The formulations of the finite-field approach to calculate the linear and non-linear optical coefficients (i, (ij, (ijk and (ijkl of a molecular system with different symmetries have been deduced and summarized. The p...The formulations of the finite-field approach to calculate the linear and non-linear optical coefficients (i, (ij, (ijk and (ijkl of a molecular system with different symmetries have been deduced and summarized. The possible choices of the energy sets of the 48 frequent point groups have been optimized and categorized into 11 classes. With the restriction of symmetry operators, a minimum of 9, no more than 21 energy points have to be calculated in order to determine the coefficients, except in the case of the first class to which C1 point group belongs and in which the 34 non-relative energy points selected in our uniform and general scheme are all needed. The symmetric operators that cause some of the tensor components to vanish have been demonstrated as well.展开更多
Let V be a vector space over a field F and G a group of linear transformations in V. It is proved in this note that for any subspace U (V, if dimU/(U∩ g(U))≤ 1, for any g∈G, then there is a g∈ G such that U∩g(U) ...Let V be a vector space over a field F and G a group of linear transformations in V. It is proved in this note that for any subspace U (V, if dimU/(U∩ g(U))≤ 1, for any g∈G, then there is a g∈ G such that U∩g(U) is a G-invariant subspace, or there is an x∈ V\U such that U + <x> is a G-invariant subspace. So a vector-space analog of Brailovsky's results on quasi-invariant sets is given.展开更多
In this paper,we show that the McKay quiver of a finite subgroup of a general linear group is a regular covering of the McKay quiver of its intersection with the special linear group.Using this and our results on &quo...In this paper,we show that the McKay quiver of a finite subgroup of a general linear group is a regular covering of the McKay quiver of its intersection with the special linear group.Using this and our results on "returning arrows" in McKay quiver,we give an algorithm to construct the McKay quiver of a finite abelian group.Using this construction,we show how the cone and cylinder of an(n-1)-Auslander absolute n-complete algebra are truncated from the McKay quivers of abelian groups.展开更多
Let G be a finite group. The degree(vertex) graph Γ(G) attached to G is a character degree graph.Its vertices are the degrees of the nonlinear irreducible complex characters of G, and different vertices m, n are adja...Let G be a finite group. The degree(vertex) graph Γ(G) attached to G is a character degree graph.Its vertices are the degrees of the nonlinear irreducible complex characters of G, and different vertices m, n are adjacent if the greatest common divisor(m, n) > 1. In this paper, we classify all graphs with four vertices that occur as Γ(G) for nonsolvable groups G.展开更多
基金the National Science Foundation of China (69978021), Fujian Provincial National Science Foundation of China (E9910030) and State
文摘The formulations of the finite-field approach to calculate the linear and non-linear optical coefficients (i, (ij, (ijk and (ijkl of a molecular system with different symmetries have been deduced and summarized. The possible choices of the energy sets of the 48 frequent point groups have been optimized and categorized into 11 classes. With the restriction of symmetry operators, a minimum of 9, no more than 21 energy points have to be calculated in order to determine the coefficients, except in the case of the first class to which C1 point group belongs and in which the 34 non-relative energy points selected in our uniform and general scheme are all needed. The symmetric operators that cause some of the tensor components to vanish have been demonstrated as well.
基金Supported by the National Natural Science Foundations of China !(19771014) and Liaoning Province! (972208)
文摘Let V be a vector space over a field F and G a group of linear transformations in V. It is proved in this note that for any subspace U (V, if dimU/(U∩ g(U))≤ 1, for any g∈G, then there is a g∈ G such that U∩g(U) is a G-invariant subspace, or there is an x∈ V\U such that U + <x> is a G-invariant subspace. So a vector-space analog of Brailovsky's results on quasi-invariant sets is given.
基金supported by National Natural Science Foundation of China (Grant No.10971172)
文摘In this paper,we show that the McKay quiver of a finite subgroup of a general linear group is a regular covering of the McKay quiver of its intersection with the special linear group.Using this and our results on "returning arrows" in McKay quiver,we give an algorithm to construct the McKay quiver of a finite abelian group.Using this construction,we show how the cone and cylinder of an(n-1)-Auslander absolute n-complete algebra are truncated from the McKay quivers of abelian groups.
基金supported by National Natural Science Foundation of China(Grant No.10871032)
文摘Let G be a finite group. The degree(vertex) graph Γ(G) attached to G is a character degree graph.Its vertices are the degrees of the nonlinear irreducible complex characters of G, and different vertices m, n are adjacent if the greatest common divisor(m, n) > 1. In this paper, we classify all graphs with four vertices that occur as Γ(G) for nonsolvable groups G.
