As a discrete analogue of Aleksandrov＇s projection theorem, it is natural to ask the following question： Can an o-symmetric convex lattice set of the integer lattice Z n be uniquely determined by its lattice projec...As a discrete analogue of Aleksandrov＇s projection theorem, it is natural to ask the following question： Can an o-symmetric convex lattice set of the integer lattice Z n be uniquely determined by its lattice projection counts？ In 2005, Gardner, Gronchi and Zong discovered a counterexample with cardinality 11 in the plane. In this paper, we will show that it is the only counterexample in Z2, up to unimodular transformations and with cardinality not larger than 17.展开更多
Rough set theory, proposed by Pawlak in 1982, is a tool for dealing with uncertainty and vagueness aspects of knowledge model. The main idea of rough sets corresponds to the lower and upper approximations based on equ...Rough set theory, proposed by Pawlak in 1982, is a tool for dealing with uncertainty and vagueness aspects of knowledge model. The main idea of rough sets corresponds to the lower and upper approximations based on equivalence relations. This paper studies the rough set and its extension. In our talk, we present a linear algebra approach to rough set and its extension, give an equivalent definition of the lower and upper approximations of rough set based on the characteristic function of sets, and then we explain the lower and upper approximations as the colinear map and linear map of sets, respectively. Finally, we define the rough sets over fuzzy lattices, which cover the rough set and fuzzy rough set,and the independent axiomatic systems are constructed to characterize the lower and upper approximations of rough set over fuzzy lattices,respectively,based on inner and outer products. The axiomatic systems unify the axiomization of Pawlak’s rough sets and fuzzy rough sets.展开更多
In this paper, we consider a discrete version of Aleksandrov＇s projection theorem. We prove that an origin-symmetric convex lattice set, whose lattice＇s y-coordinates＇ absolute values are not bigger than 2, can be ...In this paper, we consider a discrete version of Aleksandrov＇s projection theorem. We prove that an origin-symmetric convex lattice set, whose lattice＇s y-coordinates＇ absolute values are not bigger than 2, can be uniquely determined by its lattice projection counts if its cardinality is not 11. This partly answers a question on the discrete version of Aleksandrov＇s projection theorem which was proposed by Gardner, Gronchi and Zong in 2005.展开更多
文摘As a discrete analogue of Aleksandrov＇s projection theorem, it is natural to ask the following question： Can an o-symmetric convex lattice set of the integer lattice Z n be uniquely determined by its lattice projection counts？ In 2005, Gardner, Gronchi and Zong discovered a counterexample with cardinality 11 in the plane. In this paper, we will show that it is the only counterexample in Z2, up to unimodular transformations and with cardinality not larger than 17.
文摘Rough set theory, proposed by Pawlak in 1982, is a tool for dealing with uncertainty and vagueness aspects of knowledge model. The main idea of rough sets corresponds to the lower and upper approximations based on equivalence relations. This paper studies the rough set and its extension. In our talk, we present a linear algebra approach to rough set and its extension, give an equivalent definition of the lower and upper approximations of rough set based on the characteristic function of sets, and then we explain the lower and upper approximations as the colinear map and linear map of sets, respectively. Finally, we define the rough sets over fuzzy lattices, which cover the rough set and fuzzy rough set,and the independent axiomatic systems are constructed to characterize the lower and upper approximations of rough set over fuzzy lattices,respectively,based on inner and outer products. The axiomatic systems unify the axiomization of Pawlak’s rough sets and fuzzy rough sets.
基金Supported by National 973 Project (Grant No.2011CB808003)National Natural Science Foundation ofChina (Grant No.11131001)
文摘In this paper, we consider a discrete version of Aleksandrov＇s projection theorem. We prove that an origin-symmetric convex lattice set, whose lattice＇s y-coordinates＇ absolute values are not bigger than 2, can be uniquely determined by its lattice projection counts if its cardinality is not 11. This partly answers a question on the discrete version of Aleksandrov＇s projection theorem which was proposed by Gardner, Gronchi and Zong in 2005.
