Elliptic curves and Fibonacci numbers arising from Lindenmayer system with Symbolic Computation
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MI: Global COE Program Education-and-Research Hub for Mathematics-for-IndustryグローバルCOEプログラム「マス･フォア･インダストリ教育研究拠点」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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