In dieser Dissertation analysieren wir drei Aspekte von Konforme Feldtheorien (CFTs). Erstens betrachten wir Korrelationsfunktionen von sekundären Zuständen (SZ) in zweidimensionalen CFTs. Wir diskutieren eine rekursive Formel zu ihrer Berechnung und erstellen eine Computerimplementierung dieser Formel. Damit können wir jede Korrelationsfunktion von SZ des Vakuums erhalten und für Nicht-Vakuum-SZ den Korrelator als Differentialoperator, der auf den jeweiligen primären Korrelator wirkt, ausdrücken. Mit diesem Code untersuchen wir dann einige Verschränkungs- und Unterscheidbarkeitsmaße zwischen SZ, i.e. die Rényi-Entropie, den Spurquadratabstand und die Sandwich-Rényi-Divergenz. Mit unseren Ergebnissen können wir die Rényi Quanten-Null-Energie-Bedingung testen und stellen neue Werkzeuge zur Analyse der holographischen Beschreibung von SZ bereit. Zweitens untersuchen wir vierdimensionale Weyl-Fermionen auf verschiedenen Hintergründen. Unser Interesse gilt ihrer Spuranomalie, und der Frage, ob die Pontryagin-Dichte auftritt. Deshalb berechnen wir die Anomalien von Dirac-Fermionen, die an vektorielle und axiale Eichfelder gekoppelt sind, und dann auf einem metrisch-axialen Tensor Hintergrund. Geeignete Grenzwerte der Hintergründe erlauben es dann, die Anomalien von Weyl-Fermionen, die an Eichfelder gekoppelt sind, und in einer gekrümmten Raumzeit zu berechnen. Wir bestätigen das Fehlen der Pontryagin-Dichte in den Spuranomalien. Drittens liefern wir die holographische Beschreibung einer vierdimensionalen CFT mit einem irrelevanten Operator. Wenn der Operator eine ganzzahlige konforme Dimension hat, modifiziert sein Vorhandensein in der CFT die Weyl-Transformation der Metrik, was wiederum die Spuranomalie ändert. Unter Ausnutzung der Äquivalenz zwischen Diffeomorphismen im Inneren und Weyl-Transformationen auf dem Rand, berechnen wir diese Modifikationen mithilfe der dualen Gravitationstheorie. Unsere Ergebnisse repräsentieren einen weiteren Test der AdS/CFT-Korrespondenz. / Conformal field theories (CFTs) are amongst the most studied field theories and they offer a remarkable playground in modern theoretical physics. In this thesis we analyse three aspects of CFTs in different dimensions. First, we consider correlation functions of descendant states in two-dimensional CFTs. We discuss a recursive formula to calculate them and provide a computer implementation of it. This allows us to obtain any correlation function of vacuum descendants, and for non-vacuum descendants to express the correlator as a differential operator acting on the respective primary correlator. With this code, we study some entanglement and distinguishability measures between descendant states, i.e. the Rényi entropy, trace square distance and sandwiched Rényi divergence. With our results we can test the Rényi Quantum Null Energy Condition and provide new tools to analyse the holographic description of descendant states. Second, we study four-dimensional Weyl fermions on different backgrounds. Our interest is in their trace anomaly, where the Pontryagin density has been claimed to appear. To ascertain this possibility, we compute the anomalies of Dirac fermions coupled to vector and axial non-abelian gauge fields and then in a metric-axial-tensor background. Appropriate limits of the backgrounds allow to recover the anomalies of Weyl fermions coupled to non-abelian gauge fields and in a curved spacetime. In both cases we confirm the absence of the Pontryagin density in the trace anomalies. Third, we provide the holographic description of a four-dimensional CFT with an irrelevant operator. When the operator has integer conformal dimension, its presence in the CFT modifies the Weyl transformation of the metric, which in turns modifies the trace anomaly. Exploiting the equivalence between bulk diffeomorphisms and boundary Weyl transformations, we compute these modifications from the dual gravity theory. Our results represent an additional test of the AdS/CFT conjecture.

Dans ce travail, nous étudions les propriétés de le chemin le plus court entre deux sequences, et en présente trois principaux résultats: Le premier est le comportement asymptotique de le chemin le plus court comme une fonction linéaire de la taille de les cylindres. Le deuxième est un principe de grandes déviations pour cette quantitée. Et le troisième est de la convergence en distribution d\'une version re-mise à l\'échelle de cette variable aleatorie. / Definimos a função caminho mais curto como sendo a mínima quantidade de passos para que uma realização do processo com condição inicial y atinja um conjunto-alvo x. Para tal função, provamos três resultados principais: um teorema de concentração de massa, um princípio de grandes desvios, e uma convergência em distribuição.

Definimos a função caminho mais curto como sendo a mínima quantidade de passos para que uma realização do processo com condição inicial y atinja um conjunto-alvo x. Para tal função, provamos três resultados principais: um teorema de concentração de massa, um princípio de grandes desvios, e uma convergência em distribuição. / Dans ce travail, nous étudions les propriétés de le chemin le plus court entre deux sequences, et en présente trois principaux résultats: Le premier est le comportement asymptotique de le chemin le plus court comme une fonction linéaire de la taille de les cylindres. Le deuxième est un principe de grandes déviations pour cette quantitée. Et le troisième est de la convergence en distribution d\'une version re-mise à l\'échelle de cette variable aleatorie.

A caracterização de redes e o estudo de sistemas, ambos utilizando grafos, é algo muito usado por várias áreas científicas. Uma das linhas deste estudo é denominada de grafos aleatórios, que por sua vez auxilia na criação de modelos para análise de redes reais. Consideramos um modelo de grafo aleatório não homogêneo criado por Kang, Pachón e Rodríguez (2016), cuja construção é feita a partir da realização do grafo binomial G(n; p). Para este modelo, estudamos argumentos e métodos usados para encontrar resultados sobre o limiar de conectividade, importante propriedade relacionada a existência assintótica de vértices e componentes isolados. Em seguida, constatamos algumas características positivas e negativas a respeito da utilização do grafo para modelar redes reais complexas, onde usamos de simulações computacionais e medidas topológicas. / The characterization of networks and the study of systems, both using graphs, is very used by several scientific areas. One of the lines of this study is called random graphs, which in turn assists in creating models for the analysis of real networks. We consider an inhomogeneous random graph model created by Kang, Pachón e Rodríguez (2016), where its construction is made from the realization of the binomial graph G(n; p). For this model, we studied the arguments and methods used to find results on the connectivity threshold, important property related to asymptotic existence of vertices and isolated components. Then we found some positive and negative characteristics about the use of the graph to model complex real networks, using computer simulations and topological measures.

A caracterização de redes e o estudo de sistemas, ambos utilizando grafos, é algo muito usado por várias áreas científicas. Uma das linhas deste estudo é denominada de grafos aleatórios, que por sua vez auxilia na criação de modelos para análise de redes reais. Consideramos um modelo de grafo aleatório não homogêneo criado por Kang, Pachón e Rodríguez (2016), cuja construção é feita a partir da realização do grafo binomial G(n; p). Para este modelo, estudamos argumentos e métodos usados para encontrar resultados sobre o limiar de conectividade, importante propriedade relacionada a existência assintótica de vértices e componentes isolados. Em seguida, constatamos algumas características positivas e negativas a respeito da utilização do grafo para modelar redes reais complexas, onde usamos de simulações computacionais e medidas topológicas. / The characterization of networks and the study of systems, both using graphs, is very used by several scientific areas. One of the lines of this study is called random graphs, which in turn assists in creating models for the analysis of real networks. We consider an inhomogeneous random graph model created by Kang, Pachón e Rodríguez (2016), where its construction is made from the realization of the binomial graph G(n; p). For this model, we studied the arguments and methods used to find results on the connectivity threshold, important property related to asymptotic existence of vertices and isolated components. Then we found some positive and negative characteristics about the use of the graph to model complex real networks, using computer simulations and topological measures.

Le fil conducteur de cette thèse, composée de trois parties, est la notion de branchement.Le premier chapitre est consacré à l'arbre de Yule et à l'arbre binaire de recherche. Nous obtenons des résultats d'oscillations asymptotiques de l'espérance, de la variance et de la distribution de la hauteur de ces arbres, confirmant ainsi une conjecture de Drmota. Par ailleurs, l'arbre de Yule pouvant être vu comme une marche aléatoire branchante évoluant sur un réseau, nos résultats permettent de mieux comprendre ce genre de processus.Dans le second chapitre, nous étudions le nombre de particules tuées en 0 d'un mouvement brownien branchant avec dérive surcritique conditionné à s'éteindre. Nous ferons enfin apparaître une nouvelle phase de transition pour la queue de distribution de ces variables.L'objet du dernier chapitre est le graphe d'Erdős–Rényi dans le cas critique : $G(n,1/n)$. En introduisant un couplage et un changement d'échelle, nous montrerons que, lorsque $n$ augmente les composantes de ce graphe évoluent asymptotiquement selon un processus de coalescence-fragmentation qui agit sur des graphes réels. La partie coalescence sera de type multiplicatif et les fragmentations se produiront selon un processus ponctuel de Poisson sur ces objets. / This thesis is composed by three chapters and its main theme is branching processes.The first chapter is devoted to the study of the Yule tree and the binary search tree. We obtain oscillation results on the expectation, the variance and the distribution of the height of these trees and confirm a Drmota's conjecture. Moreover, the Yule tree can be seen as a particular instance of lattice branching random walk, our results thus allow a better understanding of these processes.In the second chapter, we study the number of particles killed at 0 for a Brownian motion with supercritical drift conditioned to extinction. We finally highlight a new phase transition in terms of the drift for the tail of the distributions of these variables.The main object of the last chapter is the Erdős–Rényi graph in the critical case: $G(n,1/n)$. By using coupling and scaling, we show that, when $n$ grows, the scaling process is asymptotically a coalescence-fragmentation process which acts on real graphs. The coalescent part is of multiplicative type and the fragmentations happen according a certain Poisson point process.

The incredible ability of human beings to quickly detect the prominent or salient regions in an image is often taken for granted. To be able to reproduce this intelligent ability in computer vision systems remains quite a challenge. This ability is of paramount importance to perception and image understanding since it accelerates the image analysis process, thereby allowing higher vision processes such as recognition to have a focus of attention. In addition to this, human eye fixation points occurring during the early stages of visual processing, often correspond to the loci of salient image regions. These regions provide us with assistance in determining the interesting parts of an image and they also lend support to our ability to discriminate between different objects in a scene. Salient regions attract our immediate attention without requiring an exhaustive scan of a scene. In essence, saliency can be defined as the quality of an image region that enables it to stand out in relation to its neighbors. Saliency is often approached in either one of two ways. The bottom-up saliency approach refers to mechanisms which are image-driven and independent of the knowledge in an image, whereas the top-down saliency approach refers to mechanisms which are task-oriented and make use of the prior knowledge about a scene. In this thesis, we present a bottom-up measure of saliency based on the relationships exhibited among image features. The perceived structure in an image is determined more by the relationships among features rather than the individual feature attributes. From this standpoint, we aim to capture the organization within an image by employing relational distributions derived from distance and gradient direction relationships exhibited between image primitives. The Rényi entropy of the relational distribution tends to be lower if saliency is exhibited for some image region in the local pixel neighborhood over which the distribution is defined. This notion forms the foundation of our measure. Correspondingly, results of our measure are presented in the form of a saliency map, highlighting salient image regions. We show results on a variety of real images from various datasets. We evaluate the performance of our measure in relation to a dominant saliency model and obtain comparable results. We also investigate the biological plausibility of our method by comparing our results to those captured by human fixation maps. In an effort to derive meaningful information from an image, we investigate the significance of scale relative to our saliency measure, and attempt to determine optimal scales for image analysis. In addition to this, we extend a perceptual grouping framework by using our measure as an optimization criterion for determining the organizational strength of edge groupings. As a result, the use of ground truth images is circumvented.

In the course of evolution, genomes diverge from their ancestors either via global mutations and by rearrangement of their chromosomal segments, or through local mutations within their genes. In this thesis (Chapters: 2, 3 and 4) we analyze the evolution of genomes based on different rearrangement operations including: in Chapter 2 both restricted and unrestricted double-cut-and-join (DCJ) operations, in Chapter 3 both internal and general reversal and translocation (IRT and HP, respectively) operations, and in Chapter 4 translocation, weighted reversal (WR) and maximum length reversal (MLR) operations. Based on the rearrangement operation chosen we can model the evolution of genomes as a discrete or continuous-time Markov chain process on the space of signed genomes. For each model of evolution, we study the stochastic process by investigating the time up to which the difference between the number of operations along the evolutionary trajectory and the edit distance of the genome from its ancestor is negligible, as soon as these two values starts diverging drastically from one another we say the process escapes from parsimony. One of the major parameters in the known edit distance formulas between any two genomes (such as reversal, DCJ, IRT, HP and translocation) is the number of cycles in their breakpoint graph. For DCJ, IRT and HP models by adopting the method elaborated by Berestycki and Durret, we estimate the number of cycles in the breakpoint graph of the genome at time t and its ancestor by the number of tree components of the random graph constructed from the model of evolution at time t, which is an Erdös-Rényi. We also proved that for each of the DCJ, IRT and HP models of evolution, the process on a genome of size n is bound to its parsimonious estimate up to t ≈ n/2 steps. Since the random graph constructed from the models of evolution for the translocation, WR and MLR processes are not Erdös-Rényi, the proofs of their parsimony- bound require more advanced mathematical tools, however our simulation shows for the translocation, two types of WR, and MLR (except for reversals with very short maximum length) models, the escape from parsimony do not occur before n/2 steps, where n is the number of genes in the genome. A basic result in this field is due to Berestycki and Durrett, from 2006, who found that a random transposition (pairwise exchange of the elements in the corresponding permutation of the genome) evolves along its parsimonious path of evolution up to n/2 steps, where n is the number of the genes. Although, this transposition model is applicable solely for evolution of a unichromosomal ancestor which remains unichromosomal at each step t of the process; however for the DCJ, IRT, HP and translocation models the genomes are multichromosomal which increases the difficulty of the problem at hand. The models studied in Chapters 2 - 4 are all based on signed permutation representations of genomes, where each "gene" occurs exactly once, with either positive or negative polarity. The same genes occur in all the genomes being considered. There is no distinction between the same gene in two different genomes. In Chapter 5 we generalize our representation to genes that may have several copies of a gene, which differ only by a few point mutations. This leads to problems of identifying copies in two genomes that are primary orthologs, under the assumptions of differentials in point mutation rate. We provide algorithms, software and test examples.

Many real-world network data can be formulated as graphs, where a binary relation exists between nodes. One of the fundamental problems in network data analysis is community detection, clustering the nodes into different groups. Statistically, this problem can be formulated as hypothesis testing: under the null hypothesis, there is no community structure, while under the alternative hypothesis, community structure exists. One is of the method is to use the largest eigenvalues of the scaled adjacency matrix proposed by Bickel and Sarkar (2016), which works for dense graph. Another one is the subgraph counting method proposed by Gao and Lafferty (2017a), valid for sparse network. In this paper, firstly, we empirically study the BS or GL methods to see whether either of them works for moderately sparse network; secondly, we propose a subsampling method to reduce the computation of the BS method and run simulations to evaluate the performance.

L'objectif principal de ce mémoire est de généraliser du cas discret au cas continu plusieurs quantités, inégalités et preuves qui surviennent en théorie de l'information. Dans plusieurs cas, à la place de transposer la preuve ou les quantités d'intérêts au continu, le cas discret est étendu à l'extrême en prenant un très grand nombre de probabilités discrètes. Nous espérons que ce mémoire puisse servir de ressource pour faciliter la transition du discret au continu et que les différentes quantités trouvées puissent servir de fondation à toute autre preuve concernant les variables continues en théorie de l'information. Les premières sections présenteront un survol des fondements de la théorie de l'information, une introduction aux probabilités ainsi que des fondements mathématiques requis pour la compréhension du reste du document. Les sections subséquentes introduiront les analogues continus à la théorie de l'information classique, en plus de différentes inégalités et preuves en rapport avec ces quantités. / This document's main goal is to generalize multiple quantities, inequalities, and proofs that arise in information theory. Many of these proofs use discrete variables. We seek here to generalize these proofs to the continuous case. In many instances, instead of transposing the proofs to the continuous case, the discrete case is taken to the extreme by taking very large pools of discrete possibilities. We hope that this thesis can serve as a tool to ease the transition from the discrete case to the continuous case and that the various quantities and bounds found herein will help in establishing a framework to prove statements regarding continuous variables in information theory. The first few sections will present a review of elementary information theory, as well as a primer on probabilities and fundamental mathematical concepts required for the rest of the document. The later sections will introduce the continuous counterparts of classical information theory, as well as various inequalities and proofs with respect to these new quantities.