In the quotient space theory of granular computing,the universe structure is assumed to be a topology,therefore,its application is still limited.In this study,based on the quotient space model,the universe structure i...In the quotient space theory of granular computing,the universe structure is assumed to be a topology,therefore,its application is still limited.In this study,based on the quotient space model,the universe structure is assumed as an algebra instead of a topology.As to obtain the algebraic quotient operator,the granulation must be uniquely determined by a congruence relation,and all the congruence relations form a complete semi-order lattice,which is the theoretical basis of granularities ' completeness.When the given equivalence relation is not a congruence relation,it defines the concepts of upper quotient and lower quotient,and discusses some of their properties which demonstrate that falsity preserving principle and truth preserving principle are still valid.Finally,it presents the algorithms and example of upper quotient and lower quotient.The work extends the quotient space theory from structure,and provides theoretical basis for the combination of the quotient space theory and the algebra theory.展开更多
We assume that X is a normed linear space, W and M are subspaces of X. We develop a theory of best simultaneous approximation in quotient spaces and introduce equivalent assertions between the subspaces W and W ＋ M a...We assume that X is a normed linear space, W and M are subspaces of X. We develop a theory of best simultaneous approximation in quotient spaces and introduce equivalent assertions between the subspaces W and W ＋ M and the quotient space W/M.展开更多
Granular computing is a very hot research field in recent years. In our previous work an algebraic quotient space model was proposed,where the quotient structure could not be deduced if the granulation was based on an...Granular computing is a very hot research field in recent years. In our previous work an algebraic quotient space model was proposed,where the quotient structure could not be deduced if the granulation was based on an equivalence relation. In this paper,definitions were given and formulas of the lower quotient congruence and upper quotient congruence were calculated to roughly represent the quotient structure. Then the accuracy and roughness were defined to measure the quotient structure in quantification. Finally,a numerical example was given to demonstrate that the rough representation and measuring methods are efficient and applicable. The work has greatly enriched the algebraic quotient space model and granular computing theory.展开更多
The left multiplicative continuous compactification is the universal semigroup compactification of a semitopological semigroup.In this paper an internal construction of a quotient space of the left multiplicative cont...The left multiplicative continuous compactification is the universal semigroup compactification of a semitopological semigroup.In this paper an internal construction of a quotient space of the left multiplicative continuous compactification of a semitopological semigroup is constructed as a space of z-filters展开更多
A fuzzy clustering analysis model based on the quotient space is proposed. Firstly, the conversion from coarse to fine granularity and the hierarchical structure are used to reduce the multidimensional samples. Second...A fuzzy clustering analysis model based on the quotient space is proposed. Firstly, the conversion from coarse to fine granularity and the hierarchical structure are used to reduce the multidimensional samples. Secondly, the fuzzy compatibility relation matrix of the model is converted into fuzzy equivalence relation matrix. Finally, the diagram of clustering genealogy is generated according to the fuzzy equivalence relation matrix, which enables the dynamic selection of different thresholds to effectively solve the problem of cluster analysis of the samples with multi-dimensional attributes.展开更多
In this paper, we present a quotient space approximation model of multiresolution signal analysis and discuss the properties and characteristics of the model. Then the comparison between wavelet transform and the quot...In this paper, we present a quotient space approximation model of multiresolution signal analysis and discuss the properties and characteristics of the model. Then the comparison between wavelet transform and the quotient space approximation is made. First, when wavelet transform is viewed from the new quotient space approximation perspective, it may help us to gain an insight into the essence of multiresolution signal analysis. Second, from the similarity between wavelet and quotient space approximations, it is possible to transfer the rich wavelet techniques into the latter so that a new way for multiresolution analysis may be found.展开更多
A soitable data model and data structure make underground survey objects maintained and operated easier. This paper gives a formal definition for underground survey objects. By making use of the quotient topological s...A soitable data model and data structure make underground survey objects maintained and operated easier. This paper gives a formal definition for underground survey objects. By making use of the quotient topological space, the author studies the logical relations among underground survey objects, a partiallyordered space uuder some conditions. An example is given to show the data model’s possible applications.展开更多
This paper discusses the special properties of the spectrum of linear operators （in particular, bounded linear operators） on quotient indecomposable Banach spaces; shows that in such spaces generators of C0-groups a...This paper discusses the special properties of the spectrum of linear operators （in particular, bounded linear operators） on quotient indecomposable Banach spaces; shows that in such spaces generators of C0-groups are always bounded linear operators, and that generators of C0-semigroups satisfy the spectral mapping theorem; and gives an example to show that the generators of C0-semigroups in quotient indecomposable spaces are not necessarily bounded.展开更多
The problem whether every infinite dimensional Banach space has an infinite dimensional separable quotient space has remained unsolved for a long time. In this paper we prove: the Banach space X has an infinite dimens...The problem whether every infinite dimensional Banach space has an infinite dimensional separable quotient space has remained unsolved for a long time. In this paper we prove: the Banach space X has an infinite dimensional separable quotient if and only if X has an infinite dimensional separable quasicomplemented subspace, also if and only if there exists a Banach space Y and a bounded linear operator T is an element of B(Y,X such that the range of T is nonclosed and dense in X. Besides, the other relevant questions for such spaces e.g. the question on operator ideals that on H.I.(hereditarily indecomposable) spaces, that on invariant subspaces of operators, etc, are also discussed.展开更多
Let G be a graph and f: G→ G be a continuous map with at least one periodic point. Using the quote space method, the paper addresses that f is an equicontinuous map if and only if one of the following End(G)+2k+1 con...Let G be a graph and f: G→ G be a continuous map with at least one periodic point. Using the quote space method, the paper addresses that f is an equicontinuous map if and only if one of the following End(G)+2k+1 conditions holds: 1) {f jm(End(G)+2k)!}∞j=1 is uniformly convergent, in which m=1,2,…, End(G)+2k; and 2) There is a positive integer n esuring that {f jn}∞j=1 is uniformly convergent.展开更多
We prove the existence of both final L-double gradation fuzzy topological spaces and final L-double fuzzy closure spaces. From this fact,we define and study the notions of the L-double quotient spaces and the sum of L...We prove the existence of both final L-double gradation fuzzy topological spaces and final L-double fuzzy closure spaces. From this fact,we define and study the notions of the L-double quotient spaces and the sum of L-double fuzzy closure spaces. Finally,we study the additivity of two kinds of L-double fuzzy closure spaces.展开更多
This paper gives internal characterizations of some sequence covering compact images and compact covering compact images of paracompact locally compact spaces, which improve some results on compact images of locally...This paper gives internal characterizations of some sequence covering compact images and compact covering compact images of paracompact locally compact spaces, which improve some results on compact images of locally compact metric spaces.展开更多
In this note,we give some conditions concerning that the countably bi-quotient closed mappings, and the closed mappings with each fiber having a σ-closure preserving outer base, preserve M1-spaces.
基金Supported by the National Natural Science Foundation of China(No.61173052)the Natural Science Foundation of Hunan Province(No.14JJ4007)
文摘In the quotient space theory of granular computing,the universe structure is assumed to be a topology,therefore,its application is still limited.In this study,based on the quotient space model,the universe structure is assumed as an algebra instead of a topology.As to obtain the algebraic quotient operator,the granulation must be uniquely determined by a congruence relation,and all the congruence relations form a complete semi-order lattice,which is the theoretical basis of granularities ' completeness.When the given equivalence relation is not a congruence relation,it defines the concepts of upper quotient and lower quotient,and discusses some of their properties which demonstrate that falsity preserving principle and truth preserving principle are still valid.Finally,it presents the algorithms and example of upper quotient and lower quotient.The work extends the quotient space theory from structure,and provides theoretical basis for the combination of the quotient space theory and the algebra theory.
文摘We assume that X is a normed linear space, W and M are subspaces of X. We develop a theory of best simultaneous approximation in quotient spaces and introduce equivalent assertions between the subspaces W and W ＋ M and the quotient space W/M.
基金Supported by the National Natural Science Foundation of China(No.61772031)the Special Energy Saving Foundation of Changsha,Hunan Province in 2017
文摘Granular computing is a very hot research field in recent years. In our previous work an algebraic quotient space model was proposed,where the quotient structure could not be deduced if the granulation was based on an equivalence relation. In this paper,definitions were given and formulas of the lower quotient congruence and upper quotient congruence were calculated to roughly represent the quotient structure. Then the accuracy and roughness were defined to measure the quotient structure in quantification. Finally,a numerical example was given to demonstrate that the rough representation and measuring methods are efficient and applicable. The work has greatly enriched the algebraic quotient space model and granular computing theory.
文摘The left multiplicative continuous compactification is the universal semigroup compactification of a semitopological semigroup.In this paper an internal construction of a quotient space of the left multiplicative continuous compactification of a semitopological semigroup is constructed as a space of z-filters
文摘A fuzzy clustering analysis model based on the quotient space is proposed. Firstly, the conversion from coarse to fine granularity and the hierarchical structure are used to reduce the multidimensional samples. Secondly, the fuzzy compatibility relation matrix of the model is converted into fuzzy equivalence relation matrix. Finally, the diagram of clustering genealogy is generated according to the fuzzy equivalence relation matrix, which enables the dynamic selection of different thresholds to effectively solve the problem of cluster analysis of the samples with multi-dimensional attributes.
文摘In this paper, we present a quotient space approximation model of multiresolution signal analysis and discuss the properties and characteristics of the model. Then the comparison between wavelet transform and the quotient space approximation is made. First, when wavelet transform is viewed from the new quotient space approximation perspective, it may help us to gain an insight into the essence of multiresolution signal analysis. Second, from the similarity between wavelet and quotient space approximations, it is possible to transfer the rich wavelet techniques into the latter so that a new way for multiresolution analysis may be found.
文摘A soitable data model and data structure make underground survey objects maintained and operated easier. This paper gives a formal definition for underground survey objects. By making use of the quotient topological space, the author studies the logical relations among underground survey objects, a partiallyordered space uuder some conditions. An example is given to show the data model’s possible applications.
基金The NSF (10471025) of China and the NSF (Z0511019) of Fujian Province in China.
文摘This paper discusses the special properties of the spectrum of linear operators （in particular, bounded linear operators） on quotient indecomposable Banach spaces; shows that in such spaces generators of C0-groups are always bounded linear operators, and that generators of C0-semigroups satisfy the spectral mapping theorem; and gives an example to show that the generators of C0-semigroups in quotient indecomposable spaces are not necessarily bounded.
文摘The problem whether every infinite dimensional Banach space has an infinite dimensional separable quotient space has remained unsolved for a long time. In this paper we prove: the Banach space X has an infinite dimensional separable quotient if and only if X has an infinite dimensional separable quasicomplemented subspace, also if and only if there exists a Banach space Y and a bounded linear operator T is an element of B(Y,X such that the range of T is nonclosed and dense in X. Besides, the other relevant questions for such spaces e.g. the question on operator ideals that on H.I.(hereditarily indecomposable) spaces, that on invariant subspaces of operators, etc, are also discussed.
基金the Natural Science Foundation for Youth at Higher Educational Institution of Anhui Province (No: 2005jql153) and the Natural science Foundation of Anhui (2003kj080).
文摘Let G be a graph and f: G→ G be a continuous map with at least one periodic point. Using the quote space method, the paper addresses that f is an equicontinuous map if and only if one of the following End(G)+2k+1 conditions holds: 1) {f jm(End(G)+2k)!}∞j=1 is uniformly convergent, in which m=1,2,…, End(G)+2k; and 2) There is a positive integer n esuring that {f jn}∞j=1 is uniformly convergent.
文摘We prove the existence of both final L-double gradation fuzzy topological spaces and final L-double fuzzy closure spaces. From this fact,we define and study the notions of the L-double quotient spaces and the sum of L-double fuzzy closure spaces. Finally,we study the additivity of two kinds of L-double fuzzy closure spaces.
文摘This paper gives internal characterizations of some sequence covering compact images and compact covering compact images of paracompact locally compact spaces, which improve some results on compact images of locally compact metric spaces.
文摘In this note,we give some conditions concerning that the countably bi-quotient closed mappings, and the closed mappings with each fiber having a σ-closure preserving outer base, preserve M1-spaces.
