The eigenfunction method put forward by Chen Jin-quan is illustrated. We apply this theory to the space group D1_ 6h. The selection rules of this space are worked out in the points of higher symmetry A,K,H in the firs...The eigenfunction method put forward by Chen Jin-quan is illustrated. We apply this theory to the space group D1_ 6h. The selection rules of this space are worked out in the points of higher symmetry A,K,H in the first Brillion Zone. The C-G coefficients are calculated for K.HA.展开更多
In this paper, we investigate the positive solutions to the following integral system with a polyharmonic extension operator on R^+_n:{u(x)=c_n,a∫_?R_+~n(x_n^(1-a_v)(y)/|x-y|^(n-a))dy,x∈R_+~n,v(y)=c_n,a∫_R_+~n(x_n^...In this paper, we investigate the positive solutions to the following integral system with a polyharmonic extension operator on R^+_n:{u(x)=c_n,a∫_?R_+~n(x_n^(1-a_v)(y)/|x-y|^(n-a))dy,x∈R_+~n,v(y)=c_n,a∫_R_+~n(x_n^(1-a_uθ)(x)/|x-y|^(n-a))dx,y∈ ?R_+~n,where n 2, 2-n < a < 1, κ, θ > 0. This integral system arises from the Euler-Lagrange equation corresponding to an integral inequality on the upper half space established by Chen(2014). The explicit formulations of positive solutions are obtained by the method of moving spheres for the critical case κ =n-2+a/n-a,θ =n+2-a/ n-2+a. Moreover,we also give the nonexistence of positive solutions in the subcritical case for the above system.展开更多
It is known that there is a very closed connection between the set of non-isomorphic indecomposable basic Nakayama algebras and the set of admissible sequences.To determine the cardinal number of all nonisomorphic ind...It is known that there is a very closed connection between the set of non-isomorphic indecomposable basic Nakayama algebras and the set of admissible sequences.To determine the cardinal number of all nonisomorphic indecomposable basic Nakayama algebras,we describe the cardinal number of the set of all t-length admissible sequences using a new type of integers called quasi-binomial coefficients.Furthermore,we find some intrinsic relations among binomial coefficients and quasi-binomial coefficients.展开更多
In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of un...In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method, such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives. The required solutions are obtained by solving the resulting linear system. The proposed method gives a very satisfactory solution,which can be performed by any symbolic mathematical packages such as Maple, Mathematica, etc. Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m. We present an error analysis for the proposed method to emphasize its reliability. Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.展开更多
文摘The eigenfunction method put forward by Chen Jin-quan is illustrated. We apply this theory to the space group D1_ 6h. The selection rules of this space are worked out in the points of higher symmetry A,K,H in the first Brillion Zone. The C-G coefficients are calculated for K.HA.
基金supported by National Natural Science Foundation of China (Grant No. 11571268)Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2017JQ1022)the Fundamental Research Funds for the Central Universities (Grant No. GK201802015)
文摘In this paper, we investigate the positive solutions to the following integral system with a polyharmonic extension operator on R^+_n:{u(x)=c_n,a∫_?R_+~n(x_n^(1-a_v)(y)/|x-y|^(n-a))dy,x∈R_+~n,v(y)=c_n,a∫_R_+~n(x_n^(1-a_uθ)(x)/|x-y|^(n-a))dx,y∈ ?R_+~n,where n 2, 2-n < a < 1, κ, θ > 0. This integral system arises from the Euler-Lagrange equation corresponding to an integral inequality on the upper half space established by Chen(2014). The explicit formulations of positive solutions are obtained by the method of moving spheres for the critical case κ =n-2+a/n-a,θ =n+2-a/ n-2+a. Moreover,we also give the nonexistence of positive solutions in the subcritical case for the above system.
基金supported by Shandong Provincial Natural Science Foundation of China (Grant No.ZR2011AM005)National Natural Science Foundation of China (Grant No.10931006)Shanghai Municipal Natural Science Foundation (Grant No.12ZR1413200)
文摘It is known that there is a very closed connection between the set of non-isomorphic indecomposable basic Nakayama algebras and the set of admissible sequences.To determine the cardinal number of all nonisomorphic indecomposable basic Nakayama algebras,we describe the cardinal number of the set of all t-length admissible sequences using a new type of integers called quasi-binomial coefficients.Furthermore,we find some intrinsic relations among binomial coefficients and quasi-binomial coefficients.
文摘In this article, an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems. The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method, such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives. The required solutions are obtained by solving the resulting linear system. The proposed method gives a very satisfactory solution,which can be performed by any symbolic mathematical packages such as Maple, Mathematica, etc. Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m. We present an error analysis for the proposed method to emphasize its reliability. Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.
