This thesis consists of an introduction and five papers, of which two contribute to the theory of directed random graphs and three to the theory of greedy walks on point processes.           We consider a directed random graph on a partially ordered vertex set, with an edge between any two comparable vertices present with probability p, independently of all other edges, and each edge is directed from the vertex with smaller label to the vertex with larger label. In Paper I we consider a directed random graph on ℤ2 with the vertices ordered according to the product order and we show that the limiting distribution of the centered and rescaled length of the longest path from (0,0) to (n, [na] ), a<3/14, is the Tracy-Widom distribution. In Paper II we show that, under a suitable rescaling, the closure of vertex 0 of a directed random graph on ℤ with edge probability n−1 converges in distribution to the Poisson-weighted infinite tree. Moreover, we derive limit theorems for the length of the longest path of the Poisson-weighted infinite tree.           The greedy walk is a deterministic walk on a point process that always moves from its current position to the nearest not yet visited point. Since the greedy walk on a homogeneous Poisson process on the real line, starting from 0, almost surely does not visit all points, in Paper III we find the distribution of the number of visited points on the negative half-line and the distribution of the index at which the walk achieves its minimum. In Paper IV we place homogeneous Poisson processes first on two intersecting lines and then on two parallel lines and we study whether the greedy walk visits all points of the processes. In Paper V we consider the greedy walk on an inhomogeneous Poisson process on the real line and we determine sufficient and necessary conditions on the mean measure of the process for the walk to visit all points.

We give a definition of the scanning map for configuration spaces that is equivariant under the action of the diffeomorphism group of the underlying manifold. We use this to extend the Bödigheimer-Madsen result for the stable splittings of the Borel constructions of certain mapping spaces from compact Lie group actions to all smooth actions. Moreover, we construct a stable splitting of configuration spaces which is equivariant under smooth group actions, completing a zig-zag of equivariant stable homotopy equivalences between mapping spaces and certain wedge sums of spaces. Finally we generalise these results to configuration spaces with twisted labels (labels in a fibre bundle subject to certain conditions) and extend the Bödigheimer-Madsen result to more mapping spaces.

Following the financial crisis of 2008, the need for more robust techniques to quantify the capital charge for risk management has become a pressing problem. Under Basel II/III, banks are allowed to calculate the capital charge using internally developed models subject to regulatory approval. An interesting problem for the regulator is to compare the resulting figures against the required capital under worst case scenarios. The existing literature on the latter problem, which is based on the marginal problem, assumes that no a-priori information is known about the dependencies of contributing risks. These problems are linear optimisation problems over a constrained set of probability measures, discretisation of which leads to large scale LPs. But this approach is very conservative and cannot be implemented robustly in practice, due to the scarcity of historical data. In our approach, we take a less conservative strategy by incorporating dependence information contained in the data in a form that still leads to LPs, an important feature of such problems due to their high dimensionality. Conceptually, our model is the discretisation of an infinite dimensional linear optimisation problem over a set of probability measures. For some specific cases we can prove strong duality, opening up the approach of discretising the dual instead of the primal. This approach is preferable, as it yields better numerical results. In this work we also apply our model to model-free path-dependent option pricing. Use of delayed column generation techniques allows us to solve problems several orders of magnitude larger than via the standard simplex algorithm. For high-dimensional LPs we also implement Nesterov's smoothing technique to solve the problems.

Esta pesquisa tem como núcleo uma investigação das condições que permitiram a Leibniz mobilizar o conceito de análise infinita como solução para o problema posto pela dupla e simultânea afirmação da existência de uma estrutura lógica do mundo, que subjaz a todo acontecimento e a toda substância, e da contingência desses mesmos acontecimentos e das substâncias que os produzem dinamicamente na ordem do tempo. Trata-se, portanto, de discutir e elucidar aquilo que, para o filósofo de Hannover, se constitui como raiz da contingência. Assim, sobre o cerne do problema, materializado nas noções de infinito e de análise infinita, coloca-se ainda, como uma segunda camada da pesquisa, a questão relativa à possibilidade, aberta ou não aos seres racionais criados, de produzirem alguma modificação em sua própria existência, e, mais profundamente, na ordem geral da natureza, pela ação de uma vontade livre. Com efeito, a posição da ideia de análise infinita parece bastar para excluir o princípio de contradição como razão suficiente do mundo e dos fatos que nele têm lugar, mas à primeira vista não é o bastante para assegurar aos seres humanos e a outros agentes racionais finitos que porventura haja a possibilidade real de escolha, donde a fragilidade, ainda persistente, da afirmação da contingência como componente do universo. Daí que um estudo do conceito de vontade em Leibniz se mostre tão necessário como etapa da construção da ideia de contingência quanto uma elucidação da noção de análise infinita. Tal estudo é realizado com base na leitura da correspondência com Clarke e no exame do conceito de leis da natureza, as quais para Leibniz nada mais são que atos da vontade divina. Há, enfim, um último elemento que parece se apresentar como base da contingência na filosofia leibniziana: trata-se da possibilidade de afirmar uma existência infinitesimal das substâncias possíveis, graças à qual é dado às substâncias inteligentes criadas participarem, por atos de vontade, da formação dos eventos que constituem sua realidade e a do conjunto das coisas. Dito isso, a dissertação tem como ordem de temas uma breve apresentação do problema do contingência em Leibniz, e do lugar que ele ocupa no tema mais amplo da liberdade tal como pensada pelo filósofo, seguida por estudo da definição leibniziana de proposição verdadeira que põe o primeiro obstáculo à afirmação da contingência , uma análise das conexões entre esse conceito e a tese das noções individuais completas, o exame do nexo entre essa tese e os conceitos de infinito e de expressão, uma breve elucidação dos princípios de contradição e de razão suficiente, e, por último, um estudo do conceito de vontade em Leibniz e de seus conceitos de possível e existente. / The core to this research is an investigation of the conditions that allowed Leibniz to mobilise the concept of infinite analysis as a solution for the issue posed by the double and simultaneous statement of the existence of a logical structure of the world, subjacent to every event and every substance, and of the contingency of these same events produced dynamically in the order of time by these substances. This dissertation, in that sense, will discuss and elucidate what, for Leibniz, constitutes as the root of contingency. Thus, concerning the centre of the problem materialised in the notions of infinite and infinite analysis, we further place, as a second research layer, the question related to the possibility, be it open or not to created rational beings, to produce any modification in the order of their own existences, and, plunging even deeper, in existing natural order, by act of free will. In effect, the position of the idea of infinite analysis seems enough to exclude the principle of contradiction as sufficient reason for the world and for the facts that take place in it, but at a first glance this is not enough to ensure human beings and other created rational agents there might be a real possibility of choice, where the fragility of the statement of contingency as a defining feature of the universe still persists. From there, the study of will in Leibniz presents itself as much necessary as a step for the construction of the concept contingency in the author as it is an elucidation of the notion of infinite analysis. That investigation is here made possible above all based on his correspondence with Clarke, one of the last exchanges of letters in which Leibniz engaged, and in examining the concept of laws of nature, which for Leibniz are acts of the divine will. Lastly, there is one element that presents itself as a base for contingency in Leibnizian Philosophy; that is the possibility of affirming na infinitesimal existence of possible substances, permitting created intelligent substances to participate, by acts of will, in the shaping of events that form their own reality and the universe. That said, the order of themes in this dissertation is as follows: a brief presentation of the problem of contingency in Leibniz, followed by a study around Leibnizian definition of true proposition which establishes the first barrier to the statement of contingency , an analysis of the connections between this concept and the thesis of complete individual notions, the examination of nexus between that thesis and the concepts of infinite and expression, a brief elucidation around the principles of contradiction and sufficient reason, and finally a study of the concept of will in Leibniz and of its concepts of possible and existing.

Cette thèse porte sur la question du fini et de l’infini dans la philosophie de Hegel. L’objectif est double. En premier lieu, elle vise à retracer l’influence exercée par la philosophie antique (principalement Platon et Aristote) et par la philosophie moderne (pour l’essentiel Kant et certains postkantiens) sur l’élaboration hégélienne des catégories de la finité et de l’infinité. En second lieu, elle étudie le développement systématique de la logique de l’infinité hégélienne à la lumière de cette influence. Il s’agit d’étudier, à travers une approche historique et critique, comment Hegel résout l’opposition traditionnelle du fini et de l’infini par sa théorie des deux infinis. A l’aune des conceptions de l’infinité-finie (la mauvaise infinité) et de l’infinité véritablement infinie (la véritable infinité), Hegel montre que le processus de la détermination du fini est un processus d’idéalisation qui supprime la contradiction du fini et de l’infini. Ainsi, l’enquête sur des concepts de finité et d’infinité permet de découvrir que l’idéalité spéculative est pour Hegel une réponse non seulement au problème traditionnel de leur articulation, mais aussi, plus généralement, aux problèmes soulevés par la caractérisation des idéalismes antiques et modernes. / This dissertation aims at investigating the problem of the finite and the infinite within Hegel’s philosophy. Its objective is twofold. Firstly, it begins with an examination of the impact of the ancient Greek philosophy (especially that of Plato and Aristotle) and Modern philosophy (especially that of Kant and the post-Kantians) on Hegel’s own understanding of the categories of the finite and the infinite. Secondly, it attempts to analyze the systematic development of the logic of Hegelian infinity in relation to the ancient and modern influences. By adopting an historical and critical approach, this work therefore focuses on the question of how Hegel comes to solve the traditional opposition between the finite and the infinite with the help of his theory of two infinities. With the distinction between a finite infinity (namely a spurious infinity) and an infinity which is itself infinite (namely a true infinity), Hegel shows that the process of determination of the finite is itself a process of idealization which overcomes the very contradiction of the finite and the infinite. The inquiry into the concepts of the finite and infinite thus enables us to understand that the speculative ideality is an answer not only to the traditional opposition between the finite and infinite, but also to the problems raised by the definition of ancient and modern idealisms.

Cette thèse porte sur un certain nombre de problèmes algorithmiques motivés par la planification temporelle automatisée et la vérification formelle des systèmes réactifs et finis. Nous nous sommes concentrés sur les méthodes théoriques des jeux pour obtenir de nouvelles connaissances, des limites de complexité améliorées et des algorithmes plus rapides pour les modèles suivants: réseaux temporels hyper, réseaux conditionnels Simples / Hyper temporels, jeux de mise à jour, jeux Muller McNaughton et jeux Mean Payoff / This dissertation deals with a number of algorithmic problems motivated by automated temporal planning and formal verification of reactive and finite state systems. We focused on game theoretical methods to obtain novel insights, improved complexity bounds, and faster algorithms for the following models: Hyper Temporal Networks, Conditional Simple/Hyper Temporal Networks, Update Games, Muller McNaughton Games, and Mean Payoff Games

Ce travail concerne l’écoulement incompressible et stationnaire autour d’un cylindre semi-infini en rotation, et ses propriétés de stabilité linéaire. L’effet de la courbure et de la rotation sur la stabilité de cet écoulement est étudié de manière systématique. Avant d’étudier la stabilité, nous calculons d’abord l’écoulement de base. A grand nombre de Reynolds, une couche limite se développe le long du cylindre, ce qui permet d’utiliser l’approximation de couche limite des équations de Navier–Stokes. Ces équations dépendent de deux paramètres de contrôle sans dimension, le nombre de Reynolds (Re) et le taux de rotation (S), et sont résolues numériquement pour obtenir les profils de vitesse et de pression pour une large gamme des paramètres de contrôle. Une couche limite initialement mince s’épaissit avec la distance axiale; ainsi, son épaisseur devient comparable et finalement plus importante que le rayon du cylindre. Au-delà d’un certain taux de rotation, les effets centrifuges conduisent `a un jet de paroi le long d’une portion du cylindre. L’extension axiale de ce jet augmente avec le taux de rotation. L’intensité du jet augmente aussi avec S. Des analyses asymptotiques de l’écoulement à grande distance axiale et à fort taux de rotation sont aussi présentées. L’analyse de stabilité linéaire du précédent écoulement est effectuée dans l’approximation locale. Après une décomposition en modes normaux, les équations des perturbations sont transformées en un problème de valeur propre `a fréquence complexe (ω). Ce problème dépend de cinq paramètres sans dimension: Re, S, la distance axiale normalisée (Z), le nombre d’onde axial (α) et le nombre d’onde azimutal (m). Les équations de stabilité sont résolues numériquement pour étudier les régions instables dans l’espace des paramètres. On observe que de faibles taux de rotation ont un effet important sur la stabilité de l’écoulement. Cette forte déstabilisation est associée à la présence d’un mode quasi-marginal pour le cylindre fixe et qui devient instable pour de petites valeurs de S. Ce phénomène est confirmé par une analyse en perturbation `a petit S. Sans rotation, l’écoulement est stable pour tout Re < 1060, et pour Z > 0.81. Mais, en présence d’une faible rotation, l’instabilité n’est plus limitée par une valeur minimale de Re ou un seuil en Z. Les courbes critiques dans le plan (Z, Re) sont calculées pour une large gamme de S et les conséquences pour la stabilité de l’écoulement discutées. Enfin, un développement asymptotique pour le nombre de Reynolds critique est obtenu, valable aux grandes valeurs de Z. / This work concerns the steady, incompressible flow around a semi-infinite, rotating cylinder and its linear-stability properties. The effect of cylinder curvature and rotation on the stability of this flow is investigated in a systematic manner. Prior to studying its stability, we first compute the basic flow. At large Reynolds numbers, a boundary layer develops along the cylinder. The governing equations are obtained using a boundary-layer approximation to the Navier–Stokes equations. These equations contain two non-dimensional control parameters: the Reynolds number (Re) and the rotation rate (S), and are numerically solved to obtain the velocity and pressure profiles for a wide range of control parameters. The initially thin boundary layer grows in thickness with axial distance, becoming comparable and eventually larger than the cylinder radius. Above a threshold rotation rate, a centrifugal effect leads to the presence of a wall jet for a certain range of streamwise distances. This range widens as the rotation rate increases. Furthermore, the wall jet strengthens as S increases. Asymptotic analyses of the flow at large streamwise distances and at large rotation rates are presented. A linear stability analysis of the above flow is carried out using a local-flow approximation. Upon normal-mode decomposition, the perturbation equations are transformed to an eigenvalue problem in complex frequency (ω). The problem depends on five non-dimensional parameters: Re, S, scaled streamwise direction (Z), streamwise wavenumber (α) and azimuthal wavenumber m. The stability equations are numerically solved to investigate the unstable regions in parameter space. It is found that small amounts of rotation have strong effects on flow stability. Strong destabilization by small rotation is associated with the presence of a nearly neutral mode of the non-rotating cylinder, which becomes unstable at small S. This is further quantified using smallS perturbation theory. In the absence of rotation, the flow is stable for all Re below 1060, and for Z above 0.81. However, in the presence of small rotation, the instability becomes unconstrained by a minimum Re or a threshold in Z. The critical curves in the (Z, Re) plane are computed for a wide range of S and the consequences for stability of the flow described. Finally, a large-Z asymptotic expansion of the critical Reynolds number is obtained.

The offshore wind potential of the Caribbean Sea has barely been exploited. Currently, the offshore wind power industry in Latin America and the Caribbean region is still at very early stages, leaving aside an important resource that otherwise could contribute to satisfy the growing energy demand of the zone. In this study the possibilities arising from a massive exploitation of the wind resource in the Caribbean Sea are assessed. The objective is to investigate if the resources contained in it would be sufficient to satisfy the energy demand of Latin America and the Caribbean, which is foreseen to be about 1900 TWh/year by 2020. To address this question, the “Infinite wind farm” concept is used as a simple way to model the meteorological behaviour and the wind speed in the area. The model is utilized in combination with the bathymetric data of the Caribbean Sea and with a simple economic analysis, to evaluate what the requirements to satisfy the energy demand would be in terms of area, number of turbines, and levelized cost of energy (LCoE). The assessment is performed utilizing different turbine sizes, and inter-turbine separations to find the combination that minimizes the LCoE. It is found that the energy demand of Latin America and the Caribbean could be satisfied using only 125000 km2 (4.5% of the total Caribbean Sea area) of waters shallower than 25m at a cost of 69 €/MWh, if the turbines were separated 6.5D from one another and if they had a rotor diameter of 250m. In that case, 47760 turbines should be installed using only conventional monopile foundations.

Pretendemos entender a filosofia de Espinosa, em especial, a sua Ética ordine geometrico demonstrata, a partir de uma operação conflituosa bem específica entre, por um lado, a perspectiva do transcendente (ou a teologia racional) e, por outro, um desejo de salvação mundana; entre o projeto da filosofia imanentista de Espinosa e um mundo submetido ao poder teológico-político; e entre o texto teológico e o método da escrita da filosofia de Espinosa. Tais operações estruturam o cerne de nosso trabalho, no qual visamos entender o nexo causal na passagem de um Deus sive natura absolutamente infinito para nós, os modos finitos desta mesma natureza, de maneira a chegarmos a um entendimento que possa nos garantir não apenas ser, mas tomar parte ativamente neste absolutamente infinito. Não só procuraremos caminhar neste solo conflituoso, mas ainda proporemos tratá-lo com um procedimento que em si enfatiza conflitos, pois visamos responder às nossas questões acerca da filosofia da imanência, de Deus, da passagem do infinito ao finito a partir de uma aproximação entre a obra de Espinosa e o complexo universo artístico da literatura, das artes plásticas e da música do século XVII barroco. Além disto, procuramos demonstrar a hipótese de que a singularidade da Ética enquanto texto, expressa por uma forma textual filosófica sem precedentes, produz uma questão conceitual extremamente complexa que se funde à própria idéia do absolutamente infinito. Pois se a síntese da geometria dos indivisíveis, do século XVII, fornece-nos uma nova idéia de infinito (como amplamente discutiremos) e se a ordem geométrica da demonstração da Ética é fruto desta mesma síntese, então o livro deve necessariamente trazer, já, em sua fartura textual esta idéia de infinito. (Continua) / We intended to understand Espinosa\'s philosophy, especially, his Ethics ordine geometrico demonstrata, starting from a very specific conflicting operation against, on one side, the perspective of the transcendent (or the rational theology) and, on other, a desire for a mundane salvation; between the project of Espinosa\'s immanentist philosophy and a world submitted to the theological-political power; and between the theological text and the method of writing of Espinosa\'s philosophy. Such operations structure the core of our work, in which we seek to understand the causal connection in the passage from a God sive natura, absolutely infinite, to us, the finite manners of his same nature, in way that we can arrive to an understanding that can guarantee to us not to be a part, but to take part actively in this absolutely infinite. Not only we will try to walk in this conflicting path, but we intend to treat it with a procedure that emphasizes conflicts in itself, for we aim to answer our subjects - concerning the philosophy of the immanence, God, and the passage from the infinite to the finite - dealing with an approach between Espinosa\'s work and the complex artistic universe of literature, visual arts and music from the Baroque XVII century. Farther, we intend to demonstrate the hypothesis that the singularity of the Ethics while a text, expressed by an unprecedented philosophical textual form, produces an extremely complex conceptual subject that merges to the same idea of the absolutely infinite present in the Ethics. For if the synthesis from the geometry of the indivisibles, of the XVII century, provide us a new idea of the infinite (as we will extensively discuss) and if the geometric order on the demonstration of the Ethics is a fruit of this same synthesis, then the book should necessarily bring, already, in its textual profusion 7 this idea of the infinite. In other words, the idea of the geometric-synthetic order, key to the formulation of the absolutely infinite, already takes place in the textual structure ordine geometric demonstrata of the Ethics. Thus, we look forward to demonstrate that the order of exposition of the text in the Ethics operates with the same idea expressed by its ontology (the idea that is also expressed in mathematics by the geometrical synthesis). Farther on, we will insist that the formal articulation of the Ethics renders to us patent the fruition of the infinite, because we believe that such work while a text and as text, already expresses to its reader the experience of this new synthesis of an indivisible absolutely infinite.

Hasdai Crescas (1340 -1411), foi filosofo, rabino e homem público, que viveu em um período muito turbulento para as comunidade judaicas ibéricas e provençais, do final da Idade Média. Crescas fez uma crítica veemente ao paradigma aristotélico recebido da falsifa, que foi usado por Maimônides para embasar e provar a existência, unidade e incorporeidade de Deus, conceituado no Guia dos Perplexos como o ser necessário absolutamente transcendente em relação aos seres contingentes. Crescas elabora, em Or Hashem ( Luz do Nome Divino), um conceito alternativo de ser necessário, no qual as duas noções antitéticas de imanência e transcendência divinas se relacionam à distinção, no seio do ser necessário entre sua essência simples e os seus infinitos atributos. A essência simples e inefável do ser necessário se expressa em infinitos atributos no ato eterno e constante de doar na univocidade do ser, seu bem, sua atualidade, aos infinitos entes contingentes. Crescas, advoga que universo apesar de ontologicamente contingente é infinito em sua atualidade. Deus é assim concebido como causa primeira eterna e constante, a enteléquia e Lugar do Mundo. / Hasdai Crescas (1340-1411) was a philosopher, rabbi and public man, who lived in a very turbulent period for the Iberian and Provençal Jewish communities of the late Middle Ages. Crescas made a vehement criticism of the Aristotelian paradigm received from falsifa, which was used by Maimonides to ground and prove the existence, unity, and incorporeality of God, which was conceptualized in the Guide of the Perplexed as the absolutely necessary transcendent being in relation to contingent beings. Crescas elaborates, in Or Hashem (Light of the Divine Name), an alternative concept of being necessary, in which the two antithetical notions of divine immanence and transcendence relate to the distinction, within the necessary being between its simple essence and its infinite attributes . The simple and ineffable essence of the necessary being is expressed in infinite attributes in the eternal and constant act of giving in the univocity of being, its good, its actuality, to the infinite contingent entities. Crescas, advocates that universe although ontologically contingent is infinite in its actuality. God is thus conceived as the eternal and constant first cause, the entelechy and Place of the World.