Mohyla Mathematical Journal最新文献

{"title":"Central limit theorem for the number of records in Fα-scheme","authors":"Oleksandr Kolesnik","doi":"10.18523/2617-70806202335-38","DOIUrl":"https://doi.org/10.18523/2617-70806202335-38","url":null,"abstract":"Consider the sequence {Xk, k ≥ 1} of independent identically distributed random variables whose distribution function is continuous. Then events of the type {Xi = Xj} have probability 0 if i ≠ j. Let L(1) = 1. For n ≥ 2, we define random variablesL(n) = inf{k > L(n − 1) : Xk > XL(n-1)}assuming that inf ∅ := +∞. The members of the sequence L = {L(n), n ≥ 1} are called moments of records constructed for {Xk, k ≥ 1}. Consider the sequence of random variables μ = {μ(n), n ≥ 1}, defined by the relationμ(n) = #{k : L(k) ≤ n}, n ≥ 1.It is clear that μ(n) – is the number of records that happened up to the moment n inclusive.In the work [10], the so-called Fα-scheme is considered for the first time, which is built using a given distribution function and a sequence of positive numbers {αk}. It is clear that Fαn(x) is the distribution function for each n ≥ 1. The set of independent random variables {Xn} is called the Fα scheme, if the distribution function of the random variable Xn is Fαn(x). If all αn are equal to each other, then the Fα scheme – is a set independent identically distributed random variables. If not all αn are equal to each other, then the Fα scheme – is a generalization of the classical case.This paper examines the assertions related to the fulfillment of the central limit theorem (CLT) for the number of records in the Fα-scheme of records. The method of finding exact asymptotic expressions for mathematical expectation and variance, which can be used to replace the real characteristics in CLT, is given.A specific example of power-law growth of exponents of the Fα-scheme was considered, and CLT is constructed only in terms of the moment of observation and the power of growth.The article contains 4 theorems with complete proof. Theorem 1 relates the mathematical expectation and variance to the accumulated intensity of the Fα-scheme. Theorem 2 establishes the implementation of CLT in general, and theorem 4 – for a specific case.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":" 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140686949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Interpolation problems for random fields on Sierpinski’s carpet","authors":"Viktoriia Boichenko, Nataly Shchestyuk, Anastasiia Florenko","doi":"10.18523/2617-70806202328-34","DOIUrl":"https://doi.org/10.18523/2617-70806202328-34","url":null,"abstract":"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.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":" 9","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140686197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Expected credit loss modeling","authors":"Svitlana Drin, Fedir Serdiuk","doi":"10.18523/2617-70806202314-19","DOIUrl":"https://doi.org/10.18523/2617-70806202314-19","url":null,"abstract":"This article proposes a method for modeling the probability of default, describes the statistical evaluation of the model, and presents a model of the software implementation algorithm. The algorithm automatically selects from the group of regression models where the models are both linear regression and various modifications of semi-logarithmic models and lag models for macro factors Xi,t,Xi,t-1, ...,Xi,t-TStatistical analysis is carried out using the coefficient of determination R-squared, p-value, VIF (variance inflation factor).The relevance of this topic is determined by the need for banking organizations to comply with international standards, such as International Financial Reporting Standards (IFRS 9) and the Agreement on Banking Supervision and Capital (Basel 3). These standards define credit risk assessment requirements and capital requirements. Adherence to these standards is important not only for ensuring the stability and reliability of the financial system, but also for maintaining the trust of clients and investors. Compliance with international standards also makes banks competitive in the global market and promotes investment inflows and the development of the financial sector.IFRS 9 can be presented in various mathematical models. The article proposes an approach to choosing the appropriate model for forecasting the probability of default. The described model selection method allows banks to choose the optimal default forecast assessment model within the framework of the given standard. This contributes to a more accurate and reliable assessment of credit risk, in accordance with regulatory requirements, which will provide banks with the means for better forecasting and management of financial resources, as well as risk reduction.The model selection methodology saves a significant amount of time and resources, since the search for the optimal model occurs automatically. This allows us to react more quickly to changes in the economic environment, improve decision-making strategies and manage credit risks, which is of great importance for financial institutions in a competitive environment.There is currently a war going on in Ukraine, and forecasting using current methods becomes a difficult task due to unpredictable stressful situations for the economy. In such conditions, standard models may not be sufficiently adapted to account for increased risk and volatility. The proposed approach allows finding more conservative forecasting models that can be useful in unstable periods and war.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":" 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140689748","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Weakly nonlinear models of stochastic wave propagation in two-layer hydrodynamic systems","authors":"O. Avramenko, V. Naradovyi","doi":"10.18523/2617-70806202339-44","DOIUrl":"https://doi.org/10.18523/2617-70806202339-44","url":null,"abstract":"The paper discusses three-dimensional models of the propagation of stochastic internal waves in hydrodynamic systems: ’half-space - half-space’, ’half-space - layer with rigid lid’, and ’layer with solid bottom - layer with rigid lid’. In constructing the models, the layers are considered to be ideal fluids separated by a contact surface. The main objective of the modeling is to obtain a dynamic equation for the stochastic amplitude of surface waves. A comparative analysis of the obtained results has been conducted. In order to control the contribution of nonlinear terms, a dimensionless non-numerical parameter has been introduced. The models are distinguished by boundary conditions that determine the general form of solutions. As a result, a dynamic equation for the stochastic amplitude of internal waves has been derived. After ensemble averaging of the amplitudes, the dynamic equation is formulated in integral form using Fourier-Stieltjes integrals. The dynamic equation reveals two-wave and three-wave interactions, as well as the contribution of dispersion to wave dynamics. An investigation of the boundary case of the transition of internal waves in the ’half-space - half-space’ system to surface waves in the absence of an upper liquid layer confirms the validity of the results.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":" 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140689962","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Likelihood function derivatives for a linear mixed model with compound symmetry assumption","authors":"Sofia Lukashevych, R. Yamnenko","doi":"10.18523/2617-70806202324-27","DOIUrl":"https://doi.org/10.18523/2617-70806202324-27","url":null,"abstract":"The paper explores the properties of linear mixed models with simple random effects of the form: yi = Xiβ + ZiYi + εi, i = 1, . . . ,M, Yi ∼ N(0, Ψ), εi ∼ Т(0, σ2I), where M is the number of distinct groups, each consisting of ni observations. Random effects Yi and within-group errors εi are independent across different groups and within the same group. β is a p-dimensional vector of fixed effects, Yi is a q-dimensional vector of random effects, and Xi and Zi are known design matrices of dimensions nixp and nixq, of fixed and random effects respectively. Vectors εi represent within-group errors with a spherically Gaussian distribution.Assuming a compound symmetry in the correlation structure of the matrix Ψ governing the dependence among within-group errors, analytical formulas for the first two partial derivatives of the profile restricted maximum likelihood function with respect to the correlation parameters of the model are derived. The analytical representation of derivatives facilitates the effective utilization of numerical algorithms like Newton-Raphson or Levenberg-Marquardt.The restricted maximum likelihood (REML) estimation is a statistical technique employed to estimate the parameters within a mixed-effects model, particularly in the realm of linear mixed models. It serves as an extension of the maximum likelihood estimation method, aiming to furnish unbiased and efficient parameter estimates, especially in scenarios involving correlated data. Within the framework of the REML approach, the likelihood function undergoes adjustments to remove the nuisance parameters linked to fixed effects. This modification contributes to enhancing the efficiency of parameter estimation, particularly in situations where the primary focus is on estimating variance components or when the model encompasses both fixed and random effects.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":" 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140689199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Balance function generated by limiting conditions","authors":"Denys Morozov","doi":"10.18523/2617-70806202320-23","DOIUrl":"https://doi.org/10.18523/2617-70806202320-23","url":null,"abstract":"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.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":" 18","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140687622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Predictive model for a product without history using LightGBM. Pricing model for a new product","authors":"Anastasiia Kriuchkova, Varvara Toloknova, Svitlana Drin","doi":"10.18523/2617-7080620236-13","DOIUrl":"https://doi.org/10.18523/2617-7080620236-13","url":null,"abstract":"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.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":" 42","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140687132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Properties of the ideal-intersection graph of the ring Zn","authors":"Yelizaveta Utenko","doi":"10.18523/2617-7080620233-5","DOIUrl":"https://doi.org/10.18523/2617-7080620233-5","url":null,"abstract":"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.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":" 23","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140689126","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Remarks on My Algebraic Problem of Determining Similarities Between Certain Quotient Boolean Algebras","authors":"R. Frankiewicz","doi":"10.18523/2617-7080520223-5","DOIUrl":"https://doi.org/10.18523/2617-7080520223-5","url":null,"abstract":"Remarks on my algebraic problem of determining similarities between certain quotient boolean algebras.In this paper we survey results about quotient boolean algebras of type P(κ)/fin(κ) and condition for them to be or not to be isomorphic for different cardinals κ. Our consideration have their root in the classical result of Parovicenko and a less classical, nevertheless really considerable result about non-existence of P-points by S Shellah. Our main point of interest are the algebras P(ω)/fin(ω) and P(ℵ1)/fin(ℵ1).","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126566684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Inverse Spectral Problems for Weighted Graphs","authors":"Oleksandra Pylypiva, L. Tymoshkevych","doi":"10.18523/2617-70805202226-32","DOIUrl":"https://doi.org/10.18523/2617-70805202226-32","url":null,"abstract":"The paper is devoted to inverse spectral problems for weighted graphs. We give the sharp upper bound for spectral reconstruction number of trees and unicyclic graphs.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132554333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}