An integer distance graph is a graph G(Z, D) with the integer set Z as vertexset, in which an edge joining two vertices u and v if and only if | u - v | ∈ D, where D is a setof natural numbers. Using a related theore...An integer distance graph is a graph G(Z, D) with the integer set Z as vertexset, in which an edge joining two vertices u and v if and only if | u - v | ∈ D, where D is a setof natural numbers. Using a related theorem in combinatorics and some conclusions known to us in thecoloring of the distance graph, the chromatic number _X(G) is determined in this paper that is ofthe distance graph G(Z, D) for some finite distance sets D containing {2, 3} with D = 4 andcontaining {2, 3, 5} with | D | = 5 by the method in which the combination of a few periodiccolorings.展开更多
Over a fieldF of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector spaceA[D] =A?F[D] from any pair of a commutative associative algebra,A with an identity element...Over a fieldF of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector spaceA[D] =A?F[D] from any pair of a commutative associative algebra,A with an identity element and the polynomial algebraF[D] of a commutative derivation subalgebraD ofA We prove thatA[D], as a Lie algebra (modulo its center) or as an associative algebra, is simple if and only ifA isD-simple andA[D] acts faithfully onA. Thus we obtain a lot of simple algebras.展开更多
文摘An integer distance graph is a graph G(Z, D) with the integer set Z as vertexset, in which an edge joining two vertices u and v if and only if | u - v | ∈ D, where D is a setof natural numbers. Using a related theorem in combinatorics and some conclusions known to us in thecoloring of the distance graph, the chromatic number _X(G) is determined in this paper that is ofthe distance graph G(Z, D) for some finite distance sets D containing {2, 3} with D = 4 andcontaining {2, 3, 5} with | D | = 5 by the method in which the combination of a few periodiccolorings.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 19801037) a Fund from National Education Ministry of China.
文摘Over a fieldF of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector spaceA[D] =A?F[D] from any pair of a commutative associative algebra,A with an identity element and the polynomial algebraF[D] of a commutative derivation subalgebraD ofA We prove thatA[D], as a Lie algebra (modulo its center) or as an associative algebra, is simple if and only ifA isD-simple andA[D] acts faithfully onA. Thus we obtain a lot of simple algebras.
