La vitesse de convergence dans la loi forte des grands nombres de Kolmogorov est généralement quantifiée par des majorations fines de la queue de la fonction de répartition des sommes partielles. Une autre approche, à laquelle nous nous intéressons dans ce travail, consiste à considérer ce problème de vitesse de convergence sous un aspect de martingale généralisée (amart ou quasimartingale). Nous considérons successivement la loi des grands nombres de Kolmogorov pour des variables aléatoires indépendantes équidistribuées et deux de ses généralisations : la loi des grands nombres de Marcinkiewicz-Zygmund d'ordre p (1< p<2) et celle de Cesàro d'ordre α (0<α<1). Nous exhibons, pour chacune de ces lois des grands nombres, des conditions nécessaires et suffisantes d'intégrabilité pour que les sommes partielles aient un comportement d'amart ou de quasimartingale. Nous remarquons en particulier que la généralisation de certains résultats scalaires aux variables aléatoires à valeurs dans un espace de Banach nécessite de se placer dans un espace de type p. Nous concluons notre travail par quelques résultats dans le cas non équidistribué.

In this thesis we use the syntactic-semantic method of constructive type theory to give meaning to classical logic, in particular Gentzen's LK. We interpret a derivation of a classical sequent as a derivation of a contradiction from the assumptions that the antecedent formulas are true and that the succedent formulas are false, where the concepts of truth and falsity are taken to conform to the corresponding constructive concepts, using function types to encode falsity. This representation brings LK to a manageable form that allows us to split the succedent rules into parts. In this way, every succedent rule gives rise to a natural deduction style introduction rule. These introduction rules, taken together with the antecedent rules adapted to natural deduction, yield a natural deduction calculus whose subsequent interpretation in constructive type theory gives meaning to classical logic. The Gentzen-Prawitz inversion principle holds for the introduction and elimination rules of the natural deduction calculus and allows for a corresponding notion of convertibility. We take the introduction rules to determine the meanings of the logical constants of classical logic and use the induced type-theoretic elimination rules to interpret the elimination rules of the natural deduction calculus. This produces an interpretation injective with respect to convertibility, contrary to an analogous translation into intuitionistic predicate logic. From the interpretation in constructive type theory and the interpretation of cut by explicit substitution, we derive a full precision contraction relation for a natural deduction version of LK. We use a term notation to formalize the contraction relation and the corresponding cut-elimination procedure. The interpretation can be read as a Brouwer-Heyting-Kolmogorov (BHK) semantics that justifies classical logic. The BHK semantics utilizes a notion of classical proof and a corresponding notion of classical truth akin to Kolmogorov's notion of pseudotruth. We also consider a second BHK semantics, more closely connected with Kolmogorov's double-negation translation. The first interpretation reinterprets the consequence relation while keeping the constructive interpretation of truth, whereas the second interpretation reinterprets the notion of truth while keeping the constructive interpretation of the consequence relation. The first and second interpretations act on derivations in much the same way as Plotkin's call-by-value and call-by-name continuation-passing-style translations, respectively. We conclude that classical logic can be given a constructive semantics by laying down introduction rules for the classical logical constants. This semantics constitutes a proof interpretation of classical logic.

<p>This study examines the behaviour of the South African financial markets with regards to the Geometric Brownian motion process. It uses the daily, weekly, and monthly stock returns time series of some major securities trading in the South African financial market, more specifically the US dollar/Euro, JSE ALSI Total Returns Index, South African All Bond Index, Anglo American Corporation, Standard Bank, Sasol, US dollar Gold Price , Brent spot oil price, and South African white maize near future. The assumptions underlying the&nbsp / Geometric Brownian motion in finance, namely the stationarity, the normality and the independence of stock returns, are tested using both graphical (histograms and normal plots)&nbsp / and statistical test (Kolmogorov-Simirnov test, Box-Ljung statistic and Augmented Dickey-Fuller test) methods to check whether or not the Brownian motion as a model for South&nbsp / African financial markets holds. The Hurst exponent or independence index is also applied to support the results from the previous test. Theoretically, the independent or Geometric&nbsp / Brownian motion time series should be characterised by the Hurst exponent of &frac12 / . A value of a Hurst exponent different from that would indicate the presence of long memory or&nbsp / fractional Brownian motion in a time series. The study shows that at least one assumption is violated when the Geometric Brownian motion process is examined assumption by&nbsp / assumption. It also reveals the presence of both long memory and random walk or Geometric Brownian motion in the South African financial markets returns when the Hurst index analysis is used and finds that the Currency market is the most efficient of the South African financial markets. The study concludes that although some assumptions underlying the&nbsp / rocess are violated, the Brownian motion as a model in South African financial markets can not be rejected. It can be accepted in some instances if some parameters such as the Hurst exponent are added.</p>

La Tesis Doctoral está dedicada al estudio de ciertas propiedades de los radios sucesivos de los cuerpos convexos (funcionales definidos a partir de circunradios e inradios de proyecciones o secciones del cuerpo). Comenzamos estableciendo las nociones básicas necesarias para el desarrollo de los contenidos. A continuación calculamos los radios sucesivos de familias particulares de conjuntos (p-bolas, anchura constante, cuerpos tangenciales), y estudiamos la conexión existente entre estos funcionales y los números de Gelfand y Kolmogorov. En el tercer capítulo consideramos el problema de Pukhov-Perel'man sobre la mejor cota superior para un cierto cociente de radios, determinando desigualdades para problemas de este tipo que van a permitir mejorar los resultados existentes en ciertos casos. Finalmente, estudiamos cómo se relacionan los radios sucesivos de la suma de Minkowski (Firey) de dos cuerpos convexos con los correspondientes funcionales de los conjuntos, obteniendo los resultados óptimos en todos los casos. / The Doctoral Thesis is focused in the study of some properties of the successive radii of convex bodies (functionals defined by means of circumradii and inradii of projections or sections of the set). We start establishing the basic notions that will be needed further on. Next, we compute the successive radii of particular families of sets (p-balls, constant width sets and tangential bodies), and study the connection between these functionals and the Gelfand and Kolmogorov numbers. In the third chapter we consider the Pukhov-Perel'man problem on the best upper bound for a particular ratio of radii, determining inequalities for some problems of this type which will allow to improve the known results in particular cases. Finally we study how the successive radii of the (Firey)-Minkowski addition of two convex bodies are related with the corresponding functionals of the sets, obtaining the optimal results in all cases.

In an effort to minimize the harmful effects of intracellular ice formation (IIF) during cryopreservation of confluent tissues, computer simulations based on Monte Carlo methods were performed to predict the probability of IIF in confluent monolayers during various freezing procedures. To overcome the prohibitive computational costs of such simulations for large tissues, the well-known Johnson-Mehl-Avrami (JMA) model of crystallization kinetics was implemented as a continuum approximation of IIF in tissues. This model, which describes nucleation, growth, and impingement of crystals in a supercooled melt, is analogous to the process of intracellular ice formation and propagation in biological tissues. Based on the work of Weinberg and Kapral (1989), the JMA model was modified to account for finite-size effects, and was shown to predict accurately the results of freezing simulations in 1-D tissue constructs, for various propagation rates and tissue sizes. An initial analysis of IIF kinetics in 2-D tissues is also presented. The probability of IIF in 2-D liver tissue was measured experimentally during freezing of HepG2 cells cultured in monolayers, and compared to Monte Carlo simulations and predictions of the continuum model. The Avrami coefficient and exponent for IIF in HepG2 tissue were estimated to be k = 0.19 and n = 0.45.

<p>This study examines the behaviour of the South African financial markets with regards to the Geometric Brownian motion process. It uses the daily, weekly, and monthly stock returns time series of some major securities trading in the South African financial market, more specifically the US dollar/Euro, JSE ALSI Total Returns Index, South African All Bond Index, Anglo American Corporation, Standard Bank, Sasol, US dollar Gold Price , Brent spot oil price, and South African white maize near future. The assumptions underlying the&nbsp / Geometric Brownian motion in finance, namely the stationarity, the normality and the independence of stock returns, are tested using both graphical (histograms and normal plots)&nbsp / and statistical test (Kolmogorov-Simirnov test, Box-Ljung statistic and Augmented Dickey-Fuller test) methods to check whether or not the Brownian motion as a model for South&nbsp / African financial markets holds. The Hurst exponent or independence index is also applied to support the results from the previous test. Theoretically, the independent or Geometric&nbsp / Brownian motion time series should be characterised by the Hurst exponent of &frac12 / . A value of a Hurst exponent different from that would indicate the presence of long memory or&nbsp / fractional Brownian motion in a time series. The study shows that at least one assumption is violated when the Geometric Brownian motion process is examined assumption by&nbsp / assumption. It also reveals the presence of both long memory and random walk or Geometric Brownian motion in the South African financial markets returns when the Hurst index analysis is used and finds that the Currency market is the most efficient of the South African financial markets. The study concludes that although some assumptions underlying the&nbsp / rocess are violated, the Brownian motion as a model in South African financial markets can not be rejected. It can be accepted in some instances if some parameters such as the Hurst exponent are added.</p>

Que ce soit à des fins spéculatives ou pour financer des conflits armés, une grande opacité entoure les filières des concentrés de matières premières minérales dont la demande ne cesse d'augmenter. Compte-tenu de l'éloignement entre les sites primaires d'extraction et les sites de production de produits finis, il est difficile d'identifier l'origine de ces produits. Dans un souci de traçabilité des concentrés, l'établissement d'une carte d'identité du minerai permettrait le contrôle des échanges dans l'industrie minérale. Le problème peut être posé en termes d'inversion: remonter au minerai d'origine en étudiant le produit vendu. Deux stades doivent être distingués: 1) la caractérisation du minerai brut et 2) la " perte de mémoire " des caractéristiques du tout-venant au cours du traitement minéralurgique. Les paramètres retenus sont la composition minéralogique, l'identification de microfaciès caractéristiques des minéraux cibles, la pseudo-succession paragénétique, le contenu et la distribution en éléments mineurs de minéraux cibles. Les minéraux cibles retenus sont la pyrite pour son ubiquité, la sphalérite car elle est susceptible d'incorporer une grande variété d'éléments mineurs, éventuellement valorisants ainsi que la chalcopyrite car elle est souvent liée aux deux autres minéraux. La comparaison de la composition chimique des phases minérales est effectuée en calculant la distance de Kolmogorov-Smirnov et de Colin-White. Des tests ont été réalisés sur les gîtes de type amas sulfuré volcanogène. Ils ont montré que les caractéristiques retenues permettaient de distinguer les pyrites, les sphalérites et les chalcopyrites de deux gisements de la province Sud-Ibérique (IPB), de sept gisements de la province d'Oural et du fumeur noir actuel de Rainbow. Les cartes d'identité obtenues permettent de discriminer les différents sites (IPB, Oural et Rainbow) et les gisements d'une même province. Les paramètres minéralogiques et microtexturaux ont également été suivis au cours du traitement minéralurgique de la mine de Neves Corvo. Pour une chaîne de traitement donnée, le paramètre " perte de mémoire " est une estimation de l'erreur commise lors de l'inversion, mais aussi une façon de caractériser une succession d'opérations minéralurgiques.

Cette thèse s'interroge sur les notions d'aléa et de régularité des variations boursières. Nous démontrons sur le plan théorique, la compatibilité des principales théories financières (cf. efficience informationnelle, finance comportementale et approche conventionnaliste) avec l'impossibilité de battre la stratégie "buy and hold". Cette impossibilité est confirmée par les études statistiques dans la mesure où les régularités identifiées dans les séries financières ne permettent pas de prédire le sens des variations futures. Les modèles économétriques disponibles à présent offrent souvent un "hit score" insuffisant (<60%) pour réussir des tentatives fructueuses de "market timing". Une contribution de ce travail se trouve dans l'introduction du concept de complexité algorithmique en finance. Une approche générale est proposée pour estimer la "complexité de Kolmogorov" des séries de rentabilités: après un processus "discrétisation-effacement", des algorithmes de compression sans perte sont utilisés pour détecter des structures régulières qui ne sont pas toujours visibles aux yeux des tests statistiques. En étudiant le degré d'aléa des principaux marchés internationaux à une fréquence "tick-by-tick", on constate une complexité plus élevée pour Euronext-Paris que pour le NYSE et le NASDAQ. Nous expliquons ce résultat par une auto-corrélation plus élevée des volatilités inter-journalières aux Etats-Unis qu'en France. L'inefficacité de "market timing" étant soutenue aussi bien par les théories financières que par les observations empiriques, nous définissons la notion de "battre le marché" dans ce sens spécifique avec un modèle mathématique qui s'inscrit dans le cadre de la calculabilité.

Reinforcement Learning (RL) is a learning framework in which an agent learns a policy from continual interaction with the environment. A policy is a mapping from states to actions. The agent receives rewards as feedback on the actions performed. The objective of RL is to design autonomous agents to search for the policy that maximizes the expectation of the cumulative reward. When the environment is partially observable, the agent cannot determine the states with certainty. These states are called hidden in the literature. An agent that relies exclusively on the current observations will not always find the optimal policy. For example, a mobile robot needs to remember the number of doors went by in order to reach a specific door, down a corridor of identical doors. To overcome the problem of partial observability, an agent uses both current and past (memory) observations to construct an internal state representation, which is treated as an abstraction of the environment. This research focuses on how features of past events are extracted with variable granularity regarding the internal state construction. The project introduces a new method that applies Information Theory and decision tree technique to derive a tree structure, which represents the state and the policy. The relevance, of a candidate feature, is assessed by the Information Gain Ratio ranking with respect to the cumulative expected reward. Experiments carried out on three different RL tasks have shown that our variant of the U-Tree (McCallum, 1995) produces a more robust state representation and faster learning. This better performance can be explained by the fact that the Information Gain Ratio exhibits a lower variance in return prediction than the Kolmogorov-Smirnov statistical test used in the original U-Tree algorithm.

Made available in DSpace on 2016-06-02T20:05:59Z (GMT). No. of bitstreams: 1 DissLEBS.pdf: 1626270 bytes, checksum: c18112f89ed0a1eea09a198885cf2c2c (MD5) Previous issue date: 2007-06-14 / Financiadora de Estudos e Projetos / Abstract In this dissertation we will study how extended Poisson process can be applied to construct discrete probabilistic models. An Extended Poisson Process is a continuous time stochastic process with the state space being the natural numbers, it is obtained as a generalization of homogeneous Poisson process where transition rates depend on the current state of the process. From its transition rates and Chapman-Kolmogorov di¤erential equations, we can determine the probability distribution at any &#133;xed time of the process. Conversely, given any probability distribution on the natural numbers, it is possible to determine uniquely a sequence of transition rates of an extended Poisson process such that, for some instant, the unidimensional probability distribution coincides with the provided probability distribution. Therefore, we can conclude that extended Poisson process is as a very &#135;exible framework on the analysis of discrete data, since it generalizes all probabilistic discrete models. We will present transition rates of extended Poisson process which generate Poisson, Binomial and Negative Binomial distributions and determine maximum likelihood estima- tors, con&#133;dence intervals, and hypothesis tests for parameters of the proposed models. We will also perform a bayesian analysis of such models with informative and noninformative prioris, presenting posteriori summaries and comparing these results to those obtained by means of classic inference. / Nesta dissertação veremos como o proceso de Poisson estendido pode ser aplicado à construção de modelos probabilísticos discretos. Um processo de Poisson estendido é um processo estocástico a tempo contínuo com espaço de estados igual ao conjunto dos números naturais, obtido a partir de uma generalização do processo de Poisson homogê- neo onde as taxas de transição dependem do estado atual do processo. A partir das taxas de transição e das equações diferenciais de Chapman-Kolmogorov pode-se determinar a distribuição de probabilidades para qualquer tempo &#133;xado do processo. Reciprocamente, dada qualquer distribuição de probabilidades sobre o conjunto dos números naturais é pos- sível determinar, de maneira única, uma seqüência de taxas de transição de um processo de Poisson estendido tal que, para algum instante, a distribução unidimensional do processo coincide com a dada distribuição de probabilidades. Portanto, o processo de Poisson es- tendido se apresenta como uma ferramenta bastante &#135;exível na análise de dados discretos, pois generaliza todos os modelos probabilísticos discretos. Apresentaremos as taxas de transição dos processos de Poisson estendido que ori- ginam as distribuições de Poisson, Binomial e Binomial Negativa e determinaremos os estimadores de máxima verossimilhança, intervalos de con&#133;ança e testes de hipóteses dos parâmetros dos modelos propostos. Faremos também uma análise bayesiana destes mod- elos com prioris informativas e não informativas, apresentando os resumos a posteriori e comparando estes resultados com aqueles obtidos via inferência clássica.