Les sommes partielles de Taylor $S_n, n$ ≥ 0, sont des opérateurs de rang fini dans n'importe quel espace de Banach de fonctions analytiques sur le disque unité ouvert. Dans le cadre classique de l'algèbre du disque, la valeur précise de la norme de $S_n$ n'est pas connue et donc dans la littérature, on les appelle les constantes de Lebesgue. Dans ce cadre, nous savons seulement qu'elles croissent comme $\log n$, modulo une constante multiplicative, lorsque $n$ tend vers l'infini. Cependant, dans les espaces de Dirichlet pondérés $\mathcal {D_w}$, nous évaluons précisément la norme de $S_n$. En fait, il existe différentes façons de mettre une norme sur $\mathcal {D_w}$. Bien que ces normes soient équivalentes, elles conduisent à des valeurs différentes pour la norme de $S_n$ en tant qu'opérateur sur $\mathcal {D_w}$. Nous présentons trois normes différentes sur $\mathcal {D_w}$ et dans chaque cas, nous obtenons la valeur précise de la norme de l'opérateur $S_n$. Ces résultats sont en contraste marqué avec le cadre classique de l'algèbre du disque. Les sommes partielles de Taylor $S_n, n$ ≥ 0, sont des opérateurs de rang fini dans n'importe quel espace de Banach de fonctions analytiques sur le disque unité. Dans le cadre classique de l'algèbre du disque $\mathcal {A}$, la valeur précise de $|S_n|_\mathcal {A{\to}A}$ n'est pas connue. Ces nombres sont appelés les constantes de Lebesgue et ils croissent comme $\log n$, modulo une constante multiplicative, lorsque $n$ tend vers l'infini. Dans cette thèse, nous étudions $|S_n|$ lorsqu'il agit sur l'espace de Dirichlet local $\mathcal {D_\zeta}$. Il existe plusieurs façons distinguées de mettre une norme sur $\mathcal {D_\zeta}$ et chaque choix conduit naturellement à une norme d'opérateur différente pour $S_n$, en tant qu'opérateur sur $\mathcal {D_\zeta}$. Nous considérons trois normes différentes sur $\mathcal {D_\zeta}$ et, dans chaque cas, évaluons la valeur précise de $|S_n|_\mathcal {D_\zeta{\to}D_\zeta}$. Dans tous les cas, nous montrons également que la fonction maximisante est unique. Ces formules indiquent que $|S_n|_\mathcal {D_\zeta{\to}D_\zeta}≍\sqrt{n}$ lorsque $n$ croît. Ainsi, à la lumière du principe de borne uniforme, il existe une fonction $f ∈ \mathcal {D_\zeta}$ telle que la suite locale $|S_nf|_\mathcal {D_\zeta},n$ ≥ 1, n'est pas bornée. Nous fournissons deux constructions explicites. Ensuite, nous obtenons les valeurs précises de la norme de l'opérateur de moyennes de Cesàro $σ_n$ et montrons que contrairement à la somme partielle, $|σ_nf|_\mathcal {D_\zeta},n$ ≥ 1, est bornée. / The partial Taylor sums $S_n, n$ ≥ 0, are finite rank operators on any Banach space of analytic functions on the open unit disc. In the classical setting of disc algebra, the precise value of the norm of $S_n$ is not known and thus in the literature they are referred as the Lebesgue constants. In this setting, we just know that the grow like $\log n$, modulo a multiplicative constant, as $n$ tends to infinity. However, on the weighted Dirichlet spaces $\mathcal {D_w}$, we precisely evaluate the norm of $S_n$. As a matter of fact, there are different ways to put a norm on $\mathcal {D_w}$. Even though these norms are equivalent, they lead to different values for the norm of $S_n$, as an operator on $\mathcal {D_w}$. We present three different norms on $\mathcal {D_w}$, and in each case we obtain the precise value of the operator norm of $S_n$. These results are in sharp contrast to the classical setting of the disc algebraThe partial Taylor sums $S_n, n$ ≥ 0, are finite rank operators on any Banach space of analytic functions on the open unit disc. In the classical setting of disc algebra $\mathcal {A}$, the precise value of $\|S_n\|_\mathcal {A{\to}A}$ is not known. These numbers are referred as the Lebesgue constants and they grow like $\log n$, modulo a multiplicative constant, when $n$ tends to infinity. In this note, we study $\|S_n\|$ when it acts on the local Dirichlet space $\mathcal {D_\zeta}$. There are several distinguished ways to put a norm on $\mathcal {D_\zeta}$ and each choice naturally leads to a different operator norm for $S_n$, as an operator on $\mathcal {D_\zeta}$. We consider three different norms on $\mathcal {D_\zeta}$ and, in each case, evaluate the precise value of $\|S_n\|_\mathcal {D_\zeta{\to}D_\zeta}$. In all cases, we also show that the maximizing function is unique. These formulas indicate that $\|S_n\|_\mathcal {D_\zeta{\to}D_\zeta}≍\sqrt{n}$ as $n$ grows. Hence, in the light of uniform boundedness principle, there is a function $f ∈ \mathcal {D_\zeta}$ such that the local sequence $\|S_nf\|_\mathcal {D_\zeta},n$≥1, is unbounded. We provide two explicit constructions. Next we obtain the precise values of the operator norm of Cesaro means $σ_n$ and show that contrary to the partial sum, we do have $\|σ_nf\|_\mathcal {D_\zeta},n$ ≥ 1, is bounded.

In this paper we consider an existence of a solution for a nonlinear nonmonotone wave equation in [0,π]xR and an existence of a positive solution for a non-positone Dirichlet problem in a bounded subset of R^n.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Apresentamos uma extensÃo do Teorema de Barta devido a G. P. Bessa and J. F. Montenegro e fazemos algumas aplicaÃÃes geomÃtricas do resultado obtido. A primeira aplicaÃÃo geomÃtrica da extensÃo do Teorema de Barta à uma extensÃo do Teorema de Cheng sobre estimativas inferiores de autovalores do Laplaciano em bolas geodÃsicas normais. A segunda aplicaÃÃo geomÃtrica à uma generalizaÃÃo do Teorema de Cheng-Li-Yau de estimativas de autovalores para uma subvariedade mÃnima do espaÃo forma. / We present an extension to Barta's Theorem due to G. P. Bessa and J. F. Montenegro and we show some geometric applications of the obtained result. As first application, we extend Chang's lower eigenvalue estimates of the Laplacian in normal geodesic balls. As second application, we generalize Cheng-Li-Yau's eigenvalue estimates to a minimal submanifold of the space forms.

My thesis work started in the summer of 2005 as a three way joint project by Professor Castro and Mr. John Kwon and myself. A paper from this joint project was written and the content now forms my thesis.

Thesis (M.A.)--University of North Carolina at Greensboro, 2006. / Title from PDF title page screen. Advisor: Maya Chhetri; submitted to the Dept. of Mathematical Sciences. Includes bibliographical references (p. 39-40).

Nonparametric Bayesian models have been researched extensively in the past 10 years following the work of Escobar and West (1995) on sampling schemes for Dirichlet processes. The infinite mixture representation of the Dirichlet process makes it useful for clustering problems where the number of clusters is unknown. We develop nonparametric Bayesian models for two different clustering problems, namely functional and graphical clustering. We propose a nonparametric Bayes wavelet model for clustering of functional or longitudinal data. The wavelet modelling is aimed at the resolution of global and local features during clustering. The model also allows the elicitation of prior belief about the regularity of the functions and has the ability to adapt to a wide range of functional regularity. Posterior inference is carried out by Gibbs sampling with conjugate priors for fast computation. We use simulated as well as real datasets to illustrate the suitability of the approach over other alternatives. The functional clustering model is extended to analyze splice microarray data. New microarray technologies probe consecutive segments along genes to observe alternative splicing (AS) mechanisms that produce multiple proteins from a single gene. Clues regarding the number of splice forms can be obtained by clustering the functional expression profiles from different tissues. The analysis was carried out on the Rosetta dataset (Johnson et al., 2003) to obtain a splice variant by tissue distribution for all the 10,000 genes. We were able to identify a number of splice forms that appear to be unique to cancer. We propose a Bayesian model for partitioning graphs depicting dependencies in a collection of objects. After suitable transformations and modelling techniques, the problem of graph cutting can be approached by nonparametric Bayes clustering. We draw motivation from a recent work (Dhillon, 2001) showing the equivalence of kernel k-means clustering and certain graph cutting algorithms. It is shown that loss functions similar to the kernel k-means naturally arise in this model, and the minimization of associated posterior risk comprises an effective graph cutting strategy. We present here results from the analysis of two microarray datasets, namely the melanoma dataset (Bittner et al., 2000) and the sarcoma dataset (Nykter et al., 2006).

Thèses (Ph.D.)--Université de Sherbrooke (Canada), 1999. / Titre de l'écran-titre (visionné le 20 juin 2006). Publié aussi en version papier.

Thesis--Copenhagen.

Thesis--Copenhagen.

Th. 3e cycle--Math. pures--Grenoble 1, 1980. N°: 87.