Mohyla Mathematical Journal最新文献

{"title":"Research of Stochastic Behaviour of Cellular Automata","authors":"Serhii Hlushenkov, R. Chornei","doi":"10.18523/2617-70805202246-52","DOIUrl":"https://doi.org/10.18523/2617-70805202246-52","url":null,"abstract":"A wide range of complex systems with local interactions could be described by cellular automata. Despite the fact, that in general, behaviour of each cellular automata on its own could be quite simple, their effective combination, or setting unusual interaction rules may result in extraordinary system with much more complicated behaviour, or unexpected and ambiguous observation results. Stochasticity in interactions between cells approximates simulated environment to real conditions and helps finding optimal strategy, which would be more stable under all circumstances and events, especially unlikely ones. Stochastic cellular automata are often used for modelling natural phenomena and processes, simulating traffic flows, cryptography, and so on. Finding an optimal strategy – is a key problem in managing environments with available outside influence.This article shows existence of optimal strategies for stochastic cellular automata systems, gives an example of applying improving strategy algorithm in case of extinguishing forest fires, analyses chosen strategy optimality.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"54 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":"127886925","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":"Classification of Infinite Coxeter Graphs Relative to the Value of the Index in the Interval (√√5 + 2; 3/√2]","authors":"L. Tymoshkevych, M. Kohut","doi":"10.18523/2617-70805202219-25","DOIUrl":"https://doi.org/10.18523/2617-70805202219-25","url":null,"abstract":"The structure of infinite Coxeter graphs whose largest eigenvalue belongs to the interval from √√5 + 2 to 3 3/√2] is investigated. In particular, such a graph is a tree, can have at most one label greater than 3 on its edges and such label does not exceed 6, can have only vertices with degree strictly less than 5, and among edges which are incident with vertex with degree 4 can be only one that is not incident with leaf. A number of other properties are also given for infinite Coxeter graphs with largest eigenvalue in the specified interval.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"80 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":"122688120","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":"A Model of a System of Simultaneous Equations with a Lag Effect for Estimating the Quality of an Advertising Campaign","authors":"S. Drin, Yehor Reznichenko","doi":"10.18523/2617-70805202233-37","DOIUrl":"https://doi.org/10.18523/2617-70805202233-37","url":null,"abstract":"This article describes the creation of a more generalized system of simultaneous equations for forecasting the level of sales depending on advertising campaigns on different channels and other factors. RStudio (R programming language) and Google Colab (Python programming language) environments describe the creation of a model based on real data of a product. The Hausman specification test was applied to determine the model estimation method. As a conclusion, the indicators of advertising campaigns turned out to be endogenous variables, which indicates the importance of using the 2MNK method. It was found that the volume of advertising is the cause of the volume of sales according to Granger, which cannot be said about the reverse assumption - the causality of the volume of advertising from sales according to Granger. The ”depth” of lags is also determined, namely, one lag for both advertising channels. The dependence of sales volumes on various factors, including product distribution, the price index, the influence of advertising and its lags, and the influence of competitors’ advertising activities, was evaluated. The coefficients of the resulting more generalized system of simultaneous equations were estimated using the two-step least squares method. All statistical indicators testify to the adequacy of the model. Performance indicators (ROI - return on investment) of advertising campaigns showed that advertising both on television and on the Internet is profitable for the company’s product in question. The relevance of this article lies in the creation of a more general system of simultaneous equations with the inclusion of a product sales forecast model taking into account the influence of advertising.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"41 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":"123747415","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":"A Solution of a Finitely Dimensional Harrington Problem for Cantor Set","authors":"S. Kusinski","doi":"10.18523/2617-7080520226-8","DOIUrl":"https://doi.org/10.18523/2617-7080520226-8","url":null,"abstract":"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.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"46 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":"121679960","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":"Ǭ-representation of Real Numbers as a Generalization of Cantor Numeral Systems","authors":"M. Pratsiovytyi, O. Bondarenko, S. Ratushniak, Kateryna Franchuk","doi":"10.18523/2617-7080520229-18","DOIUrl":"https://doi.org/10.18523/2617-7080520229-18","url":null,"abstract":"We consider generalization of Cantor numeral system, which is determined by the sequence of bases (sn), 1 < sn ∈ N, and the sequence of alphabets An = {0, 1, ..., sn − 1}:[0; 1] ∋ x = ∞∑ n=1 αn/s1s2...sn, αn ∈ An,the so-called Ǭ-representation. It is defined by an infinite “matrix” ||qik||, where i ∈ Ai, k ∈ N, having the properties0 < qik < 1, mk ∑ i=0 qik = 1, k ∈ N, ∞∏ n=1 max i {qik} = 0,namely[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.The applications of this representation of numbers in the metric theory of numbers, the theory of distributions of random variables, the theory of locally complicated functions, and fractal analysis are studied.For the set C[Ǭ; Vn] = {x : x = Δα1...αn..., αn ∈ Vn ⊂ An}, we study its topological and metric structure and derive a formula for calculating its Lebesgue measure:λ(C) = ∞∏ n=1 λ(Fn) / λ(Fn−1) = ∞∏ n=1 (1 − λ(Fn) / λ(Fn−1)),where F0 = [0; 1], Fn is the union of Ǭ-cylinders of rank n, such that there are points of the set C among their interior points of the set C, Fn ≡ Fn−1 Fn.A criterion and some sufficient conditions for this set to be a set of zero measure are found. Under additional conditions on the “matrix” ||qik||, the normal property for Ǭ-representation of numbers is found ((i.e., almost all in the sense of Lebesgue measure numbers have this property). The obtained results are used to establish the Lebesgue structure and the type of distribution of a random variable whose digits of Ǭ-representation are independent random variables. It is proved that the digits of the Ǭ-representation of a random variable uniformly distributed on [0; 1] are independent, and their distribution is given.If the cardinalities of the alphabets are finite and the elements of the “matrix” ||qik|| are bounded away from zero, it is proved that to calculate the Hausdorff-Besicovitch fractal dimension of subsets of the segment [0; 1], it is sufficient to cover them with Ǭ-cylinders: Δc1...cm = {x : x = Deltac1...cki1...in..., in ∈ ∈ Ak+n}.For inversor of digits of Ǭ-representation of numbers, that is, the function defined by equality I(x = = Δi1...in...) = Δ[m1−i1]...[mn−in]..., mn ≡ sn − 1 it is proved its continuity, strict monotonicity, and for certain cases, its singularity (the equality of the derivative to zero almost everywhere in the sense of the Lebesgue measure).","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"26 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":"114850018","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":"Speech Audio Modeling by Means of Causal Moving Average Equipped Gated Attention","authors":"A. Ivaniuk","doi":"10.18523/2617-70805202253-56","DOIUrl":"https://doi.org/10.18523/2617-70805202253-56","url":null,"abstract":"In the paper we compare different attention mechanisms on the task of audio generation using unsupervised approaches following previous work in language modeling. It is important problem, as far as speech synthesis technology could be used to convert textual information into acoustic waveform signals. These representations can be conveniently integrated into mobile devices and used in such applications as voice messengers or email apps. Sometimes it is difficult to understand and read important messages when being abroad. The lack of appropriate computer systems or some security problems may arise. With this technology, e-mail messages can be listened quickly and efficiently on smartphones, boosting productivity. Apart from that, it is used to assist visually impaired people, so that, for instance, the screen content can be automatically read aloud to a blind user. Nowadays, home appliances, like slow cookers can use this system too for reading culinary recipes, automobiles for voice navigation to the destination spot, or language learners for pronunciation teaching. Speech generation is the opposite problem of automatic speech recognition (ASR) and is researched since the second half of the eighteen's century. Also, this technology also helps vocally handicapped people find a way to communicate with others who do not understand sign language. However, there is a problem, related to the fact that the audio sampling rate is very high, thus leading to very long sequences which are computationally difficult to model. Second challenge is that speech signals with the same semantic meaning can be represented by a lot of signals with significant variability, which is caused by channel environment, pronunciation or speaker timbre characteristics. To overcome these problems, we train an autoencoder model to discretize continuous audio signal into a finite set of discriminative audio tokens which have a lower sampling rate. Subsequently, autoregressive models, which are not conditioned on text, are trained on this representation space to predict the next token, based on previous sequence elements. Hence, this modeling approach resembles causal language modeling. In our study, we show that unlike in the original MEGA work, traditional attention outperforms moving average equipped gated attention, which shows that EMA gated attention is not stable yet and requires careful hyper-parameter optimization.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"23 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":"114913387","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":"Two Approaches for Option Pricing under Illiquidity","authors":"Viktoriia Pauk, Oksana Petrenko, N. Shchestyuk","doi":"10.18523/2617-70805202238-45","DOIUrl":"https://doi.org/10.18523/2617-70805202238-45","url":null,"abstract":"The paper focuses on option pricing under unusual behaviour of the market, when the price may not be changed for some time what is quite a common situation on the modern financial markets. There are some patterns that can cause permanent price gaps to form and lead to illiquidity. For example, global changes that have a negative impact on financial activity, or a small number of market participants, or the market is quite young and is just in the process of developing, etc.In the paper discrete and continuous time approaches for modelling market with illiquidity and evaluation option pricing were considered.Trinomial discrete time model improves upon the binomial model by allowing a stock price not only to move up, down but stay the same with certain probabilities, what is a desirable feature for the illiquid modelling. In the paper parameters for real financial data were identified and the backward induction algorithm for building call option price trinomial tree was applied.Subdiffusive continuous time model allows successfully apply the physical models for describing the trapping events to model financial data stagnation's periods. In this paper the Inverse Gaussian process IG was proposed as a subordinator for the subdiffusive modelling of illiquidity and option pricing. The simulation of the trajectories for subordinator, inverse subordinator and subdiffusive GBM were performed. The Monte Carlo method for option evaluation was applied.Our aim was not only to compare these two models each with other, but also to show that both models adequately describe the illiquid market and can be used for option pricing on this market. For this purpose absolute relative percentage (ARPE) and root mean squared error (RMSE) for both models were computed and analysed.Thanks to the proposed approaches, the investor gets a tools, which allows him to take into account the illiquidity.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"17 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":"127126697","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":"Regularization by Denoising for Inverse Problems in Imaging","authors":"Oleg Kravchuk, G. Kriukova","doi":"10.18523/2617-70805202257-61","DOIUrl":"https://doi.org/10.18523/2617-70805202257-61","url":null,"abstract":"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.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"26 9 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":"125683822","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":"Risk Modelling Approaches for Student-like Models with Fractal Activity Time","authors":"Georgiy Solomanchuk, N. Shchestyuk","doi":"10.18523/2617-70804202128-33","DOIUrl":"https://doi.org/10.18523/2617-70804202128-33","url":null,"abstract":"The paper focuses on value at risk (V@R) measuring for Student-like models of markets with fractal activity time (FAT). The fractal activity time models were introduced by Heyde to try to encompass the empirically found characteristics of real data and elaborated on for Variance Gamma, normal inverse Gaussian and skewed Student distributions. But problem of evaluating an value at risk for this model was not researched. It is worth to mention that if we use normal or symmetric Student`s models than V@R can be computed using standard statistical packages. For calculating V@R for Student-like models we need Monte Carlo method and the iterative scheme for simulating N scenarios of stock prices. We model stock prices as a diffusion processes with the fractal activity time and for modeling increments of fractal activity time we use another diffusion process, which has a given marginal inverse gamma distribution. \u0000The aim of the paper is to perform and compare V@R Monte Carlo approach and Markowitz approach for Student-like models in terms of portfolio risk. For this purpose we propose procedure of calculating V@R for two types of investor portfolios. The first one is uniform portfolio, where d assets are equally distributed. The second is optimal Markowitz portfolio, for which variance of return is the smallest out of all other portfolios with the same mean return. \u0000The programmed model which was built using R-statistics can be used as to the simulations for any asset and for construct optimal portfolios for any given amount of assets and then can be used for understanding how this optimal portfolio behaves compared to other portfolios for Student-like models of markets with fractal activity time. \u0000Also we present numerical results for evaluating V@R for both types of investor portfolio. We show that optimal Markovitz portfolio demonstrates in the most of cases the smallest possible Value at Risk comparing with other portfolios. Thus, for making investor decisions under uncertainty we recommend to apply portfolio optimization and value at risk approach jointly.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116961822","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":"Diameter Search Algorithms for Directed Cayley Graphs","authors":"M. Olshevskyi","doi":"10.18523/2617-7080420217-19","DOIUrl":"https://doi.org/10.18523/2617-7080420217-19","url":null,"abstract":"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.","PeriodicalId":404986,"journal":{"name":"Mohyla Mathematical Journal","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126398570","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}