张量谱半径的界及张量广义逆研究

发布时间：2018-05-07 07:45

本文选题：张量 + 特征值　；参考：《哈尔滨工业大学》2017年博士论文

【摘要】：随着意大利数学家、理论物理学家、张量分析创始人之一R.Gregorio在1890年的《绝对微分学方法及其应用》经典著作的出版,张量为众多数学家所熟知。1915年左右,爱因斯坦在广义相对论的研究中引入张量,之后,张量的研究受到重视。随着量子计算、机器学习、人工智能等领域的兴起,张量理论中涌现出一批新问题,如张量的特征值、超图张量表示下的谱、张量方程的解等。2005年,香港理工大学祁力群教授和芝加哥大学Lek-Heng Lim教授分别提出了张量特征值的概念,引起了学术界的广泛关注,吸引了众多国内外学者参与到张量特征值及相关问题的研究中。张量方程广泛应用于控制系统、理论物理学等工程问题中,如爱因斯坦引力场方程,三维粘性绕流计算方程、高维Poisson方程等均为张量方程,其中张量方程的求解是重要的研究问题。本文研究非负弱不可约张量的谱半径的界;张量Hadamard乘积的谱性质;M-张量的判定;定义并研究张量广义乘积和张量爱因斯坦乘积下的广义逆;用张量广义逆给出几类张量方程解的表示,成果如下。1.用张量对应的有向图这一组合数学的方法给出张量谱半径的界和张量谱半径组合形式的Collatz-Wielandt公式。用超图的度序列刻画k一致连通超图的邻接张量和无符号Laplacian张量谱半径的新的界,改进已有的结果。2.给出张量Hadamard乘积的谱半径的界;刻画Z-张量、M-张量Hadamard乘积的特征值最小实部的下界及M-张量的若干判定定理;给出张量Hadamard乘积的行列式的不等式和有向加权超图的谱半径的估计。3.在张量广义乘积和张量爱因斯坦乘积下分别定义张量的广义逆。刻画张量广义逆的存在性;给出几类分块张量广义逆的表达式;用张量广义逆给出几类张量方程可解的充分必要条件、通解形式和极小范数最小二乘解。
[Abstract]:With the publication of the classical book "the method of absolute differential calculus and its application" published in 1890 by R.Gregorio, an Italian mathematician, a theoretical physicist and one of the founders of Zhang Liang's analysis, Zhang Liang was well known to many mathematicians. After Einstein introduced Zhang Liang into the study of general relativity, Zhang Liang received much attention. With the rise of quantum computing, machine learning, artificial intelligence and other fields, a number of new problems have emerged in Zhang Liang's theory, such as the eigenvalue of Zhang Liang, the spectrum expressed by the hypergraph Zhang Liang, the solution of Zhang Liang equation, and so on. Professor Qiliqun of Hong Kong Polytechnic University and Professor Lek-Heng Lim of University of Chicago put forward the concept of Zhang Liang eigenvalue, which has attracted a lot of scholars from home and abroad to participate in the study of Zhang Liang eigenvalue and related problems. Zhang Liang equation is widely used in control system, theoretical physics and other engineering problems, such as Einstein gravitational field equation, three-dimensional viscous flow calculation equation, high-dimensional Poisson equation, etc. The solution of Zhang Liang equation is an important research problem. In this paper, we study the bounds of spectral radius of nonnegative weakly irreducible Zhang Liang, the spectral properties of Zhang Liang Hadamard product and the determination of M Zhang Liang, define and study the generalized inverse of Zhang Liang generalized product and Zhang Liang Einstein product. By using Zhang Liang's generalized inverse, the expressions of solutions of several kinds of Zhang Liang equations are given. The results are as follows. 1. By using the combinatorial mathematics method of Zhang Liang's corresponding digraph, the paper gives the Collatz-Wielandt formula of the boundary of Zhang Liang's spectral radius and the combination form of Zhang Liang's spectral radius. In this paper, we use the degree sequence of hypergraphs to characterize the new bounds of the spectral radius of k-uniformly connected hypergraphs adjacent to Zhang Liang and unsigned Laplacian Zhang Liang, and improve the existing results .2. This paper gives the bounds of spectral radius of Zhang Liang Hadamard product, characterizes the lower bound of the minimum real part of the eigenvalue of the Hadamard product of Z- Zhang Liang and M- Zhang Liang, and gives some judgment theorems of M. The inequality of determinant of Zhang Liang Hadamard product and the estimate of spectral radius of directed weighted hypergraph. Under Zhang Liang's generalized product and Zhang Liang Einstein's product, the generalized inverse of Zhang Liang is defined respectively. This paper describes the existence of Zhang Liang generalized inverse, gives the expressions of several classes of block Zhang Liang generalized inverse, and gives the necessary and sufficient conditions for some Zhang Liang equations to be solvable by Zhang Liang generalized inverse, the form of general solution and the minimal norm least square solution.
【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O183.2

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