We present an extension of Geodesics in Lorentzian Manifolds (Semi-Riemannian Manifolds or pseudo-Riemannian Manifolds ). A geodesic on a Riemannian manifold is, locally, a length minimizing curve. On the other hand, geodesics in Lorentzian manifolds can be viewed as a distance between ``events''. They are no longer distance minimizing (instead, some are distance maximizing) and our goal is to illustrate over what time parameter geodesics in Lorentzian manifolds are defined. If all geodesics in timelike or spacelike or lightlike are defined for infinite time, then the manifold is called ``geodesically complete'', or simply, ``complete''. It is easy to show that the magnitude of a geodesic is constant, so one can characterize geodesics in terms of their causal character: if this magnitude is negative, the geodesic is called timelike. If this magnitude is positive, then it is spacelike. If this magnitude is 0, then it is called lightlike or null. Geodesic completeness can be considered by only considering one causal character to produce the notions of spacelike complete, timelike complete, and null or lightlike complete. We illustrate that some of the notions are inequivalent.

Nesse trabalho apresentaremos alguns exemplos clássicos que evidenciam as diferenças entre a geometria Riemanniana e a semi-Riemanniana (Lorentziana) quanto à completude de geodésicas. Para isso, revisitaremos conceitos básicos de Geometria, seguido de uma introdução aos espaços vetoriais de Lorentz e um estudo inicial sobre o grupo de Lorentz. Nos capítulos finais discutiremos sobre completude de geodésicas e como se distanciam do caso Riemanniano. / In this work we intend to present some classical examples that display the differences between Riemannian and semi-Riemannian (Lorentzian) geometry in relation to the completeness of geodesics. For this, we will revisit basic Geometry concepts followed by an introduction to the vector spaces of Lorentz and a simple study on the Lorentz group. In the final chapters we will discuss about the completeness of geodesics and how it distances itself from the Riemannian case.

We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance. We provide a selfcontained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawson's relative index theorem and a generalization of the cobordism theorem.

Mixed regimes are often viewed as inherently less stable and more war prone than fully democratic or autocratic systems due to their low levels of institutionalization. I ask, are certain mixed regimes more or less war prone than other mixed regimes, based on the strength and orientation (more democratic or autocratic) of their political institutions? At ends with previous research, my findings suggest that institutionalization levels play little, if any role in the onset of interstate war.

Neste trabalho, estudamos algumas formas de combinar sistemas de Lógica Modal, analisando quando a combinação preserva propriedades como correção, completude e decidibilidade. Estendemos um estudo já realizado sobre combinações de sistemas de Lógica Modal Normal para sistemas de Lógica Modal Não-normal. O principal resultado deste trabalho é a preservação de completude da aplicação externa de um sistema de Lógica Modal Não-normal M em um sistema lógico L. Outro resultado importante é um exemplo de interação forte na combinação independente, ou fusão, de dois sistemas de Lógica Modal Não-normal. / In this work, we study a few ways of combining Modal Logic systems, analysing when the combination preserves properties like soundness, completeness and decidability. We extend a study of the combination of Normal Modal Logic systems to Non-normal Modal Logic systems. The main result of this work is the completeness preservation in the external application of a Non-normal Modal Logic system M to a logic system L. Another important result is an example of strong interations arising in the fusion of two Non-normal Modal Logic system.

Philosophiae Doctor - PhD / The advantage of a well-developed health information system is the significant role played by records produced by such a system beyond recording medical history of individuals. They are the foundation for birth registrations which when fully complete is an important tool for acquiring data necessary for planning and monitoring child and maternal health in a country. This study aimed to investigate the potential of hospital birth records to estimate the number of births in the country and supplement birth registrations data. Data was abstracted from public facilities where births occur in two municipalities; Germiston in Gauteng and Nkomazi in Mpumalanga for the period 2014 to 2016. Modified version of the BORN Data Quality Framework (BORN-DQF) of the Ontario Agency for Health Protection and Promotion (2016) was used to assess the contents and quality of hospital birth records. / 2022

The completeness of eigenfunctions for linearized equations is critical for many applications, such as the study of stability of solitary waves. In this thesis, we work with the Nonlinear Schr{\"o}dinger (NLS) equation, associated with the Zakharov-Shabat spectral problem. Firstly, we construct a complete set of eigenfunctions for the spectral problem. Our method involves working with an adjoint spectral problem and deriving completeness and orthogonality relations between eigenfunctions and adjoint eigenfunctions. Furthermore, we prove completeness of squared eigenfunctions, which are used to represent solutions of the linearized NLS equation. For this, we find relations between the variation of potential and the variation of scattering data. Moreover, we show the connection between the squared eigenfunctions of the Zakharov-Shabat spectral problem and solutions of the linearized NLS equation. / Thesis / Master of Science (MSc)

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In this thesis, we study problems related to the reconstruction (up to bi-interpretability) of first-order theories from various functorial invariants: automorphism groups, endomorphism monoids, (categories of) countable models, and (ultra)categories of models. / Thesis / Master of Science (MSc)