Boundary inner and outer operators are introduced, and union, intersection, complement operators of approximations are redefined. The approximation operators have a good property of maintaining union, intersection, co...Boundary inner and outer operators are introduced, and union, intersection, complement operators of approximations are redefined. The approximation operators have a good property of maintaining union, intersection, complement operators, so the rough set theory has been enriched from the operator-oriented and set-oriented views. Approximate power set spaces are defined, and it is proved that the approximation operators are epimorphisms from power set space to approximate power set spaces. Some basic properties of approximate power set space are got by epimorphisms in contrast to power set space.展开更多
According to the report of China Daily, influenced by the nuclear accidents in Japan, the expansion of China's nuclear power industry will slow from the rapid rate of the 11 th
The more unambiguous statement of the P versus NP problem and the judgement of its hardness, are the key ways to find the full proof of the P versus NP problem. There are two sub-problems in the P versus NP problem. T...The more unambiguous statement of the P versus NP problem and the judgement of its hardness, are the key ways to find the full proof of the P versus NP problem. There are two sub-problems in the P versus NP problem. The first is the classifications of different mathematical problems (languages), and the second is the distinction between a non-deterministic Turing machine (NTM) and a deterministic Turing machine (DTM). The process of an NTM can be a power set of the corresponding DTM, which proves that the states of an NTM can be a power set of the corresponding DTM. If combining this viewpoint with Cantor's theorem, it is shown that an NTM is not equipotent to a DTM. This means that "generating the power set P(A) of a set A" is a non-canonical example to support that P is not equal to NP.展开更多
The concept of an HX-group is an upgrade of the concept of a group,in which a new operation is defined on the family of non-empty subsets of a group.If this new support set together with the new operation is a group,t...The concept of an HX-group is an upgrade of the concept of a group,in which a new operation is defined on the family of non-empty subsets of a group.If this new support set together with the new operation is a group,then we call it an HX-group.On the other hand,a hyperoperation is a mapping having the same codomain as the operation of an HX-group,i.e.,the family of non-empty subsets of the initial set,but a different domain-the set itself.This could be(and was indeed)a source of confusion,which is clarified in this paper.Moreover,HX-groups naturally lead to constructions of hypergroups.The links between these two algebraic concepts are presented,with the aim of reviving the old notion of an HX-group in the current research on algebraic hyperstructures.One of such existing links and one newly established link are also discussed.展开更多
For an element A in a unital C^(*)-algebra B,the operator-valued 1-formωA(z)=(z-A)^(-1) dz is analytic on the resolvent setρ(A),which plays an important role in the functional calculus of A.This paper defines a clas...For an element A in a unital C^(*)-algebra B,the operator-valued 1-formωA(z)=(z-A)^(-1) dz is analytic on the resolvent setρ(A),which plays an important role in the functional calculus of A.This paper defines a class of Hermitian metrics onρ(A)through the coupling of the operator-valued(1,1)-formΩA=-ωA^(*)∧ωA with tracial and vector states.Its main goal is to study the connection between A and the properties of the metric concerning curvature,arc length,completeness and singularity.A particular example is when A is quasi-nilpotent,in which case the metric lives on the punctured complex plane C\{0}.The notion of the power set is defined to gauge the"blow-up"rate of the metric at 0,and examples are given to indicate a likely link with A’s hyper-invariant subspaces.展开更多
基金Supported by the National Natural Science Foundation of China (No.69803007)
文摘Boundary inner and outer operators are introduced, and union, intersection, complement operators of approximations are redefined. The approximation operators have a good property of maintaining union, intersection, complement operators, so the rough set theory has been enriched from the operator-oriented and set-oriented views. Approximate power set spaces are defined, and it is proved that the approximation operators are epimorphisms from power set space to approximate power set spaces. Some basic properties of approximate power set space are got by epimorphisms in contrast to power set space.
文摘According to the report of China Daily, influenced by the nuclear accidents in Japan, the expansion of China's nuclear power industry will slow from the rapid rate of the 11 th
文摘The more unambiguous statement of the P versus NP problem and the judgement of its hardness, are the key ways to find the full proof of the P versus NP problem. There are two sub-problems in the P versus NP problem. The first is the classifications of different mathematical problems (languages), and the second is the distinction between a non-deterministic Turing machine (NTM) and a deterministic Turing machine (DTM). The process of an NTM can be a power set of the corresponding DTM, which proves that the states of an NTM can be a power set of the corresponding DTM. If combining this viewpoint with Cantor's theorem, it is shown that an NTM is not equipotent to a DTM. This means that "generating the power set P(A) of a set A" is a non-canonical example to support that P is not equal to NP.
基金The first author acknowledges the financial support from the Slovenian Research Agency(research core funding No.P1-0285)The second author was supported by the FEKT-S-17-4225 grant of Brno University of Technology.
文摘The concept of an HX-group is an upgrade of the concept of a group,in which a new operation is defined on the family of non-empty subsets of a group.If this new support set together with the new operation is a group,then we call it an HX-group.On the other hand,a hyperoperation is a mapping having the same codomain as the operation of an HX-group,i.e.,the family of non-empty subsets of the initial set,but a different domain-the set itself.This could be(and was indeed)a source of confusion,which is clarified in this paper.Moreover,HX-groups naturally lead to constructions of hypergroups.The links between these two algebraic concepts are presented,with the aim of reviving the old notion of an HX-group in the current research on algebraic hyperstructures.One of such existing links and one newly established link are also discussed.
文摘For an element A in a unital C^(*)-algebra B,the operator-valued 1-formωA(z)=(z-A)^(-1) dz is analytic on the resolvent setρ(A),which plays an important role in the functional calculus of A.This paper defines a class of Hermitian metrics onρ(A)through the coupling of the operator-valued(1,1)-formΩA=-ωA^(*)∧ωA with tracial and vector states.Its main goal is to study the connection between A and the properties of the metric concerning curvature,arc length,completeness and singularity.A particular example is when A is quasi-nilpotent,in which case the metric lives on the punctured complex plane C\{0}.The notion of the power set is defined to gauge the"blow-up"rate of the metric at 0,and examples are given to indicate a likely link with A’s hyper-invariant subspaces.
