二維與三維網格模型上的花樣生成問題

Abstract

本論文主要研究二維與三維的花樣生成問題。不論在二維或三維狀況中，適當給定局部花樣的次序可定義出次序矩陣，此次序矩陣的特殊結構可使得較大有限花樣所對應的高次次序矩陣被有系統的生成出來。給定某局部花樣子集合，由次序矩陣可定義出轉化矩陣。利用低次與高次次序矩陣的特殊關係，可得到相對應轉換矩陣的遞迴公式。空間熵的正則性是判斷包含所有可允許的全局花樣集合複雜性的重要指標，而在此論文中，空間熵可藉由一組轉換矩陣的最大特徵值計算出，一般而言，因為轉換矩陣的大小呈指數增長，使得空間熵不易準確的計算出。因此定義所謂的連接算子，並利用其來估計空間熵的下界，進而來驗證空間熵的正則性。另外，在二維情況中，可定義跡矩陣，利用其估計更好的空間熵上界。在三維情況中，將以三維細胞類神經網路為例，呈現三維花樣生成問題的應用。此博士論文所建立的理論，在研究網格動態系統及類神經網路中全局解的複雜性上有極大的幫助。

This dissertation investigates two and three-dimensional patterns generation problems. Both in two and three-dimensional cases, an ordering matrix for the set of all local patterns is established to derive a recursive formula for the ordering matrix for a larger finite lattice. For a given admissible set of local patterns, the transition matrix is defined and the recursive formula of high order transition matrix is presented. Then, the spatial entropy is obtained by computing the maximum eigenvalues of a sequence of transition matrices. The connecting operators are used to verify the positivity of the spatial entropy, which is important in determining the complexity of the set of admissible global patterns. Moreover, trace operator can be also introduced to give a good estimate of the upper bound of spatial entropy. In three-dimensional case, applications to three-dimensional Cellular Neural Networks is presented. The results are useful in studying a set of global stationary solutions in various Lattice Dynamical Systems and Cellular Neural Networks.