关于模糊数空间若干度量性质的研究
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摘要
本文主要是对模糊数的度量性质进行研究,主要内容如下:
1.证明了非紧模糊数空间E~中Endograph度量关于模糊数的序是有限逼近的。本文给出的证明方法是构造性的,从而说明了非紧模糊数值积分如M-积分和G-积分等是可计算的。最后给出了E~中关于Endograph度量的一些分析性质。
2.模糊数空间关于Endograph和Sendograph度量是可分的,但都不是完备的。本文给出了Endograph和Sendograph度量下模糊数空间的完备化,它们分别是E~和{([a,b]×{0})∪send(u):u_0(?)[a,b],u∈E~1}。E~1关于Endograph度量的完备化使我们首次在非紧模糊数空间E~上引入了一种完备可分度量;而对Sendograph度量的完备化则引入了Sendograph度量的一个一致等价刻化,这种刻化体现了Sendograph和Endograph度量之间的某种内部联系。
3.给出了在Sendograph度量下可以用具有连续截集函数的模糊数来逼近任意的模糊数。我们的逼近方法是有限逼近,也是非常自然的。最后讨论在扩张原理下模糊数之间的关系。
In this thesis,we investigate the metric properties of fuzzy number spaces.The content of this thesis are as followings:
1.First,it is proved that the endogragh metric is approximative with respect to orders on noncompact fuzzy number space E~.It is also shown that the endograph metric approach on orders on E~ is constructive,this shows that noncompact fuzzy-number-valued integrals such as M-integral and G-integral are computable.Finally some analytic properties with respect to the endograph metric on E~ are given.
2.The endograph and sendograph metrics on the spaces of fuzzy numbers are known to be separable,but neither of them is complete.This paper deals with the completions of the endograph and sendograph metrics.It is proved that E~ and {([a,b]×{0})∪send(u):u_0(?)[a,b],u(?)E~1} are the completions of E~1 with respect to the endograph and sendograph metrics respectively.The completion of E~1 with respect to the endograph metric enables us for the first time to consider a separable and complete metric on E~, the space of noncompact fuzzy numbers;While in considering completion with respect to the sendograph metric,a uniformly equivalent description of the sendograph metric is given which reveals that there are some internal relations between the sendograph and endograph metrics.
3.It is proved that fuzzy numbers can be approximated via sendograph metric by fuzzy numbers with continuousλ-cut functions to any accuracy.Our methods of approximation is finite in nature.Relations between fuzzy numbers with Zadeh's extension principle are discussed.
1.证明了非紧模糊数空间E~中Endograph度量关于模糊数的序是有限逼近的。本文给出的证明方法是构造性的,从而说明了非紧模糊数值积分如M-积分和G-积分等是可计算的。最后给出了E~中关于Endograph度量的一些分析性质。
2.模糊数空间关于Endograph和Sendograph度量是可分的,但都不是完备的。本文给出了Endograph和Sendograph度量下模糊数空间的完备化,它们分别是E~和{([a,b]×{0})∪send(u):u_0(?)[a,b],u∈E~1}。E~1关于Endograph度量的完备化使我们首次在非紧模糊数空间E~上引入了一种完备可分度量;而对Sendograph度量的完备化则引入了Sendograph度量的一个一致等价刻化,这种刻化体现了Sendograph和Endograph度量之间的某种内部联系。
3.给出了在Sendograph度量下可以用具有连续截集函数的模糊数来逼近任意的模糊数。我们的逼近方法是有限逼近,也是非常自然的。最后讨论在扩张原理下模糊数之间的关系。
In this thesis,we investigate the metric properties of fuzzy number spaces.The content of this thesis are as followings:
1.First,it is proved that the endogragh metric is approximative with respect to orders on noncompact fuzzy number space E~.It is also shown that the endograph metric approach on orders on E~ is constructive,this shows that noncompact fuzzy-number-valued integrals such as M-integral and G-integral are computable.Finally some analytic properties with respect to the endograph metric on E~ are given.
2.The endograph and sendograph metrics on the spaces of fuzzy numbers are known to be separable,but neither of them is complete.This paper deals with the completions of the endograph and sendograph metrics.It is proved that E~ and {([a,b]×{0})∪send(u):u_0(?)[a,b],u(?)E~1} are the completions of E~1 with respect to the endograph and sendograph metrics respectively.The completion of E~1 with respect to the endograph metric enables us for the first time to consider a separable and complete metric on E~, the space of noncompact fuzzy numbers;While in considering completion with respect to the sendograph metric,a uniformly equivalent description of the sendograph metric is given which reveals that there are some internal relations between the sendograph and endograph metrics.
3.It is proved that fuzzy numbers can be approximated via sendograph metric by fuzzy numbers with continuousλ-cut functions to any accuracy.Our methods of approximation is finite in nature.Relations between fuzzy numbers with Zadeh's extension principle are discussed.
引文
[1] Zadeh L A. Fuzzy Sets[J]. Information and Control, 1965, 8: 338-353.
[2] Chang S S L, Zadeh L A. On fuzzy mapping and Control. IEEE Trans, Systems Man Cybernet.
[3] Mizumoto M, Tanaka K. The four operations of arithmetric on fuzzy nmubers. System Compute, Control, 1976, 7: 73-81.
[4] Dubois D, Prade H. Operations on fuzzy numbers. Internat. Jour. Systems Sci., 1978, 9:613-626.
[5] Dubois D, Prade H. Towards fuzzy differential calculus, Part 1. Fuzzy Sets and Systems,1982, 8: 1-17.
[6] Dubois D, Prade H. Towards fuzzy differential calculus, Part 2. Fuzzy Sets and Systems,1982, 8: 105-116.
[7] Dubois D, Prade H. Towards fuzzy differential calculus, Part 3. Fuzzy Sets and Systems,1982, 8: 225-233.
[8] Bezdek J C. Fuzzy numbers: An overview. The theory of Fuzzy Information, Vol.1: Mathematics and Logic[M], Boca Roton: CRC Press, 1987.
[9] Heilpern S. Fuzzy mappings and fixed point theorem. J. Math. Anal. Appl., 1981, 83:566-569.
[10] Rodabaugh S E. Complete fuzzy topological heperfields and fuzzy multiplication in the fuzzy real line. Fuzzy Sets and Systems, 1985, 15: 285—310.
[11] Rodabaugh S E. Fuzzy addition in the L-fuzzy real line. Fuzzy Sets and Systems, 1982, 8:39-52.
[12] Nanda S. On fuzzy integrals. Fuzzy Sets and Systems, 1989, 32: 95-101.
[13] Wu C X, Wu C. The supremum and infimum of the set of fuzzy numbers and its applications.J. Math. Anal. Appl., 1997, 210: 499-511.
[14] Wu C X, Wu C. Some notes on the supremum and infimum of the set of fuzzy numbers.Fuzzy Sets and Systems, 1999, 103: 183-187.
[15] Fan T H, Wang G J. Endographic approach on supremum and infimum of fuzzy numbers. Information Sciences, 2004, 159(3-4): 221-231.
[16] Negoita C V, Ralescu D A. Application of fuzzy sets to system analysis. Wiley, New York,1975.
[17] Goetschel R, Voxman W. Element fuzzy calculus. Fuzzy Sets and Systems, 1986, 18: 31—43.
[18] Diamond P, Kloeden P. Metric spaces of fuzzy sets. Fuzzy Sets and Systems, 1990, 35: 241-249.
[19]Diamond P.A note on the star shaped fuzzy set.Fuzzy Sets and Systems,1990,37:193-199.
[20]Kloeden P.Compact supported endographs and fuzzy sets.Fuzzy Sets and Systems,1980,4:193-201.
[21]Rojas-Medar M,Roman-Flores H,On the equivalence of convergences of fuzzy sets.Fuzzy Sets and Systems,1996,80(2):217-224.
[22]Kaleva O,Seikkala S.On fuzzy metric spaces.Fuzzy Sets and Systems,1984,12:215-229.
[23]Fang J X,Huang H.Some properties of the level convergence topology on fuzzy number space.Fuzzy Sets and Systems,2003,140(3):509-517.
[24]Fang J X,Huang H.On the level convergence of a sequence of fuzzy numbers.Fuzzy Sets and Systems,2004,147(3):417-435.
[25]黄欢,方锦暄.关于模糊数理论的一些重要性质.模糊系统与数学,2001,15(4):58-60.
[26]Fan T H.Remarks on Continuous Fuzzy numbers.J.Fuzzy mathematics,2004(12),3:747-753.
[27]Collings,Kloeden P.Continuous appximation of fuzzy numbers.J.Fuzzy Mathematics,1995,3:449-453.
[28]Wu C X,Zhang B K.Existence of supremum and infimum in E~ and relations of(M)integral and(G)integral.Journal of Harbin Institute of Technology(New Series),2000,7(3):58-61.
[29]Wu C X,Zhang B K.Embedding problem of noncompact fuzzy number space E~(Ⅰ).Fuzzy Sets and Systems,1999,105(1):165-169.
[30]Wu C X,Zhang B K.Embedding problem of noncompact fuzzy number space E~(Ⅱ).Fuzzy Sets and Systems,2000,110(1):135-142.
[31]张博侃,任丽伟,李雷.关于模糊数空间的几点注记.哈尔滨理工大学学报,1998.3:74-76.
[32]Liu Y M,Luo M K.Fuzzy Topology[M].World Scientific,Singapore,1997.
[33]王国俊.L-Fuzzy拓扑空间论[M].陕西师范大学出版社,西安,1998.
[34]Kr(a|¨)tschmer V.Some complete metrics on spaces of fuzzy subsets.Fuzzy Sets and Systems,2002,130:357-365.
[35]Goetschel R,Voxman W.Topological properties of fuzzy numbers,Fuzzy Sets and Systems,1983,10:87-99.
[36]Fan T H.On the compactness of fuzzy numbers with sendograph metric.Fuzzy Sets and Systems, 2004, 143: 471-477.
[37] Diamond P, Kloeden P. Metric spaces of fuzzy sets[M]. World Scientific, Singapore, 1994.
[38] Beer G. Topologies on closed and closed convex sets[M]. Kluwer Academic Publishers,Dordrecht, 1993.
[39] Gahler W, Gahler S. Contributions to fuzzy analysis. Fuzzy Sets and Systems, 1999, 105:201-224.
[40] Joo S Y, Kim Y K. The Skorokhod topology on spaces of fuzzy numbers. Fuzzy Sets and Systems, 2000, 111: 497-501.
[41] Ghil B M, Joo S Y, Kim Y K. A characterization of compact subsets of fuzzy number space.Fuzzy Sets and Systems, 2001, 123: 191-195.
[42] Saade J. Mapping convex and normal fuzzy sets. Fuzzy Sets and Systems, 1996, 81(2):251-256.
[2] Chang S S L, Zadeh L A. On fuzzy mapping and Control. IEEE Trans, Systems Man Cybernet.
[3] Mizumoto M, Tanaka K. The four operations of arithmetric on fuzzy nmubers. System Compute, Control, 1976, 7: 73-81.
[4] Dubois D, Prade H. Operations on fuzzy numbers. Internat. Jour. Systems Sci., 1978, 9:613-626.
[5] Dubois D, Prade H. Towards fuzzy differential calculus, Part 1. Fuzzy Sets and Systems,1982, 8: 1-17.
[6] Dubois D, Prade H. Towards fuzzy differential calculus, Part 2. Fuzzy Sets and Systems,1982, 8: 105-116.
[7] Dubois D, Prade H. Towards fuzzy differential calculus, Part 3. Fuzzy Sets and Systems,1982, 8: 225-233.
[8] Bezdek J C. Fuzzy numbers: An overview. The theory of Fuzzy Information, Vol.1: Mathematics and Logic[M], Boca Roton: CRC Press, 1987.
[9] Heilpern S. Fuzzy mappings and fixed point theorem. J. Math. Anal. Appl., 1981, 83:566-569.
[10] Rodabaugh S E. Complete fuzzy topological heperfields and fuzzy multiplication in the fuzzy real line. Fuzzy Sets and Systems, 1985, 15: 285—310.
[11] Rodabaugh S E. Fuzzy addition in the L-fuzzy real line. Fuzzy Sets and Systems, 1982, 8:39-52.
[12] Nanda S. On fuzzy integrals. Fuzzy Sets and Systems, 1989, 32: 95-101.
[13] Wu C X, Wu C. The supremum and infimum of the set of fuzzy numbers and its applications.J. Math. Anal. Appl., 1997, 210: 499-511.
[14] Wu C X, Wu C. Some notes on the supremum and infimum of the set of fuzzy numbers.Fuzzy Sets and Systems, 1999, 103: 183-187.
[15] Fan T H, Wang G J. Endographic approach on supremum and infimum of fuzzy numbers. Information Sciences, 2004, 159(3-4): 221-231.
[16] Negoita C V, Ralescu D A. Application of fuzzy sets to system analysis. Wiley, New York,1975.
[17] Goetschel R, Voxman W. Element fuzzy calculus. Fuzzy Sets and Systems, 1986, 18: 31—43.
[18] Diamond P, Kloeden P. Metric spaces of fuzzy sets. Fuzzy Sets and Systems, 1990, 35: 241-249.
[19]Diamond P.A note on the star shaped fuzzy set.Fuzzy Sets and Systems,1990,37:193-199.
[20]Kloeden P.Compact supported endographs and fuzzy sets.Fuzzy Sets and Systems,1980,4:193-201.
[21]Rojas-Medar M,Roman-Flores H,On the equivalence of convergences of fuzzy sets.Fuzzy Sets and Systems,1996,80(2):217-224.
[22]Kaleva O,Seikkala S.On fuzzy metric spaces.Fuzzy Sets and Systems,1984,12:215-229.
[23]Fang J X,Huang H.Some properties of the level convergence topology on fuzzy number space.Fuzzy Sets and Systems,2003,140(3):509-517.
[24]Fang J X,Huang H.On the level convergence of a sequence of fuzzy numbers.Fuzzy Sets and Systems,2004,147(3):417-435.
[25]黄欢,方锦暄.关于模糊数理论的一些重要性质.模糊系统与数学,2001,15(4):58-60.
[26]Fan T H.Remarks on Continuous Fuzzy numbers.J.Fuzzy mathematics,2004(12),3:747-753.
[27]Collings,Kloeden P.Continuous appximation of fuzzy numbers.J.Fuzzy Mathematics,1995,3:449-453.
[28]Wu C X,Zhang B K.Existence of supremum and infimum in E~ and relations of(M)integral and(G)integral.Journal of Harbin Institute of Technology(New Series),2000,7(3):58-61.
[29]Wu C X,Zhang B K.Embedding problem of noncompact fuzzy number space E~(Ⅰ).Fuzzy Sets and Systems,1999,105(1):165-169.
[30]Wu C X,Zhang B K.Embedding problem of noncompact fuzzy number space E~(Ⅱ).Fuzzy Sets and Systems,2000,110(1):135-142.
[31]张博侃,任丽伟,李雷.关于模糊数空间的几点注记.哈尔滨理工大学学报,1998.3:74-76.
[32]Liu Y M,Luo M K.Fuzzy Topology[M].World Scientific,Singapore,1997.
[33]王国俊.L-Fuzzy拓扑空间论[M].陕西师范大学出版社,西安,1998.
[34]Kr(a|¨)tschmer V.Some complete metrics on spaces of fuzzy subsets.Fuzzy Sets and Systems,2002,130:357-365.
[35]Goetschel R,Voxman W.Topological properties of fuzzy numbers,Fuzzy Sets and Systems,1983,10:87-99.
[36]Fan T H.On the compactness of fuzzy numbers with sendograph metric.Fuzzy Sets and Systems, 2004, 143: 471-477.
[37] Diamond P, Kloeden P. Metric spaces of fuzzy sets[M]. World Scientific, Singapore, 1994.
[38] Beer G. Topologies on closed and closed convex sets[M]. Kluwer Academic Publishers,Dordrecht, 1993.
[39] Gahler W, Gahler S. Contributions to fuzzy analysis. Fuzzy Sets and Systems, 1999, 105:201-224.
[40] Joo S Y, Kim Y K. The Skorokhod topology on spaces of fuzzy numbers. Fuzzy Sets and Systems, 2000, 111: 497-501.
[41] Ghil B M, Joo S Y, Kim Y K. A characterization of compact subsets of fuzzy number space.Fuzzy Sets and Systems, 2001, 123: 191-195.
[42] Saade J. Mapping convex and normal fuzzy sets. Fuzzy Sets and Systems, 1996, 81(2):251-256.