Fundamental Cycle of a Periodic Box-Ball System: A Number Theoretical Aspect

Tokihiro, Tetsuji; Mada, Jun

doi:10.1017/s0017089505002405

Cited by 6 publications

(4 citation statements)

References 12 publications

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“…The key of the method is that one can obtain not only the equations but also their solutions simultaneously. It also allows us to understand the underlying mathematical structures of the ultradiscrete systems [2,3,5,8,11,13,14,15,16,25,33].…”

Section: Introductionmentioning

confidence: 99%

Ultradiscretization of solvable one-dimensional chaotic maps

Kajiwara

Nobe

Tsuda

2008

J. Phys. A: Math. Theor.

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We consider the ultradiscretization of a solvable one-dimensional chaotic map which arises from the duplication formula of the elliptic functions. It is shown that ultradiscrete limit of the map and its solution yield the tent map and its solution simultaneously. A geometric interpretation of the dynamics of the tent map is given in terms of the tropical Jacobian of a certain tropical curve. Generalization to the maps corresponding to the m-th multiplication formula of the elliptic functions is also discussed.

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Section: Introductionmentioning

confidence: 99%

Ultradiscretization of solvable one-dimensional chaotic maps

Kajiwara

Nobe

Tsuda

2008

J. Phys. A: Math. Theor.

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“…Its explicit formula as well as statistical distribution was obtained and its relation to the celebrated Riemann hypothesis was clarified [7,8,9]. To prove the formula for fundamental cycle, one of the key steps is to compare a state with its 'reduced states' constructed by the '10-elimination'.…”

Section: Introductionmentioning

confidence: 99%

“…An important property which characterizes a state of the PBBS is the fundamental cycle of the state, i.e., the length of the trajectory to which it belongs. Its explicit formula as well as statistical distribution was obtained and its relation to the celebrated Riemann hypothesis was clarified [7,8,9]. To prove the formula for fundamental cycle, one of the key steps is to compare a state with its 'reduced states' constructed by the '10-elimination'.…”

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confidence: 99%

On the initial value problem of a periodic box-ball system

Mada¹,

Idzumi²,

Tokihiro³

2006

J. Phys. A: Math. Gen.

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We show that the initial value problem of a periodic box-ball system can be solved in an elementary way using simple combinatorial methods.A periodic box-ball system (PBBS) is a dynamical system of balls in an array of boxes with a periodic boundary condition [1,2]. The PBBS is obtained from the discrete KdV equation and the discrete Toda equation, both of which are known as typical integrable nonlinear discrete equations, through a limiting procedure called ultradiscretization [3,4]. Since the ultradiscretization preserves the main properties of the original discrete equations, and the solvability of the initial value problem being an important property of integrable equations, we expect that the initial value problem of the PBBS can also be solved. In fact, the initial value problem for the PBBS was first solved by inverse ultradiscretization combined with the method of inverse scattering transform of the discrete Toda equation [5] and recently by the Bethe ansatz for an integrable lattice model with quantum group symmetry at the deformation parameter q = 0 and q = 1 [6]. These two methods, however, require fairly specialized mathematical knowledge on algebraic curves or representation theory of quantum algebras.An important property which characterizes a state of the PBBS is the fundamental cycle of the state, i.e., the length of the trajectory to which it belongs. Its explicit formula as well as statistical distribution was obtained and its relation to the celebrated Riemann hypothesis was clarified [7,8,9]. To prove the formula for fundamental cycle, one of the key steps is to compare a state with its 'reduced states' constructed by the '10-elimination'. In this article, we show that the initial value problem of the PBBS is solved by simple combinatorial arguments -essentially given in Ref.[7] -with some remarkable features of the reduced states.First we quickly review the definition of the PBBS and its conserved quantities. Consider a one-dimensional array of boxes each with a capacity of one ball. A periodic boundary condition is imposed by assuming that the last box is adjacent to the first one. Let the number of boxes be N and that of balls be M . We assume M < N/2. An arrangement of M balls in N boxes is called a state of the PBBS. Denoting a vacant box by 0 and a filled box by 1, a state 1

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“…Furthermore, using these formulae, we can estimate the asymptotic behaviour of the fundamental cycles which shows an important number theoretical aspect of the PBBS [14,15]. One of the key elements underlying these results is that we can construct the conserved quantities of the PBBS explicitly.…”

Section: Introductionmentioning

confidence: 99%