# Кафедра математики

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Item 9th Workshop on Cloud Technologies in Education: Report(2022) Kiv, Arnold; Semerikov, Serhiy; Shyshkina, Mariya; Striuk, Andrii; Striuk, Mykola; Yechkalo, Yuliia; Mintii, Iryna; Nechypurenko, Pavlo; Kalinichenko, Olga; Kolgatina, Larisa; Vlasenko, Kateryna; Amelina, Svitlana; Semenikhina, OlenaThis is an introductory text to a collection of selected papers from the 9th Workshop on Cloud Technologies in Education (CTE 2021) which held in Kryvyi Rih, Ukraine, on the December 17, 2021. It consists of short introduction, papers’ review and some observations about the event and its future.Item A competency-based approach to the systematization of mathematical problems in a specialized school(2021) Vlasenko, Catherine; Lovianova, Iryna; Armash, Tatiana; Sitak, Iryna; Kovalenko, DariaThe issue of searching for new methodological approaches to the systematization that will encourage the increase of students motivation to learn mathematics under the competency-based approach is considered in this article. The research analyzes the existing works on the increase in students motivation to learn Mathematics, in particular, the use of cross-curricular connections while forming students competency. Competency-based problems, systematized according to the topic of the 10th grade Functions, their features and graphics, were determined as the tools to measure students competency and a method to form their motivation to learn mathematics. The use of such methods as the initial research and information gathering, systematization and structural analysis of the problems, data processing allowed the authors of the article to systematize the problems for school subjects of the 10th grade that demonstrate cross-curricular connections of Mathematics with other learning subjects and allow showing the advantages of the mathematical modeling in researching real processes. The research shows the realization of cross-curricular connections in time and such connections as parallel learning, perspective connections, use of the mathematical modeling method are shown. An experiment was held in order to prove the e ciency of implementing a system of the problems to demonstrate the use of the function in di erent tasks of natural subjects. The results proved that the implemented system of problems considerably in uences the increase in students motivation to learn mathematics.Item A Comprehensive Program of activities to develop sustainable core skills in novice scientists(2021) Vlasenko, Catherine; Rovenska, Olga; Chumak, Olena; Lovianova, Iryna; Achkan, VitalyThis paper is aimed at studying scientific communication as an integral part of a scientists activity. The authors of this article analysed the development of informational technologies, which gave rise to a new paradigm of scientific communication Research 2.0. In the present study the analysis of research papers, describing models of scientific communication is done. The findings allow to define the structure and content of a comprehensive program of activities, connected to scientific communication in compliance with the Scientific Communication Life Cycle Model. In order to implement the program, aimed at developing core skills through scientific communication of scientists, a target audience, comprising postgraduate students and young researchers in Mathematics and TeachingMethods was engaged. A five-stage program of activities, which was developed, prompted scientific activity of young researchers and gave them an opportunity to learn about means of presenting research results, elements of management, mechanisms for applying the findings. A constructive description of each module of the program is done, actions and a strategy are described, communication between participants and tutors through the platform Higher School Mathematics Teacher is arranged in this research. In order to assess the efficiency of implementing the program, Researcher Development Framework (RDF) is used. The study also presents the results of the activity of young researchers, who were engaged in the program. Following the change in the phase of the development of researchers characteristic features and in compliance with RDF, a conclusion is made about a positive impact of the program on the development of career skills of young scientists, their interaction skills, awareness of professional behavior procedure.Item About one problem for equation of fractal diffusion with argument deviation(2017) Drin, Iryna; Drin, Svitlana; Drin, YaroslavМатеріал VI-ї Міжнародної науково-практичної конференції "Проблеми інформатики та комп'ютерної техніки (ПІКТ-2017)", 5-8 жовтня 2017 року.Item Accurate classification for Automatic Vehicle Type Recognition based on ensemble classifiers(2019) Shvai, Nadiya; Hasnat, Abul; Meicler, Antoine; Nakib, AmirIn this work, a real world problem of the vehicle type classification for Automatic Toll Collection (ATC) is considered. This problem is very challenging because any loss of accuracy even of the order of 1% quickly turns into a significant economic loss. To deal with such problem, many companies currently use Optical Sensors (OS) and human observers to correct the classification errors. Herein, a novel vehicle classification method is proposed. It consists in regularizing the problem using one camera to obtain vehicle class probabilities using a set of Convolutional Neural Networks (CNN), then, uses the Gradient Boosting based classifier to fuse the continuous class probabilities with the discrete class labels obtained from OS. The method is evaluated on a real world dataset collected from the toll collection points of the VINCI Autoroutes French network. Results show that it performs significantly better than the existing ATC system and, hence will vastly reduce the workload of human operators.Item The algorithm for knowledge assessment based on the Rusch model(2022) Kostikov, Alexander; Vlasenko, Kateryna; Lovianova, Iryna; Volkov, Sergii; Avramov, EvgenyIn this paper the algorithm for adaptive testing of students’ knowledge in distance learning and an assessment of its effectiveness in the educational process has been proposed. The paper provides an overview of the results of the application of modern test theory, a description and block diagram of the proposed algorithm and the results of its application in the real educational process. The effectiveness of using this algorithm for the objective assessment of students’ knowledge has been experimentally shown.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 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 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 The boundary problem by variable t for equation of fractal diffusion with argument deviation(2017) Drin, I.; Drin, Svitlana; Drin, Y.For a quasilinear pseudodifferential equation with fractional derivative by time variable t with order a e (0,1), the second derivative by space variable x and the argument deviation with the help of the step method we prove the solvability of the boundary problem with two unknown functions by variable t.Item Commuting sets for topological set operators(2021) Antoshyna, Kateryna; Kozerenko, SergiyProceedings of the "Algebraic and Geometric Methods of Analysis", May 25-28, 2021 Odesa, Ukraine.Item Conjugacy in finite state wreath powers of finite permutation groups(2019) Oliynyk, Andriy; Russyev, AndriyIt is proved that conjugated periodic elements of the infinite wreath power of a finite permutation group are conjugated in the finite state wreath power of this group. Counter-examples for non-periodic elements are given.Item The construction of riskless portdolio for Student-like FAT models(2016) Musienko, J.; Shchestyuk, NataliiaIn this paper we set up a riskless portfolio for alternative to Black-Scholes GBM model that incorporates the Student-like distribution of the returns.Item Development of the online course for training master students majoring in mathematics(2021) Vlasenko, Catherine; Lovianova, Iryna; Rovenska, Olga; Armash, Tatiana; Achkan, VitalyThis article considers the issue of implementing a model of blended learning to prepare master students majoring in Mathematics (speciality code in Ukrainian educational system "014 Secondary Education. Mathematics"). The research analyses the existing developments of the issue about the use of blended learning while training would-be mathematics teachers. The researchers determined and explained the stages of work on developing the online course "Methods for Teaching Mathematics to Students at Technical Universities", that is used when students learn methodological subjects of the curriculum. The research describes the development of the theoretical online course model and methodological recommendations on the learning materials and preparing papers for the course. The article offers recommendations related to the course structure. The course developers de ned the usability criteria of the educational platforms, determined the stages of course users' activity, their content, and organization. The research describes the areas of online course activity management, the course tutors' and moderators' teamwork is de ned as the main condition of its development and support. In order to prove the e ciency of implementing blended learning of the methodological subjects, an experiment was carried out during the assistant practice in technical universities that master students of the specialization \Mathematics" had. The results allowed con rming the e ciency of students' practical training during blended learning of the methodological subjects that in its turn encouraged the improvement of the assistant practice results.Item DEvS: Data Distillation Algorithm Based on Evolution Strategy(2022) Shvai, Nadiya; Llanza, Arcadi; Hasnat, Abul; Nakib, AmirThe development of machine learning solutions often relies on training using large labeled datasets. This raises challenges in terms of data storage, data privacy protection, and longer model training time. One of the possible solutions to overcome these problems is called dataset distillation – a process of creating a smaller dataset while maximizing the preservation of its task-related information. In this paper, a new dataset distillation algorithm is proposed, called DEvS, which uses an evolutionary strategy approach to condense the training samples initially available for an image classification task, while minimizing the loss of classification accuracy. Experiments on CIFAR-10 demonstrate the competitiveness of the proposed approach. Also, contrary to recent trends, DEvS is derivative-free image generation, and therefore has greater scalability on larger input image sizes.Item Diffusion model in Image Transforms Inversion tasks(2022) Kravchuk, Oleg; Kriukova, GalynaМатеріали 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 A discrete regularization method for hidden Markov models embedded into reproducing kernel Hilbert space(2018) Kriukova, GalynaHidden Markov models are a well-known probabilistic graphical model for time series of discrete, partially observable stochastic processes. We consider the method to extend the application of hidden Markov models to non-Gaussian continuous distributions by embedding a priori probability distribution of the state space into reproducing kernel Hilbert space. Corresponding regularization techniques are proposed to reduce the tendency to overfitting and computational complexity of the algorithm, i.e. Nystr¨om subsampling and the general regularization family for inversion of feature and kernel matrices. This method may be applied to various statistical inference and learning problems, including classification, prediction, identification, segmentation, and as an online algorithm it may be used for dynamic data mining and data stream mining. We investigate, both theoretically and empirically, the regularization and approximation bounds of the discrete regularization method. Furthermore, we discuss applications of the method to real-world problems, comparing the approach to several state-of-the-art algorithms.Item Distributed System for Sampling and Analysis of Electroencephalograms(2017) Sudakov, Oleksandr; Kriukova, Galyna; Natarov, Roman; Gaidar, Viktoriia; Maximyuk, Oleksandr; Radchenko, Sergiy; Isaev, DmytroDistributed system for sampling and analysis of electroencephalograms is proposed and implemented in alpha state. The system is based on the previously developed database for archiving of the electroencephalograms in Ukrainian National Grid infrastructure. The new components of the system include EEG sensors for laboratory animals, simulations software and data procession algorithms. The first application of the system for data sampling, analysis and simulations of epileptic seizures is performed.Item Eccentric digraphs of unique point eccentric graphs(2021) Hak, Artem; Haponenko, V.; Kozerenko, SergiyТези виступу на X Всеукраїнській науковій конференції молодих математиків, Київ, 16-17 квітня 2021.Item Edge Imbalance Sequences and Their Graphicness: [preprint](2019) Kozerenko, SergiyThe main focus of combinatorial dynamics is put on the structure of periodic points (and the corresponding orbits) of topological dynamical systems. The first result in this area is the famous Sharkovsky’s theorem which completely describes the coexistence of periods of periodic points for a continuous map from the closed unit interval to itself. One feature of this theorem is that it can be proved using digraphs of a special type (the so-called periodic graphs). In this paper we use Markov graphs (which are the natural generalization of periodic graphs in case of dynamical systems on trees) as a tool to study several classes of maps on trees. The emphasis is put on linear and metric maps.