Continuous transformations preserving tails of negabinary representations of numbers from unit interval
DOI:
https://doi.org/10.17721/1812-5409.2025/2.5Keywords:
Negabinary representation of numbers, cylinder, group of interval transformations, continuous transformations of the interval, tail set, left and right shift operatorsAbstract
For the negabinary representation of numbers from the [0; 1] interval, we consider continuous transformations of this interval (i.e., bijective mappings of the set onto itself) that preserve the tails of number representations. These transformations are constructed using left and right shift operators. It is proven that the set of such transformations, under the operation of composition, forms a non-commutative group, which contains a continuous subgroup of increasing transformations.
Pages of the article in the issue: 36 - 40
Language of the article: English
References
Albeverio, S., Pratsiovytyi, M. V., & Torbin, G. M. (2004). Fractal probability distributions and transformations preserving the Hausdorff–Besicovitch dimension. Ergodic Theory and Dynamical Systems, 24(1), 1–16. https://doi.org/10.1017/S0143385703000397
Cherri, A., & Kamal, H. (2004). Parallel negabinary signed-digit arithmetic operations: One-step negabinary, one-step trinary, and one-step quaternary addition algorithms. Proceedings of SPIE, 5484, 35–45. https://doi.org/10.1117/12.568845
Galambos, J. (2006). Representations of real numbers by infinite series. Springer.
Grover, D., Madan, V. K., Nanda, N. K., & Singh, H. (1983). Some hardware realizations of negabinary arithmetic. International Journal of Electronics, 55(2), 235–241. https://doi.org/10.1080/00207218308961584
Isaieva, T. M., & Pratsiovytyi, M. V. (2016). Transformations of (0,1) preserving tails of ∆µ-representation of numbers. Algebra and Discrete Mathematics, 22(1), 102–115.
Kasatkin, V. N. (1982). New insights into number systems. Vyshcha Shkola [in Russian].
Osaulenko, R. (2016). The group of continuous transformations of the interval [0,1] preserving the digit frequencies of Qs-expansions of numbers. Collection of Works of the Institute of Mathematics of the NAS of Ukraine, 13(3), 191–204 [in Ukrainian].
Pratsiovytyi, M. V. (2017). Nega-cantor representations of real numbers as trivial recodings of Cantor representations (nega-s-adic as recodings of s-adic representations). Proceedings of the Institute of Mathematics of the NAS of Ukraine, 14(4), 167–177 [in Ukrainian].
Pratsiovytyi, M. V. (2022). Two-symbol encoding systems of real numbers and their application. Naukova dumka [in Ukrainian].
Pratsiovytyi, M. V., Chuikov, A. S., & Skrypnyk, S. V. (2019). The chain D2-representation of real numbers and some related functions. Proceedings of the Institute of Mathematics of the NAS of Ukraine, 16(3), 101–114 [in Ukrainian].
Pratsiovytyi, M. V., Goncharenko, Y. V., & Lysenko, I. M. (2015). Negabinary representation of real numbers and its applications. Collection of scientific papers of the National Pedagogical Dragomanov University. Series 1: Physical and Mathematical Sciences, 17, 83–106 [in Ukrainian].
Pratsiovytyi, M. V., Lysenko, I. M., & Maslova, Y. P. (2020). Group of continuous transformations of real interval preserving tails of G2-representation of numbers. Algebra and Discrete Mathematics, 29(1), 99–108. https://doi.org/10.12958/adm1498
Pratsiovytyi, M. V., Lysenko, I. M., & Ratushniak, S. P. (2024). Uncountable group of continuous transformations of unit segment preserving tails of Q2-representation of numbers. Proceedings of the International Geometry Center, 17(2), 133–142. https://doi.org/10.15673/pigc.v17i2.2755
Rao, G. S., Rao, M. N., & Krishnamurthy, M. N. (1974). A variable-shift nega-binary multiplier. International Journal of Electronics, 36(6), 749–751. https://doi.org/10.1080/00207217408900472
Schweiger, F. (1995). Ergodic theory of fibred systems and metric number theory. Oxford University Press.
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