A Soliton Cellular Automaton

Takahashi, Daisuke; Satsuma, Junkichi

doi:10.1143/jpsj.59.3514

Cited by 291 publications

(349 citation statements)

References 2 publications

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“…In view of ϕ i k (u |k−1 ) (26) = t k (u) = 0 and (13), it also follows that z |k =ẽ max i k · · ·ẽ max i1 z. Therefore we conclude ϕ i k (z |k−1 ) = t k (z) from Definition 4.…”

Section: Thus We Havementioning

confidence: 65%

“…To show the former, set x = b and apply (26) in q ′ appearing in (22). The left side of the latter reads…”

Section: Applying (25) We Obtain T(σv (D) ) = T(y)mentioning

confidence: 99%

“…The box-ball systems [24][25][26][27] are important examples of soliton cellular automata. They are discrete dynamical systems whose time evolution is expressed as a certain motion of balls along the one dimensional array of boxes.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, the box-ball systems are identified with a q → 0 limit of some two-dimensional solvable vertex models, where the role of time evolution is played by the action of a row transfer matrix. The simplest example is the original Takahashi-Satsuma automaton, 26 whose classical origin is the discrete Lotka-Volterra equation 28 and the quantum origin is the fusion six-vertex model. Here is an example of the automaton time evolution.…”

Section: Introductionmentioning

confidence: 99%

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Hatayama

Kuniba

Такаги

2001

Journal of Statistical Physics

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Solvable vertex models in statistical mechanics give rise to soliton cellular automata at q = 0 in a ferromagnetic regime. By means of the crystal base theory we study a class of such automata associated with non-exceptional quantum affine algebras U ′ q (ĝn). Let B l be the crystal of the U ′ q (ĝn)-module corresponding to the l-fold symmetric fusion of the vector representation. For any crystal of the form B = B l 1 ⊗ · · · ⊗ B l N , we prove that the combinatorial R matrix BM ⊗ B ∼ − → B ⊗ BM is factorized into a product of Weyl group operators in a certain domain if M is sufficiently large. It implies the factorization of certain transfer matrix at q = 0, hence the time evolution in the associated cellular automata. The result generalizes the ball-moving algorithm in the box-ball systems.

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Section: Thus We Havementioning

confidence: 65%

“…To show the former, set x = b and apply (26) in q ′ appearing in (22). The left side of the latter reads…”

Section: Applying (25) We Obtain T(σv (D) ) = T(y)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hatayama

Kuniba

Такаги

2001

Journal of Statistical Physics

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“…However, denoting the bijection Φ from B 1 ⊗ B 2 by Φ B1⊗B2 , R : B 1 ⊗ B 2 → B 2 ⊗ B 1 is simply realized as Φ −1 B2⊗B1 • Φ B1⊗B2 thanks to the R-invariance. As a second reason, we mention the application to box-ball systems [TS90]. These are certain integrable dynamical systems formulated on the tensor product of KR crystals.…”

Section: Introductionmentioning

confidence: 99%

Type $${{\varvec{D}}}_{{\varvec{n}}}^\mathbf{(1)}$$ rigged configuration bijection

et al. 2017

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Abstract. We establish a bijection between the set of rigged configurations and the set of tensor products of Kirillov-Reshetikhin crystals of type D (1) n in full generality. We prove the invariance of rigged configurations under the action of the combinatorial R-matrix on tensor products and show that the bijection preserves certain statistics (cocharge and energy). As a result, we establish the fermionic formula for type D (1) n . In addition, we establish that the bijection is a classical crystal isomorphism.

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Computing in Verotoxin

Adamatzky

2017

ChemPhysChem

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We develop an excitable automata model of a protein verotoxin and demonstrate that logic gates and circuits are realised in the model via interacting patterns of excitation. By sampling potential input pairs of nodes, we calculate frequencies of logic gates which occurred in the verotoxin model for various parameters of node excitation rules. We show that overall the gates can be arranged in the following hierarchy of descending frequencies: AND>OR>AND‐NOT>XOR. We demonstrate realisations of one‐bit half‐adder and controlled‐not gates and estimate memory capacity of the verotoxin molecule.

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A Soliton Cellular Automaton

Cited by 291 publications

References 2 publications

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Type $${{\varvec{D}}}_{{\varvec{n}}}^\mathbf{(1)}$$ rigged configuration bijection

Computing in Verotoxin

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