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研究生: 吳樹恆
Wu, Shu-Han
論文名稱: 推廣施-董的組合固定點定理至有限分配格
Generalization of Shih-Dong's combinational fixed point theorem to finite distributive lattices
指導教授: 陳界山
Chen, Jein-Shan
施茂祥
Shih, Mau-Hsiang
學位類別: 博士
Doctor
系所名稱: 數學系
Department of Mathematics
論文出版年: 2015
畢業學年度: 103
語文別: 英文
論文頁數: 33
中文關鍵詞: 離散動態系統有限分配格固定點廣義布爾雅可比矩陣負迴路正迴路
英文關鍵詞: Discrete dynamical system, Finite distributive lattice, Fixed point, Generalized Boolean Jacobian matrix, Negative circuit, Positive circuit
論文種類: 學術論文
相關次數: 點閱:167下載:21
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  • 施-董的組合不動點定理證明,如果從n維超立方體到自身的函數滿足了每個在超立方體的元素其布爾雅可比矩陣的特徵值是零,那麼該函數有唯一的固定點。該定理等價對偶敘述具有生物學意義。我們的目標是推廣施-董定理到所有的有限分配格。我們的證明方法是基於施-董的“集體影響法”以及G.伯克霍夫的有限分配格表現定理。

    Shih-Dong's combinational fixed point theorem asserts that if a map from the n-dimensional hypercube into itself satisfies that all the Boolean eigenvalues of the Boolean Jacobian matrix are zero for each element in the hypercube, then it has a unique fixed point. Its equivalent contrapositive form has biological implications. Our goal is to provide an extension of Shih-Dong's theorem into all finite distributive lattices. Our method of proof is based on Shih-Dong's “collective effect method” as well as G. Birkhoff's representation theorem for finite distributive lattices.

    1 Introduction ... 1 2 De nitions and Notations ... 3 3 Fixed points and circuit-free ... 8 4 Fixed points and positive circuits ... 17 5 Fixed points and negative circuits ... 19 6 Concluding remarks ... 25 7 Related open questions ... 30 Reference ... 32

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