Inversor of digits $Q^∗_2$-representative of numbers

  • M. V. Pratsiovytyi National Pedagogical Dragomanov University, Kyiv, Ukraine
  • Ya. V. Goncharenko National Pedagogical Dragomanov University, Kyiv, Ukraine
  • N. V. Dyvliash National Pedagogical Dragomanov University, Kyiv, Ukraine
  • S. P. Ratushniak National Pedagogical Dragomanov University, Kyiv, Ukraine

Анотація

We consider structural, integral, differential properties of function defined by equality
$$I(\Delta^{Q_2^*}_{\alpha_1\alpha_2...\alpha_n...})=\Delta^{Q_2^*}_{[1-\alpha_1][1-\alpha_2]...[1-\alpha_n]...}, \quad \alpha_n\in A\equiv\{0,1\}$$
for two-symbol polybasic non-self-similar representation of numbers of closed interval $[0;1]$ that is a generalization of classic binary representation and self-similar two-base $Q_2$-representation.
For additional conditions on the sequence of bases, singularity of the function and self-affinity of the graph are proved.
Namely, the derivative is equal to zero almost everywhere in the sense of Lebesgue measure.
The integral of the function is calculated.

Посилання

1. O.M. Baranovskyi, M.V. Pratsiovytyi, G.M. Torbin, Ostrogradsky–Sierpiński–Pierce series and their applications. Kyiv: Nauk. Dumka, 2013, 288 p.
2. M. Pratsiovytyi, A.Chuikov, Continuous distributions whose functions preserve tails of an A2-continued fraction representation of numbers, Random Operators and Stochastic Equations, 27 (2019), No3, 199–206.
3. Peter R. Massopust, Fractal functions, fractal surfaces, and wavelets. Academic Press; 1 edition (January 18, 1995), 383 p.
4. F. Schweiger, Ergodic theory of fibred systems and metric number theory. Oxford: Clarendon Press, 1995, 320 p.
5. Ya.V. Goncharenko, N.V. Dyvliash, Testing of the statistical hypothesis about the parameter of the Salem-type probability distribution, Nauk. Chasop. Nats. Pedagog. Univ. Mykhaila Dragomanova. Ser 1. Fiz.-Mat. Nauky, 16 (2014), No2, 100–108. (in Ukrainian)
6. I.V. Zamriy, M.V. Pratsiovytyi, Singularity of inversor of the digit for the Q3-representation of the fractional part of a real number, its fractal and integral properties, J. Math. Sci. (N.Y.), 215 (2016), No3, 323–340.
7. I.M. Lysenko, Yu.P. Maslova, M.V. Pratsiovytyi, Two-symbol numerical system with two bases having different signs and related functions, Proc. Inst. Math. Nat. Acad. Sc. Ukraine, 16 (2019), No2, 50–62.
8. M.V. Pratsiovytyi, Random variables with independent Q2 -symbols (in Russian), in: Asymptotic Methods in the Study of Stochastic Models, Inst. Math. Natl. Acad. Sci. Ukraine, Kyiv, 1987, 92–102.
9. M.V. Pratsiovytyi, S.V. Skrypnyk, Q2-representation of the fractional part of a real number and the inversor of its digits, Nauk. Chasop. Nats. Pedagog. Univ. M. Dragomanov. Ser 1. Fiz.-Mat. Nauky, 15 (2013), 134–143. (in Ukrainian)
10. M.V. Pratsiovytyi, Fractal approach in investigation of singular probability distributions, Natl. Pedagog. Dragomanov Univ. Publ., Kyiv, 1998. (in Ukrainian)
11. M.V. Pratsiovytyi, Fractal properties of distributions of random variables whose Q-digits form a homogeneous Markov chain, in: Asymptotic Methods in the Study of Stochastic Models, Inst. Math. Natl. Acad. Sci. Ukraine, Kyiv, 1994, 245–254. (in Russian)
12. G.M. Torbin, M.V. Pratsiovytyi, Random variables with independent $Q^∗_2$-digits, in: Random Evolutions: Theoretical and Applied Problems, Inst. Math. Natl. Acad. Sci. Ukraine, Kyiv, 1992, 95–104. (in Russian)
13. A.F. Turbin, M.V. Pratsiovytyi, Fractal Sets, Functions, and Probability Distributions, Nauk. Dumka, Kyiv, 1992. (in Russian)
Опубліковано
2021-03-06
Як цитувати
Pratsiovytyi, M. V., Goncharenko, Y. V., Dyvliash, N. V., & Ratushniak, S. P. (2021). Inversor of digits $Q^∗_2$-representative of numbers. Математичні студії, 55(1), 37-43. https://doi.org/10.30970/ms.55.1.37-43
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