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Frames in Hilbert spacesShaman, Itamar 01 July 2002 (has links)
No description available.
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Stabilisation de quelques équations d’évolution du second ordrepar des lois de rétroaction / Stabilization of second order evolution equations with dynamical feedbacksAbbas, Zainab 02 October 2014 (has links)
Dans cette thèse, nous étudions la stabilisation de certaines équations d’évolution par des lois de rétroaction. Dans le premier chapitre nous étudions l’équation des ondes dans R avec conditions aux limites dynamiques appliquées sur une partie du bord et une condition de Dirichlet sur la partie restante. Nous fournissons des conditions suffisantes qui garantissent une stabilité polynomiale en utilisant une méthode qui combine une inégalité d’observabilité pour le problème non amorti associé avec des résultats de régularité du problème non amorti. L’optimalité de la décroissance est montrée dans certains cas à l’aide des résultats spectraux précis de l’opérateur associé. Dans le deuxième chapitre nous considérons le système sur un domaine de Rd, d ≥ 2. On trouve des conditions suffisantes qui permettent la stabilité forte. Ensuite, nous discutons de la stabilité non uniforme ainsi que de la stabilité polynomiale. L’approche en domaine fréquentiel nous permet d’établir une décroissance polynomiale sur des domaines pour lesquels l’équation des ondes avec l’amortissement standard est exponentiellement ou polynomialement stable. Dans le troisième chapitre nous considérons un cadre général d’équations d’évolution avec une dissipation dynamique. Sous une hypothèse de régularité, nous montrons que les propriétés d’observabilité pour le problème non amorti impliquent des estimations de décroissance pour le problème amorti. / In this thesis, we study the stabilization of some evolution equations by feedback laws. In the first chapter we study the wave equation in R with dynamical boundary control applied on a part of the boundary and a Dirichlet boundary condition on the remaining part. We furnish sufficient conditions that guarantee a polynomial stability proved using a method that combines an observability inequality for the associated undamped problem with regularity results of the solution of the undamped problem. In addition, the optimality of the decay is shown in some cases with the help of precise spectral results of the operator associated with the damped problem. Then in the second chapter we consider the system on a domain of Rd, d ≥ 2. In this case, the domain of the associated operator is not compactly embedded into the energy space. Nevertheless, we find sufficient conditions that give the strong stability. Then, we discuss the non uniform stability as well as the polynomial stability by two methods. The frequency domain approach allows us to establish a polynomial decay on some domains for which the wave equation with the standard damping is exponentially or polynomially stable. Finally, in the third chapter we consider a general framework of second order evolution equations with dynamical feedbacks. Under a regularity assumption we show that observability properties for the undamped problem imply decay estimates for the damped problem. We finally illustrate our general results by a variety of examples.
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Analysis of Micro-Expressions based on the Riesz Pyramid : Application to Spotting and Recognition / Analyse des micro-expressions exploitant la pyramide de Riesz : application à la détection et à la reconnaissanceArango Duque, Carlos 06 December 2018 (has links)
Les micro-expressions sont des expressions faciales brèves et subtiles qui apparaissent et disparaissent en une fraction de seconde. Ce type d'expressions reflèterait "l'intention réelle" de l'être humain. Elles ont été étudiées pour mieux comprendre les communications non verbales et dans un contexte médicale lorsqu'il devient presque impossible d'engager une conversation ou d'essayer de traduire les émotions du visage ou le langage corporel d'un patient. Cependant, détecter et reconnaître les micro-expressions est une tâche difficile pour l'homme. Il peut donc être pertinent de développer des systèmes d'aide à la communication exploitant les micro-expressions. De nombreux travaux ont été réalisés dans les domaines de l'informatique affective et de la vision par ordinateur pour analyser les micro-expressions, mais une grande majorité de ces méthodes repose essentiellement sur des méthodes de vision par ordinateur classiques telles que les motifs binaires locaux, les histogrammes de gradients orientés et le flux optique. Étant donné que ce domaine de recherche est relativement nouveau, d'autres pistes restent à explorer. Dans cette thèse, nous présentons une nouvelle méthodologie pour l'analyse des petits mouvements (que nous appellerons par la suite mouvements subtils) et des micro-expressions. Nous proposons d'utiliser la pyramide de Riesz, une approximation multi-échelle et directionnelle de la transformation de Riesz qui a été utilisée pour l'amplification du mouvement dans les vidéos à l'aide de l'estimation de la phase 2D locale. Pour l'étape générale d'analyse de mouvements subtils, nous transformons une séquence d'images avec la pyramide de Riesz, extrayons et filtrons les variations de phase de l'image. Ces variations de phase sont en lien avec le mouvement. De plus, nous isolons les régions d'intérêt où des mouvements subtils pourraient avoir lieu en masquant les zones de bruit à l'aide de l'amplitude locale. La séquence d'image est transformée en un signal ID utilisé pour l'analyse temporelle et la détection de mouvement subtils. Nous avons créé notre propre base de données de séquences de mouvements subtils pour tester notre méthode. Pour l'étape de détection de micro-expressions, nous adaptons la méthode précédente au traitement de certaines régions d'intérêt du visage. Nous développons également une méthode heuristique pour détecter les micro-événements faciaux qui sépare les micro-expressions réelles des clignotements et des mouvements subtils des yeux. Pour la classification des micro-expressions, nous exploitons l'invariance, sur de courtes durées, de l'orientation dominante issue de la transformation de Riesz afin de moyenner la séquence d'une micro-expression en une paire d'images. A partir de ces images, nous définissons le descripteur MORF (Mean Oriented Riesz Feature) constitué d'histogrammes d'orientation. Les performances de nos méthodes sont évaluées à l'aide de deux bases de données de micro-expressions spontanées. / Micro-expressions are brief and subtle facial expressions that go on and off the face in a fraction of a second. This kind of facial expressions usually occurs in high stake situations and is considered to reflect a humans real intent. They have been studied to better understand non-verbal communications and in medical applications where is almost impossible to engage in a conversation or try to read the facial emotions or body language of a patient. There has been some interest works in micro-expression analysis, however, a great majority of these methods are based on classically established computer vision methods such as local binary patterns, histogram of gradients and optical flow. Considering the fact that this area of research is relatively new, much contributions remains to be made. ln this thesis, we present a novel methodology for subtle motion and micro-expression analysis. We propose to use the Riesz pyramid, a multi-scale steerable Hilbert transformer which has been used for 2-D phase representation and video amplification, as the basis for our methodology. For the general subtle motion analysis step, we transform an image sequence with the Riesz pyramid, extract and lifter the image phase variations as proxies for motion. Furthermore, we isolate regions of intcrcst where subtle motion might take place and mask noisy areas by thresholding the local amplitude. The total sequence is transformed into a ID signal which is used fo temporal analysis and subtle motion spotting. We create our own database of subtle motion sequences to test our method. For the micro-expression spotting step, we adapt the previous method to process some facial regions of interest. We also develop a heuristic method to detect facial micro-events that separates real micro-expressions from eye blinkings and subtle eye movements. For the micro-expression classification step, we exploit the dominant orientation constancy fom the Riesz transform to average the micro-expression sequence into an image pair. Based on that, we introduce the Mean Oriented Riesz Feature descriptor. The accuracy of our methods are tested in Iwo spontaneous micro-expressions databases. Furthermore, wc analyse the parameter variations and their effect in our results.
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Generalizações do teorema de representação de Riesz / Generalizations of the Riesz Representation TheoremBatista, Cesar Adriano 19 June 2009 (has links)
Dados um espaço de medida (X;A;m) e números reais p,q>1 com 1/p+1/q=1, o Teorema de Representação de Riesz afirma que Lq(X;A;m) é o dual topológico de Lp(X;A;m) e que Loo(X;A; m) é o dual topológico de L1(X;A;m) se o espaço (X;A;m) for sigma-finito. Observamos que a sigma-finitude de (X;A;m) é condição suficiente mas não necessária para que Loo(X;A;m) seja o dual de L1(X;A;m). Os contra-exemplos tipicamente apresentados para essa última identificação são \"triviais\", no sentido de que desaparecem se \"consertarmos\" a medida , transformando-a numa medida perfeita. Neste trabalho apresentamos condições sufcientes mais fracas que sigma-finitude a fim de que Loo(X;A;m) e o dual de L1(X;A;m) possam ser isometricamente identificados. Além disso, introduzimos um invariante cardinal para espaços de medida que chamaremos a dimensão do espaço e mostramos que se o espaço (X;A;m) for de medida perfeita e tiver dimensão menor ou igual à cardinalidade do continuum então uma condição necessária e suficiente para Loo(X;A;m) seja o dual de L1(X;A;m) é que X admita uma decomposição. / Given a measure space (X;A;m) and real numbers p,q>1 with 1/p+1/q=1, the Riesz Representation Theorem states that Lq(X;A;m) is the topological dual space of Lp(X;A;m) and that Loo(X;A; m) is the topological dual space of L1(X;A;m) if (X;A; m) is sigma-finite. We observe that the sigma-finiteness of (X;A;m) is a suficient but not necessary condition for Loo(X;A;m) to be the dual of L1(X;A;m). The counter-examples that are typically presented for Loo(X;A;m) = L1(X;A;m)* are \"trivial\", in the sense that they vanish if we fix the measure , making it into a perfect measure. In this work we present suficient conditions weaker than sigma-finiteness in order that Loo(X;A; m) and L1(X;A;m)* can be isometrically identified. Moreover, we introduce a cardinal invariant for measure spaces which we call the dimension of the space and we show that if the space (X;A;m) has perfect measure and dimension less than or equal to the cardinal of the continuum then a necessary and suficient condition for Loo(X;A;m) = L1(X;A;m)* is that X admits a decomposition.
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Quasi transformées de Riesz, espaces de Hardy et estimations sous-gaussiennes du noyau de la chaleur / Quasi Riesz transforms, Hardy spaces and generalized sub-Gaussian heat kernel estimatesChen, Li 24 April 2014 (has links)
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<p<2. Dans le Chapitre 2, on montre que les quasi transformées de Riesz est aussi de type faible (1,1) si la variété satisfait la propriété de doublement du volume et l'estimation sous-gaussienne du noyau de la chaleur. On obtient des résultats analogues sur les graphes dans le Chapitre 4. Dans le Chapitre 3, on développe la théorie des espaces de Hardy sur les espaces métriques mesurés avec des estimations différentes localement et globalement du noyau de la chaleur. On définit les espaces de Hardy par les molécules et par les fonctions quadratiques. On montre tout d'abord que ces deux espaces H1 sont les mêmes. Puis, on compare l'espace Hp défini par par les fonctions quadratiques et Lp. On montre qu'ils sont équivalents. Mais on trouve des exemples tels que l'équivalence entre Lp et Hp défini par les fonctions quadratiques avec l'homogénéité t2 n'est pas vraie. Finalement, comme application, on montre que les quasi transformées de Riesz sont bornées de H1 dans L1 sur les variétés fractales. Dans le Chapitre 5, on prouve des inégalités généralisées de Poincaré et de Sobolev sur les graphes de Vicsek. On montre aussi qu'elles sont optimales. / In this thesis, we mainly study Riesz transforms and Hardy spaces associated to operators. The two subjects are closely related to volume growth and heat kernel estimates. In Chapter 1, 2 and 4, we study Riesz transforms on Riemannian manifold and on graphs. In Chapter 1, we prove that on a complete Riemannian manifold, the quasi Riesz transform is always Lp bounded on for p strictly large than 1 and no less than 2. In Chapter 2, we prove that the quasi Riesz transform is also weak L1 bounded if the manifold satisfies the doubling volume property and the sub-Gaussian heat kernel estimate. Similarly, we show in Chapter 4 the same results on graphs. In Chapter 3, we develop a Hardy space theory on metric measure spaces satisfying the doubling volume property and different local and global heat kernel estimates. Firstly we define Hardy spaces via molecules and via square functions which are adapted to the heat kernel estimates. Then we show that the two H1 spaces via molecules and via square functions are the same. Also, we compare the Hp space defined via square functions with Lp. The corresponding Hp space for p large than 1 defined via square functions is equivalent to the Lebesgue space Lp. However, it is shown that in this situation, the Hp space corresponding to Gaussian estimates does not coincide with Lp any more. Finally, as an application of this Hardy space theory, we proved that quasi Riesz transforms are bounded from H1 to L1 on fractal manifolds. In Chapter 5, we consider Vicsek graphs. We prove generalised Poincaré inequalities and Sobolev inequalities on Vicsek graphs and we show that they are optimal.
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Scharfe Ungleichungen für Normen von Kommutatoren endlicher MatrizenWenzel, David 30 July 2011 (has links) (PDF)
In der Dissertation werden Schranken für Abschätzungen des Kommutators in verschiedenen Normen gegeben. Den Ausgangspunkt bildet die Frobenius-Norm, für die eine überraschend kleine Schranke bewiesen werden kann. Auf diesem Resultat aufbauend lassen sich über eine spezielle Adaption der Interpolationsmethode von Riesz-Thorin scharfe Schranken bei Verwendung von Schatten- und Vektornormen weitestgehend bestimmen. Es werden ferner die Fälle untersucht, in denen die obere Abschätzung erreicht wird (sog. Maximalität). Eine wichtige Rolle spielen verschiedene Darstellungen der Ungleichung, welche vielfältige Interpretationsmöglichkeiten eröffen und Verbindungen der algebraischen Abschätzung zu einem wichtigen Satz der Differentialgeometrie über die Krümmung von Mannigfaltigkeiten aufzeigen.
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Estrutura eletrônica de cristais: generalização mediante o cálculo fracionário / Electronic structure of crystal: generalization through fractional calculusGomes, Arianne Vellasco 17 April 2018 (has links)
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Previous issue date: 2018-04-17 / 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.
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Generalizações do teorema de representação de Riesz / Generalizations of the Riesz Representation TheoremCesar Adriano Batista 19 June 2009 (has links)
Dados um espaço de medida (X;A;m) e números reais p,q>1 com 1/p+1/q=1, o Teorema de Representação de Riesz afirma que Lq(X;A;m) é o dual topológico de Lp(X;A;m) e que Loo(X;A; m) é o dual topológico de L1(X;A;m) se o espaço (X;A;m) for sigma-finito. Observamos que a sigma-finitude de (X;A;m) é condição suficiente mas não necessária para que Loo(X;A;m) seja o dual de L1(X;A;m). Os contra-exemplos tipicamente apresentados para essa última identificação são \"triviais\", no sentido de que desaparecem se \"consertarmos\" a medida , transformando-a numa medida perfeita. Neste trabalho apresentamos condições sufcientes mais fracas que sigma-finitude a fim de que Loo(X;A;m) e o dual de L1(X;A;m) possam ser isometricamente identificados. Além disso, introduzimos um invariante cardinal para espaços de medida que chamaremos a dimensão do espaço e mostramos que se o espaço (X;A;m) for de medida perfeita e tiver dimensão menor ou igual à cardinalidade do continuum então uma condição necessária e suficiente para Loo(X;A;m) seja o dual de L1(X;A;m) é que X admita uma decomposição. / Given a measure space (X;A;m) and real numbers p,q>1 with 1/p+1/q=1, the Riesz Representation Theorem states that Lq(X;A;m) is the topological dual space of Lp(X;A;m) and that Loo(X;A; m) is the topological dual space of L1(X;A;m) if (X;A; m) is sigma-finite. We observe that the sigma-finiteness of (X;A;m) is a suficient but not necessary condition for Loo(X;A;m) to be the dual of L1(X;A;m). The counter-examples that are typically presented for Loo(X;A;m) = L1(X;A;m)* are \"trivial\", in the sense that they vanish if we fix the measure , making it into a perfect measure. In this work we present suficient conditions weaker than sigma-finiteness in order that Loo(X;A; m) and L1(X;A;m)* can be isometrically identified. Moreover, we introduce a cardinal invariant for measure spaces which we call the dimension of the space and we show that if the space (X;A;m) has perfect measure and dimension less than or equal to the cardinal of the continuum then a necessary and suficient condition for Loo(X;A;m) = L1(X;A;m)* is that X admits a decomposition.
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Scharfe Ungleichungen für Normen von Kommutatoren endlicher MatrizenWenzel, David 21 March 2011 (has links)
In der Dissertation werden Schranken für Abschätzungen des Kommutators in verschiedenen Normen gegeben. Den Ausgangspunkt bildet die Frobenius-Norm, für die eine überraschend kleine Schranke bewiesen werden kann. Auf diesem Resultat aufbauend lassen sich über eine spezielle Adaption der Interpolationsmethode von Riesz-Thorin scharfe Schranken bei Verwendung von Schatten- und Vektornormen weitestgehend bestimmen. Es werden ferner die Fälle untersucht, in denen die obere Abschätzung erreicht wird (sog. Maximalität). Eine wichtige Rolle spielen verschiedene Darstellungen der Ungleichung, welche vielfältige Interpretationsmöglichkeiten eröffen und Verbindungen der algebraischen Abschätzung zu einem wichtigen Satz der Differentialgeometrie über die Krümmung von Mannigfaltigkeiten aufzeigen.
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Inequalities associated to Riesz potentials and non-doubling measures with applicationsBhandari, Mukta Bahadur January 1900 (has links)
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.
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