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

Web-based Sports Scoring System for Trampoline Gymnastics Competitions

This work arises within the context of the Master’s Project in Computer Engineering - Mobile Computing course, at Escola Superior de Tecnologia e Gest˜ao do Instituto Polit´ecnico de Leiria. The aim of this work is to describe the work carried out over several months, in partnership with the Trampolins Clube de Leiria. This is a local club, and as such, with somewhat limited resources, such as the competitions they manage and organize their judging solution were in need of improvements in terms of organization, speed, data acquisition, trust in the data, and resilience. This work begins by presenting the methodology used, followed by a contextualization of the sport, and a brief presentation of some software available in the market. Next, a comparison between the various current technological solutions is presented, as a means of gathering the necessary information to make informed decisions for the presentation of a solution. There is also a chapter with the details of the proposed solution, a description of the technologies used, and the implementation details. Finally, various execution scenarios, tests, and questionnaires are analyzed, concluding with the project’s findings. The main goal of the project was to propose an architecture for a solution to be made available in trampoline competitions, in order to minimize manual processes and perform calculations automatically, thus reducing human error and accelerating the entire end-toend process. There was also the objective of, during the development of the proposed solution, having it tested in actual events, including regional, national, and international competitions. The project was successfully completed, with the club’s main goals achieved. The experience gained in the interactions of requirement gathering was enriching for everyone. In the real-life context, the application’s performance and output were accepted in all possible categories