广义逆理论与偏序结构

Moore-Penrose inverse, group inverse and Drazin inverse are important, classical generalized inverses, which have been widely used in many fields. Many authors have investigated generalized inverses in Banach algebra, C*- algebra, ring, semigroup, and, bounded linear operators on Banach space (Hilbert space). With the deepening of the research, some new generalized inverses appear. With the emergence of new generalized inverses, there appeared various partial orders based on these new generalized inverses. These partial orders play an important role in characterizing generalized inverses. This project aims to investigate generalized inverses and related partial orders. The first part of the context is investigating the structures and expressions of generalized inverses, such as the Moore-Penrose inverse, the group inverse and the Drazin inverse in rings (semigroups) and categories. Secondly, we investigate the existence criteria and computations of some new generalized inverse, such as the core inverse (dual core inverse), Mary inverse and (b, c)-inverse. Then, we investigate the minus partial order, star partial order, #-partial order, core partial order and diamond partial order based on corresponding generalized inverses, and eventually create the partial order theory. Finally, The recurrent neural network is used to compute several new generalized inverses and the new generalized inverses are used to calculate the extreme value problem of quadratic form.

Moore-Penrose逆,群逆及Drazin逆是十分重要的经典广义逆,已经在许多领域有着重要的应用,许多作者在复矩阵,Banach空间(Hilbert空间)上有界线性算子,Banach代数,C*-代数,环与半群,范畴中的态射等方面展开研究.随着研究的深入,出现了一些新型广义逆.伴随广义逆的出现,相继出现了基于广义逆的各种偏序,这些偏序在刻画广义逆的性质等方面起着重要作用.本项目拟结合广义逆理论和由此产生的偏序关系对其展开研究.主要研究:环(半群)上元素,范畴中态射的经典广义逆,如Moore-Penrose逆,群逆及Drazin逆的结构与表达式.几类新型广义逆,如核逆(对偶核逆),Mary逆及(b,c)-逆的存在性及计算方法,以及由上述广义逆引出的减偏序,星偏序,#-偏序,核偏序及钻石偏序,建立基于广义逆的偏序理论.利用递归神经网络来计算新型广义逆，并利用新型广义逆计算二次型的极值问题.

Moore-Penrose逆和Drazin逆作为两类经典广义逆，在许多领域中有着重要的应用。这极大地推动了广义逆理论的发展，随之出现了几类新型广义逆（如伪核逆，（单边）Mary逆，(b,c)-逆，弱群逆）以及基于广义逆的偏序。本项目讨论了几类广义逆以及相关偏序问题，主要研究了（1）环和半群中元素，环上矩阵，范畴中态射的经典广义逆和新型广义逆的存在性与表达式；（2）基于广义逆的偏序理论及应用；（3）利用神经网络来计算新型广义逆. 本项目采用了环与模理论，范畴论，算子理论，神经网络算法等工具和方法，对几类广义逆及相关偏序展开研究，并且给出了计算新型广义逆的算法，在SCI期刊上正式发表39篇论文，在线发表3篇论文。这些成果丰富了广义逆理论及基于广义逆的偏序理论。