Кафедра математики

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Modulational stability of wave packets at fluid interface of layer and half-space
(2025) Avramenko, Olha; Naradovyi, Volodymyr
The modulational stability of internal wave packets propagated along the surface of a hydrodynamic system consisting of a lower half-space and an upper layer covered with a rigid lid is investigated. The study is conducted within the framework of a nonlinear low-dimensional model incorporating surface tension on an interface using the method of multi-scale expansions implemented via symbolic computation. The evolution equation of the envelope of the wave packet takes the form of the Schrodinger equation. Conditions ¨ for the modulational stability of the solution of the evolution equation are identified for various physical and geometrical characteristics of the system. Significant influence on the modulational stability of the system’s geometrical characteristics and surface tension is observed for relatively small liquid layer thicknesses. For large layer thicknesses, the stability diagram degenerates to that of a system composed of two half-spaces.
Knowledge Transfer in Model-Based Reinforcement Learning Agents for Efficient Multi-Task Learning
(2025) Kuzmenko, Dmytro; Shvai, Nadiya
We propose an efficient knowledge transfer approach for modelbased reinforcement learning, addressing the challenge of deploying large world models in resource-constrained environments. Our method distills a high-capacity multi-task agent (317M parameters) into a compact 1M parameter model, achieving state-of-the-art performance on the MT30 benchmark with a normalized score of 28.45, a substantial improvement over the original 1M parameter model’s score of 18.93. This demonstrates the ability of our distillation technique to consolidate complex multi-task knowledge effectively. Additionally, we apply FP16 post-training quantization, reducing the model size by 50% while maintaining performance. Our work bridges the gap between the power of large models and practical deployment constraints, offering a scalable solution for efficient and accessible multi-task reinforcement learning in robotics and other resource-limited domains.
Nonlinear systems of PDEs admitting infinite-dimensional Lie algebras and their connection with Ricci flows
(2025) Cherniha, Roman; King, John
A wide class of two-component evolution systems is constructed admitting an infinite-dimensional Lie algebra. Some examples of such systems that are relevant to reaction–diffusion systems with cross-diffusion are highlighted. It is shown that a nonlinear evolution system related to the Ricci flow on warped product manifold, which has been extensively studied by several authors, follows from the above-mentioned class as a very particular case. The Lie symmetry properties of this system and its natural generalization are identified and a wide range of exact solutions is constructed using the Lie symmetry obtained. Moreover, a special case is identified when the system in question is reducible to the fast diffusion equation in one space dimension. Finally, another class of two-component evolution systems with an infinite-dimensional Lie symmetry that possess essentially different structures is presented.
An Age-Structured Diffusive Model for Epidemic Modelling: Lie Symmetries and Exact Solutions
(2025) Cherniha, Roman; Davydovych, Vasyl’
A new age-structured diffusive model for the mathematical modelling of epidemics is suggested. The model can be considered as a generalization of two models suggested earlier for similar purposes. The Lie symmetry classification of the model is derived. It is shown that themodel admits an infinite-dimensional Lie algebra of invariance. Using the Lie symmetries, exact solutions, in particular those of the travelling wave types and in terms of special functions, are constructed. Examples of application of exact solutions with the correctly-specified parameters for calculation of the total number of infected individuals during an epidemic are presented.
The cauchy problem for one class of parabolic pseudodifferential equation with deviation of the argument
(2015) Drin, Yaroslav; Drin, Iryna; Drin, Svitlana; Kotsur, Maksym
In this paper, we study solvability of the Cauchy problem for a parabolic pseudodifferential equation with the deviation of the argument. Parabolic pseudodifferential operator with non-smooth symbols introduced by Eidel’man and Drin’ for the first time. For such equations, the initial condition is set on a certain interval. Technical and physical reasons for delays can be transport delays, delays in decision-making, delays in information transmission, etc. The most natural are delays when modeling objects in medicine, population dynamics, ecology, etc. Other physical and technical interpretations are also possible, for example, the molecular distribution of thermal energy in various media (liquids, solid bodies, etc.) is modeled by heat conduction equations. Features of the dynamics of vehicles in different environments (water, land, air) can also be taken into account by introducing a delay. The formula for the solution of the Cauchy problem is constructed for the nonlinear equation of heat conduction with a deviation of the argument, its properties are investigated.
Benjamin-Feir Instability of Interfacial Gravity–Capillary Waves in a Two-Layer Fluid. Part I
(2025) Avramenko, Olha; Naradovyi, Volodymyr
This study presents a detailed investigation of the modulational stability of interfacialwave packets in a two-layer inviscid incompressible fluid with finite layer thicknesses and interfacial surface tension. The stability analysis is carried out for a broad range of density ratios and geometric configurations, enabling the construction of stability diagrams in the (𝜌, 𝑘)-plane, where 𝜌 is the density ratio and 𝑘 is the carrier wavenumber. The Benjamin-–Feir index is used as the stability criterion, and its interplay with the curvature of the dispersion relation is examined to determine the onset of modulational instability. The topology of the stability diagrams reveals several characteristic structures: a localized loop of stability within an instability zone, a global upper stability domain, an elongated corridor bounded by resonance and dispersion curves, and a degenerate cut structure arising in strongly asymmetric configurations. Each of these structures is associated with a distinct physical mechanism involving the balance between focusing/defocusing nonlinearity and anomalous/normal dispersion. Systematic variation of layer thicknesses allows us to track the formation, deformation, and disappearance of these regions, as well as their merging or segmentation due to resonance effects. Limiting cases of semi-infinite layers are analyzed to connect the results with known configurations, including the "half-space–layer", "layer–half-space’" and "half-space–half-space" systems. The influence of symmetry and asymmetry in layer geometry is examined in detail, showing how it governs the arrangement and connectivity of stable and unstable regions in parameter space. The results provide a unified framework for interpreting modulational stability in layered fluids with interfacial tension, highlighting both global dispersion-controlled regimes and localized stability islands. This work constitutes Part I of the study; Part II will address the role of varying surface tension, which is expected to deform existing stability domains and modify the associated nonlinear–dispersive mechanisms.
A Reaction-Diffusion System with Nonconstant Diffusion Coefficients: Exact and Numerical Solutions
(2025) Cherniha, Roman; Kriukova, Galyna
A Lotka–Volterra-type system with porous diffusion, which can be used as an alternative model to the classical Lotka–Volterra system, is under study. Multiparameter families of exact solutions of the system in question are constructed and their properties are established. It is shown that the solutions obtained can satisfy the zero Neumann conditions, which are typical conditions for mathematical models describing real-world processes. It is proved that the system possesses two stable steady-state points provided its coefficients are correctly specified. In particular, this occurs when the system models the prey–predator interaction. The exact solutions are used for solving boundary-value problems. The analytical results are compared with numerical solutions of the same boundary-value problems but perturbed initial profiles. It is demonstrated that the numerical solutions coincide with the relevant exact solutions with high exactness in the case of sufficiently small perturbations of the initial profiles.
Методичні рекомендації до виробничої практики для здобувачів другого (магістерського) рівня вищої освіти галузі знань 11 "Математика та статистика", спеціальність 113 "Прикладна математика", освітньо-наукова програма "Прикладна математика"
(Національний університет "Києво-Могилянська академія", 2023) Авраменко, Ольга; Власенко, Катерина; Чорней, Руслан
У методичних рекомендаціях висвітлено мету, завдання, зміст і особливості організації та проведення виробничої практики магістрантів. Визначено перелік знань, практичних умінь і навичок магістрантів, розкрито особливості оцінювання різних видів діяльності здобувачів під час проходження практики; надано рекомендації по розробці фрагменту конспекту лекції, гурткового та практичного занять, наведено рекомендації по проведенню виховних заходів з основних напрямів виховання; окреслено основні вимоги до оформлення звітної документації магістрантів. Розроблені матеріали повністю співпадають з рекомендаціями поданими на дистанційному курсі "Практика асистентська ПМ (ukma.edu.ua)", відповідають вимогам, викладеним в положенні НаУКМА про практику та забезпечують навчальний план та освітньо-наукову програму "Прикладна математика".
Локальне керування в мережах Ґордона — Ньюелла
(2024) Чорней, Руслан
Запропоновано 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.
Схема розподiлу секрету, що базується на криптосистемi Голдвассер-Голдрiха-Халевi
(2024) Ліхачов, Артемій; Олійник, Богдана
З 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.
Вiдновлююче спектральне число графа K4
(2024) Аверкін, Олександр; Тимошкевич, Лариса
Статтю 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.
Про деякi застосування керованих випадкових полiв з локальною структурою взаємодiї
(2024) Чорней, Руслан
У 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.
Fractional calculus and its application in financial mathematics
(2024) Zubritska, Dariia; Shchestyuk, Nataliya; Sluchynskyi, Dmytro
Fractional 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.
GAN-generated strokes extension for Paint Transformer
(2024) Poliakov, Mykhailo; Shvai, Nadiya
Neural 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].
Robust Bayesian regression model in Bernstein form
(2024) Mytnyk, Oleh
In 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.
Deviation of the interface between two liquid half-spaces with surface tension: multiscale approach
(2024) Avramenko, Olha
This 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.
Peculiarities of initial condition specification in a problem of wave packet propagation in layered fluid
(Дніпровський національний університет імені Олеся Гончара, 2024) Avramenko, Olha
The 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.
On strongly connected Markov graphs of maps on combinatorial trees
(2025) Kozerenko, Sergiy
Markov 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.
Guided inverse problems
(Національний університет "Києво-Могилянська академія", 2024) Ivaniuk, Andrii; Kravchuk, Oleg; Kriukova, Galyna
The 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.
Прогнозування нестаціонарних фінансових процесів в умовах інформаційної волатильності
(2024) Митник, Олег; Бідюк, Петро
Метою 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.

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