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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
Chapter
Part of the Understanding Complex Systems book series (UCS)

Abstract

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.

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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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