Neste trabalho estudamos as singularidades de projeções no plano de curvas genéricas, introduzindo uma nova relação de equivalência para germes e multigermes de curvas planas, denominada A_h-equivalência. / In this work singularities of projections to the plane of curves are studied. We introduce a new equivalence relation for germs of plane curves, called A_h-equivalence.

Cette thèse porte sur l'analyse de la singularité d'un manipulateur plan pour l'application d'une XY-Théta plate-forme. Cette plate-forme possède une cinématique brevetée, conçue pour garder l’erreur finale de position en dessous de 2mμ dans son espace de travail de dimensions 300 mm × 300 mm. Ces performances de haute précision s'expliquent par la proximité des singularités. Certains inconvénients peuvent survenir lorsque la trajectoire se rapproche des singularités, notamment si une vitesse articulaire élevée est atteinte. Par conséquent, l'objectif principal de cette thèse est d'identifier les lieux des singularités. Habituellement, quand un robot non-redondant se déplace dans un espace à trois dimensions, le lieu de singularité est défini par une surface. Une contribution majeure de ce travail de thèse réside dans l'identification d'une ligne hélicoïdale pour définir le lieu de la singularité au sein de l'espace de travail. Une autre partie du travail réalisé a consisté à prendre en compte la redondance du robot à identifier les lieux des singularités et dans ce cas à analyser les problèmes de contrôle liés à la traversée de surfaces de singularités. En dernier lieu, une attention a été portée sur l'indice de maniabilité afin d'évaluer la distance entre le manipulateur et la singularité. / This thesis deals with the singularity analysis of a planar robotic manipulator for the application of an XY-Theta platform. This XY-Theta platform has a patented kinematics designed to keep the final position error below 2 μm in its 300 mm × 300 mm workspace. But as the high precision performances are due to the proximity of singularities, some drawbacksmay also appear when the trajectory is too close to singularities, such as large joint velocities, high forces and torques. Therefore, the main objective of this thesis is to identify the singularity loci. Usually, when a non-redundant robot operates in a 3D space, the singularity locus is represented by a surface. Here, one contribution is the identification of an helicoidal line for the singularity locus within the workspace. Another contribution is to take into account the redundancy of the robot, identify the singularity loci in this case and analyze the control problems linked to the crossing of singularity surfaces. Finally, the manipulability index is calculated to show how far the manipulator is from the singularity configuration.

The abstract boundary construction of Scott and Szekeres has proven a practical classification scheme for boundary points of pseudo-Riemannian manifolds. It has also proved its utility in problems associated with the re-embedding of exact solutions containing directional singularities in space-time. Moreover it provides a model for singularities in space-time - essential singularities. However the literature has been devoid of abstract boundary results which have results of direct physical applicability.¶ This thesis presents several theorems on the existence of essential singularities in space-time and on how the abstract boundary allows definition of optimal em- beddings for depicting space-time. Firstly, a review of other boundary constructions for space-time is made with particular emphasis on the deficiencies they possess for describing singularities. The abstract boundary construction is then pedagogically defined and an overview of previous research provided.¶ We prove that strongly causal, maximally extended space-times possess essential singularities if and only if they possess incomplete causal geodesics. This result creates a link between the Hawking-Penrose incompleteness theorems and the existence of essential singularities. Using this result again together with the work of Beem on the stability of geodesic incompleteness it is possible to prove the stability of existence for essential singularities.¶ Invariant topological contact properties of abstract boundary points are presented for the first time and used to define partial cross sections, which are an generalization of the notion of embedding for boundary points. Partial cross sections are then used to define a model for an optimal embedding of space-time.¶ Finally we end with a presentation of the current research into the relationship between curvature singularities and the abstract boundary. This work proposes that the abstract boundary may provide the correct framework to prove curvature singularity theorems for General Relativity. This exciting development would culminate over 30 years of research into the physical conditions required for curvature singularities in space-time.

Estudaremos as singularidades genéricas de superfécies regradas em R3. O objetivo do trabalho é mostrar que as singularidades genéricas que ocorrem no conjunto das superfícies regradas são as mesmas que ocorrem no conjunto das aplicações diferenciáveis de R2 em R3, enquanto que as singularidades genéricas das superfícies desenvolvíveis, que formam um subconjunto das superfícies regradas, são mais degeneradas. / We study generic singularities of ruled surfaces in R3. In this work we show that generic singularities appearing in the set of ruled surfaces are the same that occur in the set of map germs from R2 to R3, while the generic singularities of developable surfaces are more degenerate.

Neste trabalho estudamos as singularidades de projeções no plano de curvas genéricas, introduzindo uma nova relação de equivalência para germes e multigermes de curvas planas, denominada A_h-equivalência. / In this work singularities of projections to the plane of curves are studied. We introduce a new equivalence relation for germs of plane curves, called A_h-equivalence.

Estudamos singularidades de famílias de matrizes simétricas. O objetivo é classificar as singularidades simples de tais famílias e estudar a geometria de alguns objetos associados a elas / We study the singularities of families of symmetric matrices. The aim of this work is to classify simple singularities of such families and study the geometry of some objects associated to them

Neste trabalho, nós abordamos alguns resultados de T. Fukuda e de N. Dutertre e T. Fukui sobre a topologia das singularidades de Morin. Em particular, apresentamos uma nova prova para o Teorema de Dutertre-Fukui [2, Theorem 6.2], para o caso em que N = Rn, usando a Teoria de Morse para variedades com bordo. Baseados nas propriedades de um n-campo de vetores gradiente (∇ f1; : : : ∇fn) de uma aplicação de Morin f : M → Rn, com dim M ≥ n, na segunda parte deste trabalho, nós introduzimos o conceito de n-campos de Morin para n-campos de vetores que não são necessariamente gradientes. Nós também generalizamos o resultado de T. Fukuda [3, Theorem 1], que estabelece uma equivalência módulo 2 entre a característica de Euler de uma variedade diferenciável M e a característica de Euler dos conjuntos singulares de uma aplicação de Morin definida sobre M, para o contexto dos n-campos de Morin. / In this work, we revisit results of T. Fukuda and N. Dutertre and T. Fukui on the topology of Morin maps. In particular, we give a new proof for Dutertre-Fukui\'s Theorem [2, Theorem 6.2] when N = Rn, using Morse Theory for manifolds with boundary. Based on the properties of a gradient n-vector field (∇ f1; : : : ∇ fn) of a Morin map f : M → Rn, where dim M ≥ n, in the second part of this work, we introduce the concept of Morin n-vector field for n-vector fields V = (V1; : : : ; Vn) that are not necessarily gradients. We also generalize the result of T. Fukuda [3, Theorem 1], which establishes a module 2 equivalence between Euler\'s characteristic of a manifold M and Euler\'s characteristic of the singular sets of a Morin map defined on M, to the context of Morin n-vector fields.

Neste trabalho, nós abordamos alguns resultados de T. Fukuda e de N. Dutertre e T. Fukui sobre a topologia das singularidades de Morin. Em particular, apresentamos uma nova prova para o Teorema de Dutertre-Fukui [2, Theorem 6.2], para o caso em que N = Rn, usando a Teoria de Morse para variedades com bordo. Baseados nas propriedades de um n-campo de vetores gradiente (∇ f1; : : : ∇fn) de uma aplicação de Morin f : M → Rn, com dim M ≥ n, na segunda parte deste trabalho, nós introduzimos o conceito de n-campos de Morin para n-campos de vetores que não são necessariamente gradientes. Nós também generalizamos o resultado de T. Fukuda [3, Theorem 1], que estabelece uma equivalência módulo 2 entre a característica de Euler de uma variedade diferenciável M e a característica de Euler dos conjuntos singulares de uma aplicação de Morin definida sobre M, para o contexto dos n-campos de Morin. / In this work, we revisit results of T. Fukuda and N. Dutertre and T. Fukui on the topology of Morin maps. In particular, we give a new proof for Dutertre-Fukui\'s Theorem [2, Theorem 6.2] when N = Rn, using Morse Theory for manifolds with boundary. Based on the properties of a gradient n-vector field (∇ f1; : : : ∇ fn) of a Morin map f : M → Rn, where dim M ≥ n, in the second part of this work, we introduce the concept of Morin n-vector field for n-vector fields V = (V1; : : : ; Vn) that are not necessarily gradients. We also generalize the result of T. Fukuda [3, Theorem 1], which establishes a module 2 equivalence between Euler\'s characteristic of a manifold M and Euler\'s characteristic of the singular sets of a Morin map defined on M, to the context of Morin n-vector fields.

Estudamos singularidades de famílias de matrizes simétricas. O objetivo é classificar as singularidades simples de tais famílias e estudar a geometria de alguns objetos associados a elas / We study the singularities of families of symmetric matrices. The aim of this work is to classify simple singularities of such families and study the geometry of some objects associated to them

Inspired by studies on the regularity of solutions to the fractional Navier-Stokes system and the impact of noise on singularity formation in hydrodynamic models, we investigated these issues within the framework of the fractional 1D Burgers equation. Initially, our research concentrated on the deterministic scenario, where we conducted precise numerical computations to understand the dynamics in both subcritical and supercritical regimes. We utilized a pseudo-spectral approach with automated resolution refinement for discretization in space combined with a hybrid Crank-Nicolson/ Runge-Kutta method for time discretization.We estimated the blow-up time by analyzing the evolution of enstrophy (H1 seminorm) and the width of the analyticity strip. Our findings in the deterministic case highlighted the interplay between dissipative and nonlinear components, leading to distinct dynamics and the formation of shocks and finite-time singularities. In the second part of our study, we explored the fractional Burgers equation under the influence of linear multiplicative noise. To tackle this problem, we employed the Milstein Monte Carlo approach to approximate stochastic effects. Our statistical analysis of stochastic solutions for various noise magnitudes showed that as noise amplitude increases, the distribution of blow-up times becomes more non-Gaussian. Specifically, higher noise levels result in extended mean blow-up time and increase its variability, indicating a regularizing effect of multiplicative noise on the solution. This highlights the crucial role of stochastic perturbations in influencing the behavior of singularities in such systems. Although the trends are rather weak, they nevertheless are consistent with the predictions of the theorem of [41]. However, there is no evidence for a complete elimination of blow-up, which is probably due to the fact that the noise amplitudes considered were not sufficiently large. This highlights the crucial role of stochastic perturbations in influencing the behavior of singularities in such systems. / Thesis / Master of Science (MSc)