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
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.
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.
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.
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.
Remarks on my algebraic problem of determining similarities between certain quotient boolean algebras
(2022) Frankiewicz, Ryszard
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 к. 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) i P(ℵ1)/fin(ℵ1).
Risk modelling approaches for student-like models with fractal activity time
(2021) Solomanchuk, Georgiy; Shchestyuk, Nataliia
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 read 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. The 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. The 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. Also we present numerical results for evaluating V@R for both types of investor portfolio. We show that optimal Markowitz 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.
Simulating stochastic diffusion processes and processes with "market" time
(2020) Boluh, Kateryna; Shchestyuk, Nataliia
The paper focuses on modelling, simulation techniques and numerical methods concerned stochastic processes in subject such as financial mathematics and financial engineering. The main result of this work is simulation of a stochastic process with new market active time using Monte Carlo techniques. The processes with market time is a new vision of how stock price behavior can be modeled so that the nature of the process is more real. The iterative scheme for computer modelling of this process was proposed. It includes the modeling of diffusion processes with a given marginal inverse gamma distribution. Graphs of simulation of the Ornstein-Uhlenbeck random walk for different parameters, a simulation of the diffusion process with a gamma-inverse distribution and simulation of the process with market active time are presented.
Solvable Lie algebras of derviations of rank one
(2019) Petravchuk, Anatolii; Sysak, Kateryna
Let K be a ﬁeld of characteristic zero, A = K[x1,...,xn] the polynomial ring and R = K(x1,...,xn) the ﬁeld of rational functions in n variables over K. The Lie algebra Wn(K) of all K-derivations on A is of great interest since its elements may be considered as vector ﬁelds on Kn with polynomial coeﬃcients. If L is a subalgebra of Wn(K), then one can deﬁne the rank rkAL of L over A as the dimension of the vector space RL over the ﬁeld R. Finite dimensional (over K) subalgebras of Wn(K) of rank 1 over A were studied by the ﬁrst author jointly with I. Arzhantsev and E. Makedonskiy. We study solvable subalgebras L of Wn(K) with rkAL = 1, without restrictions on dimension over K. Such Lie algebras are described in terms of Darboux polynomials.
Speech audio modeling by means of causal moving average equipped gated attention
(2022) Ivaniuk, Andrii
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 lea,ding 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.
Two approaches for option pricing under illiquidity
(2022) Pauk, Viktoriia; Petrenko, Oksana; Shchestyuk, Nataliya
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.
Weakly nonlinear models of stochastic wave propagation in two-layer hydrodynamic systems
(2023) Avramenko, Olga; Naradovy, Volodymyr
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.
Володимир Васильович Кириченко
(2019) Олійник, Андрій; Олійник, Богдана
Статтю присвячено пам’ятi видатного українського математика i педагога, одного iз засновникiв київської школи теорiї зображень i теорiї кiлець, професора Київського нацiонального унiверситету iменi Тараса Шевченка Володимира Васильовича Кириченка.
Метрична розмiрнiсть кiстякових дерев унiциклiчних графiв
(2019) Дуденко, Маргарита
Найменшу за потужнiстю множину M ∈ V скiнченного графа G = (V,E) таку, що для будьякої пари вершин u,v ∈ V iснує принаймнi одна вершина t ∈ M, для якої має мiсце нерiвнiсть dG(t,v) 6= dG(t,u), називають метричним базисом, а потужнiсть множини M – метричною розмiрнiстю. Оскiльки, як вiдомо, пошук метричної розмiрностi для довiльного графа є NP-важкою проблемою, то пошук метричної розмiрностi графiв обмежують пошуком для певних родин графiв. Для унiциклiчних графiв, тобто графiв, що мiстять рiвно один цикл, пiсля вилучення ребра можна отримати дерево. Метою статтi є встановлення зв’язку мiж унiциклiчним графом, що має метричну розмiрнiсть 2, та метричними розмiрностями його кiстякових дерев залежно вiд способу вилучення ребра

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