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Title: Patterns generation and spatial entropy in two-dimensional lattice models
Authors: 班榮超
Ban, Jung-Chao
Lin, Song-Sun
Lin, Yin-Heng
Contributors: 應數系
Keywords: Lattice dynamical systems;Spatial entropy;Patterns generation;Connecting operator;Trace operator
Date: 2007-09
Issue Date: 2020-06-22 13:42:51 (UTC+8)
Abstract: Patterns generation problems in two-dimensional lattice models are studied. Let S be the set of p symbols and ℤ2ℓ×2ℓ, ℓ ≥ 1, be a fixed finite square sublattice of ℤ2. Function U: ℤ 2ℓ×2ℓ → S is called local pattern. Given a basic set B of local patterns, a unique transition matrix A2 which is a q2 × q2 matrix, q = pℓ2, can be defined. The recursive formulae of higher transition matrix An on ℤ 2ℓ×nℓ have already been derived. Now Anm, m ≥ 1, contains all admissible patterns on ℤ(m+1)ℓ×nℓ which can be generated by B. In this paper, the connecting operator ℂ m, which comprises all admissible patterns on ℤ (m+1)ℓ×2ℓ, is care fully arranged. ℂm can be used to extend Anm to Anm+1 recursively for n ≥ 2. Furthermore, the lower bound of spatial entropy h(A2) can be derived through the diagonal part of ℂ m. This yields a powerful method for verifying the positivity of spatial entropy which is important in examining the complexity of the set of admissible global patterns. The trace operator Tm of ℂm can also be introduced. In the case of symmetric A2, T2m gives a good estimate of the upper bound on spatial entropy. Combining ℂm with Tm helps to understand the patterns generation problems more systematically
Relation: Asian Journal of Mathematics, Vol.11, No.3, pp.497-534
Data Type: article
DOI 連結: 10.4310/AJM.2007.v11.n3.a7
Appears in Collections:[應用數學系] 期刊論文

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