反对称张量空间中可合元素的一些性质

<正> 本文讨论反对称张量空间可合元素的必要条件,用可合元素的 Plücker坐标表示相应子空间的方法以及子空间正交与可合元素之间的关系.设 V 是一个特征为 0 的域 R 上的 n 维向量空间.对于1≤m≤n,(?)V 是 m 阶     

广义矩阵函数及其指标

本文推广了广义矩阵函数及对称张量的一些定理，给出了广义矩阵函数指标的若干关系式．     

基于块循环矩阵的对称张量的最佳秩-1逼近

对称张量的最佳秩-1问题是张量研究中非常重要的部分.首先,基于三阶张量的块循环矩阵,提出了求解对称张量最佳秩-1逼近问题的一个新方法.其次,针对求解对称张量的最佳秩-1逼近方法,给出了对称张量的最佳秩-1逼近不变性的一个充要条件,以及逼近误差上界的估计.最后,数值算例表明了上述方法的可行性和误差上界的正确性.     

四阶不完全对称张量M-特征值的新包含域及应用

四阶不完全对称张量的M-特征值在非线性弹性材料分析中有着广泛的应用.本文的目的是给出四阶不完全对称张量M-特征值的新包含域,得到最大M-特征值上界更精确的估计,并将得到的上界估计值应用到计算最大M-特征值的WQZ算法中,数值例子验证了结果的有效性.最后,基于得到的包含域,给出了四阶不完全对称张量正定性判定的充分条件.     

实对称张量正定性的判定

实对称张量的正定性在自动控制系统稳定性、多项式全局优化、医疗影像降噪等问题中具有重要的应用价值.通过构造不同的正对角阵和运用不等式的放缩技巧,给出了H-张量新的判别条件.作为应用,给出了偶数阶实对称张量,即偶次齐次多项式正定性的新实用判定方法.相应数值算例表明了结果的有效性.     

H-张量的新判定及其应用

<正>1引言实对称张量具有广泛的应用背景~([1-3]),众多学者对其进行研究~([4-8]).研究实对称张量的主要问题之一是其正定性的判定~([9]),其应用在于解决多元偶次齐次多项式f(x)≡Ax~m=sum from (i_1,...,i_m=1) to n(a_(i_1…i_m)x_(i_1)…x_i_m)(m为偶数,A为实对称张量)正定性的判定问题~([10]),即f(x)0,?x=(x_1,x_2,…,x_n)~T∈R~n,x≠0(1)是否成立?问题(1)不仅是一个重要的问题,而且在许多领域具有应用.如在自动控制系     

S—R分解定理的唯一性,存在性和客观性

对于连续体的一切物理可能的变形场，其变形梯度张量F可被分解为一个对称张量S和一个正交张量R的直和，这便是S-R分解定理．本文通过矩阵方法和张量方法证明了S-R分解定理的唯一性、存在性和客观性．     

关于各向同性二阶对称张量函数势的一个定理

本文提出并证明了这样一个定理:二阶张量H为二阶对称张量T的各向同性函数,若H存在着势,则T也一定存在着势.     

利用对称正交逼近求解对称张量的MSOA算法

本文在Pan等工作的基础上,提出了一种修正的对称正交分解方法(MSOA)来逼近实对称张量.为讨论实对称张量的对称正交逼近,首先将其转化为具有等式约束的极小化问题来进行理论分析,在算法中使用自适应带位移的乘幂法来求解特征向量,同时给出了该算法的收敛性分析.最后通过数值实验验证了对该算法所做的理论分析.数值结果表明,我们提出的算法是稳健和有效的.     

求解对称张量绝对值方程问题的非光滑牛顿法

提出一类对称张量绝对值方程问题,给出了求解此类问题的一类非光滑牛顿法,并且在一般的假设条件下,给出了算法的局部收敛性.最后给出相关的数值实验表明了算法的有效性.     

Orthogonal bases of Brauer symmetry classes of tensors for the dihedral group

In this article necessary and sufficient conditions are given for the existence of an orthogonal basis consisting of standard (decomposable) symmetrized tensors for the class of tensors symmetrized using a Brauer character of the dihedral group.     

On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group

A necessary and sufficient condition for the existence of orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with the dicyclic group is given. In particular we apply these conditions to the generalized quaternion group, for which the dimensions of the symmetry classes of tensors are computed.     

On the orthogonal basis of the symmetry classes of tensors associated with the dicyclic group

A necessary and sufficient condition for the existence of orthogonal basis of decomposable symmetrized tensors for the symmetry classes of tensors associated with the dicyclic group is given. In particular we apply these conditions to the generalized quaternion group, for which the dimensions of the symmetry classes of tensors are computed.     

Lower bounds for the norms of decomposable symmetrized tensors

Lower bounds are given for the difference of two decomposable symmetrized tensors. The first bound uses a norm which makes the component vectors in a decomposable symmetrized tensor part of an orthonormal basis. The second bound holds only for decomposable elements of symmetry classes whose associated characters are linear.     

Orthogonality of cosets relative to irreducible characters of finite groups

Studied is an assumption on a group that ensures that no matter how the group is embedded in a symmetric group, the corresponding symmetrized tensor space has an orthogonal basis of standard (decomposable) symmetrized tensors.     

On the definability of completely decomposable torsion-free Abelian groups by endomorphism rings and some groups of homomorphisms

Mathematical Notes - In the present paper,we describe conditions on a vector group C which are necessary and sufficient for the class of completely decomposable torsion-free Abelian groups to be a...     

Super-Wavelets and Decomposable Wavelet Frames

A wavelet frame is called decomposable whenever it is equivalent to a superwavelet frame of length greater than one. Decomposable wavelet frames are closely related to some problems on super-wavelets. In this article we first obtain some necessary or sufficient conditions for decomposable Parseval wavelet frames. As an application of these conditions, we prove that for each n > 1 there exists a Parseval wavelet frame which is m-decomposable for any 1 < m ≤ n, but not k-decomposable for any k > n. Moreover, there exists a super-wavelet whose components are non-decomposable. Similarly we also prove that for each n > 1, there exists a Parseval wavelet frame that can be extended to a super-wavelet of length m for any 1 < m ≤ n, but can not be extended to any super-wavelet of length k with k > n. The connection between decomposable Parseval wavelet frames and super-wavelets is investigated, and some necessary or sufficient conditions for extendable Parseval wavelet frames are given.     

A high symmetry class of tensors with an orthogonal basis of decomposable symmetrized tensors

We present a high symmetry class of tensors with an orthogonal basis of decomposable symmetrized tensors, and this is a counter-example of the claim presented in [1].     

The question of the definability of Abelian groups by the centers of their endomorphism rings

We obtain necessary and sufficient conditions for an Abelian group in the class of completely decomposable torsion-free Abelian groups of a chosen finite rank to be definable in this class by the center of the endomorphism ring of the group.     

Orthogonal decomposable symmetrized tensors

In this article we present a sufficient condition for orthogonality of decompassable symmetrized tensors.