K e y w o r d s I n d e x , Drazin inverse, Group inverse, Per turbat ion bound, Core rank. 1. I N T R O D U C T I O N A necessary and sufficient condition for the continuity of the Drazin inverse (to be defined in the next section) was established by Campbell and Meyer in 1975 [1]. They stated the main result: suppose tha t Aj, j = 1, 2 , . . . , and A are n × n matrices such tha t A j ~ A . T...

We summarize results on the Penrose inequality bounding the ADM-mass or the Bondi mass in terms of the area of an outermost apparent horizon for asymptotically flat initial data of Einstein’s equations. We first recall the proof, due to Geroch and to Jang and Wald, of monotonicity of the Geroch-Hawking mass under a smooth inverse mean curvature flow for data with non-negative Ricci scalar, whic...

We investigate the analytic perturbation of generalized inverses. Firstly we analyze the analytic perturbation of the Drazin generalized inverse (also known as reduced resolvent in operator theory). Our approach is based on spectral theory of linear operators as well as on a new notion of group reduced resolvent. It allows us to treat regular and singular perturbations in a unified framework. W...

We introduce a method and algorithm for computing the weighted MoorePenrose inverse of multiple-variable polynomial matrix and the related algorithm which is appropriated for sparse polynomial matrices. These methods and algorithms are generalizations of algorithms developed in [24] to multiple variable rational and polynomial matrices and improvements of these algorithms on sparse matrices. Al...

If A(z) is a function of a real or complex variable with values in the space B(X) of all bounded linear operators on a Banach space X with each A(z) g-Drazin invertible, we study conditions under which the g-Drazin inverse AD(z) is differentiable. From our results we recover a theorem due to Campbell on the differentiability of the Drazin inverse of a matrix valued function and a result on diff...