This paper deals with the class of Q-tensors, that is, a Q-tensor is a real tensor ,4 such that the tensor complementarity problem （q, A）： finding an x ∈R^n such that x ≥ 0, q＋Axm-1 ≥ 0, and xT（q＋Ax^m-1） = 0...This paper deals with the class of Q-tensors, that is, a Q-tensor is a real tensor ,4 such that the tensor complementarity problem （q, A）： finding an x ∈R^n such that x ≥ 0, q＋Axm-1 ≥ 0, and xT（q＋Ax^m-1） = 0, has a solution for each vector q ∈R^n. Several subclasses of Q-tensors are given： F-tensors, R-tensors, strictly semi-positive tensors and semi-positive R0-tensors. We prove that a nonnegative tensor is a Q-tensor if and only if all of its principal diagonal entries are positive, and so the equivalence of Q-tensor, R-tensors, strictly semi-positive tensors was showed if they are nonnegative tensors. We also show that a tensor is an R0-tensor if and only if the tensor complementarity problem （0, A） has no non-zero vector solution, and a tensor is a R-tensor if and only if it is an R0-tensor and the tensor complementarity problem （e,A） has no non-zero vector solution, where e = （1, 1…. , 1）T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
基金supported by the National Natural Science Foundation of China(Grant No.11571095,11601134)the Hong Kong Research Grant Council(Grant No.PolyU502111,501212,501913 and 15302114)
文摘This paper deals with the class of Q-tensors, that is, a Q-tensor is a real tensor ,4 such that the tensor complementarity problem （q, A）： finding an x ∈R^n such that x ≥ 0, q＋Axm-1 ≥ 0, and xT（q＋Ax^m-1） = 0, has a solution for each vector q ∈R^n. Several subclasses of Q-tensors are given： F-tensors, R-tensors, strictly semi-positive tensors and semi-positive R0-tensors. We prove that a nonnegative tensor is a Q-tensor if and only if all of its principal diagonal entries are positive, and so the equivalence of Q-tensor, R-tensors, strictly semi-positive tensors was showed if they are nonnegative tensors. We also show that a tensor is an R0-tensor if and only if the tensor complementarity problem （0, A） has no non-zero vector solution, and a tensor is a R-tensor if and only if it is an R0-tensor and the tensor complementarity problem （e,A） has no non-zero vector solution, where e = （1, 1…. , 1）T
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.
