$Transformations of $(0,1]$ preserving tails of $\Delta^{\mu}$-representation of numbers$

Transformations of $(0,1]$ preserving tails of $\Delta^{\mu}$-representation of numbers

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 \Delt...

Full description

Saved in:

Bibliographic Details
Date:	2016
Main Authors:	Isaieva, Tetiana M., Pratsiovytyi, Mykola V.
Format:	Article
Language:	English
Published:	Lugansk National Taras Shevchenko University 2016
Subjects:	$\Delta^{\mu}$-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 11H71 26A46 93B17
Online Access:	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/202
Tags:	Add Tag No Tags, Be the first to tag this record!
Journal Title:	Algebra and Discrete Mathematics

Institution

Algebra and Discrete Mathematics

id	oai:ojs.admjournal.luguniv.edu.ua:article-202
record_format	ojs
spelling	oai:ojs.admjournal.luguniv.edu.ua:article-2022016-11-15T13:03:03Z Transformations of $(0,1]$ preserving tails of $\Delta^{\mu}$-representation of numbers Isaieva, Tetiana M. Pratsiovytyi, Mykola V. $\Delta^{\mu}$-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 11H71, 26A46, 93B17 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. Lugansk National Taras Shevchenko University 2016-11-15 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/202 Algebra and Discrete Mathematics; Vol 22, No 1 (2016) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/202/pdf Copyright (c) 2016 Algebra and Discrete Mathematics
institution	Algebra and Discrete Mathematics
baseUrl_str
datestamp_date	2016-11-15T13:03:03Z
collection	OJS
language	English
topic	$\Delta^{\mu}$-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 11H71 26A46 93B17
spellingShingle	$\Delta^{\mu}$-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 11H71 26A46 93B17 Isaieva, Tetiana M. Pratsiovytyi, Mykola V. Transformations of $(0,1]$ preserving tails of $\Delta^{\mu}$-representation of numbers
topic_facet	$\Delta^{\mu}$-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 11H71 26A46 93B17
format	Article
author	Isaieva, Tetiana M. Pratsiovytyi, Mykola V.
author_facet	Isaieva, Tetiana M. Pratsiovytyi, Mykola V.
author_sort	Isaieva, Tetiana M.
title	Transformations of $(0,1]$ preserving tails of $\Delta^{\mu}$-representation of numbers
title_short	Transformations of $(0,1]$ preserving tails of $\Delta^{\mu}$-representation of numbers
title_full	Transformations of $(0,1]$ preserving tails of $\Delta^{\mu}$-representation of numbers
title_fullStr	Transformations of $(0,1]$ preserving tails of $\Delta^{\mu}$-representation of numbers
title_full_unstemmed	Transformations of $(0,1]$ preserving tails of $\Delta^{\mu}$-representation of numbers
title_sort	transformations of $(0,1]$ preserving tails of $\delta^{\mu}$-representation of numbers
description	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.
publisher	Lugansk National Taras Shevchenko University
publishDate	2016
url	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/202
work_keys_str_mv	AT isaievatetianam transformationsof01preservingtailsofdeltamurepresentationofnumbers AT pratsiovytyimykolav transformationsof01preservingtailsofdeltamurepresentationofnumbers
first_indexed	2025-07-17T10:31:06Z
last_indexed	2025-07-17T10:31:06Z
_version_	1837890133861335040

Transformations of \((0,1]\) preserving tails of \(\Delta^{\mu}\)-representation of numbers

Institution

Similar Items