# Кафедра математики

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Item All-path convexity: two characterizations, general position number, and one algorithm(2024) Haponenko, Vladyslav; Kozerenko, SergiyWe present two characterizations for the all-path convex sets in graphs. Using the first criterion, we obtain a new characterization of connected block graphs and compute the general position number in a graph with respect to the all-path convexity. The second criterion allows us to provide a new algorithm for testing a set on all-path convexity.Item Predictive model for a product without history using LightGBM. pricing model for a new product(2023) Drin, Svitlana; Kriuchkova, Anastasiia; Toloknova, VarvaraThe article focuses on developing a predictive product pricing model using LightGBM. Also, the goal was to adapt the LightGBM method for regression problems and, especially, in the problems of forecasting the price of a product without history, that is, with a cold start. The article contains the necessary concepts to understand the working principles of the light gradient boosting machine, such as decision trees, boosting, random forests, gradient descent, GBM (Gradient Boosting Machine), GBDT (Gradient Boosting Decision Trees). The article provides detailed insights into the algorithms used for identifying split points, with a focus on the histogram-based approach. LightGBM enhances the gradient boosting algorithm by introducing an automated feature selection mechanism and giving special attention to boosting instances characterized by more substantial gradients. This can lead to significantly faster training and improved prediction performance. The Gradient-based One-Side Sampling (GOSS) and Exclusive Feature Bundling (EFB) techniques used as enhancements to LightGBM are vividly described. The article presents the algorithms for both techniques and the complete LightGBM algorithm. This work contains an experimental result. To test the lightGBM, a real dataset of one Japanese C2C marketplace from the Kaggle site was taken. In the practical part, a performance comparison between LightGBM and XGBoost (Extreme Gradient Boosting Machine) was performed. As a result, only a slight increase in estimation performance (RMSE, MAE, R-squard) was found by applying LightGBM over XGBoost, however, there exists a notable contrast in the training procedure’s time efficiency. LightGBM exhibits an almost threefold increase in speed compared to XGBoost, making it a superior choice for handling extensive datasets. This article is dedicated to the development and implementation of machine learning models for product pricing using LightGBM. The incorporation of automatic feature selection, a focus on highgradient examples, and techniques like GOSS and EFB demonstrate the model’s versatility and efficiency. Such predictive models will help companies improve their pricing models for a new product. The speed of obtaining a forecast for each element of the database is extremely relevant at a time of rapid data accumulation.Item Моделювання очiкуваних кредитних збиткiв(2023) Дрінь, Світлана; Сердюк, ФедірУ цiй статтi запропоновано метод моделювання ймовiрностi дефолту, описано статистичну оцiнку моделi та представлено модель алгоритму програмної реалiзацiї. Алгоритм автоматично обирає з групи регресiйних моделей, де моделями є як лiнiйна регресiя, так i рiзнi модифiкацiї напiвлогарифмiчних моделей та лаговi моделi для макрофакторiв 𝑋𝑖,𝑡,𝑋𝑖,𝑡−1, ...,𝑋𝑖,𝑡−𝑇 Cтатистичний аналiз проводиться за використання коефiцiєнта детермiнацiї R-квадрат, p-value, VIF (variance inflation factor). Актуальнiсть цiєї теми визначається необхiднiстю дотримання банкiвськими органiзацiями мiжнародних стандартiв, таких як Мiжнароднi стандарти фiнансової звiтностi (МСФЗ 9) та Угода про банкiвський нагляд та капiтал (Базель 3). Цi стандарти визначають вимоги щодо оцiнки кредитного ризику та вимоги до розмiрiв капiталу. Дотримання цих стандартiв є важливим не тiльки для забезпечення стабiльностi та надiйностi фiнансової системи, а й для збереження довiри клiєнтiв та iнвесторiв. Вiдповiднiсть мiжнародним нормам також робить банки конкурентоспроможними на свiтовому ринку та сприяє припливу iнвестицiй та розвитку фiнансового сектору. МСФЗ 9 може бути iнтерпретований рiзними моделями. В статтi запропоновано пiдхiд щодо вибору вiдповiдної моделi для прогнозування ймовiрностi дефолту. Описана методика вибору моделi дає змогу банкам вибрати оптимальну модель оцiнки прогнозу дефолту в рамках наведеного стандарту. Це сприяє бiльш точнiй та надiйнiй оцiнцi кредитного ризику, вiдповiдно регуляторним вимогам, що забезпечить банки засобами для кращого прогнозування та управлiння фiнансовими ресурсами, а також зменшення ризикiв. Методологiя вибору моделi економить значну кiлькiсть часу та ресурсiв, оскiльки, пошук оптимальної моделi вiдбувається автоматично. Це дає змогу швидше реагувати на змiни в економiчному середовищi, вдосконалювати стратегiї прийняття рiшення та управляти кредитними ризиками, що має велике значення для фiнансових установ у конкурентному середовищi. В Українi в цей час триває вiйна, i прогнозування за допомогою чинних методiв стає складним завданням через непередбачуванi стресовi ситуацiї для економiки. У таких умовах стандартнi моделi можуть бути недостатньо адаптованими для врахування пiдвищеного ризику та нестабiльностi. Запропонований пiдхiд допоможе знайти бiльш консервативнi моделi прогнозування, якi можуть бути корисними в умовах нестабiльних перiодiв i вiйни.Item Balance function generated by limiting conditions(2023) Morozov, DenysThis article conducts an analysis of the inherent constraints governing the formation of the price function that describes the interaction between two markets. The research not only identifies these constraints but also obtains an explicit form of the specified function. The key factors considered in constructing the price function are defined in the article. Through analyzing these constraints and their impact on market interaction, a formula for the price function is provided. This approach not only reveals the essence of natural constraints in forming the price function but also provides a contextual foundation for negotiations shaping a fair exchange price for the interaction process between two markets. This offers a theoretical basis for modeling and solving similar problems arising during practical economic activities. Two economies, Economy 1 and Economy 2, producing goods X and Y with linear Production Possibility Curve (PPC) graphs, are under consideration. The cost of producing one unit of good X relative to Y is denoted as 𝑅1 for Economy 1 and 𝑅2 for Economy 2. Exchange between economies occurs in a market, where the possible exchange is Δ𝑥 units of X for Δ𝑦 = 𝑅market ·Δ𝑥 units of Y, and vice versa. If 𝑅1 is less than 𝑅2, Economy 1 specializes in the production of X, and Economy 2 specializes in Y, fostering mutually beneficial trade. For mutually beneficial exchange on the market with a price 𝑅market, it is necessary and sufficient that 𝑅1 ≤ 𝑅market ≤ 𝑅2. The article also explores the concept of a fair exchange price, specifying conditions for symmetry, reciprocity, and scale invariance. Notably, it indicates that the unique solution satisfying these conditions is 𝑓(𝑅1,𝑅2) = √ 𝑅1 · 𝑅2. In the context of balanced exchange, where economies gain equal profit per unit of the acquired good, the balanced exchange price 𝑅market[𝑏𝑎𝑙𝑎𝑛𝑐𝑒] is determined as 𝑅market = √ 𝑅1 · 𝑅2. This serves as a fair price, meeting the aforementioned conditions of symmetry, reciprocity, and scale invariance. In the provided example with 𝑅1 = 2 and 𝑅2 = 8, the article examines the mutually beneficial interval for 𝑅market and computes the balanced and fair exchange price.Item Interpolation problems for random fields on Sierpinski's carpet(2023) Boichenko, Viktoriia; Shchestyuk, Nataliia; Florenko, AnastasiiaThe prediction of stochastic processes and the estimation of random fields of different natures is becoming an increasingly common field of research among scientists of various specialties. However, an analysis of papers across different estimating problems shows that a dynamic approach over an iterative and recursive interpolation of random fields on fractal is still an open area of investigation. There are many papers related to the interpolation problems of stationary sequences, estimation of random fields, even on the perforated planes, but all of this still provides a place for an investigation of a more complicated structure like a fractal, which might be more beneficial in appliances of certain industry fields. For example, there has been a development of mobile phone and WiFi fractal antennas based on a first few iterations of the Sierpinski carpet. In this paper, we introduce an estimation for random fields on the Sierpinski carpet, based on the usage of the known spectral density, and calculation of the spectral characteristic that allows an estimation of the optimal linear functional of the omitted points in the field. We give coverage of an idea of stationary sequence estimating that is necessary to provide a basic understanding of the approach of the interpolation of one or a set of omitted values. After that, the expansion to random fields allows us to deduce a dynamic approach on the iteration steps of the Sierpinski carpet. We describe the numerical results of the initial iteration steps and demonstrate a recurring pattern in both the matrix of Fourier series coefficients of the spectral density and the result of the optimal linear functional estimation. So that it provides a dependency between formulas of the different initial sizes of the field as well as a possible generalizing of the solution for N-steps in the Sierpinski carpet. We expect that further evaluation of the mean squared error of this estimation can be used to identify the possible iteration step when further estimation becomes irrelevant, hence allowing us to reduce the cost of calculations and make the process viable.Item Weakly nonlinear models of stochastic wave propagation in two-layer hydrodynamic systems(2023) Avramenko, Olga; Naradovy, VolodymyrThe paper discusses three-dimensional models of the propagation of stochastic internal waves in hydrodynamic systems: ’half-space - half-space’, ’half-space - layer with rigid lid’, and ’layer with solid bottom - layer with rigid lid’. In constructing the models, the layers are considered to be ideal fluids separated by a contact surface. The main objective of the modeling is to obtain a dynamic equation for the stochastic amplitude of surface waves. A comparative analysis of the obtained results has been conducted. In order to control the contribution of nonlinear terms, a dimensionless non-numerical parameter has been introduced. The models are distinguished by boundary conditions that determine the general form of solutions. As a result, a dynamic equation for the stochastic amplitude of internal waves has been derived. After ensemble averaging of the amplitudes, the dynamic equation is formulated in integral form using Fourier-Stieltjes integrals. The dynamic equation reveals two-wave and three-wave interactions, as well as the contribution of dispersion to wave dynamics. An investigation of the boundary case of the transition of internal waves in the ’half-space - half-space’ system to surface waves in the absence of an upper liquid layer confirms the validity of the results.Item StairNet: visual recognition of stairs for human–robot locomotion(2024) Kurbis, Andrew Garrett; Kuzmenko, Dmytro; Ivanyuk-Skulskiy, Bogdan; Mihailidis, Alex; Laschowski, BrokoslawHuman–robot walking with prosthetic legs and exoskeletons, especially over complex terrains, such as stairs, remains a significant challenge. Egocentric vision has the unique potential to detect the walking environment prior to physical interactions, which can improve transitions to and from stairs. This motivated us to develop the StairNet initiative to support the development of new deep learning models for visual perception of real-world stair environments. In this study, we present a comprehensive overview of the StairNet initiative and key research to date. First, we summarize the development of our large-scale data set with over 515,000 manually labeled images. We then provide a summary and detailed comparison of the performances achieved with different algorithms (i.e., 2D and 3D CNN, hybrid CNN and LSTM, and ViT networks), training methods (i.e., supervised learning with and without temporal data, and semi-supervised learning with unlabeled images), and deployment methods (i.e., mobile and embedded computing), using the StairNet data set. Finally, we discuss the challenges and future directions. To date, our StairNet models have consistently achieved high classification accuracy (i.e., up to 98.8%) with different designs, offering trade-offs between model accuracy and size. When deployed on mobile devices with GPU and NPU accelerators, our deep learning models achieved inference speeds up to 2.8 ms. In comparison, when deployed on our custom-designed CPU-powered smart glasses, our models yielded slower inference speeds of 1.5 s, presenting a trade-off between human-centered design and performance. Overall, the results of numerous experiments presented herein provide consistent evidence that StairNet can be an effective platform to develop and study new deep learning models for visual perception of human–robot walking environments, with an emphasis on stair recognition. This research aims to support the development of next-generation vision-based control systems for robotic prosthetic legs, exoskeletons, and other mobility assistive technologies.Item Верхня оцiнка вiдновлюючого спектрального числа для зважених графiв(Чернівецький національний університет імені Юрія Федьковича, 2022) Тимошкевич, Лариса; Пилипіва, ОлександраМатеріали доповіді учасників міжнародної наукової конференції, присвяченої 60-річчю кафедри прикладної математики та інформаційних технологій "Прикладна математика та інформаційні технології", 22-24 вересня 2022 р.Item Класифікація зліченних графів Кокстера відносно значення індексу у проміжку (√(√5+2),3/√2](Чернівецький національний університет імені Юрія Федьковича, 2022) Когут, Марія; Тимошкевич, ЛарисаМатеріали доповіді учасників міжнародної наукової конференції, присвяченої 60-річчю кафедри прикладної математики та інформаційних технологій "Прикладна математика та інформаційні технології", 22-24 вересня 2022 р.Item 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, YanaThis 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.Item 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, TetianaThe 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.Item 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, OlenaThis 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.Item The Wheel of Mathematics Learning Methods(2023) Vlasenko, Kateryna; Lovianova, Iryna; Chumak, Olena; Sitak, Iryna; Kovalenko, DariaThe 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.Item An optimal lower bound for the size of periodic digraphs(2023) Kozerenko, SergiyA 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.Item Unique eccentric point graphs and their eccentric digraphs(2023) Hak, Artem; Haponenko, Vladyslav; Kozerenko, Sergiy; Serdiuk, AndriiWe 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.Item 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, YuriyThe 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.Item Analysis of the Shape of Wave Packets in the "Half Space–Layer–Layer with Rigid Lid" Three-Layer Hydrodynamic System(2022) Avramenko, Olga; Lunyova, МariiaWe 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.Item The nonlocal problem for fractal diffusion equation(2022) Drin, Yaroslav; Drin, I.; Drin, SvitlanaOver 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.Item Відновлююче спектральне число зважених графів-циклів(Молодий вчений, 2022) Пилипіва, Олександра; Тимошкевич, ЛарисаМатеріали доповіді учасників науково-практичної конференції "Наука в контексті глобальної трансформації суспільства", м. Полтава, 26-27 серпня 2022 р.Item Multiple auxiliary classifiers GAN for controllable imagegeneration: Application to license plate recognition(2021) Shvai, Nadiya; Hasnat, Abul; Nakib, AmirOne 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.