases or external files. This allows developers to create various testing scenarios and execute them on different data sets.</em></p> <p><em>One of the key advantages of this platform is the ability to configure testing parameters for different projects and environments. Developers can set different configurations, such as browser types, versions, extensions, and utilities used, for each individual project. This allows for customization of testing according to specific needs and ensures maximum compatibility.</em></p> <p><em>The platform also provides additional utilities to facilitate the testing process, such as integration with code repositories, version control systems, build automation, and deployment tools. This contributes to fast and efficient test execution, simplifies their integration into the development process, and improves overall software quality.</em></p> <p><em>Overall, this web application testing automation platform provides developers with convenient tools that simplify the testing process, ensure high-quality software, and facilitate the rapid deployment of web applications.</em></p> <p><em><strong>Pages of the article in the issue</strong></em>: 89 - 93</p> <p><em><strong>Language of the article</strong>: Ukrainian</em></p>2023-07-13T00:00:00+03:00Copyright (c) 2023 D. O. Lipskyihttps://bphm.knu.ua/index.php/bphm/article/view/376Web applications as one of the modern ways of implementing decision support systems2023-07-31T18:32:15+03:00O. P. Ilkunalexander.ilkun@gmail.com<p><em>Decision support systems have a wide range of applications in various fields, including but not limited to commercial activities, medical institutions, educational institutions, scientific research, engineering, and government agencies. These systems support efficient processing of information from various sources using a variety of methodological approaches, including statistical analysis, data analysis, machine learning methods, optimization, and many others.</em></p> <p><em>In view of the above-mentioned features, there is an important issue of developing effective decision support systems that would meet the needs of modern organizations. This article reviews current approaches to software development, including analyzing a potential architecture for implementing a decision support system focused on patient diagnosis using fuzzy logic.</em></p> <p><em><strong>Pages of the article in the issue</strong></em>: 94 - 100</p> <p><em><strong>Language of the article</strong>: Ukrainian</em></p>2023-07-13T00:00:00+03:00Copyright (c) 2023 O. P. Ilkunhttps://bphm.knu.ua/index.php/bphm/article/view/357Geometric recurrence of inhomogeneous Gaussian autoregression process2023-04-09T23:46:45+03:00Olga Moskanovaolga.moskanova@knu.ua<p><em>In this paper we study Gaussian autoregression model of the form X_{n+1} = α_{n+1} X_n + W_{n+1}. It has time-inhomogeneous centered normal increments W_n and control ratios α_n. We obtained upper bounds for expectation of exponential return time to the compact [−c; c] and for expectation of the function of compressing ratios and the mentioned moment.</em></p> <p><em><strong>Pages of the article in the issue</strong></em>: 101 - 105</p> <p><em><strong>Language of the article</strong>: English</em></p>2023-07-13T00:00:00+03:00Copyright (c) 2023 Olga Moskanova