环中几类新型广义逆及其在神经网络和Banach代数中的应用

Generalized inverses are indispensable research tools in the study of the least squares problems, nonlinear problems, statistical problems, and neural network problems. And at the same time, generalized inverses (Moore-Penrose inverse, Drazin inverse) are associated with regular rings (*-regular rings, strong π-regular rings), which have important applications in many fields. The emergence of several new classes of generalized inverses such as generalized Drazin inverse, Mary inverse and (b, c)-inverse in rings injects new vitality into the study of generalized inverse theory, which is of great significance to the spectral theory research of Banach algebras. The project is planned to investigate in the following aspects: (1)By investigating the existence and related properties of classical generalized inverses in rings, we will study the existence and related properties of several new types of generalized inverses, especially Mary inverse, generalized Drazin inverse, (pseudo, one-sided, extended) (b, c)-inverse, and find out the intrinsic relationship between them; (2)For special rings (such as commutative ring, automorphism ring, Bezout ring, regular ring), we will give the decomposition and computation of these new type of generalized inverses, then applying the related results to Banach algebra; (3) Construct a neural network model to compute the generalized inverse of a matrix. It is hoped that the results obtained can unify some classical results, and enrich the generalized inverse theory.

在研究最小二乘问题，非线性问题，统计问题，神经网络问题的过程中，广义逆都是不可或缺的研究工具。广义逆（Moore-Penrose逆、Drazin逆）与正则环（*-正则环、强π正则环）相联系，在许多领域有重要应用。 环上广义Drazin逆、Mary逆和(b,c)-逆等新型广义逆的出现，为广义逆理论的研究注入了新的活力，对Banach代数的谱理论研究具有重要意义。本项目拟从以下方面展开研究：(1)通过研究环上经典广义逆的存在性及相关性质，研究几类新型广义逆，特别是Mary逆、广义Drazin逆、（伪、单边、延伸）(b,c)-逆的存在性及相关性质，找出它们之间的内在联系；(2)特殊环(交换环、自同态环、Bezout整环、正则环)上新型广义逆的矩阵分解及计算，将环上相关成果应用到Banach代数；(3)构造神经网络模型求矩阵的广义逆。相信本项目所得结论将统一已知的经典结果，进一步丰富广义逆理论。

广义逆在研究最小二乘问题、非线性问题、统计问题、神经网络问题的过程中都是不可或缺的。围绕环上新型广义逆和（b, c）-逆展开研究，并应用于Banach代数和神经网络上，这对算子理论、环论和神经网络理论的研究具有重要意义。本项目首先研究了单边（b,c）-逆，给出了左右零因子（b,c）-逆的定义、性质和等价刻画，左π正则，右π正则的刻画，元素乘积的单边（b,c）-逆的刻画及性质；其次研究了(b,c)-逆的一些新的性质，给出了（b,c）可逆元的（b,c）可逆的刻画，并证明了当b, c是正则的，且1是(b, c)-可逆的，则环上所有(b, c)-可逆元的集合关于乘法构成群；再次研究了核逆、伪核逆、DMP逆，定义了*-环上的加权右核逆和加权右伪核逆，利用加权右核逆的刻画给出了加权核逆的一些新刻画，同时给出加权右伪核逆的等价刻画；最后，在应用方面，我们研究了Banach代数中的伪Drazin逆和广义Drazin逆问题。这些结果为经典广义逆的研究提供了新的思路和方法，丰富了广义逆的研究内容，统一了一些经典结果，进一步丰富了广义逆理论。