{"title":"On some generalizations of multiplicative operators on the space h(g)","authors":"Yu. S. Linchuk","doi":"10.31861/bmj2019.01.056","DOIUrl":"https://doi.org/10.31861/bmj2019.01.056","url":null,"abstract":"","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115704118","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}
Multifrequency systems of dierential equations were studied with the help of averaging method in the works by R.I. Arnold, Ye.O. Grebenikov, Yu.O. Mitropolsky, A.M. Samoilenko and many other scientists. The complexity of the study of such systems is their inherent resonant phenomena, which consist in the rational complete or almost complete commensurability of frequencies. As a result, the solution of the system of equations averaged over fast variables in the general case may deviate from the solution of the exact problem by the quantity O (1). The approach to the study of such systems, which was based on the estimation of the corresponding oscillating integrals, was proposed by A.M. Samoilenko, which allowed to obtain in the works by A.M. Samoilenko and R.I. Petryshyn a number of important results for multifrequency systems with initial , boundary and integral conditions. For multifrequency systems with an argument delay, the averaging method is substantiated in the works by Ya.Y. Bihun, R.I. Petryshyn, I.V. Krasnokutska and other authors. In this paper, the averaging method is used to study the solvability of a multifrequency system with an arbitrary nite number of linearly transformed arguments in slow and fast variables and integral conditions for slow and fast variables on parts of the interval [0, L] of the system of equations. An unimproved estimate of the error of the averaging method under the superimposed conditions is obtained, which clearly depends on the small parameter and the number of linearly transformed arguments in fast variables.
{"title":"AVERAGING IN MULTIFREQUENCY SYSTEMS WITH DELAY AND LOCAL INTEGRAL CONDITIONS","authors":"Ya. I. Bihun, I. Skutar","doi":"10.31861/bmj2020.02.02","DOIUrl":"https://doi.org/10.31861/bmj2020.02.02","url":null,"abstract":"Multifrequency systems of dierential equations were studied with the help of averaging\u0000method in the works by R.I. Arnold, Ye.O. Grebenikov, Yu.O. Mitropolsky, A.M. Samoilenko\u0000and many other scientists. The complexity of the study of such systems is their inherent resonant\u0000phenomena, which consist in the rational complete or almost complete commensurability of\u0000frequencies. As a result, the solution of the system of equations averaged over fast variables in\u0000the general case may deviate from the solution of the exact problem by the quantity O (1). The\u0000approach to the study of such systems, which was based on the estimation of the corresponding\u0000oscillating integrals, was proposed by A.M. Samoilenko, which allowed to obtain in the works by\u0000A.M. Samoilenko and R.I. Petryshyn a number of important results for multifrequency systems\u0000with initial , boundary and integral conditions.\u0000For multifrequency systems with an argument delay, the averaging method is substantiated\u0000in the works by Ya.Y. Bihun, R.I. Petryshyn, I.V. Krasnokutska and other authors.\u0000In this paper, the averaging method is used to study the solvability of a multifrequency\u0000system with an arbitrary nite number of linearly transformed arguments in slow and fast\u0000variables and integral conditions for slow and fast variables on parts of the interval [0, L] of\u0000the system of equations. An unimproved estimate of the error of the averaging method under\u0000the superimposed conditions is obtained, which clearly depends on the small parameter and\u0000the number of linearly transformed arguments in fast variables.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115917190","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":"ON SPLITTING AND STABILITY OF LINEAR STATIONARY SINGULARLY PERTURBED DIFFERENTIAL EQUATIONS","authors":"O. Osypova, I. Cherevko","doi":"10.31861/bmj2019.02.076","DOIUrl":"https://doi.org/10.31861/bmj2019.02.076","url":null,"abstract":"","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117097420","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}
The paper considers the extension of the CMA-ES algorithm using mixtures of distributions for finding optimal hyperparameters of neural networks. Hyperparameter optimization, formulated as the optimization of the black box objective function, which is a necessary condition for automation and high performance of machine learning approaches. CMA-ES is an efficient optimization algorithm without derivatives, one of the alternatives in the combination of hyperparameter optimization methods. The developed algorithm is based on the assumption of a multi-peak density distribution of the parameters of complex systems. Compared to other optimization methods, CMA-ES is computationally inexpensive and supports parallel computations. Research results show that CMA-ES can be competitive, especially in the concurrent assessment mode. However, a much broader and more detailed comparison is still needed, which will include more test tasks and various modifications, such as adding constraints. Based on the Monte Carlo method, it was shown that the new algorithm will improve the search for optimal hyperparameters by an average of 12%.
{"title":"ADVANCED ALGORITHM OF EVOLUTION STRATEGIES OF COVARIATION MATRIX ADAPTATION","authors":"Yu. A. Litvinchuk, I. Malyk","doi":"10.31861/bmj2022.02.09","DOIUrl":"https://doi.org/10.31861/bmj2022.02.09","url":null,"abstract":"The paper considers the extension of the CMA-ES algorithm using mixtures of distributions for finding optimal hyperparameters of neural networks. Hyperparameter optimization, formulated as the optimization of the black box objective function, which is a necessary condition for automation and high performance of machine learning approaches. CMA-ES is an efficient optimization algorithm without derivatives, one of the alternatives in the combination of hyperparameter optimization methods. The developed algorithm is based on the assumption of a multi-peak density distribution of the parameters of complex systems. Compared to other optimization methods, CMA-ES is computationally inexpensive and supports parallel computations. Research results show that CMA-ES can be competitive, especially in the concurrent assessment mode. However, a much broader and more detailed comparison is still needed, which will include more test tasks and various modifications, such as adding constraints. Based on the Monte Carlo method, it was shown that the new algorithm will improve the search for optimal hyperparameters by an average of 12%.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129492930","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}
In this paper we study topological and metric properties of the set of incomplete sums for positive series $sum {a_k}$, where $a_{2n-1}=3/4^n+3/4^{in}$ and $a_{2n}=2/4^n+2/4^{in}$, $n in N$. The series depends on positive integer parameter $i geq 2$ and it is some perturbation of the known Guthrie-Nymann series. We prove that the set of incomplete sums of this series is a Cantorval (which is a specific union of a perfect nowhere dense set of zero Lebesgue measure and an infinite union of intervals), and its Lebesgue measure is given by formula: $lambda(X^+_i)=1+frac{1}{4^i-3}.$ The main idea of ??proving the theorem is based on the well-known Kakey theorem, the closedness of sets of incomplete sums of the series and the density of the set everywhere in a certain segment. The work provides a full justification of the facts for the case $i=2$. To justify the main facts, the ratio between the members and the remainders of the series is used. For $i=2$ we have $r_0=sum {a_k}=2$, $a_{2n}-r_{2n}= frac{1}{3} cdot frac{1}{4^n} + frac{5}{3} cdot frac{1}{16^n}$ $r_{2n-1}-a_{2n-1}= frac{2}{3} cdot frac{ 1}{4^n}-frac{2}{3} cdot frac{1}{16^n}$. The relevance of the study of the object is dictated by the problems of the geometry of numerical series, fractal analysis and fractal geometry of one-dimensional objects and the theory of infinite Bernoulli convolutions, one of the problems of which is the problem of the singularity of the convolution of two singular distributions.
{"title":"THE SET OF INCOMPLETE SUMS OF THE MODIFIED GUTHRIE-NYMANN SERIES","authors":"M. Pratsiovytyi, D. Karvatsky","doi":"10.31861/bmj2022.02.15","DOIUrl":"https://doi.org/10.31861/bmj2022.02.15","url":null,"abstract":"In this paper we study topological and metric properties of the set of incomplete sums for positive series $sum {a_k}$, where $a_{2n-1}=3/4^n+3/4^{in}$ and $a_{2n}=2/4^n+2/4^{in}$, $n in N$. The series depends on positive integer parameter $i geq 2$ and it is some perturbation of the known Guthrie-Nymann series. We prove that the set of incomplete sums of this series is a Cantorval (which is a specific union of a perfect nowhere dense set of zero Lebesgue measure and an infinite union of intervals), and its Lebesgue measure is given by formula: $lambda(X^+_i)=1+frac{1}{4^i-3}.$ The main idea of ??proving the theorem is based on the well-known Kakey theorem, the closedness of sets of incomplete sums of the series and the density of the set everywhere in a certain segment. The work provides a full justification of the facts for the case $i=2$. To justify the main facts, the ratio between the members and the remainders of the series is used. For $i=2$ we have $r_0=sum {a_k}=2$, $a_{2n}-r_{2n}= frac{1}{3} cdot frac{1}{4^n} + frac{5}{3} cdot frac{1}{16^n}$ $r_{2n-1}-a_{2n-1}= frac{2}{3} cdot frac{ 1}{4^n}-frac{2}{3} cdot frac{1}{16^n}$. The relevance of the study of the object is dictated by the problems of the geometry of numerical series, fractal analysis and fractal geometry of one-dimensional objects and the theory of infinite Bernoulli convolutions, one of the problems of which is the problem of the singularity of the convolution of two singular distributions.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126565689","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}
Our main result asserts that, under some assumptions, the uniformly-to-order continuity of an order bounded orthogonally additive operator between vector lattices together with its horizontally-to-order continuity implies its order continuity (we say that a mapping f : E → F between vector lattices E and F is horizontally-to-order continuous provided f sends laterally increasing order convergent nets in E to order convergent nets in F, and f is uniformly-to-order continuous provided f sends uniformly convergent nets to order convergent nets).
{"title":"ON SEPARATE ORDER CONTINUITY OF ORTHOGONALLY ADDITIVE OPERATORS","authors":"I. Krasikova, O. Fotiy, M. Pliev, M. Popov","doi":"10.31861/bmj2021.01.17","DOIUrl":"https://doi.org/10.31861/bmj2021.01.17","url":null,"abstract":"Our main result asserts that, under some assumptions, the uniformly-to-order continuity of an order bounded orthogonally additive operator between vector lattices together with its horizontally-to-order continuity implies its order continuity (we say that a mapping f : E → F between vector lattices E and F is horizontally-to-order continuous provided f sends laterally increasing order convergent nets in E to order convergent nets in F, and f is uniformly-to-order continuous provided f sends uniformly convergent nets to order convergent nets).","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129147841","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}
The singular nonsymmetric rank one perturbation of a self-adjoint operator from classes ${mathcal H}_{-1}$ and ${mathcal H}_{-2}$ was considered for the first time in works by Dudkin M.E. and Vdovenko T.I. cite{k8,k9}. In the mentioned papers, some properties of the point spectrum are described, which occur during such perturbations. This paper proposes generalizations of the results presented in cite{k8,k9} and cite{k2} in the case of nonsymmetric class ${mathcal H}_{-2}$ perturbations of finite rank. That is, the formal expression of the following is considered begin{equation*} tilde A=A+sum limits_{j=1}^{n}alpha_jlanglecdot,omega_jrangledelta_j, end{equation*} where $A$ is an unperturbed self-adjoint operator on a separable Hilbert space ${mathcal H}$, $alpha_jin{mathbb C}$, $omega_j$, $delta_j$, $j=1,2, ..., n
dudkin M.E.和Vdovenko T.I. cite{k8,k9}首次考虑了${mathcal H}_{-1}$和${mathcal H}_{-2}$类自伴随算子的奇异非对称秩1摄动。在上述论文中，描述了在这种扰动中发生的点谱的一些性质。本文推广了在有限秩的非对称类${mathcal H}_{-2}$扰动情况下cite{k8,k9}和cite{k2}的结果。也就是说，下面的形式表达式被认为是begin{equation*}tilde A=A+sum limits_{j=1}^{n}alpha_jlanglecdot,omega_jrangledelta_j,end{equation*}，其中$A$是可分离希尔伯特空间上的一个非摄动自共轭算子，${mathcal H}$, $alpha_jin{mathbb C}$, $omega_j$, $delta_j$, $j=1,2, ..., n
{"title":"SINGULARLY FINITE RANK NONSYMMETRIC PERTURBATIONS ${mathcal H}_{-2}$-CLASS OF A SELF-ADJOINT OPERATOR","authors":"O. Dyuzhenkova, M. Dudkin","doi":"10.31861/bmj2021.01.11","DOIUrl":"https://doi.org/10.31861/bmj2021.01.11","url":null,"abstract":"The singular nonsymmetric rank one perturbation of\u0000a self-adjoint operator from classes ${mathcal H}_{-1}$ and ${mathcal H}_{-2}$ was considered for the first time in works by\u0000Dudkin M.E. and Vdovenko T.I. cite{k8,k9}. In the mentioned papers, some properties of the point spectrum are described,\u0000which occur during such perturbations.\u0000\u0000This paper proposes generalizations of the results presented in cite{k8,k9} and cite{k2} in the case of\u0000nonsymmetric class ${mathcal H}_{-2}$ perturbations of finite rank.\u0000That is, the formal expression of the following is considered\u0000begin{equation*}\u0000tilde A=A+sum limits_{j=1}^{n}alpha_jlanglecdot,omega_jrangledelta_j,\u0000end{equation*}\u0000where $A$ is an unperturbed self-adjoint operator on a separable Hilbert space\u0000${mathcal H}$, $alpha_jin{mathbb C}$, $omega_j$, $delta_j$, $j=1,2, ..., n<infty$ are\u0000vectors from the negative space ${mathcal H}_{-2}$ constructed by the operator $A$,\u0000$langlecdot,cdotrangle$ is the dual scalar product between positive and negative spaces.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128367904","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}
In mathematical modeling of physical and technical processes, the evolution of which depends on prehistory, we arrive at differential equations with a delay. With the help of such equations it was possible to identify and describe new effects and phenomena in physics, biology, technology. An important task for differential-functional equations is to construct and substantiate finding approximate solutions, since there are currently no universal methods for finding their precise solutions. Of particular interest are studies that allow the use of methods of the theory of ordinary differential equations for the analysis of delay differential equations. Schemes for approximating differential-difference equations by special schemes of ordinary differential equations are proposed in the works N. N. Krasovsky, A. Halanay, I. M. Cherevko, L. A. Piddubna, O. V. Matwiy in various functional spaces. The purpose of this paper is to apply approximation schemes of differential-difference equations to approximation of solutions of boundary-value problems for integro-differential equations with a delay. The paper presents sufficient conditions for the existence of a solution of the boundary value problem for integro-differential equations with many delays. The scheme of its approximation by a sequence of boundary value problems for ordinary integro-differential equations is proposed and the conditions of its convergence are investigated. A model example is considered to demonstrate the given approximation scheme.
在物理和技术过程的数学建模中，其演变取决于史前，我们到达微分方程是有延迟的。借助这些方程，可以识别和描述物理学、生物学和技术领域的新效应和新现象。微分泛函方程的一个重要任务是构造并证实其近似解，因为目前还没有找到精确解的通用方法。特别令人感兴趣的是允许使用常微分方程理论的方法来分析时滞微分方程的研究。N. N. Krasovsky, A. Halanay, I. M. Cherevko, L. A. Piddubna, O. V. Matwiy在各种泛函空间中提出了用常微分方程的特殊格式逼近微分-差分方程的格式。本文的目的是将微分-差分方程的近似格式应用于带时滞的积分-微分方程边值问题解的近似。本文给出了多时滞积分-微分方程边值问题解存在的充分条件。给出了用一组常积分-微分方程边值问题逼近它的格式，并研究了它的收敛条件。通过一个模型实例来说明所给出的近似格式。
{"title":"APPROXIMATION OF BOUNDARY VALUE PROBLEMS FOR INTEGRO-DIFFERENTIAL EQUATIONS WITH DELAY","authors":"I. Tuzyk, I. Cherevko","doi":"10.31861/bmj2022.01.11","DOIUrl":"https://doi.org/10.31861/bmj2022.01.11","url":null,"abstract":"In mathematical modeling of physical and technical processes, the evolution of which\u0000depends on prehistory, we arrive at differential equations with a delay. With the help of such equations it was possible to identify and describe new effects and phenomena in physics, biology, technology.\u0000An important task for differential-functional equations is to construct and substantiate\u0000finding approximate solutions, since there are currently no universal methods for finding their precise solutions. Of particular interest are studies that allow the use of methods of the theory of ordinary differential equations for the analysis of delay differential equations.\u0000Schemes for approximating differential-difference equations by special schemes of ordinary differential equations are proposed in the works N. N. Krasovsky, A. Halanay, I. M. Cherevko, L. A. Piddubna, O. V. Matwiy in various functional spaces.\u0000The purpose of this paper is to apply approximation schemes of differential-difference equations to approximation of solutions of boundary-value problems for integro-differential equations with a delay. The paper presents sufficient conditions for the existence of a solution of the boundary value problem for integro-differential equations with many delays. The scheme of its approximation by a sequence of boundary value problems for ordinary integro-differential equations is proposed and the conditions of its convergence are investigated. A model example is considered to demonstrate the given approximation scheme.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128783534","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}
M. Pratsiovytyi, V. Drozdenko, I. Lysenko, Y. Maslova
In the paper, we introduce a new two-symbol system of representation for numbers from segment $[0;0,5]$ with alphabet (set of digits) $A={0;1}$ and two bases 2 and $-2$: [x=dfrac{alpha_1}{2}+dfrac{1}{2}sumlimits^infty_{k=1}dfrac{alpha_{k+1}}{2^{k-(alpha_1+ldots+alpha_k)}(-2)^{alpha_1+ldots+alpha_k}}equiv Delta^{G}_{alpha_1alpha_2ldotsalpha_kldots}, ;;; alpha_kin {0;1}.] We compare this new system with classic binary system. The function $I(x=Delta^G_{alpha_1ldots alpha_nldots})=Delta^G_{1-alpha_1,ldots, 1-alpha_nldots}$, such that digits of its $G$--representation are inverse (opposite) to digits of $G$--representation of argument is considered in detail. This function is well-defined at points having two $G$--representations provided we use only one of them. We prove that inversor is a function of unbounded variation, continuous function at points having a unique $G$--representation, and right- or left-continuous at points with two representations. The values of all jumps of the function are calculated. We prove also that the function does not have monotonicity intervals and its graph has a self-similar structure.
{"title":"INVERSOR OF DIGITS OF TWO-BASE G–REPRESENTATION OF REAL NUMBERS AND ITS STRUCTURAL FRACTALITY","authors":"M. Pratsiovytyi, V. Drozdenko, I. Lysenko, Y. Maslova","doi":"10.31861/bmj2022.01.09","DOIUrl":"https://doi.org/10.31861/bmj2022.01.09","url":null,"abstract":"In the paper, we introduce a new two-symbol system of representation\u0000for numbers from segment $[0;0,5]$ with alphabet (set of digits)\u0000$A={0;1}$ and two bases 2 and $-2$:\u0000[x=dfrac{alpha_1}{2}+dfrac{1}{2}sumlimits^infty_{k=1}dfrac{alpha_{k+1}}{2^{k-(alpha_1+ldots+alpha_k)}(-2)^{alpha_1+ldots+alpha_k}}equiv\u0000Delta^{G}_{alpha_1alpha_2ldotsalpha_kldots},\u0000;;; alpha_kin {0;1}.]\u0000We compare this new system with classic binary system. The function\u0000$I(x=Delta^G_{alpha_1ldots\u0000alpha_nldots})=Delta^G_{1-alpha_1,ldots, 1-alpha_nldots}$,\u0000such that digits of its $G$--representation are inverse (opposite) to\u0000digits of $G$--representation of argument is considered in detail.\u0000This function is well-defined at points having two\u0000$G$--representations provided we use only one of them. We prove that\u0000inversor is a function of unbounded variation, continuous function at\u0000points having a unique $G$--representation, and right- or\u0000left-continuous at points with two representations. The values of all\u0000jumps of the function are calculated. We prove also that the function\u0000does not have monotonicity intervals and its graph has a self-similar\u0000structure.","PeriodicalId":196726,"journal":{"name":"Bukovinian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130669775","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}
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