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This thesis proposes a novel method for locating a p-dimensional invariant torus of an n-dimensional map. A set of non-linear equations is formulated and solved using the Newton-Raphson scheme. The method requires a set of sampled points on a guess invariant torus. An interpolant is passed through these points to compute the pointwise shift on the invariant torus, which is used to formulate the equation of invariance for the torus under the given map. The principal application of this method is to locate invariant tori of continuous systems. These tori occur for continuous dynamical systems having quasiperiodic orbits in state space. The discretization of the continuous system in terms of a map is accomplished in terms of its flow function. Results for one-dimensional invariant tori in two and three-dimensional state space and for two-dimensional invariant tori in three and four-dimensional maps are presented. / Master of Science

Probability-one homotopy algorithms have strong convergence characteristics under mild assumptions. Such algorithms for mixed complementarity problems (MCPs) have potentially wide impact because MCPs are pervasive in science and engineering. A probability-one homotopy algorithm for MCPs was developed earlier by Billups and Watson based on the default homotopy mapping. This algorithm had guaranteed global convergence under some mild conditions, and was able to solve most of the MCPs from the MCPLIB test library. This thesis extends that work by presenting some other homotopy mappings, enabling the solution of all the remaining problems from MCPLIB. The homotopy maps employed are the Newton homotopy and homotopy parameter embeddings. / Master of Science

Tableau d'honneur de la Faculté des études supérieures et postdorales, 2015-2016 / Dans ce mémoire, une méthode assistée numériquement est introduite et utilisée afin de montrer l'existence d'une connexion homocline à zéro pour l'équation du pont suspendu. Cette méthode, basée sur l'utilisation du théorème de contraction de Banach, permet d'obtenir les points fixes de l'opérateur de Newton légèrement modifié. La méthode ainsi que son cadre théorique sont introduits au premier chapitre. L'espace de Banach sur lequel sera définit l'opérateur ainsi que la manière de construire l'approximation de l'inverse utilisée pour l'opérateur sont les éléments majeurs du cadre théorique. Par la suite, la méthode est utilisée dans le Chapitre 2 pour prouver rigoureusement la validité de l'approximation numérique utilisée pour la variété stable locale. Puis cette approximation est réutilisée pour prouver l'existence de la connexion homocline. Cette preuve est à nouveau effectuée en utilisant la méthode introduite au premier chapitre. Finalement, certains résultats des calculs numériques sont présentés pour conclure ce mémoire. / In this work, a numerically assisted technique is introduced in order to prove the existence of a homoclinic connexion to zero for the suspension bridge equation. This technique, based on the use of the Banach fixed point theorem, can provide the fixed point of a slightly modified version of the Newton operator. The technique and its theorical background are introduced in the first chapter. The Banach space on which the operator is defined and the way to construct the approximation of the inverse needed to define the operator are the major parts of the theoretical background. The method is then used to rigorously validate the numerical approximation used to parametrize the local stable manifold. This parametrization is used to find the homoclinic connexion we are looking for. This proof is also completed using the technique from the first chapter. Finally, some results and numerical approximations will be presented in the last chapter.

No / The Raman spectra of two historical specimens of human hair attributed to the engineer Robert Stephenson and scientist Sir Isaac Newton, preserved in private collections are reported. Comparisons are made with the Raman spectra of modern hair specimens and with hair from archaeological excavations. The hair spectra collected with a laser excitation of 785 nm are of a better quality than those collected using 1064 nm. The historical hair specimens are remarkably well-defined spectroscopically in terms of the amide I vibrational mode and the ν(SS), ascribed to a predominantly gauche–gauche–gauche CSSC conformation. The contrast with degraded hair specimens recovered from archaeological excavations is striking. The presence of a weak feature near 2590 cm−1 in the hair samples attributed to a ν(SH) vibration could be indicative of a reduction process operative on the CSSC cystine keratotic linkages and a possible origin of this is bacterial biodegradation identified histologically. This study demonstrates the molecular information available from non-destructive Raman spectroscopic analysis from single hair shafts or small bundles of fibres which complements information available from histological and destructive analytical techniques for rare biological specimens subjected to conservation or curation procedures in museums or private collections.

A presente dissertação reflete sobre a produção de dramaturgia contemporânea brasileira a partir do horizonte desvelado por duas obras em especial, Agreste, de Newton Moreno, e BR-3, de Bernardo Carvalho. O foco nos dois textos não anula o diálogo pertinente e necessário também com a esfera espetacular quando necessário. Através da ideia de deslocamento, analisamos, primeiramente, o devir errático associado às obras em duas instâncias; errar para além de sua própria esfera de gênero em busca do outro e do surgimento de novos de campos de subjetividade; errar como parte de uma demanda interna e estrutural através de personagens em constante movimento e da exploração de um Brasil longínquo e extenso. Em seguida, empenhados na investigação da atuação crítica de tais obras, tendo como hipótese sua relação não documental com o presente, analisaremos o tratamento da subjetividade na elaboração das personagens, e o tratamento da ideia de território. Para além da dualidade entre realidade e representação, descortina-se a possibilidade de narrativas imagéticas, ou seja, em que os espaços autorais e imaginários possuem grande importância. Tais considerações guiam, por último, o questionamento a respeito da dimensão política de tais obras, percorrendo, primeiramente, algumas divergências que marcam o chamado teatro político. / This work reflects on the production of Brazilian contemporary dramaturgy from the unveiled horizon by two works in particular: Agreste, by Newton Moreno, and BR-3, by Bernardo Carvalho. The focus on both texts does not annul the relevant and necessary dialogue with the spectacular realm when required. Through the idea of displacement we analyze first the erratic becoming associated with the works in two instances; err beyond their own sphere in search of others and the emergence of new fields of subjectivity; err as part of an internal and structural demand, characters in constant motion, exploring a distant and extensive Brazil. Then engaged in research the critical role of the works under the hypothesis of their non-documentary relationship with the present, we will examine the treatment of subjectivity in the development of the characters, and the treatment of the territory ideia. In addition to the duality between reality and representation, it becomes visible the possibility of imagery narratives, where the authorial and imaginary spaces are of great importance. Such considerations guide, lastly, the question about the political dimension of such works, browsing, first, some differences that mark the so-called political theater.

A presente dissertação reflete sobre a produção de dramaturgia contemporânea brasileira a partir do horizonte desvelado por duas obras em especial, Agreste, de Newton Moreno, e BR-3, de Bernardo Carvalho. O foco nos dois textos não anula o diálogo pertinente e necessário também com a esfera espetacular quando necessário. Através da ideia de deslocamento, analisamos, primeiramente, o devir errático associado às obras em duas instâncias; errar para além de sua própria esfera de gênero em busca do outro e do surgimento de novos de campos de subjetividade; errar como parte de uma demanda interna e estrutural através de personagens em constante movimento e da exploração de um Brasil longínquo e extenso. Em seguida, empenhados na investigação da atuação crítica de tais obras, tendo como hipótese sua relação não documental com o presente, analisaremos o tratamento da subjetividade na elaboração das personagens, e o tratamento da ideia de território. Para além da dualidade entre realidade e representação, descortina-se a possibilidade de narrativas imagéticas, ou seja, em que os espaços autorais e imaginários possuem grande importância. Tais considerações guiam, por último, o questionamento a respeito da dimensão política de tais obras, percorrendo, primeiramente, algumas divergências que marcam o chamado teatro político. / This work reflects on the production of Brazilian contemporary dramaturgy from the unveiled horizon by two works in particular: Agreste, by Newton Moreno, and BR-3, by Bernardo Carvalho. The focus on both texts does not annul the relevant and necessary dialogue with the spectacular realm when required. Through the idea of displacement we analyze first the erratic becoming associated with the works in two instances; err beyond their own sphere in search of others and the emergence of new fields of subjectivity; err as part of an internal and structural demand, characters in constant motion, exploring a distant and extensive Brazil. Then engaged in research the critical role of the works under the hypothesis of their non-documentary relationship with the present, we will examine the treatment of subjectivity in the development of the characters, and the treatment of the territory ideia. In addition to the duality between reality and representation, it becomes visible the possibility of imagery narratives, where the authorial and imaginary spaces are of great importance. Such considerations guide, lastly, the question about the political dimension of such works, browsing, first, some differences that mark the so-called political theater.