模糊分析中的若干问题及与粗糙集理论的结合研究
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摘要
模糊分析学是模糊数学的重要组成部分，一直以来都是学术界最为重视的课题之一．本文对模糊分析学中几个相互联系十分紧密的问题进行了研究．此外，作为模糊分析学的应用，也给出了模糊分析与粗糙集理论的结合研究．全文共六章，主要研究工作分四个部分：
第一部分研究了模糊实数的推广和模糊复分析．首先给出了模糊实数的一些性质．接下来，由反例说明模糊复分析中最基本概念的定义(模糊复数和一般模糊复数)存在的一些问题．重新修正模糊复分析中这两个基本概念后，一些重要的结论再一次被得到．最后，分析了现有的几种模糊复变函数的导数定义，改进了一些已有的重要结论．所有的这些事实说明尽管对模糊复分析Buckley1989年就开始了研究，但是其仍然处于起步阶段，需要进一步的深入研究．
第二部分研究了模糊凸分析及其推广．利用模糊单形，给出了模糊凸包的几个等价刻画．作为模糊凸性的推广，从模糊拟凸集和模糊凸集中衍生出一般模糊星状集和模糊伪星状集这两个新的概念．接着讨论了模糊星状集、模糊拟星状集、模糊伪星状集和一般模糊星状集之间的关系．同时，详细地给出了这些不同类型的星状性的基本性质．讨论了模糊星状集的落影问题并得到了一些重要结论．澄清了两个关于模糊凸过程的定义并得到了模糊凸过程的一些基本性质．这些性质改进了关于这两类模糊映射与其图像之间的关系的已有结论．最后，指出了两种s-凸模糊过程之间的联系和区别并给出了s-凸模糊过程和s-凸模糊映射的一些充要条件．
第三部分研究了模糊集组成的距离空间和模糊映射族的公共不动点理论．首先推广了关于全体非空紧集在Hausdorff距离下形成的空间中的一个经典结果，并运用此结果建立了紧距离空间的全体具有非空紧截集的模糊集合以d_∞-距离形成的空间(?)(X)的完备性．为了进一步推广这一结果，在推广了关于全体非空有界闭集在Hausdorff距离下形成的空间中的一个经典结果后，用类似的方法得到对完备距离空间X的全体具有非空有界闭截集的模糊集合引入由Hausdorff距离诱导出的上确界距离，即d_∞-距离形成的空间(?)(X)的完备性．此外，也分别讨论了这两个空间中模糊映射族的公共不动点问题，并举例说明了不动点理论的有效性．
第四部分是模糊分析和粗糙集理论的结合研究．一方面，运用粗糙集理论推广粗糙函数为粗糙映射并给出了粗糙映射的各种理论性质；另一方面，从映射的角度出发，考虑对粗糙集理论的推广使其能解决在信息变化下对集合的近似问题．最后给出了新近似算子的一些基本性质．
Fuzzy analysis is an important part in the theory of fuzzy mathematics.It is one ofthe most focused problems in the fuzzy research community.This doctoral dissertationstudies several problems which are closely related with each other in fuzzy analysis.Inaddition,a cross-disciplinary research between fuzzy analysis and rough set theory is alsogiven as the application of fuzzy analysis.The dissertation consists of four parts with sixchapters:
Part 1 contributes to the generalizations of fuzzy real numbers and fuzzy complexanalysis.Firstly some properties of fuzzy real numbers are given.Next,by counterexam-ples,it is shown that there are some errors in the definitions of the fundamental concepts offuzzy complex analysis,fuzzy complex number and generalized complex fuzzy number.After modifying the two definitions,some important results of fuzzy complex analysis areobtained again.At last,the existing definitions of derivatives for fuzzy complex functionsare investigated,and some main results in literature are improved.All of these facts showthat although the study of fuzzy complex analysis was launched by Buckley in 1989,it isstill in the initiatory phase,and needs further study.
Part 2 is to study fuzzy convex analysis and its generalizations.By using fuzzysimplex,several equivalent descriptions for the fuzzy convex hull are presented.As thegeneralizations of the fuzzy convexity,fuzzy general starshaped sets and fuzzy pseudo-starshaped sets are derived from the concepts of quasi-convex fuzzy sets and convex fuzzysets.Then,the exact relationships among the concepts of fuzzy starshaped sets,fuzzyquasi-starshaped sets,fuzzy pseudo-starshaped sets and fuzzy general starshaped sets areclarified.In the meantime,a detailed study on the basic properties of these different typesof starshapedness is also given.The shadows of fuzzy sets are investigated and severalimportant results on the shadows of starshaped fuzzy sets are yielded.Two conceptsconcerning convex fuzzy process are clarified,and some basic properties of convex fuzzyprocesses are presented.These results improve some known results about the connectionbetween these fuzzy mappings and their graphs.Finally,the correlation and difference oftwo definitions of s-convex fuzzy process are investigated.Some necessary and sufficientconditions for s-convex fuzzy processes and for s-convex fuzzy mappings are given.
Part 3 is devoted to investigating the metric spaces of fuzzy sets and the commonfixed point theory for fuzzy mappings.Firstly,a classical result about the space ofnonempty compact sets with the Hausdorff metric is generalized.By using this result,the completeness of(?)(X) with respect to the compactness of the metric space X isestablished,where(?)(X) is the class of fuzzy sets with nonempty compactα-cut sets,equipped with the supremum metric d_∞which takes the supremum on the Hausdorff dis-tances between the correspondingα-cut sets.To extend the result,after generalizing aclassical result about the space of nonempty bounded closed sets with the Hausdorff met-ric,the completeness of(?)(X) with respect to the completeness of the metric space Xis established in a similar way,where(?)(X) is the class of fuzzy sets with nonemptybounded closedα-cut sets,equipped with the supremum metric d_∞which takes the supre-mum on the Hausdorff distances between the correspondingα-cut sets.In addition,somecommon fixed point theorems for fuzzy mappings in the two spaces are proved and anexample is given to illustrate the validity of the main results in fixed point theory.
Part 4 is a cross-disciplinary research between fuzzy analysis and rough set theory.On the one hand,by using rough set theory,the concept of rough function is generalizedto rough mapping and various theoretic properties are exploited to characterize the roughmappings.On the other hand,from the mapping view,the rough set theory is extended inorder to solve the approximation problem under the changing information.Finally someproperties about the approximation operators are discussed.
第一部分研究了模糊实数的推广和模糊复分析．首先给出了模糊实数的一些性质．接下来，由反例说明模糊复分析中最基本概念的定义(模糊复数和一般模糊复数)存在的一些问题．重新修正模糊复分析中这两个基本概念后，一些重要的结论再一次被得到．最后，分析了现有的几种模糊复变函数的导数定义，改进了一些已有的重要结论．所有的这些事实说明尽管对模糊复分析Buckley1989年就开始了研究，但是其仍然处于起步阶段，需要进一步的深入研究．
第二部分研究了模糊凸分析及其推广．利用模糊单形，给出了模糊凸包的几个等价刻画．作为模糊凸性的推广，从模糊拟凸集和模糊凸集中衍生出一般模糊星状集和模糊伪星状集这两个新的概念．接着讨论了模糊星状集、模糊拟星状集、模糊伪星状集和一般模糊星状集之间的关系．同时，详细地给出了这些不同类型的星状性的基本性质．讨论了模糊星状集的落影问题并得到了一些重要结论．澄清了两个关于模糊凸过程的定义并得到了模糊凸过程的一些基本性质．这些性质改进了关于这两类模糊映射与其图像之间的关系的已有结论．最后，指出了两种s-凸模糊过程之间的联系和区别并给出了s-凸模糊过程和s-凸模糊映射的一些充要条件．
第三部分研究了模糊集组成的距离空间和模糊映射族的公共不动点理论．首先推广了关于全体非空紧集在Hausdorff距离下形成的空间中的一个经典结果，并运用此结果建立了紧距离空间的全体具有非空紧截集的模糊集合以d_∞-距离形成的空间(?)(X)的完备性．为了进一步推广这一结果，在推广了关于全体非空有界闭集在Hausdorff距离下形成的空间中的一个经典结果后，用类似的方法得到对完备距离空间X的全体具有非空有界闭截集的模糊集合引入由Hausdorff距离诱导出的上确界距离，即d_∞-距离形成的空间(?)(X)的完备性．此外，也分别讨论了这两个空间中模糊映射族的公共不动点问题，并举例说明了不动点理论的有效性．
第四部分是模糊分析和粗糙集理论的结合研究．一方面，运用粗糙集理论推广粗糙函数为粗糙映射并给出了粗糙映射的各种理论性质；另一方面，从映射的角度出发，考虑对粗糙集理论的推广使其能解决在信息变化下对集合的近似问题．最后给出了新近似算子的一些基本性质．
Fuzzy analysis is an important part in the theory of fuzzy mathematics.It is one ofthe most focused problems in the fuzzy research community.This doctoral dissertationstudies several problems which are closely related with each other in fuzzy analysis.Inaddition,a cross-disciplinary research between fuzzy analysis and rough set theory is alsogiven as the application of fuzzy analysis.The dissertation consists of four parts with sixchapters:
Part 1 contributes to the generalizations of fuzzy real numbers and fuzzy complexanalysis.Firstly some properties of fuzzy real numbers are given.Next,by counterexam-ples,it is shown that there are some errors in the definitions of the fundamental concepts offuzzy complex analysis,fuzzy complex number and generalized complex fuzzy number.After modifying the two definitions,some important results of fuzzy complex analysis areobtained again.At last,the existing definitions of derivatives for fuzzy complex functionsare investigated,and some main results in literature are improved.All of these facts showthat although the study of fuzzy complex analysis was launched by Buckley in 1989,it isstill in the initiatory phase,and needs further study.
Part 2 is to study fuzzy convex analysis and its generalizations.By using fuzzysimplex,several equivalent descriptions for the fuzzy convex hull are presented.As thegeneralizations of the fuzzy convexity,fuzzy general starshaped sets and fuzzy pseudo-starshaped sets are derived from the concepts of quasi-convex fuzzy sets and convex fuzzysets.Then,the exact relationships among the concepts of fuzzy starshaped sets,fuzzyquasi-starshaped sets,fuzzy pseudo-starshaped sets and fuzzy general starshaped sets areclarified.In the meantime,a detailed study on the basic properties of these different typesof starshapedness is also given.The shadows of fuzzy sets are investigated and severalimportant results on the shadows of starshaped fuzzy sets are yielded.Two conceptsconcerning convex fuzzy process are clarified,and some basic properties of convex fuzzyprocesses are presented.These results improve some known results about the connectionbetween these fuzzy mappings and their graphs.Finally,the correlation and difference oftwo definitions of s-convex fuzzy process are investigated.Some necessary and sufficientconditions for s-convex fuzzy processes and for s-convex fuzzy mappings are given.
Part 3 is devoted to investigating the metric spaces of fuzzy sets and the commonfixed point theory for fuzzy mappings.Firstly,a classical result about the space ofnonempty compact sets with the Hausdorff metric is generalized.By using this result,the completeness of(?)(X) with respect to the compactness of the metric space X isestablished,where(?)(X) is the class of fuzzy sets with nonempty compactα-cut sets,equipped with the supremum metric d_∞which takes the supremum on the Hausdorff dis-tances between the correspondingα-cut sets.To extend the result,after generalizing aclassical result about the space of nonempty bounded closed sets with the Hausdorff met-ric,the completeness of(?)(X) with respect to the completeness of the metric space Xis established in a similar way,where(?)(X) is the class of fuzzy sets with nonemptybounded closedα-cut sets,equipped with the supremum metric d_∞which takes the supre-mum on the Hausdorff distances between the correspondingα-cut sets.In addition,somecommon fixed point theorems for fuzzy mappings in the two spaces are proved and anexample is given to illustrate the validity of the main results in fixed point theory.
Part 4 is a cross-disciplinary research between fuzzy analysis and rough set theory.On the one hand,by using rough set theory,the concept of rough function is generalizedto rough mapping and various theoretic properties are exploited to characterize the roughmappings.On the other hand,from the mapping view,the rough set theory is extended inorder to solve the approximation problem under the changing information.Finally someproperties about the approximation operators are discussed.
引文
[1] Y. Abe, T. Kubota, H. Yoneguchi. Some properties of a set of points in Euclidean space. K(?)dai Math. Sem. Rept., 1950, 2: 117-119
[2] H. Adibi, Y.J. Cho, D. O'Regan, R. Saadati. Common fixed point theorems in (?)-fuzzy metric spaces. Appl. Math. Comput., 2006, 182: 820-828
[3] L. V. Ahlfors. Complex Analysis, 3rd edition. New York: McGraw-Hill, 1979
[4] M. Amemiya, W. Takahashi. Generalization of shadows and fixed point theorems for fuzzy sets.Fuzzy Sets and Systems, 2000, 114: 469-476
[5] E. Ammar, J. Metz. On fuzzy convexity and parametric fuzzy optimization. Fuzzy Sets and Systems, 1992,49: 135-141
[6] E. Ammar. Some properties of convex fuzzy sets and convex fuzzy cones. Fuzzy Sets and Systems, 1999, 106: 381-386
[7] J.P. Aubin. Applied Abstract Analysis. New York: Wiley, 1977
[8] J.P. Aubin, I. Ekeland. Applied Nonlinear Analysis. New York: Wiley, 1984
[9] J. P. Aubin, H. Frankowska. Set-valued analysis. Boston: Birkhauser, 1990
[10] D. Az(?), J.N.Corvellec, R.E. Lucchetti. Variational pairs and applications to stability in nons-mooth analysis. Nonlinear Analysis TMA, 2002,49: 643-670
[11] T. Bag, S.K. Samanta. Fixed point theorems on fuzzy normed linear spaces. Information Sciences, 2004, 176: 2910-2931
[12] T. Bag, S.K. Samanta. Some fixed point theorems in fuzzy normed linear spaces. Information Sciences, 2007, 177: 3271-3289
[13] B. Bede, S.G. Gal. Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets and Systems, 2005, 151: 581-599
[14] B. Bede. A note on "two-point boundary value problems associated with non-linear fuzzy differential equations". Fuzzy Sets and Systems, 2006, 157: 986-989
[15] P. Bhattacharya, A. Rosenfeld. α-Convexity. Pattern Recognition Letters, 2000, 21: 955-957
[16] I. Bose. Deciding the financial health of dot-coms using rough sets. Information and Management, 2006, 43: 835-846
[17]M.Breen.A Krasnosel'ski(?)-type theorem for unions of two starshaped sets in the plane.Pacific J.Math.,1985,120:19-31
[18]J.G.Brown.A note on fuzzy sets.Inform.and Control,1971,18:32-39
[19]J.J.Buckley.Fuzzy complex numbers.Fuzzy Sets and Systems,1989,33:333-345
[20]J.J.Buckley,Y.Qu.Solving fuzzy equations:A new solution concept.Fuzzy Sets and Systems,1991,39:291-301
[21]J.J.Buckley,Y.Qu.Fuzzy complex analysis Ⅰ:Differentiation.Fuzzy Sets and Systems,1991,41:269-284
[22]J.J.Buckley.Fuzzy complex analysis Ⅱ:Integration.Fuzzy Sets and Systems,1992,49:171-179
[23]M.Burgin.Neoclassical analysis:fuzzy continuity and convergence.Fuzzy Sets and Systems,1995,75:291-299
[24]M.Burgin.General approach to continuity measures.Fuzzy Sets and Systems,1999,105:225-231
[25]M.Burgin,M.Kalina.Fuzzy conditional convergence and nearness relations.Fuzzy Sets and Systems,2005,149:383-398
[26]蔡井伟，陈世联．集值映射的近似算子及其性质．昆明理工大学学报，2006，31：122-124
[27]J.Cel.Representations of starshaped sets in normed linear spaces.Journal of Functional Analysis,2000,174:264-273
[28]Y.Chalco-Cano,M.A.Rojas-Medar,R.Osuna-Gomez.s-Convex fuzzy processes.Computers Math.Appl.,2004,47:1411-1418
[29]C.C.Chan.Incremental learning of production rules from examples under uncertainty:a rough set approach.International Journal of Software Engineering and Knowledge Engineering,1991,1:439-461
[30]J.C.Chang,H.J.Chen,S.M.Shyu,W.C.Lian.Fixed-point theorems in fuzzy real line.Computers Math.Appl.,2004,47:845-851
[31]S.L.Chang,L.A.Zadeh.On fuzzy mappings and control.IEEE Trans.Systems and Cybernet,1972,2:30-34
[32]S.S.Chang,Y.J.Cho,B.S.Lee,J.S.Jung,S.M.Kang.Coincidence point and minimization theorems in fuzzy metric spaces.Fuzzy Sets and Systems,1997,88:119-128
[33]J.Chanussot,I.Nystr(?)m,N.Sladoje.Shape signatures of fuzzy star-shaped sets based on distance from the centroid.Pattern Recognition Letters,2005,26:735-746
[34] J. Chen, Y.R. Syau, C.J. Ting. Convexity and semicontinuity of fuzzy sets. Fuzzy Sets and Systems, 2004, 143: 459-469
[35] C.B. Cheng. Fuzzy process control: construction of control charts with fuzzy numbers. Fuzzy Sets and Systems, 2005,154: 287-303
[36] G.P. Crespi, I. Ginchev, M. Rocca. Minty variational inequalities, increase-along-rays property and optimization. J. Optim. Theory Appl., 2004, 123: 479-496
[37] G.P. Crespi, I. Ginchev, M. Rocca. Existence of solutions and star-shapedness in Minty variational inequalities. J. Global Optim., 2005, 32: 485-494
[38] G. Debreu. Integration of correspondences. Proc 5th Berkeley Symp. Math. Statist. Probab.,1967,2:351-372
[39] Z.K. Deng. Fuzzy pseudo-metric spaces. J. Math. Anal. Appl., 1982, 86: 74-95
[40] P. Diamond, P. Kloeden. Characterization of compact subsets of fuzzy sets. Fuzzy Sets and Systems, 1989, 29: 341-348
[41] P. Diamond, P. Kloeden. Metric spaces of fuzzy sets. Fuzzy Sets and Systems, 1990,35: 241-249
[42] P. Diamond. A note on fuzzy starshaped sets. Fuzzy Sets and Systems, 1990, 37: 193-199
[43] P. Diamond, P. Kloeden, A. Pokrovskii. Absolute retracts and a general fixed point theorem for fuzzy sets. Fuzzy Sets and Systems, 1997, 86: 377-380
[44] P. Diamond. Brief note on the variation of constants formula for fuzzy differential equations.Fuzzy Sets and Systems, 2002, 129: 65-71
[45] D. Dubois, H. Prade. Fuzzy sets and systems: Theory and applications. New York: Academic Press, 1980
[46] D. Dubois, H. Prade. Towards fuzzy differential calculus. Part 1: Integration of fuzzy mappings. Fuzzy Sets and Systems, 1982, 8: 1-17
[47] D. Dubois, H. Prade. Towards fuzzy differential calculus. Part 3: Differentiation. Fuzzy Sets and Systems, 1982, 8: 225-233
[48] D. Dubois, H. Prade. On several definitions of the differential of a fuzzy mapping. Fuzzy Sets and Systems, 1987,24: 117-120
[49] D. Dubois, H. Prade. Putting rough sets and fuzzy sets together. In R. Slowinski (ed.), Intelligent Decision Support: Handbook of Applications and Advances of the Rough Sets Theory, Boston: Kluwer Academic Publishers, 1992, 203-232
[50] F. Durante, P. Sarkoci. A note on the convex combinations of triangular norms. Fuzzy Sets and Systems, 2008, 159: 77-80
[51]Y.P.Fang,N.J.Huang.Increasing-along-rays property,vector optimization and well-posedness.Math.Methods Oper.Res.,2007,65:99-114
[52]H.R.Flores,A.F.Franulic.A note on projection of fuzzy sets on hyperplanes.Proyecciones,2001,20:339-349
[53]I.Friedman,T.Hastie,R.Tihshirani.The Elements of Statistical Learning:Data Mining,Inference,and Prediction.Heidelberg:Springer-Verlag,2001
[54]W.Fulton.Algebraic Topology:A First Course.New York:Srpinger-Verlag,1995
[55]P.Goodey,W.Weil.Average section functions for star-shaped sets.Advances in Applied Mathematics,2006,36:70-84
[56]G.H.Greco.A characterization of relatively compact sets of fuzzy sets.Nonlinear Analysis TMA,2006,64:518-529
[57]G.H.Greco,M.P.Moschen.Supremum metric and relatively compact sets of fuzzy sets.Nonlinear Analysis TMA,2006,64:1325-1335
[58]V.Gregori,A.Sapena.On fixed point theorem in fuzzy metric spaces.Fuzzy Sets and Systems,2002,125:245-252
[59]郭嗣琮．基于结构元理论的模糊数学分析原理．沈阳：东北大学出版社，2004
[60]哈明虎，吴从炘．模糊测度与模糊积分理论．北京：科学出版社，1998
[61]J.Harding,C.Walker,E.Walker.Lattices of convex normal functions.Fuzzy Sets and Systems,2008,159:1061-1071
[62]F.Hausdorff.Set Theory.New York:Chelsea,1957
[63]H.J.AM.Heijmans.Morphological Image Operators.Boston:Academic Press,1994
[64]J.B.Hiriart-Urruty,C.Lemar(?)chal.Fundamentals of convex analysis.Berlin:Springer-Verlag,2001
[65]N.V.Hop.Solving fuzzy (stochastic) linear programming problems using superiority and inferiority measures.Information Sciences,2007,177:1977-1991
[66]N.V.Hop.Solving linear programming problems under fuzziness and randomness environment using attainment values.Information Sciences,2007,177:2971-2984
[67]W.L.Hung,M.S.Yang.Fuzzy clustering on LR-type fuzzy numbers with an application in Taiwanese tea evaluation.Fuzzy Sets and Systems,2005,150:561-577
[68]K.D.Jamison,W.A.Lodwick.Fuzzy linear programming using a penalty method.Fuzzy Sets and Systems,2001,119:97-110
[69]S.Y.Joo,Y.K.Kim.The Skorokhodtopology on space of fuzzy numbers.Fuzzy Sets and Systems,2000,111:497-501
[70]S.Y.Joo,Y.K.Kim.Topological properties on the space of fuzzy sets.J.Math.Anal.Appl.,2000,246:576-590
[71]O.Kaleva,S.Seikkala.On fuzzy metric spaces.Fuzzy Sets and Systems,1984,12:215-229
[72]O.Kaleva.On the convergence of fuzzy sets.Fuzzy Sets and Systems,1985,17:53-65
[73]O.Kaleva.Fuzzy differential equations.Fuzzy Sets and Systems,1987,24:301-317
[74]M.Kalina.Fuzzy smoothness and sequences of fuzzy smooth functions.Fuzzy Sets and Systems,1999,105:233-239
[75]A.K.Katsaras,D.B.Liu.Fuzzy vector spaces and fuzzy topological vector spaces.J.Math.Anal.Appl.,1977,58:135-147
[76]Y.K.Kim.Compactness and convexity on the space of fuzzy sets.J.Math.Anal.Appl.,2001,264:122-132
[77]Y.K.Kim.Compactness and convexity on the space of fuzzy sets Ⅱ.Nonlinear Analysis TMA,2004,57:639-653
[78]E.P.Klement,M.L.Puri,D.A.Ralescu.Limit theorems for fuzzy random variables.Proc.Roy.Soc.London Ser.A,1986,407:171-182
[79]P.Kloeden.Compact supported endographs and fuzzy sets.Fuzzy Sets and Systems,1980,4:193-201
[80]P.E.Kloeden.Remarks on Peano theorem for fuzzy differential equations.Fuzzy Sets and Systems,1991,44:161-163
[81]W.Kloesgen,J.Zylkow.Handbook of Knowledge Discovery and Data Mining.Oxford:Oxford University Press,2002
[82]M.A.Krasnosel'ski(?).Sur un Crit(?)re pour pu'un domain soit (?)toi(?).Mat.Sb.,N.S.,1946,19:309-310
[83]B.Kuipers.Qualitative simulation.In R.E.Meyers (ed.),Encyclopedia of Physical Science and Technology,3rd edition,New York:Academic Press,2001
[84]C.Kuratoski.Topologie I.Warsaw:Monografie Matematyczne,1975
[85]李雷，吴从炘．集值分析．北京：科学出版社，2003
[86]Y.M.Liu.Some properties of convex fuzzy sets,J.Math.Anal.Appl.,1985,111:119-129
[87] W.A. Lodwick. Analysis of structure in fuzzy linear programs. Fuzzy Sets and Systems, 1990,38: 15-26
[88] W.A. Lodwick, K.D. Jamison. Theoretical and semantic distinctions of fuzzy, possibilistic, and mixed fuzzy/possibilistic optimization. Fuzzy Sets and Systems, 2007,158: 1861-1872
[89] R. Lowen. Convex fuzzy sets. Fuzzy Sets and Systems, 1980, 3: 291-310
[90] 马生全,曹纯。模糊复分析法。北京：民族出版社,2001
[91] G. Matheron. Random Sets and Integral Geometry. New York: Wiley, 1975
[92] M. Mat/oka. Convex fuzzy processes. Fuzzy Sets and Systems, 2000, 110: 109-114
[93] B.S. Mordukhovich, Y. Shao. Fuzzy calculus for coderivatives of multifunctions. Nonlinear Analysis TMA, 1997, 29: 605-626
[94] S.B. Nadler Jr. Multi-valued contraction mappings. Pacific J. Math., 1969, 30: 475-487
[95] A. Nakamura, A. Rosenfeld. Digital calculus. Information Sciences, 1997, 98: 63-98
[96] C.V. Negoita, D. Ralescu. Representation theorems for fuzzy concepts. Kybernetes, 1975, 4:169-174
[97] C.V. Negoita, D. Ralescu. Applications of fuzzy sets to systems analysis. Basel: Birkauser, 1975
[98] M. Ousmane, C. Wu. Semi-continuity of complex fuzzy function. Tsinghua Science and Technology, 2003, 8: 65-70
[99] M. Papa, H.M. Tai, S. Shenoi. Design and evaluation of control systems using cell mapping techniques. IEEE Control Systems, 1997, 17: 52-45
[100] Z. Pawlak. Rough sets. Internal Journal of Computer and Information Science, 1982, 11: 341-356
[101] Z. Pawlak. Rough functions. Bull. Polish Acad. Sci. Tech., 1987, 35: 249-251
[102] Z. Pawlak. Rough sets: Theoretical Aspects of Reasoning about Data. Boston: Kluwer Academic Publishers, 1991
[103] Z. Pawlak. Rough Sets, Rough Real Functions. Volume 50 of ICS WUT Reports, Institute of Computer Science, Warsaw University of Technology, Warsaw, 1994
[104] Z. Pawlak. Rough Real Functions and Rough Controllers. Volume 1 of ICS WUT Reports, Institute of Computer Science, Warsaw University of Technology, Warsaw, 1995
[105] Z. Pawlak. On Some Issues Connected with Roughly Continuous Functions. Volume 21 of ICS WUT Reports, Institute of Computer Science, Warsaw University of Technology, Warsaw, 1995
[106] Z. Pawlak. On Rough Derivatives, Rough Integrals and Rough Differential Equations. Volume 41 of ICS WUT Reports, Institute of Computer Science, Warsaw University of Technology, Warsaw, 1995
[107] Z. Pawlak. Rough Calculus. Volume 58 of ICS WUT Reports, Institute of Computer Science,Warsaw University of Technology, Warsaw, 1995
[108] Z. Pawlak. Rough sets, rough relations and rough functions. Fundamenta Informaticae, 1996,27:103-108
[109] Z. Pawlak. Rough real functions and rough controllers. In T.Y. Lin, N. Cercone (ed.), Rough Sets and Data Mining: Analysis of Imprecise Daia, pages 139-147, Kluwer Academic Publishers, Boston, Dordrecht, 1997
[110] Z. Pawlak. Rough sets, rough functions and rough calculus. In S.K. Pal, A. Skowron (ed.),Rough Fuzzy Hybridization: A New Trend in Decision Making, pages 99-109, Springer Verlag,Singapore, 1999
[111] Z. Pawlak, J. Peters, A. Skowron. A rough set approach to measuring information granules.In IEEE COMPSAC'OZ Workshop on Data Mining and Rough Computing, pages 1135-1139,Oxford, GB, 2002
[112] Z. Pawlak, J. Peters, A. Skowron, Z. Suraj, S. Ramanna, M. Borkowski. Rough measures, rough integrals and sensor fusion. In M. Inuiguchi, S. Hirano, S. Tsumoto (ed.), Rough Set Theory and Granular Computing, volume 125 of Studies in Fuzziness and Soft Computing, pages 263-272,Springer-Verlag, Heidelberg, 2003
[113] Z. Pawlak, A. Skowron. Rudiments of rough sets. Information Sciences, 2007, 177: 3-27
[114] Z. Pawlak, A. Skowron. Rough sets: Some extensions. Information Sciences, 2007, 177: 28-40
[115] T. Poggio, S. Smale. The mathematics of learning: Dealing with data. Notices of the AMs,2003, 50: 537-544
[116] A.T. Popov. Convexity indicators based on fuzzy morphology. Pattern Recognition Letters,1997, 18: 259-267
[117] P.M. Pu, Y.M. Liu. Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore-Smith convergence. J. Math. Anal. Appl., 1980, 76: 571-599
[118] M.L. Puri, D.A. Ralescu. Differentials of fuzzy functions. J. Math. Anal. Appl., 1983, 91: 552-558
[119] M.L. Puri, D.A. Ralescu. The concept of normality for fuzzy random variables. Ann. Probab.,1985, 13: 1373-1379
[120] M.L. Puri, D.A. Ralescu. Fuzzy random variables. J. Math. Anal. AppI., 1986, 114: 409-422
[121] J. Qiu, C. Wu, F. Li. On the restudy of fuzzy complex analysis: Part I. The sequence and series of fuzzy complex numbers and their convergences. Fuzzy Sets and Systems, 2000,115: 445-450
[122] J. Qiu, C. Wu, F. Li. On the restudy of fuzzy complex analysis: Part n. The continuity and differentiation of fuzzy complex functions. Fuzzy Sets and Systems, 2001, 120: 517-521
[123] J. Radon. Mengen konconvexer K(?)rper die einen gemeinsamen Punkt enthalten. Math. Ann.,1921,83: 113-115
[124] R.T. Rockafellar. Convex Analysis, second printing. Princeton: Princeton University Press,1972
[125] R. Rodr(?)guez-L(?)pez, Periodic boundary value problems for impulsive fuzzy differential equations. Fuzzy Sets and Systems, 2008, 159: 1384-1409
[126] A. Rosenfeld. Continuouis functions on digital pictures. Pattern Recognition Letters, 1986, 4:177-184
[127] G. Salvatore, M. Benedetto, R. Slowinski. Rough approximation by dominance relations. International Journal of Intelligent Systems, 2002, 17: 153-171
[128] F.G. Shi, C.Y. Zheng. Metrization theorems in L-topological spaces. Fuzzy Sets and Systems,2005, 149: 455-471
[129] A. Shveidel. Separability of star-shaped sets and its application to an optimization problem.Optimization, 1997,40: 207-227 [130] R. Slowinski, D. Vanderpooten. A Generalized definition of rough approximations based on similarity. IEEE Trans. Knowledge and Data Engineering, 2000, 12: 331-336
[131] S. Song, C. Wu. Existence and uniqueness of solutions to the Cauchy problem of fuzzy differential equations. Fuzzy Sets and Systems, 2000,110: 55-67
[132] Y.R. Syau. Closed and convex fuzzy sets. Fuzzy Sets and Systems, 2000, 110: 287-291
[133] P. V. Subrahmanyan. A common fixed point theorem in fuzzy metric spaces. Information Sciences, 1995, 83: 109-112
[134] P. Ter(?)n. An embedding theorem for convex fuzzy sets. Fuzzy Sets and Systems, 2005, 152:191-208
[135] F.A. Toranzos. The points of local nonconvexity of starshaped sets. Pacific J. Math., 1982, 101:209-213
[136] E. Trillas. On the use of words and fuzzy sets. Information Sciences, 2006, 176: 1463-1487
[137]D.Vorobiev,S.Seikkala.Towards the theory of fuzzy differential equations.Fuzzy Sets and Systems,2002,125:231-237
[138]G.Wang,X.Li.Generalized Lebesgue integrals of fuzzy complex valued functions.Fuzzy Sets and Systems,2002,127:363-370
[139]G.Wang,C.Wu.Fuzzy n-cell numbers and the differential of fuzzy n-cell number value mappings.Fuzzy Sets and Systems,2002,130:367-381
[140]G.Wang,Y.Li,C.Wen.On fuzzy n-cell numbers and n-dimension fuzzy vectors.Fuzzy Sets and Systems,2007,158:71-84
[141]吴从炘，马明．模糊分析学基础．北京：国防工业出版社，1991
[142]吴从炘，马明，方锦暄．模糊分析学的结构理论．贵阳：贵州科技出版社，1994
[143]C.Wu,S.Song,E.Stanley Lee.Approximate solutions,existence and uniqueness of the Cauchy problem of fuzzy differential equations.J.Math.Anal.Appl.,1996,202:629-644
[144]C.Wu,J.Qiu.Some remarks for fuzzy complex analysis.Fuzzy Sets and Systems,1999,106:231-238
[145]C.Wu,Z.Zhao.Some notes on the characterization of compact sets of fuzzy sets with L_p metric.Fuzzy Sets and Systems,2008,159:2104-2115
[146]S.Y.Wu,W.h.Cheng.A note on fuzzy convexity.Applied Mathematics Letters,2004,17:1127-1133
[147]X.M.Yang,F.M.Yang.A property on convex fuzzy sets.Fuzzy Sets and Systems,2002,126:269-271
[148]X.H.Yuan,E.S.Lee.The definition of convex fuzzy subset.Computers Math.Appl.,2004,47:101-113
[149]L.A.Zadeh.Fuzzy sets.Inform.and Control,1965,8:338-353
[150]L.A.Zadeh.Shadows of fuzzy sets.Problems Inform.Transmission,1966,2:37-44
[151]L.A.Zadeh.The concept of linguistic variable and its application to approximate teasoning-Part Ⅰ.Information Sciences,1975,8:199-249
[152]张文修，吴伟志，梁吉业，李德玉．粗糙集理论与方法．北京：科学出版社，2001
[2] H. Adibi, Y.J. Cho, D. O'Regan, R. Saadati. Common fixed point theorems in (?)-fuzzy metric spaces. Appl. Math. Comput., 2006, 182: 820-828
[3] L. V. Ahlfors. Complex Analysis, 3rd edition. New York: McGraw-Hill, 1979
[4] M. Amemiya, W. Takahashi. Generalization of shadows and fixed point theorems for fuzzy sets.Fuzzy Sets and Systems, 2000, 114: 469-476
[5] E. Ammar, J. Metz. On fuzzy convexity and parametric fuzzy optimization. Fuzzy Sets and Systems, 1992,49: 135-141
[6] E. Ammar. Some properties of convex fuzzy sets and convex fuzzy cones. Fuzzy Sets and Systems, 1999, 106: 381-386
[7] J.P. Aubin. Applied Abstract Analysis. New York: Wiley, 1977
[8] J.P. Aubin, I. Ekeland. Applied Nonlinear Analysis. New York: Wiley, 1984
[9] J. P. Aubin, H. Frankowska. Set-valued analysis. Boston: Birkhauser, 1990
[10] D. Az(?), J.N.Corvellec, R.E. Lucchetti. Variational pairs and applications to stability in nons-mooth analysis. Nonlinear Analysis TMA, 2002,49: 643-670
[11] T. Bag, S.K. Samanta. Fixed point theorems on fuzzy normed linear spaces. Information Sciences, 2004, 176: 2910-2931
[12] T. Bag, S.K. Samanta. Some fixed point theorems in fuzzy normed linear spaces. Information Sciences, 2007, 177: 3271-3289
[13] B. Bede, S.G. Gal. Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets and Systems, 2005, 151: 581-599
[14] B. Bede. A note on "two-point boundary value problems associated with non-linear fuzzy differential equations". Fuzzy Sets and Systems, 2006, 157: 986-989
[15] P. Bhattacharya, A. Rosenfeld. α-Convexity. Pattern Recognition Letters, 2000, 21: 955-957
[16] I. Bose. Deciding the financial health of dot-coms using rough sets. Information and Management, 2006, 43: 835-846
[17]M.Breen.A Krasnosel'ski(?)-type theorem for unions of two starshaped sets in the plane.Pacific J.Math.,1985,120:19-31
[18]J.G.Brown.A note on fuzzy sets.Inform.and Control,1971,18:32-39
[19]J.J.Buckley.Fuzzy complex numbers.Fuzzy Sets and Systems,1989,33:333-345
[20]J.J.Buckley,Y.Qu.Solving fuzzy equations:A new solution concept.Fuzzy Sets and Systems,1991,39:291-301
[21]J.J.Buckley,Y.Qu.Fuzzy complex analysis Ⅰ:Differentiation.Fuzzy Sets and Systems,1991,41:269-284
[22]J.J.Buckley.Fuzzy complex analysis Ⅱ:Integration.Fuzzy Sets and Systems,1992,49:171-179
[23]M.Burgin.Neoclassical analysis:fuzzy continuity and convergence.Fuzzy Sets and Systems,1995,75:291-299
[24]M.Burgin.General approach to continuity measures.Fuzzy Sets and Systems,1999,105:225-231
[25]M.Burgin,M.Kalina.Fuzzy conditional convergence and nearness relations.Fuzzy Sets and Systems,2005,149:383-398
[26]蔡井伟，陈世联．集值映射的近似算子及其性质．昆明理工大学学报，2006，31：122-124
[27]J.Cel.Representations of starshaped sets in normed linear spaces.Journal of Functional Analysis,2000,174:264-273
[28]Y.Chalco-Cano,M.A.Rojas-Medar,R.Osuna-Gomez.s-Convex fuzzy processes.Computers Math.Appl.,2004,47:1411-1418
[29]C.C.Chan.Incremental learning of production rules from examples under uncertainty:a rough set approach.International Journal of Software Engineering and Knowledge Engineering,1991,1:439-461
[30]J.C.Chang,H.J.Chen,S.M.Shyu,W.C.Lian.Fixed-point theorems in fuzzy real line.Computers Math.Appl.,2004,47:845-851
[31]S.L.Chang,L.A.Zadeh.On fuzzy mappings and control.IEEE Trans.Systems and Cybernet,1972,2:30-34
[32]S.S.Chang,Y.J.Cho,B.S.Lee,J.S.Jung,S.M.Kang.Coincidence point and minimization theorems in fuzzy metric spaces.Fuzzy Sets and Systems,1997,88:119-128
[33]J.Chanussot,I.Nystr(?)m,N.Sladoje.Shape signatures of fuzzy star-shaped sets based on distance from the centroid.Pattern Recognition Letters,2005,26:735-746
[34] J. Chen, Y.R. Syau, C.J. Ting. Convexity and semicontinuity of fuzzy sets. Fuzzy Sets and Systems, 2004, 143: 459-469
[35] C.B. Cheng. Fuzzy process control: construction of control charts with fuzzy numbers. Fuzzy Sets and Systems, 2005,154: 287-303
[36] G.P. Crespi, I. Ginchev, M. Rocca. Minty variational inequalities, increase-along-rays property and optimization. J. Optim. Theory Appl., 2004, 123: 479-496
[37] G.P. Crespi, I. Ginchev, M. Rocca. Existence of solutions and star-shapedness in Minty variational inequalities. J. Global Optim., 2005, 32: 485-494
[38] G. Debreu. Integration of correspondences. Proc 5th Berkeley Symp. Math. Statist. Probab.,1967,2:351-372
[39] Z.K. Deng. Fuzzy pseudo-metric spaces. J. Math. Anal. Appl., 1982, 86: 74-95
[40] P. Diamond, P. Kloeden. Characterization of compact subsets of fuzzy sets. Fuzzy Sets and Systems, 1989, 29: 341-348
[41] P. Diamond, P. Kloeden. Metric spaces of fuzzy sets. Fuzzy Sets and Systems, 1990,35: 241-249
[42] P. Diamond. A note on fuzzy starshaped sets. Fuzzy Sets and Systems, 1990, 37: 193-199
[43] P. Diamond, P. Kloeden, A. Pokrovskii. Absolute retracts and a general fixed point theorem for fuzzy sets. Fuzzy Sets and Systems, 1997, 86: 377-380
[44] P. Diamond. Brief note on the variation of constants formula for fuzzy differential equations.Fuzzy Sets and Systems, 2002, 129: 65-71
[45] D. Dubois, H. Prade. Fuzzy sets and systems: Theory and applications. New York: Academic Press, 1980
[46] D. Dubois, H. Prade. Towards fuzzy differential calculus. Part 1: Integration of fuzzy mappings. Fuzzy Sets and Systems, 1982, 8: 1-17
[47] D. Dubois, H. Prade. Towards fuzzy differential calculus. Part 3: Differentiation. Fuzzy Sets and Systems, 1982, 8: 225-233
[48] D. Dubois, H. Prade. On several definitions of the differential of a fuzzy mapping. Fuzzy Sets and Systems, 1987,24: 117-120
[49] D. Dubois, H. Prade. Putting rough sets and fuzzy sets together. In R. Slowinski (ed.), Intelligent Decision Support: Handbook of Applications and Advances of the Rough Sets Theory, Boston: Kluwer Academic Publishers, 1992, 203-232
[50] F. Durante, P. Sarkoci. A note on the convex combinations of triangular norms. Fuzzy Sets and Systems, 2008, 159: 77-80
[51]Y.P.Fang,N.J.Huang.Increasing-along-rays property,vector optimization and well-posedness.Math.Methods Oper.Res.,2007,65:99-114
[52]H.R.Flores,A.F.Franulic.A note on projection of fuzzy sets on hyperplanes.Proyecciones,2001,20:339-349
[53]I.Friedman,T.Hastie,R.Tihshirani.The Elements of Statistical Learning:Data Mining,Inference,and Prediction.Heidelberg:Springer-Verlag,2001
[54]W.Fulton.Algebraic Topology:A First Course.New York:Srpinger-Verlag,1995
[55]P.Goodey,W.Weil.Average section functions for star-shaped sets.Advances in Applied Mathematics,2006,36:70-84
[56]G.H.Greco.A characterization of relatively compact sets of fuzzy sets.Nonlinear Analysis TMA,2006,64:518-529
[57]G.H.Greco,M.P.Moschen.Supremum metric and relatively compact sets of fuzzy sets.Nonlinear Analysis TMA,2006,64:1325-1335
[58]V.Gregori,A.Sapena.On fixed point theorem in fuzzy metric spaces.Fuzzy Sets and Systems,2002,125:245-252
[59]郭嗣琮．基于结构元理论的模糊数学分析原理．沈阳：东北大学出版社，2004
[60]哈明虎，吴从炘．模糊测度与模糊积分理论．北京：科学出版社，1998
[61]J.Harding,C.Walker,E.Walker.Lattices of convex normal functions.Fuzzy Sets and Systems,2008,159:1061-1071
[62]F.Hausdorff.Set Theory.New York:Chelsea,1957
[63]H.J.AM.Heijmans.Morphological Image Operators.Boston:Academic Press,1994
[64]J.B.Hiriart-Urruty,C.Lemar(?)chal.Fundamentals of convex analysis.Berlin:Springer-Verlag,2001
[65]N.V.Hop.Solving fuzzy (stochastic) linear programming problems using superiority and inferiority measures.Information Sciences,2007,177:1977-1991
[66]N.V.Hop.Solving linear programming problems under fuzziness and randomness environment using attainment values.Information Sciences,2007,177:2971-2984
[67]W.L.Hung,M.S.Yang.Fuzzy clustering on LR-type fuzzy numbers with an application in Taiwanese tea evaluation.Fuzzy Sets and Systems,2005,150:561-577
[68]K.D.Jamison,W.A.Lodwick.Fuzzy linear programming using a penalty method.Fuzzy Sets and Systems,2001,119:97-110
[69]S.Y.Joo,Y.K.Kim.The Skorokhodtopology on space of fuzzy numbers.Fuzzy Sets and Systems,2000,111:497-501
[70]S.Y.Joo,Y.K.Kim.Topological properties on the space of fuzzy sets.J.Math.Anal.Appl.,2000,246:576-590
[71]O.Kaleva,S.Seikkala.On fuzzy metric spaces.Fuzzy Sets and Systems,1984,12:215-229
[72]O.Kaleva.On the convergence of fuzzy sets.Fuzzy Sets and Systems,1985,17:53-65
[73]O.Kaleva.Fuzzy differential equations.Fuzzy Sets and Systems,1987,24:301-317
[74]M.Kalina.Fuzzy smoothness and sequences of fuzzy smooth functions.Fuzzy Sets and Systems,1999,105:233-239
[75]A.K.Katsaras,D.B.Liu.Fuzzy vector spaces and fuzzy topological vector spaces.J.Math.Anal.Appl.,1977,58:135-147
[76]Y.K.Kim.Compactness and convexity on the space of fuzzy sets.J.Math.Anal.Appl.,2001,264:122-132
[77]Y.K.Kim.Compactness and convexity on the space of fuzzy sets Ⅱ.Nonlinear Analysis TMA,2004,57:639-653
[78]E.P.Klement,M.L.Puri,D.A.Ralescu.Limit theorems for fuzzy random variables.Proc.Roy.Soc.London Ser.A,1986,407:171-182
[79]P.Kloeden.Compact supported endographs and fuzzy sets.Fuzzy Sets and Systems,1980,4:193-201
[80]P.E.Kloeden.Remarks on Peano theorem for fuzzy differential equations.Fuzzy Sets and Systems,1991,44:161-163
[81]W.Kloesgen,J.Zylkow.Handbook of Knowledge Discovery and Data Mining.Oxford:Oxford University Press,2002
[82]M.A.Krasnosel'ski(?).Sur un Crit(?)re pour pu'un domain soit (?)toi(?).Mat.Sb.,N.S.,1946,19:309-310
[83]B.Kuipers.Qualitative simulation.In R.E.Meyers (ed.),Encyclopedia of Physical Science and Technology,3rd edition,New York:Academic Press,2001
[84]C.Kuratoski.Topologie I.Warsaw:Monografie Matematyczne,1975
[85]李雷，吴从炘．集值分析．北京：科学出版社，2003
[86]Y.M.Liu.Some properties of convex fuzzy sets,J.Math.Anal.Appl.,1985,111:119-129
[87] W.A. Lodwick. Analysis of structure in fuzzy linear programs. Fuzzy Sets and Systems, 1990,38: 15-26
[88] W.A. Lodwick, K.D. Jamison. Theoretical and semantic distinctions of fuzzy, possibilistic, and mixed fuzzy/possibilistic optimization. Fuzzy Sets and Systems, 2007,158: 1861-1872
[89] R. Lowen. Convex fuzzy sets. Fuzzy Sets and Systems, 1980, 3: 291-310
[90] 马生全,曹纯。模糊复分析法。北京：民族出版社,2001
[91] G. Matheron. Random Sets and Integral Geometry. New York: Wiley, 1975
[92] M. Mat/oka. Convex fuzzy processes. Fuzzy Sets and Systems, 2000, 110: 109-114
[93] B.S. Mordukhovich, Y. Shao. Fuzzy calculus for coderivatives of multifunctions. Nonlinear Analysis TMA, 1997, 29: 605-626
[94] S.B. Nadler Jr. Multi-valued contraction mappings. Pacific J. Math., 1969, 30: 475-487
[95] A. Nakamura, A. Rosenfeld. Digital calculus. Information Sciences, 1997, 98: 63-98
[96] C.V. Negoita, D. Ralescu. Representation theorems for fuzzy concepts. Kybernetes, 1975, 4:169-174
[97] C.V. Negoita, D. Ralescu. Applications of fuzzy sets to systems analysis. Basel: Birkauser, 1975
[98] M. Ousmane, C. Wu. Semi-continuity of complex fuzzy function. Tsinghua Science and Technology, 2003, 8: 65-70
[99] M. Papa, H.M. Tai, S. Shenoi. Design and evaluation of control systems using cell mapping techniques. IEEE Control Systems, 1997, 17: 52-45
[100] Z. Pawlak. Rough sets. Internal Journal of Computer and Information Science, 1982, 11: 341-356
[101] Z. Pawlak. Rough functions. Bull. Polish Acad. Sci. Tech., 1987, 35: 249-251
[102] Z. Pawlak. Rough sets: Theoretical Aspects of Reasoning about Data. Boston: Kluwer Academic Publishers, 1991
[103] Z. Pawlak. Rough Sets, Rough Real Functions. Volume 50 of ICS WUT Reports, Institute of Computer Science, Warsaw University of Technology, Warsaw, 1994
[104] Z. Pawlak. Rough Real Functions and Rough Controllers. Volume 1 of ICS WUT Reports, Institute of Computer Science, Warsaw University of Technology, Warsaw, 1995
[105] Z. Pawlak. On Some Issues Connected with Roughly Continuous Functions. Volume 21 of ICS WUT Reports, Institute of Computer Science, Warsaw University of Technology, Warsaw, 1995
[106] Z. Pawlak. On Rough Derivatives, Rough Integrals and Rough Differential Equations. Volume 41 of ICS WUT Reports, Institute of Computer Science, Warsaw University of Technology, Warsaw, 1995
[107] Z. Pawlak. Rough Calculus. Volume 58 of ICS WUT Reports, Institute of Computer Science,Warsaw University of Technology, Warsaw, 1995
[108] Z. Pawlak. Rough sets, rough relations and rough functions. Fundamenta Informaticae, 1996,27:103-108
[109] Z. Pawlak. Rough real functions and rough controllers. In T.Y. Lin, N. Cercone (ed.), Rough Sets and Data Mining: Analysis of Imprecise Daia, pages 139-147, Kluwer Academic Publishers, Boston, Dordrecht, 1997
[110] Z. Pawlak. Rough sets, rough functions and rough calculus. In S.K. Pal, A. Skowron (ed.),Rough Fuzzy Hybridization: A New Trend in Decision Making, pages 99-109, Springer Verlag,Singapore, 1999
[111] Z. Pawlak, J. Peters, A. Skowron. A rough set approach to measuring information granules.In IEEE COMPSAC'OZ Workshop on Data Mining and Rough Computing, pages 1135-1139,Oxford, GB, 2002
[112] Z. Pawlak, J. Peters, A. Skowron, Z. Suraj, S. Ramanna, M. Borkowski. Rough measures, rough integrals and sensor fusion. In M. Inuiguchi, S. Hirano, S. Tsumoto (ed.), Rough Set Theory and Granular Computing, volume 125 of Studies in Fuzziness and Soft Computing, pages 263-272,Springer-Verlag, Heidelberg, 2003
[113] Z. Pawlak, A. Skowron. Rudiments of rough sets. Information Sciences, 2007, 177: 3-27
[114] Z. Pawlak, A. Skowron. Rough sets: Some extensions. Information Sciences, 2007, 177: 28-40
[115] T. Poggio, S. Smale. The mathematics of learning: Dealing with data. Notices of the AMs,2003, 50: 537-544
[116] A.T. Popov. Convexity indicators based on fuzzy morphology. Pattern Recognition Letters,1997, 18: 259-267
[117] P.M. Pu, Y.M. Liu. Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore-Smith convergence. J. Math. Anal. Appl., 1980, 76: 571-599
[118] M.L. Puri, D.A. Ralescu. Differentials of fuzzy functions. J. Math. Anal. Appl., 1983, 91: 552-558
[119] M.L. Puri, D.A. Ralescu. The concept of normality for fuzzy random variables. Ann. Probab.,1985, 13: 1373-1379
[120] M.L. Puri, D.A. Ralescu. Fuzzy random variables. J. Math. Anal. AppI., 1986, 114: 409-422
[121] J. Qiu, C. Wu, F. Li. On the restudy of fuzzy complex analysis: Part I. The sequence and series of fuzzy complex numbers and their convergences. Fuzzy Sets and Systems, 2000,115: 445-450
[122] J. Qiu, C. Wu, F. Li. On the restudy of fuzzy complex analysis: Part n. The continuity and differentiation of fuzzy complex functions. Fuzzy Sets and Systems, 2001, 120: 517-521
[123] J. Radon. Mengen konconvexer K(?)rper die einen gemeinsamen Punkt enthalten. Math. Ann.,1921,83: 113-115
[124] R.T. Rockafellar. Convex Analysis, second printing. Princeton: Princeton University Press,1972
[125] R. Rodr(?)guez-L(?)pez, Periodic boundary value problems for impulsive fuzzy differential equations. Fuzzy Sets and Systems, 2008, 159: 1384-1409
[126] A. Rosenfeld. Continuouis functions on digital pictures. Pattern Recognition Letters, 1986, 4:177-184
[127] G. Salvatore, M. Benedetto, R. Slowinski. Rough approximation by dominance relations. International Journal of Intelligent Systems, 2002, 17: 153-171
[128] F.G. Shi, C.Y. Zheng. Metrization theorems in L-topological spaces. Fuzzy Sets and Systems,2005, 149: 455-471
[129] A. Shveidel. Separability of star-shaped sets and its application to an optimization problem.Optimization, 1997,40: 207-227 [130] R. Slowinski, D. Vanderpooten. A Generalized definition of rough approximations based on similarity. IEEE Trans. Knowledge and Data Engineering, 2000, 12: 331-336
[131] S. Song, C. Wu. Existence and uniqueness of solutions to the Cauchy problem of fuzzy differential equations. Fuzzy Sets and Systems, 2000,110: 55-67
[132] Y.R. Syau. Closed and convex fuzzy sets. Fuzzy Sets and Systems, 2000, 110: 287-291
[133] P. V. Subrahmanyan. A common fixed point theorem in fuzzy metric spaces. Information Sciences, 1995, 83: 109-112
[134] P. Ter(?)n. An embedding theorem for convex fuzzy sets. Fuzzy Sets and Systems, 2005, 152:191-208
[135] F.A. Toranzos. The points of local nonconvexity of starshaped sets. Pacific J. Math., 1982, 101:209-213
[136] E. Trillas. On the use of words and fuzzy sets. Information Sciences, 2006, 176: 1463-1487
[137]D.Vorobiev,S.Seikkala.Towards the theory of fuzzy differential equations.Fuzzy Sets and Systems,2002,125:231-237
[138]G.Wang,X.Li.Generalized Lebesgue integrals of fuzzy complex valued functions.Fuzzy Sets and Systems,2002,127:363-370
[139]G.Wang,C.Wu.Fuzzy n-cell numbers and the differential of fuzzy n-cell number value mappings.Fuzzy Sets and Systems,2002,130:367-381
[140]G.Wang,Y.Li,C.Wen.On fuzzy n-cell numbers and n-dimension fuzzy vectors.Fuzzy Sets and Systems,2007,158:71-84
[141]吴从炘，马明．模糊分析学基础．北京：国防工业出版社，1991
[142]吴从炘，马明，方锦暄．模糊分析学的结构理论．贵阳：贵州科技出版社，1994
[143]C.Wu,S.Song,E.Stanley Lee.Approximate solutions,existence and uniqueness of the Cauchy problem of fuzzy differential equations.J.Math.Anal.Appl.,1996,202:629-644
[144]C.Wu,J.Qiu.Some remarks for fuzzy complex analysis.Fuzzy Sets and Systems,1999,106:231-238
[145]C.Wu,Z.Zhao.Some notes on the characterization of compact sets of fuzzy sets with L_p metric.Fuzzy Sets and Systems,2008,159:2104-2115
[146]S.Y.Wu,W.h.Cheng.A note on fuzzy convexity.Applied Mathematics Letters,2004,17:1127-1133
[147]X.M.Yang,F.M.Yang.A property on convex fuzzy sets.Fuzzy Sets and Systems,2002,126:269-271
[148]X.H.Yuan,E.S.Lee.The definition of convex fuzzy subset.Computers Math.Appl.,2004,47:101-113
[149]L.A.Zadeh.Fuzzy sets.Inform.and Control,1965,8:338-353
[150]L.A.Zadeh.Shadows of fuzzy sets.Problems Inform.Transmission,1966,2:37-44
[151]L.A.Zadeh.The concept of linguistic variable and its application to approximate teasoning-Part Ⅰ.Information Sciences,1975,8:199-249
[152]张文修，吴伟志，梁吉业，李德玉．粗糙集理论与方法．北京：科学出版社，2001
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