Fraktal and Differential Properties of the Inversor of Digits of Qs-Representation of Real Number

  • Oleg BarabashEmail author
  • Oleg Kopiika
  • Iryna Zamrii
  • Valentyn Sobchuk
  • Andrey Musienko
Part of the Understanding Complex Systems book series (UCS)


This paper aims at introducing and studying a continuous function I(x) that depends on the s − 1 parameters, I(x) is called inversor of digits of Qs-representation of real number. This representation is determined by stochastic vector (q0, q1, …, qs−1) with positive entries and for an arbitrary x ∈ [0;1] there exists a sequence (αn), αn ∈{0, 1, …, s − 1}≡ As, such that
$$\displaystyle x=\beta _{\alpha _1} + \sum ^{\infty }_{k=2}\left [\beta _{\alpha _k}\prod ^{k-1}_{j=1}q_{\alpha _j}\right ]=\varDelta ^{Q_s}_{\alpha _1 \alpha _2 \ldots \alpha _n \ldots }, $$
where β0 = 0, \(\beta _k =\sum \limits _{i=0}^{k-1}q_i\), it is generalization of the classical s-adic representation (because it coincides with the last-mentioned if \(q_{i}=\frac {1}{s}\), i ∈ As).

The differential and fractal properties of the inversor of digits of Qs-representation of real number are described.


Real Numbers Fractal Dimension Stochastic Vector Complex Local Structure Singular Function 
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  1. 1.
    Agadzhanov, A.N.: Singular functions that do not have intervals of monotonicity in problems of finite control of distributed systems. Rep. Acad. Sci. 454(5), 503–506 (2014) (in Russian)MathSciNetGoogle Scholar
  2. 2.
    Albeverio, S., Baranovskyi, O., Kondratiev, Y., Pratsiovytyi, M.: On one class of functions related to Ostrogradsky series and containing singular and nowhere monotonic functions. Sci. J. Natl. Pedagogical Dragomanov Univ. Serya 1. Phys. Math. Sci. 15, 35–55 (2013) (National Pedagogical Dragomanov University, Kiev)Google Scholar
  3. 3.
    Bernstein, D.: Algorithmic Definitions of Singular Functions. Davidson College, Davidson (2013)Google Scholar
  4. 4.
    Freilich, G.: Increasing continuous singular functions. Am. Math. Mon. 80, 918–919 (1973)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gelbaum, R.B., Olmsted, J.M.H.: Counterexamples in Analysis. HoldenDay, San Francisco (1964)zbMATHGoogle Scholar
  6. 6.
    Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer, New York (1965)zbMATHGoogle Scholar
  7. 7.
    Kalpazidou, S., Knopfmacher, A., Knopfmacher, J.: L\(\ddot {u}\)roth-type alternating series representations for real numbers. Acta Arith. 55, 311–322 (1990)Google Scholar
  8. 8.
    Kapustyan, O.V., Kapustyan, O.A., Sukretna, A.V.: Approximate bounded synthesis for one weakly nonlinear boundary-value problem. Nonlinear Oscil. 12(3), 297–304 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kawamura, K.: On the set of points where lebesgues singular function has the derivative zero. Proc. Jpn. Acad. 87(A), 162–166 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kyoungsoo, P., Jeronymo, P.P., Armando Duarte, C., Paulino, G.H.: Integration of singular enrichment functions in the generalized/extended finite element method for three-dimensional problems. Int. J. Numer. Methods Eng. 78, 1220–1257 (2009)CrossRefGoogle Scholar
  11. 11.
    Luroth, J.: \(\ddot {U}\)eber eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe. Math. Ann. 21, 411–423 (1883)Google Scholar
  12. 12.
    Massopust, P.R.: Fractal Functions, Fractal Surfaces, and Wavelets, 1st edn., 383 pp. Academic, Cambridge (1995)Google Scholar
  13. 13.
    Mironovsky, L.A., Petrova, X.Y.: Singular functions of a nonlinear pendulum on finite time intervals. In: Conference: Physics and Control, Proceedings. 2003 International Conference, vol. 4 (2003)Google Scholar
  14. 14.
    Pratsiovytyi, M.V.: The fractal approach in the research of singular distributions, 296 pp. (1998). View of the National Pedagogical Dragomanov University, Kyiv (in Ukrainian)Google Scholar
  15. 15.
    Pratsiovytyi, M.V., Hetman, B.I.: Engel’s series and their application. Sci. J. Natl. Pedagogical Dragomanov Univ. Serya 1. Phys. Math. Sci. 7, 105–116 (2006) (National Pedagogical Dragomanov University, Kiev (in Ukrainian))Google Scholar
  16. 16.
    Pratsiovytyi, M.V., Skrypnyk, S.V.: Q 2-representation for fractional part of real number and the inversor of its digits. Sci. J. Natl. Pedagogical Dragomanov Univ. Serya 1. Phys. Math. Sci. 15, 134–143 (2013) (National Pedagogical Dragomanov University, Kiev (in Ukrainian))Google Scholar
  17. 17.
    Pratsiovytyi, M.V., Zamrii, I.V.: Continuous functions preserving digit 1 in the Q 3-representation of number. Bukovinsky Math. J. 3(3–4), 142–159 (2015). Chernivtsi: Chernivtsi National University (in Ukrainian)Google Scholar
  18. 18.
    Pratsovyta, I.M., Zadniprianyi, M.V.: Schedules of numbers in the Sylvester series and their application. Sci. J. Natl. Pedagogical Dragomanov Univ. Serya 1. Phys. Math. Sci. 10, 73–87 (2009) (National Pedagogical Dragomanov University, Kiev (in Ukrainian))Google Scholar
  19. 19.
    Ricardo, E., Fulling, S.A.: How singular functions define distributions. J. Phys. A Math. Gen. 35(13), 3079–3089 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Riesco, A., Rodriguez-Hortala, J.: Singular and plural functions for functional logic programming: detailed proofs. Technical report SIC-9/11, Dpto. Sistemas Informaticos y Computacion, Universidad Complutense de Madrid (2011)Google Scholar
  21. 21.
    Riesz, F., Nagy, B.Sz.: Functional Analysis. Ungar, New York (1965)Google Scholar
  22. 22.
    Salem, R.: On some singular monotonic function which are strictly increasing. Trans. Am. Math. Soc. 53(3), 427–439 (1943)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Takacs, L.: An increasing continuous singular function. Am. Math. Mon. 85, 35–37 (1978)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Turbin, A.F., Pratsiovytyi, M.V.: Fractal Sets, Functions, Distributions, 208 pp. Naukova dumka, Kiev (1992) (in Russian)Google Scholar
  25. 25.
    Wen, L.: An approach to construct the singular monotone functions by using markov chains. Taiwan. J. Math. 2(3), 361–368 (1998)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Zamrii, I.V.: Lebesgue structure and properties of the inversor of digits of Q s-representation for fractional part of real number. Sci. Educ. New Dimens. Nat. Tech. Sci. 16(148), 47–49 (2017). BudapestGoogle Scholar
  27. 27.
    Zamrii, I.V., Pratsiovytyi, M.V.: The singularity of the inversor of digits of Q 3-representation of the fractional part of the real number, its fractal and integral properties. Nonlinear oscil. 18(1), 55–70 (2015). ISSN 1562-3076, Institute of Mathematics, National Academy of Sciences of UkraineGoogle Scholar
  28. 28.
    Zhiqiang, C., Seokchan, K., Sangdong, K., Sooryun, K.: A finite element method using singular functions for Poisson equations: mixed boundary conditions. Comput. Methods Appl. Mech. Eng. 195, 2635–2648 (2006)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Zhykharieva, Yu.I., Pratsovytyi, M.V.: Representation of numbers by the applicable Liurot’s series: the basis of the metric theories. Sci. J. Natl. Pedagogical Dragomanov Univ. Serya 1. Phys. Math. Sci. 9, 200–211 (2008) (National Pedagogical Dragomanov University, Kiev (in Ukrainian))Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Oleg Barabash
    • 1
    Email author
  • Oleg Kopiika
    • 2
  • Iryna Zamrii
    • 1
  • Valentyn Sobchuk
    • 3
  • Andrey Musienko
    • 4
  1. 1.State University of TelecommunicationsKyivUkraine
  2. 2.Institute of Telecommunications and Global Information SpaceKyivUkraine
  3. 3.East-European National University of Lesya UkrainkaLutskUkraine
  4. 4.Taras Shevchenko National University of KyivKyivUkraine

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