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
Keywords: two-symbol system of encoding of numbers of unit interval; singular function; inversor of digits of Q∗ 2 -representation of numbers; self-affine set; self-affine dimension; fractal graphs of functions.

Abstract

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.

References

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Published
2021-03-06
How to Cite
1.
Pratsiovytyi MV, Goncharenko YV, Dyvliash NV, Ratushniak SP. Inversor of digits $Q^∗_2$-representative of numbers. Mat. Stud. [Internet]. 2021Mar.6 [cited 2021Oct.16];55(1):37-3. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/192
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Articles