Dissertations / Theses on the topic 'Riesz Functional'

The present work consists of two main parts. In the first part, invariant subspaces of positive operators or operator families on locally convex solid Riesz spaces are examined. The concept of a weakly-quasinilpotent operator on a locally convex solid Riesz space has been introduced and several results that are known for a single operator on Banach lattices have been generalized to families of positive or close-to-them operators on these spaces. In the second part, the so-called generalized Alexandroff duplicates are studied and CDsigma, gamma(K, E)-type spaces are investigated. It has then been shown that the space CDsigma, gamma(K, E) can be represented as the space of E-valued continuous functions on the generalized Alexandroff duplicate of K.

Inverse problems naturally occur in many branches of science and mathematics. An inverse problem entails finding the values of one or more parameters using the values obtained from observed data. A typical example of an inverse problem is the inversion of the Radon transform. Here a function (for example of two variables) is deduced from its integrals along all possible lines. This problem is intimately connected with image reconstruction for X-ray computerized tomography. Moment problems are a special class of inverse problems. While the classical theory of moments dates back to the beginning of the 20th century, the systematic study of truncated moment problems began only a few years ago. In this dissertation we will first survey the elementary theory of truncated moment problems, and then focus on those problems with cubic column relations. For a degree 2n real d-dimensional multisequence β ≡ β (2n) ={β i}i∈Zd+,|i|≤2n to have a representing measure μ, it is necessary for the associated moment matrix Μ(n) to be positive semidefinite, and for the algebraic variety associated to β, Vβ, to satisfy rank Μ(n)≤ card Vβ as well as the following consistency condition: if a polynomial p(x)≡ ∑|i|≤2naixi vanishes on Vβ, then Λ(p):=∑|i|≤2naiβi=0. In 2005, Professor Raúl Curto collaborated with L. Fialkow and M. Möller to prove that for the extremal case (Μ(n)= Vβ), positivity and consistency are sufficient for the existence of a (unique, rank Μ(n)-atomic) representing measure. In joint work with Professor Raúl Curto we have considered cubic column relations in M(3) of the form (in complex notation) Z3=itZ+ubar Z, where u and t are real numbers. For (u,t) in the interior of a real cone, we prove that the algebraic variety Vβ consists of exactly 7 points, and we then apply the above mentioned solution of the extremal moment problem to obtain a necessary and sufficient condition for the existence of a representing measure. This requires a new representation theorem for sextic polynomials in Z and bar Z which vanish in the 7-point set Vβ. Our proof of this representation theorem relies on two successive applications of the Fundamental Theorem of Linear Algebra. Finally, we use the Division Algorithm from algebraic geometry to extend this result to other situations involving cubic column relations.

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.

L'objectif de cette thèse est de contribuer à l'optimisation de problèmes dynamiques en présence de retard. Le point de vue qui nous intéressera est celui de Pontryagin qui dans son ouvrage publié en 1962 a donné les conditions nécessaires d'existence de solutions pour ce type de problème. Warga dans son ouvrage publié en 1972 a fait un catalogue des solutions possible, Li et al. ont étudié le cas de contrôle périodique. Notre méthode de démonstration est directement inspirée de la démonstration de P. Michel du cas des systèmes gouvernés par des équations différentielles ordinaires. La principale difficulté pour cette approche est l'utilisation de la résolvante de l'équation différentielle fonctionnelle linéarisée de l'équation différentielle fonctionnelle d'évolution qui gouverne le système. Nous traitons aussi de condition d'Euler-Lagrange dans le cadre d'un problème de calcul variationnel avec retard
In this thesis, we have attempted to contribute to the optimization of dynamical problems with delay in state space. We are specifically interested in the viewpoint of Pontryagin who outlined in his book published in 1962 the necessary conditions required for solving such problems. In his work published in 1972, Warga catalogued the possible solutions. Li and al. analyzed the case of periodic control. We will treat an optimal control problem governed by a Delay Functional Differential Equation. Our method is close to the one of P. Michel on dynamical system governed by Ordinary Differential Equations. The main problem ariving out in this approach is the use of the resolvent of the Delay Functional Differential Equation. We also consider with Euler-Lagrange condition in the framework of variational problems with delay

Doctor of Philosophy
Department of Mathematics
Charles N. Moore
The main focus of this work is to study the classical Calder\'n-Zygmund theory and its recent developments. An attempt has been made to study some of its theory in more generality in the context of a nonhomogeneous space equipped with a measure which is not necessarily doubling. We establish a Hedberg type inequality associated to a non-doubling measure which connects two famous theorems of Harmonic Analysis-the Hardy-Littlewood-Weiner maximal theorem and the Hardy-Sobolev integral theorem. Hedberg inequalities give pointwise estimates of the Riesz potentials in terms of an appropriate maximal function. We also establish a good lambda inequality relating the distribution function of the Riesz potential and the fractional maximal function in $(\rn, d\mu)$, where $\mu$ is a positive Radon measure which is not necessarily doubling. Finally, we also derive potential inequalities as an application.

If T:C(H,X)-->Y is a bounded linear operator then there exists a unique weakly regular finitely additive set function m:-->L(X,Y**) so that T(f) = ∫Hfdm. In this paper, bounded linear operators on C(H,X) are studied in terms the measure given by this representation theorem. The first chapter provides a brief history of representation theorems of these classes of operators. In the second chapter the represenation theorem used in the remainder of the paper is presented. If T is a weakly compact operator on C(H,X) with representing measure m, then m(A) is a weakly compact operator for every Borel set A. Furthermore, m is strongly bounded. Analogous statements may be made for many interesting classes of operators. In chapter III, two classes of operators, weakly precompact and QSP, are studied. Examples are provided to show that if T is weakly precompact (QSP) then m(A) need not be weakly precompact (QSP), for every Borel set A. In addition, it will be shown that weakly precompact and GSP operators need not have strongly bounded representing measures. Sufficient conditions are provided which guarantee that a weakly precompact (QSP) operator has weakly precompact (QSP) values. A sufficient condition for a weakly precomact operator to be strongly bounded is given. In chapter IV, weakly precompact subsets of L1(μ,X) are examined. For a Banach space X whose dual has the Radon-Nikodym property, it is shown that the weakly precompact subsets of L1(μ,X) are exactly the uniformly integrable subsets of L1(μ,X). Furthermore, it is shown that this characterization does not hold in Banach spaces X for which X* does not have the weak Radon-Nikodym property.

Cette thèse s'inscrit dans le domaine de l'analyse harmonique et plus exactement, des estimations à poids. Un intérêt particulier est porté aux estimations Lp à poids des transformées de Riesz sur des variétés Riemanniennes complètes ainsi qu'à l'optimalité des résultats en terme de la puissance de la caractéristique des poids. On obtient un premier résultat (en terme de la linéarité et de la non dépendance de la dimension) sur des espaces pas nécessairement de type homogène, lorsque p = 2 et la courbure de Bakry-Emery est positive. On utilise pour cela une approche analytique en exhibant une fonction de Bellman concrète. Puis, en utilisant des techniques stochastiques et une domination éparse, on démontre que les transformées de Riesz sont bornées sur Lp, pour p ∈ (1, +∞) et on déduit également le résultat précèdent. Enfin, on utilise un changement élégant dans la preuve précèdente pour affaiblir l'hypothèse sur la courbure et la supposer minorée
The topics addressed in this thesis lie in the field of harmonic analysis and more pre- cisely, weighted inequalities. Our main interests are the weighted Lp-bounds of the Riesz transforms on complete Riemannian manifolds and the sharpness of the bounds in terms of the power of the characteristic of the weights. We first obtain a linear and dimensionless result on non necessarily homogeneous spaces, when p = 2 and the Bakry-Emery curvature is non-negative. We use here an analytical approach by exhibiting a concrete Bellman function. Next, using stochastic techniques and sparse domination, we prove that the Riesz transforms are Lp-bounded for p ∈ (1, +∞) and obtain the previous result for free. Finally, we use an elegant change in the precedent proof to weaken the condition on the curvature and assume it is bounded from below

In this dissertation we study the Laplace operator acting on functions on a smooth, compact Riemannian manifold. Our approach is based on the study of the spectrum of the aforementioned operator. The main objects of our interest are the counting function of the Laplacian and its Riesz means. We discuss the asymptotics of aforementioned functions when the argument approaches infinity.

Dans cette thèse, nous étudions l'interpolation réelle des espaces de Sobolev et ses applications. Le manuscrit est constitué de deux parties. Dans la première partie, nous démontrons au premier chapitre que les espaces de Sobolev non homogènes W^1_p (resp. homogènes ) sur les variétés Riemanniennes complètes vérifiant la propriété de doublement et une inégalité de Poincaré forment une échelle d'interpolation réelle pour un intervalle de valeurs de p. Nous étendons ce résultat à d'autres cadres géométriques. Dans un deuxième court chapitre, nous comparons différents espaces de Sobolev sur le cone Euclidien et nous regardons le lien de ces espaces avec l'interpolation. Nous montrons sur cet exemple que l'hypothèse de Poincaré n'est pas une condition nécessaire pour pouvoir interpoler les espaces de Sobolev. Dans le dernier chapitre de cette partie, nous définissons les espaces de Sobolev non homog'nes W^1_p,V (resp. homogènes ) associés à un potentiel positif V sur une variété Riemannienne. Nous démontrons que si la variété véifie la propriété de doublement et une inégalité de Poincaré et si de plus V est dans une classe de Holder inverse, ces espaces forment aussi une échelle d'interpolation réelle pour un intervalle de valeurs de p. Nous étendons ce résultat aux cas des groupes de Lie. Dans la deuxième partie, dans un premier chapitre en collaboration avec E. Russ, nous étudions sur un graphe vérifiant la propriété de doublement et une inégalité de Poincaré, la Lp bornitude de la transformée de Riesz pour p > 2 et son inégalité inverse pour p < 2. Pour notre but, nous démontrons aussi des résultats d'interpolation des espaces de Sobolev et des inégalités de Littlewood-Paley. Dans le deuxième chapitre, nous démontrons en utilisant notre résultat d'interpolation, des inégalités de Gagliardo-Nirenberg sur les variétés Riemanniennes complètes vérifiant le doublement, des inégalités de Poincaré et pseudo-Poincaré. Ce résultat s'applique aussi dans le cadre des groupes de Lie et des graphes.

In questa tesi presentiamo alcune delle più importanti tecniche di convergenza debole che permettono di studiare minimi vincolati di funzionali. In particolare, nella prima parte della tesi sviluppiamo le basi teoriche riguardanti la teoria della misura e degli spazi di Sobolev; poi, ci concentriamo su strumenti di misura di non compattezza che ci permettono di comprendere in che modo una successione di funzioni debolmente convergente può non essere fortemente convergente. In particolare, studiamo problemi di concentrazione e oscillazione e li applichiamo a minimi vincolati di funzionali nel caso di crescita critica.

On étudie la continuité en norme Lp de certaines fonctionnelles liées à des équations d'évolution. Les fonctionnelles qui nous intéressent sont les fonctions de Littlewood-Paley-Stein et sont à l'origine définies pour le laplacien sur {R}N par H(f)(x) = left( int_0^infty |e^{-tDelta} f|^2 {d}tight)^{1/2}. La fonctionnelle H est bornée sur L^p(RR) pour tout p in (1,+infty), mais ce n'est pas le cas sur les variétés. Plus précisément, on s'intéresse dans cette thèse à l'étude des fonctionnelles de Littlewood-Paley-Stein pour les opérateurs de Schrödinger et le laplacien de Hodge-de Rham sur les variétés riemanniennes non compactes. Elles sont définies par des formules analogues à celle introduite par Stein. Nous nous intéressons aussi au problème qui a motivé l'étude de ces fonctions, celui de la continuité en norme L^p de la transformée de Riesz L^{-1/2} et d^* LF^{-1/2} et aux interactions entre ces deux problèmes.Nous étudions d'abord les fonctionnelles associées aux opérateurs de Schrödinger ou au laplacien de Hodge-de Rham en dehors du cadre habituel de l'estimation gaussienne du noyau de la chaleur et des variétés doublantes. Nous obtenons un résultat positif analogue à la continuité inconditionnelle de H sur L^p pour p in (1,2]. Dans un second temps, nous étudions les liens entre la bornétude de ces fonctions de Littlewood-Paley-Stein pour l'opérateur de Schrödinger et celle de la transformée de Riesz e^{-tL}. Nous montrons que la {R}-born étude des familles d'opérateurs { sqrt{t} sqrt{V} e^{-tL} ,t geq 0} et { sqrt{t}e^{-tL} , t geq 0 } est équivalente à la bornétude de H_L, et implique aussi des estimations de Littlewood-Paley-Stein généralisées.Enfin, nous étudions la bornétude de fonctions carrés coniques dans le cadre d'opérateurs de Schrödinger sur les variétés. Ces fonctions ont un comportement différent sur Lp selon si p in (1,2] ou si p in [2,infty). Nous comparons aussi les fonctions coniques aux fonctions de Littlewood-Paley-Stein classiques
We study the boundedness in Lp norm of some functionals linked to evolution equations. The functions we are interested in are the Littlewood-Paley-Stein functionals and are originally defined for the Laplacian on {R}^N by H(f)(x) = left( int_0^infty | e^{-tDelta} f|^2 {d}tight)^{1/2}. The functional H is bounded on Lp for any p in (1,+infty), but this is not the case on manifolds. More precisely, we are interested in the study of Littlewood-Paley-Stein functionals for Schrödinger's operators and Hodge-de Rham's laplacian on non-compact Riemannian manifolds. They are defined by formulas similar to the one introduced by Stein.We are also interested in the problem which motivated the study of these functions, that of the continuity in standard Lp of the Riesz transform L^{-1/2} and d^* LF^{-1/2} and the interactions between these two problems.We first study the functionals associated with Schrödinger's operators or Hodge-de Rham's laplacian outside the usual framework of Gaussian kernel estimation of heat and doubling varieties. We obtain a positive result analogous to the unconditional boundedness of H over L^p for p in (1.2]. In a second step, we study the links between the boundedness of these Littlewood-Paley-Stein functions for the Schrödinger operator and that of the Riesz transform e^{-tL}. We show that the {R}-boundedness of the families of operators { sqrt{t} sqrt{V} e^{-tL}, t geq 0} and { sqrt{t} abla e^{-tL}, tgeq 0 } is equivalent to the boundedness of H_L, and also implies generalized Littlewood-Paley-Stein estimates. Finally, we study the boundedness of conical square functions within the framework of Schrödinger operators on manifolds

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

Ce mémoire est consacré à des résultats d'analyse harmonique réelle dans des cadres géométriques discrets (graphes) ou continus (groupes de Lie).Soit $\Gamma$ un graphe (ensemble de sommets et d'arêtes) muni d'un laplacien discret $\Delta=I-P$, où $P$ est un opérateur de Markov.Sous des hypothèses géométriques convenables sur $\Gamma$, nous montrons la continuité $L^p$ de fonctionnelles de Littlewood-Paley fractionnaires. Nous introduisons des espaces de Hardy $H^1$ de fonctions et de $1$-formes différentielles sur $\Gamma$, dont nous donnons plusieurs caractérisations, en supposant seulement la propriété de doublement pour le volume des boules de $\Gamma$. Nous en déduisons la continuité de la transformée de Riesz sur $H^1$. En supposant de plus des estimations supérieures ponctuelles (gaussiennes ou sous-gaussiennes) sur les itérées du noyau de l'opérateur $P$, nous obtenons aussi la continuité de la transformée de Riesz sur $L^p$ pour $10$, $1\leq p\leq+\infty$ et $1\leq q\leq +\infty$. Les résultats sont valables en croissance polynomiale ou exponentielle du volume des boules
This thesis is devoted to results in real harmonic analysis in discrete (graphs) or continuous (Lie groups) geometric contexts.Let $\Gamma$ be a graph (a set of vertices and edges) equipped with a discrete laplacian $\Delta=I-P$, where $P$ is a Markov operator.Under suitable geometric assumptions on $\Gamma$, we show the $L^p$ boundedness of fractional Littlewood-Paley functionals. We introduce $H^1$ Hardy spaces of functions and of $1$-differential forms on $\Gamma$, giving several characterizations of these spaces, only assuming the doubling property for the volumes of balls in $\Gamma$. As a consequence, we derive the $H^1$ boundedness of the Riesz transform. Assuming furthermore pointwise upper bounds for the kernel (Gaussian of subgaussian upper bounds) on the iterates of the kernel of $P$, we also establish the $L^p$ boundedness of the Riesz transform for $10$, $1\leq p\leq+\infty$ and $1\leq q\leq +\infty$.These results hold for polynomial as well as for exponential volume growth of balls

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
Tópicos fundamentais da estrutura eletrônica de materiais cristalinos, são investigados de forma generalizada mediante o Cálculo Fracionário. São calculadas as bandas de energia, as funções de Bloch e as funções de Wannier, para a equação de Schrödinger fracionária com derivada de Riesz. É apresentado um estudo detalhado do caráter não local desse tipo de derivada fracionária. Resolve-se a equação de Schrödinger fracionária para o modelo de Kronig-Penney e estuda-se os efeitos da ordem da derivada e da intensidade do potencial. Verificou-se que, ao passar da derivada de segunda ordem para derivadas fracionárias, o comportamento assintótico das funções de Wannier muda apreciavelmente. Elas perdem o decaimento exponencial, e exibem um decaimento acentuado em forma de potência. Fórmulas simples foram dadas para as caudas das funções de Wannier. A banda de energia mais baixa mostrou-se estar relacionada ao estado ligado de um único poço quântico. Sua função de onda também apresentou decaimento em lei de potência. As bandas de energia superiores mudam de comportamento em função da intensidade do potencial. No caso inteiro, a largura de cada uma dessas bandas diminui. No caso fracionário, diminui inicialmente e depois volta a aumentar, aproximando-se de um valor infinito à medida que a intensidade do potencial tende ao infinito. O grau de localização das funções de Wannier, expresso pelo desvio padrão da posição, mostra um comportamento similar ao da largura das bandas de energia. Além dos cristais perfeitos a Ciência de Materiais estuda cristais com defeito. Os defeitos são responsáveis por muitas propriedades de interesse tecnológico e podem induzir estados localizados. Neste trabalho, calculado o estado localizado de menor energia no modelo de Kronig-Penney fracionário com defeito, mediante método das transformadas de Fourier e das funções de Wannier. Verificou-se que este estado também decai em forma de lei de potência.
Basics topics on the electronic structure of crystalline materials are investigated in a generalized fashion through Fractional Calculus. The energy bands, the Bloch and Wannier functions for the fractional Schr odinger equation with Riesz derivative are calculated. The non-locality of the Riesz fractional derivative is analyzed. The fractional Schr odinger equation is solved for the Kronig-Penney model and the e ects of the derivative order and the potential intensity are studied. It was shown that moving from the integer to the fractional order strongly a ects the asymptotic behavior of the Wannier functions. They lose the exponential decay, gaining a strong power-law decay. Simple formulas have been given for the tails of the Wannier functions. A close relatim between the lowest energy band and the bound state of a single quantum well was found. The wavefunction of the latter decays as a power law. Higher energy bands change their behavior as the periodic potential gets stronger. In the integer case, the width of each one of those bands decreases. In the fractional case, it initially decreases and then increases. The width approaching a nite value as the strength tends to in nity. The degree of localization of the Wannier functions, as expressed by the position standard deviation, behaves similarly to the width of the energy bands. In addition to perfect crystals, Materials Science studies defective crystals. Defects are responsible for many properties of technological interest and can induce localized states. In this work, the localized state of lowest energy in the fractional Kronig-Penney model with defect is calculated through of the Fourier transform method and the Wannier functions. It was shown that is decays as a power law.

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.

Die Reichhaltigkeit und breite Anwendbarkeit der Theorie der holomorphen Funktionen in der komplexen Ebene ist stark motivierend eine ähnliche Theorie für höhere Dimensionen zu entwickeln. Viele Forscher waren und sind in diese Aufgaben involviert, insbesondere in der Entwicklung der Quaternionenanalysis. In den letzten Jahren wurde die Quaternionenanalysis bereits erfolgreich auf eine Vielzahl von Problemen der mathematischen Physik angewandt. Das Ziel der Dissertation besteht darin, holomorphe Strukturen in höheren Dimensionen zu studieren. Zunächst wird ein neues Holomorphiekonzept vorgelegt, was auf der Theorie rechtsinvertierbarer Operatoren basiert und nicht auf Verallgemeinerungen des Cauchy-Riemann-Systems wie üblich. Dieser Begriff umfasst die meisten der gut bekannten holomorphen Strukturen in höheren Dimensionen. Unter anderem sind die üblichen Modelle für reelle und komplexe quaternionenwertige Funktionen sowie Clifford-algebra-wertige Funktionen enthalten. Außerdem werden holomorphe Funktionen mittels einer geeignete Formel vom Taylor-Typ durch spezielle Funktionen lokal approximiert. Um globale Approximationen für holomorphe Funktionen zu erhalten, werden im zweiten Teil der Arbeit verschiedene Systeme holomorpher Basisfunktionen in drei und vier Dimensionen mittels geeigneter Fourier-Entwicklungen explizit konstruiert. Das Konzept der Holomorphie ist verbunden mit der Lösung verallgemeinerter Cauchy-Riemann Systeme, deren Funktionswerte reellen Quaternionen bzw. reduzierte Quaternionen sind. In expliziter Form werden orthogonale holomorphe Funktionensysteme konstruiert, die Lösungen des Riesz-Systems bzw. des Moisil-Teodorescu Systems über zylindrischen Gebieten im R3, sowie Lösungen des Riesz-Systems in Kugeln des R4 sind. Um konkrete Anwendungen auf Randwertprobleme realisieren zu können wird eine orthogonale Zerlegung eines Rechts-Quasi-Hilbert-Moduls komplex-quaternionischer Funktionen unter gegebenen Bedingungen studiert. Die Ergebnisse werden auf die Behandlung von Maxwell-Gleichungen mit zeitvariabler elektrischer Dielektrizitätskonstante und magnetischer Permeabilität angewandt
The richness and widely applicability of the theory of holomorphic functions in complex analysis requires to perform a similar theory in higher dimensions. It has been developed by many researchers so far, especially in quaternionic analysis. Over the last years, it has been successfully applied to a vast array of problems in mathematical physics. The aim of this thesis is to study the structure of holomorphy in higher dimensions. First, a new concept of holomorphy is introduced based on the theory of right invertible operators, and not by means of an analogue of the Cauchy-Riemann operator as usual. This notion covers most of the well-known holomorphic structures in higher dimensions including real, complex, quaternionic, Clifford analysis, among others. In addition, from our operators a local approximation of a holomorphic function is attained by the Taylor type formula. In order to obtain the global approximation for holomorphic functions, the second part of the thesis deals with the construction of different systems of basis holomorphic functions in three and four dimensions by means of Fourier analysis. The concept of holomorphy is related to the null-solutions of generalized Cauchy-Riemann systems, which take either values in the reduced quaternions or real quaternions. We obtain several explicit orthogonal holomorphic function systems: solutions to the Riesz and Moisil-Teodorescu systems over cylindrical domains in R3, and solutions to the Riesz system over spherical domains in R4. Having in mind concrete applications to boundary value problems, we investigate an orthogonal decomposition of complex-quaternionic functions over a right quasi-Hilbert module under given conditions. It is then applied to the treatment of Maxwell’s equations with electric permittivity and magnetic permeability depending on the time variable

Banachverbände spielen sowohl in der Theorie als auch in der Anwendung von geordneten normierten Räume eine bedeutende Rolle. Einerseits erweisen sich viele in der Praxis relevanten Räume als Banachverbände, andererseits ermöglichen die Vektorverbandsstruktur und die enge Beziehung zwischen Ordnung und Norm ein tiefes Verständnis solcher normierter Räume. An dieser Stelle setzen folgende Überlegungen an: - Die genaue Untersuchung einiger Resultate der reichhaltigen Banachverbandstheorie ließ (zu Recht) vermuten, dass in manchen Fällen die Verbandsnormeigenschaft keine notwendige Voraussetzung ist. In der Literatur gibt es bereits einige interessante Untersuchungen allgemeiner geordneter normierter Räume mit qualifizierten positiven Kegeln und in dem Zusammenhang eine Reihe wertvoller Dualitätsaussagen. An dieser Stelle sind die Eigenschaften der Normalität, der Nichtabgeflachtheit und der Regularität eines Kegels erwähnt, welche selbst im Falle eines mit einer Norm versehenen Vektorverbandes eine schwächere Relation zwischen Ordnung und Norm ergeben als die Verbandsnormeigenschaft. - In einer neueren Arbeit wurde der aus der Theorie der Vektorverbände gut bekannte Begriff der Disjunktheit bereits auf beliebige geordnete Räume verallgemeinert, wobei viele Eigenschaften disjunkter Vektoren, des disjunkten Komplements einer Menge usw., welche aus der Verbandstheorie bekannt sind, erhalten bleiben. Auf entsprechende Weise, d.h. durch das Ersetzen exakter Infima und Suprema durch Mengen unterer bzw. oberer Schranken, können der Modul eines Vektors sowie der Begriff der Solidität einer Menge für geordnete (normierte) Räume eingeführt werden. An solchen Überlegungen knüpft die vorliegende Arbeit an. Im Kapitel m-Normen ======== werden verallgemeinerte Formen der M-Norm Eigenschaft eingeführt und untersucht. AM-Räume und (approximative) Ordnungseinheit-Räume sind Beispiele für geordnete normierte Räume mit m-Norm. Die Schwerpunkte dieses Kapitels sind zum Einen Kegel- und Normeigenschaften dieser Räume und deren Charakterisierung mit Hilfe solcher Eigenschaften und zum Anderen Dualitätsaussagen, wie sie zum Teil bereits aus der Theorie der AM- und AL-Räume bekannt sind. Minimal totale Mengen ===================== Ziel dieses Kapitels ist es, den oben erwähnten verallgemeinerten Disjunktheitsbegiff für geordnete normierte Räume zu untersuchen. Eine zentrale Rolle spielen dabei totale Mengen im Dualraum und insbesondere minimal totale Mengen sowie deren Zusammenhang mit der Disjunktheit von Elementen des Ausgangsraumes. Normierte pre-Riesz Räume ========================= Wie bereits bekannt, lässt sich jeder pre-Riesz Raum ordnungsdicht in einen (bis auf Isomorphie) eindeutigen minimalen Vektorverband einbetten, die so genannte Riesz Vervollständigung. Ist der pre-Riesz Raum normiert und sein positiver Kegel abgeschlossen, dann kann eine Verbandsnorm auf der Riesz Vervollständigung eingeführt werden, welche sich in vielen Fällen als äquivalent zur Ausgangsnorm auf dem pre-Riesz Raum erweist. Es ist allgemein bekannt, dass sich dann auch stetige lineare Funktionale fortsetzen lassen. In diesem Kapitel wird nun untersucht, inwiefern sich Ordnungsrelationen auf einer Menge stetiger linearer Funktionale beim Übergang zur Menge der Fortsetzungen erhalten lassen. Die gewonnenen Erkenntnisse kommen anschließend bei Untersuchungen zur schwachen bzw. schwach*-Topologie auf geordneten normierten Räumen zur Anwendung. Hierbei werden zwei Fragestellungen behandelt. Zum Einen gilt das Augenmerk disjunkten Folgen in geordneten normierten Räumen. Als Beispiel seien ordnungsbeschränkte disjunkte Folgen in geordneten normierten Räumen mit halbmonotoner mNorm genannt, welche stets schwach gegen Null konvergieren. Zum Anderen werden monoton fallende Folgen und Netze bzw. disjunkte Folgen von stetigen linearen Funktionalen auf einem geordneten normierten Raum betrachtet
Banach lattices play an important role in the theory of ordered normed spaces. One reason is, that many ordered normed vector spaces, that are important in practice, turn out to be Banach lattices, on the other hand, the lattice structure and strong relations between order and norm allow a deep understanding of such ordered normed spaces. At this point the following is to be considered. - The analysis of some results in the rich Banach lattice theory leads to the conjecture, that sometimes the lattice norm property is no necessary supposition. General ordered normed spaces with a convenient positive cone were already examined, where some valuable duality properties could be achieved. We point out the properties of normality, non-flatness and regularity of a cone, which are a weaker relation between order and norm than the lattice norm property in normed vector lattices. - The notion of disjointness in vector lattices has already been generalized to arbitrary ordered vector spaces. Many properties of disjoint elements, the disjoint complement of a set etc., well known from the vector lattice theory, are preserved. The modulus of a vector as well as the concept of the solidness of a set can be introduced in a similar way, namely by replacing suprema and infima by sets of upper and lower bounds, respectively. We take such ideas up in the present thesis. A generalized version of the M-norm property is introduced and examined in section m-norms. ======= AM-spaces and approximate order unit spaces are examples of ordered normed spaces with m-norm. The main points of this section are the special properties of the positive cone and the norm of such spaces and the duality properties of spaces with m-norm. Minimal total sets ================== In this section we examine the mentioned generalized disjointness in ordered normed spaces. Total sets as well as minimal total sets and their relation to disjoint elements play an inportant at this. Normed pre-Riesz spaces ======================= As already known, every pre-Riesz space can be order densely embedded into an (up to isomorphism) unique vector lattice, the so called Riesz completion. If, in addition, the pre-Riesz space is normed and its positive cone is closed, then a lattice norm can be introduced on the Riesz completion, that turns out to be equivalent to the primary norm on the pre-Riesz space in many cases. Positive linear continuous functionals on the pre-Riesz space are extendable to positive linear continuous functionals in this setting. Here we investigate, how some order relations on a set of continuous functionals can be preserved to the set of the extension. In the last paragraph of this section the obtained results are applied for investigations of some questions concerning the weak and the weak* topology on ordered normed vector spaces. On the one hand, we focus on disjoint sequences in ordered normed spaces. On the other hand, we deal with decreasing sequences and nets and disjoint sequences of linear continuous functionals on ordered normed spaces

Banachverbände spielen sowohl in der Theorie als auch in der Anwendung von geordneten normierten Räume eine bedeutende Rolle. Einerseits erweisen sich viele in der Praxis relevanten Räume als Banachverbände, andererseits ermöglichen die Vektorverbandsstruktur und die enge Beziehung zwischen Ordnung und Norm ein tiefes Verständnis solcher normierter Räume. An dieser Stelle setzen folgende Überlegungen an: - Die genaue Untersuchung einiger Resultate der reichhaltigen Banachverbandstheorie ließ (zu Recht) vermuten, dass in manchen Fällen die Verbandsnormeigenschaft keine notwendige Voraussetzung ist. In der Literatur gibt es bereits einige interessante Untersuchungen allgemeiner geordneter normierter Räume mit qualifizierten positiven Kegeln und in dem Zusammenhang eine Reihe wertvoller Dualitätsaussagen. An dieser Stelle sind die Eigenschaften der Normalität, der Nichtabgeflachtheit und der Regularität eines Kegels erwähnt, welche selbst im Falle eines mit einer Norm versehenen Vektorverbandes eine schwächere Relation zwischen Ordnung und Norm ergeben als die Verbandsnormeigenschaft. - In einer neueren Arbeit wurde der aus der Theorie der Vektorverbände gut bekannte Begriff der Disjunktheit bereits auf beliebige geordnete Räume verallgemeinert, wobei viele Eigenschaften disjunkter Vektoren, des disjunkten Komplements einer Menge usw., welche aus der Verbandstheorie bekannt sind, erhalten bleiben. Auf entsprechende Weise, d.h. durch das Ersetzen exakter Infima und Suprema durch Mengen unterer bzw. oberer Schranken, können der Modul eines Vektors sowie der Begriff der Solidität einer Menge für geordnete (normierte) Räume eingeführt werden. An solchen Überlegungen knüpft die vorliegende Arbeit an. Im Kapitel m-Normen ======== werden verallgemeinerte Formen der M-Norm Eigenschaft eingeführt und untersucht. AM-Räume und (approximative) Ordnungseinheit-Räume sind Beispiele für geordnete normierte Räume mit m-Norm. Die Schwerpunkte dieses Kapitels sind zum Einen Kegel- und Normeigenschaften dieser Räume und deren Charakterisierung mit Hilfe solcher Eigenschaften und zum Anderen Dualitätsaussagen, wie sie zum Teil bereits aus der Theorie der AM- und AL-Räume bekannt sind. Minimal totale Mengen ===================== Ziel dieses Kapitels ist es, den oben erwähnten verallgemeinerten Disjunktheitsbegiff für geordnete normierte Räume zu untersuchen. Eine zentrale Rolle spielen dabei totale Mengen im Dualraum und insbesondere minimal totale Mengen sowie deren Zusammenhang mit der Disjunktheit von Elementen des Ausgangsraumes. Normierte pre-Riesz Räume ========================= Wie bereits bekannt, lässt sich jeder pre-Riesz Raum ordnungsdicht in einen (bis auf Isomorphie) eindeutigen minimalen Vektorverband einbetten, die so genannte Riesz Vervollständigung. Ist der pre-Riesz Raum normiert und sein positiver Kegel abgeschlossen, dann kann eine Verbandsnorm auf der Riesz Vervollständigung eingeführt werden, welche sich in vielen Fällen als äquivalent zur Ausgangsnorm auf dem pre-Riesz Raum erweist. Es ist allgemein bekannt, dass sich dann auch stetige lineare Funktionale fortsetzen lassen. In diesem Kapitel wird nun untersucht, inwiefern sich Ordnungsrelationen auf einer Menge stetiger linearer Funktionale beim Übergang zur Menge der Fortsetzungen erhalten lassen. Die gewonnenen Erkenntnisse kommen anschließend bei Untersuchungen zur schwachen bzw. schwach*-Topologie auf geordneten normierten Räumen zur Anwendung. Hierbei werden zwei Fragestellungen behandelt. Zum Einen gilt das Augenmerk disjunkten Folgen in geordneten normierten Räumen. Als Beispiel seien ordnungsbeschränkte disjunkte Folgen in geordneten normierten Räumen mit halbmonotoner mNorm genannt, welche stets schwach gegen Null konvergieren. Zum Anderen werden monoton fallende Folgen und Netze bzw. disjunkte Folgen von stetigen linearen Funktionalen auf einem geordneten normierten Raum betrachtet.
Banach lattices play an important role in the theory of ordered normed spaces. One reason is, that many ordered normed vector spaces, that are important in practice, turn out to be Banach lattices, on the other hand, the lattice structure and strong relations between order and norm allow a deep understanding of such ordered normed spaces. At this point the following is to be considered. - The analysis of some results in the rich Banach lattice theory leads to the conjecture, that sometimes the lattice norm property is no necessary supposition. General ordered normed spaces with a convenient positive cone were already examined, where some valuable duality properties could be achieved. We point out the properties of normality, non-flatness and regularity of a cone, which are a weaker relation between order and norm than the lattice norm property in normed vector lattices. - The notion of disjointness in vector lattices has already been generalized to arbitrary ordered vector spaces. Many properties of disjoint elements, the disjoint complement of a set etc., well known from the vector lattice theory, are preserved. The modulus of a vector as well as the concept of the solidness of a set can be introduced in a similar way, namely by replacing suprema and infima by sets of upper and lower bounds, respectively. We take such ideas up in the present thesis. A generalized version of the M-norm property is introduced and examined in section m-norms. ======= AM-spaces and approximate order unit spaces are examples of ordered normed spaces with m-norm. The main points of this section are the special properties of the positive cone and the norm of such spaces and the duality properties of spaces with m-norm. Minimal total sets ================== In this section we examine the mentioned generalized disjointness in ordered normed spaces. Total sets as well as minimal total sets and their relation to disjoint elements play an inportant at this. Normed pre-Riesz spaces ======================= As already known, every pre-Riesz space can be order densely embedded into an (up to isomorphism) unique vector lattice, the so called Riesz completion. If, in addition, the pre-Riesz space is normed and its positive cone is closed, then a lattice norm can be introduced on the Riesz completion, that turns out to be equivalent to the primary norm on the pre-Riesz space in many cases. Positive linear continuous functionals on the pre-Riesz space are extendable to positive linear continuous functionals in this setting. Here we investigate, how some order relations on a set of continuous functionals can be preserved to the set of the extension. In the last paragraph of this section the obtained results are applied for investigations of some questions concerning the weak and the weak* topology on ordered normed vector spaces. On the one hand, we focus on disjoint sequences in ordered normed spaces. On the other hand, we deal with decreasing sequences and nets and disjoint sequences of linear continuous functionals on ordered normed spaces.

Cette thèse est consacrée à l'évaluation des familles exponentielles pour les problèmes de la modélisation des bruits et de la segmentation des images couleurs. Dans un premier temps, nous avons développé une nouvelle caractérisation des familles exponentielles naturelles infiniment divisible basée sur la fonction trace de la matrice de variance covariance associée. Au niveau application, cette nouvelle caractérisation a permis de détecter la nature de la loi d'un bruit additif associé à un signal où à une image couleur. Dans un deuxième temps, nous avons proposé un nouveau modèle statistique paramétrique mulltivarié basé sur la loi de Riesz. La loi de ce nouveau modèle est appelée loi de la diagonale modifiée de Riesz. Ensuite, nous avons généralisé ce modèle au cas de mélange fini de lois. Enfin, nous avons introduit un algorithme de segmentation statistique d'image ouleur, à travers l'intégration de la méthode des centres mobiles (K-means) au niveau de l'initialisation pour une meilleure définition des classes de l'image et l'algorithme EM pour l'estimation des différents paramètres de chaque classe qui suit la loi de la diagonale modifiée de la loi de Riesz.

This thesis consists of a comprehensive summary and six scientific papers dealing with the boundary behaviour of functions related to parabolic partial differential equations and several complex variables. Paper I concerns solutions to non-linear parabolic equations of linear growth. The main results include a backward Harnack inequality, and the Hölder continuity up to the boundary of quotients of non-negative solutions vanishing on the lateral boundary of an NTA cylinder. It is also shown that the Riesz measure associated with such solutions has the doubling property. Paper II is concerned with solutions to linear degenerate parabolic equations, where the degeneracy is controlled by a weight in the Muckenhoupt class 1+2/n. Two main results are that non-negative solutions which vanish continuously on the lateral boundary of an NTA cylinder satisfy a backward Harnack inequality and that the quotient of two such functions is Hölder continuous up to the boundary. Another result is that the parabolic measure associated to such equations has the doubling property. In Paper III, it is shown that a bounded pseudoconvex domain whose boundary is α-Hölder for each 0<α<1, is hyperconvex. Global estimates of the exhaustion function are given. In Paper IV, it is shown that on the closure of a domain whose boundary locally is the graph of a continuous function, all plurisubharmonic functions with continuous boundary values can be uniformly approximated by smooth plurisubharmonic functions defined in neighbourhoods of the closure of the domain. Paper V studies  Poletsky’s notion of plurisubharmonicity on compact sets. It is shown that a function is plurisubharmonic on a given compact set if, and only if, it can be pointwise approximated by a decreasing sequence of smooth plurisubharmonic functions defined in neighbourhoods of the set. Paper VI introduces the notion of a P-hyperconvex domain. It is shown that in such a domain, both the Dirichlet problem with respect to functions plurisubharmonic on the closure of the domain, and the problem of approximation by smooth plurisubharmoinc functions in neighbourhoods of the closure of the domain have satisfactory answers in terms of plurisubharmonicity on the boundary.

Cette thèse est consacrée à l’étude de la stabilisation de certains systèmes distribués avec contrôle frontière de type dynamique. Nous considérons, d’abord, la stabilisation de l’équation de la poutre de Rayleigh avec un seul contrôle frontière dynamique moment ou force. Nous montrons que le système n’est pas uniformément (autrement dit exponentiellement) stable; mais par une méthode spectrale, nous établissons le taux polynomial optimal de décroissance de l’énergie du système. Ensuite, nous étudions la stabilisation indirecte de l’équation des ondes avec un amortissement frontière de type dynamique fractionnel. Nous montrons que le taux de décroissance de l’énergie dépend de la nature géométrique du domaine. En utilisant la méthode fréquentielle et une méthode spectrale, nous montrons la non stabilité exponentielle et nous établissons, plusieurs résultats de stabilité polynomiale. Enfin, nous considérons l’approximation de l’équation des ondes mono-dimensionnelle avec un seul amortissement frontière de type dynamique par un schéma de différence finie. Par une méthode spectrale, nous montrons que l’énergie discrétisée ne décroit pas uniformément (par rapport au pas du maillage) polynomialement vers zéro comme l’énergie du système continu. Nous introduisons, alors, un terme de viscosité numérique et nous montrons la décroissance polynomiale uniforme de l’énergie de notre schéma discret avec ce terme de viscosité
This thesis is devoted to the study of the stabilization of some distributed systems with dynamic boundary control. First, we consider the stabilization of the Rayleigh beam equation with only one dynamic boundary control moment or force. We show that the system is not uniformly (exponentially) stable. However, using a spectral method, we establish the optimal polynomial decay rate of the energy of the system. Next, we study the indirect stability of the wave equation with a fractional dynamic boundary control. We show that the decay rate of the energy depends on the nature of the geometry of the domain. Using a frequency approach and a spectral method, we show the non exponential stability of the system and we establish, different polynomial stability results. Finally, we consider the finite difference space discretization of the 1-d wave equation with dynamic boundary control. First, using a spectral approach, we show that the polynomial decay of the discretized energy is not uniform with respect to the mesh size, as the energy of the continuous system. Next, we introduce a viscosity term and we establish the uniform (with respect to the mesh size) polynomial energy decay of our discrete scheme

Cette thèse est consacrée à l'étude de la régularité des solutions des équations de Monge-Ampère complexes ainsi que des équations hessiennes complexes dans un domaine borné de Cn. Dans le premier chapitre, on donne des rappels sur la théorie du pluripotentiel. Dans le deuxième chapitre, on étudie le module de continuité des solutions du problème de Dirichlet pour les équations de Monge-Ampère lorsque le second membre est une mesure à densité continue par rapport à la mesure de Lebesgue dans un domaine strictement hyperconvexe lipschitzien. Dans le troisième chapitre, on prouve la continuité hölderienne des solutions de ce problème pour certaines mesures générales. Dans le quatrième chapitre, on considère le problème de Dirichlet pour les équations hessiennes complexes plus générales où le second membre dépend de la fonction inconnue. On donne une estimation précise du module de continuité de la solution lorsque la densité est continue. De plus, si la densité est dans Lp , on démontre que la solution est Hölder-continue jusqu'au bord
In this thesis we study the regularity of solutions to the Dirichlet problem for complex Monge-Ampère equations and also for complex Hessian equations in a bounded domain of Cn. In the first chapter, we give basic facts in pluripotential theory. In the second chapter, we study the modulus of continuity of solutions to the Dirichlet problem for complex Monge-Ampère equations when the right hand side is a measure with continuous density with respect to the Lebesgue measure in a bounded strongly hyperconvex Lipschitz domain. In the third chapter, we prove the Hölder continuity of solutions to this problem for some general measures. In the fourth chapter, we consider the Dirichlet problem for complex Hessian equations when the right hand side depends on the unknown function. We give a sharp estimate of the modulus of continuity of the solution as the density is continuous. Moreover, for the case of Lp-density we demonstrate that the solution is Hölder continuous up to the boundary

This PhD thesis aims to study financial processes which have semi-heavy-tailed marginal distributions and may exhibit memory. The traditional Black-Scholes model is expanded to incorporate memory via an integral operator, resulting in a class of market models which still preserve the completeness and arbitragefree conditions needed for replication of contingent claims. This approach is used to estimate the implied volatility of the resulting model. The first part of the thesis investigates the semi-heavy-tailed behaviour of financial processes. We treat these processes as continuous-time random walks characterised by a transition probability density governed by a fractional Riesz- Bessel equation. This equation extends the Feller fractional heat equation which generates a-stable processes. These latter processes have heavy tails, while those processes generated by the fractional Riesz-Bessel equation have semi-heavy tails, which are more suitable to model financial data. We propose a quasi-likelihood method to estimate the parameters of the fractional Riesz- Bessel equation based on the empirical characteristic function. The second part considers a dynamic model of complete financial markets in which the prices of European calls and puts are given by the Black-Scholes formula. The model has memory and can distinguish between historical volatility and implied volatility. A new method is then provided to estimate the implied volatility from the model. The third part of the thesis considers the problem of classification of financial markets using high-frequency data. The classification is based on the measure representation of high-frequency data, which is then modelled as a recurrent iterated function system. The new methodology developed is applied to some stock prices, stock indices, foreign exchange rates and other financial time series of some major markets. In particular, the models and techniques are used to analyse the SET index, the SET50 index and the MAI index of the Stock Exchange of Thailand.

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This work has as main objective to find mathematical models based on linear parametric estimation techniques applied to the problem of calculating the grow of gas in oil wells. In particular we focus on achieving grow models applied to the case of wells that produce by plunger-lift technique on oil rigs, in which case, there are high peaks in the grow values that hinder their direct measurement by instruments. For this, we have developed estimators based on recursive least squares and make an analysis of statistical measures such as autocorrelation, cross-correlation, variogram and the cumulative periodogram, which are calculated recursively as data are obtained in real time from the plant in operation; the values obtained for these measures tell us how accurate the used model is and how it can be changed to better fit the measured values. The models have been tested in a pilot plant which emulates the process gas production in oil wells
Este trabalho teve como objetivo principal encontrar modelos matem?ticos baseados em t?cnicas de estima??o param?trica lineares aplicado ao problema do c?lculo da vaz?o de g?s em po?os de petr?leo. Em particular se concentrou na obten??o de modelos de vaz?o aplicados ao caso de po?os que produzem pela t?cnica de plunger-lift em plataformas de petr?leo, pois nesse caso, ocorrem picos elevados nos valores da vaz?o que dificultam sua medi- ??o direta atrav?s de instrumentos. Para isso, desenvolveram-se estimadores baseados em m?nimos quadrados recursivos e fez-se uma an?lise das medidas estat?sticas tais como autocorrela??o, correla??o cruzada, variograma e o periodograma acumulado, que s?o calculadas recursivamente ? medida que dados s?o obtidos em tempo real da planta em opera??o; os valores obtidos para essas medidas indicaram o qu?o exato ? o modelo utilizado e de que forma ele pode ser alterado para melhor se adequar aos valores medidos. Os modelos obtidos foram testados em uma planta piloto que emula o processo de produ??o de g?s em po?os de petr?leo

We characterise the higher order Riesz transforms on the Heisenberg group and also show that they satisfy dimension-free bounds under some assumptions on the multipliers. We also prove the boundedness of the higher order Riesz transforms associated to the Hermite operator. Using transference theorems, we deduce boundedness theorems for Riesz transforms on the reduced Heisenberg group and hence also for the Riesz transforms associated to special Hermite and Laguerre expansions. Next we study the Riesz transforms associated to the Grushin operator G = - Δ - |x|2@t2 on Rn+1. We prove that both the first order and higher order Riesz transforms are bounded on Lp(Rn+1): We also prove that norms of the first order Riesz transforms are independent of the dimension n.

This thesis is divided into three parts. In the first part we study mixed norm estimates for Riesz transforms associated with various differential operators. First we prove the mixed norm estimates for the Riesz transforms associated with Dunkl harmonic oscillator by means of vector valued inequalities for sequences of operators defined in terms of Laguerre function expansions. In certain cases, the result can be deduced from the corresponding result for Hermite Riesz transforms, for which we give a simple and an independent proof. The mixed norm estimates for Riesz transforms associated with other operators, namely the sub-Laplacian on Heisenberg group, special Hermite operator on C^d and Laplace-Beltrami operator on the group SU(2) are obtained using their L^pestimates and by making use of a lemma of Herz and Riviere along with an idea of Rubio de Francia. Applying these results to functions expanded in terms of spherical harmonics, we deduce certain vector valued inequalities for sequences of operators defined in terms of radial parts of the corresponding operators. In the second part, we study the chaotic behavior of the heat semigroup generated by the Dunkl-Laplacian ∆_κ on weighted L^P-spaces. In the general case, for the chaotic behavior of the Dunkl-heat semigroup on weighted L^p-spaces, we only have partial results, but in the case of the heat semigroup generated by the standard Laplacian, a complete picture of the chaotic behavior is obtained on the spaces L^p ( R^d,〖 (φ_iρ (x ))〗^2 dx) where φ_iρ the Euclidean spherical function is. The behavior is very similar to the case of the Laplace-Beltrami operator on non-compact Riemannian symmetric spaces studied by Pramanik and Sarkar. In the last part, we study mixed norm estimates for the Cesáro means associated with Dunkl-Hermite expansions on〖 R〗^d. These expansions arise when one considers the Dunkl-Hermite operator (or Dunkl harmonic oscillator)〖 H〗_κ:=-Δ_κ+|x|^2. It is shown that the desired mixed norm estimates are equivalent to vector-valued inequalities for a sequence of Cesáro means for Laguerre expansions with shifted parameter. In order to obtain the latter, we develop an argument to extend these operators for complex values of the parameters involved and apply a version of Three Lines Lemma.