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arXiv · 0708.0705

Creation of ballot sequences in a periodic cellular automaton

Taichiro Takagi
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Abstract

Motivated by an attempt to develop a method for solving initial value problems in a class of one dimensional periodic cellular automata (CA) associated with crystal bases and soliton equations, we consider a generalization of a simple proposition in elementary mathematics. The original proposition says that any sequence of letters 1 and 2, having no less 1's than 2's, can be changed into a ballot sequence via cyclic shifts only. We generalize it to treat sequences of cells of common capacity s > 1, each of them containing consecutive 2's (left) and 1's (right), and show that these sequences can be changed into a ballot sequence via two manipulations, cyclic and "quasi-cyclic" shifts. The latter is a new CA rule and we find that various kink-like structures are traveling along the system like particles under the time evolution of this rule.

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Taichiro Takagi. 2009-02-10. Creation of ballot sequences in a periodic cellular automaton. https://doi.org/10.1143/jpsj.78.024003

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