Кафедра математики

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The analytical view of solution of the first boundary value problem for the nonlinear equation of heat conduction with deviation of the argument
(2023) Drin, Yaroslav; Drin, Iryna; Drin, Svitlana
In this article, for the first time, the first boundary value problem for the equation of thermal conductivity with a variable diffusion coefficient and with a nonlinear term, which depends on the sought function with the deviation of the argument, is solved. For such equations, the initial condition is set on a certain interval. Physical and technical reasons for delays can be transport delays, delays in information transmission, delays in decision-making, etc. The most natural are delays when modeling objects in ecology, medicine, population dynamics, etc. Features of the dynamics of vehicles in different environments (water, land, air) can also be taken into account by introducing a delay. Other physical and technical interpretations are also possible, for example, the molecular distribution of thermal energy in various media (solid bodies, liquids, etc.) is modeled by heat conduction equations. The Green’s function of the first boundary value problem is constructed for the nonlinear equation of heat conduction with a deviation of the argument, its properties are investigated, and the formula for the solution is established.
The cauchy problem for quasilinear equation with nonstationary diffusion coefficient
(2023) Drin, Yaroslav; Drin, Iryna; Drin, Svitlana
The initial problem for the heat conduction equation with the inversion of the argument are considered and Green’s function are determined. The theorem declared that the Poisson’s formula determines the solution of the Cauchy problem considered and proved.
The first boundary value problem for the nonlinear equation of heat conduction with deviation of the argument
(2022) Drin, Yaroslav; Drin, Iryna; Drin, Svitlana; Stetsko, Yuriy
The initial-boundary problem for the heat conduction equation with the inversion of the argument are considered. The Green’s function of considered problem are determined. The theorem about the Poisson integral limitation is proved. The theorem declared that the Poisson integral determine the solution of the first boundary problem considered and proved.
The nonlocal problem for fractal diffusion equation
(2022) Drin, Yaroslav; Drin, I.; Drin, Svitlana
Over the past few decades, the theory of pseudodifferential operators (PDO) and equations with such operators (PDE) has been intensively developed. The authors of a new direction in the theory of PDE, which they called parabolic PDE with non-smooth homogeneous symbols (PPDE), are Yaroslav Drin and Samuil Eidelman. In the early 1970s, they constructed an example of the Cauchy problem for a modified heat equation containing, instead of the Laplace operator, PDO, which is its square root. Such a PDO has a homogeneous symbol |σ|, which is not smooth at the origin. The fundamental solution of the Cauchy problem (FSCP) for such an equation is an exact power function. For the heat equation, FSCP is an exact exponential function. The Laplace operator can be interpreted as a PDO with a smooth homogeneous symbol |σ|^2, σ ∈ Rn. A generalization of the heat equation is PPDE containing PDO with homogeneous non-smooth symbols. They have an important application in the theory of random processes, in particular, in the construction of discontinuous Markov processes with generators of integro-differential operators, which are related to PDO; in the modern theory of fractals, which has recently been rapidly developing. If the PDO symbol does not depend on spatial coordinates, then the Cauchy problem for PPDE is correctly solvable in the space of distribution-type generalized functions. In this case, the solution is written as a convolution of the FSCP with an initial generalized function. These results belong to a number of domestic and foreign mathematicians, in particular S. Eidelman and Y. Drin (who were the first to define PPDO with non-smooth symbols and began the study of the Cauchy problem for the corresponding PPDE), M. Fedoruk, A. Kochubey, V. Gorodetsky, V . Litovchenko and others. For certain new classes of PPDE, the correct solvability of the Cauchy problem in the space of Hölder functions has been proved, classical FSCP have been constructed, and exact estimates of their power-law derivatives have been obtained [1–4]. Of fundamental importance is the interpretation of PDO proposed by A. Kochubey in terms of hypersingular integrals (HSI). At the same time, the HSI symbol is constructed from the known PDO symbol and vice versa [6]. The theory of HSI, which significantly extend the class of PDO, was developed by S. Samko [7]. We extends this concept to matrix HSI [5]. Generalizations of the Cauchy problem are non-local multipoint problems with respect to the time variable and the problem with argument deviation. Here we prove the solvability of a nonlocal problem using the method of steps. We consider an evolutionary nonlinear equation with a regularized fractal fractional derivative α ∈ (0, 1] with respect to the time variable and a general elliptic operator with variable coefficients with respect to the second-order spatial variable. Such equations describe fractal properties in real processes characterized by turbulence, in hydrology, ecology, geophysics, environment pollution, economics and finance.
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.
A recommendation system with reinforcement for determining the price of a product
(2023) Drin, Svitlana; Kriuchkova, Anastasia
Recommender systems belong to a fairly new concept with a wide scope of application and the property of adaptation to non-classical optimization problems. Thus, recommender systems can be used in e-commerce to predict demand and price for a new product in a chain of stores. LightGBM is a reinforcement recommendation system based on regression trees. The Boosting technology for the new tree ˆ f(x) ← ˆ f(x) + λfk(x) is selected with the specified parameter value λ. A typical value of λ is 0.001 to 0.1. But despite its simplicity and many obvious advantages, for many real big data with long-term dependence, classic Boosting techniques should not be used in real models. For example, if there are 900 stores and 7000 SKUs in the chain, then we have more than 6 million combinations per day in the event horizon. In the LightGBM technique, trees of different structure q are combined, where q : Rm → T. And in the known tree structure q(x), the optimal weight of each leaf is obtained by minimizing the loss function L′(t) = PT i=1 [ Gjwj + 1 2 (Hj + λ)w2 j ] + γT. It is worth noting that the LightGBM algorithm, which is characterized by its efficiency, accuracy and speed, creates histograms and uses the generated classes instead of the entire range of values of each variable, achieving a significant reduction in training time. The generated classes are the result of the Gradient One Side Sampling (GOSS) method. The main idea of the GOSS methodology focuses on the fact that not all observations contribute equally to the learning of the algorithm, since those with a small first derivative of the loss function learn better than those with a large first derivative of the loss function. In the case of forecasting the price of goods that do not have a sales history, we will use embedding in NLP in our model. The main purpose of using embedding is the distance between vectors of products to a new product that have a similar nature of sales, and this distance should be minimal. Results will be evaluated using the MAE performance metric.
A test on the location of tangency portfolio for small sample size and singular covariance matrix
(2023) Drin, Svitlana; Mazur, Stepan; Muhinyuza, Stanislas
In this paper, we propose the test for the location of the tangency portfolio on the set of feasible portfolios when both the population and the sample covariance matrices of asset returns are singular. We derive the exact distribution of the test statistic under both the null and alternative hypotheses. Furthermore, we establish the high-dimensional asymptotic distribution of that test statistic when both the portfolio dimension and the sample size increase to infinity. We complement our theoretical findings by comparing the high-dimensional asymptotic test with an exact finite sample test in the numerical study. A good performance of the obtained results is documented.

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