Могилянський математичний журнал

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Перiодичне наукове видання "Могилянський математичний журнал" засновано з метою публiкацiї результатiв науково-дослiдних робiт, теоретичних дослiджень учених, науково-педагогiчних працiвникiв, аспiрантiв, магiстрiв i студентiв, присвячених широкому колу питань сучасної математичної науки. Тематичне розмаїття статей охоплює iсторiю математики, виклад результатiв теоретичних дослiджень з математики i статистики, а також їх застосувань. Засновник (1996 р.) i видавець журналу - Нацiональний унiверситет "Києво-Могилянська академiя" Виходив як частина багатосерiйного видання "Науковi записки НаУКМА" ("Фiзико-математичнi науки") З 2018 р. - окреме видання, що має назву "Могилянський математичний журнал" (англ. "Mohyla Mathematical Journal")

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A solution of a finitely dimensional Harrington problem for Cantor set
(2022) Kusinski, Slawomir
In this paper we are exploring application of fusion lemma - a result about perfect trees, having its origin in forcing theory - to some special cases of a problem proposed by Leo Harrington in a book Analytic Sets. In all generality the problem ask whether given a sequence of functions from Rω to [0; 1] one can find a subsequence of it that is pointwise convergent on a product of perfect subsets of R. We restrict our attention mainly to binary functions on the Cantor set as well as outline the possible direction of generalization of result to other topological spaces and different notions of measurablity.
About the approximate solutions to linear and non-linear pseudodifferential reaction diffusion equations
(2019) Drin, Yaroslav; Ushenko, Yuri; Drin, Iryna; Drin, Svitlana
Background. The concept of fractal is one of the main paradigms of modern theoretical and experimental physics, radiophysics and radar, and fractional calculus is the mathematical basis of fractal physics, geothermal energy and space electrodynamics. We investigate the solvability of the Cauchy problem for linear and nonlinear inhomogeneous pseudodiﬀerential diﬀusion equations. The equation contains a fractional derivative of a Riemann–Liouville time variable deﬁned by Caputo and a pseudodiﬀerential operator that acts on spatial variables and is constructed in a homogeneous, non-negative homogeneous order, a non-smooth character at the origin, smooth enough outside. The heterogeneity of the equation depends on the temporal and spatial variables and permits the Laplace transform of the temporal variable. The initial condition contains a restricted function. Objective. To show that the homotopy perturbation transform method (HPTM) is easily applied to linear and nonlinear inhomogeneous pseudodiﬀerential diﬀusion equations. To prove the solvability and obtain the solution formula for the Cauchy problem series for the given linear and nonlinear diﬀusion equations. Methods. The problem is solved by the NPTM method, which combines a Laplace transform with a time variable and a homotopy perturbation method (HPM). After the Laplace transform, we obtain an integral equation which is solved as a series by degrees of the entered parameter with unknown coeﬃcients. Substituting the input formula for the solution into the integral equation, we equate the expressions to equal parameter degrees and obtain formulas for unknown coeﬃcients. When solving the nonlinear equation, we use a special polynomial which is included in the decomposition coeﬃcients of the nonlinear function and allows the homotopy perturbation method to be applied as well for nonlinear non-uniform pseudodiﬀerential diﬀusion equation. Results. The result is a solution of the Cauchy problem for the investigated diﬀusion equation, which is represented as a series of terms whose functions are found from the parametric series. Conclusions. In this paper we ﬁrst prove the solvability and obtain the formula for solving the Cauchy problem as a series for linear and nonlinear inhomogeneous pseudodiﬀerential equations
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.
Balance function generated by limiting conditions
(2023) Morozov, Denys
This article conducts an analysis of the inherent constraints governing the formation of the price function that describes the interaction between two markets. The research not only identifies these constraints but also obtains an explicit form of the specified function. The key factors considered in constructing the price function are defined in the article. Through analyzing these constraints and their impact on market interaction, a formula for the price function is provided. This approach not only reveals the essence of natural constraints in forming the price function but also provides a contextual foundation for negotiations shaping a fair exchange price for the interaction process between two markets. This offers a theoretical basis for modeling and solving similar problems arising during practical economic activities. Two economies, Economy 1 and Economy 2, producing goods X and Y with linear Production Possibility Curve (PPC) graphs, are under consideration. The cost of producing one unit of good X relative to Y is denoted as 𝑅1 for Economy 1 and 𝑅2 for Economy 2. Exchange between economies occurs in a market, where the possible exchange is Δ𝑥 units of X for Δ𝑦 = 𝑅market ·Δ𝑥 units of Y, and vice versa. If 𝑅1 is less than 𝑅2, Economy 1 specializes in the production of X, and Economy 2 specializes in Y, fostering mutually beneficial trade. For mutually beneficial exchange on the market with a price 𝑅market, it is necessary and sufficient that 𝑅1 ≤ 𝑅market ≤ 𝑅2. The article also explores the concept of a fair exchange price, specifying conditions for symmetry, reciprocity, and scale invariance. Notably, it indicates that the unique solution satisfying these conditions is 𝑓(𝑅1,𝑅2) = √ 𝑅1 · 𝑅2. In the context of balanced exchange, where economies gain equal profit per unit of the acquired good, the balanced exchange price 𝑅market[𝑏𝑎𝑙𝑎𝑛𝑐𝑒] is determined as 𝑅market = √ 𝑅1 · 𝑅2. This serves as a fair price, meeting the aforementioned conditions of symmetry, reciprocity, and scale invariance. In the provided example with 𝑅1 = 2 and 𝑅2 = 8, the article examines the mutually beneficial interval for 𝑅market and computes the balanced and fair exchange price.
Computing the Moore-Penrose inverse for bidiagonal matrices
(2019) Hakopian, Yuri
The Moore-Penrose inverse is the most popular type of matrix generalized inverses which has many applications both in matrix theory and numerical linear algebra. It is well known that the Moore-Penrose inverse can be found via singular value decomposition. In this regard, there is the most eﬀective algorithm which consists of two stages. In the ﬁrst stage, through the use of the Householder reﬂections, an initial matrix is reduced to the upper bidiagonal form (the Golub-Kahan bidiagonalization algorithm). The second stage is known in scientiﬁc literature as the Golub-Reinsch algorithm. This is an iterative procedure which with the help of the Givens rotations generates a sequence of bidiagonal matrices converging to a diagonal form. This allows to obtain an iterative approximation to the singular value decomposition of the bidiagonal matrix. The principal intention of the present paper is to develop a method which can be considered as an alternative to the Golub-Reinsch iterative algorithm. Realizing the approach proposed in the study, the following two main results have been achieved. First, we obtain explicit expressions for the entries of the Moore-Penrose inverse of bidigonal matrices. Secondly, based on the closed form formulas, we get a ﬁnite recursive numerical algorithm of optimal computational complexity. Thus, we can compute the Moore-Penrose inverse of bidiagonal matrices without using the singular value decomposition.
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.
Diameter Search Algorithms for Directed Cayley Graphs
(2021-05) Olshevskyi, Maksym
It is considered a well known diameter search problem for finite groups. It can be formulated as follows: find the maximum possible diameter of the group over its system of generators. The diameter of a group over a specific system of generators is the diameter of the corresponding Cayley graph. In the paper a closely related problem is considered. For a specific system of generators find the diameter of corresponding Cayley graph. It is shown that the last problem is polynomially reduced to the problem of searching the minimal decomposition of elements over a system of generators. It is proposed five algorithms to solve the diameter search problem: simple down search algorithm, fast down search algorithm, middle down search algorithms, homogeneous down search algorithm and homogeneous middle down search algorithm.
A discrete regularization method for hidden Markov models embedded into reproducing kernel Hilbert space
(2018) Kriukova, Galyna
Hidden Markov models are a well-known probabilistic graphical model for time series of discrete, partially observable stochastic processes. We consider the method to extend the application of hidden Markov models to non-Gaussian continuous distributions by embedding a priori probability distribution of the state space into reproducing kernel Hilbert space. Corresponding regularization techniques are proposed to reduce the tendency to overfitting and computational complexity of the algorithm, i.e. Nystr¨om subsampling and the general regularization family for inversion of feature and kernel matrices. This method may be applied to various statistical inference and learning problems, including classification, prediction, identification, segmentation, and as an online algorithm it may be used for dynamic data mining and data stream mining. We investigate, both theoretically and empirically, the regularization and approximation bounds of the discrete regularization method. Furthermore, we discuss applications of the method to real-world problems, comparing the approach to several state-of-the-art algorithms.
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].
Generalization of cross-entropy loss function for image classification
(2020) Andreieva, Valeria; Shvai, Nadiya
Classification task is one of the most common tasks in machine learning. This supervised learning problem consists in assigning each input to one of a finite number of discrete categories. Classification task appears naturally in numerous applications, such as medical image processing, speech recognition, maintenance systems, accident detection, autonomous driving etc. In the last decade methods of deep learning have proven to be extremely efficient in multiple machine learning problems, including classification. Whereas the neural network architecture might depend a lot on data type and restrictions posed by the nature of the problem (for example, real-time applications), the process of its training (i.e. finding model’s parameters) is almost always presented as loss function optimization problem. Cross-entropy is a loss function often used for multiclass classification problems, as it allows to achieve high accuracy results. Here we propose to use a generalized version of this loss based on Renyi divergence and entropy. We remark that in case of binary labels proposed generalization is reduced to cross-entropy, thus we work in the context of soft labels. Specifically, we consider a problem of image classification being solved by application of convolution neural networks with mixup regularizer. The latter expands the training set by taking convex combination of pairs of data samples and corresponding labels. Consequently, labels are no longer binary (corresponding to single class), but have a form of vector of probabilities. In such settings cross-entropy and proposed generalization with Renyi divergence and entropy are distinct, and their comparison makes sense. To measure effectiveness of the proposed loss function we consider image classification problem on benchmark CIFAR-10 dataset. This dataset consists of 60000 images belonging to 10 classes, where images are color and have the size of 32 x 32. Training set consists of 50000 images, and the test set contains 10000 images. For the convolution neural network, we follow [1] where the same classification task was studied with respect to different loss functions and consider the same neural network architecture in order to obtain comparable results. Experiments demonstrate superiority of the proposed method over cross-entropy for loss function parameter value a < 1. For parameter value a > 1 proposed method shows worse results than crossentropy loss function. Finally, parameter value a = 1 corresponds to cross-entropy.
Interpolation problems for random fields on Sierpinski's carpet
(2023) Boichenko, Viktoriia; Shchestyuk, Nataliia; Florenko, Anastasiia
The prediction of stochastic processes and the estimation of random fields of different natures is becoming an increasingly common field of research among scientists of various specialties. However, an analysis of papers across different estimating problems shows that a dynamic approach over an iterative and recursive interpolation of random fields on fractal is still an open area of investigation. There are many papers related to the interpolation problems of stationary sequences, estimation of random fields, even on the perforated planes, but all of this still provides a place for an investigation of a more complicated structure like a fractal, which might be more beneficial in appliances of certain industry fields. For example, there has been a development of mobile phone and WiFi fractal antennas based on a first few iterations of the Sierpinski carpet. In this paper, we introduce an estimation for random fields on the Sierpinski carpet, based on the usage of the known spectral density, and calculation of the spectral characteristic that allows an estimation of the optimal linear functional of the omitted points in the field. We give coverage of an idea of stationary sequence estimating that is necessary to provide a basic understanding of the approach of the interpolation of one or a set of omitted values. After that, the expansion to random fields allows us to deduce a dynamic approach on the iteration steps of the Sierpinski carpet. We describe the numerical results of the initial iteration steps and demonstrate a recurring pattern in both the matrix of Fourier series coefficients of the spectral density and the result of the optimal linear functional estimation. So that it provides a dependency between formulas of the different initial sizes of the field as well as a possible generalizing of the solution for N-steps in the Sierpinski carpet. We expect that further evaluation of the mean squared error of this estimation can be used to identify the possible iteration step when further estimation becomes irrelevant, hence allowing us to reduce the cost of calculations and make the process viable.
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.
Ǭ-зображення дiйсних чисел як узагальнення канторiвських систем числення
(2022) Працьовитий, Микола; Бондаренко, Ольга; Ратушняк, Софія; Франчук, Катерина
Роботу присвячено узагальненню канторівської системи числення, яка визначається послідовністю основ( sn), 1 < sn ∈ N і послідовністю алфавітів An = {0, 1, ..., sn − 1}: [0; 1] ∋ x = ∞∑ n=1 αn / s1s2...sn, αn ∈ An, яке назване Ǭ-зображенням. Воно визначається нескінченною матрицею ||qik||, де i ∈ Ai, k ∈ N, що має властивості 0 < qik < 1, mk ∑ i=0 qik = 1, k ∈ N, ∞∏ n=1 max i {qik} = 0, а саме [0; 1] ∋ x = ai11 + ∞∑ k=2 [aikk k−1 ∏ j=1 qij (x)j ] ≡ Δi1i2...ik..., where ainn = in−1 ∑ j=0 qjn, in ∈ An, n ∈ N. У роботі розглянуто застосування вказаного зображення чисел у метричній теорії чисел, теорії розподілів випадкових величин, теорії локально складних функцій та фрактальному аналізі. Вивчено тополого-метричну структуру множини C[Ǭ; Vn] = {x : x = Δα1...αn..., αn ∈ Vn ⊂ An}. Виведено формулу для обчислення її міри Лебега: λ(C) = ∞∏ n=1 λ(Fn) / λ(Fn−1) = ∞∏ n=1 (1 − λ(Fn) / λ(Fn−1)), де F0 = [0; 1], Fn - об'єднання Ǭ-циліндрів рангу n, серед внутрішніх точок яких є точки множини C, Fn ≡ Fn−1 \ Fn. Знайдено критерій і деякі достатні умови нуль-мірності цієї множини. За додаткових умов на "матрицю" ||qik|| знайдено нормальну властивість Ǭ-зображення чисел (властивість, яку мають майже всі у розумінні міри Лебега числа). Отримані результати використано для встановлення лебегівської структури і типу розподілу випадкової величини, Ǭ-зображення якої є незалежними випадковими величинами. Доведено, що цифри Ǭ-зображення рівномірно розподіленої на [0; 1] випадкової величини є незалежними, і вказано їх розподіл. Доведено, що при обчисленні фрактальної розмірності Гаусдорфа Безиковича підмножин відрізка [0; 1] можна обмежитись покриттями Ǭ-циліндрами: Δc1...cm = {x : x = Deltac1...cki1...in..., in ∈ ∈ Ak+n}, якщо потужності алфавітів обмежені, а елементи "матриці" ||qik|| відокремлені від нуля. Для інферсора цифр Ǭ-зображення чисел, тобто функції, означеної рівністю I(x = = Δi1...in...) = Δ[m1−i1]...[mn−in]..., mn ≡ sn − 1 доведено неперервність, строгу монотонність, а для окремих випадків її сингулярність (рівність похідної нулю майже скрізь у розумінні міри Лебега).
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.
Predictive model for a product without history using LightGBM. pricing model for a new product
(2023) Drin, Svitlana; Kriuchkova, Anastasiia; Toloknova, Varvara
The article focuses on developing a predictive product pricing model using LightGBM. Also, the goal was to adapt the LightGBM method for regression problems and, especially, in the problems of forecasting the price of a product without history, that is, with a cold start. The article contains the necessary concepts to understand the working principles of the light gradient boosting machine, such as decision trees, boosting, random forests, gradient descent, GBM (Gradient Boosting Machine), GBDT (Gradient Boosting Decision Trees). The article provides detailed insights into the algorithms used for identifying split points, with a focus on the histogram-based approach. LightGBM enhances the gradient boosting algorithm by introducing an automated feature selection mechanism and giving special attention to boosting instances characterized by more substantial gradients. This can lead to significantly faster training and improved prediction performance. The Gradient-based One-Side Sampling (GOSS) and Exclusive Feature Bundling (EFB) techniques used as enhancements to LightGBM are vividly described. The article presents the algorithms for both techniques and the complete LightGBM algorithm. This work contains an experimental result. To test the lightGBM, a real dataset of one Japanese C2C marketplace from the Kaggle site was taken. In the practical part, a performance comparison between LightGBM and XGBoost (Extreme Gradient Boosting Machine) was performed. As a result, only a slight increase in estimation performance (RMSE, MAE, R-squard) was found by applying LightGBM over XGBoost, however, there exists a notable contrast in the training procedure’s time efficiency. LightGBM exhibits an almost threefold increase in speed compared to XGBoost, making it a superior choice for handling extensive datasets. This article is dedicated to the development and implementation of machine learning models for product pricing using LightGBM. The incorporation of automatic feature selection, a focus on highgradient examples, and techniques like GOSS and EFB demonstrate the model’s versatility and efficiency. Such predictive models will help companies improve their pricing models for a new product. The speed of obtaining a forecast for each element of the database is extremely relevant at a time of rapid data accumulation.
Properties of the ideal-intersection graph of the ring Zn
(2023) Utenko, Yelizaveta
In this paper we study properties of the ideal-intersection graph of the ring 𝑍𝑛. The graph of ideal intersections is a simple graph in which the vertices are non-zero ideals of the ring, and two vertices (ideals) are adjacent if their intersection is also a non-zero ideal of the ring. These graphs can be referred to as the intersection scheme of equivalence classes (See: Laxman Saha, Mithun Basak Kalishankar Tiwary "Metric dimension of ideal-intersection graph of the ring 𝑍𝑛" [1] ). In this article we prove that the triameter of graph is equal to six or less than six. We also describe maximal clique of the ideal-intersection graph of the ring 𝑍𝑛. We prove that the chromatic number of this graph is equal to the sum of the number of elements in the zero equivalence class and the class with the largest number of element. In addition, we demonstrate that eccentricity is equal to 1 or it is equal to 2. And in the end we describe the central vertices in the ideal-intersection graph of the ring 𝑍𝑛.
Regularization by Denoising for Inverse Problems in Imaging
(2022) Kravchuk, Oleg; Kriukova, Galyna
In this work, a generalized scheme of regularization of inverse problems is considered, where a priori knowledge about the smoothness of the solution is given by means of some self-adjoint operator in the solution space. The formulation of the problem is considered, namely, in addition to the main inverse problem, an additional problem is defined, in which the solution is the right-hand side of the equation. Thus, for the regularization of the main inverse problem, an additional inverse problem is used, which brings information about the smoothness of the solution to the initial problem. This formulation of the problem makes it possible to use operators of high complexity for regularization of inverse problems, which is an urgent need in modern machine learning problems, in particular, in image processing problems. The paper examines the approximation error of the solution of the initial problem using an additional problem.

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