BII 2010

In 2011 four grants were awarded. These grants were initiated on 22nd June 2011. The work carried out by the students was presented in a public session on October 17th, 2012.

Student: Ivo Carrajola (Lic. Ciências de Computação)

Title of the project: Semigrupos Regulares com um subsemigrupo inverso associado

Supervisors: Paula Marques Smith and Paula Mendes Martins

**Summary:** During the last decade, the study of regular semigroups has proved to be fundamental for the development of the algebraic theory of semigroups. The presentation of examples is crucial for the evolution of the study of this class of semigroups. Computational methods are certainly a useful tool for the construction of those examples. GAP system was created and developed to study topics of computational discrete algebra and provides a programming language and a broad file of algebraic objects and functions that implement algebraic algoritms. This software is used for both teaching and

research on group representation, rings, vector spaces, semigroups and many other algebraic and combinatoric structures. It is a flexible system that can be readily modified/extended to permit analysis of specific situations.

With support from a BII grant, in 2009, an undergraduate student created new functions that allowed the construction, using GAP, of examples of regular semigroups with associate inverse subsemigroup. The performance of the algoritm that he created was unable to generate examples with more than 20 elements. With the support from this new BII grant, an improved algoritm that overcomes the limitations and the difficulties presented by the first algoritm was established.

Student: Victor Cacciari Miraldo (Lic. Ciências da Computação)

Title of the project: Interpretações do lambda-calculus em redes de interação

Supervisors: José Carlos Espírito Santo e Jorge Sousa Pinto

**Summary: **The interaction nets are a graphical formalism, invented by Yves Lafont in 1990 as a proof nets generalization. Since it constitutes a universal model of computation and the ease of a parallel implementation, they showed to be of direct interest in the implementation of functional languages.

The lambda-calculus, on the other hand, is the theoretical basis of functional programming languages, and can be translated into interaction nets. The project consists of studying interpretations of the known lambda-calculus on interaction nets, considering to what extent such interpretations are sensitive to variations of the lambda-calculus. Known translations were divided into two classes: direct and logical, and in this paper we present both a short survey of the translation process behind the logical interpretations and a detailed study of a direct interpretation. The project was supervised by Professors Jorge Sousa Pinto and José Carlos Espírito Santo as part of an Integration into Research Scholarship, provided by the Center for Mathematics, University of Minho.

Student: João Marcelo Pires Silva (MI Eng. Civil)

Title of the project: Método dos elementos finitos para determinar o coeficiente de segurança à rotura de maciços terrosos heterogéneos com sobrecargas

Supervisors: Gaspar Machado and Júlio Barreiros Martins

**Summary: **The most usual methods for calculating the Safety Factor of soil are the so-called "plastic-rigid" methods (Slice Methods). However, the Safety Factor calculation can also be made using the so-called "elastic-plastic" methods, where the Finite Element Method is usually the support tool.

Observing the available commercial programs that apply the "elastic-plastic" methods, it is not easy to calculate the Safety Factor, since normally these programs present only "spots", generally in two-dimensional space, where the material plastify, and never the rupture mechanism.

In this work we intend to develop a MATLAB program that uses the Finite Element Method to directly determine the calculation of the Safety Factor, starting from systems of structures of elastic elements connected with each other and the exterior, conceived inside the potentially sliding terrain mass.

Keywords: Safety Factor, Finite Element Method, Monte-Carlo Method, and MATLAB.

Student: Lara Maria Lopes Teixeira (Mestrado em Estatística)

Title of the project: Análise Estatística das Desigualdades em Saúde na região Norte de Portugal

Supervisors: Arminda Manuela Gonçalves, Susana Faria and Vasco Machado

**Summary: **Different public health needs demand differentiated interventions by the Health services in order to attain efficiency and thus reducing Health inequalities.

Developing effective means to identify and analyze Health Inequalities among populations is thus one of the greatest challenges in Public Health at the regional and local levels.

For this purpose a statistical model was developed, which allows the identification and analysis of health inequalities in the Northern Region of Portugal, at the regional and local levels, as well as a prognostic evaluation aimed at guiding interventions that could be carried out to reduce those inequalities. As health status summary measure it was used life expectancy at birth, which uses specific mortality rates by age group, calculated by trienniums, refering the used data to the to trienniums 1996-1998 to 2008-2010.

In order to identify Health Inequalities we used the relative gap in life expectancy, which is the absolute difference between life expectancy of a Reference area and the life expectancy of a Study area, presented as percentage of life expectancy of the Reference area. For a better understanding of the relative gap in life expectancy it is determined the contribution of each death cause and of each age group for that gap, as well as potential gains in life expectancy. It is also possible to analyze the impact of interventions in life expectancy, (in terms of a percentual reduction of the main death causes in specific age groups), specifically in the reduction of existing inequalities.

The statistical model which was developed will allow the Public Health Services to conduct a diagnostic evaluation of Health inequalities, with a specification of the death causes and age groups that most contribute to those inequalities. It will also allow a prognostic evaluation by analyzing the impact of some interventions in reducing the relative gap in life expectancy, in terms of health gains.