各向同性超对称张量和一般实张量的特征问题-基础数学专业论文.docxVIP

中 文 摘 要 张量分析是研究理论物理、连续介质力学、科学与工程等领域的一个重要工 具. 论文考察各向同性超对称 Descartes 张量, 张量的特征值与特征多项式以及 超对称张量的对称超行列式. 全文共分为六个部分. 第一部分简要介绍了张量的研究背景、研究现状和论文的主要内容. 第二部分主要介绍了研究对象张量的定义和基本性质. 第二部分讨论了各向同性超对称 Descartes 张量. 首先介绍了坐标变换以及 己有的结论进行介绍, 进而推导了二阶至六阶各向同性超对称 Descartes 张量的 表达式. 第四部分主要利用结式研究了超对称张量的对称超行列式的问题. 第五部分把超对称张量的特征值和特征向量的研究推广到一般实张量. 并且 对祁所提出关于超对称张量的正则性和奇异性的关系给出了回答并证明. 第六部分是结束语:对本文进行了概括总结及对未来的展望. 关键词: 张量 ; 各向同性超对称 Descartes 张量 ; 超对称张量; 特征值 ; E-特 征值 ; 对称超行列式 . Abstract Tensor analysis is an important tool in theoretical physics, continuum mechanics and many other areas of science and engineering. The paper mainly investigated the isotropic super-symmetric Descartes tensor, the eigenvalues of super-symmetric tensor and its characteristic polynomial and super-determinant of super-symmetric tensor. The full text is divided into six parts. The first part briefly introduce the research background and current situation of the tensor, and point out the main content of the paper. The second part introduces the basic concepts of tensor.The third part consider the isotropic super-symmetric Descartes tensor. At first, we introduce coordinate transformations and some conclusions, and thus deduce the expression of isotropic super-symmetric Descartes tensor from second-order to sixth-order. The fourth part mainly consider the symmetry of the super-symmetric tensor de- terminant. This part bases the study of the nature of symmetry super-determinant resultant.The fifth part studies the eigenvalue problem of the general real tensor eigen- values and E - characteristic values. And we give a positive answer in the literature Qi [24] of Question 2.The sixth part is the Conclusion. This part summarizes the essay and makes a vision of the future. Key Words: Tensor; Isotropic super-symmetric Descartes tensor; Super-symmetric tensor; Characteristic values; E - characteristic values; Symmetric super-determinant. 目 录 第 章 J êl . . . . . . . . . . . . . . . . . . . . . . . . . . .