Кафедра математики

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    Inquiry-based learning for enhancing students’ interest in mathematical research: a case study on approximation theory and Fourier series
    (2023) Vlasenko, Kateryna; Rovenska, Olha; Lovianova, Iryna; Kondratyeva, Oksana; Achkan, Vitaliy; Tkachenko, Yana
    This paper investigates how to develop students’ interest in mathematical research by using inquiry-based learning (IBL) as a pedagogical approach. We conducted a case study on the application of IBL to the teaching of approximation theory and Fourier series, which are important topics in mathematics and computer science. We surveyed the students who participated in the IBL workshops and measured their emotional state using the Differential Emotion Scale (DES) by Izard. The results showed that the IBL environment reduced the students’ negative emotions and increased their positive emotions, which in turn enhanced their engagement and motivation in the mathematical research activities. We conclude that IBL is an effective method for fostering students’ interest in mathematical research and suggest some implications for future practice and research.
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    The method of using the online course "Creative Thinking through Learning Elementary Maths" in the Mathematics teacher training system
    (2023) Achkan, Vitaliy; Vlasenko, Kateryna; Lovianova, Iryna; Sitak, Iryna; Armash, Tetiana
    The article offers the method of using the online "Creative Thinking through Learning Elementary Maths" and the results of checking the effectiveness of the implementation of this course in the mathematics teacher training system. The research substantiates the possibility of three methodological options for combining educational materials of the online course with the study of the educational discipline "Elementary Mathematics". The methodological options are chosen depending on the number of credits assigned to the study of the elementary mathematics course at the university. Practical tasks of the course were focused on the organization of students' activities (recognition, classi cation, solving and creation) with Rich tasks of three levels of complexity. The results of control tests in Elementary Mathematics give grounds for concluding about the effectiveness of the developed method.
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    Interdisciplinary connections of Mathematics and Literature in the preparation for External Independent Assessment of Humanities students
    (2023) Vlasenko, Kateryna; Lovianova, Iryna; Achkan, Vitaliy; Armash, Tetiana; Chumak, Olena
    This article examines the issue of finding new methodological approaches to the development of problems that can contribute to the preparation of ‘Humanities’ students to pass the state final certification in the format of External independent assessment (EIA). The study analyzes the existing development of methods of qualitative mathematical training of students of humanities classes who study Mathematics according to the basic level program. In particular, the authors of the article were interested in the use of interdisciplinary connections between Mathematics and humanitarian disciplines. The authors of the article identified crosscurricular problems as a means of forming students’ motivation to study Mathematics, as well as a toolkit for measuring their level of readiness to solve typical problems of EIA. The researchers set a goal to develop cross-curricular problems following the EIA program based on the plot of Bulgakov’s novel ‘The Master and Margarita’. An experiment was conducted to confirm the effectiveness of the use of the system of interdisciplinary problems in the preparation for the EIA of humanitarian students. The obtained results confirmed that the implementation of problems in which the intersubject connections of Mathematics and Literature are realized has a more significant effect on both increasing the motivation of students to study Mathematics and their mathematical preparation.
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    The Wheel of Mathematics Learning Methods
    (2023) Vlasenko, Kateryna; Lovianova, Iryna; Chumak, Olena; Sitak, Iryna; Kovalenko, Daria
    The article is devoted to the creation of a wheel of mathematics learning methods. The study analyzed how Bloom's taxonomy is used by the educational community. Special attention is paid to the study of the issue of bringing to a single system a set of disparate goals and tasks of learning in new models of education, which provide for the openness of learning to new technologies. Analysis of scienti c works and resources helped to determine the structuring of the Wheel of Mathematics Learning Methods (WMLM). The authors of the article identi ed the areas of didactics that should ll the wheel: Competencies, Motivation, Bloom's Taxonomy, Activities, and Technologies. A structured set of hints for the teacher is presented in the form of 5 sectors of the wheel: Apply, Analyze, Evaluate, Create, Remember Understand. Each sector is a set of constituent components Methods, Forms, Tools, Activities and Active Verbs. The analysis of the results of the survey of 58 respondents proved that Bloom's taxonomy is important both in the selection of teaching methods and means and in the determination of e ective forms of organization of educational activities. It was concluded that the idea of developing a methodology wheel should be based on the structuring of a set of tips for the teacher at all stages of activity from planning to implementation of mathematics education.
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    An optimal lower bound for the size of periodic digraphs
    (2023) Kozerenko, Sergiy
    A periodic digraph is the digraph associated with a periodic point of a continuous map from the unit interval to itself. This digraph encodes "covering" relation between minimal intervals in the corresponding orbit, which allows the application of purely combinatorial arguments in establishing results on the existence and co-existence of periods of periodic points (for example, in proving the famous Sharkovsky’s theorem). In this article, an optimal lower bound for the size of periodic digraphs is provided and thus some previous results of Pavlenko are tightened.
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    Unique eccentric point graphs and their eccentric digraphs
    (2023) Hak, Artem; Haponenko, Vladyslav; Kozerenko, Sergiy; Serdiuk, Andrii
    We study graph-theoretic properties of eccentric digraphs of unique eccentric point graphs (shortly, uep-graphs). The latter are the connected graphs in which every vertex has a unique eccentric vertex. In particular, we characterize uep-graphs and the corresponding eccentric digraphs in the following classes: self-centered graphs having the number of vertices twice as diameter, block graphs, and graphs with diameter three. Also, we obtain non-trivial properties of weak components in eccentric digraphs of uep-graphs with diameter four and pose several open questions in this direction.
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    The first boundary value problem for the nonlinear equation of heat conduction with deviation of the argument
    (2022) Drin, Yaroslav; Drin, Iryna; Drin, Svitlana; Stetsko, Yuriy
    The initial-boundary problem for the heat conduction equation with the inversion of the argument are considered. The Green’s function of considered problem are determined. The theorem about the Poisson integral limitation is proved. The theorem declared that the Poisson integral determine the solution of the first boundary problem considered and proved.
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    Analysis of the Shape of Wave Packets in the "Half Space–Layer–Layer with Rigid Lid" Three-Layer Hydrodynamic System
    (2022) Avramenko, Olga; Lunyova, Мariia
    We study the process of propagation of weakly nonlinear wave packets on the contact surfaces of a "half space–layer–layer with rigid lid" hydrodynamic system by the method of multiscale expansions. The solutions of the weakly nonlinear problem are obtained in the second approximation. The condition of solvability of this problem is established. For each frequency of the wave packet, we construct the domains of sign constancy for the coefficient for the second harmonic on the bottom and top contact surfaces. The regularities of wave formation are determined depending on the geometric and physical parameters of the hydrodynamic system. We also analyze the plots of the shapes of deviations of the bottom and top contact surfaces typical of the constructed domains of sign-constancy of the coefficient. We discover the domains where the waves become ∪ - and ∩ -shaped and reveal a significant influence of wavelength on the shapes of deviations of the contact surfaces of the analyzed hydrodynamic system.
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    The nonlocal problem for fractal diffusion equation
    (2022) Drin, Yaroslav; Drin, I.; Drin, Svitlana
    Over the past few decades, the theory of pseudodifferential operators (PDO) and equations with such operators (PDE) has been intensively developed. The authors of a new direction in the theory of PDE, which they called parabolic PDE with non-smooth homogeneous symbols (PPDE), are Yaroslav Drin and Samuil Eidelman. In the early 1970s, they constructed an example of the Cauchy problem for a modified heat equation containing, instead of the Laplace operator, PDO, which is its square root. Such a PDO has a homogeneous symbol |σ|, which is not smooth at the origin. The fundamental solution of the Cauchy problem (FSCP) for such an equation is an exact power function. For the heat equation, FSCP is an exact exponential function. The Laplace operator can be interpreted as a PDO with a smooth homogeneous symbol |σ|^2, σ ∈ Rn. A generalization of the heat equation is PPDE containing PDO with homogeneous non-smooth symbols. They have an important application in the theory of random processes, in particular, in the construction of discontinuous Markov processes with generators of integro-differential operators, which are related to PDO; in the modern theory of fractals, which has recently been rapidly developing. If the PDO symbol does not depend on spatial coordinates, then the Cauchy problem for PPDE is correctly solvable in the space of distribution-type generalized functions. In this case, the solution is written as a convolution of the FSCP with an initial generalized function. These results belong to a number of domestic and foreign mathematicians, in particular S. Eidelman and Y. Drin (who were the first to define PPDO with non-smooth symbols and began the study of the Cauchy problem for the corresponding PPDE), M. Fedoruk, A. Kochubey, V. Gorodetsky, V . Litovchenko and others. For certain new classes of PPDE, the correct solvability of the Cauchy problem in the space of Hölder functions has been proved, classical FSCP have been constructed, and exact estimates of their power-law derivatives have been obtained [1–4]. Of fundamental importance is the interpretation of PDO proposed by A. Kochubey in terms of hypersingular integrals (HSI). At the same time, the HSI symbol is constructed from the known PDO symbol and vice versa [6]. The theory of HSI, which significantly extend the class of PDO, was developed by S. Samko [7]. We extends this concept to matrix HSI [5]. Generalizations of the Cauchy problem are non-local multipoint problems with respect to the time variable and the problem with argument deviation. Here we prove the solvability of a nonlocal problem using the method of steps. We consider an evolutionary nonlinear equation with a regularized fractal fractional derivative α ∈ (0, 1] with respect to the time variable and a general elliptic operator with variable coefficients with respect to the second-order spatial variable. Such equations describe fractal properties in real processes characterized by turbulence, in hydrology, ecology, geophysics, environment pollution, economics and finance.
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    Відновлююче спектральне число зважених графів-циклів
    (Молодий вчений, 2022) Пилипіва, Олександра; Тимошкевич, Лариса
    Матеріали доповіді учасників науково-практичної конференції "Наука в контексті глобальної трансформації суспільства", м. Полтава, 26-27 серпня 2022 р.
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    Multiple auxiliary classifiers GAN for controllable imagegeneration: Application to license plate recognition
    (2021) Shvai, Nadiya; Hasnat, Abul; Nakib, Amir
    One of the main challenges in developing machine learning (ML) applications is the lack of labeled and balanced datasets. In the literature, different techniques tackle this problem via augmentation, rendering, and over-sampling. Still, these methods produce datasets that appear less natural, exhibit poor balance, and have less variation. One potential solution is to leverage the Generative Adversarial Network (GAN) which achieves remarkable results in the generation of high-fidelity natural images. However, expanding the ability of GANs’ to control generated image attributes with supervisory information remains a challenge. This research aims to propose an efficient method to generate high-fidelity natural images with total control of its main attributes. Therefore, this paper proposes a novel Multiple Auxiliary Classifiers GAN (MAC-GAN) framework based on Auxiliary Classifier GAN (AC-GAN), multi-conditioning, Wasserstein distance, gradient penalty, and dynamic loss. It is therefore presented as an efficient solution for highly controllable image synthesis red that allows to enrich and re-balance datasets beyond data augmentation. Furthermore, the effectiveness of MAC-GAN images on a target ML application called Automatic License Plate Recognition (ALPR) under limited resource constraints is probed. The improvement achieved is over 5% accuracy, which is mainly due to the ability of the MAC-GAN to create a balanced dataset with controllable synthesis and produce multiple (different) images with the same attributes, thus increasing the variation of the dataset in a more elaborate way than data augmentation techniques.
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    Класифiкацiя злiченних графiв Кокстера вiдносно iндексу у промiжку (√√5 + 2; 3/√2]
    (2022) Тимошкевич, Лариса; Когут, Марія
    Досліджено структуру зліченних графів Кокстера зі значенням індексу в проміжку від √√5 + 2 до 3/√2. Зокрема, такі графи є деревами, можуть мати щонайбільше одну позначку на ребрах, більшу за 3, і такі позначки не перевищують 6, можуть мати лише вершини степеня строго меншого за 5, і серед ребер, інцидентних вершині степеня 4, може бути лише одне, що інцидентне не висячій вершині. Також наведено ряд інших властивостей зліченних графів Кокстера з індексами у квазаному проміжку.
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    Оберненi спектральнi задачi для зважених графiв
    (2022) Пилипіва, Олександра; Тимошкевич, Лариса
    Роботу присвячено оберненим спектральним задачам для зважених графiв. Наведено верхню оцiнку спектрального вiдновлюючого числа для дерев та унiциклiчних графiв.
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    Модель системи одночасних рiвнянь з лаговим ефектом для оцiнки якостi рекламної кампанiї
    (2022) Дрінь, Світлана; Резнiченко, Єгор
    У цiй статтi описано бiльш узагальнену систему одночасних рiвнянь для прогнозування рiвня продажiв залежно вiд рекламних кампанiй за рiзними каналами та iнших факторiв. У середовищах RStudio та Google Colab описано створення моделi на основi реальних даних деякого товару. Застосовано специфiкацiйний тест Хаусмана для визначення методу оцiнки моделi. Оскiльки показники рекламних кампанiй виявились ендогенними змiнними, зроблено висновок про важливiсть використання двокрокового методу найменших квадратiв (2МНК). Виявлено, що обсяг реклами є причиною для обсягу продажiв за Гренжером, що не можна сказати про зворотне припущення, а саме, причиновiсть обсягу реклами вiд продажiв за Гренжером. Також визначено "глибину" лагiв. Пiсля проведеного дослiдження виявили вагомi лаги, тобто по одному лагу для обох каналiв реклами. Було оцiнено залежностi обсягiв продажiв вiд рiзних факторiв, у тому числi дистрибуцiї товару, цiнового iндексу, впливу реклами та її лагiв, вплив рекламної дiяльностi конкурентiв. Коефiцiєнти утвореної бiльш узагальненої системи одночасних рiвнянь були оцiненi за допомогою двокрокового методу найменших квадратiв. Усi статистичнi показники свiдчать про адекватнiсть моделi. Показники ефективностi (ROI – return on investment) рекламних кампанiй показали, що реклама i на телебаченнi, i в iнтернетi є прибутковою для розглянутого товару фiрми. Актуальнiсть цiєї статтi полягає у створеннi бiльш узагальненої системи одночасних рiвнянь iз включенням моделi прогнозу продажiв товару з урахуванням впливу реклами.
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    Two approaches for option pricing under illiquidity
    (2022) Pauk, Viktoriia; Petrenko, Oksana; Shchestyuk, Nataliya
    The paper focuses on option pricing under unusual behaviour of the market, when the price may not be changed for some time what is quite a common situation on the modern financial markets. There are some patterns that can cause permanent price gaps to form and lead to illiquidity. For example, global changes that have a negative impact on financial activity, or a small number of market participants, or the market is quite young and is just in the process of developing, etc. In the paper discrete and continuous time approaches for modelling market with illiquidity and evaluation option pricing were considered. Trinomial discrete time model improves upon the binomial model by allowing a stock price not only to move up, down but stay the same with certain probabilities, what is a desirable feature for the illiquid modelling. In the paper parameters for real financial data were identified and the backward induction algorithm for building call option price trinomial tree was applied. Subdiffusive continuous time model allows successfully apply the physical models for describing the trapping events to model financial data stagnation’s periods. In this paper the Inverse Gaussian process IG was proposed as a subordinator for the subdiffusive modelling of illiquidity and option pricing. The simulation of the trajectories for subordinator, inverse subordinator and subdiffusive GBM were performed. The Monte Carlo method for option evaluation was applied. Our aim was not only to compare these two models each with other, but also to show that both models adequately describe the illiquid market and can be used for option pricing on this market. For this purpose absolute relative percentage (ARPE) and root mean squared error (RMSE) for both models were computed and analysed. Thanks to the proposed approaches, the investor gets a tools, which allows him to take into account the illiquidity.
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    Дослiдження стохастичної поведiнки клiтинних автоматiв
    (2022) Глушенков, Сергій; Чорней, Руслан
    Клітинні автомати дають змогу моделювати широкий спектр складних систем із локальною взаємодією. Попри те, що загалом поведінка окремо взятих клітинних автоматів може бути дуже простою, вдала їх комбінація або задання нестандартних правил взаємодії може значно ускладнити поведінку системи і призвести до доволі неоднозначних та непередбачуваних результатів спостережень. Стохастичність допомагає наблизити симульоване середовище до реальних умов і знайти оптимальну стратегію, яка буде більш стійкою до усіх можливих видів подій, в тому числі малоймовірних. Саме стохастичні клітинні автомати широко використовують у відтворенні природних явищ та процесів, симуляції транспортних потоків, криптографії тощо. У середовищах з наявним зовнішнім впливом стає актуальною задача пошуку оптимального керування системою. У цій статті розглянуто оптимальні стратегії керування для систем стохастичних клітинних автоматів, наведено приклад використання алгоритму покращення стратегії в задачі гасіння лісових пожеж, проаналізовано оптимальність вибраної стратегії.
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    Regularization by Denoising for Inverse Problems in Imaging
    (2022) Kravchuk, Oleg; Kriukova, Galyna
    In this work, a generalized scheme of regularization of inverse problems is considered, where a priori knowledge about the smoothness of the solution is given by means of some self-adjoint operator in the solution space. The formulation of the problem is considered, namely, in addition to the main inverse problem, an additional problem is defined, in which the solution is the right-hand side of the equation. Thus, for the regularization of the main inverse problem, an additional inverse problem is used, which brings information about the smoothness of the solution to the initial problem. This formulation of the problem makes it possible to use operators of high complexity for regularization of inverse problems, which is an urgent need in modern machine learning problems, in particular, in image processing problems. The paper examines the approximation error of the solution of the initial problem using an additional problem.
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    Two approaches for evaluation of option prices under illiquidity
    (2022) Pauk, Victoria; Petrenko, Oksana; Shchestyuk, Nataliia
    Abstracts of the report of the participants of the XХXVII International Conference "Problems of decision making under uncertainties" (PDMU-2022), November 23-25, 2022, Sheki-Lankaran, Republic of Azerbaijan.
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    Diffusion model in Image Transforms Inversion tasks
    (2022) Kravchuk, Oleg; Kriukova, Galyna
    Матеріали доповіді учасників міжнародної наукової конференції, присвяченої 60-річчю кафедри прикладної математики та інформаційних технологій "Прикладна математика та інформаційні технології", 22-24 вересня 2022 р.
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    Regularization by means of denoising in imaging
    (2022) Kravchuk, Oleg; Kriukova, Galyna
    Abstracts of the report of the participants of the XXXVII International Conference "Problems of decision making under uncertainties" (PDMU-2022), November 23-25, 2022, Sheki-Lankaran, Republic of Azerbaijan.