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Elliptic curves and Fibonacci numbers arising from Lindenmayer system with symbolic computation

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Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

Starting from an egg, the multicell becomes a set of cells comprising a variety of types to serve functions. This phenomenon brings us a bio-motivated Lindenmayer system. To investigate conditions for a variety of cell types, we have constructed a stochastic model over Lindenmayer systems. This model considers interactive behaviors among cells, yielding complicated polynomials. Using symbolic computation, we have derived explicit relations between cell-type diversity and cell-type ratio constraint. These relations exhibit elliptic curve- and Fibonacci number-related patterns. This is the first example of elliptic curves to appear in the Lindenmayer context. A survey of the rational points and the quadratic irrational numbers on the derived curves has revealed Fibonacci-related periodic and quasiperiodic patterns. Further we have found that in some region, there are only two elliptic curve-related periodic patterns.

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Authors and Affiliations

  1. Faculty of Mathematics, Kyushu University, Itou, Motouoka 744, Nishi-ku, Fukuoka, 819-0395, Japan

    Hiroshi Yoshida & Masanobu Kaneko

  2. Graduate School of Comprehensive Human Sciences, University of Tsukuba, Tennoudai 1-1-1, Tsukuba-shi, Ibaraki, 305-8575, Japan

    Yoshihiro Miwa

Authors
  1. Hiroshi Yoshida
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  2. Yoshihiro Miwa
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  3. Masanobu Kaneko
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Correspondence to Hiroshi Yoshida.

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Yoshida, H., Miwa, Y. & Kaneko, M. Elliptic curves and Fibonacci numbers arising from Lindenmayer system with symbolic computation. AAECC 22, 147–164 (2011). https://doi.org/10.1007/s00200-011-0143-7

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  • DOI: https://doi.org/10.1007/s00200-011-0143-7

Keywords

Mathematics Subject Classification (2000)