Stimulated by odd-bipartite and even-bipartite hypergraphs, we define odd-bipartite （weakly odd-bipartie） and even-bipartite （weakly even- bipartite） tensors. It is verified that all even order odd-bipartite tenso...Stimulated by odd-bipartite and even-bipartite hypergraphs, we define odd-bipartite （weakly odd-bipartie） and even-bipartite （weakly even- bipartite） tensors. It is verified that all even order odd-bipartite tensors are irreducible tensors, while all even-bipartite tensors are reducible no matter the parity of the order. Based on properties of odd-bipartite tensors, we study the relationship between the largest H-eigenvalue of a Z-tensor with nonnegative diagonal elements, and the largest H-eigenvalue of absolute tensor of that Z- tensor. When the order is even and the Z-tensor is weakly irreducible, we prove that the largest H-eigenvalue of the Z-tensor and the largest H-eigenvalue of the absolute tensor of that Z-tensor are equal, if and only if the Z-tensor is weakly odd-bipartite. Examples show the authenticity of the conclusions. Then, we prove that a symmetric Z-tensor with nonnegative diagonal entries and the absolute tensor of the Z-tensor are diagonal similar, if and only if the Z-tensor has even order and it is weakly odd-bipartite. After that, it is proved that, when an even order symmetric Z-tensor with nonnegative diagonal entries is weakly irreducible, the equality of the spectrum of the Z-tensor and the spectrum of absolute tensor of that Z-tensor, can be characterized by the equality of their spectral radii.展开更多
首先,将求解不同阶对称张量组的Z-特征值问题转化为非线性函数的极小值问题.当Newton方向与非线性函数负梯度方向夹角的余弦值小于取定的某一固定值时,对下降方向进行改进,从而提出改进的Newton-法求解不同阶对称张量组的Z-特征值.其次...首先,将求解不同阶对称张量组的Z-特征值问题转化为非线性函数的极小值问题.当Newton方向与非线性函数负梯度方向夹角的余弦值小于取定的某一固定值时,对下降方向进行改进,从而提出改进的Newton-法求解不同阶对称张量组的Z-特征值.其次,理论证明改进Newton-法是全局超线性收敛的.最后,数值实例表明,与带位移对称高阶幂法(shifted symmetric high order power method,SS-HOPM)相比,改进Newton-法能够计算出更多的Z-特征值和特征向量,且所用的时间更短.展开更多
The positive definiteness of elasticity tensors plays an important role in the elasticity theory.In this paper,we consider the bi-block symmetric tensors,which contain elasticity tensors as a subclass.First,we define ...The positive definiteness of elasticity tensors plays an important role in the elasticity theory.In this paper,we consider the bi-block symmetric tensors,which contain elasticity tensors as a subclass.First,we define the bi-block M-eigenvalue of a bi-block symmetric tensor,and show that a bi-block symmetric tensor is bi-block positive(semi)definite if and only if its smallest bi-block M-eigenvalue is(nonnegative)positive.Then,we discuss the distribution of bi-block M-eigenvalues,by which we get a sufficient condition for judging bi-block positive(semi)definiteness of the bi-block symmetric tensor involved.Particularly,we show that several classes of bi-block symmetric tensors are bi-block positive definite or bi-block positive semidefinite,including bi-block(strictly)diagonally dominant symmetric tensors and bi-block symmetric(B)B0-tensors.These give easily checkable sufficient conditions for judging bi-block positive(semi)definiteness of a bi-block symmetric tensor.As a byproduct,we also obtain two easily checkable sufficient conditions for the strong ellipticity of elasticity tensors.展开更多
We show that a connected uniform hypergraph G is odd-bipartite if and only if G has the same Laplacian and signless Laplacian Z-eigenvalues. We obtain some bounds for the largest （signless） Laplacian Z-eigenvalue of...We show that a connected uniform hypergraph G is odd-bipartite if and only if G has the same Laplacian and signless Laplacian Z-eigenvalues. We obtain some bounds for the largest （signless） Laplacian Z-eigenvalue of a hypergraph. For a k-uniform hyperstar with d edges （2d ≥ k ≥ 3）, we show that its largest （signless） Laplacian Z-eigenvalue is d.展开更多
基金Acknowledgements The authors were thankful to the anonymous relerees for the meaningful suggestions to improve the paper. The first author＇s work was supported in part by the National Natural Science Foundation of China （Grant No. 11171180） and the second author＇s work was supported by the Hong Kong Research Grant Council （Grant Nos. PolyU 502111, 501212, 501913, 15302114）.
文摘Stimulated by odd-bipartite and even-bipartite hypergraphs, we define odd-bipartite （weakly odd-bipartie） and even-bipartite （weakly even- bipartite） tensors. It is verified that all even order odd-bipartite tensors are irreducible tensors, while all even-bipartite tensors are reducible no matter the parity of the order. Based on properties of odd-bipartite tensors, we study the relationship between the largest H-eigenvalue of a Z-tensor with nonnegative diagonal elements, and the largest H-eigenvalue of absolute tensor of that Z- tensor. When the order is even and the Z-tensor is weakly irreducible, we prove that the largest H-eigenvalue of the Z-tensor and the largest H-eigenvalue of the absolute tensor of that Z-tensor are equal, if and only if the Z-tensor is weakly odd-bipartite. Examples show the authenticity of the conclusions. Then, we prove that a symmetric Z-tensor with nonnegative diagonal entries and the absolute tensor of the Z-tensor are diagonal similar, if and only if the Z-tensor has even order and it is weakly odd-bipartite. After that, it is proved that, when an even order symmetric Z-tensor with nonnegative diagonal entries is weakly irreducible, the equality of the spectrum of the Z-tensor and the spectrum of absolute tensor of that Z-tensor, can be characterized by the equality of their spectral radii.
文摘首先,将求解不同阶对称张量组的Z-特征值问题转化为非线性函数的极小值问题.当Newton方向与非线性函数负梯度方向夹角的余弦值小于取定的某一固定值时,对下降方向进行改进,从而提出改进的Newton-法求解不同阶对称张量组的Z-特征值.其次,理论证明改进Newton-法是全局超线性收敛的.最后,数值实例表明,与带位移对称高阶幂法(shifted symmetric high order power method,SS-HOPM)相比,改进Newton-法能够计算出更多的Z-特征值和特征向量,且所用的时间更短.
基金The first author’s work was supported by the National Natural Science Foundation of China(Grant No.11871051).
文摘The positive definiteness of elasticity tensors plays an important role in the elasticity theory.In this paper,we consider the bi-block symmetric tensors,which contain elasticity tensors as a subclass.First,we define the bi-block M-eigenvalue of a bi-block symmetric tensor,and show that a bi-block symmetric tensor is bi-block positive(semi)definite if and only if its smallest bi-block M-eigenvalue is(nonnegative)positive.Then,we discuss the distribution of bi-block M-eigenvalues,by which we get a sufficient condition for judging bi-block positive(semi)definiteness of the bi-block symmetric tensor involved.Particularly,we show that several classes of bi-block symmetric tensors are bi-block positive definite or bi-block positive semidefinite,including bi-block(strictly)diagonally dominant symmetric tensors and bi-block symmetric(B)B0-tensors.These give easily checkable sufficient conditions for judging bi-block positive(semi)definiteness of a bi-block symmetric tensor.As a byproduct,we also obtain two easily checkable sufficient conditions for the strong ellipticity of elasticity tensors.
基金Acknowledgements Foundation of China This work was supported by the National Natural Science （Grant Nos. 11371109, 11426075）, the Natural Science Foundation of tteilongjiang Province （No. QC2014C001）, ：and the Fundamental Research Funds for the Central Universities
文摘We show that a connected uniform hypergraph G is odd-bipartite if and only if G has the same Laplacian and signless Laplacian Z-eigenvalues. We obtain some bounds for the largest （signless） Laplacian Z-eigenvalue of a hypergraph. For a k-uniform hyperstar with d edges （2d ≥ k ≥ 3）, we show that its largest （signless） Laplacian Z-eigenvalue is d.
