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Item Локальне керування в мережах Ґордона — Ньюелла(2024) Чорней, РусланЗапропоновано модифікацію мережі Ґордона — Ньюелла з локальною та синхронною взаємодією, яка обслуговує клієнтів у замкнутому режимі. Система околів задається за допомогою деякого скінченного графа вузлів системи. Запропоновано процедуру знаходження оптимальних нерандомізованих стратегій керування для систем із критерієм усереднених в одиницю часу витрат.Item Схема розподiлу секрету, що базується на криптосистемi Голдвассер-Голдрiха-Халевi(2024) Ліхачов, Артемій; Олійник, БогданаЗ розвитком квантових технологiй стає актуальним питання про дослiдження та впровадження криптографiчних примiтивiв, що базуються на складних задачах для квантових обчислень. Такi криптографiчнi примiтиви є стiйкими щодо квантового криптоаналiзу. Прикладом задач, що мають експоненцiйну складнiсть для квантових обчислень, є задачi на решiтках, такi як пошук найкоротшого вектора або пошук найближчого вектора. Однiєю з перших i найвiдомiших квантово-стiйких криптосистем, що в основi свого математичного апарату використовує задачi на решiтках, є криптосистема Голдвасcер-Голдрiха-Халевi. Схема розподiлення секрету є фундаментальним криптографiчним примiтивом, що допускає розподiлення секрету мiж множиною учасникiв, при цьому вiдновлення секрету можливе тiльки при авторизацiї всiх або певної частини учасникiв (порогу учасникiв). Також необхiдною умовою схеми розподiлення секрету є неможливiсть окремих учасникiв, або груп учасникiв, кiлькiсть яких менша за порiг, вiдновити секрет. Варiанти побудови схем розподiлу секрету на рiзних математичних моделях, у тому числi на решiтках, наразi активно дослiджуються, оскiльки вони дозволяють проводити надiйнi багатостороннi обчислення, безпечно поширювати iнформацiю шляхом поширення i розподiлення оригiналу даних мiж рiзними серверами, для побудови компiляторов схем iз захистом вiд витоку тощо. У цiй роботi запропоновано нову квантово-стiйку n-порогову схему розподiлу секрету для n учасникiв, що базується на криптосистемi Голдвасcер-Голдрiха-Халевi.Item Вiдновлююче спектральне число графа K4(2024) Аверкін, Олександр; Тимошкевич, ЛарисаСтаттю присвячено дослiдженню обернених спектральних задач для зважених графiв. Розглянуто задачу щодо вiдновлення ваг на множинi ребер графа за спектрами його iндукованих пiдграфiв. Завдяки широкому колу застосувань, оберненi спектральнi задачi активно вивчають для рiзних класiв матриць: зазвичай вони зводяться до вiдновлення матрицi (або її частини) за спектром самої матрицi чи її пiдматриць. Наша задача стосується класу нерозкладних симетричних матриць з невiд’ємними елементами та нулями на головнiй дiагоналi — матриць сумiжностi зв’язних зважених графiв. Ключовим поняттям цiєї роботи є вiдновлююче спектральне число графа Srn(G) — мiнiмальна кiлькiсть спектрiв iндукованих пiдграфiв, необхiдних для однозначного вiдновлення всiх ваг ребер графа G. Головним результатом дослiдження є знаходження точного значення Srn(K4) для повного графа на чотирьох вершинах. Одержанi результати та використанi у роботi методи можуть бути застосованi в подальших дослiдженнях, зокрема для визначення точних значень вiдновлюючого спектрального числа iнших графiв.Item Про деякi застосування керованих випадкових полiв з локальною структурою взаємодiї(2024) Чорней, РусланУ статтi розглянуто керованi випадковi поля з локальною структурою взаємодiї та їхнi застосування. Основну увагу придiлено питанням застосування оптимального керування випадковими системами на графах, зокрема в аналiзi ризику катастроф, моделюваннi соцiальних мереж та психометричному мережевому аналiзi. Описано математичнi пiдходи, що дозволяють формалiзувати та вирiшувати задачi стохастичної оптимiзацiї в таких системах. Результати роботи можуть бути застосованi в економiцi, кiбербезпецi, соцiальних науках та iнших сферах.Item Fractional calculus and its application in financial mathematics(2024) Zubritska, Dariia; Shchestyuk, Nataliya; Sluchynskyi, DmytroFractional calculus extends classical calculus by allowing differentiation and integration of non-integer orders, providing valuable tools for analyzing complex systems. In this part of the paper we demonstrate the main methods of fractional calculus, including Euler’s, Riemann-Liouville, and Caputo approaches. The behavior of functions such as xn, eλx, and sin(x) is analyzed for fractional orders, demonstrating how fractional differentiation results in varying patterns of growth and decay. The second part explores the application of fractal derivatives in financial mathematics. We present the use of the Riemann-Liouville derivative to model stock prices in illiquid markets, where the price of an asset may remain unchanged for some time. For this, subdiffusion processes and a fractal integrodifferential equation with the Riemann-Liouville derivative are used. The idea of subdiffusion models is to replace the calendar time t in the risk-free bond motion and classical GBM by some stochastic process Ht, which represents a hitting time, which is interpreted as the first time at which Gt hits the barrier t. Next, we focus on the pricing of a European option when the underlying asset is illiquid. The option price is found as a solution to a fractal Dupire integro-differential equation, in which the time derivative is replaced by the Dzerbayshan–Caputo (D-K) derivative. The D–K derivative is a generalization of the Caputo approach. The form of the D–K derivative depends on a random process Gt, called the subordinate. We take a standard inverse Gaussian process with parameters (1,1) as the subordinate Gt and formulate the Proposition about the form of the fractal Dupire equation for the chosen subordinate. These approaches provide tools that allow the investor to take into account the illiquidity of the financial markets.Item GAN-generated strokes extension for Paint Transformer(2024) Poliakov, Mykhailo; Shvai, NadiyaNeural painting produces a sequence of strokes for a given image and artistically recreates it using neural networks. In this paper, we explore a novel Transformer-based framework named the Paint Transformer to predict the parameters of a stroke set with a feed-forward neural network. The Paint Transformer achieves better painting results than previous methods with more inexpensive training and inference costs. The paper proposes a novel extension to the Paint Transformer that adds more complex GAN-generated strokes to achieve a more artistically abstract painting style than the original method. This research was originally published as a Master’s thesis [1].Item Robust Bayesian regression model in Bernstein form(2024) Mytnyk, OlehIn this paper, we present an inductive method for constructing robust Bayesian Polynomial Regression (BPR) models in Bernstein form, referred to as PRIAM (Polynomial Regression Inductive AlgorithM). PRIAM is an algorithm designed to determine stochastic dependence between variables. The triple nature of PRIAM combines the advantages of Bayesian inference, the interpretability of neurofuzzy models in Bernstein form, and the robustness of the support vector approach. This combination facilitates the integration of state-of-the-art machine learning techniques in decision support systems. We conduct experiments using well-known datasets and real-world economic, ecological, and meteorological models. Furthermore, we compare the forecast errors of PRIAM against several competitive algorithms.Item Deviation of the interface between two liquid half-spaces with surface tension: multiscale approach(2024) Avramenko, OlhaThis paper investigates the deviation of the interface between two semi-infinite liquid media under the influence of surface tension and gravity using a multiscale analysis. The initial-boundary value problem is formulated based on key dimensionless parameters, such as the density ratio and the surface tension coefficient, to describe the generation and propagation of wave packets along the interface. A weakly nonlinear model is employed to examine initial deviations of the interface, enabling the derivation of integral solutions for both linear and nonlinear approximations. The linear approximation captures the fundamental structure of forward and backward waves, while nonlinear corrections account for higherorder effects derived through multiscale expansions. These corrections describe the evolution of the wave packet envelope, highlighting the interplay between dispersion, nonlinearity, and surface tension. Integral expressions are provided for both linear and nonlinear solutions, including those illustrating the role of even and odd initial deviations of the interface. Comparisons between linear and nonlinear approximations emphasize their interconnectedness. The linear model defines the primary wave dynamics, while the nonlinear terms contribute higher harmonics, refining the solutions and facilitating stability analysis. The results reveal significant contributions from higher-order harmonics in determining the dynamics of the interface. Furthermore, the study explores the conditions under which the nonlinear envelope remains stable, including constraints on initial amplitudes to prevent instability. This research opens new perspectives for further analysis of stability and wave dynamics at fluid interfaces using symbolic computations. Potential applications include the study of wave behavior under various geometric configurations and fluid properties. The findings contribute to advancing hydrodynamic wave modeling and establish a foundation for future research in this field.Item Peculiarities of initial condition specification in a problem of wave packet propagation in layered fluid(Дніпровський національний університет імені Олеся Гончара, 2024) Avramenko, OlhaThe problem of wave packet propagation along the interface of two semiinfinite fluids with different densities is considered within the framework of a weakly nonlinear model, taking surface tension into account. The method of multiple scales expansions is applied. The analytical analysis of admissible initial conditions is carried out in two stages. In the first stage, the initial perturbation of the free surface is specified as a smooth function symmetric about the central point. This function is expanded into a series of the first harmonics, taking into account the dispersion relation. In the second stage, a sequence of second harmonics is constructed that satisfies the evolution equation, namely, the nonlinear Schrödinger equation.Item On strongly connected Markov graphs of maps on combinatorial trees(2025) Kozerenko, SergiyMarkov graphs form a special class of digraphs constructed from self-maps on the vertex sets of combinatorial trees. In this paper, the trees that admit cyclic permutations of their vertex sets with non-strongly connected Markov graphs in terms of the existence of a special subset of edges are characterized. Additionally, the structure of self-maps of finite sets, which produce strongly connected Markov graphs for all trees, is described. A similar question, concerning which self-maps produce strongly connected Markov graphs for some trees, is answered for the class of permutations.Item Guided inverse problems(Національний університет "Києво-Могилянська академія", 2024) Ivaniuk, Andrii; Kravchuk, Oleg; Kriukova, GalynaThe given work proposes a novel approach for solving inverse problems in machine learning leveraging Physics-Guided Neural Networks (PGNNs). This method incorporates domain knowledge through an additional inverse problem, leading to significant improvements in model performance and accuracy.Item Прогнозування нестаціонарних фінансових процесів в умовах інформаційної волатильності(2024) Митник, Олег; Бідюк, ПетроМетою дослідження є аналіз гаусівських процесів як непараметричного методу машинного навчання з вчителем для побудови регресійної моделі нестаціонарих фінансових процесів в умовах інформаційної волатильності. Показано, що викривлення вхідного часу-простору відповідно до рівня волатильності додає нестаціонарність в функцію коваріації і покращує прогнозуючі властивості регресії гаусівського процесу. В якості прикладу досліджена динаміка курсу акцій GME в період її сильної волатильності.Item Graphs with odd and even distances between non-cut vertices(2025) Antoshyna, Kateryna; Kozerenko, SergiyWe prove that in a connected graph, the distances between non-cut vertices are odd if and only if it is the line graph of a strong unique independence tree. We then show that any such tree can be inductively constructed from stars using a simple operation. Further, we study the connected graphs in which the distances between non-cut vertices are even (shortly, NCE-graphs). Our main results on NCE-graphs are the following: we give a criterion of NCE-graphs, show that any bipartite graph is an induced subgraph of an NCE-graph, characterize NCE-graphs with exactly two leaves, characterize graphs that can be subdivided to NCE-graphs, and provide a characterization for NCE-graphs which are maximal with respect to the edge addition operation.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.