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 appreciate the comment from Horne,Miller and Wang on the paper‘A numerical investigation of the acoustic mode waves in a deviated borehole penetrating a transversely isotropic formation’by Liu et al.（2015）[1],f...We appreciate the comment from Horne,Miller and Wang on the paper‘A numerical investigation of the acoustic mode waves in a deviated borehole penetrating a transversely isotropic formation’by Liu et al.（2015）[1],for the opportunity to consider and clarify the group and phase velocity issue in sonic logging with a deviated borehole penetrating a VTI formation.We re-read the paper by Liu et al.[1]and did some numerical tests and analyses.展开更多
We argue that more cosmological solutions in massive gravity can be obtained if the metric tensor and the tensor ∑μv defined by Stiickelberg fields take the homogeneous and isotropic form. The standard cosmology wit...We argue that more cosmological solutions in massive gravity can be obtained if the metric tensor and the tensor ∑μv defined by Stiickelberg fields take the homogeneous and isotropic form. The standard cosmology with matter and radiation dominations in the past can be recovered and ACDM model is easily obtained. The dynamical evolution of the universe is modified at very early times.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.
文摘We appreciate the comment from Horne,Miller and Wang on the paper‘A numerical investigation of the acoustic mode waves in a deviated borehole penetrating a transversely isotropic formation’by Liu et al.（2015）[1],for the opportunity to consider and clarify the group and phase velocity issue in sonic logging with a deviated borehole penetrating a VTI formation.We re-read the paper by Liu et al.[1]and did some numerical tests and analyses.
基金Supported by the National Basic Science Program (Project 973) of China under Grant No. 2010CB833004the National Natural Science Foundation of China under Grant Nos. 10935013 and 11175270the Fundamental Research Funds for the Central Universities
文摘We argue that more cosmological solutions in massive gravity can be obtained if the metric tensor and the tensor ∑μv defined by Stiickelberg fields take the homogeneous and isotropic form. The standard cosmology with matter and radiation dominations in the past can be recovered and ACDM model is easily obtained. The dynamical evolution of the universe is modified at very early times.
