Em decorrência do grande avanço alcançado pela mecânica estatística devido a introdução das idéias de invariância conforme às teorias de escala para sistemas finitos, retomamos, neste trabalho, o estudo do modelo XY em campo transverso. A princípio, apresentamos uma análise detalhada do comportamento \"crossover\" característico do modelo, onde incluímos cálculos melhorados dos expoentes da susceptibilidade e do gap de energia anteriormente apresentados por dos Santos e Stinch-combe. Em seguida, uma análise numérica do espectro foi desenvolvida, considerando-se condições livres de contorno, e comparada com as previsões da invariância conforme. Finalmente, as correções à energia do estado fundamental de cadeias finitas foram utilizadas para obter o parâmetro que caracteriza as classes de universalidade (a carga central c). / In view of the great advance attached from statistical mechanics due to the conformal invariance ideas introduced to the scale theories, we take over at this work, the study of the XY model in a transverse field. At first, we present a detailed analysis on the sample\'s typical crossover behavior. An improved calculation of the susceptibility and gap exponents early presented by dos Santos and Stinchcombe is included. Nest, a numerical analysis of the spectrum, regarding free boundary condi¬tions was developed and compared with conformal invariance predictions. Finally, the fundamental state energy corrections of finite chains were used to obtain the parameter which ,distingoishes the universality classes (the central charge c).

The conformal bootstrap method is a non-perturbative method that uses the symmetry in a conformal field theory to constrain and solve for the observables in the theory. We consider a conformal field theory with the symmetry group SU(N) and four general scalar fields as the only low dimensional operators. The four-point correlation function of a quartic interaction of four general scalar fields in a conformal field theory can be written as a sum over primary operators. In order to study the four-point correlator a large-N expansion is made, where N comes from the symmetry group SU(N). Using the conformal bootstrap method the anomalous dimension of the primary operators in the four-point correlator is calculated. Using the AdS/CFT correspondence the anomalous dimension of the primary operators is also calculated using Witten diagrams. / Konform fältteori är en kvantfältteori med konform symmetri. Konform symmetri är en symmetri som bevarar vinklar och lokalt ser ut som en kombination av en rotation och en förändring i skala. En metod för att beräkna de observerbara kvantiteterna i en konform fältteori är metoden "conformal boostrap". Denna metod går ut på att använda symmetrin i en konform fältteori för att begränsa och beräkna värdet på de observerbara kvantiteterna i teorin.En av de observerbar kvantiteterna i en fältteori är en korrelationsfunktion. Korrelationsfunktioner beskriver interaktionerna mellan partiklarna i en fältteori.  I detta arbete studerar vi en interaktion mellan fyra skalärfält genom att studera fyra-punkts korrelationsfunktionen för denna interaktion. Metoden vi använder är "conformal bootstrap" men vi testar också om AdS/CFT dualiteten håller för våra beräkningar. AdS/CFT dualiteten är en ekvivalens av två olika teorier, en strängteori i ett (d+1)-dimensionellt anti-de Sitter (AdS) rum och en konform fältteori (CFT) i den d-dimensionella gränsen av anti-de Sitter rummet. Enligt denna dualitet kan en observerbar kvantitet beräknas från båda dessa två teorier och ge samma resultat. Teorin vi studerar har symmetrigrupp SU(N) och vi arbetar i dimension två. Vi arbetar också med att N, matrisrangen i teorin, är stort då detta är den gräns där AdS/CFT dualiteten gäller. Enligt konform fältteori så kan en fyra-punkts korrelationsfunktion av fyra skalärer beskrivas som en summa över vad som kallas primära fält. Genom att använda "conformal bootstrap" metoden beräknas den anormala dimensionen, vilket är en korrektion av första icke-triviala ordning till dimensionen, av dessa primära fält. Samma kvantitet beräknas också från strängteorisidan av AdS/CFT dualiteten genom användandet av så kallade Witten diagram. Resultatet från båda sidor av dualiteten visas stämma överens.

Ce travail de thèse concerne l’analyse de surfaces faciales en 3D, ainsi que leur traitement, dans le récent cadre de la modalité de reconnaissance de visages en 3D,basé sur des techniques d’appariement. Le traitement de la surface faciale et son analyse constituent une étape importante dans les algorithmes de reconnaissance de visage en 3D. La localisation de points d’intérêt anthropométriques du visage joue par ailleurs un rôle important dans les techniques de localisation du visage, de reconnaissance d’expression, de recalage, etc. Ainsi, leur localisation automatique joue un rôle crucial dans les algorithmes de traitement du visage 3D. Dans ce travail, nous avons mis l’accent sur la localisation précise et invariante en rotation des points d’intérêt, qui seront utilisés plus tard pour la reconnaissance de visages. Ces points d’intérêt sont localisés en combinant les propriétés locales de la surface faciale, exprimées en termes de géométrie différentielle, et un modèle global et générique du visage. Etant donné que la sensibilité des courbures, qui sont des propriétés de géométrie différentielle, au bruit, une des contributions de cette thèse est la modification d’une méthode de calcul de courbures. Cette modification incorpore le bruit de la surface dans la méthode de calcul, et permet de contrôler la progressivité des courbures. Par conséquent, nous pouvons localiser les points d’intérêt de la surface faciale avec précision et fiabilité (100% de bonnes localisation du bout du nez avec une erreur maximale de 8mmpar exemple) y compris en présence de rotations et de bruit. La modification de la méthode de calcul de courbure a été également testée pour différentes résolutions de visage, présentant des valeurs de courbure stables. Enfin, étant donné que donné que l’analyse de courbures mène à de nombreux candidats de points d’intérêt du visage, dont la validation est coûteuse, nous proposons de localiser les points d’intérêt grâce à une méthode d’apprentissage. Cette méthode permet de rejeter précocement des faux candidats avec une grande confiance, accélérant d’autant la localisation des points d’intérêt. La reconnaissance de visages à l’aide de modèles 3D est un sujet relativement nouveau, qui a été propose pour palier aux insuffisantes de la modalité de reconnaissance de visages en 2D. Cependant, les algorithmes de reconnaissance de visage en 3D sont généralement plus complexes. De plus, étant donné que les modèles de visage 3D décrivent la géométrie du visage, ils sont plus sensibles que les images 2Dde texture aux expressions faciales. Notre contribution est de réduire la dimensionnalité des données de départ en appariant les modèles de visage 3D au domaine 2Dà l’aide de méthodes, non rigides, d’appariement conformal. L’existence de modèles2D représentant les visages permet alors d’utiliser les techniques précédemment développées dans le domaine de la reconnaissance de visages en 2D. Dans nos travaux, nous avons utilisé les cartes conformales de visages 3D en conjonction avec l’algorithme2D2 PCA, atteignant le score de 86% en reconnaissance de rang 1 sur la base de données FRGC. L’efficacité de toutes les méthodes a été évaluée sur les bases FRGC et Bosphorus. / This Ph.D thesis work is dedicated to 3D facial surface analysis, processing as well as to the newly proposed 3D face recognition modality, which is based on mapping techniques. Facial surface processing and analysis is one of the most important steps for 3Dface recognition algorithms. Automatic anthropometric facial features localization also plays an important role for face localization, face expression recognition, face registration ect., thus its automatic localization is a crucial step for 3D face processing algorithms. In this work we focused on precise and rotation invariant landmarks localization, which are later used directly for face recognition. The landmarks are localized combining local surface properties expressed in terms of differential geometry tools and global facial generic model, used for face validation. Since curvatures, which are differential geometry properties, are sensitive to surface noise, one of the main contributions of this thesis is a modification of curvatures calculation method. The modification incorporates the surface noise into the calculation method and helps to control smoothness of the curvatures. Therefore the main facial points can be reliably and precisely localized (100% nose tip localization using 8 mm precision)under the influence of rotations and surface noise. The modification of the curvatures calculation method was also tested under different face model resolutions, resulting in stable curvature values. Finally, since curvatures analysis leads to many facial landmark candidates, the validation of which is time consuming, facial landmarks localization based on learning technique was proposed. The learning technique helps to reject incorrect landmark candidates with a high probability, thus accelerating landmarks localization. Face recognition using 3D models is a relatively new subject, which has been proposed to overcome shortcomings of 2D face recognition modality. However, 3Dface recognition algorithms are likely more complicated. Additionally, since 3D face models describe facial surface geometry, they are more sensitive to facial expression changes. Our contribution is reducing dimensionality of the input data by mapping3D facial models on to 2D domain using non-rigid, conformal mapping techniques. Having 2D images which represent facial models, all previously developed 2D face recognition algorithms can be used. In our work, conformal shape images of 3Dfacial surfaces were fed in to 2D2 PCA, achieving more than 86% recognition rate rank-one using the FRGC data set. The effectiveness of all the methods has been evaluated using the FRGC and Bosphorus datasets.

The work done in this thesis includes an exploration of both the conformal field theory techniques and holographic techniques of the Gauge/Gravity duality. From the field theory, we have analyzed the analytical aspects of the Conformal Bootstrap program to gain handle on at least a part of the CFT spectrum. The program applies equally to the strongly coupled as well as the weakly coupled theories. We have considered both the regimes of interest in this thesis. In the strongly coupled sector, as we have shown that it is possible to extract information about the anomalous dimensions, of a particular subset of large spin operators in the spectrum, as a function of the spin and twist of these operators. The holographic analog of the anomalous dimensions from CFT are the binding energies of generalized free fields in the bulk, which has also been analyzed in this thesis. On the contrary, in the weakly coupled sector, the same idea can be used to calculate the anomalous dimensions of operators, with any spin and dimension in an expansion. We have considered a simple set of scalar operators, whose anomalous dimensions are reproduced correctly up to O( 2). In another holographic calculation, we have analyzed generic higher derivative theories of gravity, which corresponds to boundary theories with in finite colors but finite `t Hooft coupling. Certain universal aspects of these theories, such as anomalies and correlation functions are also calculated. The three point functions for these higher derivative theories will serve as a building block for considering four point functions for finitely coupled boundary CFTs. In the conclusion, we have pointed out the directions of interest which could be locating the bulk duals of large N finitely coupled theories, or that of an intermediate theory with both finite `t Hooft coupling as well as finite gauge group, with a speculative string theory dual.

Every forecast is valid only if proper prediction intervals are stated. Currently models focus mainly on point forecast and neglect the area of prediction intervals. The estimation of the error of the model is made and is applied to every prediction in the same way, whereas we could identify that every case is different and different error measure should be applied to every instance. One of the state-of-the-art techniques which can address this behaviour is conformal prediction with its variant of normalized conformal prediction. In this thesis we apply this technique into time series problems. The special focus is put to examine the technique of estimating the difficulty of every instance using the error of neighbouring instances. This thesis describes the entire process of adjusting time series data into normalized conformal prediction framework and the comparison with other techniques will be made. The final results do not show that aforementioned method is superior over an existing techniques in various setups different method performed the best. However, it is similar in terms of performance. Therefore, it is an interesting add-on to data science forecasting toolkit. / Varje prognos är endast giltig om korrekt förutsägningsintervall anges. För närvarande fokuserar modeller huvudsakligen på punktprognos och försummar området med förutsägelsesintervall. Uppskattningen av modellens fel görs och tillämpas på varje förutsägelse på samma sätt, medan vi kunde identifiera att varje fall är annorlunda och olika felmått bör tillämpas på varje instans. En av de senaste teknikerna som kan hantera detta beteende är konform förutsägelse med dess variant av normaliserad konform förutsägelse. I denna avhandling tillämpar vi denna teknik i tidsserieproblem. Det speciella fokus ligger på att undersöka tekniken för att uppskatta svårigheten för varje instans med hjälp av felet i angränsande instanser. Den här avhandlingen beskriver hela processen för att anpassa tidsseriedata till normaliserat konformitetsprognosram och jämförelsen med andra tekniker kommer att göras. De slutliga resultaten visar inte att ovannämnda metod är överlägsen jämfört med en befintlig teknik - i olika uppsättningar utförde olika metoder bäst. Men det är liknande vad gäller prestanda. Därför är det ett intressant tillägg till datavetenskapens prognosverktygssats.

This master thesis is framed in the striking correspondence between gravity theories in Anti-de Sitter spacetime (AdS) and Conformal Field Theories (CFT). This is usually known as AdS/CFT duality and relates gravity theories in the bulk with CFTs that live in their conformal boundary. We start by presenting the notion of CFTs and some of the results and techniques that are widely used in this field. This includes conformal correlators for scalar and spin operators, the state-operator correspondence and the operator product expansion (OPE) of operators. The embedding formalism and the index-free notation to encode tensors in polynomials are also discussed and used throughout this work. The basic notions of AdS are outlined and CFT at finite temperature is then introduced. We include a review of thermal blocks and thermal coefficients for a thermal two-point function between scalar fields in mean field theory. We then analyse the thermal two-point function for conserved currents, which was not known in the literature. Finally, we start a study of its thermal blocks and thermal coefficients for the mean field theory application.

Thesis: M.S., Massachusetts Institute of Technology, Department of Ocean Engineering, 1978 / Includes bibliographical references. / M.S. / M.S. Massachusetts Institute of Technology, Department of Ocean Engineering

Cette thèse porte sur les relations entre les processus SLE, les ensembles CLE et le champ libre Gaussien. Dans le chapitre 2, nous donnons une construction des processus SLE(k,r) à partir des boucles des CLE(k) et d'échantillons de restriction chordale. Sheffield et Werner ont prouvé que les CLE(k) peuvent être construits à partir des processus d'exploration symétriques des SLE(k,r).Nous montrons dans le chapitre 3 que la configuration des boucles construites à partir du processus d'exploration asymétrique des SLE(k,k-6) donne la même loi CLE(k). Le processus SLE(4) peut être considéré comme les lignes de niveau du champ libre Gaussien et l'ensemble CLE(4) correspond à la collection des lignes de niveau de ce champ libre Gaussien. Dans la deuxième partie du chapitre 3, nous définissons un paramètre de temps invariant conforme pour chaque boucle appartenant à CLE(4) et nous donnons ensuite dans le chapitre 4 un couplage entre le champ libre Gaussien et l'ensemble CLE(4) à l'aide du paramètre de temps. Les processus SLE(k) peuvent être considérés comme les lignes de flot du champ libre Gaussien. Nous explicitons la dimension de Hausdorff de l'intersection de deux lignes de flot du champ libre Gaussien. Cela nous permet d'obtenir la dimension de l'ensemble des points de coupure et des points doubles de la courbe SLE, voir le chapitre 5. Dans le chapitre 6, nous définissons la mesure de restriction radiale, prouvons la caractérisation de ces mesures, et montrons la condition nécessaire et suffisante de l'existence des mesures de restriction radiale. / This thesis focuses on various relations between SLE, CLE and GFF. In Chapter 2, we give a construction of SLE(k,r) processes from CLE(k) loop configuration and chordal restriction samples. Sheffield and Werner has proved that CLE(k) can be constructed from symmetric SLE(k,k-6) exploration processes. We prove in Chapter 3 that the loop configuration constructed from the asymmetric SLE(k,k-6) exploration processes also give the same law CLE(k). SLE(4) can be viewed as level lines of GFF and CLE(4) can be viewed as the collection of level lines of GFF. We define a conformally invariant time parameter for each loop in CLE(4) in the second part of Chapter 3 and then give a coupling between GFF and CLE(4) with time parameter in Chapter 4. SLE(k,r) can be viewed as flow lines of GFF. We derive the Hausdorff dimension of the intersection of two flow lines in GFF. Then, from there, we obtain the dimension of the cut and double point set of SLE curve in Chapter 5. In Chapter 6, we define the radial restriction measure, prove the characterization of these measures, and show the if and only if condition for the existence of radial restriction measure.

A identificação de observações influentes e/ou aberrantes de um conjunto de dados é conhecida como uma parte das análises de diagnóstico. Esta técnica de diagnóstico têm como uma das finalidades verificar a robustez de um modelo estatístico, pois a não identificação dos dados influentes pode afetar a análise ou obter resultados incorretos. As metodologias comumente utilizadas para o diagnóstico de observações influentes em modelos de regressão são métodos de influência global (Belsey et al., 1980). Cook (1986) introduziu um método geral para avaliar a influência local de pequenas perturbações no modelo estatístico ou nos dados, usando diferentes tipos de perturbações. Como complemento às técnicas de detecção de observações discrepantes, é proposto o método procura \\forward\", por Atkinson e Riani (2000), que é uma metodologia para detectar observações atípicas mascaradas. Neste trabalho, propomos o uso da influência local com procura \"forward\" na obtenção de observações mascaradas influentes considerando modelos de regressão linear. / The identification of influential and/or atypical observations in a data set is known as a part of the diagnostic analysis. One of the purposes of the diagnostic analysis is to verify the robustness of a statistical model, as the non-identification of influential observations can affect the analysis or may cause the obtainment of incorrect results. The most commonly used methodology for the diagnostic of influential observations in regression models are the global influence (Belsey et al., 1980). Cook (1986) introduced a general method to evaluate the local influence of small perturbations in the statistical model or in the data set using different perturbation schemes. As a complement to the techniques of detection atypical observations, it is proposed the forward search procedure by Atkinson e Riani (2000), which is a methodology to detect the masked atypical observations in a data set. In this work we propose the use of the local influence approach together with the forward search to obtain the masked influential observations in linear regression models.

<p>This thesis deals with the Open Mapping Theorem for analytic functions on domains in the complex plane: A non-constant analytic function on an open subset of the complex plane is an open map.</p><p>As applications of this fundamental theorem we study Schwarz’s Lemma and its consequences concerning the groups of conformal automorphisms of the unit disk and of the upper halfplane.</p><p>In the last part of the thesis we indicate the first steps in hyperbolic geometry.</p> / <p>Denna uppsats behandlar satsen om öppna avbildningar för analytiska funktioner på domäner i det komplexa talplanet: En icke-konstant analytisk funktion på en öppen delmängd av det komplexa talplanet är en öppen avbildning.</p><p>Som tillämpningar på denna fundamentala sats studeras Schwarz’s lemma och dess konsekvenser för grupperna av konforma automorfismer på enhetsdisken och på det övre halvplanet.</p><p>I uppsatsens sista del antyds de första stegen inom hyperbolisk geometri.</p>