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Algebra and Discrete Mathematics, 2016, Volume 22, Issue 1, Pages 102–115
(Mi adm577)
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This article is cited in 2 scientific papers (total in 2 papers)
RESEARCH ARTICLE
Transformations of (0,1](0,1] preserving tails of ΔμΔμ-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 μ∈(0,1)μ∈(0,1) be a given parameter, ν≡1−μν≡1−μ. We consider ΔμΔμ-representation of numbers x=Δμa1a2…an…x=Δμa1a2…an… belonging to (0,1](0,1] based on their expansion in alternating series or finite sum in the form:
x=∑n(Bn−B′n)≡Δμa1a2…an…,
where Bn=νa1+a3+…+a2n−1−1μa2+a4+…+a2n−2, B′n=νa1+a3+…+a2n−1−1μa2+a4+…+a2n, ai∈N. This representation has an infinite alphabet {1,2,…}, zero redundancy and N-self-similar geometry.
In the paper, classes of continuous strictly increasing functions preserving “tails” of Δμ-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:
Δμ-representation, cylinder, tail set, function preserving “tails” of Δμ-representation of numbers, continuous transformation of (0,1] preserving “tails” of Δμ-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 Δμ-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
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Abstract page: | 192 | Full-text PDF : | 75 | References: | 56 |
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