A direct proof of uniqueness of square-root of a positive semi-definite tensor

Understanding of the basic properties of the positive semi-definite tensor is a prerequisite for its extensive applications in theoretical and practical fields, especially for its square-root. Uniqueness of the square-root of a positive semi-definite tensor is proven in this paper without resorting to the notion of eigenvalues, eigenvectors and the spectral decomposition of the second-order symmetric tensor.

Positive definiteness of fourth-order partially symmetric tensors

In this paper,we consider the positive definiteness of fourth-order partially symmetric tensors.First,two analytically sufficient and necessary conditions of positive definiteness are provided for fourth-order two dimensional partially symmetric tensors.Then,we obtain several sufficient conditions for rank-one positive definiteness of fourth-order three dimensional partially symmetric tensors.

ON SHRINKING GRADIENT RICCI SOLITONS WITH POSITIVE RICCI CURVATURE AND SMALL WEYL TENSOR

We show that closed shrinking gradient Ricci solitons with positive Ricci curvature and sufficiently pinched Weyl tensor are Einstein. When Weyl tensor vanishes, this has been proved before but our proof here is much simpler.

<i>H</i>-Singular Value of a Positive Tensor

In this paper we study properties of H-singular values of a positive tensor and present an iterative algorithm for computing the largest H-singular value of the positive tensor. We prove that this method?converges for any positive tensors.

The Path-Positive Property on the Products of Graphs

The products of graphs discussed in this paper are the following four kinds: the Cartesian product of graphs, the tensor product of graphs, the lexicographic product of graphs and the strong direct product of graphs. It is proved that:① If the graphs G 1 and G 2 are the connected graphs, then the Cartesian product, the lexicographic product and the strong direct product in the products of graphs, are the path positive graphs. ② If the tensor product is a path positive graph if and only if the graph G 1 and G 2 are the connected graphs, and the graph G 1 or G 2 has an odd cycle and max{ λ 1μ 1,λ nμ m}≥2 in which λ 1 and λ n [ or μ 1 and μ m] are maximum and minimum characteristic values of graph G 1 [ or G 2 ], respectively.

Calculation and correction of magnetic object positioning error caused by magnetic field gradient tensor measurement

Magnetic field gradient tensor measurement is an important technique to obtain position information of magnetic objects. When using magnetic field sensors to measure magnetic field gradient as the coefficients of tensor, field differentiation is generally approximated by field difference. As a result, magnetic objects positioning by magnetic field gradient tensor measurement always involves an inherent error caused by sensor sizes, leading to a reduction in detectable distance and detectable angle. In this paper, the inherent positioning error caused by magnetic field gradient tensor measurement is calculated and corrected by iterations based on the systematic position error distribution patterns. The results show that, the detectable distance range and the angle range of an ac magnetic object（2.44 Am^2@1 kHz） can be increased from（0.45 m, 0.75 m）,（0？, 25？） to（0.30 m, 0.80 m）,（0？,80？）, respectively.

基于伪磁梯度张量混合方向tilt梯度的磁源定位方法及其应用:以内蒙古塔木素铀矿床为例

斜磁化磁源的准确定位是磁法勘探有效圈定场源分布及划分断裂构造的重要依据.但受磁化方向影响,总磁异常与地质体分布并无明显对应关系.为此,本文从总磁异常垂向积分出发,推导出了伪磁梯度张量,定义了伪磁梯度张量的混合方向tilt梯度及混合方向解析信号振幅,提出了基于伪磁梯度张量混合方向tilt梯度水平导数组合的场源定位方法.模型试验表明,相对于解析信号振幅、tilt梯度的总水平导数及方向tilt梯度的总水平导数模这3种受磁化方向影响较小的方法,本文提出的磁源定位方法能够更有效地反映出不同磁化方向的磁源位置,具有更强的识别能力和定位精度.在内蒙古塔木素铀矿床的航磁资料应用中,新方法较好地展示出了已知断裂带的展布及出露岩浆岩的分布,同时有效地提取了塔木素铀矿床上方的弱磁异常信息,处理结果为研究区圈定隐伏岩体、划分隐伏断裂及扩大铀矿找矿靶区提供了依据.

POSITIVE DEFINITE AND SEMI-DEFINITE SPLITTING METHODS FOR NON-HERMITIAN POSITIVE DEFINITE LINEAR SYSTEMS

In this paper, we further generalize the technique for constructing the normal （or pos- itive definite） and skew-Hermitian splitting iteration method for solving large sparse non- Hermitian positive definite system of linear equations. By introducing a new splitting, we establish a class of efficient iteration methods, called positive definite and semi-definite splitting （PPS） methods, and prove that the sequence produced by the PPS method con- verges unconditionally to the unique solution of the system. Moreover, we propose two kinds of typical practical choices of the PPS method and study the upper bound of the spectral radius of the iteration matrix. In addition, we show the optimal parameters such that the spectral radius achieves the minimum under certain conditions. Finally, some numerical examples are given to demonstrate the effectiveness of the considered methods.

THE PRIMAL-DUAL POTENTIAL REDUCTION ALGORITHM FOR POSITIVE SEMI-DEFINITE PROGRAMMING

In this paper we introduce a primal-dual potential reduction algorithm for positive semi-definite programming. Using the symetric preserving scalings for both primal and dual interior matrices, we can construct an algorithm which is very similar to the primal-dual potential reduction algorithm of Huang and Kortanek [6] for linear programming. The complexity of the algorithm is either O(nlog(X0 · S0/ε) or O(nlog(X0· S0/ε) depends on the value of ρ in the primal-dual potential function, where X0 and S0 is the initial interior matrices of the positive semi-definite programming.

Generalized Inversions of Hadamard and Tensor Products for Matrices

We shall give natural generalized solutions of Hadamard and tensor products equations for matrices by the concept of the Tikhonov regularization combined with the theory of reproducing kernels.

基于张量不变量及避免椭圆误差的定位方法研究

针对目前广泛使用的基于磁梯度张量不变量的磁性目标定位方法未考虑椭圆系数的变化而导致定位精度降低这一问题,提出一种新的单位距离矢量求解方法和距离矢量模求解方法。通过分析磁性目标与测量系统之间的位置关系,构建新的单位距离矢量,并通过麦克劳林公式和磁梯度张量不变量分别构建新的存在一阶误差与二阶误差的距离矢量模,避免未考虑椭圆系数变化而导致定位精度降低。仿真试验验证了所提方法较以前方法定位精度有明显提升,存在一阶误差与二阶误差的距离矢量模对定位精度的影响基本相同。该方法对进行探潜和未爆炸物的检测等方面具有较高的研究和应用价值。

磁梯度张量不变量约束条件下的两点定位方法

文中针对单点磁梯度张量定位方法受地磁场估计误差影响较大,同时多点磁梯度张量定位方法容易陷入局部最优解等问题,提出一种两点磁梯度张量定位方法。该方法在单点磁梯度张量定位算法的基础上,采用两点磁梯度张量测量数据,叠加张量几何不变量的约束条件,构建关于目标位置坐标的非线性目标函数,采用基于自然选择的粒子群算法对目标位置坐标进行求解。仿真实验表明,文中提出的方法受地磁场估计误差影响较小,且能实现对全局最优解的搜索,定位精度较高。仿真分析不同系统基线长度和磁力仪灵敏度条件下文中方法对磁性目标的定位效果可知,当系统的基线长度越大,磁力仪的灵敏度越高,磁性目标定位误差越小。

基于交变信号源的磁定位技术研究

针对磁偶极子定位系统中静态磁信号源产生的静态磁场容易受到背景磁场和环境噪声的干扰的问题,提出了以交变磁场作为磁信号源进行梯度张量定位,研究了其稳定性。以正弦交变磁场信号为发射源,并利用锁相放大器实现对交流信号的解调,通过单点定位算法完成定位。实验结果表明:当噪声的信噪比逐渐减小时,静态磁源的绝对定位误差明显增大,而交变磁源定位系统受环境噪声的影响较小,具有较强的稳定性。

LINEAR CONVERGENCE OF THE LZI ALGORITHM FOR WEAKLY POSITIVE TENSORS

We define weakly positive tensors and study the relations among essentially positive tensors, weakly positive tensors, and primitive tensors. In particular, an explicit linear convergence rate of the Liu-Zhou-Ibrahim（LZI） algorithm for finding the largest eigenvalue of an irreducible nonnegative tensor, is established for weakly positive tensors. Numerical results are given to demonstrate linear convergence of the LZI algorithm for weakly positive tensors.

Linear operators and positive semidefiniteness of symmetric tensor spaces

We study symmetric tensor spaces and cones arising from polynomial optimization and physical sciences.We prove a decomposition invariance theorem for linear operators over the symmetric tensor space,which leads to several other interesting properties in symmetric tensor spaces.We then consider the positive semidefiniteness of linear operators which deduces the convexity of the Frobenius norm function of a symmetric tensor.Furthermore,we characterize the symmetric positive semidefinite tensor(SDT)cone by employing the properties of linear operators,design some face structures of its dual cone,and analyze its relationship to many other tensor cones.In particular,we show that the cone is self-dual if and only if the polynomial is quadratic,give specific characterizations of tensors that are in the primal cone but not in the dual for higher order cases,and develop a complete relationship map among the tensor cones appeared in the literature.

Hypergraph characterizations of copositive tensors

A real symmetric tensor A=(ai_(1…im))∈R^([m,n]) is copositive(resp.,strictly copositive)if Ax^(m)≥0(resp.,Ax^(m)>0)for any nonzero nonnegative vector x∈ℝ^(n).By using the associated hypergraph of A,we give necessary and sufficient conditions for the copositivity of A.For a real symmetric tensor A satisfying the associated negative hypergraph H−(A)and associated positive hypergraph H+(A)are edge disjoint subhypergraphs of a supertree or cored hypergraph,we derive criteria for the copositivity of A.We also use copositive tensors to study the positivity of tensor systems.

Separability for Positive Operators on Tensor Product of Hilbert Spaces

The separability and the entanglement(that is,inseparability)of the composite quantum states play important roles in quantum information theory.Mathematically,a quantum state is a trace-class positive operator with trace one acting on a complex separable Hilbert space.In this paper,in more general frame,the notion of separability for quantum states is generalized to bounded positive operators acting on tensor product of Hilbert spaces.However,not like the quantum state case,there are different kinds of separability for positive operators with different operator topologies.Four types of such separability are discussed;several criteria such as the finite rank entanglement witness criterion,the positive elementary operator criterion and PPT criterion to detect the separability of the positive operators are established;some methods to construct separable positive operators by operator matrices are provided.These may also make us to understand the separability and entanglement of quantum states better,and may be applied to find new separable quantum states.

On the Bound of the Eigenvalue in Module for a Positive Tensor

In this paper,we propose a bound for ratio of the largest eigenvalue and second largest eigenvalue in module for a higher-order tensor.From this bound,one may deduce the bound of the second largest eigenvalue in module for a positive tensor,and the bound can reduce to the matrix cases.

Completely positive tensors in the complex field

In this paper, we introduce the complex completely positive tensor, which has a symmetric complex decomposition with all real and imaginary parts of the decomposition vectors being non-negative. Some properties of the complex completely positive tensor are given. A semidefinite algorithm is also proposed for checking whether a complex tensor is complex completely positive or not. If a tensor is not complex completely positive, a certificate for it can be obtained;if it is complex completely positive, a complex completely positive decomposition can be obtained.

基于差分的磁偶极子单点张量定位方法

采用单点测量的磁场矢量及磁张量数据进行磁偶极子定位的方法,受地磁场干扰大,极大地限制了其实用性。为此,提出基于差分的磁偶极子单点张量定位方法,设计出磁张量测量系统。该方法采用差分方式,消除地磁场及其他共模干扰。仿真结果表明:在地磁场存在的情况下原方法基本上无法实现定位,而该方法则具有较高的定位精度,当探测距离为17.320 5m时,定位误差仅有0.002 3m。