Let A be an ruth order n-dimensional tensor, where m, n are some positive integers and N ：= re（n-1）. Then A is called a Hankel tensor associated with a vector v ∈ R^N＋1 if Aσ = Vk for each k = 0,1,...,N whenever...Let A be an ruth order n-dimensional tensor, where m, n are some positive integers and N ：= re（n-1）. Then A is called a Hankel tensor associated with a vector v ∈ R^N＋1 if Aσ = Vk for each k = 0,1,...,N whenever σ= （i1,..., im） satisfies i1 ＋... ＋ im - m ＋ k. We introduce the elementary Hankel tensors which are some special Hankel tensors, and present all the eigenvalues of the elementary Hankel tensors for k = 0, 1, 2. We also show that a convolution can be expressed as the product of some third-order elementary Hankel tensors, and a Hankel tensor can be decomposed as a convolution of two Vandermonde matrices following the definition of the convolution of tensors. Finally, we use the properties of the convolution to characterize Hankel tensors and （0,1） Hankel tensors. Keywords Tensor, convolution, Hankel tensor, elementary Hankel tensor, symmetric tensor展开更多
A general method to construct symmetric orbital (SO) is proposed. For all point groups, the corresponding SOs can be uniquely formulated. Due to the closure of SOs with regard to the multiplication operation, the dire...A general method to construct symmetric orbital (SO) is proposed. For all point groups, the corresponding SOs can be uniquely formulated. Due to the closure of SOs with regard to the multiplication operation, the direct products of SOs form an N th rank SO tensor (SOT) group. Moreover, the matrix elements between SOTs can be automatically divided into physical factors and geometric factors. Consequently, the traditional coupling coefficients can be discarded and thus the present method significantly reduces the computation efforts as compared with the irreducible tensor method.展开更多
Based on the generalized characteristic polynomial introduced by J. Canny in Generalized characteristic polynomials [J. Symbolic Comput., 1990, 9（3）： 241-250], it is immediate that for any m-order n-dimensional rea...Based on the generalized characteristic polynomial introduced by J. Canny in Generalized characteristic polynomials [J. Symbolic Comput., 1990, 9（3）： 241-250], it is immediate that for any m-order n-dimensional real tensor, the number of distinct H-eigenvalues is less than or equal to n（m-1）n-1. However, there is no known bounds on the maximal number of distinct H- eigenvectors in general. We prove that for any m ~〉 2, an m-order 2-dimensional tensor sd exists such that d has 2（m - 1） distinct H-eigenpairs. We give examples of 4-order 2-dimensional tensors with six distinct H-eigenvalues as well as six distinct H-eigenvectors. We demonstrate the structure of eigenpairs for a higher order tensor is far more complicated than that of a matrix. Further- more, we introduce a new class of weakly symmetric tensors, called p-symmetric tensors, and show under certain conditions, p-symmetry will effectively reduce the maximal number of distinct H-eigenveetors for a given two-dimensional tensor. Lastly, we provide a complete classification of the H-eigenvectors of a given 4-order 2-dimensional nonnegative p-symmetric tensor. Additionally, we give sufficient conditions which prevent a given 4-order 2-dimensional nonnegative irreducible weakly symmetric tensor from possessing six pairwise distinct H-eigenveetors.展开更多
We show that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric. Furthermore, a symmetric best rank one approximation to a symmetric tensor is uni...We show that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric. Furthermore, a symmetric best rank one approximation to a symmetric tensor is unique if the tensor does not lie on a certain real algebraic variety.展开更多
Some new criteria for identifying H-tensors are obtained. As applications, some sufficient conditions of the positive definiteness for an even- order real symmetric tensor are given, as well as a new eigenvalue inclus...Some new criteria for identifying H-tensors are obtained. As applications, some sufficient conditions of the positive definiteness for an even- order real symmetric tensor are given, as well as a new eigenvalue inclusion region for tensors is established. It is proved that the new eigenvalue inclusion region is tighter than that of Y. Yang and Q. Yang [SIAM J. Matrix Anal. Appl., 2010, 31： 2517-2530]. Numerical examples are reported to demonstrate the corresponding results.展开更多
对于球面约束下的四次型极小化问题,可使用DC(difference of convex)规划来求解。现基于近端算法对DC算法进行了一些改进,提出pDCA和aDCA两种算法,并证明了算法局部收敛以及收敛速度至少达到了次线性收敛。数值实验结果表明,与一般的DC...对于球面约束下的四次型极小化问题,可使用DC(difference of convex)规划来求解。现基于近端算法对DC算法进行了一些改进,提出pDCA和aDCA两种算法,并证明了算法局部收敛以及收敛速度至少达到了次线性收敛。数值实验结果表明,与一般的DC算法和对称移位高阶幂法相比,在计算时间和解的最优性方面都得到了很大提升。展开更多
An algorithm is presented for decomposing a symmetric tensor into a sum of rank-1 symmetric tensors. For a given tensor, by using apolarity, catalecticant matrices and the condition that the mapping matrices are commu...An algorithm is presented for decomposing a symmetric tensor into a sum of rank-1 symmetric tensors. For a given tensor, by using apolarity, catalecticant matrices and the condition that the mapping matrices are commutative, the rank of the tensor can be obtained by iteration. Then we can find the generating polynomials under a selected basis set. The decomposition can be constructed by the solutions of generating polynomials under the condition that the solutions are all distinct which can be guaranteed by the commutative property of the matrices. Numerical examples demonstrate the efficiency and accuracy of the proposed method.展开更多
The symmetry orbital-symmetry orbital tensor method is applied to the evaluation of molecular integrals (one-electron and two-electron integrals) and the symmetry-orbital-tensor and self-consistent-field (SOT-SCF) cal...The symmetry orbital-symmetry orbital tensor method is applied to the evaluation of molecular integrals (one-electron and two-electron integrals) and the symmetry-orbital-tensor and self-consistent-field (SOT-SCF) calculations. A calculation scheme is proposed to simplify the evaluation of integrals and a key equation is derived to reduce the computation efforts in SCF iterations. According to the key equation, compared with the traditional SCF method, the computation efficiencies including CPU timing and external disk (or internal memory) requirement increase in the magnitude of the square of the order of a point group. The new SOT method is expected to be useful in the theoretical calculations of large molecular systems of high point group symmetries.展开更多
In this paper some properties of a symmetric tensor field T（X,Y） = g（A（X）, Y） on a Riemannian manifold （M, g） without boundary which satisfies the S quasi-Einstein equation Rij-S/2gij=Tij＋bξiξj are given. ...In this paper some properties of a symmetric tensor field T（X,Y） = g（A（X）, Y） on a Riemannian manifold （M, g） without boundary which satisfies the S quasi-Einstein equation Rij-S/2gij=Tij＋bξiξj are given. The necessary and sufficient conditions for this tensor to satisfy the quasi-Einstein equation are also obtained.展开更多
文摘Let A be an ruth order n-dimensional tensor, where m, n are some positive integers and N ：= re（n-1）. Then A is called a Hankel tensor associated with a vector v ∈ R^N＋1 if Aσ = Vk for each k = 0,1,...,N whenever σ= （i1,..., im） satisfies i1 ＋... ＋ im - m ＋ k. We introduce the elementary Hankel tensors which are some special Hankel tensors, and present all the eigenvalues of the elementary Hankel tensors for k = 0, 1, 2. We also show that a convolution can be expressed as the product of some third-order elementary Hankel tensors, and a Hankel tensor can be decomposed as a convolution of two Vandermonde matrices following the definition of the convolution of tensors. Finally, we use the properties of the convolution to characterize Hankel tensors and （0,1） Hankel tensors. Keywords Tensor, convolution, Hankel tensor, elementary Hankel tensor, symmetric tensor
基金Supported by the National Natural Science Foundation of China(11361074)the Foundation of Science and Technology Department of Guizhou Province([2015]7206)+1 种基金the Natural Science Programs of Education Department of Guizhou Province([2015]420)the Research Foundation of Guizhou Minzu University(15XRY004)
文摘A general method to construct symmetric orbital (SO) is proposed. For all point groups, the corresponding SOs can be uniquely formulated. Due to the closure of SOs with regard to the multiplication operation, the direct products of SOs form an N th rank SO tensor (SOT) group. Moreover, the matrix elements between SOTs can be automatically divided into physical factors and geometric factors. Consequently, the traditional coupling coefficients can be discarded and thus the present method significantly reduces the computation efforts as compared with the irreducible tensor method.
文摘Based on the generalized characteristic polynomial introduced by J. Canny in Generalized characteristic polynomials [J. Symbolic Comput., 1990, 9（3）： 241-250], it is immediate that for any m-order n-dimensional real tensor, the number of distinct H-eigenvalues is less than or equal to n（m-1）n-1. However, there is no known bounds on the maximal number of distinct H- eigenvectors in general. We prove that for any m ~〉 2, an m-order 2-dimensional tensor sd exists such that d has 2（m - 1） distinct H-eigenpairs. We give examples of 4-order 2-dimensional tensors with six distinct H-eigenvalues as well as six distinct H-eigenvectors. We demonstrate the structure of eigenpairs for a higher order tensor is far more complicated than that of a matrix. Further- more, we introduce a new class of weakly symmetric tensors, called p-symmetric tensors, and show under certain conditions, p-symmetry will effectively reduce the maximal number of distinct H-eigenveetors for a given two-dimensional tensor. Lastly, we provide a complete classification of the H-eigenvectors of a given 4-order 2-dimensional nonnegative p-symmetric tensor. Additionally, we give sufficient conditions which prevent a given 4-order 2-dimensional nonnegative irreducible weakly symmetric tensor from possessing six pairwise distinct H-eigenveetors.
文摘We show that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric. Furthermore, a symmetric best rank one approximation to a symmetric tensor is unique if the tensor does not lie on a certain real algebraic variety.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant Nos. 11361074, 11326242） and the Science Foundation of the Education Department of Yunnan Province （Grant No. 2013FD002）.
文摘Some new criteria for identifying H-tensors are obtained. As applications, some sufficient conditions of the positive definiteness for an even- order real symmetric tensor are given, as well as a new eigenvalue inclusion region for tensors is established. It is proved that the new eigenvalue inclusion region is tighter than that of Y. Yang and Q. Yang [SIAM J. Matrix Anal. Appl., 2010, 31： 2517-2530]. Numerical examples are reported to demonstrate the corresponding results.
文摘对于球面约束下的四次型极小化问题,可使用DC(difference of convex)规划来求解。现基于近端算法对DC算法进行了一些改进,提出pDCA和aDCA两种算法,并证明了算法局部收敛以及收敛速度至少达到了次线性收敛。数值实验结果表明,与一般的DC算法和对称移位高阶幂法相比,在计算时间和解的最优性方面都得到了很大提升。
基金This work was supported by the National Natural Science Foundation of China （Grants Nos. 11471159, 11571169, 61661136001） and the Natural Science Foundation of Jiangsu Province （No. BK20141409）.
文摘An algorithm is presented for decomposing a symmetric tensor into a sum of rank-1 symmetric tensors. For a given tensor, by using apolarity, catalecticant matrices and the condition that the mapping matrices are commutative, the rank of the tensor can be obtained by iteration. Then we can find the generating polynomials under a selected basis set. The decomposition can be constructed by the solutions of generating polynomials under the condition that the solutions are all distinct which can be guaranteed by the commutative property of the matrices. Numerical examples demonstrate the efficiency and accuracy of the proposed method.
基金Project supported by the National Natural Science Foundation of China (Grant No. 29473119)
文摘The symmetry orbital-symmetry orbital tensor method is applied to the evaluation of molecular integrals (one-electron and two-electron integrals) and the symmetry-orbital-tensor and self-consistent-field (SOT-SCF) calculations. A calculation scheme is proposed to simplify the evaluation of integrals and a key equation is derived to reduce the computation efforts in SCF iterations. According to the key equation, compared with the traditional SCF method, the computation efficiencies including CPU timing and external disk (or internal memory) requirement increase in the magnitude of the square of the order of a point group. The new SOT method is expected to be useful in the theoretical calculations of large molecular systems of high point group symmetries.
基金The Grant-in-Aid for Scientific Research from Nanjing University of ScienceTechnology (AB41409) the NNSF (19771048) of China partly.
文摘In this paper some properties of a symmetric tensor field T（X,Y） = g（A（X）, Y） on a Riemannian manifold （M, g） without boundary which satisfies the S quasi-Einstein equation Rij-S/2gij=Tij＋bξiξj are given. The necessary and sufficient conditions for this tensor to satisfy the quasi-Einstein equation are also obtained.
