Made available in DSpace on 2014-06-11T19:27:08Z (GMT). No. of bitstreams: 0 Previous issue date: 2005-12-16Bitstream added on 2014-06-13T20:08:14Z : No. of bitstreams: 1 godoy_r_me_sjrp.pdf: 1009803 bytes, checksum: 156b74c6b6bc9b086abf40743673384e (MD5) / Em considerando-se dois subsistemas A e B com espaço de estados HA e HB e com o espaço do sistema total ( A+B ) associado ao produto tensorial HA ? HB, alguns vetores desse sistema total podem ser decompostos em um produto tensorial de dois vetores descrevendo o estado do sistema A e B. Quando essa decomposição não é possível, diz-se que os subsistemas estão emaranhados. Uma medida de emaranhamento utilizada é a entropia de von Neumann de um dos subsistemas. Neste trabalho utiliza-se a entropia de Tsallis, uma generalização da entropia de von Neumann, como medida de emaranhamento.Faz-se uma comparação entre essas duas entropias como medida do emaranhamento entre campos emergentes de um divisor de feixes óticos. / Let A and B be two subsystems with space of states HA and HB respectively, being the space of the total system (A + B) associated to the tensorial product HA ? HB; some vectors of the total system may be decomposed in a tensorial product of two vectors describing the state of system A and B . When this decomposition is not possible, we say that the subsystems are entangled. An usual measure of entanglement used in each one of the subsystems is called von Neumann entropy. In this work we use Tsallis' entropy, a generalization of the von Neumann's measure to entanglement. We compare the two entropies as a measure of the entanglement between emerging fields of an optical beam splitter.

In recent years, there has been an exponential growth in sequencing projects due to accelerated technological advances, leading to a significant increase in the amount of data and resulting in new challenges for biological sequence analysis. Consequently, the use of techniques capable of analyzing large amounts of data has been explored, such as machine learning (ML) algorithms. ML algorithms are being used to analyze and classify biological sequences, despite the intrinsic difficulty in extracting and finding representative biological sequence methods suitable for them. Thereby, extracting numerical features to represent sequences makes it statistically feasible to use universal concepts from Information Theory, such as Tsallis and Shannon entropy. In this study, we propose a novel Tsallis entropy-based feature extractor to provide useful information to classify biological sequences. To assess its relevance, we prepared five case studies: (1) an analysis of the entropic index q; (2) performance testing of the best entropic indices on new datasets; (3) a comparison made with Shannon entropy and (4) generalized entropies; (5) an investigation of the Tsallis entropy in the context of dimensionality reduction. As a result, our proposal proved to be effective, being superior to Shannon entropy and robust in terms of generalization, and also potentially representative for collecting information in fewer dimensions compared with methods such as Singular Value Decomposition and Uniform Manifold Approximation and Projection.

Os avanços na pesquisa em biologia molecular e bioquímica permitiram o desenvolvimento de técnicas capazes de extrair informações moleculares de milhares de genes simultaneamente, como DNA Microarrays, SAGE e, mais recentemente RNA-Seq, gerando um volume massivo de dados biológicos. O mapeamento dos níveis de transcrição dos genes em larga escala é motivado pela proposição de que o estado funcional de um organismo é amplamente determinado pela expressão de seus genes. No entanto, o grande desafio enfrentado é o pequeno número de amostras (experimentos) com enorme dimensionalidade (genes). Dessa forma, se faz necessário o desenvolvimento de novas técnicas computacionais e estatísticas que reduzam o erro de estimação intrínseco cometido na presença de um pequeno número de amostras com enorme dimensionalidade. Neste contexto, um foco importante de pesquisa é a modelagem e identificação de redes de regulação gênica (GRNs) a partir desses dados de expressão. O objetivo central nesta pesquisa é inferir como os genes estão regulados, trazendo conhecimento sobre as interações moleculares e atividades metabólicas de um organismo. Tal conhecimento é fundamental para muitas aplicações, tais como o tratamento de doenças, estratégias de intervenção terapêutica e criação de novas drogas, bem como para o planejamento de novos experimentos. Nessa direção, este trabalho apresenta algumas contribuições: (1) software de seleção de características; (2) nova abordagem para a geração de Redes Gênicas Artificiais (AGNs); (3) função critério baseada na entropia de Tsallis; (4) estratégias alternativas de busca para a inferência de GRNs: SFFS-MR e SFFS-BA; (5) investigação biológica das redes gênicas envolvidas na biossíntese de tiamina, usando a Arabidopsis thaliana como planta modelo. O software de seleção de características consiste de um ambiente de código livre, gráfico e multiplataforma para problemas de bioinformática, que disponibiliza alguns algoritmos de seleção de características, funções critério e ferramentas de visualização gráfica. Em particular, implementa um método de inferência de GRNs baseado em seleção de características. Embora existam vários métodos propostos na literatura para a modelagem e identificação de GRNs, ainda há um problema muito importante em aberto: como validar as redes identificadas por esses métodos computacionais? Este trabalho apresenta uma nova abordagem para validação de tais algoritmos, considerando três aspectos principais: (a) Modelo para geração de Redes Gênicas Artificiais (AGNs), baseada em modelos teóricos de redes complexas, os quais são usados para simular perfis temporais de expressão gênica; (b) Método computacional para identificação de redes gênicas a partir de dados temporais de expressão; e (c) Validação das redes identificadas por meio do modelo AGN. O desenvolvimento do modelo AGN permitiu a análise e investigação das características de métodos de inferência de GRNs, levando ao desenvolvimento de um estudo comparativo entre quatro métodos disponíveis na literatura. A avaliação dos métodos de inferência levou ao desenvolvimento de novas metodologias para essa tarefa: (a) uma função critério, baseada na entropia de Tsallis, com objetivo de inferir os inter-relacionamentos gênicos com maior precisão; (b) uma estratégia alternativa de busca para a inferência de GRNs, chamada SFFS-MR, a qual tenta explorar uma característica local das interdependências regulatórias dos genes, conhecida como predição intrinsecamente multivariada; e (c) uma estratégia de busca, interativa e flutuante, que baseia-se na topologia de redes scale-free, como uma característica global das GRNs, considerada como uma informação a priori, com objetivo de oferecer um método mais adequado para essa classe de problemas e, com isso, obter resultados com maior precisão. Também é objetivo deste trabalho aplicar a metodologia desenvolvida em dados biológicos, em particular na identificação de GRNs relacionadas a funções específicas de Arabidopsis thaliana. Os resultados experimentais, obtidos a partir da aplicação das metodologias propostas, mostraram que os respectivos ganhos de desempenho foram significativos e adequados para os problemas a que foram propostos. / Thanks to recent advances in molecular biology and biochemistry, allied to an ever increasing amount of experimental data, the functional state of thousands of genes can now be extracted simultaneously by using methods such as DNA microarrays, SAGE, and more recently RNA-Seq, generating a massive volume of biological data. The mapping of gene transcription levels at large scale is motivated by the proposition that information of the functional state of an organism is broadly determined by its gene expression. However, the main limitation faced is the small number of samples (experiments) with huge dimensionalities (genes). Thus, it is necessary to develop new computational and statistics techniques to reduce the inherent estimation error committed in the presence of a small number of samples with large dimensionality. In this context, particularly important related investigations are the modeling and identification of gene regulatory networks from expression data sets. The main objective of this research is to infer how genes are regulated, bringing knowledge about the molecular interactions and metabolic activities of an organism. Such a knowledge is fundamental for many applications, such as disease treatment, therapeutic intervention strategies and drugs design, as well as for planning high-throughput new experiments. In this direction, this work presents some contributions: (1) feature selection software; (2) new approach for the generation of artificial gene networks (AGN); (3) criterion function based on Tsallis entropy; (4) alternative search strategies for GRNs inference: SFFS-MR and SFFS-BA; (5) biological investigation of GRNs involved in the thiamine biosynthesis by adopting the Arabidopsis thaliana as a model plant. The feature selection software is an open-source multiplataform graphical environment for bioinformatics problems, which supports many feature selection algorithms, criterion functions and graphic visualization tools. In particular, a feature selection method for GRNs inference is also implemented in the software. Although there are several methods proposed in the literature for the modeling and identification of GRNs, an important open problem regards: how to validate such methods and its results? This work presents a new approach for validation of such algorithms by considering three main aspects: (a) Artificial Gene Networks (AGNs) model generation through theoretical models of complex networks, which is used to simulate temporal expression data; (b) computational method for GRNs identification from temporal expression data; and (c) Validation of the identified AGN-based network through comparison with the original network. Through the development of the AGN model was possible the analysis and investigation of the characteristics of GRNs inference methods, leading to the development of a comparative study of four inference methods available in literature. The evaluation of inference methods led to the development of new methodologies for this task: (a) a new criterion function based on Tsallis entropy, in order to infer the genetic inter-relationships with better precision; (b) an alternative search strategy for the GRNs inference, called SFFS-MR, which tries to exploit a local property of the regulatory gene interdependencies, which is known as intrinsically multivariate prediction; and (c) a search strategy, interactive and floating, which is based on scale-free network topology, as a global property of the GRNs, which is considered as a priori information, in order to provide a more appropriate method for this class of problems and thereby achieve results with better precision. It is also an objective of this work, to apply the developed methodology in biological data, particularly in identifying GRNs related to specific functions of the Arabidopsis thaliana. The experimental results, obtained from the application of the proposed methodologies, indicate that the respective performances of each methodology were significant and adequate to the problems that have been proposed.

Os avanços na pesquisa em biologia molecular e bioquímica permitiram o desenvolvimento de técnicas capazes de extrair informações moleculares de milhares de genes simultaneamente, como DNA Microarrays, SAGE e, mais recentemente RNA-Seq, gerando um volume massivo de dados biológicos. O mapeamento dos níveis de transcrição dos genes em larga escala é motivado pela proposição de que o estado funcional de um organismo é amplamente determinado pela expressão de seus genes. No entanto, o grande desafio enfrentado é o pequeno número de amostras (experimentos) com enorme dimensionalidade (genes). Dessa forma, se faz necessário o desenvolvimento de novas técnicas computacionais e estatísticas que reduzam o erro de estimação intrínseco cometido na presença de um pequeno número de amostras com enorme dimensionalidade. Neste contexto, um foco importante de pesquisa é a modelagem e identificação de redes de regulação gênica (GRNs) a partir desses dados de expressão. O objetivo central nesta pesquisa é inferir como os genes estão regulados, trazendo conhecimento sobre as interações moleculares e atividades metabólicas de um organismo. Tal conhecimento é fundamental para muitas aplicações, tais como o tratamento de doenças, estratégias de intervenção terapêutica e criação de novas drogas, bem como para o planejamento de novos experimentos. Nessa direção, este trabalho apresenta algumas contribuições: (1) software de seleção de características; (2) nova abordagem para a geração de Redes Gênicas Artificiais (AGNs); (3) função critério baseada na entropia de Tsallis; (4) estratégias alternativas de busca para a inferência de GRNs: SFFS-MR e SFFS-BA; (5) investigação biológica das redes gênicas envolvidas na biossíntese de tiamina, usando a Arabidopsis thaliana como planta modelo. O software de seleção de características consiste de um ambiente de código livre, gráfico e multiplataforma para problemas de bioinformática, que disponibiliza alguns algoritmos de seleção de características, funções critério e ferramentas de visualização gráfica. Em particular, implementa um método de inferência de GRNs baseado em seleção de características. Embora existam vários métodos propostos na literatura para a modelagem e identificação de GRNs, ainda há um problema muito importante em aberto: como validar as redes identificadas por esses métodos computacionais? Este trabalho apresenta uma nova abordagem para validação de tais algoritmos, considerando três aspectos principais: (a) Modelo para geração de Redes Gênicas Artificiais (AGNs), baseada em modelos teóricos de redes complexas, os quais são usados para simular perfis temporais de expressão gênica; (b) Método computacional para identificação de redes gênicas a partir de dados temporais de expressão; e (c) Validação das redes identificadas por meio do modelo AGN. O desenvolvimento do modelo AGN permitiu a análise e investigação das características de métodos de inferência de GRNs, levando ao desenvolvimento de um estudo comparativo entre quatro métodos disponíveis na literatura. A avaliação dos métodos de inferência levou ao desenvolvimento de novas metodologias para essa tarefa: (a) uma função critério, baseada na entropia de Tsallis, com objetivo de inferir os inter-relacionamentos gênicos com maior precisão; (b) uma estratégia alternativa de busca para a inferência de GRNs, chamada SFFS-MR, a qual tenta explorar uma característica local das interdependências regulatórias dos genes, conhecida como predição intrinsecamente multivariada; e (c) uma estratégia de busca, interativa e flutuante, que baseia-se na topologia de redes scale-free, como uma característica global das GRNs, considerada como uma informação a priori, com objetivo de oferecer um método mais adequado para essa classe de problemas e, com isso, obter resultados com maior precisão. Também é objetivo deste trabalho aplicar a metodologia desenvolvida em dados biológicos, em particular na identificação de GRNs relacionadas a funções específicas de Arabidopsis thaliana. Os resultados experimentais, obtidos a partir da aplicação das metodologias propostas, mostraram que os respectivos ganhos de desempenho foram significativos e adequados para os problemas a que foram propostos. / Thanks to recent advances in molecular biology and biochemistry, allied to an ever increasing amount of experimental data, the functional state of thousands of genes can now be extracted simultaneously by using methods such as DNA microarrays, SAGE, and more recently RNA-Seq, generating a massive volume of biological data. The mapping of gene transcription levels at large scale is motivated by the proposition that information of the functional state of an organism is broadly determined by its gene expression. However, the main limitation faced is the small number of samples (experiments) with huge dimensionalities (genes). Thus, it is necessary to develop new computational and statistics techniques to reduce the inherent estimation error committed in the presence of a small number of samples with large dimensionality. In this context, particularly important related investigations are the modeling and identification of gene regulatory networks from expression data sets. The main objective of this research is to infer how genes are regulated, bringing knowledge about the molecular interactions and metabolic activities of an organism. Such a knowledge is fundamental for many applications, such as disease treatment, therapeutic intervention strategies and drugs design, as well as for planning high-throughput new experiments. In this direction, this work presents some contributions: (1) feature selection software; (2) new approach for the generation of artificial gene networks (AGN); (3) criterion function based on Tsallis entropy; (4) alternative search strategies for GRNs inference: SFFS-MR and SFFS-BA; (5) biological investigation of GRNs involved in the thiamine biosynthesis by adopting the Arabidopsis thaliana as a model plant. The feature selection software is an open-source multiplataform graphical environment for bioinformatics problems, which supports many feature selection algorithms, criterion functions and graphic visualization tools. In particular, a feature selection method for GRNs inference is also implemented in the software. Although there are several methods proposed in the literature for the modeling and identification of GRNs, an important open problem regards: how to validate such methods and its results? This work presents a new approach for validation of such algorithms by considering three main aspects: (a) Artificial Gene Networks (AGNs) model generation through theoretical models of complex networks, which is used to simulate temporal expression data; (b) computational method for GRNs identification from temporal expression data; and (c) Validation of the identified AGN-based network through comparison with the original network. Through the development of the AGN model was possible the analysis and investigation of the characteristics of GRNs inference methods, leading to the development of a comparative study of four inference methods available in literature. The evaluation of inference methods led to the development of new methodologies for this task: (a) a new criterion function based on Tsallis entropy, in order to infer the genetic inter-relationships with better precision; (b) an alternative search strategy for the GRNs inference, called SFFS-MR, which tries to exploit a local property of the regulatory gene interdependencies, which is known as intrinsically multivariate prediction; and (c) a search strategy, interactive and floating, which is based on scale-free network topology, as a global property of the GRNs, which is considered as a priori information, in order to provide a more appropriate method for this class of problems and thereby achieve results with better precision. It is also an objective of this work, to apply the developed methodology in biological data, particularly in identifying GRNs related to specific functions of the Arabidopsis thaliana. The experimental results, obtained from the application of the proposed methodologies, indicate that the respective performances of each methodology were significant and adequate to the problems that have been proposed.

Cette thèse porte sur le lien entre la tomographie et la notion de copule. La tomographie à rayons X consiste à (re)construire la structure cachée d'un objet (une densité de matière, la distribution d'une quantité physique, ou une densité de loi conjointe) à partir de certaines données obtenues ou mesurées de l'objet (les projections, les radiographies, les densités marginales). Le lien entre les mesures et l'objet se modélise mathématiquement par la Transformée à Rayons X ou la Transformée de Radon. Par exemple, dans les problèmes d'imagerie en géométrie parallèle, lorsqu'on a seulement deux projections à deux angles de 0 et pi/2 (horizontale et verticale), le problème peut être identifié comme un autre problème très important en mathématique qui est la détermination d'une densité conjointe à partir de ses marginales. En se limitant à deux projections, les deux problèmes sont des problèmes mal posés au sens de Hadamard. Il faut alors ajouter de l'information a priori, ou bien des contraintes supplémentaires. L'apport principal de cette thèse est l'utilisation des critères de plusieurs entropies (Rényi, Tsallis, Burg, Shannon) permettant d'aboutir à une solution régularisée. Ce travail couvre alors différents domaines. Les aspects mathématiques de la tomographie via l'élément fondamental qui est la transformée de Radon. En probabilité sur la recherche d'une loi conjointe connaissant ses lois marginales d'où la notion de ``copule'' via le théorème de Sklar. Avec seulement deux projections, ce problème est extrêmement difficile. Mais en assimilant les deux projections (normalisées) aux densités marginales et l'image à reconstruire à une densité de probabilité, le lien se fait et les deux problèmes sont équivalents et peuvent se transposer dans le cadre statistique. Pour caractériser toutes les images possibles à reconstruire on a choisi alors l'outil de la théorie de probabilité, c'est-à-dire les copules. Et pour faire notre choix parmi les copules ou les images nous avons imposé le critère d'information a priori qui se base sur différentes entropies. L'entropie est une quantité scientifique importante car elle est utilisée dans divers domaines (en Thermodynamique, en théorie de l'information, etc). Ainsi, en utilisant par exemple l'entropie de Rényi nous avons découvert de nouvelles classes de copules. Cette thèse apporte de nouvelles contributions à l'imagerie, par l'interaction entre les domaines qui sont la tomographie et la théorie des probabilités et statistiques. / This thesis studies the relationship between Computed Tomography (CT) and the notion of copula. In X-ray tomography the objective is to (re)construct an image representing the distribution of a physical quantity (density of matter) inside of an object from the radiographs obtained all around the object called projections. The link between these images and the object is described by the X-ray transform or the Radon transform. In 2D, when only two projections at two angles 0 and pi/2 (horizontal and vertical) are available, the problem can be identified as another problem in mathematics which is the determination of a joint density from its marginals, hence the notion of copula. Both problems are ill-posed in the sense of Hadamard. It requires prior information or additional criteria or constraints. The main contribution of this thesis is the use of entropy as a constraint that provides a regularized solution to this ill-posed inverse problem. Our work covers different areas. The mathematics aspects of X-ray tomography where the fundamental model to obtain projections is based mainly on the Radon transform. In general this transform does not provide all necessary projections which need to be associated with certain regularization techniques. We have two projections, which makes the problem extremely difficult, and ill-posed but noting that if a link can be done, that is, if the two projections can be equated with marginal densities and the image to reconstruct to a probability density, the problem translates into the statistical framework via Sklar's theorem. And the tool of probability theory called "copula" that characterizes all possible reconstructed images is suitable. Hence the choice of the image that will be the best and most reliable arises. Then we must find techniques or a criterion of a priori information, one of the criteria most often used, we have chosen is a criterion of entropy. Entropy is an important scientific quantity because it is used in various areas, originally in thermodynamics, but also in information theory. Different types of entropy exist (Rényi, Tsallis, Burg, Shannon), we have chosen some as criteria. Using the Rényi entropy we have discovered new copulas. This thesis provides new contributions to CT imaging, the interaction between areas that are tomography and probability theory and statistics.

Kullback-Leibler relative-entropy or KL-entropy of P with respect to R defined as ∫xlnddPRdP , where P and R are probability measures on a measurable space (X, ), plays a basic role in the definitions of classical information measures. It overcomes a shortcoming of Shannon entropy – discrete case definition of which cannot be extended to nondiscrete case naturally. Further, entropy and other classical information measures can be expressed in terms of KL-entropy and hence properties of their measure-theoretic analogs will follow from those of measure-theoretic KL-entropy. An important theorem in this respect is the Gelfand-Yaglom-Perez (GYP) Theorem which equips KL-entropy with a fundamental definition and can be stated as: measure-theoretic KL-entropy equals the supremum of KL-entropies over all measurable partitions of X . In this thesis we provide the measure-theoretic formulations for ‘generalized’ information measures, and state and prove the corresponding GYP-theorem – the ‘generalizations’ being in the sense of R ´enyi and nonextensive, both of which are explained below. Kolmogorov-Nagumo average or quasilinear mean of a vector x = (x1, . . . , xn) with respect to a pmf p= (p1, . . . , pn)is defined ashxiψ=ψ−1nk=1pkψ(xk), whereψis an arbitrarycontinuous and strictly monotone function. Replacing linear averaging in Shannon entropy with Kolmogorov-Nagumo averages (KN-averages) and further imposing the additivity constraint – a characteristic property of underlying information associated with single event, which is logarithmic – leads to the definition of α-entropy or R ´enyi entropy. This is the first formal well-known generalization of Shannon entropy. Using this recipe of R´enyi’s generalization, one can prepare only two information measures: Shannon and R´enyi entropy. Indeed, using this formalism R´enyi characterized these additive entropies in terms of axioms of KN-averages. On the other hand, if one generalizes the information of a single event in the definition of Shannon entropy, by replacing the logarithm with the so called q-logarithm, which is defined as lnqx =x1− 1 −1 −q , one gets what is known as Tsallis entropy. Tsallis entropy is also a generalization of Shannon entropy but it does not satisfy the additivity property. Instead, it satisfies pseudo-additivity of the form x ⊕qy = x + y + (1 − q)xy, and hence it is also known as nonextensive entropy. One can apply R´enyi’s recipe in the nonextensive case by replacing the linear averaging in Tsallis entropy with KN-averages and thereby imposing the constraint of pseudo-additivity. A natural question that arises is what are the various pseudo-additive information measures that can be prepared with this recipe? We prove that Tsallis entropy is the only one. Here, we mention that one of the important characteristics of this generalized entropy is that while canonical distributions resulting from ‘maximization’ of Shannon entropy are exponential in nature, in the Tsallis case they result in power-law distributions. The concept of maximum entropy (ME), originally from physics, has been promoted to a general principle of inference primarily by the works of Jaynes and (later on) Kullback. This connects information theory and statistical mechanics via the principle: the states of thermodynamic equi- librium are states of maximum entropy, and further connects to statistical inference via select the probability distribution that maximizes the entropy. The two fundamental principles related to the concept of maximum entropy are Jaynes maximum entropy principle, which involves maximizing Shannon entropy and the Kullback minimum entropy principle that involves minimizing relative-entropy, with respect to appropriate moment constraints. Though relative-entropy is not a metric, in cases involving distributions resulting from relative-entropy minimization, one can bring forth certain geometrical formulations. These are reminiscent of squared Euclidean distance and satisfy an analogue of the Pythagoras’ theorem. This property is referred to as Pythagoras’ theorem of relative-entropy minimization or triangle equality and plays a fundamental role in geometrical approaches to statistical estimation theory like information geometry. In this thesis we state and prove the equivalent of Pythagoras’ theorem in the nonextensive formalism. For this purpose we study relative-entropy minimization in detail and present some results. Finally, we demonstrate the use of power-law distributions, resulting from ME-rescriptions of Tsallis entropy, in evolutionary algorithms. This work is motivated by the recently proposed generalized simulated annealing algorithm based on Tsallis statistics. To sum up, in light of their well-known axiomatic and operational justifications, this thesis establishes some results pertaining to the mathematical significance of generalized measures of information. We believe that these results represent an important contribution towards the ongoing research on understanding the phenomina of information. (For formulas pl see the original document) ii

In this thesis random walks similar to the Metropolis algorithm are investigated. Special emphasis is laid on different types of acceptance probabilities, namely Metropolis, Tsallis and Threshold Accepting. Equilibrium and relaxation properties as well as performance aspects in stochastic optimization are investigated. Analytical investigation of a simple system mimicking an harmonic oscillator yields that a variety of acceptance probabilities, including the abovementioned, result in an equilibrium distribution that is widely dominated by an exponential function. In the last chapter an optimal optimization schedule for the Tsallis acceptance probability for the idealized barrier is investigated. / In dieser Dissertation werden Random Walks ähnlich dem Metropolis Algorithmus untersucht. Es werden verschiedene Akzeptanzwahrscheinlichkeiten untersucht, dabei werden Metropolis, Tsallis und Threshold Accepting besonders betrachtet. Gleichgewichts- und Relaxationseigenschaften sowie Performanceaspekte im Bereich der stochastischen Optimierung werden untersucht. Die Analytische Betrachtung eines simplen, dem harmonischen Oszillator ähnlichen Systems zeigt, dass eine Reihe von Akzeptanzwahrscheinlichkeiten, eingeschlossen die oben Erwähnten, eine Gleichgewichtsverteilung ausbilden, die von einer Exponentialfunktion dominiert wird. Im letzten Kapitel wird der optimale Schedule für die Tsallis Akzeptanzwahrscheinlichkeit für eine idealisierte Barriere untersucht.

Ziel dieser Arbeit ist die Untersuchung der Sub- und Superdiffusion in fraktalen Strukturen. Der Fokus liegt auf zwei separaten Ansätzen, die entsprechend des Diffusionbereiches gewählt und variiert werden. Dadurch erhält man ein tieferes Verständnis und eine bessere Beschreibungsweise für beide Bereiche. Im ersten Teil betrachten wir subdiffusive Prozesse, die vor allem bei Transportvorgängen, z. B. in lebenden Geweben, eine grundlegende Rolle spielen. Hierbei modellieren wir den fraktalen Zustandsraum durch endliche Sierpinski Teppiche mit absorbierenden Randbedingungen und lösen dann die Mastergleichung zur Berechnung der Zeitentwicklung der Wahrscheinlichkeitsverteilung. Zur Charakterisierung der Diffusion auf regelmäßigen und zufälligen Teppichen bestimmen wir die Abfallzeit der Wahrscheinlichkeitsverteilung, die mittlere Austrittszeit und die Random Walk Dimension. Somit können wir den Einfluss zufälliger Strukturen auf die Diffusion aufzeigen. Superdiffusive Prozesse werden im zweiten Teil der Arbeit mit Hilfe der Diffusionsgleichung untersucht. Deren zweite Ableitung im Ort erweitern wir auf nichtganzzahlige Ordnungen, um die fraktalen Eigenschaften der Umgebung darzustellen. Die resultierende raum-fraktionale Diffusionsgleichung spannt ein Übergangsregime von der irreversiblen Diffusionsgleichung zur reversiblen Wellengleichung auf. Deren Lösungen untersuchen wir mittels verschiedener Entropien, wie Shannon, Tsallis oder Rényi Entropien, und deren Entropieproduktionsraten, welche natürliche Maße für die Irreversibilität sind. Das dabei gefundene Entropieproduktions-Paradoxon, d. h. ein unerwarteter Anstieg der Entropieproduktionsrate bei sinkender Irreversibilität des Prozesses, können wir nach geeigneter Reskalierung der Entropien auflösen. / The aim of this thesis is the examination of sub- and superdiffusive processes in fractal structures. The focus of the work concentrates on two separate approaches that are chosen and varied according to the corresponding regime. Thus, we obtain new insights about the underlying mechanisms and a more appropriate way of description for both regimes. In the first part subdiffusion is considered, which plays a crucial role for transport processes, as in living tissues. First, we model the fractal state space via finite Sierpinski carpets with absorbing boundary conditions and we solve the master equation to compute the time development of the probability distribution. To characterize the diffusion on regular as well as random carpets we determine the longest decay time of the probability distribution, the mean exit time and the Random walk dimension. Thus, we can verify the influence of random structures on the diffusive dynamics. In the second part of this thesis superdiffusive processes are studied by means of the diffusion equation. Its second order space derivative is extended to fractional order, which represents the fractal properties of the surrounding media. The resulting space-fractional diffusion equations span a linking regime from the irreversible diffusion equation to the reversible (half) wave equation. The corresponding solutions are analyzed by different entropies, as the Shannon, Tsallis or Rényi entropies and their entropy production rates, which are natural measures of irreversibility. We find an entropy production paradox, i. e. an unexpected increase of the entropy production rate by decreasing irreversibility of the processes. Due to an appropriate rescaling of the entropy we are able to resolve the paradox.

Cette thèse porte sur le lien entre la tomographie et la notion de copule. La tomographie à rayons X consiste à (re)construire la structure cachée d'un objet (une densité de matière, la distribution d'une quantité physique, ou une densité de loi conjointe) à partir de certaines données obtenues ou mesurées de l'objet (les projections, les radiographies, les densités marginales). Le lien entre les mesures et l'objet se modélise mathématiquement par la Transformée à Rayons X ou la Transformée de Radon. Par exemple, dans les problèmes d'imagerie en géométrie parallèle, lorsqu'on a seulement deux projections à deux angles de 0 et pi/2 (horizontale et verticale), le problème peut être identifié comme un autre problème très important en mathématique qui est la détermination d'une densité conjointe à partir de ses marginales. En se limitant à deux projections, les deux problèmes sont des problèmes mal posés au sens de Hadamard. Il faut alors ajouter de l'information a priori, ou bien des contraintes supplémentaires. L'apport principal de cette thèse est l'utilisation des critères de plusieurs entropies (Rényi, Tsallis, Burg, Shannon) permettant d'aboutir à une solution régularisée. Ce travail couvre alors différents domaines. Les aspects mathématiques de la tomographie via l'élément fondamental qui est la transformée de Radon. En probabilité sur la recherche d'une loi conjointe connaissant ses lois marginales d'où la notion de ''copule'' via le théorème de Sklar. Avec seulement deux projections, ce problème est extrêmement difficile. Mais en assimilant les deux projections (normalisées) aux densités marginales et l'image à reconstruire à une densité de probabilité, le lien se fait et les deux problèmes sont équivalents et peuvent se transposer dans le cadre statistique. Pour caractériser toutes les images possibles à reconstruire on a choisi alors l'outil de la théorie de probabilité, c'est-à-dire les copules. Et pour faire notre choix parmi les copules ou les images nous avons imposé le critère d'information a priori qui se base sur différentes entropies. L'entropie est une quantité scientifique importante car elle est utilisée dans divers domaines (en Thermodynamique, en théorie de l'information, etc). Ainsi, en utilisant par exemple l'entropie de Rényi nous avons découvert de nouvelles classes de copules. Cette thèse apporte de nouvelles contributions à l'imagerie, par l'interaction entre les domaines qui sont la tomographie et la théorie des probabilités et statistiques.

Orientador: Manoel Ferreira Borges Neto / Banca: Waldir Leite Roque / Banca: José Márcio Machado / Resumo: Nesta dissertação, além da apresentação de um ensaio teórico sobre a fundamentação da Mecânica Quântica, Computação, Informação Quântica, Criptografia e Entropias Quânticas, serão mostradas, de forma inédita, algumas implementações sobre o efeito da Entropia no Emaranhamento Quântico, importante para processos de transmissão da Informação Quântica, com o uso dos programas Mathematica e Matlab. Primeiramente e apresentado um breve histórico sobre a Computação Quântica e a Informação Quântica, junto com uma perspectiva do futuro. Logo em seguida uma breve introdu cão sobre a Mecânica Quântica, com o estudo de autovetores e autovalores e seus postulados, produtos tensoriais e o micro-universo. Na sequência um texto sucinto com os conceitos fundamentais da Computação Quântica como os bits quânticos, e portas lógicas. Além dos principais algoritmos quânticos. Depois passa-se a estudar a Informa ção Quântica, as operações quânticas, canais de inversão e polarização, para então chegar-se a Entropia, quando e feito um estudo comparativo entre as entropias de Von Neumann e Tsallis. E por fim um pouco de Criptografia Quântica. / Abstract: In this dissertation, beyond the presentation of a theoretical essay on the basis of the Quantum Mechanics, Computation, Quantum information, Quantum Criptografy and Entropies, it will also be shown, for rst time, some implementations on the e ect of the Entropy tests on Quantum Entanglement for processes of transmission of Quantum Information, through the uses Mathematica and Matlab Programs. First I present a historical brie ng on the Quantum Computation and Quantum Information, together with a perspective of the future. Afterwards it will shown on introduction on the Quantum Mechanics, and its postulates, and the micro-universe. In sequence, a brief text with the fundamental concepts of the Quantum Computation, as the quantum bits, logic gates, and the main quantum algorithms. Later we will start to study Quantum Information, the quantum operations, channels of inversion and polarization. Furthermore we will go to discuss Entropy, where it is made a comparative study of Entropies of Von Neumann and Tsallis. And nally a little of Quantum Criptografy will be worked out. / Mestre