格矩阵幂序列的图论方法
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摘要
本文利用图论方法研究了格矩阵幂序列的性质。我们从模糊集与模糊关系的概念开始，引出模糊矩阵幂序列的收敛性问题。然后针对中外学者研究成果中区间问题，把模糊矩阵幂序列的研究推广到格矩阵上。
格矩阵的分解定理，是联系格矩阵和布尔矩阵之间的桥梁。本文主要研究格矩阵的有向伴随图，得到了收敛格矩阵的图论特征，进而得到了格矩阵收敛的一些充要条件以及幂敛指数的计算方法。
本文建立的有关结果可以视为对以往关于模糊矩阵幂序列的收敛性的研究成果的发展，为格矩阵应用于神经网络等领域解决了收敛性问题。
This paper researched on the convergence of lattice matrices using graphtheoretical approach. At first, both the concepts of fuzzy set and fuzzy re-lation are given, and the convergence of fuzzy matrices is introduced. Thenwe stressed the interval question of the previous research and extended theresearch of fuzzy matrix to the research of lattice matrix.
The decomposition theorem of lattice matrix builds the bridge betweenlattice matrices and Boolean matrices.This paper mainly researched on thedirected graph of lattice matrices,and got the gragh-theoritical characteris-tic of the convergent lattice matrices, then got some sufficient and necessaryconvergent conditions of lattice matrices and the method of computing theconvergence index.
The results presented could be regarded as the development of the previ-ous research on fuzzy matrix.And the convergence problem is basically resolvedfor lattice matrix applying to the fields of nerve network.
格矩阵的分解定理，是联系格矩阵和布尔矩阵之间的桥梁。本文主要研究格矩阵的有向伴随图，得到了收敛格矩阵的图论特征，进而得到了格矩阵收敛的一些充要条件以及幂敛指数的计算方法。
本文建立的有关结果可以视为对以往关于模糊矩阵幂序列的收敛性的研究成果的发展，为格矩阵应用于神经网络等领域解决了收敛性问题。
This paper researched on the convergence of lattice matrices using graphtheoretical approach. At first, both the concepts of fuzzy set and fuzzy re-lation are given, and the convergence of fuzzy matrices is introduced. Thenwe stressed the interval question of the previous research and extended theresearch of fuzzy matrix to the research of lattice matrix.
The decomposition theorem of lattice matrix builds the bridge betweenlattice matrices and Boolean matrices.This paper mainly researched on thedirected graph of lattice matrices,and got the gragh-theoritical characteris-tic of the convergent lattice matrices, then got some sufficient and necessaryconvergent conditions of lattice matrices and the method of computing theconvergence index.
The results presented could be regarded as the development of the previ-ous research on fuzzy matrix.And the convergence problem is basically resolvedfor lattice matrix applying to the fields of nerve network.
引文
1 Zhou-Tian Fan, De-Fu Liu, Convergency of Power Sequence of Monotone Increasing Fuzzy Matrix, Fuzzy Sets And Systems, 1997 (88):363-372
2 Zhou-Tian Fan, De-Fu Liu, On the Power Sequence of a Fuzzy Matrix Convergency Power Sequence 、The Korea J. Comput. Appl. Math., 1997(4): 147-165
3 Zhou-Tian Fan, De-Fu Liu, On the Oscillating Power Sequence of a Fuzzy Matrix, Fuzzy Sets And Systems, 1998 (93): 75-85
4 Zhou-Tian Fan and De-Fu Liu, On the Power Sequence of a Fuzzy Matrix Ⅲ A Detailed Study on the Power Sequence of Matrices of Commonly Used Types, Fuzzy Sets And Systems, 1998(99):197-203
5 Zhou-Tian Fan, A Note On the Power Sequence of a Fuzzy Matrix, Fuzzy Sets And Systems, 1999(102):281-286
6 Zhou-Tian Fan, On the Convergence of a Fuzzy Matrix in the Sense of Triangular Norms, Fuzzy Sets And Systems, 2000(109):409-417
7 Zhou-Tian Fan, The Convergence Index of An Implication Matrix, Fuzzy Sets And Systems, 2001(123):265-269
8 Abraham Kandel, Fuzzy Mathematical Techniques with Applications Ad-dison Wesley Publishing, Tokyo, 1986:113-125
9 H.Hashimoto, Convergence of Powers of a Fuzzy Transitive Matrix, Fuzzy Sets and Systems, 1983 (9): 153-160
10 H.Hashimoto, Szpilrajn's Theorem on Fuzzy Matrix, Fuzzy Sets and Systems, 1983 (10): 101-108
11 W.Kolodzicjczyk, Convergence of Powers of S-Transitive Fuzzy Matrices. Fuzzy Sets and Systems, 1988 (26): 127-130
12 李建新，关于幂零矩阵的若干结果，模糊系统与数学1，1989：52-55
13 J.-X. Li, Convergence of Powers of Controllable Fuzzy Matrices, Fuzzy Sets and Systems, 1994 (62): 83-99
14 J.J. Buckley, Note On "Convergence of Powers of a Fuzzy Matrix", Fuzzy Sets and Systems, 2001 (121): 363-364
15 范周田，模糊矩阵幂序列理论与FBAM的动态分析，清华大学博士学位论文，1999：11-42
16 H, Hashimoto, Canonical Form of a Transitive Fuzzy Matrix, Fuzzy Sets and Systemsll (1983) 157-162
17 Li Jian-Xin, Controllable Fuzzy Matrices, Fuzzy Sets and Systems, 1992 (45): 313-319
18 李乔，矩阵论八讲，上海科技出版社，1988：66-80
19 邵学才，蒋强荣，石嘉明，张秀珍，离散数学，清华大学出版社，2001：33-36
20 左孝凌，李为辚，刘永才，离散数学理论·分析·题解，上海科学技术文献出版社，2002，306-308
21 胡宝清，模糊理论基础，武汉大学出版社，2004：103-139
22 Yung-Yih, Lur, Chin, Tzong Pang, Sy-Ming Guu, On Simultaneously Nilpotent Puzzy Matrices, Linear Algebra and its Application, 2003 (367): 37-45
23 Yung-Yih Lur, Chin, Tzong Pang, Sy-Ming Guu, On Nilpotent Puzzy Matrices, Fuzzy Sets and Systems, 2004(145): 287-299
24 Chin-Tzong Pang, On the Sequence of Consecutive Powers of a Fuzzy Matrix with Max-Archimedean-t-norms, Fuzzy Sets and Systems, 2003 (138): 643-656
25 Sy-Ming Guu, Yung-Yinlur, Chin-Tzong Pang, On Infinite Products of Fuzzy Matrices, SIAM J.MATRIX ANAL.APPL., 2001(22):1190-1203
26 焦烨，李永明，max-zero t-模复合意义下的模糊矩阵的收敛性，模糊系统与数学，2005(19)：80-85
27 焦烨，李永明，基于max-product复合的模糊矩阵幂序列的收敛性，工程数学学报，2005(22)：357-360
28 正传递模糊矩阵，谷云东，李洪兴：北京师范大学学报(自然科学版)，2001 (37)：596-601
29 拟收敛模糊矩阵的幂敛性，张光军．青岛大学学报(自然科学版)，2005(18)：14-16
30 Jicheng Li, Wenxiu Zhang, On Convergence of the Min-Max Compositions of Fuzzy Matrices, Southeast Asian Bulletin of Mathematics, 2000 (24): 389-393
31 Zhou Ji-tuan, On the Convergence of Lattice Matrices in the ∨-∧ Composition, J.of Math, 2005(25):379-382
32 姜超，格上的T-幂等矩阵，模糊系统与数学，2004(3)：50-58
33 姜超，格上矩阵的合成与T-传递矩阵，淮海工学院学报，2004(3)：9-12
34 Yi-jia Tan, On Nilpotent Matrices over Distributive Lattices, Fuzzy Sets and Systems, 2005 (151): 421-433
35 Kun-Lun Zhang, On the Nilpotent Matrices over D01-Lattices, Fuzzy Sets and systems, 2001 (117): 403-406
36 Yi-jia Tan, On Composition of Lattice Matrices, Fuzzy Sets and Systems, 2002 (129): 19-28
37 Yi-jia Tan, On the Powers of Matrices over a Distributive Lattice, Linear Algebra and its Applications, 2001 (336): 1-14
38 李盘林，离散数学，人民邮电出版社，2002：157-167
39 许蔓苓，离散数学，北京航空航天大学出版社，2004：291-314
40 何善(译)，离散数学，知识出版社，1987：1-24
41 Shao Jiayu, Li Qiao, The Index Set for the Class of Irreducible Boolan Matrices with Given Period, Linear and Multilinear Algebra, 1988(22): 285-303
42 耿素云，屈婉玲，离散数学，高等教育出版社，1998：288-317
43 李群，不确定性数学方法及其在社会科学中的应用，中国社会科学出版社，2005：1-50
44 Katarina Cechlarova, Powers of Matrices over Distributive Lattices-A Re-view, Fuzzy Sets and Systems, 2003(138): 627-641
45 范周田，模糊矩阵理论与应用，科学出版社，2006：100-115
2 Zhou-Tian Fan, De-Fu Liu, On the Power Sequence of a Fuzzy Matrix Convergency Power Sequence 、The Korea J. Comput. Appl. Math., 1997(4): 147-165
3 Zhou-Tian Fan, De-Fu Liu, On the Oscillating Power Sequence of a Fuzzy Matrix, Fuzzy Sets And Systems, 1998 (93): 75-85
4 Zhou-Tian Fan and De-Fu Liu, On the Power Sequence of a Fuzzy Matrix Ⅲ A Detailed Study on the Power Sequence of Matrices of Commonly Used Types, Fuzzy Sets And Systems, 1998(99):197-203
5 Zhou-Tian Fan, A Note On the Power Sequence of a Fuzzy Matrix, Fuzzy Sets And Systems, 1999(102):281-286
6 Zhou-Tian Fan, On the Convergence of a Fuzzy Matrix in the Sense of Triangular Norms, Fuzzy Sets And Systems, 2000(109):409-417
7 Zhou-Tian Fan, The Convergence Index of An Implication Matrix, Fuzzy Sets And Systems, 2001(123):265-269
8 Abraham Kandel, Fuzzy Mathematical Techniques with Applications Ad-dison Wesley Publishing, Tokyo, 1986:113-125
9 H.Hashimoto, Convergence of Powers of a Fuzzy Transitive Matrix, Fuzzy Sets and Systems, 1983 (9): 153-160
10 H.Hashimoto, Szpilrajn's Theorem on Fuzzy Matrix, Fuzzy Sets and Systems, 1983 (10): 101-108
11 W.Kolodzicjczyk, Convergence of Powers of S-Transitive Fuzzy Matrices. Fuzzy Sets and Systems, 1988 (26): 127-130
12 李建新，关于幂零矩阵的若干结果，模糊系统与数学1，1989：52-55
13 J.-X. Li, Convergence of Powers of Controllable Fuzzy Matrices, Fuzzy Sets and Systems, 1994 (62): 83-99
14 J.J. Buckley, Note On "Convergence of Powers of a Fuzzy Matrix", Fuzzy Sets and Systems, 2001 (121): 363-364
15 范周田，模糊矩阵幂序列理论与FBAM的动态分析，清华大学博士学位论文，1999：11-42
16 H, Hashimoto, Canonical Form of a Transitive Fuzzy Matrix, Fuzzy Sets and Systemsll (1983) 157-162
17 Li Jian-Xin, Controllable Fuzzy Matrices, Fuzzy Sets and Systems, 1992 (45): 313-319
18 李乔，矩阵论八讲，上海科技出版社，1988：66-80
19 邵学才，蒋强荣，石嘉明，张秀珍，离散数学，清华大学出版社，2001：33-36
20 左孝凌，李为辚，刘永才，离散数学理论·分析·题解，上海科学技术文献出版社，2002，306-308
21 胡宝清，模糊理论基础，武汉大学出版社，2004：103-139
22 Yung-Yih, Lur, Chin, Tzong Pang, Sy-Ming Guu, On Simultaneously Nilpotent Puzzy Matrices, Linear Algebra and its Application, 2003 (367): 37-45
23 Yung-Yih Lur, Chin, Tzong Pang, Sy-Ming Guu, On Nilpotent Puzzy Matrices, Fuzzy Sets and Systems, 2004(145): 287-299
24 Chin-Tzong Pang, On the Sequence of Consecutive Powers of a Fuzzy Matrix with Max-Archimedean-t-norms, Fuzzy Sets and Systems, 2003 (138): 643-656
25 Sy-Ming Guu, Yung-Yinlur, Chin-Tzong Pang, On Infinite Products of Fuzzy Matrices, SIAM J.MATRIX ANAL.APPL., 2001(22):1190-1203
26 焦烨，李永明，max-zero t-模复合意义下的模糊矩阵的收敛性，模糊系统与数学，2005(19)：80-85
27 焦烨，李永明，基于max-product复合的模糊矩阵幂序列的收敛性，工程数学学报，2005(22)：357-360
28 正传递模糊矩阵，谷云东，李洪兴：北京师范大学学报(自然科学版)，2001 (37)：596-601
29 拟收敛模糊矩阵的幂敛性，张光军．青岛大学学报(自然科学版)，2005(18)：14-16
30 Jicheng Li, Wenxiu Zhang, On Convergence of the Min-Max Compositions of Fuzzy Matrices, Southeast Asian Bulletin of Mathematics, 2000 (24): 389-393
31 Zhou Ji-tuan, On the Convergence of Lattice Matrices in the ∨-∧ Composition, J.of Math, 2005(25):379-382
32 姜超，格上的T-幂等矩阵，模糊系统与数学，2004(3)：50-58
33 姜超，格上矩阵的合成与T-传递矩阵，淮海工学院学报，2004(3)：9-12
34 Yi-jia Tan, On Nilpotent Matrices over Distributive Lattices, Fuzzy Sets and Systems, 2005 (151): 421-433
35 Kun-Lun Zhang, On the Nilpotent Matrices over D01-Lattices, Fuzzy Sets and systems, 2001 (117): 403-406
36 Yi-jia Tan, On Composition of Lattice Matrices, Fuzzy Sets and Systems, 2002 (129): 19-28
37 Yi-jia Tan, On the Powers of Matrices over a Distributive Lattice, Linear Algebra and its Applications, 2001 (336): 1-14
38 李盘林，离散数学，人民邮电出版社，2002：157-167
39 许蔓苓，离散数学，北京航空航天大学出版社，2004：291-314
40 何善(译)，离散数学，知识出版社，1987：1-24
41 Shao Jiayu, Li Qiao, The Index Set for the Class of Irreducible Boolan Matrices with Given Period, Linear and Multilinear Algebra, 1988(22): 285-303
42 耿素云，屈婉玲，离散数学，高等教育出版社，1998：288-317
43 李群，不确定性数学方法及其在社会科学中的应用，中国社会科学出版社，2005：1-50
44 Katarina Cechlarova, Powers of Matrices over Distributive Lattices-A Re-view, Fuzzy Sets and Systems, 2003(138): 627-641
45 范周田，模糊矩阵理论与应用，科学出版社，2006：100-115