Let G(V,E) be a connected graph and W{w 1,w 2,…,w k} an ordered set of V. Given v∈V, the representation of v with respect to W is the k-vector r(v|W)(d(v,w 1),d(v,w 2),…,d(v,w k)). The set W is a resolving set of G...Let G(V,E) be a connected graph and W{w 1,w 2,…,w k} an ordered set of V. Given v∈V, the representation of v with respect to W is the k-vector r(v|W)(d(v,w 1),d(v,w 2),…,d(v,w k)). The set W is a resolving set of G if r(u|W)r(v|W) implies that uv for all pairs {u,v} of vertices of G. The resolving set of G with the smallest cardinality is called a basis of G. The dimension of G, dim (G), is the cardinality of a basis for G. The bound of a Cartesian product of a connected graph H and a path P k was reached: dim(H)≤dim(H×P k)≤dim(H)+1. Then, the dimension value of some graphs was given. At last, the constructions of some graphs’ bases were showed.展开更多
Let K〈X〉 = K（X1,..., Xn） be the free K-algebra on X = {X1,..., Xn} over a field K, which is equipped with a weight N-gradation （i.e., each Xi is assigned a positive degree）, and let G be a finite homogeneous GrS...Let K〈X〉 = K（X1,..., Xn） be the free K-algebra on X = {X1,..., Xn} over a field K, which is equipped with a weight N-gradation （i.e., each Xi is assigned a positive degree）, and let G be a finite homogeneous GrSbner basis for the ideal I = （G） of K（X） with respect to some monomial ordering 〈 on K（X）. It is shown that if the monomial algebra K（X）/（LM（6）） is semiprime, where LM（6） is the set of leading monomials of 6 with respect to 〈, then the N-graded algebra A ： K（X）/I is semiprimitive in the sense of Jacobson. In the case that G is a finite nonhomogeneous Gr6bner basis with respect to a graded monomial ordering 〈gr, and the N-filtration FA of the algebra A = K（X）/I induced by the N-grading filtration FK（X） of K（X） is considered, if the monomial algebra K（X）/（LM（6）） is semiprime, then it is shown that the associated N-graded algebra G（A） and the Rees algebra A of A determined by FA are all semiprimitive.展开更多
Quantum degenerate code may improve the hashing bound of quantum capacity. We propose a family of quantum degenerate codes derived from two-colorable graphs. The coherent information of the codes is analyticaJly obtai...Quantum degenerate code may improve the hashing bound of quantum capacity. We propose a family of quantum degenerate codes derived from two-colorable graphs. The coherent information of the codes is analyticaJly obtained as a function of the channd noise for the depolarizing channel. We find a new code which has a higher noise threshold than that of the repetition code.展开更多
文摘Let G(V,E) be a connected graph and W{w 1,w 2,…,w k} an ordered set of V. Given v∈V, the representation of v with respect to W is the k-vector r(v|W)(d(v,w 1),d(v,w 2),…,d(v,w k)). The set W is a resolving set of G if r(u|W)r(v|W) implies that uv for all pairs {u,v} of vertices of G. The resolving set of G with the smallest cardinality is called a basis of G. The dimension of G, dim (G), is the cardinality of a basis for G. The bound of a Cartesian product of a connected graph H and a path P k was reached: dim(H)≤dim(H×P k)≤dim(H)+1. Then, the dimension value of some graphs was given. At last, the constructions of some graphs’ bases were showed.
基金Project supported by the National Natural Science Foundation of China （10971044）.
文摘Let K〈X〉 = K（X1,..., Xn） be the free K-algebra on X = {X1,..., Xn} over a field K, which is equipped with a weight N-gradation （i.e., each Xi is assigned a positive degree）, and let G be a finite homogeneous GrSbner basis for the ideal I = （G） of K（X） with respect to some monomial ordering 〈 on K（X）. It is shown that if the monomial algebra K（X）/（LM（6）） is semiprime, where LM（6） is the set of leading monomials of 6 with respect to 〈, then the N-graded algebra A ： K（X）/I is semiprimitive in the sense of Jacobson. In the case that G is a finite nonhomogeneous Gr6bner basis with respect to a graded monomial ordering 〈gr, and the N-filtration FA of the algebra A = K（X）/I induced by the N-grading filtration FK（X） of K（X） is considered, if the monomial algebra K（X）/（LM（6）） is semiprime, then it is shown that the associated N-graded algebra G（A） and the Rees algebra A of A determined by FA are all semiprimitive.
基金Supported by the National Natural Science Foundation of China under Grant Nos.60972071 and 11375152
文摘Quantum degenerate code may improve the hashing bound of quantum capacity. We propose a family of quantum degenerate codes derived from two-colorable graphs. The coherent information of the codes is analyticaJly obtained as a function of the channd noise for the depolarizing channel. We find a new code which has a higher noise threshold than that of the repetition code.
