Rozprawy doktorskie na temat „Spectral spaces”

In the paper [14] Hochster gave a topological characterisation of those spaces X which arise as the prime spectrum of a commutative ring: they are the spectral spaces, defined as those topological spaces which are T_0, quasi-compact and sober, whose quasi-compact and open subsets form a basis for the topology and are closed under finite intersections. It is well known that the prime spectrum of a ring is always spectral; Hochster proved the converse by describing a construction which, starting from such a space X, builds a ring having the desired prime spectrum; however the construction given is (in Hochster's own words) very intricate, and has not been further exploited in the literature (a passing exception perhaps being the use of [14] Theorem 4 in the example on page 272 of [24]). In the finite setting, alternative constructions of a ring having a given spectrum are provided by Lewis [17] and Ershov [7], which, particularly in light of the work of Fontana in [8], appear to be more tractable, and at least more readily understood. This insight into the methods of constructing rings with a given spectrum is used to prove a result about which spaces may arise as the prime spectrum of a Noetherian ring: it is shown that every 1-dimensional Noetherian spectral space may be realised as the prime spectrum of a Noetherian ring. A close analysis of the two finite constructions considered here reveals considerable similarities between their underlying operation, despite their radically different presentations. Furthermore, we generalise the framework of Ershov's construction beyond the finite setting, finding that the ring we thus define on a space X contains the ring defined by Hochster's construction on X as a subring. We find that in certain examples these rings coincide, but that in general the containment is proper, and that the spectrum of the ring provided by our generalised construction is not necessarily homeomorphic to our original space. We then offer an additional condition on our ring which may (-and indeed in certain examples does) serve to repair this disparity. It is hoped that the analysis of the constructions presented herein, and the demonstration of the heretofore unrecognised connections between the disparate ring constructions proposed by various authors, will facilitate further investigation into the prime ideal structure of commutative rings.

The principal objects of study in this thesis are arbitrary spectral families, E, on a complex Banach space X. The central theme is the relationship between the geometry of X and the p-variation of E. We show that provided X is super- reflexive, then given any E, there exists a value 1 · p < 1, depending only on E and X, such that var p(E) < 1. If X is uniformly smooth we provide an explicit range of such values p, which depends only on E and the modulus of convexity of X*, delta X*(.).

Spectral theory is the study of Mark Kac's famous question [K], "can one hear the shape of a drum?" That is, can we determine the geometrical or topological properties of a manifold by using its Laplace Spectrum? In recent years, the problem has been extended to include the study of Riemannian orbifolds within the same context. In this thesis, on the one hand, we answer Kac's question in the negative for orbifolds that are spherical space forms of dimension higher than eight. On the other hand, for the three-dimensional and four-dimensional cases, we answer Kac's question in the affirmative for orbifold lens spaces, which are spherical space forms with cyclic fundamental groups. We also show that the isotropy types and the topology of the singularities of Riemannian orbifolds are not determined by the Laplace spectrum. This is done in a joint work with E. Stanhope and D. Webb by using P. Berard's generalization of T. Sunada's theorem to obtain isospectral orbifolds. Finally, we construct a technique to get examples of orbifold lens spaces that are not isospectral, but have the same asymptotic expansion of the heat kernel. There are several examples of such pairs in the manifold setting, but to the author's knowledge, the examples developed in this thesis are among the first such examples in the orbifold setting.

Spectral multiplicity theory solves the problem of unitary equivalence of normal operate on a Hilbert space ${ cal H}$ by associating with each normal operator N a multiplicity function, such that two operators are unitarily equivalent if and only if their multiplicity functions are equal. This problem was first solved in the classical case in which ${ cal H}$ is separable by Hellinger in 1907, and in the general case in which ${ cal H}$ is nonseparable by Wecken in 1939. This thesis develops the later versions of multiplicity theory in the nonseparable case given by Halmos and Brown, and gives the simplification of Brown's version to the classical theory. Then the versions of Halmos and Brown are shown directly to be equivalent. Also, the multiplicity function of Brown is expressed in terms of the multiplicity function of Halmos.

In this thesis we consider a connected locally finite graph G that possesses the Cheeger isoperimetric property. We define a decreasing one parameter family of Hardy-type spaces associated with the standard nearest neighbour Laplacian on G. We show that the space with parameter ½ is the space of all integrable functions whose Riesz transform is integrable. We show that if G has bounded geometry and the parameter is an integer, the corresponding Hardy-type space admits an atomic decomposition. We also show that if G is a homogeneous tree and the parameter is not an integer, the corresponding Hardy-type space does not admit an atomic decomposition. Furthermore, we consider the Hardy-type spaces defined in terms of the heat and Poisson maximal operators, and we analyse their relationships with the family of spaces defined previously. We also show that the space associated with the heat maximal operator is properly contained in the one associated with the heat maximal operator, a phenomenon which has no counterpart in the euclidean setting. Applications to the purely imaginary powers of the Laplacian are also given. Finally, we characterise, for every p, the class of spherical multipliers on the p-integrable functions on homogeneous trees in terms of Fourier multipliers on the torus. Furthermore we give a sharp sufficient condition on spherical multipliers on the product of homogeneous trees.

Thesis (Ph.D.) - University of Glasgow, 2000.<br>Ph.D. thesis submitted to the Department of Mathematics, University of Glasgow, 2000. Includes bibliographical references. Print version also available.

Let (X, μ) be a measure space, and let L1, . . . ,Ln be (possibly unbounded) selfadjoint operators on L2(X, μ), which commute strongly pairwise, i.e., which admit a joint spectral resolution E on Rn. A joint functional calculus is then defined via spectral integration: for every Borel function m : Rn → C, m(L) = m(L1, . . . ,Ln) = ∫ Rn m(λ) dE(λ) is a normal operator on L2(X, μ), which is bounded if and only if m - called the joint spectral multiplier associated to m(L) - is (E-essentially) bounded. However, the abstract theory of spectral integrals does not tackle the following problem: to find conditions on the multiplier m ensuring the boundedness of m(L) on Lp(X, μ) for some p ≠ 2. We are interested in this problem when the measure space is a connected Lie group G with a right Haar measure, and L1, . . . ,Ln are left-invariant differential operators on G. In fact, the question has been studied quite extensively in the case of a single operator, namely, a sublaplacian or a higher-order analogue. On the other hand, for multiple operators, only specific classes of groups and specific choices of operators have been considered in the literature. Suppose that L1, . . . ,Ln are formally self-adjoint, left-invariant differential operators on a connected Lie group G, which commute pairwise (as operators on smooth functions). Under the assumption that the algebra generated by L1, . . . ,Ln contains a weighted subcoercive operator --- a notion due to [ER98], including positive elliptic operators, sublaplacians and Rockland operators---we prove that L1, . . . ,Ln are (essentially) self-adjoint and strongly commuting on L2(G). Moreover, we perform an abstract study of such a system of operators, in connection with the algebraic structure and the representation theory of G, similarly as what is done in the literature for the algebras of differential operators associated with Gelfand pairs. Under the additional assumption that G has polynomial volume growth, weighted L1 estimates are obtained for the convolution kernel of the operator m(L) corresponding to a compactly supported multiplier m satisfying some smoothness condition. The order of smoothness which we require on m is related to the degree of polynomial growth of G. Some techniques are presented, which allow, for some specific groups and operators, to lower the smoothness requirement on the multiplier. In the case G is a homogeneous Lie group and L1, . . . ,Ln are homogeneous operators, a multiplier theorem of Mihlin-H\"ormander type is proved, extending the result for a single operator of [Chr91] and [MM90]. Further, a product theory is developed, by considering several homogeneous groups Gj , each of which with its own system of operators; a non-conventional use of transference techniques then yields a multiplier theorem of Marcinkiewicz type, not only on the direct product of the Gj , but also on other (possibly non-homogeneous) groups, containing homomorphic images of the Gj . Consequently, for certain non-nilpotent groups of polynomial growth and for some distinguished sublaplacians, we are able to improve the general result of [Ale94].

PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO<br>COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR<br>PROGRAMA DE EXCELENCIA ACADEMICA<br>Com o desenvolvimento tecnológico no setor de telecomunicações, o espectro radioelétrico está quase totalmente ocupado com um grande número de múltiplas atribuições para os muitos serviços sem fio de aplicação comercial e, também, não comercial, tais como defesa, controle de tráfego aéreo e exploração científica. O espectro eletromagnético é um recurso natural precioso e escasso, por isso, importantes esforços estão sendo direcionados para o desenvolvimento de rádios cognitivos, com capacidade de sensoriar o uso do espectro e utilizar frequências momentaneamente disponíveis de forma oportunista. O rastreamento e a utilização de intervalos espectrais, ou white spaces, através da tecnologia de rádios cognitivos, permitirá aumentar a eficiência de uso do espectro com a introdução de novos serviços de telecomunicações a serem explorados por usuários secundários, obrigados a não interferir ou a provocar interferência muito limitada nos usuários primários. O objetivo geral deste trabalho é avaliar os principais algoritmos de detecção dos intervalos espectrais (Detector de Energia, Detecção do Valor Absoluto de Covariância, Sensoriamento de Covariância Espectral) por meio de simulações com dados experimentais obtidos em campanhas de medições e testes em laboratório. Os algoritmos foram testados para avaliar o seu desempenho em termos de probabilidade de detecção dada uma probabilidade de falso alarme requerida, complexidade computacional e robustez quanto a relações sinal-a-ruído baixas. Os dados experimentais utilizados provêm de campanhas de medidas realizadas em ambiente urbano na faixa de 3.5 GHz.<br>With the technological development of the telecommunications industry, the radio spectrum is almost fully occupied with a large number of multiple assignments for wireless services for both commercial and non-commercial applications, such as defense, air traffic control and scientific exploration. The electromagnetic spectrum is a precious and scarce natural resource. Therefore, a considerable effort is being directed at the development of cognitive radios, capable of sensoring the spectrum and using momentarily available frequency bands in an opportunistic way. The tracking and using of these spectral intervals, or white spaces, using cognitive radio technology will enhance the efficiency of the spectrum use and allow the introduction of new telecommunications services to be exploited by secondary users, obliged not to interfere or produce very limited interference to primary users. The aim of this study is to evaluate the main algorithms for detection of spectral intervals (Energy Detector, Detection of Covariance Absolute Value, Spectral Covariance Sensing) through simulations with experimental data obtained in field measurements campaigns. The algorithms were tested to evaluate their performance in terms of detection probability given a required false alarm probability, computational complexity and robustness in low signal-to-noise conditions. The experimental data used comes from the measurements campaigns in urban environments at the 3.5 GHz band.

In this thesis, we prove a variety of discrete Agmon Kolmogorov inequalities and apply them to prove Lieb Thirring inequalities for discrete Schrodinger operators on ℓ[superscript 2](ℤ). We generalise these results in two ways: Firstly, to higher order difference operators, leading to spectral bounds for Tri-, Penta- and Polydiagonal Jacobi-type matrix operators. Secondly, to ℓ[superscript 2]-spaces on higher dimensional domains, specifically on ℓ[superscript 2](ℤ[superscript 2]), ℓ[superscript 2](ℤ[superscript 3]) and finally ℓ[superscript 2](ℤ[superscript d]). In the Introduction we discuss previous work on Landau Kolmogorov inequalities on a variety of Banach Spaces, Lieb Thirring inequalities in ℓ[superscript 2](ℝ[superscript d]), and the use of Jacobi Matrices in relation to the discrete Schrodinger Operator. We additionally give our main results with some introduction to the notation at hand. Chapters 2, 3 and 4 follow a similar structure. We first introduce the relevant difference operators and examine their properties. We then move on to prove the Agmon Kolmogorov and Generalised Sobolev inequalities over ℤ of order 1, 2 and σ respectively. Furthermore, we prove the Lieb Thirring inequality for the respective discrete Schrodinger-type operators, which we subsequently lift to arbitrary moments. Finally we apply this inequality to obtain spectral bounds for tri-, penta- and polydiagonal matrices. In Chapter 5, we prove a variety of Agmon Kolmogorov inequalities on ℓ[superscript 2](ℤ[superscript 2]) and ℓ[superscript 2](ℤ[superscript 3]). We use these intuitive ideas to obtain 2[superscript d-1] Agmon Kolmogorov inequalities on ℓ[superscript 2](ℤ[superscript d]). We continue from here in the same manner as before and prove the discrete Generalised Sobolev and Lieb Thirring inequalities for a variety of exponent combinations on ℓ[superscript 2](ℤ[superscript d]).

Thesis (Ph. D.)--University of Oregon, 2008.<br>Typescript. Includes vita and abstract. Includes bibliographical references (leaves 71-72). Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.

L’objectif de la thèse est de présenter de nouveaux résultats d’analyse sur les espaces métriques mesurés. Nous étendons d’abord à une certaine classe d’espaces avec doublement et Poincaré des inégalités de Sobolev pondérées introduites par V. Minerbe en 2009 dans le cadre des variétés riemanniennes à courbure de Ricci positives. Dans le contexte des espaces RCD(0,N), nous en déduisons une inégalité de Nash pondérée et un contrôle uniforme du noyau de la chaleur pondéré associé. Puis nous démontrons la loi de Weyl sur les espaces RCD(K,N) compactes à l’aide d’un théorème de convergence ponctuelle des noyaux de la chaleur associés à une suite mGH-convergente d’espaces RCD(K,N). Enfin nous abordons dans le contexte RCD(K,N) un théorème de Bérard, Besson et Gallot fournissant, à l’aide du noyau de la chaleur, une famille de plongements asymptotiquement isométriques d’une variété riemannienne fermée dans l’espace de ses fonctions de carré intégrable. Nous introduisons notamment les notions de métrique RCD, de métrique pull-back, et de convergence faible/forte de métriques RCD sur un espace RCD(K,N) compacte, et nous prouvons un résultat de convergence analogue à celui de Bérard, Besson et Gallot<br>The aim of this thesis is to present new results in the analysis of metric measure spaces. We first extend to a certain class of spaces with doubling and Poincaré some weighted Sobolev inequalities introduced by V. Minerbe in 2009 in the context of Riemannian manifolds with non-negative Ricci curvature. In the context of RCD(0,N) spaces, we deduce a weighted Nash inequality and a uniform control of the associated weighted heat kernel. Then we prove Weyl’s law for compact RCD(K,N) spaces thanks to a pointwise convergence theorem for the heat kernels associated with a mGH-convergent sequence of RCD(K,N) spaces. Finally we address in the RCD(K,N) context a theorem from Bérard, Besson and Gallot which provides, by means of the heat kernel, an asymptotically isometric family of embeddings for a closed Riemannian manifold into its space of square integrable functions. We notably introduce the notions of RCD metrics, pull-back metrics, weak/strong convergence of RCD metrics, and we prove a convergence theorem analog to the one of Bérard, Besson and Gallot

[from the introduction]: The aim of this thesis is to study metric measure spaces with a synthetic notion of Ricci curvature bounded below. We study them from the point of view of Sobolev/Nash type functional inequalities in the non-compact case, and from the point of view of spectral analysis in the compact case. The heat kernel links the two cases: in the first one, the goal is to get new estimates on the heat kernel of some associated weighted structure; in the second one, the heat kernel is the basic tool to establish our results. The topic of synthetic Ricci curvature bounds has known a constant development over the past few years. In this introduction, we shall give some historical account on this theory, before explaining in few words the content of this work. The letter K will refer to an arbitrary real number and N will refer to any finite number greater or equal than 1.

Der Laplace-Operator auf kompakten Riemannschen Mannigfaltigkeiten besitzt eine natürliche Verallgemeinerung auf kompakte Riemannsche Orbifolds und das Spektrum des so gewonnenen Operators besteht ausschließlich aus Eigenwerten endlicher Vielfachheit. Die Feststellung, dass das Spektrum Informationen über die Geometrie einer Mannigfaltigkeit (oder, allgemeiner, einer Orbifold) enthält, begründete ein ganzes Teilgebiet der Mathematik. Es ist eine offene Frage der sogenannten Spektralgeometrie, ob eine Mannigfaltigkeit und eine singuläre Orbifold isospektral sein (d.h., dasselbe Spektrum mitsamt den Vielfachheiten der Eigenwerte besitzen) können. Angesichts diverser Obstruktionen zur Existenz eines solchen Beispiels für die bekannten Beispiele isospektraler guter Orbifolds, soll diese Arbeit die Spektralgeometrie schlechter Orbifolds erhellen. Zu diesem Zweck geben wir die ersten Beispiele für isospektrale Metriken auf schlechten Orbifolds an. Diese basieren auf bestimmten gewichteten projektiven Räumen, auf denen wir mittels einer Verallgemeinerung von Schüths Version der Torus-Methode nicht-trivial isospektrale Metriken konstruieren.<br>The Laplace Operator on compact Riemannian manifolds naturally generalizes to compact Riemannian orbifolds and the spectrum of the resulting operator consists only of eigenvalues with finite multiplicities. The observation that the spectrum contains information about the geometry of a manifold (and, more generally, an orbifold) gave rise to a whole field of mathematics. It is an open question of so-called spectral geometry, whether a manifold and a singular orbifold can be isospectral (i.e., have the same spectrum with the same multiplicities of the eigenvalues). Given the various obstructions to the existence of such an example for the known examples of isospectral good orbifolds, this work is an attempt to shed light on the spectral geometry of bad orbifolds by giving the first examples of isospectral Riemannian metrics on bad orbifolds. In our case these are particular fixed weighted projective spaces equipped with non-trivially isospectral metrics obtained by a generalization of Schüth''s version of the torus method.

This PhD thesis consists of a summary and five papers which all deal with asymptotic problems on certain homogeneous spaces. In Paper I we prove asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of arbitrary dimension. All our results are given with precise estimates on the rates of convergence to equidistribution. Papers II and III are concerned with statistical problems on the space of n-dimensional lattices of covolume one. In Paper II we study the distribution of lengths of non-zero lattice vectors in a random lattice of large dimension. We prove that these lengths, when properly normalized, determine a stochastic process that, as the dimension n tends to infinity, converges weakly to a Poisson process on the positive real line with intensity 1/2. In Paper III we complement this result by proving that the asymptotic distribution of the angles between the shortest non-zero vectors in a random lattice is that of a family of independent Gaussians. In Papers IV and V we investigate the value distribution of the Epstein zeta function along the real axis. In Paper IV we determine the asymptotic value distribution and moments of the Epstein zeta function to the right of the critical strip as the dimension of the underlying space of lattices tends to infinity. In Paper V we determine the asymptotic value distribution of the Epstein zeta function also in the critical strip. As a special case we deduce a result on the asymptotic value distribution of the height function for flat tori. Furthermore, applying our results we discuss a question posed by Sarnak and Strömbergsson as to whether there in large dimensions exist lattices for which the Epstein zeta function has no zeros on the positive real line.

In this thesis we study the essential spectrum of composition operators on the Hardy space of the unit disc and of the upper half-plane. Our starting point is the spectral analysis of the composition operators induced by translations of the upper half-plane. We completely characterize the essential spectrum of a class of composition operators that are induced by perturbations of translations

Ce travail de thèse porte sur l'obtention de bornes de généralisation pour des échantillons statistiques à valeur dans des espaces de Hilbert définis par des noyaux reproduisants. L'approche consiste à obtenir des bornes non asymptotiques indépendantes de la dimension dans des espaces de dimension finie, en utilisant des inégalités PAC-Bayesiennes liées à une perturbation Gaussienne du paramètre et à les étendre ensuite aux espaces de Hilbert séparables. On se pose dans un premier temps la question de l'estimation de l'opérateur de Gram à partir d'un échantillon i. i. d. par un estimateur robuste et on propose des bornes uniformes, sous des hypothèses faibles de moments. Ces résultats permettent de caractériser l'analyse en composantes principales indépendamment de la dimension et d'en proposer des variantes robustes. On propose ensuite un nouvel algorithme de clustering spectral. Au lieu de ne garder que la projection sur les premiers vecteurs propres, on calcule une itérée du Laplacian normalisé. Cette itération, justifiée par l'analyse du clustering en termes de chaînes de Markov, opère comme une version régularisée de la projection sur les premiers vecteurs propres et permet d'obtenir un algorithme dans lequel le nombre de clusters est déterminé automatiquement. On présente des bornes non asymptotiques concernant la convergence de cet algorithme, lorsque les points à classer forment un échantillon i. i. d. d'une loi à support compact dans un espace de Hilbert. Ces bornes sont déduites des bornes obtenues pour l'estimation d'un opérateur de Gram dans un espace de Hilbert. On termine par un aperçu de l'intérêt du clustering spectral dans le cadre de l'analyse d'images<br>This thesis focuses on obtaining generalization bounds for random samples in reproducing kernel Hilbert spaces. The approach consists in first obtaining non-asymptotic dimension-free bounds in finite-dimensional spaces using some PAC-Bayesian inequalities related to Gaussian perturbations and then in generalizing the results in a separable Hilbert space. We first investigate the question of estimating the Gram operator by a robust estimator from an i. i. d. sample and we present uniform bounds that hold under weak moment assumptions. These results allow us to qualify principal component analysis independently of the dimension of the ambient space and to propose stable versions of it. In the last part of the thesis we present a new algorithm for spectral clustering. It consists in replacing the projection on the eigenvectors associated with the largest eigenvalues of the Laplacian matrix by a power of the normalized Laplacian. This iteration, justified by the analysis of clustering in terms of Markov chains, performs a smooth truncation. We prove nonasymptotic bounds for the convergence of our spectral clustering algorithm applied to a random sample of points in a Hilbert space that are deduced from the bounds for the Gram operator in a Hilbert space. Experiments are done in the context of image analysis

Les équations différentielles constituent un important outil pour l'étude des variétés algébriques et analytiques, sur les nombres complexes et $p$-adiques. Dans le cas $p$-adique, elles présentent des phénomènes qui n'apparaissent pas dans le cas complexe. En effet, le rayon de convergence des solutions d'une équation différentielle linéaire peut être fini, et cela même en l'absence des pôles.La connaissance de ce rayon permet d’obtenir de nombreuses informations intéressantes sur l’équation. Plus précisément, depuis les travaux de F. Baldassarri, on sait associer un rayon de convergence à tout point d’une courbe p-adique au sens de Berkovich munie d’une connexion. Des travaux récents de F. Baldassarri, K. Kedlaya, J. Poineauet A. Pulita ont révélé que ce rayon se comporte de manière très contrainte. Afin de pousser l'étude, on introduit un objet géométrique qui raffine ce rayon, le spectre au sens de Berekovich d'une équation différentielle.Dans ce mémoire de thèse, nous définissons le spectre d'un module différentiel et donnons ses premières propriétés. Nous calculons aussi les spectres de quelques classes de modules différentiels: modules différentiels d'une équations différentielles à coefficients constants, modules différentiels singuliers réguliers et enfin modules différentiels sur un corps des séries de Laurent<br>Differential equations constitute an important tool for theinvestigation of algebraic and analytic varieties, over thecomplex and the $p$-adic numbers. In the $p$-adic setting, theypresent phenomena that do not appear in the complex case. Indeed, theradius of convergence of the solutions of a linear differential equation,even without presence of poles.The knowledge of that radius permits to obtain several interestinginformations about the equation. More precisely, since the works ofF. Baldassarri, we know how to associate a radius of convergece to allpoint of a p-adic curve in the sense of Berkovich endowed with aconnexion. Recent works of F. Baldassarri, K.S. Kedlaya, J. Poineau, etA. Pulita have proved that this radius behave in a very controlledway. The radius of convergence can be refined using subsidiary radii,that are known to have similar properties. In order to push forward the study, we introduce a geometric object that refine this radius, thespectrum in the sense of Berkovich of a differential equation.In the present thesis, we define the spectrum of a differentialequation and provide its first properties. We also compute the spectraof some classes of differential modules: differential modules ofa differential équation with constant coefficients, singular regulardifferential modules and at last differential modules over the field ofLaurent power series

Bayesian inference occurs when prior knowledge about uncertain parameters in mathematical models is merged with new observational data related to the model outcome. In this thesis we focus on models given by partial differential equations where the uncertain parameters are coefficient functions belonging to infinite dimensional function spaces. The result of the Bayesian inference is then a well-defined posterior probability measure on a function space describing the updated knowledge about the uncertain coefficient. For decision making and post-processing it is often required to sample or integrate wit resprect to the posterior measure. This calls for sampling or numerical methods which are suitable for infinite dimensional spaces. In this work we focus on Kalman filter techniques based on ensembles or polynomial chaos expansions as well as Markov chain Monte Carlo methods. We analyze the Kalman filters by proving convergence and discussing their applicability in the context of Bayesian inference. Moreover, we develop and study an improved dimension-independent Metropolis-Hastings algorithm. Here, we show geometric ergodicity of the new method by a spectral gap approach using a novel comparison result for spectral gaps. Besides that, we observe and further analyze the robustness of the proposed algorithm with respect to decreasing observational noise. This robustness is another desirable property of numerical methods for Bayesian inference. The work concludes with the application of the discussed methods to a real-world groundwater flow problem illustrating, in particular, the Bayesian approach for uncertainty quantification in practice<br>Bayessche Inferenz besteht daraus, vorhandenes a-priori Wissen über unsichere Parameter in mathematischen Modellen mit neuen Beobachtungen messbarer Modellgrößen zusammenzuführen. In dieser Dissertation beschäftigen wir uns mit Modellen, die durch partielle Differentialgleichungen beschrieben sind. Die unbekannten Parameter sind dabei Koeffizientenfunktionen, die aus einem unendlich dimensionalen Funktionenraum kommen. Das Resultat der Bayesschen Inferenz ist dann eine wohldefinierte a-posteriori Wahrscheinlichkeitsverteilung auf diesem Funktionenraum, welche das aktualisierte Wissen über den unsicheren Koeffizienten beschreibt. Für Entscheidungsverfahren oder Postprocessing ist es oft notwendig die a-posteriori Verteilung zu simulieren oder bzgl. dieser zu integrieren. Dies verlangt nach numerischen Verfahren, welche sich zur Simulation in unendlich dimensionalen Räumen eignen. In dieser Arbeit betrachten wir Kalmanfiltertechniken, die auf Ensembles oder polynomiellen Chaosentwicklungen basieren, sowie Markowketten-Monte-Carlo-Methoden. Wir analysieren die erwähnte Kalmanfilter, indem wir deren Konvergenz zeigen und ihre Anwendbarkeit im Kontext Bayesscher Inferenz diskutieren. Weiterhin entwickeln und studieren wir einen verbesserten dimensionsunabhängigen Metropolis-Hastings-Algorithmus. Hierbei weisen wir geometrische Ergodizität mit Hilfe eines neuen Resultates zum Vergleich der Spektrallücken von Markowketten nach. Zusätzlich beobachten und analysieren wir die Robustheit der neuen Methode bzgl. eines fallenden Beobachtungsfehlers. Diese Robustheit ist eine weitere wünschenswerte Eigenschaft numerischer Methoden für Bayessche Inferenz. Den Abschluss der Arbeit bildet die Anwendung der diskutierten Methoden auf ein reales Grundwasserproblem, was insbesondere den Bayesschen Zugang zur Unsicherheitsquantifizierung in der Praxis illustriert

Bayesian inference occurs when prior knowledge about uncertain parameters in mathematical models is merged with new observational data related to the model outcome. In this thesis we focus on models given by partial differential equations where the uncertain parameters are coefficient functions belonging to infinite dimensional function spaces. The result of the Bayesian inference is then a well-defined posterior probability measure on a function space describing the updated knowledge about the uncertain coefficient. For decision making and post-processing it is often required to sample or integrate wit resprect to the posterior measure. This calls for sampling or numerical methods which are suitable for infinite dimensional spaces. In this work we focus on Kalman filter techniques based on ensembles or polynomial chaos expansions as well as Markov chain Monte Carlo methods. We analyze the Kalman filters by proving convergence and discussing their applicability in the context of Bayesian inference. Moreover, we develop and study an improved dimension-independent Metropolis-Hastings algorithm. Here, we show geometric ergodicity of the new method by a spectral gap approach using a novel comparison result for spectral gaps. Besides that, we observe and further analyze the robustness of the proposed algorithm with respect to decreasing observational noise. This robustness is another desirable property of numerical methods for Bayesian inference. The work concludes with the application of the discussed methods to a real-world groundwater flow problem illustrating, in particular, the Bayesian approach for uncertainty quantification in practice.<br>Bayessche Inferenz besteht daraus, vorhandenes a-priori Wissen über unsichere Parameter in mathematischen Modellen mit neuen Beobachtungen messbarer Modellgrößen zusammenzuführen. In dieser Dissertation beschäftigen wir uns mit Modellen, die durch partielle Differentialgleichungen beschrieben sind. Die unbekannten Parameter sind dabei Koeffizientenfunktionen, die aus einem unendlich dimensionalen Funktionenraum kommen. Das Resultat der Bayesschen Inferenz ist dann eine wohldefinierte a-posteriori Wahrscheinlichkeitsverteilung auf diesem Funktionenraum, welche das aktualisierte Wissen über den unsicheren Koeffizienten beschreibt. Für Entscheidungsverfahren oder Postprocessing ist es oft notwendig die a-posteriori Verteilung zu simulieren oder bzgl. dieser zu integrieren. Dies verlangt nach numerischen Verfahren, welche sich zur Simulation in unendlich dimensionalen Räumen eignen. In dieser Arbeit betrachten wir Kalmanfiltertechniken, die auf Ensembles oder polynomiellen Chaosentwicklungen basieren, sowie Markowketten-Monte-Carlo-Methoden. Wir analysieren die erwähnte Kalmanfilter, indem wir deren Konvergenz zeigen und ihre Anwendbarkeit im Kontext Bayesscher Inferenz diskutieren. Weiterhin entwickeln und studieren wir einen verbesserten dimensionsunabhängigen Metropolis-Hastings-Algorithmus. Hierbei weisen wir geometrische Ergodizität mit Hilfe eines neuen Resultates zum Vergleich der Spektrallücken von Markowketten nach. Zusätzlich beobachten und analysieren wir die Robustheit der neuen Methode bzgl. eines fallenden Beobachtungsfehlers. Diese Robustheit ist eine weitere wünschenswerte Eigenschaft numerischer Methoden für Bayessche Inferenz. Den Abschluss der Arbeit bildet die Anwendung der diskutierten Methoden auf ein reales Grundwasserproblem, was insbesondere den Bayesschen Zugang zur Unsicherheitsquantifizierung in der Praxis illustriert.

O objetivo principal desta dissertação é apresentar o fluxo espectral de um caminho de operadores de Fredholm auto-adjuntos em um espaço de Hilbert e suas propriedades. Pelos resultados clássicos de teoria espectral, sabemos que se H é um espaço de Hilbert e L : H &#8594 H é um operador linear, limitado e auto-adjunto, H pode ser escrito como soma direta ortogonal H+(L)&#8853 H-(L)&#8853 Ker L, onde H+(L) e H-(L) são os subespaços espectrais positivo e negativo de L, respectivamente. No trabalho damos uma definição de fluxo espectral baseada na decomposição acima, aprofundando as conexões deste conceito com a teoria espectral dos operadores de Fredholm em espaços de Hilbert. Entre as propriedades do fluxo espectral, será analisada a invariância homotópica que se apresenta em várias formas. Veremos o conceito de índice de Morse relativo, que estende o clássico índice de Morse, e sua relação com o fluxo espectral. A construção do fluxo espectral dada neste trabalho segue a abordagem de P. M. Fitzpatrick, J. Pejsachowicz e L. Recht em [9].<br>The main purpose of this dissertation is to present the spectral flow of a path of selfadjoint Fredholm operators in a Hilbert space and its properties. By classical results in spectral theory, we know that, if H is a Hilbert space and L : H &#8594 H is a bounded self-adjoint linear operator, H may be written as the following orthogonal direct sum H = H+(L)&#8853 H-(L)&#8853 Ker L, where H+(L) and H-(L) are the positive and negative spectral subspaces of L, respectively. In this work we give a definition of spectral flow which is based on the above splitting, examining in depth the connection between this concept and the spectral theory of Fredholm operators in Hilbert spaces. Among the properties of the spectral flow we will analyze the homotopic invariance, which appears on different ways. We will see the concept of relative Morse index, which generalize the classical Morse index, and its relation with the spectral flow. The construction of the spectral flow given in this work follows the approach of P. M. Fitzpatrick, J. Pejsachowicz and L. Recht in [9].

<p>This thesis consists of two papers on the spectral geometry of locally symmetric spaces of Riemannian and Lorentzian signature. Both works are concerned with the idea of relating analysis on such spaces to structures on their boundaries.</p><p>The first paper is motivated by a conjecture of Patterson on the Selberg zeta function of Kleinian groups. We consider geometrically finite hyperbolic cylinders with non-compact Riemann surfaces of finite area as cross sections. For these cylinders, we present a detailed investigation of the Bunke-Olbrich extension operator under the assumption that the cross section of the cylinder has one cusp. We establish the meromorphic continuation of the extension of Eisenstein series and incomplete theta series through the limit set. Furthermore, we derive explicit formulas for the residues of the extension operator in terms of boundary values of automorphic eigenfunctions.</p><p>The motivation for the second paper comes from conformal geometry in Lorentzian signature. We prove the existence and uniqueness of a sequence of differential intertwining operators for spherical principal series representations, which are realized on boundaries of anti de Sitter spaces. Algebraically, these operators correspond to homomorphisms of generalized Verma modules. We relate these families to the asymptotics of eigenfunctions on anti de Sitter spaces.</p>

[ES] Las labores de documentación de arte rupestre son arduas y delicadas, donde el color desempeña un papel fundamental, proporcionando información vital a nivel descriptivo, técnico y cuantitativo . Tradicionalmente los métodos de documentación en arqueología quedaban restringidos a procedimientos estrictamente subjetivos. Sin embargo, esta metodología conlleva limitaciones prácticas y técnicas, afectando a los resultados obtenidos en la determinación del color. El empleo combinado de técnicas geomáticas, como la fotogrametría o el láser escáner, junto con técnicas de procesamiento de imágenes digitales, ha supuesto un notable avance. El problema es que, aunque las imágenes digitales permiten capturar el color de forma rápida, sencilla, y no invasiva, los datos RGB registrados por la cámara no tienen un sentido colorimétrico riguroso. Se requiere la aplicación de un proceso riguroso de tranformación que permita obtener datos fidedignos del color a través de imágenes digitales. En esta tesis se propone una solución científica novedosa y de vanguardia, en la que se persigue integrar el análisis espectrofotométrico y colorimétrico como complemento a técnicas fotogramétricas que permitan una mejora en la identificación del color y representación de pigmentos con máxima fiabilidad en levantamientos, modelos y reconstrucciones tridimensionales (3D). La metodología propuesta se basa en la caracterización colorimétrica de sensores digitales, que es de novel aplicación en pinturas rupestres. La caracterización pretende obtener las ecuaciones de transformación entre los datos de color registrados por la cámara, dependientes del dispositivo, y espacios de color independientes, de base física, como los establecidos por la Commission Internationale de l'Éclairage (CIE). Para el tratamiento de datos colorimétricos y espectrales se requiere disponer de un software de características técnicas muy específicas. Aunque existe software comercial, lo cierto es que realizan por separado el tratamiento digital de imágenes y las operaciones colorimétricas. No existe software que integre ambas, ni que además permita llevar a cabo la caracterización. Como aspecto fundamental, presentamos en esta tesis el software propio desarrollado, denominado pyColourimetry, siguiendo las recomendaciones publicadas por la CIE, de código abierto, y adaptado al flujo metodológico propuesto, de modo que facilite la independencia y el progreso científico sin ataduras comerciales, permitiendo el tratamiento de datos colorimétricos y espectrales, y confiriendo al usuario pleno control del proceso y la gestión de los datos obtenidos. Adicinalmente, en este estudio se expone el análisis de los principales factores que afectan a la caracterización tales como el sensor empleado, los parámetros de la cámara durante la toma, la iluminación, el modelo de regresión, y el conjunto de datos empleados como entrenamiento del modelo. Se ha aplicado un modelo de regresión basado en procesos Gaussianos, y se ha comparado con los resultados obtenidos mediante polinomios. También presentamos un nuevo esquema de trabajo que permite la selección automática de muestras de color, adaptado al rango cromático de la escena, que se ha denominado P-ASK, basado en el algoritmo de clasificación K-means. Los resultados obtenidos en esta tesis demuestran que el proceso metodológico de caracterización propuesto es altamente aplicable en tareas de documentación y preservación del patrimonio cultural en general, y en arte rupestre en particular. Se trata de una metodología de bajo coste, no invasiva, que permite obtener el registro colorimétrico de escenas completas. Una vez caracterizada, una cámara digital convencional puede emplearse para la determinación del color de forma rigurosa, simulando un colorímetro, lo que permitirá trabajar en un espacio de color de base física, independiente del dispositivo y comparable con<br>[CA] Les tasques de documentació gràfica d'art rupestre són àrdues i delicades, on el color compleix un paper fonamental, proporcionant informació vital a nivell descriptiu, t\`ecnic i quantitatiu.Tradicionalment els mètodes de documentació en arqueologia quedaven restringits a procediments estrictament subjectius, comportant limitacions pràctiques i tècniques, afectant els resultats obtinguts en la determinació de la color. L'ús combinat de tècniques geomàtiques, com la fotogrametria o el làser escàner, juntament amb tècniques de processament i realç d'imatges digitals, ha suposat un notable avanç. Tot i que les imatges digitals permeten capturar el color de forma ràpida, senzilla, i no invasiva, les dades RGB proporcionades per la càmera no tenen un sentit colorimètric rigorós. Es requereix l'aplicació d'un procés rigorós de transformació que permeti obtenir dades fidedignes de la color a través d'imatges digitals. En aquesta tesi es proposa una solució científica innovadora i d'avantguarda, en la qual es persegueix integrar l'anàlisi espectrofotomètric i colorimètric com a complement a tècniques fotogramètriques que permetin una millora en la identificació de la color i representació de pigments amb màxima fiabilitat en aixecaments, models i reconstruccions tridimensionals 3D. La metodologia proposada es basa en la caracterització colorimètrica de sensors digitals, que és de novell aplicació en pintures rupestres. La caracterització pretén obtenir les equacions de transformació entre les dades de color registrats per la càmera, dependents d'el dispositiu, i espais de color independents, de base física, com els establerts per la Commission Internationale de l'Éclairage (CIE). Per al tractament de dades colorimètriques i espectrals de forma rigorosa es requereix disposar d'un programari de característiques tècniques molt específiques. Encara que hi ha programari comercial, fan per separat el tractament digital d'imatges i les operacions colorimètriques. No hi ha programari que integri totes dues, ni que permeti dur a terme la caracterització. Com a aspecte addicional i fonamental, vam presentar el programari propi que s'ha desenvolupat, denominat pyColourimetry, segons les recomanacions publicades per la CIE, de codi obert, i adaptat al flux metodológic proposat, de manera que faciliti la independència i el progrés científic sense lligams comercials, permetent el tractament de dades colorimètriques i espectrals, i conferint a l'usuari ple control del procés i la gestió de les dades obtingudes. A més, s'exposa l'anàlisi dels principals factors que afecten la caracterització tals com el sensor emprat, els paràmetres de la càmera durant la presa, il¿luminació, el model de regressió, i el conjunt de dades emprades com a entrenament d'el model. S'ha aplicat un model de regressió basat en processos Gaussians, i s'han comparat els resultats obtinguts mitjançant polinomis. També vam presentar un nou esquema de treball que permet la selecció automàtica de mostres de color, adaptat a la franja cromàtica de l'escena, que s'ha anomenat P-ASK, basat en l'algoritme de classificació K-means. Els resultats obtinguts en aquesta tesi demostren que el procés metodològic de caracterització proposat és altament aplicable en tasques de documentació i preservació de el patrimoni cultural en general, i en art rupestre en particular. Es tracta d'una metodologia de baix cost, no invasiva, que permet obtenir el registre colorimètric d'escenes completes. Un cop caracteritzada, una càmera digital convencional pot emprar-se per a la determinació de la color de forma rigorosa, simulant un colorímetre, el que permetrà treballar en un espai de color de base física, independent d'el dispositiu i comparable amb dades obtingudes mitjançant altres càmeres que tambè estiguin caracteritzades.<br>[EN] Cultural heritage documentation and preservation is an arduous and delicate task in which color plays a fundamental role. The correct determination of color provides vital information on a descriptive, technical and quantitative level. Classical color documentation methods in archaeology were usually restricted to strictly subjective procedures. However, this methodology has practical and technical limitations, affecting the results obtained in the determination of color. Nowadays, it is frequent to support classical methods with geomatics techniques, such as photogrammetry or laser scanning, together with digital image processing. Although digital images allow color to be captured quickly, easily, and in a non-invasive way, the RGB data provided by the camera does not itself have a rigorous colorimetric sense. Therefore, a rigorous transformation process to obtain reliable color data from digital images is required. This thesis proposes a novel technical solution, in which the integration of spectrophotometric and colorimetric analysis is intended as a complement to photogrammetric techniques that allow an improvement in color identification and representation of pigments with maximum reliability in 3D surveys, models and reconstructions. The proposed methodology is based on the colorimetric characterization of digital sensors, which is of novel application in cave paintings. The characterization aims to obtain the transformation equations between the device-dependent color data recorded by the camera and the independent, physically-based color spaces, such as those established by the Commission Internationale de l'Éclairage (CIE). The rigorous processing of color and spectral data requires software packages with specific colorimetric functionalities. Although there are different commercial software options, they do not integrate the digital image processing and colorimetric computations together. And more importantly, they do not allow the camera characterization to be carried out. Therefore, as a key aspect in this thesis is our in-house pyColourimetry software that was developed and tested taking into account the recommendations published by the CIE. pyColourimetry is an open-source code, independent without commercial ties; it allows the treatment of colorimetric and spectral data and the digital image processing, and gives full control of the characterization process and the management of the obtained data to the user. On the other hand, this study presents a further analysis of the main factors affecting the characterization, such as the camera built-in sensor, the camera parameters, the illuminant, the regression model, and the data set used for model training. For computing the transformation equations, the literature recommends the use of polynomial equations as a regression model. Thus, polynomial models are considered as a starting point in this thesis. Additionally, a regression model based on Gaussian processes has been applied, and the results obtained by means of polynomials have been compared. Also, a new working scheme was reported which allows the automatic selection of color samples, adapted to the chromatic range of the scene. This scheme is called P-ASK, based on the K-means classification algorithm. The results achieved in this thesis show that the proposed framework for camera characterization is highly applicable in documentation and conservation tasks in general cultural heritage applications, and particularly in rock art painting. It is a low-cost and non-invasive methodology that allows for the colorimetric recording from complete image scenes. Once characterized, a conventional digital camera can be used for rigorous color determination, simulating a colorimeter. Thus, it is possible to work in a physical color space, independent of the device used, and comparable with data obtained from other cameras that are also characterized.<br>Thanks to the Universitat Politècnica de València for the FPI scholarship<br>Molada Tebar, A. (2020). Colorimetric and spectral analysis of rock art by means of the characterization of digital sensors [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/160386<br>TESIS

Cette thèse est dédiée à l'étude d'opérateurs de composition pondérés sur plusieurs espaces fonctionnels sous fond du problème du sous-espace invariant. Cet important problème ouvert pose la question de l'existence pour tout opérateur sur un espace de Hilbert, complexe, séparable de dimension infinie, d'un sous-espace fermé, non-trivial et invariant par cet opérateur. La première partie est consacrée à l'étude spectrale et à la caractérisation des vecteurs cycliques d'un opérateur de composition à poids particulier sur L^2([0,1]^d) : l'opérateur de type Bishop, introduit comme possible contre-exemple au problème du sous-espace invariant. Les seconde, troisième et quatrième parties abordent ce problème sous un autre aspect : celui de l'universalité d'un opérateur. Ces opérateurs universels possèdent la propriété d'universalité : la description complète des sous-espaces invariants d'un opérateur universel permettrait de répondre au problème du sous-espace invariant. Déterminer l'universalité d'un opérateur repose sur l'établissement de propriétés spectrales fines de l’opérateur considéré (théorème de Caradus). Dans ce but, nous établissons des propriétés spectrales ad-hoc de classes d’opérateurs de composition à poids sur L^2([0,1]), les espaces de Sobolev d’ordre n, sur les espaces de Hardy du disque unité et du demi-plan supérieur, permettant de déduire des résultats d’universalité. Nous obtenons aussi une généralisation du critère d’universalité. Dans la dernière partie, nous donnons une caractérisation des opérateurs de composition rsid16415432 inversibles et une caractérisation partielle des opérateurs de composition isométriques sur les espaces de Hardy de l’anneau<br>In this thesis, we study classes of weighted composition operators on several functional spaces in the background of the invariant subspace problem. This important open problem asks the question of the existence for every operator, defined on a complex and separable infinite dimensional Hilbert space, of a non trivial closed subspace invariant under the operator. The first part is dedicated to the establishment of the spectrum and the characterization of cyclic vectors of a special weighted composition operator defined on L^2([0,1]^d) : the Bishop type operator, introduced as possible counter-example of the invariant subspace problem. The second, third and fourth part approach the problem from the point of view of universal operators. More precisely, universal operators have the universal property in the sense of the complete description of all the invariant subspaces of a universal operator could solve the invariant subspace problem. A sufficient condition for an operator to be universal (Caradus’theorem) is given in terms of spectral properties. To this aim, we establish ad-hoc spectral properties of a class of weighted composition operators on L^2([0,1]) and Sobolev spaces of order n, of composition operator in the Hardy space of the unit disc and of the upper half-plane, which lead us to deduce universality results. We also obtain a generalization of the universality criteria mentioned above. In the last part, we give a characterization of invertible composition operators and a partial characterization of composition operators on the Hardy space of the annulus

Spectrum sensing is an essential pre-processing step of cognitive radio technology for dynamic radio spectrum management. One of the main functions of Cognitive radios is to detect the unused spectrum and share it without harmful interference with other users. The detection of signal components present within a determined frequency band is an important requirement of any sensing technique. Most methods are restricted to the detection of the spectral lines. However, these methods may not comply with the needs imposed by practical applications.  This master thesis work presents a novel method to detect significant spectral components in measured non-flat spectra by classifying them in two groups: signal and noise frequency lines. The algorithm based on Fisher’s discriminant analysis, aside from the detection of spectral lines, estimates the magnitude of the spectral lines and provides a measure of the quality of classification to determine if a spectral line was incorrectly classified. Furthermore, the frequency lines with higher probability of misclassification are regrouped and the validation process recomputed, which results in lower probabilities of misclassification. The proposed automatic detection algorithm requires no user interaction since any prior knowledge about the measured signal and the noise power is needed. The presence or absence of a signal regardless of the shape of the spectrum can be detected. Hence, this method becomes a strong basis for high-quality operation mode of cognitive radios. Simulation and measurement results prove the advantages of the presented technique. The performance of the technique is evaluated for different signal-to-noise ratios (SNR) ranging from 0 to -21dB as required by the IEEE standard for smart radios. The method is compared with previous signal detection methods.

L’irradiance spectrale solaire (SSI) dans la bande ultraviolette est un paramètre-clé pour la spécification de la moyenne et la haute atmosphère terrestre. Elle est requise dans de nombreuses applications en météorologie de l’espace, et aussi pour l’étude du climat. Or les observations souffrent de plusieurs défauts : manque de couverture spectrale et temporelle permanente, dégradation des capteurs, désaccords entre les instruments, etc. Plusieurs modèles de reconstruction de la SSI ont été développés pour pallier à ces difficultés. Chacun souffre de défauts, et la reconstruction du spectre en-dessous de 120nm est un réel défi. C’est dans ce contexte que nous avons développé un modèle empirique, qui recourt au champ magnétique photosphérique pour reconstruire les variations du spectre solaire. Ce modèle décompose les magnétogrammes solaires en différentes structures qui sont classées à partir de leur aire (et non sur la base de leur intensité, comme dans la plupart des autres modèles). La signature spectrale de ces structures est déduite des observations, et non pas imposée par des modèles de l’atmosphère solaire. La qualité de la reconstruction s’avère être comparable à celle d’autres modèles. Parmi les principaux résultats, relevons que deux classes seulement de structures solaires suffisent à reproduire correctement la variabilité spectrale solaire. En outre, seule une faible résolution radiale suffit pour reproduire les variations de centre-bord. Enfin, nous montrons que l’amélioration apportée par la décomposition du modèle en deux constantes de temps peut être attribuée à l’effet des raies optiquement minces<br>The spectrally-resolved radiative output of the Sun (SSI) in the UV band, i.e. at wavelengths below 300 nm, is a key quantity for specifying the state of the middle and upper terrestrial atmosphere. This quantity is required in numerous space weather applications, and also for climate studies. Unfortunately, SSI observations suffer from several problems : they have numerous spectral and temporal gaps, instruments are prone to degradation and often disagree, etc. This has stimulated the development of various types of SSI models. Proxy-based models suffer from lack of the physical interpretation and are as good as the proxies are. Semi-empirical models do not perform well below 300 nm, where the local thermodynamic equilibrium approximation does not hold anymore. We have developed an empirical model, which assumes that variations in the SSI are driven by solar surface magnetic flux. This model proceeds by segmenting solar magnetograms into different structures. In contrast to existing models, these features are classified by their area (and not their intensity), and their spectral signatures are derived from the observations (and not from models). The quality of the reconstruction is comparable to that of other models. More importantly, we find that two classes only of solar features are required to properly reproduce the spectral variability. Furthermore, we find that a coarse radial resolution suffices to account for geometrical line-of-sight effects. Finally, we show how the performance of the model on different time-scales is related to the optical thickness of the emission lines

O objetivo principal deste trabalho será apresentar um estudo detalhado dos espaços de configurações. Dissertaremos sobre: espaços de configurações clássicos, invariância do bordo, espaço de configurações para superfícies, fibração de Fadell e Neuwirth e espaços de configurações do espaço Euclideano, da esfera e do espaço projetivo complexo.<br>The main objective of this work will be to present a detailed study of the configuration spaces. We will study: classical configuration spaces, invariance of the boundary, configuration spaces of surfaces, Fadell and Neuwirth fibration and configuration spaces of the Euclidean space and spheres.

<p>KIM, HYON-JUNG. Variance Estimation in Spatial Regression Using a NonparametricSemivariogram Based on Residuals. (Under the direction of Professor Dennis D. Boos.)The empirical semivariogram of residuals from a regression model withstationary errors may be used to estimate the covariance structure of the underlyingprocess.For prediction (Kriging) the bias of the semivariogram estimate induced byusing residuals instead of errors has only a minor effect because thebias is small for small lags. However, for estimating the variance of estimatedregression coefficients and of predictions,the bias due to using residuals can be quite substantial. Thus wepropose a method for reducing the bias in empirical semivariogram estimatesbased on residuals. The adjusted empirical semivariogram is then isotonizedand made positive definite and used to estimate the variance of estimatedregression coefficients in a general estimating equations setup.Simulation results for least squares and robust regression show that theproposed method works well in linear models withstationary correlated errors. Spectral Analysis with Spatial Periodogram and Data Tapers.(Under the direction of Professor Montserrat Fuentes.)The spatial periodogram is a nonparametric estimate of the spectral density, which is the Fourier Transform of the covariance function. The periodogram is a useful tool to explain the dependence structure of aspatial process.Tapering (data filtering) is an effective technique to remove the edge effects even inhigh dimensional problemsand can be applied to the spatial data in order to reduce the bias of the periodogram.However, the variance of the periodogram increases as the bias is reduced.We present a method to choose an appropriate smoothing parameter for datatapers and obtain better estimates of the spectral densityby improving the properties of the periodogram.The smoothing parameter is selected taking intoaccount the trade-off between bias and variance of the taperedperiodogram. We introduce a new asymptotic approach for spatial datacalled `shrinking asymptotics', which combines theincreasing-domain and the fixed-domain asymptotics.With this approach, the tapered spatial periodogram can be usedto determine uniquely the spectral density of the stationary process,avoiding the aliasing problem. <P>

In this thesis I discuss some results on the noncommutative (spin) geometry of quantum principal G-bundles. The first part of the thesis is devoted to the study of spectral triples over toral bundles; extending some recent results by L. Dabrowski and A. Sitarz, we introduce the notion of projectable spectral triple for T^n-bundles. Moreover, we work out twisted Dirac operators. We discuss, in particular, the application of these results to noncommutative tori. In the second part of the thesis, instead, we work out a method for constructing real spectral triples over cleft quantum principal G-bundles and we study the properties of these triples and their behaviour under gauge transformations. Some of the results discussed in this thesis can also be found in the following papers: arXiv:1305.6185 arXiv:1308.4738

Il est bien connu que la convergence fortement stable est une condition suffisante de convergence des éléments spectraux approchées, i. E. Les valeurs propres non nulles, isolées et de multiplicité algébrique finie et les sous-espaces invariants maximaux qui leur sont associés, d'opérateurs linéaires bornés définis sur des espaces de Banach complexes. Dans ce travail, nous commençons par proposer une nouvelle notion de convergence : la convergence spectrale, que l'on montre être une condition nécessaire de convergence fortement stable et suffisante de convergence des éléments spectraux approchés. Nous donnons ensuite des conditions suffisantes de convergence spectrale moins restrictives que celles habituellement utilisées. Nous montrons également la convergence de quelques schémas de raffinement itératif pour l'approximation des bases de sous-espaces invariants maximaux, dans le cadre des méthodes de Newton inexactes et des séries de Rayleigh-Schrodinger, sous certaines des conditions suffisantes de convergence spectrale proposées. Nous donnons ensuite les résultats de quelques essais numériques

ITC/USA 2009 Conference Proceedings / The Forty-Fifth Annual International Telemetering Conference and Technical Exhibition / October 26-29, 2009 / Riviera Hotel & Convention Center, Las Vegas, Nevada<br>Prognostic technology uses a series of algorithms, combined forms a prognostic-based inference engine (PBIE) for the identification of deterministic behavior embedded in completely normal appearing telemetry from fully functional equipment. The algorithms used to define normal behavior in the PBIE from which deterministic behavior is identified can be adapted to quantify normal spacecraft telemetry behavior while in orbit about a moon or planet or during interplanetary travel. Time-series analog engineering data (telemetry) from orbiting satellites and interplanetary spacecraft are defined by harmonic and non-harmonic influences which shape it behavior. Spectrum analysis can be used to understand and quantify the fundamental behavior of spacecraft analog telemetry and relate the behavior's frequency and phase to its time-series behavior through Fourier analysis.

Dissertation for the Degree of Doctor of Philosophy in Mathematics<br>Fundação para a Ciência e a Tecnologia through the Carnegie Mellon | Portugal Program under Grant SFRH/BD/35695/2007, the Financiamento Base 20010 ISFL–1–297, PTDC/MAT/109973/2009 and UTA

Denotemos por X ~ p \'S POT. m\' x \'S POT. n\' um espaço finitístico com anel de cohomologia módulo p isomorfo ao anel de cohomologia de um produto de esferas \'S POT. m\' x \'S POT. n\', o qual admite ação livre do grupo cíclico G = Zp, com p um primo ímpar. Nosso objetivo neste trabalho é determinar o anel de cohomologia do espaço de órbitas X / G, usando como ferramenta principal a seqüência espectral de Leray-Serre associada à fibração de Borel X \'SETA\' \'imath\' X G \'SETA\' \'pi\' B G, onde BG é o espaço classificante do G-fibrado universal wG = (EG;BG; pG; G;G) e XG = EG x G X é o espaço de Borel. Este resultado foi provado por R. M. Dotzel, T. B. Singh and S. P. Tripathi em [14]<br>Let denote by X ~ p \'S POT. m\' x \'S POT. n\' finitistic space with mod p cohomology ring isomorphic to the cohomology ring of a product of spheres \'S POT. m\' x \'S POT. n\' , which admits a free action of the cyclic group G = Zp, with p an odd prime. Our goal in this work is to determine the cohomology ring of the orbit space X / G, using as main tool the Leray-Serre spectral sequence associated to the Borel fibration X \'SETA\" \'imath\' \'X G \'SETA\' \'pi\' BG, where BG is the classifying space of the G-universal bundle wG = (EG;BG; pG; G;G) and XG = EG x G X is the Borel space. This result was proved by R. M. Dotzel, T. B. Singh and S. P. Tripathi in [14]

Spectral rendering, or the synthesis of images by taking into account the wavelengths of light, allows effects otherwise impossible with other methods. One of these effects is dispersion, the phenomenon that creates a rainbow when white light shines through a prism. Spectral rendering has previously remained in the realm of off-line rendering (with a few exceptions) due to the extensive computation required to keep track of individual light wavelengths. Caustics, the focusing and de-focusing of light through a refractive medium, can be interpreted as a special case of dispersion where all the wavelengths travel together. This thesis extends Adaptive Caustic Mapping, a previously proposed caustics mapping algorithm, to handle spectral dispersion. Because ACM can display caustics in real-time, it is quite amenable to be extended to handle the more general case of dispersion. A method is presented that runs in screen-space and is fast enough to display plausible dispersion phenomena in real-time at interactive frame rates.

Magnetic confinement fusion reactors suffer severely from heat and particle losses through turbulent transport, which has inspired the construction of ever larger and more expensive reactors. Numerical simulations are vital to their design and operation, but particle collisions are too infrequent for fluid descriptions to be valid. Instead, strongly magnetised fusion plasmas are described by the gyrokinetic equations, a nonlinear integro-differential system for evolving the particle distribution functions in a five-dimensional position and velocity space, and the consequent electromagnetic field. Due to the high dimensionality, simulations of small reactor sections require hundreds of thousands of CPU hours on cutting-edge High Performance Computing platforms. We develop a Hankel-Hermite spectral representation for velocity space that exploits structural features of the particle streaming, gyroaveraging, and collision terms in the gyrokinetic system. This representation exactly conserves a discrete free energy in the absence of explicit dissipation, while our Hermite hypercollision operator captures Landau damping with as few as ten variables. Calculation of the electromagnetic fields also becomes purely local. This eliminates all inter-processor communication in, and hence vastly accelerates, searches for linear instabilities. We implement these ideas in SpectroGK, an efficient parallel code. Turbulent fusion plasmas may dissipate free energy through linear phase mixing to fine scales in velocity space, as in Landau damping, or through a nonlinear cascade to fine scales in physical space, as in hydrodynamic turbulence. Using SpectroGK to study saturated electrostatic drift-kinetic turbulence in Cartesian geometry, we find that the nonlinear cascade completely suppresses linear phase mixing at energetically-dominant scales, so the turbulence is fluid-like. We use these observations to derive Fourier-Hermite spectra for the electrostatic potential and distribution function, and confirm these spectra with SpectroGK simulations.

In this dissertation we will address two types of homogenization problems. The first one is a spectral problem in the realm of lower dimensional theories, whose physical motivation is the study of waves propagation in a domain of very small thickness and where it is introduced a very thin net of heterogeneities. Precisely, we consider an elliptic operator with "ε-periodic coefficients and the corresponding Dirichlet spectral problem in a three-dimensional bounded domain of small thickness δ. We study the asymptotic behavior of the spectrum as ε and δ tend to zero. This asymptotic behavior depends crucially on whether ε and δ are of the same order (δ ≈ ε), or ε is of order smaller than that of δ (δ = ετ , τ < 1), or ε is of order greater than that of δ (δ = ετ , τ > 1). We consider all three cases. The second problem concerns the study of multiscale homogenization problems with linear growth, aimed at the identification of effective energies for composite materials in the presence of fracture or cracks. Precisely, we characterize (n+1)-scale limit pairs (u,U) of sequences {(uεLN⌊Ω,Duε⌊Ω)}ε>0 ⊂ M(Ω;ℝd) × M(Ω;ℝd×N) whenever {uε}ε>0 is a bounded sequence in BV (Ω;ℝd). Using this characterization, we study the asymptotic behavior of periodically oscillating functionals with linear growth, defined in the space BV of functions of bounded variation and described by n ∈ ℕ microscales