Том 8

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Analysis of wave propagation conditions in a two-layer hydro-dynamic system with a free surface
(2025) Naradovyi, Volodymyr; Huriev, Vasyl; Demidov, Valerii
The study examines the problem of the propagation of internal and surface waves in a two-layerhydrodynamic system "a half-space - a layer - a layer with a free surface". A mathematical model ina linear approximation is presented. The research problem is formulated under the assumption thatthe fluids are ideal and incompressible. The mathematical formulation of the problem is given in adimensionless form. Expressions for the deviation of the contact interface η1(x,t) and the free surfaceη2(x,t) in the form of traveling waves are found. Expressions for the potentials φ1(x,z,t) and φ2(x,z,t),whose gradients describe the propagation velocities in the layers Ω1and Ω2respectively, are obtainedin an analytical form. A dispersion relation that connects the wave number and the wave propagationfrequency is derived. The roots of the dispersion relation, which are the frequencies of wave propagationon the contact interface and on the free surface, are found. An analysis of the roots of the dispersionrelation depending on the geometric and physical parameters of the system is carried out. In particular,the dependence of the wave propagation frequencies on the wave number without considering surfacetension is analyzed.The conducted research indicates that in the absence of surface tension (T1= T2= 0), the densityratio ρ acts as a defining parameter that governs both the quantitative and qualitative characteristics ofthe wave modes in the considered system. A transition from the classical state of the system with clearlyseparated fast surface and slow internal modes to a regime of their inversion was identified, which is asignificant result for a deeper understanding of the dynamics of strongly stratified fluids.The consideration of surface tension forces reveals a complex interaction between the effects of densitystratification and capillarity. Capillary forces lead to a substantial increase in wave frequencies and canbecome a dominant factor for internal modes, effectively neutralizing the influence of density changes.At the same time, it has been established that the density ratio ρ retains its role as the key parameter thatdetermines the qualitative structure of the modes, including the possibility of their complete inversionunder conditions of strong fluid stratification.
Iterative demand optimization using the discrete functional particlemethod
(2025) Drin, Svitlana; Avdieienko, Ivan; Chornei, Ruslan
This article addresses the challenge of assortment planning in retail under uncertain demand and operational constraints. It develops a hybrid methodology that integrates SARIMAX time-series forecasting with the Discrete Functional Particle Method (DFPM) for optimisation, enabling both strategic (long-term) and tactical (monthly) decision support. The proposed framework combines statistical forecasting with iterative optimisation in order to balance predictive accuracy and operational feasibility. In the forecasting stage, a SARIMAX model with exogenous regressors captures seasonality, promotions, and demand fluctuations, while a safeguard mechanism prevents excessively pessimistic predictions. In the optimisation stage, DFPM is applied to a quadratic objective under linear constraints, with parameters tuned using eigenvalue analysis of the risk matrix. A novel operational risk metric—the Inventory Efficiency Ratio—is introduced, defined as the ratio of leftover stock value to revenue, and used to construct the covariance structure for optimisation. A hybrid strategy blends the mathematically optimal allocation with a baseline derived from historical sales shares, ensuring both practical stability and data-driven improvements. Tactical adjustments refine this strategic solution by incorporating seasonal indices and business constraints such as minimum and maximum category weights. The framework is implemented in Python and evaluated on real-world retail data from a Ukrainian anti-stress toy retailer. Results demonstrate a 25% reduction in operational risk and a threefold increase in inventory turnover, while maintaining realistic revenue forecasts. Overall, the work contributes a flexible and reproducible decision-support methodology that unifies modern forecasting and optimisation techniques, providing practitioners with a tool for improving assortment decisions in dynamic retail environments.
Last time moment optimality in uniform 1-bullet silent duel with scaled exponentially-convex accuracy
(2025) Romanuk, Vadym
The uniform 1-bullet silent duel with scaled exponentially-convex accuracy of payoffs is a symmetric matrix game whose optimal value is 0, and each of the duelists has the same optimal behavior, whether it is in pure or mixed strategies. Such duels model two-side competitive interaction, where the purpose is to gain a reward by making the best possible decision through quantized time. It is proved that the last time moment is optimal in the duel with N time moments only when the accuracy factor does not exceed marginal value e−e N−2 / N−1 / N−2 e N−1 −1. If the accuracy factor is dropped below this marginal value, then the last time moment is single optimal. If the accuracy factor is exactly equal to the marginal value, the duelist has two optimal time moments: the penultimate and last one. The conditions of the last time moment optimality can be set to force the duelist to act the latest possible, which is quite useful in some blockchain settings, where participants (e. g., validators or miners) choose when to attempt block proposal or transaction insertion under uncertainty.
PINN-based machine learning for modeling internal waves insemi-infinite fluids
(2025) Avramenko, Olha; Kompan, Serhii; Sarana, Maksym
This study investigates the application of Physics-Informed Neural Networks (PINNs) for modelingwave processes at the interface between two incompressible fluids of differing densities. As a first step,the linear formulation of the problem is considered, which admits an analytical solution based on aspectral method involving Fourier decomposition of the initial perturbation. This solution serves as abenchmark for testing and validating the accuracy of the PINN predictions.The implementation is carried out in Python using specialized libraries such as TensorFlow, NumPy,SciPy, and Matplotlib, which provide both efficient deep learning frameworks and tools for solving mathe-matical physics problems numerically. The approach integrates artificial intelligence with domain-specificknowledge in hydrodynamics, enabling the construction of interpretable and physically consistent mod-els. Particular attention is given to the organization of the computational experiment, automation ofvisualizations, and storage of intermediate results for further analysis. The PINN model includes a lossfunction that encodes the governing equations and boundary conditions, and the training is conductedon randomly sampled points across the spatio-temporal domain. The influence of network architectureand training parameters on solution accuracy is examined. Visualization of loss function evolutionand predicted wave profiles provides insight into convergence behavior and physical plausibility of thesolutions.A comparative analysis between the PINN-based and analytical solutions across different time in-stances is presented, revealing phase shifts and amplitude deviations. The model demonstrates goodagreement at early times and a gradual accumulation of errors as time progresses—an expected featureof this class of methods. The results confirm the feasibility of applying the PINN framework to linearhydrodynamic problems, laying the groundwork for future extensions to weakly and strongly nonlinearregimes, including studies of wave stability and nonlinear wave dynamics.
Portfolio optimization for real data: approaches and chal-lenges
(2025) Burdym, Anastasiia; Danyliuk, Yevheniia; Shchestyuk, Nataliya
Portfolio optimization continues to be a dynamic field within finance, integrating new theories and technologies to better meet investor needs. As financial markets evolve, so too will the methodologies used to optimize portfolios, making it an area ripe for ongoing research and innovation. Classical Markowitz approach is based on the mean-variance optimization, which quantifies the tradeoff between risk (variance) and return (expected return). This approach had some limitations. It assumes investors are rational, markets are efficient, and asset returns are normally distributed. As a response to the some limitations of Markowitz theory minimum-VaR approach was appeared. This theory recognizes some assymetry, that investors are more concerned about potential losses than gains and incorporates downside risk measures like Value-at-Risk. Despite advancements of the classical Markowitz theory and minimum VaR approach, challenges remain in accurately estimating parameters, singularity of the covariance matrix and managing risks in volatile markets. In this paper we consider the mean-variance and mean-Var optimal portfolios and take into account the case when the covariance estimated matrix is singular. We use the Moore-Penrose pseudoinverse and Singular Value Decomposition (SVD) to find solutions. We apply these approaches and methodics to real financial data, construct mean-variance and mean-Var optimal portfolios and compare the dynamics of expected returns (mean), volatility and VaR for it. Thanks to the proposed approaches, the investor gets a tool that allows him to make decisions about choosing an approach to building an optimal portfolio, as well as taking into account the singularity of the covariance matrix.

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