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In Search of a Structure of Fractals by Using Membranes as Hyperedges

  • Adam Obtułowicz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8340)

Abstract

The internal structure of the iterations of Koch curve and Sierpiński gasket—the known fractals [4]—is described in terms of multi-hypergraphical membrane systems related to membrane structures [13] and whose membranes are hyperedges of multi-hypergraphs used to define gluing patterns for the components of the iterations of the considered fractals.

Keywords

Membrane System Venn Diagram Iterate Function System Lambda Calculus Membrane Computing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bruni, R., Gadducci, F., Lluch Lafuente, A.: An algebra of hierarchical graphs. In: Wirsing, M., Hofmann, M., Rauschmayer, A. (eds.) TGC 2010, LNCS, vol. 6084, pp. 205–221. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  2. 2.
    Cherkasova, L.A., Kotov, V.E.: Structured nets. In: Gruska, J., Chytil, M. (eds.) Mathematical Foundations of Computer Science. LNCS, vol. 118, pp. 242–251. Springer, Heidelberg (1981)Google Scholar
  3. 3.
    Edalat, A.: Domains for computation in mathematics, physics and exact real arithmetic. The Bulletin of Symbolic Logic 3, 401–452 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Falconer, K.: Fractal Geometry. Mathematical Foundations and Applications. Wiley, Hoboken (2003)CrossRefzbMATHGoogle Scholar
  5. 5.
    Gallo, G., Longo, G., Pallottino, S., Nguyen, S.: Directed hypergraphs and applications. Discrete Appl. Math. 42, 177–201 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Harel, D.: On Visual Formalisms. Comm. ACM 31, 514–530 (1988)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Hasuo, I., Jacobs, B., Niqui, M.: Coalgebraic representation theory of fractals. Electron. Notes Theor. Comput. Sci. 265, 351–368 (2010)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Leinster, T.: A general theory of self-similarity. Adv. Math. 226, 2935–3017 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Narici, L., Beckenstein, E.: The Hahn–Banach theorem: the life and times. Topology Appl. 77, 193–211 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Obtułowicz, A.: Multigraphical membrane systems revisited. In: Csuhaj-Varjú, E., Gheorghe, M., Rozenberg, G., Salomaa, A., Vaszil, G. (eds.) CMC 2012. LNCS, vol. 7762, pp. 311–322. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  12. 12.
    Orlarey, Y., Fober, D., Letz, S., Bilton, M.: Lambda calculus and music calculi. In: International Computer Music Conference ICMA 1994 (1994)Google Scholar
  13. 13.
    Păun, G.: Membrane Computing. An Introduction. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  14. 14.
    Riddle, L.: Classic iterated function systems, Koch curve, Sierpiński gasket, http://ecademy.agnesscott.edu/~lriddle/ifs/kcurve/kcurve.htm, http://ecademy.agnesscott.edu/~lriddle/ifs/siertri/siertri.htm
  15. 15.
    Rozenkrantz, D.J., Hunt III, H.B.: The complexity of processing hierarchical specifications. SIAM J. Comput. 22, 627–649 (1993)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Stefanescu, G.: The algebra of flownomials, Report, Technical University Munich (1994)Google Scholar
  17. 17.
    Vallée, N., Monsuez, B.: A formal model of system components using fractal hypergraphs. In: Proc. of the Int. Multiconference of Engineers and Computer Scientists, IMECS 2010, Hong Kong, vol. II (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Adam Obtułowicz
    • 1
  1. 1.Institute of MathematicsPolish Academy of SciencesWarsawPoland

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