|
Algebra and Discrete Mathematics, 2016, Volume 22, Issue 1, Pages 102–115
(Mi adm577)
|
|
|
|
This article is cited in 2 scientific papers (total in 2 papers)
RESEARCH ARTICLE
Transformations of $(0,1]$ preserving tails of $\Delta^{\mu}$-representation of numbers
Tetiana M. Isaieva, Mykola V. Pratsiovytyi Institute of Physics and Mathematics, National Pedagogical Mykhailo Drahomanov University, 9 Pyrohova St., Kyiv, 01601, Ukraine
Abstract:
Let $\mu\in (0,1)$ be a given parameter, $\nu\equiv 1-\mu$. We consider $\Delta^{\mu}$-representation of numbers $x=\Delta^{\mu}_{a_1a_2\ldots a_n\ldots}$ belonging to $(0,1]$ based on their expansion in alternating series or finite sum in the form:
$$
x=\sum_n(B_{n}-{B'_n})\equiv \Delta^{\mu}_{a_1a_2\ldots a_n\ldots},
$$
where $B_n=\nu^{a_1+a_3+\ldots+a_{2n-1}-1}{\mu}^{a_2+a_4+\ldots+a_{2n-2}}$, ${B^{\prime}_n}=\nu^{a_1+a_3+\ldots+a_{2n-1}-1}{\mu}^{a_2+a_4+\ldots+a_{2n}}$, $a_i\!\in\! \mathbb{N}$. This representation has an infinite alphabet $\{1,2,\ldots\}$, zero redundancy and $N$-self-similar geometry.
In the paper, classes of continuous strictly increasing functions preserving “tails” of $\Delta^{\mu}$-representation of numbers are constructed. Using these functions we construct also continuous transformations of $(0,1]$. We prove that the set of all such transformations is infinite and forms non-commutative group together with an composition operation.
Keywords:
$\Delta^{\mu}$-representation, cylinder, tail set, function preserving “tails” of $\Delta^{\mu}$-representation of numbers, continuous transformation of $(0,1]$ preserving “tails” of $\Delta^{\mu}$-representation of numbers, group of transformations.
Received: 10.04.2016 Revised: 10.08.2016
Citation:
Tetiana M. Isaieva, Mykola V. Pratsiovytyi, “Transformations of $(0,1]$ preserving tails of $\Delta^{\mu}$-representation of numbers”, Algebra Discrete Math., 22:1 (2016), 102–115
Linking options:
https://www.mathnet.ru/eng/adm577 https://www.mathnet.ru/eng/adm/v22/i1/p102
|
|