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Item License Plate Images Generation with Diffusion Models(2024) Shpir, Mariia; Shvai, Nadiya; Nakib, AmirDespite the evident practical importance of license plate recognition (LPR), corresponding research is limited by the volume of publicly available datasets due to privacy regulations such as the General Data Protection Regulation (GDPR). To address this challenge, synthetic data generation has emerged as a promising approach. In this paper, we propose to synthesize realistic license plates (LPs) using diffusion models, inspired by recent advances in image and video generation. In our experiments a diffusion model was successfully trained on a Ukrainian LP dataset, and 1000 synthetic images were generated for detailed analysis. Through manual classification and annotation of the generated images, we performed a thorough study of the model output, such as success rate, character distributions, and type of failures. Our contributions include experimental validation of the efficacy of diffusion models for LP synthesis, along with insights into the characteristics of the generated data. Furthermore, we have prepared a synthetic dataset consisting of 10,000 LP images, publicly available at https://zenodo.org/doi/10.5281/zenodo. 13342102. Conducted experiments empirically confirm the usefulness of synthetic data for the LPR task. Despite the initial performance gap between the model trained with real and synthetic data, the expansion of the training data set with pseudolabeled synthetic data leads to an improvement in LPR accuracy by 3% compared to baseline.Item The Unitary Cayley Graph of Upper Triangular Matrix Rings(2024) Hołubowski, Waldemar; Kozerenko, Sergiy; Oliynyk, Bogdana; Solomko, ViktoriiaThe unitary Cayley graph CR of a finite unital ring R is the simple graph with vertex set R in which two elements x and y are connected by an edge if and only if x − y is a unit of R. We characterize the unitary Cayley graph CTn(F) of the ring of all upper triangular matrices Tn(F) over a finite field F. We show that CTn(F) is isomorphic to the semistrong product of the complete graph Km and the antipodal graph of the Hamming graph A(H(n, pk)), where m = p kn(n−1) 2 and |F| = pk. In particular, if |F| = 2, then the graph CTn(F) has 2n−1 connected components, each component is isomorphic to the complete bipartite graph Km,m, where m = 2 n(n−1) 2 . We also compute the diameter, triameter, and clique number of the graph CTn(F).Item Value-at-risk measuring for subdiffusion option pricing models(2024) Shchestyuk, Nataliya; Tyshchenko, SerhiiThe value-at-risk is a useful tool for investors and can be used for understanding the past and making medium-term and strategic decisions for the future. The paper focuses on the risk measuring in the option price subdiffusive model under the unusual behavior of the market, when the price may not be changed for some time, which is quite a common situation in modern illiquid financial markets or during global crise.Item Study of numerical and analytical solutions of a generalized boundary value problem for the heat conduction equation(2024) Drin, Iryna; Drin, Svitlana; Drin, Yaroslav; Lutskiv, MykhailoThe computed values of the solution obtained by the finite difference method and the results of the numerical investigation of the analytical solution of this problem match with maximum and average relative errors of +7.03% and ±1.82%, respectively. The graphs of the numerical and analytical solutions coincide over the entire range of investigated time and space values. Further improvements in the accuracy of the numerical solution can be achieved by adjusting grid parameters – reducing spatial step size and increasing the number of computational iterations.Item Non-classical boundary value problem for the heat conduction equation(2024) Drin, Iryna; Drin, Svitlana; Drin, Yaroslav; Lutskiv, MykhailoThe first boundary value problem for the heat conduction equation was studied in. We provide the first proof of a formula for solving the non-classical boundary value problem, where the temperature is specified at the left end of a homogeneous rod and its flux at the right end.Item Hitting-time models for option pricing in illiquid markets(2024) Shchestyuk, Nataliya; Tyshchenko, SerhiiThe hitting time process arises naturally in fields such as insurance, process control and survival analysis. Hitting time models are also applied to describe the dynamic of illiquid market when relatively long periods without any trading are observed.Item A search for regular K3-irregular graphs(2024) Hak, ArtemFor a given graph F, the F-degree of a vertex v in G is the number of subgraphs of G, isomorphic to F, to which v belongs. A graph G is called F-irregular if all vertices of G have distinct F-degrees. In [1], the existence of regular K3-irregular graphs was posed as an open question. Examples of such graphs for regularities r ∈ {10, 11, 12} were constructed in [2]. We analytically prove that no such graphs exist for r ≤ 7, present such a graph for r = 9, and establish bounds on the order for r = 8. We will use t(v) to denote the K3-degree.Item Subdiffusive option price model with Inverse Gaussian subordinator(2024) Shchestyuk, Nataliya; Tyshchenko, SerhiiThe paper focuses on the option price subdiffusive model under the unusual behavior of the market, when the price may not be changed for some time, which is a quite common situation in modern illiquid financial markets or during global crises. In the model, the riskfree bond motion and classical geometrical Brownian motion (GBM) are time-changed by an inverted inverse Gaussian(IG) subordinator. We explore the correlation structure of the subdiffusive GBM stock returns process, discuss option pricing techniques based on the martingale option pricing method and the fractal Dupire equation, and demonstrate how it applies in the case of the IG subordinator.Item A test on the location of tangency portfolio for small sample size and singular covariance matrix(2025) Drin, Svitlana; Mazur, Stepan; Muhinyuza, StanislasThe test for the location of the tangency portfolio on the set of feasible portfolios is proposed when both the population and the sample covariance matrices of asset returns are singular. The particular case of investigation is when the number of observations, n, is smaller than the number of assets, k, in the portfolio, and the asset returns are i.i.d. normally distributed with singular covariance matrix Σ such that rank(Σ) = r < nItem Student Models for a Risky Asset with Dependence: Option Pricing and Greeks(2024) Leonenko, Nikolai; Liu, Anqi; Shchestyuk, NataliyaWe propose several new models in finance known as the Fractal Activity Time Geometric Brownian Motion (FATGBM) models with Student marginals. We summarize four models that construct stochastic processes of underlying prices with short-range and long-range dependencies. We derive solutions of option Greeks and compare with those in the Black-Scholes model. We analyse performance of delta hedging strategy using simulated time series data and verify that hedging errors are biased particularly for long-range dependence cases. We also apply underlying model calibration on S&P 500 index (SPX) and the U.S./Euro rate, and implement delta hedging on SPX options.Item A Space Distributed Model and Its Application for Modeling the COVID-19 Pandemic in Ukraine(2024) Cherniha, Roman; Dutka, Vasyl; Davydovych, VasylA space distributed model based on reaction–diffusion equations, which was previously developed, is generalized and applied to COVID-19 pandemic modeling in Ukraine. Theoretical analysis and a wide range of numerical simulations demonstrate that the model adequately describes the second wave of the COVID-19 pandemic in Ukraine. In particular, comparison of the numerical results obtained with the official data shows that the model produces very plausible total numbers of the COVID-19 cases and deaths. An extensive analysis of the impact of the parameters arising from the model is presented as well. It is shown that a well-founded choice of parameters plays a crucial role in the applicability of the model.Item Dynamical structure of metric and linear self-maps on combinatorial trees(2024) Kozerenko, SergiyThe dynamical structure of metric and linear self-maps on combinatorial trees is described. Specifically, the following question is addressed: given a map from a finite set to itself, under what conditions there exists a tree on this set such that the given map is either a metric or a linear map on this tree? The author proves that a necessary and sufficient condition for this is that the map has either a fixed point or a periodic point with period two, in which case all its periodic points must have even periods. The dynamical structure of tree automorphisms and endomorphisms is also described in a similar manner.Item The usage of stochastic matrices while learning the topic "eigenvalues and eigenvectors of a matrix" in the course of Higher Mathematic(2024) Vlasenko, Kateryna; Armash, Tetiana; Kostikov, Alexander; Lovianova, Iryna; Moiseienko, MykhailoThe article looks into teaching students mathematical modeling during the mastering of the topic \Eigenvalues and eigenvectors of a matrix" in the Linear Algebra. The modeling is based on the usage of matrices built based on Markov chains. The analysis of the researchers' papers shows that the usage of matrices of this type can be implemented while teaching students of different university specialties. The analysis of the prerequisites for the involvement of stochastic matrices in the consideration of students has helped to determine additional concepts and theorems that should be considered while learning \Eigenvalues and eigenvectors of a matrix". The theorem on the presence of stochastic vectors in a stochastic matrix can serve as a formula for systematizing problems that can present applications of eigenvalues and eigenvectors of the matrix in real-life situations. The results of the experiment conducted using the developed system of problems for students of different University specialties show the improvement of skills provided by mastering the topic \Eigenvalues and eigenvectors of a matrix".Item The case classification and their development for would-be mathematics teachers' training(2024) Achkan, Vitaliy ; Vlasenko, Kateryna; Lovianova, Iryna; Kaluhin, Ruslan; Armash, TetianaThe article looks into the issue of the case classification for the training of Mathematics teachers. The analysis, which was carried out, and the survey of 47 University teachers of methodical disciplines allowed to highlight the classification features of cases in the process of teaching methodical disciplines: the amount of time to complete, the way of presentation, the level of complexity and the breadth of the covered problem. Classifying cases by the amount of time for their implementation contributed to the separation of mini-cases, medium-term cases, and long-term cases. Classification by the method of presentation of cases in the process of studying methodical disciplines ensured their use in printed (handwritten) form, multimedia presentation of cases, and video cases. According to the level of complexity,the cases were divided into reproductive-training, partial-research, and creative-innovative cases. Thematic and integrated cases ensured the breadth of coverage of the problem during the teaching of methodical disciplines. The article presents general requirements for the selection and development of cases in the process of studying methodical disciplines. These requirements are based on the principles of accessibility, scientificity, contextuality, systematicity, methodological expediency, and practical orientation. The classification and specific requirements became the basis for the development of cases that can be used in the process of teaching methodological disciplines of would-be Mathematics teachers.Item A recommendation system with reinforcement for determining the price of a product(2023) Drin, Svitlana; Kriuchkova, AnastasiaRecommender systems belong to a fairly new concept with a wide scope of application and the property of adaptation to non-classical optimization problems. Thus, recommender systems can be used in e-commerce to predict demand and price for a new product in a chain of stores. LightGBM is a reinforcement recommendation system based on regression trees. The Boosting technology for the new tree ˆ f(x) ← ˆ f(x) + λfk(x) is selected with the specified parameter value λ. A typical value of λ is 0.001 to 0.1. But despite its simplicity and many obvious advantages, for many real big data with long-term dependence, classic Boosting techniques should not be used in real models. For example, if there are 900 stores and 7000 SKUs in the chain, then we have more than 6 million combinations per day in the event horizon. In the LightGBM technique, trees of different structure q are combined, where q : Rm → T. And in the known tree structure q(x), the optimal weight of each leaf is obtained by minimizing the loss function L′(t) = PT i=1 [ Gjwj + 1 2 (Hj + λ)w2 j ] + γT. It is worth noting that the LightGBM algorithm, which is characterized by its efficiency, accuracy and speed, creates histograms and uses the generated classes instead of the entire range of values of each variable, achieving a significant reduction in training time. The generated classes are the result of the Gradient One Side Sampling (GOSS) method. The main idea of the GOSS methodology focuses on the fact that not all observations contribute equally to the learning of the algorithm, since those with a small first derivative of the loss function learn better than those with a large first derivative of the loss function. In the case of forecasting the price of goods that do not have a sales history, we will use embedding in NLP in our model. The main purpose of using embedding is the distance between vectors of products to a new product that have a similar nature of sales, and this distance should be minimal. Results will be evaluated using the MAE performance metric.Item A test on the location of tangency portfolio for small sample size and singular covariance matrix(2023) Drin, Svitlana; Mazur, Stepan; Muhinyuza, StanislasIn this paper, we propose the test for the location of the tangency portfolio on the set of feasible portfolios when both the population and the sample covariance matrices of asset returns are singular. We derive the exact distribution of the test statistic under both the null and alternative hypotheses. Furthermore, we establish the high-dimensional asymptotic distribution of that test statistic when both the portfolio dimension and the sample size increase to infinity. We complement our theoretical findings by comparing the high-dimensional asymptotic test with an exact finite sample test in the numerical study. A good performance of the obtained results is documented.Item The analytical view of solution of the first boundary value problem for the nonlinear equation of heat conduction with deviation of the argument(2023) Drin, Yaroslav; Drin, Iryna; Drin, SvitlanaIn this article, for the first time, the first boundary value problem for the equation of thermal conductivity with a variable diffusion coefficient and with a nonlinear term, which depends on the sought function with the deviation of the argument, is solved. For such equations, the initial condition is set on a certain interval. Physical and technical reasons for delays can be transport delays, delays in information transmission, delays in decision-making, etc. The most natural are delays when modeling objects in ecology, medicine, population dynamics, etc. Features of the dynamics of vehicles in different environments (water, land, air) can also be taken into account by introducing a delay. Other physical and technical interpretations are also possible, for example, the molecular distribution of thermal energy in various media (solid bodies, liquids, etc.) is modeled by heat conduction equations. The Green’s function of the first boundary value problem is constructed for the nonlinear equation of heat conduction with a deviation of the argument, its properties are investigated, and the formula for the solution is established.Item The cauchy problem for quasilinear equation with nonstationary diffusion coefficient(2023) Drin, Yaroslav; Drin, Iryna; Drin, SvitlanaThe initial problem for the heat conduction equation with the inversion of the argument are considered and Green’s function are determined. The theorem declared that the Poisson’s formula determines the solution of the Cauchy problem considered and proved.Item Modeling of dynamic systems using Maple(2023) Avramenko, OlgaThis report presents the capabilities of the software product for building models of dynamic systems with subsequent computer simulation and conducting virtual experiments in Maple, as well as the results of its implementation in the educational process.Item Convolutional Neural Network Compression Based on Improved Fractal Decomposition Algorithm for Large Scale Optimization(2023) Llanza, Arcadi; Keddous, Fekhr Eddine; Shvai, Nadiya; Nakib, AmirDeep learning based methods have become the de-facto standard for various computer vision tasks. Nevertheless, they have repeatedly shown their vulnerability to various form of input perturbations such as pixels modification, region anonymization, etc. which are closely related to the adversarial attacks. This research particularly addresses the case of image anonymization, which is significantly important to preserve privacy and hence to secure digitized form of personal information from being exposed and potentially misused by different services that have captured it for various purposes. However, applying anonymization causes the classifier to provide different class decisions before and after applying it and therefore reduces the classifier’s reliability and usability. In order to achieve a robust solution to this problem we propose a novel anonymization procedure that allows the existing classifiers to become class decision invariant on the anonymized images without any modification requires to apply on the classification models. We conduct numerous experiments on the popular ImageNet benchmark as well as on a large scale industrial toll classification problem’s dataset. Obtained results confirm the efficiency and effectiveness of the proposed method as it obtained 0% rate of class decision change for both datasets compared to 15.95% on ImageNet and 0.18% on toll dataset obtained by applying the na¨ıve anonymization approaches. Moreover, it has shown a great potential to be applied to similar problems from different domains.