# Кафедра математики

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Item Accurate classification for Automatic Vehicle Type Recognition based on ensemble classifiers(2019) Shvai, Nadiya; Hasnat, Abul; Meicler, Antoine; Nakib, AmirIn this work, a real world problem of the vehicle type classification for Automatic Toll Collection (ATC) is considered. This problem is very challenging because any loss of accuracy even of the order of 1% quickly turns into a significant economic loss. To deal with such problem, many companies currently use Optical Sensors (OS) and human observers to correct the classification errors. Herein, a novel vehicle classification method is proposed. It consists in regularizing the problem using one camera to obtain vehicle class probabilities using a set of Convolutional Neural Networks (CNN), then, uses the Gradient Boosting based classifier to fuse the continuous class probabilities with the discrete class labels obtained from OS. The method is evaluated on a real world dataset collected from the toll collection points of the VINCI Autoroutes French network. Results show that it performs significantly better than the existing ATC system and, hence will vastly reduce the workload of human operators.Item Analysis of the Shape of Wave Packets in the "Half Space–Layer–Layer with Rigid Lid" Three-Layer Hydrodynamic System(2022) Avramenko, Olga; Lunyova, МariiaWe study the process of propagation of weakly nonlinear wave packets on the contact surfaces of a "half space–layer–layer with rigid lid" hydrodynamic system by the method of multiscale expansions. The solutions of the weakly nonlinear problem are obtained in the second approximation. The condition of solvability of this problem is established. For each frequency of the wave packet, we construct the domains of sign constancy for the coefficient for the second harmonic on the bottom and top contact surfaces. The regularities of wave formation are determined depending on the geometric and physical parameters of the hydrodynamic system. We also analyze the plots of the shapes of deviations of the bottom and top contact surfaces typical of the constructed domains of sign-constancy of the coefficient. We discover the domains where the waves become ∪ - and ∩ -shaped and reveal a significant influence of wavelength on the shapes of deviations of the contact surfaces of the analyzed hydrodynamic system.Item Conjugacy in finite state wreath powers of finite permutation groups(2019) Oliynyk, Andriy; Russyev, AndriyIt is proved that conjugated periodic elements of the infinite wreath power of a finite permutation group are conjugated in the finite state wreath power of this group. Counter-examples for non-periodic elements are given.Item A discrete regularization method for hidden Markov models embedded into reproducing kernel Hilbert space(2018) Kriukova, GalynaHidden 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.Item Edge Imbalance Sequences and Their Graphicness: [preprint](2019) Kozerenko, SergiyThe main focus of combinatorial dynamics is put on the structure of periodic points (and the corresponding orbits) of topological dynamical systems. The first result in this area is the famous Sharkovsky’s theorem which completely describes the coexistence of periods of periodic points for a continuous map from the closed unit interval to itself. One feature of this theorem is that it can be proved using digraphs of a special type (the so-called periodic graphs). In this paper we use Markov graphs (which are the natural generalization of periodic graphs in case of dynamical systems on trees) as a tool to study several classes of maps on trees. The emphasis is put on linear and metric maps.Item Linear and metric maps on trees via Markov graphs : [preprint](2018) Kozerenko, SergiyThe main focus of combinatorial dynamics is put on the structure of periodic points (and the corresponding orbits) of topological dynamical systems. The first result in this area is the famous Sharkovsky's theorem which completely describes the coexistence of periods of periodic points for a continuous map from the closed unit interval to itself. One feature of this theorem is that it can be proved using digraphs of a special type (the so-called periodic graphs). In this paper we use Markov graphs (which are the natural generalization of periodic graphs in case of dynamical systems on trees) as a tool to study several classes of maps on trees. The emphasis is put on linear and metric maps.Item Metric dimension of metric transform and wreath product(2019) Ponomarchuk, BohdanLet (X, d) be a metric space. A non-empty subset A of the set X is called resolving set of the metric space (X, d) if for two arbitrary not equal points u, v from X there exists an element a from A, such that d(u, a) 6= d(v, a). The smallest of cardinalities of resolving subsets of the set X is called the metric dimension md(X) of the metric space (X, d). In general, finding the metric dimension is an NP-hard problem. In this paper, metric dimension for metric transform and wreath product of metric spaces are provided. It is shown that the metric dimension of an arbitrary metric space is equal to the metric dimension of its metric transform.Item Monte-Carlo method for option pricing in sub-diffusive arithmetic models(2021) Shchestyuk, Nataliia; Tyshchenko, OksanaТНів paper focuses on applying Monte Carlo approach to option ргісіпд іп markets with illiquid assets. Anomalous sub-diffusion ів a well-known model for describing such markets, when relatively long periods without any trading are observed. For constructing sub-diffusive models we need to replace a calendar rime t with the some stochasUc processes S(t), which is called inverse subordinator. The inverse subordmator S (t) means first hitting rime and based on subordmator processes. In this paper we propose to use gamma gamma process as subordmator for BasheUe sub-diffusion model. Using wellknown properUes for gamma and mverse gamma processes we find the covariance structure of fractional BacheUer model with FBM Ume-changed by gamma process and then explore the asymptotic behavior of it. Then we apply Monte-Carlo method and propose procedure of option pricing for BasheUe subdiffusion model. For this am we use the iterative schemes for smutting N scenarios of stock prices for our models. Finally we demonstrate numerical results.Item More on linear and metric tree maps(2021) Kozerenko, SergiyWe consider linear and metric self-maps on vertex sets of finite combinatorial trees. Linear maps are maps which preserve intervals between pairs of vertices whereas metric maps are maps which do not increase distances between pairs of vertices. We obtain criteria for a given linear or a metric map to be a positive (negative) under some orientation of the edges in a tree, we characterize trees which admit maps with Markov graphs being paths and prove that the converse of any partial functional digraph is isomorphic to a Markov graph for some suitable map on a tree.Item Morita equivalent unital locally matrix algebras(2020) Bezushchak, Oksana; Oliynyk, BogdanaWe describe Morita equivalence of unital locally matrix algebras in terms of their Steinitz parametrization. Two countable-dimensional unital locally matrix algebras are Morita equivalent if and only if their Steinitz numbers are rationally connected. For an arbitrary uncountable dimension α and an arbitrary not locally finite Steinitz number s there exist unital locally matrix algebras A, B such that dimFA=dimFB=α, st(A)=st(B)=s, however, the algebras A, B are not Morita equivalent.Item Multiple auxiliary classifiers GAN for controllable imagegeneration: Application to license plate recognition(2021) Shvai, Nadiya; Hasnat, Abul; Nakib, AmirOne of the main challenges in developing machine learning (ML) applications is the lack of labeled and balanced datasets. In the literature, different techniques tackle this problem via augmentation, rendering, and over-sampling. Still, these methods produce datasets that appear less natural, exhibit poor balance, and have less variation. One potential solution is to leverage the Generative Adversarial Network (GAN) which achieves remarkable results in the generation of high-fidelity natural images. However, expanding the ability of GANs’ to control generated image attributes with supervisory information remains a challenge. This research aims to propose an efficient method to generate high-fidelity natural images with total control of its main attributes. Therefore, this paper proposes a novel Multiple Auxiliary Classifiers GAN (MAC-GAN) framework based on Auxiliary Classifier GAN (AC-GAN), multi-conditioning, Wasserstein distance, gradient penalty, and dynamic loss. It is therefore presented as an efficient solution for highly controllable image synthesis red that allows to enrich and re-balance datasets beyond data augmentation. Furthermore, the effectiveness of MAC-GAN images on a target ML application called Automatic License Plate Recognition (ALPR) under limited resource constraints is probed. The improvement achieved is over 5% accuracy, which is mainly due to the ability of the MAC-GAN to create a balanced dataset with controllable synthesis and produce multiple (different) images with the same attributes, thus increasing the variation of the dataset in a more elaborate way than data augmentation techniques.Item The nonlocal problem for fractal diffusion equation(2022) Drin, Yaroslav; Drin, I.; Drin, SvitlanaOver 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.Item On expansive and anti-expansive tree maps(2018) Kozerenko, SergiyWith every self-map on the vertex set of a finite tree one can associate the directed graph of a special type which is called the Markov graph. Expansive and anti-expansive tree maps are two extremal classes of maps with respect to the number of loops in their Markov graphs. In this paper we prove that a tree with at least two vertices has a perfect matching if and only if it admits an expansive cyclic permutation of its vertices. Also, we show that for every tree with at least three vertices there exists an expansive map with a weakly connected (strongly connected provided the tree has a perfect matching) Markov graph as well as anti-expansive map with a strongly connected Markov graph.Item On unicyclic graphs of metric dimension 2(2017) Dudenko, Marharyta; Oliynyk, BogdanaA metric basis S of a graph G is the subset of vertices of minimum cardinality such that all other vertices are uniquely determined by their distances to the vertices in S. The metric dimension of a graph G is the cardinality of the subset S. A unicyclic graph is a graph containing exactly one cycle. The construction of a knitting unicyclic graph is introduced. Using this construction all unicyclic graphs with two main vertices and metric dimensions 2 are characterized.Item On unicyclic graphs of metric dimension 2 with vertices of degree 4(2018) Dudenko, Marharyta; Oliynyk, BogdanaWe show that if G is a unicyclic graph with metric dimension 2 and {a, b} is a metric basis of G then the degree of any vertex v of G is at most 4 and degrees of both a and b are at most 2. The constructions of unispider and semiunispider graphs and their knittings are introduced. Using these constructions all unicyclic graphs of metric dimension 2 with vertices of degree 4 are characterized.Item An optimal lower bound for the size of periodic digraphs(2023) Kozerenko, SergiyA periodic digraph is the digraph associated with a periodic point of a continuous map from the unit interval to itself. This digraph encodes "covering" relation between minimal intervals in the corresponding orbit, which allows the application of purely combinatorial arguments in establishing results on the existence and co-existence of periods of periodic points (for example, in proving the famous Sharkovsky’s theorem). In this article, an optimal lower bound for the size of periodic digraphs is provided and thus some previous results of Pavlenko are tightened.Item Permutation codes over Sylow 2-subgroups Syl2(S2n) of symmetric groups S2n(2021) Olshevska, VitaThe permutation code (or the code) is well known object of research starting from 1970s. The code and its properties is used in different algorithmic domains such as error-correction, computer search, etc. It can be de ned as follows: the set of permutations with the minimum distance between every pair of them. The considered distance can be di erent. In general, there are studied codes with Hamming, Ulam, Levensteins, etc. distances. In the paper we considered permutations codes over 2-Sylow subgroups of symmetric groups with Hamming distance over them. For this approach representation of permutations by rooted labeled binary trees is used. This representation was introduced in the previous author's paper. We also study the property of the Hamming distance de ned on permutations from Sylow 2-subgroup Syl2(S2n) of symmetric group S2n and describe an algorithm for nding the Hamming distance over elements from Sylow 2-subgroup of the symmetric group with complexity O(2n). The metric properties of the codes that are de ned on permutations from Sylow 2-subgroup Syl2(S2n) of symmetric group S2n are studied. The capacity and number of codes for the maximum and the minimum nontrivial distance over codes are characterized.Item Primary decompositions of unital locally matrix algebras(2020) Bezushchak, Oksana; Oliynyk, BogdanaWe construct a unital locally matrix algebra of uncountable dimension that (1) does not admit a primary decomposition, (2) has an infinite locally finite Steinitz number. It gives negative answers to questions from [V. M. Kurochkin, On the theory of locally simple and locally normal algebras, Mat. Sb., Nov. Ser. 22(64)(3) (1948) 443–454; O. Bezushchak and B. Oliynyk, Unital locally matrix algebras and Steinitz numbers, J. Algebra Appl. (2020), online ready]. We also show that for an arbitrary infinite Steinitz number s there exists a unital locally matrix algebra A having the Steinitz number s and not isomorphic to a tensor product of finite-dimensional matrix algebras.Item Regularization by Denoising for Inverse Problems in Imaging(2022) Kravchuk, Oleg; Kriukova, GalynaIn 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.Item Two approaches for option pricing under illiquidity(2022) Pauk, Viktoriia; Petrenko, Oksana; Shchestyuk, NataliyaThe 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.