Drazin inverse: representation, approximation, continuity and illustrations

In this paper, we present some characteristics and expressions of the Drazin inverse for matrices and bounded linear operators in Banach spaces. We give a survey of some of results on the continuity of the Moore-Penrose and Drazin inverse, direct technics for computing the Drazin inverse are discussed, they are based on Euler-Knopp Method and characterized in terms of a limiting process. The examples presented are for illustrative purposes, some of which are provided for testing the considered iterative processes

We come in Peace! (Well, do they?): Human Communicative Fears in Films of an Extra-terrestrial Nature

Extra-terrestrial beings always have the potential to alter ‘the world as we know it.’ Whenever they appear in film or television, there is a debate about the possibility that we have, as human beings, to communicate with them. Linguistics and semiotics become keyelements in our attempt to bridge the cultural gap. Be it through mathematics, music or technological gadgets, these films only mimic the uncertainty that arises from all types of interculturalcontact. The fear is always the misinterpretation that can turn a ‘We come in peace’ into a fully-fledged war. This article starts from a general review of 138 films from the decade 2007-2017 to recognize the main points present in their narrative, which shows that hostile conflict remains the main theme. Then it provides an interpretive analysis of the films to describe the type of communication fears that exist incontemporary films, taking a special look at Avatar (Cameron, 2009), Arrival (Villeneuve, 2016) and District 9 (Blomkamp, 2009) as the main representations of our current understanding of communication difficulties. The humanization of extra-terrestrial beings to establish communication with them becomes an inverse representation of the dehumanization of the other, the enemy, as depicted by certain sectors of society and a few recently elected governments, with whom it is difficult to establish mechanisms of communication and dialogue.

Representations and sign pattern of the group inverse for some block matrices

Let $M= \left[ \begin{array}{cc} A& B \\ C& O \end{array} \right]$ be a complex square matrix where A is square. When BCB^{\Omega} =0, rank(BC) = rank(B) and the group inverse of $\left[ \begin{array}{cc} B^{\Omega} A B^{\Omega} & 0 \\ CB^{\Omega} & 0 \right]$ exists, the group inverse of M exists if and only if rank(BC + A)B^{\Omega}AB^{\Omega})^{\pi}B^{\Omega}A)= rank(B). In this case, a representation of $M^#$ in terms of the group inverse and Moore-Penrose inverse of its subblocks is given. Let A be a real matrix. The sign pattern of A is a (0,+,−)-matrix obtained from A by replacing each entry by its sign. The qualitative class of A is the set of the matrices with the same sign pattern as A, denoted by Q(A). The matrix A is called S^2GI, if the group inverse of each matrix \bar{A} in Q(A) exists and its sign pattern is independent of e A. By using the group inverse representation, a necessary and sufficient condition for a real block matrix to be an S^2GI-matrix is given.