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Publications and source records attributed to Symon Serbenyuk.
The present article is devoted to the description of further investigations of the author of this article. These investigations (in terms of various representations of real numbers) include the generalized Salem functions and generalizations of the Gauss-Kuzmin problem.
The present article is devoted to certain examples of functions whose argument represented in terms of Cantor series.
In the present article, modeling certain rational numbers, that are represented in terms of Cantor series, are described. The statements on relations between digits in the representations of rational numbers by Cantor series (for the case of an arbitrary sequence $(q_k)$) are proved.
In the present article, real number representations, that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are proved. The theorem on the representation of real numbers from a certain interval is formulated.
The article is devoted to investigation of applications of infinite systems of functional equations for modeling of functions with complicated local structure, that are defined in terms of the nega-$\tilde Q$-representation. The infinite system of functional equations $$ f\left(\hat φ^k(x)\right)=\tilde β_{i_{k+1},k+1}+\tilde p_{i_{k+1},k+1}f\left(\hat φ^{k+1}(x)\right), $$ where $k=0,1,...$, $\hat φ$ is a shift operator of the $\tilde Q$-expansion, $x=Δ^{-\tilde Q} _{i_1(x)i_2(x)...i_n(x)...}$, are investigated. It is proved, that the system has the unique solution in the class of determined and bounded on $[0;1]$ functions and continuity of the solution. His analytical presentation is founded. Conditions of its monotonicity and nonmonotonicity, differential, integral properties are studied. Conditions under which the solution of the functional equations system is a distribution function of random variable $η=Δ^{\tilde Q} _{ξ_1ξ_2...ξ_n...}$ with independent $\tilde Q$-symbols are discovered. The results of the article was represented in Fourth all-Ukrainian Scientific Conference of Young Scientists on Mathematics and Physics, Kyiv, April 23-25, 2015 (https://www.researchgate.net/publication/301765100). The investigation was represented at the seminar on fractal analysis of Institute of Mathematics of the National Academy of Sciences of Ukraine on, October 30, 2014 (http://www.imath.kiev.ua/events/index.php?seminarId=21&archiv=1).
In the present article, topological, metric, and fractal properties of certain sets are investigated. These sets are images of sets whose elements have restrictions on using digits or combinations of digits in own s-adic representations, under the map $f$, that is a certain distribution function.
The present article is devoted to functions from a certain subclass of non-differentiable functions. The arguments and values of considered functions represented by the s-adic representation or the nega-s-adic representation of real numbers. The technique of modeling such functions is the simplest as compared with well-known techniques of modeling non-differentiable functions. In other words, values of these functions are obtained from the s-adic or nega-s-adic representation of the argument by a certain change of digits or combinations of digits.
In the present article a new class $Υ$ of all sets represented in the following form is introduced: $$ \mathbb S_{(s,u)}\equiv\left\{x: x= Δ^{s}_{{\underbrace{u...u}_{α_1-1}} α_1{\underbrace{u...u}_{α_2 -1}}α_2 ...{\underbrace{u...u}_{ α_n -1}}α_n...}, α_n \in A_0, α_n \ne u, α_n \ne 0 \right\}, $$ where $2<s\in \mathbb N$ and $u\in A$ are fixed parameters, $A=\{0,1,\dots,s-1\}$, and $A_0=A \setminus \{0\}$. Topological, metric, and fractal properties of these sets are studied. The problem of belonging to these sets of normal numbers is investigated. The theorem on the calculation of the Hausdorff-Besicovitch dimension of an arbitrary set whose elements have restrictions on using of digits or combinations of digits in own s-adic representations is formulated and proved. In 2012, the results of this article were presented by the author in the abstracts (https://www.researchgate.net/publication/303054326, https://www.researchgate.net/publication/303053939, https://www.researchgate.net/publication/303054835). The investigation was represented at the seminar on fractal analysis of Institute of Mathematics of the National Academy of Sciences of Ukraine (29, September 2011 and 16, February 2012, available at http://www.imath.kiev.ua/events/index.php?seminarId=21&archiv=1) and such investigations for the case of the nega-s-adic representation were published by the author in Trans. Dragomanov Nat. Pedagogical Univ. Ser. 1, Physics and Mathematics 15 (2013), p. 168-187 (https://www.researchgate.net/publication/292970280, in Ukrainian).
This paper is devoted to the investigation of the following function $$ f: x=Δ^{3}_{α_{1}α_{2}...α_{n}...}{\rightarrow} Δ^{3}_{φ(α_{1})φ(α_{2})...φ(α_{n})...}=f(x)=y, $$ where $φ(i)=\frac{-3i^{2}+7i}{2}$, $ i \in N^{0}_{2}=\{0,1,2\}$, and $Δ^{3}_{α_{1}α_{2}...α_{n}...}$ is the ternary representation of $x \in [0;1]$. That is values of this function are obtained from the ternary representation of the argument by the following change of digits: 0 by 0, 1 by 2, and 2 by 1. This function preserves the ternary digit $0$. Main mapping properties and differential, integral, fractal properties of the function are studied. Equivalent representations by additionally defined auxiliary functions of this function are proved. This paper is the paper translated from Ukrainian (the Ukrainian variant available at https://www.researchgate.net/publication/292970012). In 2012, the Ukrainian variant of this paper was represented by the author in the International Scientific Conference "Asymptotic Methods in the Theory of Differential Equations" dedicated to 80th anniversary of M. I. Shkil (the conference paper available at https://www.researchgate.net/publication/301765319). In 2013, the investigations of the present article were generalized by the author in the paper "One one class of functions with complicated local structure" (https://arxiv.org/pdf/1601.06126.pdf) and in the several conference papers (available at: https://www.researchgate.net/publication/301765326, https://www.researchgate.net/publication/303052308).
The article is devoted to the investigation of representation of rational numbers by Cantor series. Necessary and sufficient conditions for a rational number to be representable by a positive Cantor series are formulated for the case of an arbitrary sequence $(q_k)$ and some its corollaries are considered. Results of this article were presented by the author of this article on the International Conference on Algebra dedicated to 100th anniversary of S. M. Chernikov (www.researchgate.net/publication/311415815, www.researchgate.net/publication/301849984). This investigation was also presented in some reports (links to the reports: www.researchgate.net/publication/303736670, www.researchgate.net/publication/303720573, etc.).
It is shown in the present article that the Minkowski function does not preserve the Hausdorff-Besicovitch dimension. Results of this article were presented by the author of this article on the Second Interuniversity Scientific Conference on Mathematics and Physics for Young Scientists in April, 2011 (Links to the conference paper (in Ukrainian): www.researchgate.net/publication/301637057, www.researchgate.net/publication/311665828. A link to the presentation (in Ukrainian): www.researchgate.net/publication/303752636).
The article is devoted to introduce and investigation of new numeral systems. These representations are generalizations of classical representations of real numbers. The main properties of investigating representations are described. The results are represented in International Conference of Yung Mathematicians, June 3-6, 2015, Kyiv.
The article is devoted to one infinite parametric class of continuous functions with complicated local structure. In the article differential, integral, self-affine and other properties of functions, that their argument is represented by the alternating Cantor series are investigated. The functional equations systems, that investigating functions are unique solution of these systems in the class of determined and bounded on $[0;1]$ functions are indicated. The investigation was represented in seminar on fractal analysis of Institute of Mathematics of the National Academy of Sciences of Ukraine on, October 16, 2014 (http://www.imath.kiev.ua/events/index.php?seminarId=21&archiv=1). The following investigations in the list of references in the article one can to find by the corresponding links: [6] (http://fmi.npu.edu.ua/images/files/publications/naukchasopys1/NZ2013_15.pdf ) [7] (http://fmi.npu.edu.ua/images/files/publications/naukchasopys1/NZ2013_14.pdf) [8] (http://fmi.npu.edu.ua/images/files/publications/naukchasopys1/NZ2012_13_2.pdf) [9](http://www.ekmair.ukma.edu.ua/bitstream/handle/123456789/7409/Serbeniuk_Funktsii_oznacheni_systemamy.pdf)
The article is devoted to modeling of the nega-$\tilde Q$-representation of real numbers. The representation is a generalization of representation by alternating Cantor series and positive $\tilde Q$-representation is a generalization of representation of real numbers by the positive Cantor series. Analytic and geometric approach are used for modeling of nega-$\tilde Q$-representation. Advantages and disadvantages of these approaches are investigated, the representation is modeled. The investigation was represented in seminar on fractal analysis of Institute of Mathematics of the National Academy of Sciences of Ukraine on, October 30, 2014 (http://www.imath.kiev.ua/events/index.php?seminarId=21&archiv=1) and International Conference "Probability, Reliability and Stochastic Optimization", Kyiv, April 7-10, 2015.
The article is devoted to the alternating Cantor series. It is proved that any real number belonging to $[a_0-1;a_0]$, where $a_0=\sum^{\infty} _{k=1} {\frac{d_{2k}-1}{d_1d_2...d_{2k}}} $, has no more than two representations by the series and only numbers from countable subset of real numbers have two representations. Geometry of the representation, properties of cylinders sets and semicylinders, simplest metric problems are investigated. Some applications of the series in fractal theory and relation between positive and alternating Cantor series are described. A shift operator and its some applications, set of incomplete sums are studied. Necessary and sufficient conditions for representations of rational numbers by the alternating Cantor series are formulated.
We introduce a class $Λ_{s}$ of functions with complicated local structure. Any function from the class belongs to one of three specifically defined types $f^s _k$, $f_+$, and $f^{-1} _+$ or is a specifically defined composition of two or three functions of these types. Differential, integral, fractal and other properties of such functions are investigated. In particular, all functions $f$ from $Λ_{s}$ such that $f(x)\ne x$, $f(x)\ne -\frac{s-1}{s+1}-x$ and $f(x)\ne 1-x$ are nonmonotonic and nowhere differentiable. The Hausdorff--Besicovitch dimension of a plot of each $f \in Λ_s$ is equal to $1,$ and the Lebesgue integral of $f$ is equal to $\frac 1 2$. The proof of these statements for the compositions and the corresponding proofs for $f^s _k$, $f_+$, and $f^{-1} _+$ are similar.