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
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
In this paper we define an equivalence relation on the set of all xj in order to form a basis for a new descent algebra of Weyl groups of type A,. By means of this, we construct a new commutative and semi-simple desce...In this paper we define an equivalence relation on the set of all xj in order to form a basis for a new descent algebra of Weyl groups of type A,. By means of this, we construct a new commutative and semi-simple descent algebra for Weyl groups of type An generated by equivalence classes arising from this equivalence relation.展开更多
Complex networks are everywhere. A typical example is software network. How to measure and control coupling interactions of software components is a largely explored research problem in software network. In terms of g...Complex networks are everywhere. A typical example is software network. How to measure and control coupling interactions of software components is a largely explored research problem in software network. In terms of graph theory and linear algebra, this paper investigates a pair of coupling metrics to evaluate coupling interactions between the classes of object-oriented systems. These metrics differ from the majority of existing metrics in three aspects： Taking into account the strength that one class depends on other ones, reflecting indirect coupling, and distinguishing various coupling interaction. An empirical comparison of the novel measures with one of the most widely used coupling metrics is described. Specifically, an experiment about the relationships of this pair metrics is conducted. The result shows that software complexity derived from coupling interaction could not be accurately reflected by one dimension of coupling metric for negative correlation.展开更多
We introduce an algebraic structure allowing us to describe subgraphs of a regular rooted tree. Its elements are called structure polynomials, and they are in a one- to-one correspondence with the set of all subgraphs...We introduce an algebraic structure allowing us to describe subgraphs of a regular rooted tree. Its elements are called structure polynomials, and they are in a one- to-one correspondence with the set of all subgraphs of the tree. We define two operations, the sum and the product of structure polynomials, giving a graph interpretation of them. Then we introduce an equivalence relation between polynomials, using the action of the full automorphism group of the tree, and we count equivalence classes of subgraphs modulo this equivalence. We also prove that this action gives rise to symmetric Gelfand pairs. Finally, when the regularity degree of the tree is a prime p, we regard each level of the tree as a finite dimensional vector space over the finite field Fp, and we are able to completely characterize structure polynomials corresponding to subgraphs whose leaf set is a vector subspace.展开更多
Let T（q, D） be a self-similar （fractal） set generated by {fi（x） = 1/q（（x ＋ di）}^Ni=1 where integer q 〉 1and D = {d1, d2 dN} C R. To show the Lipschitz equivalence of T（q, D） and a dust-iik-e T（q, C）, on...Let T（q, D） be a self-similar （fractal） set generated by {fi（x） = 1/q（（x ＋ di）}^Ni=1 where integer q 〉 1and D = {d1, d2 dN} C R. To show the Lipschitz equivalence of T（q, D） and a dust-iik-e T（q, C）, one general restriction is 79 C Q by Peres et al. [Israel] Math, 2000, 117： 353-379]. In this paper, we obtain several sufficient criterions for the Lipschitz equivalence of two self-similar sets by using dust-like graph-directed iterating function systems and combinatorial techniques. Several examples are given to illustrate our theory.展开更多
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.
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In this paper we define an equivalence relation on the set of all xj in order to form a basis for a new descent algebra of Weyl groups of type A,. By means of this, we construct a new commutative and semi-simple descent algebra for Weyl groups of type An generated by equivalence classes arising from this equivalence relation.
基金This research is supported by the National Key Basic Research and Development 973 Program of China under Grant No. 2007CB310805, Foundation for the Author of National Excellent Doctoral Dissertation of China under Grant No. 2007B4, the National Natural Science Foundation of China under Grant Nos. 60873083, 60803025, and the National High Technology Research and Development Program of China under Grant No. 2006AA04Z156.
文摘Complex networks are everywhere. A typical example is software network. How to measure and control coupling interactions of software components is a largely explored research problem in software network. In terms of graph theory and linear algebra, this paper investigates a pair of coupling metrics to evaluate coupling interactions between the classes of object-oriented systems. These metrics differ from the majority of existing metrics in three aspects： Taking into account the strength that one class depends on other ones, reflecting indirect coupling, and distinguishing various coupling interaction. An empirical comparison of the novel measures with one of the most widely used coupling metrics is described. Specifically, an experiment about the relationships of this pair metrics is conducted. The result shows that software complexity derived from coupling interaction could not be accurately reflected by one dimension of coupling metric for negative correlation.
文摘We introduce an algebraic structure allowing us to describe subgraphs of a regular rooted tree. Its elements are called structure polynomials, and they are in a one- to-one correspondence with the set of all subgraphs of the tree. We define two operations, the sum and the product of structure polynomials, giving a graph interpretation of them. Then we introduce an equivalence relation between polynomials, using the action of the full automorphism group of the tree, and we count equivalence classes of subgraphs modulo this equivalence. We also prove that this action gives rise to symmetric Gelfand pairs. Finally, when the regularity degree of the tree is a prime p, we regard each level of the tree as a finite dimensional vector space over the finite field Fp, and we are able to completely characterize structure polynomials corresponding to subgraphs whose leaf set is a vector subspace.
基金supported by National Natural Science Foundation of China (Grant No.10871180)
文摘Let T（q, D） be a self-similar （fractal） set generated by {fi（x） = 1/q（（x ＋ di）}^Ni=1 where integer q 〉 1and D = {d1, d2 dN} C R. To show the Lipschitz equivalence of T（q, D） and a dust-iik-e T（q, C）, one general restriction is 79 C Q by Peres et al. [Israel] Math, 2000, 117： 353-379]. In this paper, we obtain several sufficient criterions for the Lipschitz equivalence of two self-similar sets by using dust-like graph-directed iterating function systems and combinatorial techniques. Several examples are given to illustrate our theory.
