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Item FDS: Fractal decomposition based direct search approach for continuous dynamic optimization(2025) Llanza, Arcadi ; Shvai, Nadiya; Nakib, AmirDynamic optimization problems (DOPs) are known to be challenging due to the variability of their objective functions and constraints over time. The complexity of these problems increases further when the frequency of landscape change and the dimensionality of the search space are large. In this work, we propose a novel fractal decomposition-based method designed for DOPs, called FDS. It is a new single solution metaheuristic that introduces a new hyperspherebased space decomposition for efficient exploration, an archive for diversity control, and a pseudo-gradient-based local search (called GraILS) for fast exploitation. Extensive experiments on the well-known and the standard benchmark (the Moving Peak Benchmark: MPB) demonstrate that FDS consistently outperforms state-of-the-art competitors. Furthermore, FDS shows high robustness across diverse scenarios, maintaining superior performance despite variations in key benchmark parameters, such as the severity of landscape shifts, the number of peaks, the dimensionality of the problem, and the frequency of change. FDS achieves the highes average rank across all experiments and demonstrates dominant performance in 19 out of 23 scenarios. The implementation of FDS is available via the following GitHub repository: https:// github.com/alc1218/FDS.Item Benjamin-Feir Instability of Interfacial Gravity-Capillary Waves in a Two-Layer Fluid. Part II. Surface-Tension Effects(2026) Avramenko, Olha; Naradovyi, VolodymyrThis second part of the study develops a complete geometric and asymptotic description of how surface tension governs the modulationalstability of interfacial waves in a two-layer fluid. Extending the analytical framework of Part I, surface tension is treated as a freelyadjustable parameter, making it possible to trace the nonlinear and dispersive properties of the system across the full range of depthratios and density contrasts. Using the nonlinear Schr ̈odinger reduction together with long-wave asymptotics, the mechanisms thatshape the boundaries between stable and unstable regimes are identified and their dependence on surface tension is quantified. Thelong-wave structure is controlled by two special density values that mark the bases of the loop and the corridor on the stability diagrams.Their ordering switches at a threshold that exists only when the lower layer is deeper, and loop-type structures occur only in thisregime. A second organising parameter is the classical Bond threshold, at which the dispersive and nonlinear singularities coincide.When surface tension exceeds this value and the upper layer is sufficiently deep, the interaction between resonant and dispersive effectsproduces a capillary cut that replaces the corridor and characterises strongly capillary, upper-layer-dominated configurations. To unifythese observations, the full three-dimensional critical surfaces that separate different types of nonlinear and dispersive behaviour arecomputed. The familiar loop, corridor, and cut appear as planar sections of these surfaces, and their transitions follow directly fromthe deformation of the intersection between the resonant and dispersive sheets. Two depth ratios correspond to genuine geometricdegeneracies: equal layer depths, where the intersection reduces to a straight line, and the golden-ratio configuration, where the criticalsurface becomes horizontally tangent at the Bond threshold. Overall, Part II completes the geometric and physical classification ofmodulational stability in two-layer interfacial waves and provides a framework for future extensions incorporating shear, external forcing,flexible boundaries, or variable bathymetry.Item Benjamin–Feir Instability of Wave Packets at Interface of Liquid Half-Space and Layer(2026) Avramenko, Olha; Naradovyi, VolodymyrThe propagation of internal waves in a hydrodynamic system comprising a solid bottom and an upper half-space 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. The evolution equation of the envelope of the wave packet takes the form of the Schr ̈odinger equation. Conditions for the Benjamin–Feir stability of the solution of the evolution equation are identified for various physical and geometrical characteristics of the system.Item S4 separation and p-partition in all-path and detour convexities(2026) Haponenko, VladyslavIn this work, we consider problems of S4 and p-convex partition separations with respect to the all-path and the detour convexities. We give characterizations of p-all-path convex and p-detour convex graphs. With respect to all-path convexity S2, S3, and S4 separable graphs are characterized. Also, we present necessary and sufficient conditions for two sets to be S4 separable, for both convexities. Moreover, we prove that in all-path convexity the time complexity of those problems is linear, and it is NP-hard for detour convexity. Finally, we give an algorithm for determining whether two sets in graph are S4 separable with respect to all-path convexity.Item Using the case method while training would-be mathematics teachers(2025) Achkan, Vitaliy; Vlasenko, Kateryna; Chashechnikova, Olga; Bohdanova, Nataliia; Protsyk, N.The article looks into the issue of developing a methodology for using the case method in mathematics teacher training. The study proposes and substantiates the components of using the case method in teaching the methodological disciplines of the would-be mathematics teacher: motivation of educational activity, goal setting, content, methods, forms, and means of learning. Motivation for learning using the case method is provided by modeling the teacher's real professional activity while considering methodological problem situations (cases). To ensure goal setting, an algorithm for case selection is proposed and illustrated depending on the goal of the educational curriculum or the goal of a speci c educational class. The research method is the case method, implemented through work with a system of cases for methodological disciplines. The content of the cases covers all modules of the methodological discipline curriculums. Forms of training depend on the types of cases used in the class. Examples of cases of all kinds are proposed, suitable for use within frontal, group, and individual work. Learning means include multimedia presentations, video cases, dynamic mathematics packages, and educational cloud services. The run experiment con rmed the practicality of using the developed methodology when training would-be mathematics teachers for professional activities.Item Application of the Cobb-Douglas function for the formation of would-be economists' analytical skills(2025) Vlasenko, Kateryna; Armash, Tetiana; Achkan, Vitaliy; Kaluhin, Ruslan; Bohdanova, NataliiaTraining high-quality specialists in the economic eld is crucial. This task can be achieved by training students in economic specialties in mathematical modeling. Solving problems using production functions, particularly Cobb-Douglas functions, can become the ground for mastering mathematical modeling. The article looks into the issue of the combination of table and clustering methods to systematize such kinds of problems. The involvement of students in real production situations while mathematical training contributes to the development of analytical skills needed for analyzing economic processes, modeling decisions, working with data, and coming to conclusions. The article presents a system of problems involving the Cobb-Douglas production function. The study shows the implementation of these problems while mastering `Multivariate Functions' by economic majors. Also, the work presents the practicality of such a system when developing their analytical skills. The run experiment con rmed that the developed system of problems results in the progression of wouldbe economists' analytical skills.Item Павло Кнопов: науковий шлях і становлення школи стохастичної оптимізації(2025) Пономарьов, Вадим; Моренець, Володимир; Розора, Ірина; Шарапов, Михайло; Чорней, Руслан; Слабоспицький, ОлександрПрисвячено 85-річчю Павла Соломоновича Кнопова (нар. 21.05.1940, Київ) – доктора фізико-математичних наук, члена-кореспондента НАН України; фахівця зі стохастичної оптимізації, статистики випадкових процесів і полів, теорії ризику та керування; автора понад 250 праць і 13 монографій; від 1999 р. – завідувача відділу математичних методів дослідження операцій Інституту кібернетики ім. В. М. Глушкова НАН України, від 1987 р. – професора КНУ; лауреата Державної премії України (2009), премій ім. В. М. Глушкова (1997) та ім. В. С. Михалевича (2021), заслуженого діяча науки і техніки України (2021). Київська школа стохастичної оптимізації, до розвитку якої істотно долучився П. С. Кнопов, є науковим феноменом, що поєднує теорію ймовірностей, математичну статистику та теорію керування для розв'язання задач ухвалення рішень в умовах невизначеності. У статті також висвітлено непростий життєвий і науковий шлях ученого, що пройшов крізь випробування епохи й утвердився як один із провідних математиків свого покоління.Item Development of 3D models for implementing game environments(2025) Chornyi, Danyil; Moiseienko, Natalia; Moiseienko, Mykhailo; Vlasenko, KaterynaThis paper presents a comprehensive approach to developing 3D models for implementing game environments in computer games, with emphasis on both traditional modeling techniques and contemporary optimization strategies. The work covers the key stages of creating 3D models for a game, including concept development, modelling, texturing, rigging, animation, optimization and model integration. The popular Blender software package is used as the primary tool for 3D modelling and animation. The result is a set of futuristic-style chess piece models with neon lighting on a chessboard, with two robot chess players serving as avatars to demonstrate gameplay. Our approach is contextualized within current industry trends, including the integration of simulation data (CFD/FEA) with real-time rendering, advanced Level-of-Detail (LOD) algorithms, and emerging procedural content generation techniques using GANs and mixed-initiative systems. Performance analysis demonstrates that the implemented optimization techniques, including GPU-accelerated rendering and texture batching, achieve frame rates exceeding 60 FPS for scenes with over 100,000 polygons. The models and animations created can be used in the development of a computer game for virtual reality, supporting both traditional displays and immersive VR headsets. This work highlights the synergy between artistic design and technical optimization in transforming traditional games into immersive modern experiences, while providing a framework applicable to broader game development contexts.Item Nowcasting як сучасний підхід до оцінювання ВВП України: порівняння з традиційними моделями прогнозування(Національний університет "Києво-Могилянська академія", 2025) Болотов, Єгор; Дрінь, СвітланаМетою даного дослідження є порівняльний аналіз точності прогнозів ВВП України. До таких сучасних підходів до прогнозування на основі моделей змішаної частоти належать векторної авторегресії (MF-VAR) [1] та їх факторних розширень (MF-FAVAR) з традиційними моделями прогнозування. Особливий акцент робиться на їх здатності інтегрувати багаточастотні та нетрадиційні джерела інформації. Це дослідження є особливо актуальним, оскільки, хоча в українській науці вже існують дослідження змішаних частотних моделей на національному рівні (наприклад, U-MIDAS [2]), сучасні економічні підходи, такі як MF-FAVAR, ще не застосовувалися для регіональних прогнозів.Item Фрактально-дифузійні генеративні моделі: ієрархічний підхід до синтезу зображень(Національний університет "Києво-Могилянська академія", 2025) Шалімов, Андрій; Авраменко, ОльгаУ цьому дослідженні пропонується модифікація FGM із використанням дифузійної нейронної мережі на архітектурі U-Net як альтернативного генератора. Такий підхід підвищує паралельність процесу, скорочує час синтезу й інтегрує ієрархічний самоподібний принцип у стандартну дифузійну модель.Item PINN Modeling of Interfacial Gravity-Capillary Waves(Національний університет "Києво-Могилянська академія", 2025) Avramenko, Olha; Sontikov, MaksymThis paper presents an automated computational framework for modeling hydrodynamic processes using physics-informed neural networks (PINNs). The modular system integrates all stages of numerical experimentation — from data generation and model training to validation and accuracy evaluation — ensuring reproducibility, flexibility, and scalability. The framework was verified on the classical problem of interfacial gravity–capillary waves between two incompressible fluids, using the analytical solution as a benchmark for numerical assessment. Computational experiments showed that increasing the number of training points from 400 to 1000 improved accuracy and convergence, with the Extended configuration achieving 98.86% accuracy and a MAPE of 1.14%, while Adaptive_LR remained stable. The results confirm the reliability and efficiency of the proposed PINN-based framework for solving complex hydrodynamic problems governed by nonlinear partial differential equations.Item Moore-Penrose Pseudoinverse Matrix(Національний університет "Києво-Могилянська академія", 2025) Kravchuk, Oleg; Kriukova, GalynaThe Moore-Penrose pseudo-inverse is a foundational concept in modern numerical linear algebra, offering a principled approach to solving ill-posed and inconsistent systems arising in machine learning and other fields. This paper explores the pseudo-inverse from five distinct perspectives — axiomatic, variational, regularization, spectral, and algebraic graph theory — highlighting its theoretical depth and practical relevance across disciplines such as machine learning, signal processing, and network analysis.Item Гра з неповною інформацією на прикладі спортивного бетінгу(Національний університет "Києво-Могилянська академія", 2025) Куцалаба, Назарій; Чорней, РусланУ сучасному спортивному аналітичному середовищі прогнозування результатів матчів є однією з ключових задач, що поєднує методи статистики, машинного навчання та теорії ігор.Item DecisioNet з пропорційним розподілом обчислювальних потужностей(Національний університет "Києво-Могилянська академія", 2025) Мокрий, Михайло; Швай, НадіяУ цьому дослідженні розглянуто нейронну мережу з бінарною деревоподібною структурою DecisioNet (DN) [1], яка належить до категорії нейронних дерев рішень [2] та представлено нову версію моделі з пропорційним розподілом обчислювальних ресурсів.Item Efficient Policy Learning via Knowledge Distillation for Robotic Manipulation(Національний університет "Києво-Могилянська академія", 2025) Severhin, Oleksandr; Kuzmenko, Dmytro; Shvai, NadiyaThe work focuses on the computational intractability of large-scale Reinforcement Learning (RL) models for robotic manipulation. While world-like models like TD-MPC2 demonstrate high performance in various manipulative tasks, their immense parameter count (e.g., 317M) hinders training and deployment on resource-constrained hardware. This research investigates Knowledge Distillation (KD) with a loss function specifically described in [1] and [2] as a primary method for model compression. This involves training a lightweight "student" model to mimic the behavior of a large, pre-trained "teacher" model. Unlike in supervised learning, distilling knowledge in RL is uniquely complex; the objective is to transfer a dynamic, reward-driven policy, not a simple input-output function.Item Validating Architectural Hypotheses in Neural Decision Trees with Neural Architecture Search(2025) Mykytyshyn, Artem; Shvai, NadiiaThis article introduces an automated and unbiased framework for validating architectural hypotheses for neural network models, with a particular focus on Neural Decision Trees (NDTs). The proposed methodology employs Neural Architecture Search (NAS) as an unbiased tool to explore architectural variations and empirically assess theoretical claims. To demonstrate this framework, we investigate a hypothesis found in the literature: that the complexity of decision nodes in NDTs decreases monotonically with tree depth. This assumption, initially motivated by the task of monocular depth estimation, suggests that deeper nodes in the tree require fewer parameters due to simpler split functions. To rigorously test this hypothesis, we conduct a series of NAS campaigns over the CIFAR-10 image and fully connected layers, while all other architectural components are held constant to isolate the effect of node depth. By applying Tree-structured Parzen Estimator (TPE)-based NAS and evaluating over 300 architectures, we quantify complexity metrics across tree levels and analyze their correlations using Spearman’s rank coefficient. The results provide no statistical or visual evidence supporting the hypothesized trend: node complexity does not decrease with depth. Instead, complexity remains nearly constant across levels, regardless of tree depth or search space size. These results suggest that assumptions derived from specific applications may not generalize to other domains, underscoring the importance of empirical validation and careful searchspace design. The presented framework may serve as a foundation for verifying other structural assumptions across various neural network families and applications.Item Robustness of Neural Decision Trees to Noise in Input Data for Image Classification Tasks(2025) Mokryi, Mykhailo; Shvai, NadiiaNeural networks, particularly convolutional neural networks (CNNs), have demonstrated high effectiveness in image classification tasks. However, they are known to be vulnerable to input data perturbations and have weak interpretability due to their black-box nature. In contrast, traditional decision trees (DTs) provide transparent decision-making processes, but are limited to low-dimensional or tabular data, restricting their field of application in computer vision tasks such as image classification. To address this gap, a hybrid architecture known as Neural Decision Trees (NDTs) has emerged, combining strong generalization and learning capabilities of neural networks, with transparent hierarchical inference and interpretability of DTs. The article investigates the robustness of NDTs to noise in input data for image classification tasks. Despite the extensive studies covering the robustness of both CNNs and traditional DTs against various forms of input perturbations, the robustness of NDT models remains a largely underexplored area. This study provides two robust training methods to improve robustness: constant noise learning and incremental noise learning, originally developed for CNNs, but which can be effectively applied to NDT-based architectures and significantly improve the robustness to noisy images for models. These methods involve adding perturbed samples via a Gaussian blur during the training stage. The noisy test set consists of images perturbed by a Gaussian blur and is used to evaluate the robustness performance. A series of experiments were conducted on the CIFAR-10 dataset using the original training baseline and robust training methods. The results demonstrate that constant and incremental noise learning significantly improve the robustness of all tested NDT models to noisy images compared to their original training performance. While the ResNet18 baseline model demonstrates higher overall performance, the NDT models show comparable robustness improvements using the proposed robust training strategies. Constant noise learning offered an adjustable trade-off between performance on clean and noisy images, while incremental noise learning provided a more stable training process. The first method is considered preferable due to the simplicity of implementation. This study empirically confirms that NDT models can effectively use methods adapted from CNNs to improve their robustness against perturbations in input data. An NDT framework was developed to conduct training and validation using a standardized shared pipeline. It is available via the link: github.com/ MikhailoMokryy/NDTFramework.Item Готфрiд Лейбнiц i створення диференцiального числення(2025) Федоровська, КатеринаУ статтi розглядається внесок Готфрiда Вiльгельма Лейбнiца у створення диференцiального числення в контекстi його фiлософської системи. Дослiджено взаємозв’язок мiж метафiзичною концепцiєю монад та математичним поняттям нескiнченно малих величин. Проаналiзовано публiкацiю "Nova methodus pro maximis et minimis" 1684 року як першу друковану працю з диференцiального числення. Висвiтлено особливостi лейбнiцiвського пiдходу до математичного аналiзу, його нотацiю та методологiю. Розглянуто iсторичну суперечку щодо прiоритету винаходу числення мiж Лейбнiцем та Ньютоном. Показано вплив фiлософських iдей Лейбнiца на формування сучасного математичного аналiзу.Item Про розв’язнiсть задачi пошуку нерухомої точки вiдображення в просторах багатовимiрних послiдовностей(2025) Гончаренко, Юрiй; Ляшко, Вiктор; Тимошенко, Андрiй; Чорней, РусланУ статтi розглянуто задачу пошуку нерухомої точки для вiдображень у просторах багатовимiрних послiдовностей. Автори формулюють i доводять основну теорему, що забезпечує iснування та єдинiсть розв’язку рiвняння типу x = h + Ax, де A є лiнiйним оператором у просторi Банаха з певними властивостями. В роботi введено систему пiвнорм, яка узгоджується з нормою простору та задовольняє умови монотонностi й обмеженостi. Використовуючи метод послiдовних наближень та аналiз збiжностi вiдповiдного ряду, доведено iснування розв’язку задачi, а також отримано оцiнки для норми розв’язку. Особливу увагу придiлено доведенню обмеженостi оператора та унiкальностi розв’язку, що гарантує коректнiсть постановки задачi. Запропонованi результати є розвитком класичних пiдходiв до задачi нерухомих точок у новому контекстi багатовимiрних послiдовностей, що мають як теоретичну, так i прикладну цiннiсть.Item Про деякi властивостi майже перiодичних функцiй(2025) Кашпіровський, Олексій; Митник, ЮрійДослiджуються достатнi умови показникiв λn та коєфiцiєнтiв Фур’є, при виконаннi яких майже перiодичнi функцiї f(t) з простору Безиковича B2 неперервнi, неперервно-диференцiйовнi та голоморфнi. У випадку показникiв λn, що мають степеневу асимптотику λn = L(nα + εn), де L ∈ R1, α > 0, εn → 0 при n → +∞ отримано аналог теореми Соболєва про вкладення. Для показникiв λn, що за n → +∞ зростають повiльнiше довiльного додатного степеня n, описано клас функцiй з простору Безиковича B2, що мають аналiтичне продовження у пiвплощину Re s > a ≥ 0. До таких функцiй належить дзета-функцiя Рiмана ζ(s). Для функцiй з B2, у яких показники λn прямують до нуля, встановленi достатнi умови аналiтичного продовження до цiлих функцiй 1-го експоненцiального порядку.