花樣生成問題

Abstract

花樣生成問題是數學、統計物理、生物及網路系統經常碰到的問題。以平面網格著色為例，在平面網格上，若以黑白兩色著色，預先規定如同色不得相鄰。把在m×n的網格上，所有可能的著色法記為，其空間熵記為mn×Γ,1limlog,mnmnhmn×→∞= Γ 。則花樣生成問題的重點就在研究：如何計算mn×Γ及。在統計物理相變問題的臨界現象，就經常在計算類似問題。在研究胞狀神經網路及網格動態系統穩定花樣的複雜性問題時，也是在計算相關的及。在腦神經科學上，研究大腦皮質結構及其訊息處理及傳遞也會考慮類似問題。hmn×Γh 在之前的計畫裡，我們發現了空間轉置矩陣nΤ的遞迴公式，且證明：1limlog()nnhnρ→∞=Τ，()nρΤ是的最大固有值。但因nΤ()nρΤ的計算很困難，以致的計算更加困難。為突破此困境，我們發現了連繫算子。可用來估計h的下界。hn..n.. 在本計畫裡，我們將綜合應用nΤ及，對做精密計算，並應用數論來瞭解其特性。也希望能把這方法去處理相變問題的臨界現象。

Patterns generation problems are very common in mathematics, statistical physics, biology and network system. For example, consider the coloring problems of plane lattice：using black and white colors to cover the m×n lattice subject to certain rule. Denoted by mn×Γ the total numbers of admissible ways (patterns) on m×n lattice and spatial entropy ,1limlog,mnmnhmn×→∞=Γ. Then the patterns generation problems need to study mn×Γ and . The similar problems occurs in critical phenomena of phase-transition problems in statistical physics. It also happens in study the complexity of stable patterns in Cellular Neural Network and lattice dynamical system. The information process in brain cortex also exhibit a similar phenomena. h In our earlier project, we discovered the recursive formula for transition matrix and proved nΤ1limlog()nnhnρ→∞=Τ, where ()nρΤ is the maximum eigenvalue of . The computation of nΤ()nρΤ is difficult for large n and even harder for . Later, we discovered connecting operator . can be used to estimate the lower bound of . hn..n..h In this project, combining with nΤ and , we shall give better estimate of and find out the characteristic of from number theory. We shall also apply the method to study the critical phenomena in phase-transition problems.