This thesis consists of three papers on the following topics in functional analysis and probability theory: Riesz bases and frames, weakly stationary stochastic processes and analysis of set-valued stochastic processes. In the first paper we investigate Uniformly Bounded Linearly Stationary stochastic processes from the point of view of the theory of Riesz bases. By regarding these stochastic processes as generalized Riesz bases we are able to gain some new insight into there structure. Special attention is paid to regular UBLS processes as well as perturbations of weakly stationary processes. An infinite sequence of subspaces of a Hilbert space is called regular if it is decreasing and zero is the only element in its intersection. In the second paper we ask for conditions under which the regularity of a sequence of subspaces is preserved when the sequence undergoes a deformation by a linear and bounded operator. Linear, bounded and surjective operators are closely linked with frames and we also investigate when a frame is a regular sequence of vectors. A multiprocess is a stochastic process whose values are compact sets. As generalizations of the class of subharmonic processes and the class of subholomorphic processesas introduced by Thomas Ransford, in the third paper of this thesis we introduce the general notions of a gauge of processes and a multigauge of multiprocesses. Compositions of multiprocesses with multifunctions are discussed and the boundary crossing property, related to the intermediate-value property, is investigated for general multiprocesses. Time changes of multiprocesses are investigated in the environment of multigauges and we give a multiprocess version of the Dambis-Dubins-Schwarz Theorem.

In this thesis, we study two problems. The first problem is to introduce the general version of Abramovich-Wickstead type spaces and investigate its order properties. In particular, we study the ideals, order bounded sets, disjointness properties, Dedekind completion and the norm properties of this Riesz space. We also define a new concrete example of Riesz space-valued uniformly continuous functions, denoted by CDr0 which generalizes the original Abramovich-Wickstead space. It is also shown that similar spaces CD0 and CDw introduced earlier by Alpay and Ercan are decomposable lattice-normed spaces. The second problem is related to analytic representations of different classes of dominated operators on these spaces. Our main representation theorems say that regular linear operators on CDr0 or linear dominated operators on CD0 may be represented as the sum of integration with respect to operator-valued measure and summation operation. In the case when the operator is order continuous or bo-continuous, then these representations reduce to discrete parts.

We look at three distinct topics in analysis. In the first we give a direct and easy proof that the usual Newton-Leibniz rule implies the fundamental theorem of algebra that any nonconstant complex polynomial of one complex variable has a complex root. Next, we look at the Riesz representation theorem and show that the Riesz representing measure often can be given in the form of mini sums just like in the case of the usual Lebesgue measure on a cube. Lastly, we look at the idea of holomorphic domination and use it to define a class of complex Banach manifolds that is similar in nature and definition to the class of Stein manifolds.

Dans cette thèse nous étudions les transformées de Riesz et les espaces de Hardy associés à un opérateur sur un espace métrique mesuré. Ces deux sujets sont en lien avec des estimations du noyau de la chaleur associé à cet opérateur. Dans les Chapitres 1, 2 et 4, on étudie les transformées quasi de Riesz sur les variétés riemannienne et sur les graphes. Dans le Chapitre 1, on prouve que les quasi transformées de Riesz sont bornées dans Lp pour 1

En analyse multivariée de données de grande dimension, les lois de Wishart définies dans le contexte des modèles graphiques revêtent une grande importance car elles procurent parcimonie et modularité. Dans le contexte des modèles graphiques Gaussiens régis par un graphe G, les lois de Wishart peuvent être définies sur deux restrictions alternatives du cône des matrices symétriques définies positives : le cône PG des matrices symétriques définies positives x satisfaisant xij=0, pour tous sommets i et j non adjacents, et son cône dual QG. Dans cette thèse, nous proposons une construction harmonieuse de familles exponentielles de lois de Wishart sur les cônes PG et QG. Elle se focalise sur les modèles graphiques d'interactions des plus proches voisins qui présentent l'avantage d'être relativement simples tout en incluant des exemples de tous les cas particuliers intéressants: le cas univarié, un cas d'un cône symétrique, un cas d'un cône homogène non symétrique, et une infinité de cas de cônes non-homogènes. Notre méthode, simple, se fonde sur l'analyse sur les cônes convexes. Les lois de Wishart sur QAn sont définies à travers la fonction gamma sur QAn et les lois de Wishart sur PAn sont définies comme la famille de Diaconis- Ylvisaker conjuguée. Ensuite, les méthodes développées sont utilisées pour résoudre la conjecture de Letac- Massam sur l'ensemble des paramètres de la loi de Wishart sur QAn. Cette thèse étudie aussi les sousmodèles, paramétrés par un segment dans M, d'une famille exponentielle paramétrée par le domaine des moyennes M. / In the framework of Gaussian graphical models governed by a graph G, Wishart distributions can be defined on two alternative restrictions of the cone of symmetric positive definite matrices: the cone PG of symmetric positive definite matrices x satisfying xij=0 for all non-adjacent vertices i and j and its dual cone QG. In this thesis, we provide a harmonious construction of Wishart exponential families in graphical models. Our simple method is based on analysis on convex cones. The focus is on nearest neighbours interactions graphical models, governed by a graph An, which have the advantage of being relatively simple while including all particular cases of interest such as the univariate case, a symmetric cone case, a nonsymmetric homogeneous cone case and an infinite number of non-homogeneous cone cases. The Wishart distributions on QAn are constructed as the exponential family generated from the gamma function on QAn. The Wishart distributions on PAn are then constructed as the Diaconis- Ylvisaker conjugate family for the exponential family of Wishart distributions on QAn. The developed methods are then used to solve the Letac-Massam Conjecture on the set of parameters of type I Wishart distributions on QAn. Finally, we introduce and study exponential families of distributions parametrized by a segment of means with an emphasis on their Fisher information. The focus in on distributions with matrix parameters. The particular cases of Gaussian and Wishart exponential families are further examined.

The payoff functions of contingent claims (options) of one variable are prominent in Financial Economics and thus assume a fundamental role in option pricing theory. Some of these payoff functions are continuous, piecewise-defined and linear or affine. Such option payoff functions can be analysed in a useful way when they are represented in additive, Boolean normal, graphical and linear form. The issue of converting such payoff functions expressed in the additive, linear or graphical form into an equivalent Boolean normal form, has been considered by several authors for more than half-a-century to better-understand the role of such functions. One aspect of our study is to unify the foregoing different forms of representation, by creating algorithms that convert a payoff function expressed in graphical form into Boolean normal form and then into the additive form and vice versa. Applications of these algorithms are considered in a general theoretical sense and also in the context of specific option contracts wherever relevant. The use of these algorithms have yielded easy computation of the area enclosed by the graph of various functions using min and max operators in several ways, which, in our opinion, are important in option pricing. To summarise, this study effectively dealt with maximum and minimum operators from several perspectives

No description available.

Endometriosis is an oestrogen-dependent gynaecological condition defined as the presence of endometrial tissue outside the uterus cavity. The condition is predominantly found in women in their reproductive years, and associated with significant pelvic and abdominal chronic pain and infertility. The disease is believed to affect approximately 33% of women by a recent study. Currently, surgical intervention, often laparoscopic surgery, is the gold standard for diagnosing the disease and it remains an effective and common treatment method for all stages of endometriosis. Magnetic resonance imaging (MRI) of the patient is performed before surgery in order to locate any endometriosis lesions and to determine whether a multidisciplinary surgical team meeting is required. In this dissertation, our goal is to use image processing techniques to aid surgical planning. Specifically, we aim to improve quality of the existing images, and to automatically detect bladder endometriosis lesion in MR images as a form of bladder wall thickening. One of the main problems posed by abdominal MRI is the sparse anisotropic frequency sampling process. As a consequence, the resulting images consist of thick slices and have gaps between those slices. We have devised a method to fuse multi-view MRI consisting of axial/transverse, sagittal and coronal scans, in an attempt to restore an isotropic densely sampled frequency plane of the fused image. In addition, the proposed fusion method is steerable and is able to fuse component images in any orientation. To achieve this, we apply the Riesz transform for image decomposition and reconstruction in the frequency domain, and we propose an adaptive fusion rule to fuse multiple Riesz-components of images in different orientations. The adaptive fusion is parameterised and switches between combining frequency components via the mean and maximum rule, which is effectively a trade-off between smoothing the intrinsically noisy images while retaining the sharp delineation of features. We first validate the method using simulated images, and compare it with another fusion scheme using the discrete wavelet transform. The results show that the proposed method is better in both accuracy and computational time. Improvements of fused clinical images against unfused raw images are also illustrated. For the segmentation of the bladder wall, we investigate the level set approach. While the traditional gradient based feature detection is prone to intensity non-uniformity, we present a novel way to compute phase congruency as a reliable feature representation. In order to avoid the phase wrapping problem with inverse trigonometric functions, we devise a mathematically elegant and efficient way to combine multi-scale image features via geometric algebra. As opposed to the original phase congruency, the proposed method is more robust against noise and hence more suitable for clinical data. To address the practical issues in segmenting the bladder wall, we suggest two coupled level set frameworks to utilise information in two different MRI sequences of the same patients - the T2- and T1-weighted image. The results demonstrate a dramatic decrease in the number of failed segmentations done using a single kind of image. The resulting automated segmentations are finally validated by comparing to manual segmentations done in 2D.

Classical operators of harmonic analysis in Orlicz spaces V'ıt Musil We deal with classical operators of harmonic analysis in Orlicz spaces such as the Hardy-Littlewood maximal operator, the Hardy-type integral operators, the maximal operator of fractional order, the Riesz potential, the Laplace transform, and also with Sobolev-type embeddings on open subsets of Rn or with respect to Frostman measures and, in particular, trace embeddings on the boundary. For each operator (in case of embeddings we consider the identity operator) we investigate the question of its boundedness from an Orlicz space into another. Particular attention is paid to the sharpness of the results. We further study the question of the existence of optimal Orlicz domain and target spaces and their description. The work consists of author's published and unpublished results compiled together with material appearing in the literature.

ON ETUDIE DANS CETTE THESE CERTAINS PROBLEMES SPECTRAUX POUR DES OPERATEURS DESCHRODINGER. ON S'INTERESSE D'ABORD A LA LIMITE SEMI-CLASSIQUE POUR LE NOMBRE D'ETATS PROPRESDE L'OPERATEUR DE SCHRODINGER A N CORPS. ON UTILISE ENSUITE LE CROCHET DE DIRICHLET-NEUMANN POUR OBTENIR LA LIMITE SEMI-CLASSIQUE DES MOYENNES DE RIESZ DES VALEURS PROPRES DISCRETES POUR L'OPERATEUR DE SCHR¨ODINGER A N CORPS. ON CONSIDERE EGALEMENT LE POTENTIEL EFFECTIF DE L'OPERATEUR DE SCHRODINGER A N CORPS AVEC POTENTIEL DE COULOMB ET ON OBTIENT QU'IL A UNE DECROISSANCE CRITIQUE A L'INFINI. ON ETUDIE DONC L'OPERATEUR DE SCHRODINGER A POTENTIEL CRITIQUE. ON S'INTERESSE AU SEUIL POUR LA CONSTANTE DE COUPLAGE ET AU DEVELOPPEMENT ASYMPTOTIQUE DE LA RESOLVANTE DE L'OPERATEUR DE SCHRODINGER, PUIS ON UTILISE CE DEVELOPPEMENT POUR ETUDIER LA LIMITE A BASSE ENERGIE DE LA DERIVEE DE LA FONCTION DE DECALAGE SPECTRAL POUR UNE PERTURBATION A DECROISSANCE CRITIQUE. FINALEMENT, ON UTILISE CE RESULTAT AVEC LE RESULTAT CONNU POUR LE DEVELOPPEMENT ASYMPTOTIQUE A HAUTE ENERGIE DE CETTE FONCTION DE DECALAGE SPECTRAL POUR OBTENIR LE THEOREME DE LEVINSON.