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...
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| Date: | 2016 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Lugansk National Taras Shevchenko University
2016
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/202 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543343922642944 |
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| author | Isaieva, Tetiana M. Pratsiovytyi, Mykola V. |
| author_facet | Isaieva, Tetiana M. Pratsiovytyi, Mykola V. |
| author_sort | Isaieva, Tetiana M. |
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| collection | OJS |
| datestamp_date | 2016-11-15T13:03:03Z |
| 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. |
| first_indexed | 2026-02-08T07:57:46Z |
| format | Article |
| id | admjournalluguniveduua-article-202 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T07:57:46Z |
| publishDate | 2016 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-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 |
| 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 |
| title | 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_short | 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 |
| 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 |
| 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 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/202 |
| work_keys_str_mv | AT isaievatetianam transformationsof01preservingtailsofdeltamurepresentationofnumbers AT pratsiovytyimykolav transformationsof01preservingtailsofdeltamurepresentationofnumbers |