Thesis (Ph. D.)--University of Rhode Island, 2006. / Includes bibliographical references (leaves 175-178).

<p>The paper begins with a short survey of monotone functions. The functions of bounded variation are introduced and some basic properties of these functions are given. Finally the jump function of a function of bounded variation is defined.</p>

The paper begins with a short survey of monotone functions. The functions of bounded variation are introduced and some basic properties of these functions are given. Finally the jump function of a function of bounded variation is defined.

Thesis (M.S.)--Rochester Institute of Technology, 2005. / Typescript. Includes bibliographical references (leaf 41).

Lin, Jessey. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 79-80). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.

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A busca por grandes estruturas lineares em conjuntos de funções com propriedades pa-tológicas é um tópico que fora desenvolvido nos últimos vinte anos. Esse trabalho detalhaalguns desses resultados sobre lineabilidade e espaçabilidade de forma clara e diluida parafacilitar a introdução desses conceitos para um pesquisador não familiarizado.Veremos que os seguintes conjuntos são lineáveis: funçõesCnão analíticas, funçõescom apenas uma quantidade finita de pontos de continuidade, funções cujas derivadas sãoilimitadas num intervalo fechado, funções sobrejetoras em todo lugar que se anulam quasesempre. Também mostraremos a espaçabilidade dos seguintes conjuntos: funções de variaçãolimitada com um conjunto denso de descontinuidades em salto e funções Lebesgue integráveisem [0,1] não essencialmente limitadas em nenhum intervalo. Finalmente, veremos algunsresultados sobre a lineabilidade no conjunto dos funcionais lineares que atingem a norma. / Finding large linear structures in sets of functions with pathological properties is a topicthat has been developed in the last twenty years. This work details some of these resultsabout lineability and spaceability in a clear and diluted way to make the introduction ofthese concepts easier for an unfamiliar researcher.We show that the following sets are lineable:Cnon-analytic functions, functions witha finite number of points of continuity, functions whose derivative is unbounded on a closedinterval and everywhere surjective functions that are almost everywhere zero. We also showthe spaceability of the following sets: functions of bounded variation which have a denseset of jump discontinuities and Lebesgue integrable functions in [0,1] which are nowhereessentially bounded. At last, we show some results about lineability in the set of linearfunctionals that attain their norm.

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

Cette thèse est une étude de modèles d'image aléatoires avec des applications en synthèse de texture.Dans la première partie de la thèse, des algorithmes de synthèse de texture basés sur le modèle shot noise sont développés. Dans le cadre discret, deux processus aléatoires, à savoir le shot noise discret asymptotique et le bruit à phase aléatoire, sont étudiés. On élabore ensuite un algorithme rapide de synthèse de texture basé sur ces processus. De nombreuses expériences démontrent que cet algorithme permet de reproduire une certaine classe de textures naturelles que l'on nomme micro-textures. Dans le cadre continu, la convergence gaussienne des modèles shot noise est étudiée d'avantage et de nouvelles bornes pour la vitesse de cette convergence sont établies. Enfin, on présente un nouvel algorithme de synthèse de texture procédurale par l'exemple basé sur le récent modèle Gabor noise. Cet algorithme permet de calculer automatiquement un modèle procédural représentant des micro-textures naturelles.La deuxième partie de la thèse est consacrée à l'étude du processus feuilles mortes transparentes (FMT), un nouveau modèle germes-grains obtenu en superposant des objets semi-transparents. Le résultat principal de cette partie montre que, lorsque la transparence des objets varie, le processus FMT fournit une famille de modèles variant du modèle feuilles mortes à un champ gaussien. Dans la troisième partie de la thèse, les champs aléatoires à variation bornés sont étudiés et on établit des résultats généraux sur le calcul de la variation totale moyenne de ces champs. En particulier, ces résultats généraux permettent de calculer le périmètre moyen des ensembles aléatoires et de calculer explicitement la variation totale moyenne des modèles germes-grains classiques. / This thesis is a study of stochastic image models with applications to texture synthesis. Most of the stochastic texture models under investigation are germ-grain models. In the first part of the thesis, texture synthesis algorithms relying on the shot noise model are developed. In the discrete framework, two different random processes, namely the asymptotic discrete spot noise and the random phase noise, are theoretically and experimentally studied. A fast texture synthesis algorithm relying on these random processes is then elaborated. Numerous results demonstrate that the algorithm is able to reproduce a class of real-world textures which we call micro-textures. In the continuous framework, the Gaussian convergence of shot noise models is further studied and new bounds for the rate of this convergence are established. Finally, a new algorithm for procedural texture synthesis from example relying on the recent Gabor noise model is presented. This new algorithm permits to automatically compute procedural models for real-world micro-textures. The second part of the thesis is devoted to the introduction and study of the transparent dead leaves (TDL) process, a new germ-grain model obtained by superimposing semi-transparent objects. The main result of this part shows that, when varying the transparency of the objects, the TDL process provides a family of models varying from the dead leaves model to a Gaussian random field. In the third part of the thesis, general results on random fields with bounded variation are established with an emphasis on the computation of the mean total variation of random fields. As particular cases of interest, these general results permit the computation of the mean perimeter of random sets and of the mean total variation of classical germ-grain models.