Singular values of nonnegative rectangular tensors

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. Some properties concerning the singular values of a real rectangular tensor were discussed by K. C. Chang et al. [J. Math. Anal. Appl., 2010, 370: 284–294]. In this paper, we give some new results on the Perron-Frobenius Theorem for nonnegative rectangular tensors. We show that the weak Perron-Frobenius keeps valid and the largest singular value is really geometrically simple under some conditions. In addition, we establish the convergence of an algorithm proposed by K. C. Chang et al. for finding the largest singular value of nonnegative primitive rectangular tensors.     

Linear convergence of an algorithm for largest singular value of a nonnegative rectangular tensor

An algorithm for finding the largest singular value of a nonnegative rectangular tensor was recently proposed by Chang, Qi, and Zhou [J. Math. Anal. Appl., 2010, 370: 284–294]. In this paper, we establish a linear convergence rate of the Chang-Qi-Zhou algorithm under a reasonable assumption.     

l k,s -Singular values and spectral radius of rectangular tensors

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we study the singular values/vectors problem of real nonnegative partially symmetric rectangular tensors. We first introduce the concepts of l ^k,s -singular values/vectors of real partially symmetric rectangular tensors. Then, based upon the presented properties of l ^k,s -singular values /vectors, some properties of the related l ^k,s -spectral radius are discussed. Furthermore, we prove two analogs of Perron-Frobenius theorem and weak Perron-Frobenius theorem for real nonnegative partially symmetric rectangular tensors.     

Singular value inclusion sets for rectangular tensors

Two singular value inclusion sets for rectangular tensors are given. These sets provide two upper bounds and lower bounds for the largest singular value of nonnegative rectangular tensors, which can be taken as a parameter of an algorithm presented by Zhou et al. (Linear Algebra Appl. 2013; 438: 959–968) such that the sequences produced by this algorithm converge rapidly to the largest singular value of an irreducible nonnegative rectangular tensor.     

Singular values of matrix exponentials

The singular values of a matrix and those of its exponential are related via multiplicative majorization. Matrices giving some equalities in the majorization are characterized. As an application, a scalar inequality for the exponential function is generalized to a matrix-valued inequality and the case of equality is examined.     

Standard tensor and its applications in problem of singular values of tensors

In this paper, first we give the definition of standard tensor. Then we clarify the relationship between weakly irreducible tensors and weakly irreducible polynomial maps by the definition of standard tensor. And we prove that the singular values of rectangular tensors are the special cases of the eigen-values of standard tensors related to rectangular tensors. Based on standard tensor, we present a generalized version of the weak Perron-Frobenius Theorem of nonnegative rectangular tensors under weaker conditions. Furthermore, by studying standard tensors, we get some new results of rectangular tensors. Besides, by using the special structure of standard tensors corresponding to nonnegative rectangular tensors, we show that the largest singular value is really geometrically simple under some weaker conditions.     

Harnack Estimates for Weak Solutions to a Singular Parabolic Equation

By an interpolation method, an intrinsic Harnack estimate and some supestimates are established for nonnegative solutions to a general singular parabolic equation.     

Maximal number of distinct H-eigenpairs for a two-dimensional real tensor

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 Heigenvectors in general. We prove that for any m ? 2, an m-order 2-dimensional tensor A exists such that A 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. Furthermore, 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-eigenvectors 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-eigenvectors.     

Singular discrete (n, p) boundary value problems

This paper discusses a (n, p) singular discrete boundary value problem. Existence of a nonnegative solution is established.     

On a Kaehler hypersurface with the cyclic Ricci semi-symmetric tensor

The purpose of the present paper is to prove that a Kaehler hypersurface with the cyclic Ricci semi-symmetric tensor is locally symmetric.     

算子奇异值与迹的一些不等式

本文给出了算子奇异值与迹的一些不等式，推广了［１］的结果．     

无穷级亚纯函数的公共值与奇异方向

f（z）为无穷级亚纯函数,且σ（f）〈∞。本文证明了复平内存在奇异方向,f（z）与其k阶导函数f（k）（z）（k≥3）在这方向所对应的任意小角域内至多有两个不同的有穷公‘共值。     

On the typical rank of real binary forms

We determine the rank of a general real binary form of degree d?=?4 or d?=?5. In the case d?=?5, the possible values of the rank of such general forms are 3, 4, and 5. This is the first reported case, to our knowledge, where more than two typical ranks have been found. We prove that a real binary form of degree d with d real roots has rank?d.     

一类带奇异性的两点边值问题

对一类带有奇异性的两点边值问题讨论正解的存在性,在很一般的条件下,建立了摄动问题的可解性与原问题的可解性之间的关系,做为此结论的应用,对某些特殊情形,给出正解存在的充分必要条件。     

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.     

An always convergent algorithm for the largest eigenvalue of an irreducible nonnegative tensor

In this paper we propose an iterative method to calculate the largest eigenvalue of a nonnegative tensor. We prove this method converges for any irreducible nonnegative tensor. We also apply this method to study the positive definiteness of a multivariate form.     

Some Simple Estimates for the Singular Values of Matrices

Abstract We first provide a simple estimate for ||A~(-1)||_∞ and ||A~(-1)||_1 of a strictly diagonally dominant matrixA. On the Basis of the result, we obtain an estimate for the smallest singular value of A. Secondly, by scalingwith a positive diagonal matrix D, we obtain some simple estimates for the smallest singular value of an H-matrix, which is not necessarily positive definite. Finally, we give some examples to show the effectiveness ofthe new bounds.