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2015 | 5 | 687--692
Tytuł artykułu

Kaprekar's Transformations. Part I - Theoretical Discussion

Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The paper is devoted to discussion of the minimal cycles of the so called Kaprekar's transformations and some of its generalizations. The considered transformations are the self-maps of the sets of natural numbers possessing n digits in their decimal expansions. In the paper there are introduced several new characteristics of such maps, among others, the ones connected with the Sharkovsky's theorem and with the Erd˝os-Szekeres theorem concerning the monotonic subsequences. Because of the size the study is divided into two parts. Part I includes the considerations of strictly theoretical nature resulting from the definition of Kaprekar's transformations. We find here all the minimal orbits of Kaprekar's transformations Tn, for n = 3, ..., 7. Moreover, we define many different generalizations of the Kaprekar's transformations and we discuss their minimal orbits for the selected cases. In Part II (ibidem), which is a continuation of the current paper, the theoretical discussion will be supported by the numerical observations. For example, we notice there that each fixed point, familiar to us, of any Kaprekar's transformation generates an infinite sequence of fixed points of the other Kaprekar's transformations. The observed facts concern also several generalizations of the Kaprekar's transformations defined in Part I.(original abstract)
Słowa kluczowe
PL
EN
Rocznik
Tom
5
Strony
687--692
Opis fizyczny
Twórcy
  • Silesian University of Technology, Gliwice, Poland
  • Silesian University of Technology, Gliwice, Poland
  • Silesian University of Technology, Gliwice, Poland
  • Silesian University of Technology, Gliwice, Poland
Bibliografia
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  • K.E. Eldridge and S. Sagong, "The determination of Kaprekar convergence and loop convergence of all three-digit numbers", Amer. Math. Monthly, vol. 95 No 2, 1988, pp. 105-112, http://dx.doi.org/10.2307/2323062
  • P. Erdos, G. Szekeres, "A combinatorial problem in geometry", Compositio Math., 1935, pp. 463-470.
  • R.K. Guy, "Conway's RATS and other reversals", Amer. Math. Monthly, vol. 96 No 5, 1989, pp. 425-428, http://dx.doi.org/10.2307/2325149
  • H. Hasse and G.D. Prichett, "The determination of all four-digit Kaprekars constants", J. Reine Angew. Math., vol. 299/300, 1978, pp. 113-124.
  • J.H. Jordan, "Self producing sequences of digits", Amer. Math. Monthly, vol. 71 No 1, 1964, pp. 61-64, http://dx.doi.org/10.2307/2311308
  • D.R. Kaprekar, "Another solitarie game", Scripta Mathematica, vol. 15, 1949, pp. 244-245.
  • D.R. Kaprekar, "An interesting property of the number 6174", Scripta Mathematica, vol. 21, 1955, p. 304.
  • R.M. Krause, N. Miller and C.W. Trigg, "Kaprekar's constant", Amer. Math. Monthly, vol. 78 No 2, 1971, pp. 197-198, http://dx.doi.org/10.2307/2317638
  • E. Lundberg, "Almost all orbit types imply period- 3", Topology Appl., vol. 154, 2007, pp. 2741-2744, http://dx.doi.org/10.1016/j.topol.2007.05.009
  • T. Mansour, Combinatorics of Set Partitions, CRC Press, Boca Raton; 2013.
  • J. Miheli´c, L. Furst and U. Cibej, "Exploratory equivalence in graphs: Definition and algorithms", Proc. FedCSIS, ACSIS, vol. 2, 2014, pp. 447ij456, http://dx.doi.org/10.15439/2014F352
  • P. Minc and W.R.R. Transue, "Sarkovskii's theorem for hereditarily decomposable chainable continua", Trans. Amer. Math. Soc., vol. 315, 1989, pp. 173-188, http://dx.doi.org/10.2307/2001378
  • M. Misiurewicz, "Remarks on Sharkovsky's Theorem", Amer. Math. Monthly, vol. 104 No 9, 1997, pp. 846-847, http://dx.doi.org/10.2307/2975290
  • J. Mulvey, "A geometric algorithm to decide the forcing relation on cycles", Real Analysis Exchange, vol. 23 No 2, 1997-1998, pp. 709- 717.
  • D.J. Ryden, "The Sarkovskii order for periodic continua", Topology Appl., vol. 154 No 11, 2007, pp. 2253-2264, http://dx.doi.org/10.1016/j.topol.2007.03.001
  • D.J. Ryden, "The Sarkovskii order for periodic continua II", Topology Appl., vol. 155 No 2, 2007, pp. 92-104, http://dx.doi.org/10.1016/j.topol.2007.08.009
  • A.N. Sharkovskii, "Coexistence of cycles of a continuous map of the line into itself", Ukrain. Math. J., vol. 16, 1964, pp. 61-71 (in Russian).
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Bibliografia
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