Generalizations of Certain Representations of Real Numbers

[1] BUSH, K. A.: Continuous functions without derivatives,Amer. Math. Monthly 59 (1952), 222–225.10.1080/00029890.1952.11988110Search in Google Scholar

[2] CANTOR, G.: Ueber die einfachen Zahlensysteme,Z. Math. Phys. 14 (1869), 121–128.Search in Google Scholar

[3] FALCONER, K.: Techniques in Fractal Geometry. John Wiley & Sons, Ltd., Chichester, 1997.Search in Google Scholar

[4] FALCONER, K.: Fractal Geometry. Mathematical Foundations and Applications. 2nd edition. John Wiley & Sons, Inc., Hoboken, NJ, 2003.10.1002/0470013850Search in Google Scholar

[5] GALAMBOS,J.: Representations of Real Numbers by Infinite Series, Lecture Notes in Mathematics Vol. 502, Springer-Verlag, Berlin, 1976.10.1007/BFb0081642Search in Google Scholar

[6] ITO, S.–SADAHIRO, T.: Beta-expansions with negative bases, Integers 9 (2009), 239–259.10.1515/INTEG.2009.023Search in Google Scholar

[7] KALPAZIDOU, S.—KNOPFMACHER, A.—KNOPFMACHER, J.: Metric properties of alternating Lüroth series, Port. Math. 48 (1991), no. 3, 319–325.Search in Google Scholar

[8] RÉNYI, A. : Representations for real numbers and their ergodic properties,Acta. Math. Acad. Sci. Hungar. 8 (1957), 477–493.10.1007/BF02020331Search in Google Scholar

[9] SERBENYUK, S. O.: Quasi-nega-Q˜\tilde QQ-representation as a generalization of a representation of real numbers by certain sign-variable series. In: International Conference of Young Mathematicians: Abstracts, Kyiv, Institute of Mathematics of the National Academy of Sciences of Ukraine, 2015, p. 85. https://www.researchgate.net/publication/303255656 (In Ukrainian)Search in Google Scholar

[10] SERBENYUK, S.: Nega-Q˜\tilde Q-representation as a generalization of certain alternating representations of real numbers, Bull. Taras Shevchenko Natl. Univ. Kyiv Math. Mech. 35 (2016), no. 1, 32–39; https://www.researchgate.net/publication/308273000 (In Ukrainian)Search in Google Scholar

[11] SERBENYUK, S.: Representation of real numbers by the alternating Cantor series, Integers 17 (2017), Paper No. A15, 27 pp.Search in Google Scholar

[12] SERBENYUK, S. O.: Continuous Functions with Complicated Local Structure Defined in Terms of Alternating Cantor Series Representation of Numbers,Zh. Mat. Fiz. Anal. Geom. 13 (2017), no. 1, 57–81; https://doi.org/10.15407/mag13.01.05710.15407/mag13.01.057Search in Google Scholar

[13] SERBENYUK, S. O.: Non-differentiable functions defined in terms of classical representations of real numbers, Zh. Mat. Fiz. Anal. Geom. 14(2018), no. 2, 197–213; https://doi.org/10.15407/mag14.02.19710.15407/mag14.02.197Search in Google Scholar

[14] SERBENYUK, S.: On some generalizations of real numbers representations, arXiv:1602.07929v1 (In Ukrainian)Search in Google Scholar

[15] SERBENYUK, S.: Generalizations of certain representations of real numbers, arXiv:1801.10540Search in Google Scholar

[16] SERBENYUK, S.: On one fractal property of the Minkowski function,Rev.R. Acad. Cienc. Exactas, Fís. Nat. Ser. A Mat. 112 (2018), no. 2, 555–559; DOI:10.1007/s13398-017–0396–510.1007/s13398-017-0396-5Search in Google Scholar

[17] SERBENYUK, S.: On one application of infinite systems of functional equations in function theory, Tatra Mountains Mathematical Publications 74 (2019), 117–144; https://doi.org/10.2478/tmmp-2019–002410.2478/tmmp-2019-0024Search in Google Scholar

[18] SERBENYUK, S.: Modeling rational numbers by Cantor series, arXiv:1904.07264Search in Google Scholar

[19] STEEN, L. A. —SEEBACH, J. A. JR.: Counterexamples in Topology. Springer-Verlag, Berlin, 1978.10.1007/978-1-4612-6290-9Search in Google Scholar

[20] WIKIPEDIA CONTRIBUTORS: Pathological (mathematics), The Free Encyclopedia; https://en.wikipedia.org/wiki/Pathological_(mathematics) (accessed October 5, 2019).Search in Google Scholar

[21] WISE, G. L. —HALL, E. B.: Counterexamples in Probability and Real Analysis. The Clarendon Press, Oxford University Press, New York, 1993.Search in Google Scholar