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.

In dieser Arbeit werden die zu einem m-sektoriellen Operator assoziierten Halbgruppen betrachtet, die die Lösungen des Anfangswertproblems der zugehörigen Evolutionsgleichung beschreiben. Es wird eine 1987 von Davies veröffentlichte Methode zur Abschätzung dieser Halbgruppen verallgemeinert. Einen Schwerpunkt bilden die zu Dirichlet-Formen assoziierten Markov-Halbgruppen. Für diese werden die Resultate spezialisiert und der Zusammenhang zur intrinsischen Metrik dargelegt. Die Arbeit schließt mit verschiedenen Beispielen, die zeigen, wie mit diesen Verallgemeinerungen von Davies Methode neue Anwendungsgebiete erschlossen werden können. / This thesis is about m-sectorial operators and their associated semigroups describing the solutions of the initial value problem of the corresponding evolution equation. We generalize a method published by Davies 1987 to estimate these semigroups. A focus is set on Markov semigroups associated with Dirchlet forms. The results are applied to them and the connection to the intrinsic metric is presented. The thesis ends with different examples showing how this generalization of Davies method can be applied into new fields of application.

A growing number of 3D graphic applications have an impact on today's society. These applications are being used in several domains ranging from digital entertainment, computer aided design, to medical applications. In this context, a 3D object search engine with a good performance in time consuming and results becomes mandatory. We propose a novel approach for 3D-model retrieval based on closed curves. Then we enhance our method to handle partial 3D-model retrieval. Our method starts by the definition of an invariant mapping function. The important properties of a mapping function are its invariance to rigid and non rigid transformations, the correct description of the 3D-model, its insensitivity to noise, its robustness to topology changes, and its independance on parameters. However, current state-of-the-art methods do not respect all these properties. To respect these properties, we define our mapping function based on the diffusion and the commute-time distances. To prove the properties of this function, we compute the Reeb graph of the 3D-models. To describe the whole 3D-model, using our mapping function, we generate indexed closed curves from a source point detected automatically at the center of a 3D-model. Each curve describes a small region of the 3D-model. These curves lead to create an invariant descriptor to different transformations. To show the robustness of our method on various classes of 3D-models with different poses, we use shapes from SHREC 2012. We also compare our approach to existing methods in the state-of-the-art with a dataset from SHREC 2010. For partial 3D-model retrieval, we enhance the proposed method using the Bag-Of-Features built with all the extracted closed curves, and show the accurate performances using the same dataset

Proteins and the formation of large protein complexes are essential parts of living organisms. Proteins are present in all aspects of life processes, performing a multitude of various functions ranging from being structural components of cells, to facilitating the passage of certain molecules between various regions of cells. The 'protein docking problem' refers to the computational method of predicting the appropriate matching pair of a protein (receptor) with respect to another protein (ligand), when attempting to bind to one another to form a stable complex. Research shows that matching the three-dimensional (3D) geometric structures of candidate proteins plays a key role in determining a so-called docking pair, which is one of the key aspects of the Computer Aided Drug Design process. However, the active sites which are responsible for binding do not always present a rigid-body shape matching problem. Rather, they may undergo sufficient deformation when docking occurs, which complicates the problem of finding a match. To address this issue, we present an isometry-invariant and topologically robust partial shape matching method for finding complementary protein binding sites, which we call the ProtoDock algorithm. The ProtoDock algorithm comes in two variations. The first version performs a partial shape complementarity matching by initially segmenting the underlying protein object mesh into smaller portions using a spectral mesh segmentation approach. The Heat Kernel Signature (HKS), the underlying basis of our shape descriptor, is subsequently computed for the obtained segments. A final descriptor vector is constructed from the Heat Kernel Signatures and used as the basis for the segment matching. The three different descriptor methods employed are, the accepted Bag of Features (BoF) technique, and our two novel approaches, Closest Medoid Set (CMS) and Medoid Set Average (MSA). The second variation of our ProtoDock algorithm aims to perform the partial matching by utilizing the pointwise HKS descriptors. The use of the pointwise HKS is mainly motivated by the suggestion that, at adequate times, the Heat Kernel Signature of a point on a surface sufficiently describes its neighbourhood. Hence, the HKS of a point may serve as the representative descriptor of its given region of which it forms a part. We propose three (3) sampling methods---Uniform, Random, and Segment-based Random sampling---for selecting these points for the partial matching. Random and Segment-based Random sampling both prove superior to the Uniform sampling method. Our experimental results, run against the Protein-Protein Benchmark 4.0, demonstrate the viability of our approach, in that, it successfully returns known binding segments for known pairing proteins. Furthermore, our ProtoDock-1 algorithm still still yields good results for low resolution protein meshes. This results in even faster processing and matching times with sufficiently reduced computational requirements when obtaining the HKS.

In this dissertation we study non-negative self-adjoint Laplace type operators acting on smooth sections of a vector bundle. First, we assume base manifolds are compact, boundaryless, and Riemannian. We start from the Fourier integral operator representation of half-wave operators, continue with spectral zeta functions, heat and resolvent trace asymptotic expansions, and end with the quantitative Wodzicki residue method. In particular, all of the asymptotic coefficients of the microlocalized spectral counting function can be explicitly given and clearly interpreted. With the auxiliary pseudo-differential operators ranging all smooth endomorphisms of the given bundle, we obtain certain asymptotic estimates about the integral kernel of heat operators. As applications, we study spectral asymptotics of Dirac type operators such as characterizing those for which the second coefficient vanishes. Next, we assume vector bundles are trivial and base manifolds are Euclidean domains, and study non-negative self-adjoint extensions of the Laplace operator which acts component-wise on compactly supported smooth functions. Using finite propagation speed estimates for wave equations and explicit Fourier Tauberian theorems obtained by Yuri Safarov, we establish the principle of not feeling the boundary estimates for the heat kernel of these operators. In particular, the implied constants are independent of self-adjoint extensions. As a by-product, we affirmatively answer a question about upper estimate for the Neumann heat kernel. Finally, we study some specific values of the spectral zeta function of two-dimensional Dirichlet Laplacians such as spectral determinant and Casimir energy. For numerical purposes we substantially improve the short-time Dirichlet heat trace asymptotics for polygons. This could be used to measure the spectral determinant and Casimir energy of polygons whenever the first several hundred or one thousand Dirichlet eigenvalues are known with high precision by other means.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / FundaÃÃo Cearense de Apoio ao Desenvolvimento Cientifico e TecnolÃgico / No presente trabalho, provaremos resultados de comparaÃÃo para o nÃcleo do calor de subvariedades mÃnimas de variedades Riemannianas com curvatura seccional limitada superiormente pela curvatura de uma variedade modelo. Em seguida, iremos obter resultados sobre a propriedade L1-Liouville de submersÃes Riemannianas com fibras mÃnimas. Por Ãltimo, provaremos desigualdades para o tom fundamental ponderado de subconjuntos transversalmente folheados de variedades Riemannianas ponderadas em termos das curvaturas mÃdias ponderadas das folhas da folheaÃÃo. / In this work we will prove comparison results for the heat kernel of minimal submanifolds in Riemannian manifolds with sectional curvature bounded above by the curvature of a model manifold. Next we will obtain results about the L1-Liouville property of Riemannian submersions with minimal fibers. Finnaly, we will prove inequalities for the weighted fundamental tone of transversally foliated subsets of weighted Riemannian manifolds in terms of the weighted mean curvatures of the leaves of the foliation.

Nous considérons la fonction zêta d’Epstein des réseaux euclidiens pour étudier le problème des minima de la hauteur du tore plat associé à un réseau. La hauteur est définie comme la dérivée au point s = 0 de la fonction zêta spectrale du tore, fonction qui coïncide, à un facteur près, avec la fonction zêta d’Epstein du réseau dual du réseau donné. Nous donnons dans cette dissertation une condition suffisante pour qu’un réseau donné soit un point critique de la hauteur. En particulier, en utilisant la théorie des designs sphériques, nous montrons qu’un réseau qui a des 2-designs sphériques sur toutes ses couches est un point critique de la hauteur. Nous donnons un algorithme pour tester si un réseau donné satisfait cette condition de 2-designs, et nous donnons des tables de résultats en dimension jusqu’à 7. Ensuite, nous montrons qu’un réseau qui réalise un minimum local de la hauteur est nécessairement irréductible. Enfin, nous nous intéressons à certains tores définis sur les corps de nombres quadratiques imaginaires, et nous prouvons une formule qui donne leur hauteur comme limite d’une suite de hauteurs de tores complexes discrets. / In this thesis we consider the Epstein zeta function of Euclidean lattices, in order to study the problem of the minima of the height of the flat torus associated to a lattice. The height is defined as the first derivative at the point s = 0 of the spectral zeta function of the torus ; this function coincides, up to a factor, with the Epstein zeta function of the dual lattice of the given lattice. We describe a sufficient condition for a given lattice to be a stationary point of the height. In particular, by means of the theory of spherical designs, we show that a lattice which has a spherical 2-design on every shell is a stationary point of the height. We give an algorithm to check whether a given lattice satisfies this 2-design condition or not, and we give some tables of results in dimension up to 7. Then, we show that a lattice which realises a local minimum of the height is necessarily irreducible. Finally, we deal with some tori defined over the imaginary quadratic number fields, and we show a formula which gives their height as a limit of a sequence of heights of discrete complex tori.

Any Riemannian manifold possesses a minimal solution of the heat equation for the Dirichlet Laplacian, called the heat kernel. During the last decades many authors investigated geometric properties of the manifold such that its heat kernel fulfills a so-called Gaussian upper bound. Especially compact and non-compact manifolds with lower bounded Ricci curvature have been examined and provide such Gaussian estimates. In the compact case it ended even with integral Ricci curvature assumptions. The important techniques to obtain Gaussian bounds are the symmetrization procedure for compact manifolds and relative Faber-Krahn estimates or gradient estimates for the heat equation, where the first two base on isoperimetric properties of certain sets. In this thesis, we generalize the existing results to the following. Locally uniform integral bounds on the negative part of Ricci curvature lead to Gaussian upper bounds for the heat kernel, no matter whether the manifold is compact or not. Therefore, we show local isoperimetric inequalities under this condition and use relative Faber-Krahn estimates to derive explicit Gaussian upper bounds. If the manifold is compact, we can even generalize the integral curvature condition to the case that the negative part of Ricci curvature is in the so-called Kato class. We even obtain uniform Gaussian upper bounds using gradient estimate techniques. Apart from the geometric generalizations for obtaining Gaussian upper bounds we use those estimates to generalize Bochner’s theorem. More precisely, the estimates for the heat kernel obtained above lead to ultracontractive estimates for the heat semigroup and the semigroup generated by the Hodge Laplacian. In turn, we can formulate rigidity results for the triviality of the first cohomology group if the amount of curvature going below a certain positive threshold is small in a suitable sense. If we can only assume such smallness of the negative part of the Ricci curvature, we can bound the Betti number by explicit terms depending on the generalized curvature assumptions in a uniform manner, generalizing certain existing results from the cited literature. / Jede Riemannsche Mannigfaltigkeit besitzt eine minimale Lösung für die Wärmeleitungsgleichung des zur Mannigfaltigkeit gehörigen Dirichlet-Laplaceoperators, den Wärmeleitungskern. Während der letzten Jahrzehnte fanden viele Autoren geometrische Eigenschaften der Mannigfaltigkeiten unter welchen der Wärmeleitungskern eine sogenannte Gaußsche obere Abschätzung besitzt. Insbesondere bestizen sowohl kompakte als auch nichtkompakte Mannigfaltigkeiten mit nach unten beschränkter Ricci-Krümmung solche Gaußschen Abschätzungen. Im kompakten Fall reichten bisher sogar Integralbedingungen an die Ricci-Krümmung aus. Die wichtigen Techniken, um Gaußsche Abschätzungen zu erhalten, sind die Symmetrisierung für kompakte Mannigfaltigkeiten und relative Faber-Krahn- und Gradientenabschätzungen für die Wärmeleitungsgleichung, wobei die ersten beiden auf isoperimetrischen Eigenschaften gewisser Mengen beruhen. In dieser Arbeit verallgemeinern wir die bestehenden Resultate im folgenden Sinne. Lokal gleichmäßig beschränkte Integralschranken an den Negativteil der Ricci-Krümmung ergeben Gaußsche obere Abschätzungen sowohl im kompakten als auch nichtkompakten Fall. Dafür zeigen wir lokale isoperimetrische Ungleichungen unter dieser Voraussetzung und nutzen die relativen Faber-Krahn-Abschätzungen für eine explizite Gaußsche Schranke. Für kompakte Mannigfaltigkeiten können wir sogar die Integralschranken an den Negativteil der Ricci-Krümmung durch die sogenannte Kato-Bedingung ersetzen. In diesem Fall erhalten wir gleichmäßige Gaußsche Abschätzungen mit einer Gradientenabschätzung. Neben den geometrischen Verallgemeinerungen für Gaußsche Schranken nutzen wir unsere Ergebnisse, um Bochners Theorem zu verallgemeinern. Wärmeleitungskernabschätzungen ergeben ultrakontraktive Schranken für die Wärmeleitungshalbgruppe und die Halbgruppe, die durch den Hodge-Operator erzeugt wird. Damit können wir Starrheitseigenschaften für die erste Kohomologiegruppe zeigen, wenn der Teil der Ricci-Krümmung, welcher unter einem positiven Level liegt, in einem bestimmten Sinne klein genug ist. Wenn der Negativteil der Ricci-Krümmung nicht zu groß ist, können wir die erste Betti-Zahl noch immer explizit uniform abschätzen.

Proteins and the formation of large protein complexes are essential parts of living organisms. Proteins are present in all aspects of life processes, performing a multitude of various functions ranging from being structural components of cells, to facilitating the passage of certain molecules between various regions of cells. The 'protein docking problem' refers to the computational method of predicting the appropriate matching pair of a protein (receptor) with respect to another protein (ligand), when attempting to bind to one another to form a stable complex. Research shows that matching the three-dimensional (3D) geometric structures of candidate proteins plays a key role in determining a so-called docking pair, which is one of the key aspects of the Computer Aided Drug Design process. However, the active sites which are responsible for binding do not always present a rigid-body shape matching problem. Rather, they may undergo sufficient deformation when docking occurs, which complicates the problem of finding a match. To address this issue, we present an isometry-invariant and topologically robust partial shape matching method for finding complementary protein binding sites, which we call the ProtoDock algorithm. The ProtoDock algorithm comes in two variations. The first version performs a partial shape complementarity matching by initially segmenting the underlying protein object mesh into smaller portions using a spectral mesh segmentation approach. The Heat Kernel Signature (HKS), the underlying basis of our shape descriptor, is subsequently computed for the obtained segments. A final descriptor vector is constructed from the Heat Kernel Signatures and used as the basis for the segment matching. The three different descriptor methods employed are, the accepted Bag of Features (BoF) technique, and our two novel approaches, Closest Medoid Set (CMS) and Medoid Set Average (MSA). The second variation of our ProtoDock algorithm aims to perform the partial matching by utilizing the pointwise HKS descriptors. The use of the pointwise HKS is mainly motivated by the suggestion that, at adequate times, the Heat Kernel Signature of a point on a surface sufficiently describes its neighbourhood. Hence, the HKS of a point may serve as the representative descriptor of its given region of which it forms a part. We propose three (3) sampling methods---Uniform, Random, and Segment-based Random sampling---for selecting these points for the partial matching. Random and Segment-based Random sampling both prove superior to the Uniform sampling method. Our experimental results, run against the Protein-Protein Benchmark 4.0, demonstrate the viability of our approach, in that, it successfully returns known binding segments for known pairing proteins. Furthermore, our ProtoDock-1 algorithm still still yields good results for low resolution protein meshes. This results in even faster processing and matching times with sufficiently reduced computational requirements when obtaining the HKS.

Es sei Γ eine torsionsfreie arithmetische Untergruppe der symplektischen Gruppe Sp(n,R), die auf dem Siegelschen oberen Halbraum H_n vom Grad n wirkt. Wir betrachten den d-dimensionalen Raum der Siegelschen Spitzenformen vom Gewicht k zur Gruppe Γ, mit einer Orthonormalbasis {f_1,…,f_d}. In der vorliegenden Dissertation zeigen wir mit Hilfe des Wärmeleitungskerns, dass die Supremumsnorm von S_k(Z):=det(Y)^k (|f_1(Z)|^2+…+|f_d(Z)|^2) (Z∈H_n) für n=2 ohne zusätzliche Bedingungen und für n>2 unter Annahme einer vermuteten Determinanten-Ungleichung nach oben beschränkt ist. Wenn M:=Γ\H_n kompakt ist, dann ist die obere Schranke durch c_(n,Γ) k^{n(n+1)/2} gegeben. Wenn M nicht kompakt und von endlichem Volumen ist, dann ist die obere Schranke durch c_(n,Γ) k^{3n(n+1)/4} gegeben. In beiden Fällen ist c_(n,Γ) eine positive reelle Konstante, die nur vom Grad n und der Gruppe Γ abhängt. Wir zeigen weiter, dass die obere Schranke in dem Sinne gleichmäßig ist, dass bei fixierter Gruppe Γ_0 die Konstante c_(n,Γ) für Untergruppen Γ von endlichem Index nur vom Grad n und der Gruppe Γ_0 abhängt. / Let Γ be a torsion-free arithmetic subgroup of the symplectic group Sp(n,R) acting on the Siegel upper half-space H_n of degree n. Consider the d-dimensional space of Siegel cusp forms of weight k for Γ with an orthonormal basis {f_1,…,f_d}. In this thesis we show using the heat kernel method that for n=2 unconditionally and for n>2 subject to a conjectural determinant-inequality, the sup-norm of the quantity S_k(Z):=det(Y)^k (|f1(Z)|^2+…+|f_d(Z)|^2) (Z∈H_n) is bounded above by c_(n,Γ) k^{n(n+1)/2} when M:=Γ\H_n is compact and by c_(n,Γ) k^{3n(n+1)/4} when M is non-compact of finite volume, where c_(n,Γ) denotes a positive real constant depending only on the degree n and the group Γ. Furthermore, we show that this bound is uniform in the sense that if we fix a group Γ_0 and take Γ to be a subgroup of Γ_0 of finite index, then the constant c_(n,Γ) in these bounds depends only on the degree n and the fixed group Γ_0.