Grafos e digrafos com o sagemath

Texto de apoio a parte da Acção de Formação em Grafos, com uso de Maxima e Sagemath.Breve introdução ao digrafos com auxílio do SageMath

The Moore-Penrose inverse of a factorization

In this paper, we consider the product of matrices $PAQ$, where $A$ is von Neumann regular and there exist $P^{\prime }$ and $Q^{\prime }$ such that $P^{\prime }PA=A=AQQ^{\prime }$. We give necessary and sufficient conditions in order to $PAQ$ be Moore-Penrose invertible, extending known characterizations. Finally, an application is given to matrices over separative regular rings.Fundação para a Ciência e a Tecnologia (FCT).Programa de Desenvolvimento Educativo para Portugal III (PRODEP III) - proj. 5.3/N/189.007/00)

The Moore-Penrose inverse of von Neumann regular matrices over a ring

Necessary and sufficient conditions are given in order that a von Neumann regular matrix over an arbitrary ring and some factorizations are Moore-Penrose invertible.Fundação para a Ciência e a Tecnologia (FCT)

Teoria de números computacional visto pelos olhos do Pari/GP

Notas de apoio para Teoria de Números Computacional, LCC&LMatNestas notas pretendemos exemplificar algumas formas de aplicar a Teoria de Números Computacional num CAS, e em particular no Pari/GP. Estas notas não substituem as que por ventura retirará das aulas teóricas. Deve, sempre que possível, experimentar por si as informações descritas neste documento

The link between regularity and strong-pi-regularity

It is shown that if all powers of a ring element $a$ are regular, then $a$ will be strongly-pi-regular exactly when a suitable word in the powers of $a$ and their inner inverses is a unit.Research with financial support provided by the Research Centre of Mathematics of the University of Minho (CMAT) through the FCT Pluriannual Funding Program

The Moore-Penrose inverse of a companion matrix

Necessary and sufficient conditions are given for the Moore-Penrose inverse of a companion matrix over an arbitrary ring to exist.This research was financed by FEDER Funds through ``Programa Operacional Factores de Competitividade - COMPETE'' and by Portuguese Funds through FCT - ``Fundação para a Ciência e a Tecnologia'', within the project PEst-C/MAT/UI0013/2011

The inverse along a lower triangular matrix

In this paper, we investigate the recently defined notion of inverse along an element in the context of matrices over a ring. Precisely, we study the inverse of a matrix along a lower triangular matrix, under some conditions.This research was financed by FEDER Funds through ``Programa Operacional Factores de Competitividade - COMPETE'' and by Portuguese Funds through FCT - ``Fundação para a Ciência e a Tecnologia'', within the project PEst-C/MAT/UI0013/2011

About the von Neumann regularity of triangular block matrices

Necessary and sufficient conditions are given for the von Neumann regularity of triangular block matrices with von Neumann regular diagonal blocks over arbitrary rings. This leads to the characterization of the von Neumann regularity of a class of triangular Toeplitz matrices over arbitrary rings. Some special results and a new algorithm are derived for triangular Toeplitz matrices over commutative rings. Finally, the Drazin invertibility of some companion matrices over arbitrary rings is considered, as an application.Fundação para a Ciência e a Tecnologia (FCT)

Group-regular rings

We propose different generalizations of unit-regularity of elements in general rings (non necessarily unital rings). We then study general rings for which all elements have these properties. We notably compare them with unit-regular ideals and general rings with stable range one. We also prove that these rings are morphic rings.This research has been conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01), and was also partially supported by FCT-'Fundação para a Ciência e a Tecnologia', within the project UID-MAT-00013/2013

The group inverse of a product

In this paper, we characterize the existence and give an expression of the group inverse of a product of two regular elements by means of a ring unit.∗This research was financed by Laboratoire Modal’X, Universit´e Paris–Ouest, and by the Research Centre of Mathematics of the University of Minho with the Portuguese Funds from the “Fundação para a Ciência e a Tecnologia”, through the Project PEst-OE/MAT/UI0013/2014