標題: Patterns generation and spatial entropy in two-dimensional lattice models 作者: Ban, Jung-ChaoLin, Song-SunLin, Yin-Heng應用數學系Department of Applied Mathematics 關鍵字: lattice dynamical systems;spatial entropy;patterns generation;connecting operator;trace operator 公開日期: 1-Sep-2007 摘要: Patterns generation problems in two-dimensional lattice models are studied. Let S be the set of p symbols and Z(2lx2l), l >= 1, be a fixed finite square sublattice of Z(2). Function U : Z(2lx2l) -> S is called local pattern. Given a basic set B of local patterns, a unique transition matrix A(2) which is a q(2)xq(2) matrix, q = p(l2), can be defined. The recursive formulae of higher transition matrix A(n) on Z(2lxnl) have already been derived . Now A(n)(m), m >= 1, contains all admissible patterns on Z((m+1)lxnl) which can be generated by B. In this paper, the connecting operator C-m, which comprises all admissible patterns on Z((m+1)lx2l), is carefully arranged. C-m can be used to extend A(n)(m) to A(n+1)(m) recursively for n >= 2. Furthermore, the lower bound of spatial entropy h(A(2)) 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 T-m of C-m can also be introduced. In the case of symmetric A(2), T-2m gives a good estimate of the upper bound on spatial entropy. Combining C-m with T-m helps to understand the patterns generation problems more systematically. URI: http://hdl.handle.net/11536/10399 ISSN: 1093-6106 期刊: ASIAN JOURNAL OF MATHEMATICS Volume: 11 Issue: 3 起始頁: 497 結束頁: 534 Appears in Collections: Articles