Lokal parametrische Modelle werden häufig im Kontext der nichtparametrischen Schätzung verwendet. Bei einer punktweisen Schätzung der Zielfunktion können die parametrischen Umgebungen mithilfe von Gewichten beschrieben werden, die entweder von den Designpunkten oder (zusätzlich) von den Beobachtungen abhängen. Der Vergleich von verrauschten Beobachtungen in einzelnen Punkten leidet allerdings unter einem Mangel an Robustheit. Der Propagations-Separations-Ansatz von Polzehl und Spokoiny [2006] verwendet daher einen Multiskalen-Ansatz mit iterativ aktualisierten Gewichten. Wir präsentieren hier eine theoretische Studie und numerische Resultate, die ein besseres Verständnis des Verfahrens ermöglichen. Zu diesem Zweck definieren und untersuchen wir eine neue Strategie für die Wahl des entscheidenden Parameters des Verfahrens, der Adaptationsbandweite. Insbesondere untersuchen wir ihre Variabilität in Abhängigkeit von der unbekannten Zielfunktion. Unsere Resultate rechtfertigen eine Wahl, die unabhängig von den jeweils vorliegenden Beobachtungen ist. Die neue Parameterwahl liefert für stückweise konstante und stückweise beschränkte Funktionen theoretische Beweise der Haupteigenschaften des Algorithmus. Für den Fall eines falsch spezifizierten Modells führen wir eine spezielle Stufenfunktion ein und weisen eine punktweise Fehlerschranke im Vergleich zum Schätzer des Algorithmus nach. Des Weiteren entwickeln wir eine neue Methode zur Entrauschung von diffusionsgewichteten Magnetresonanzdaten. Unser neues Verfahren (ms)POAS basiert auf einer speziellen Beschreibung der Daten, die eine zeitgleiche Glättung bezüglich der gemessenen Positionen und der Richtungen der verwendeten Diffusionsgradienten ermöglicht. Für den kombinierten Messraum schlagen wir zwei Distanzfunktionen vor, deren Eignung wir mithilfe eines differentialgeometrischen Ansatzes nachweisen. Schließlich demonstrieren wir das große Potential von (ms)POAS auf simulierten und experimentellen Daten. / In statistics, nonparametric estimation is often based on local parametric modeling. For pointwise estimation of the target function, the parametric neighborhoods can be described by weights that depend on design points or on observations. As it turned out, the comparison of noisy observations at single points suffers from a lack of robustness. The Propagation-Separation Approach by Polzehl and Spokoiny [2006] overcomes this problem by using a multiscale approach with iteratively updated weights. The method has been successfully applied to a large variety of statistical problems. Here, we present a theoretical study and numerical results, which provide a better understanding of this versatile procedure. For this purpose, we introduce and analyse a novel strategy for the choice of the crucial parameter of the algorithm, namely the adaptation bandwidth. In particular, we study its variability with respect to the unknown target function. This justifies a choice independent of the data at hand. For piecewise constant and piecewise bounded functions, this choice enables theoretical proofs of the main heuristic properties of the algorithm. Additionally, we consider the case of a misspecified model. Here, we introduce a specific step function, and we establish a pointwise error bound between this function and the corresponding estimates of the Propagation-Separation Approach. Finally, we develop a method for the denoising of diffusion-weighted magnetic resonance data, which is based on the Propagation-Separation Approach. Our new procedure, called (ms)POAS, relies on a specific description of the data, which enables simultaneous smoothing in the measured positions and with respect to the directions of the applied diffusion-weighting magnetic field gradients. We define and justify two distance functions on the combined measurement space, where we follow a differential geometric approach. We demonstrate the capability of (ms)POAS on simulated and experimental data.

The purpose of this thesis in to introduce the systematic study of symmetries in binary differential equations (BDEs). We formalize the concept of a symmetric BDE, under the linear action of a compact Lie group. One of the main results establishes a formula that relates the algebraic and geometric effects of the occurrence of the symmetry in the problem. Using tools from invariant theory and representation theory for compact Lie groups we deduce the general forms of equivariant binary differential equations under compact subgroups of O(2). A study about the behavior of the invariant straight lines on the configuration of homogeneous BDEs of degree n is done with emphasis on cases in which n = 0 and n = 1. Also for the linear case (n = 1) the equivariant normal forms are presented. Symmetries of linear 1-forms are also studied and related with symmetries of tangent orthogonal vectors fields associated with it. / O objetivo desta tese é introduzir o estudo sistemático de simetrias em equações diferenciais binárias (EDBs). Neste trabalho formalizamos o conceito de EDB simétrica sobre a ação de um grupo de Lie compacto. Um dos principais resultados é uma fórmula que relaciona o efeito geométrico e algébrico das simetrias presentes no problema. Utilizando ferramentas da teoria invariante e de representação para grupos compactos deduzimos as formas gerais para EDBs equivariantes. Um estudo sobre o comportamento das retas invariantes na configuração de EDBs com coeficientes homogêneos de grau n é feito com ênfase nos casos de grau 0 e 1, ainda no caso de grau 1 são apresentadas suas formas normais. Simetrias de 1-formas lineares são também estudadas e relacionadas com as simetrias dos seus campos tangente e ortogonal.

Na análise global e local de sistemas dinâmicos assumimos, em geral, que as equações estão numa forma normal. Em presença de simetrias, as equações e o domínio do problema são invariantes pelo grupo formado por estas simetrias; neste caso, o campo de vetores é equivariante pela ação deste grupo. Quando, além das simetrias, temos também ocorrência de anti-simetrias - ou reversibilidades - as equações e o domínio do problema são ainda invariantes pelo grupo formado pelo conjunto de todas as simetrias e anti-simetrias; neste caso, o campo de vetores é reversível-equivariante. Existem muitos modelos físicos onde simetrias e anti-simetrias aparecem naturalmente e cujo efeito pode ser estudado de uma forma sistemática através de teoria de representação de grupos de Lie. O primeiro passo deste processo é colocar a aplicação que modela tal sistema numa forma normal e isto é feito com a dedução a priori da forma geral dos campos de vetores. Esta forma geral depende de dois componentes: da base de Hilbert do anel das funções invariantes e dos geradores do módulo das aplicações reversíveis-equivariantes. Neste projeto, nos concentramos principalmente na aplicação de resultados recentes da literatura para a construção de uma lista de formas gerais de aplicações reversíveisequivariantes sob a ação de diferentes grupos. Além disso, adaptamos ferramentas algébricas da literatura existentes no contexto equivariante para o estudo sistemático de acoplamento de células idênticas no contexto reversível-equivariante / In the global and local analysis of dynamical systems, we assume, in general, that the equations are in a normal form. In presence of symmetries, the equations and the problem domain are invariant under the group formed by these symmetries; in that case, the vector field is equivariant by the action of this group. When, in addition to the symmetries, we have the occurrence of anti-symmetries - or reversibility - the equations and the problem domain are still invariant by the group formed by the set of all symmetries and anti-symmetries; in this case, the vector field is reversible-equivariant. There are many physical models where both symmetries and anti-symmetries occur naturally and whose effect can be studied in a systematic way through group representation theory. The first step of this process is to put the mapping that model the system in a normal form, and this is done with the deduction of the general form of the vector field. This general form depends on two components: the Hilbert basis of the invariant function ring and also the generators of the module of the revesible-equivariants. In this work, we mainly focus on the applications of recent results of the literature to build a list of general forms of reversible-equivariant mappings under the action of different groups. We also adapt algebraic tools of the existing literature in the equivariant context to the systematic study of coupling of identical cells in the reversible-equivariant context

The purpose of this thesis in to introduce the systematic study of symmetries in binary differential equations (BDEs). We formalize the concept of a symmetric BDE, under the linear action of a compact Lie group. One of the main results establishes a formula that relates the algebraic and geometric effects of the occurrence of the symmetry in the problem. Using tools from invariant theory and representation theory for compact Lie groups we deduce the general forms of equivariant binary differential equations under compact subgroups of O(2). A study about the behavior of the invariant straight lines on the configuration of homogeneous BDEs of degree n is done with emphasis on cases in which n = 0 and n = 1. Also for the linear case (n = 1) the equivariant normal forms are presented. Symmetries of linear 1-forms are also studied and related with symmetries of tangent orthogonal vectors fields associated with it. / O objetivo desta tese é introduzir o estudo sistemático de simetrias em equações diferenciais binárias (EDBs). Neste trabalho formalizamos o conceito de EDB simétrica sobre a ação de um grupo de Lie compacto. Um dos principais resultados é uma fórmula que relaciona o efeito geométrico e algébrico das simetrias presentes no problema. Utilizando ferramentas da teoria invariante e de representação para grupos compactos deduzimos as formas gerais para EDBs equivariantes. Um estudo sobre o comportamento das retas invariantes na configuração de EDBs com coeficientes homogêneos de grau n é feito com ênfase nos casos de grau 0 e 1, ainda no caso de grau 1 são apresentadas suas formas normais. Simetrias de 1-formas lineares são também estudadas e relacionadas com as simetrias dos seus campos tangente e ortogonal.

In recent years, there has been increasing interest in studying time-delayed coupled networks of oscillators since these occur in many real life applications. In many cases symmetry, patterns can emerge in these networks; as a consequence, a part of the system might repeat itself, and properties of this symmetric subsystem represent the whole dynamics. In this thesis, an analysis of a second order N-node time-delay fully connected network is made. This study is carried out using symmetry groups. The existence of multiple eigenvalues forced by symmetry is shown, as well as the possibility of uncoupling the linearization at equilibria, into irreducible representations due to the symmetry. The existence of steady-state and Hopf bifurcations in each irreducible representation is also proved. Three different models are used to analyze the network dynamics, namely, the full-phase, the phase, and the phase-difference model. A finite set of frequencies ω is also determined, which might correspond to Hopf bifurcations in each case for critical values of the delay. Although we restrict our attention to second order nodes, the results could be extended to higher order networks provided the time-delay in the connections between nodes remains equal. / Nos últimos anos, tem havido um crescente interesse em estudar redes de osciladores acopladas com retardo de tempo uma vez que estes ocorrem em muitas aplicações da vida real. Em muitos casos, simetria e padrões podem surgir nessas redes; em consequência, uma parte do sistema pode repetir-se, e as propriedades deste subsistema simétrico representam a dinâmica da rede toda. Nesta tese é feita uma análise de uma rede de N nós de segunda ordem totalmente conectada com atraso de tempo. Este estudo é realizado utilizando grupos de simetria. É mostrada a existência de múltiplos valores próprios forçados por simetria, bem como a possibilidade de desacoplamento da linearização no equilíbrio, em representações irredutíveis. É também provada a existência de bifurcações de estado estacionário e Hopf em cada representação irredutível. São usados três modelos diferentes para analisar a dinâmica da rede: o modelo de fase completa, o modelo de fase, e o modelo de diferença de fase. É também determinado um conjunto finito de frequências ω, que pode corresponder a bifurcações de Hopf em cada caso, para valores críticos do atraso. Apesar de restringir a nossa atenção para nós de segunda ordem, os resultados podem ser estendido para redes de ordem superior, desde que o tempo de atraso nas conexões entre nós permanece igual.

Molecular Dynamics (MD) is the numerical simulation of a large system of interacting molecules, and one of the key components of a MD simulation is the numerical estimation of the solutions to a system of nonlinear differential equations. Such systems are very sensitive to discretization and round-off error, and correspondingly, standard techniques such as Runge-Kutta methods can lead to poor results. However, MD systems are conservative, which means that we can use Hamiltonian mechanics and symplectic transformations (also known as canonical transformations) in analyzing and approximating solutions. This is standard in MD applications, leading to numerical techniques known as symplectic integrators, and often, these techniques are developed for well-understood Hamiltonian systems such as Hill’s lunar equation. In this presentation, we explore how well symplectic techniques developed for well-understood systems (specifically, Hill’s Lunar equation) address discretization errors in MD systems which fail for one or more reasons.

Chaque année lors de la période hivernale, des phénomènes épidémiques affectent l’organisation des services d’urgences pédiatriques et dégradent la qualité de la réponse fournie. Ces phénomènes présentent une forte variabilité qui rend leur analyse difficile. Nous nous proposons d’étudier cette volatilité pour apporter une vision nouvelle et éclairante sur le comportement de ces épidémies. Pour ce faire, nous avons adopté une vision géométrique et topologique originale directement issue d’une application de la théorie de la dilation: le processus de variabilité étant périodiquement corrélé, cette théorie fournit un ensemble de matrices dites de dilations qui portent toute l’information utile sur ce processus. Cet ensemble de matrices nous permet de représenter les processus stochastiques comme des éléments d’un groupe de Lie particulier, à savoir le groupe de Lie constitué de l’ensemble des courbes sur une variété. Il est alors possible de comparer des processus par ce biais. Pour avoir une perception plus intuitive du processus de variabilité, nous nous sommes ensuite concentrés sur le nuage de points formé par l’ensemble des matrices de dilations. En effet, nous souhaitons mettre en évidence une relation entre la forme temporelle d’un processus et l’organisation de ces matrices de dilations. Nous avons utilisé et développé des outils d’homologie persistante et avons établi un lien entre la désorganisation de ce nuage de points et le type de processus sous-jacents. Enfin nous avons appliqué ces méthodes directement sur le processus de variabilité pour pouvoir détecter le déclenchement de l’épidémie. Ainsi nous avons établi un cadre complet et cohérent, à la fois théorique et appliqué pour répondre à notre problématique. / Each year emergency department are faced with epidemics that affect their organisation and deteriorate the quality of the cares. The analyse of these outbreak is tough due to their huge variability. We aim to study these phenomenon and to bring out a new paradigm in the analysis of their behavior. With this aim in mind, we propose to tackle this problem through geometry and topology: the variability process being periodically correlated, the theory of dilation exhibit a set of matrices that carry all the information about this process. This set of matrices allow to map the process into a Lie group, defined as the set of all curves on a manifold. Thus, it is possible to compare stochastic processes using properties of Lie groups. Then, we consider the point cloud formed by the set of dilation matrices, to gain more intuitions about the underlying process. We proved a relation between the temporal aspect of the signal and the structure of the set of its dilation matrices. We used and developped persistent homology tools, and were able to classify non-stationary processes. Eventually, we implement these techniques directly on the process of arrivals to detect the trigger of the epidemics. Overall we established a complete and a coherent framework, both theoretical and practical.

Conservation laws play an important role in science. The aim of this thesis is to provide an overview and develop new methods for constructing conservation laws using Lie group theory. The derivation of conservation laws for invariant variational problems is based on Noether’s theorem. It is shown that the use of Lie-Bäcklund transformation groups allows one to reduce the number of basic conserved quantities for differential equations obtained by Noether’s theorem and construct a basis of conservation laws. Several examples on constructing a basis for some well-known equations are provided. Moreover, this approach allows one to obtain new conservation laws even for equations without Lagrangians. A formal Lagrangian can be introduced and used for computing nonlocal conservation laws. For self-adjoint or quasi-self-adjoint equations nonlocal conservation laws can be transformed into local conservation laws. One of the fields of applications of this approach is electromagnetic theory, namely, nonlocal conservation laws are obtained for the generalized Maxwell-Dirac equations. The theory is also applied to the nonlinear magma equation and its nonlocal conservation laws are computed.

Conservation laws play an important role in science. The aim of this thesis is to provide an overview and develop new methods for constructing conservation laws using Lie group theory. The derivation of conservation laws for invariant variational problems is based on Noether’s theorem. It is shown that the use of Lie-Bäcklund transformation groups allows one to reduce the number of basic conserved quantities for differential equations obtained by Noether’s theorem and construct a basis of conservation laws. Several examples on constructing a basis for some well-known equations are provided. Moreover, this approach allows one to obtain new conservation laws even for equations without Lagrangians. A formal Lagrangian can be introduced and used for computing nonlocal conservation laws. For self-adjoint or quasi-self-adjoint equations nonlocal conservation laws can be transformed into local conservation laws. One of the fields of applications of this approach is electromagnetic theory, namely, nonlocal conservation laws are obtained for the generalized Maxwell-Dirac equations. The theory is also applied to the nonlinear magma equation and its nonlocal conservation laws are computed. / <p>Thesis for the degree of Doctor of Philosophy</p>

This dissertation has as base Jocelino Sato and Vicente de Souza Neto's paper called Complete and Stable O(p + 1) x O(q + 1)-Invariant Hypersurfaces with Zero Scalar Curvature in Euclidean Space Rp+q+2, published on the Annals of Global Analysis and Geometry - 29 in 2006. The main result of this dissertation is the Classi_cation Theorem, which states: The O(p+1) x O(q+1)-Invariant Hypersurfaces in Rp+q+2, p; q > 1, with zero scalar curvature belong to one of the following classes: (1) Cones with a singularity at the orign of Rp+q+2; (2) Hypersurfaces having one orbit of singularity and asymptoting both of the cones C&#945; and C&#946;; (3) Regular hypersurfaces asymptoting the cone C&#945;; (4) Regular hypersurfaces asymptoting the cone C&#946;; (5) Regular hypersurfaces asymptoting both of the cones C&#945; and C&#946;. It was reached by the studies of the ordinary differential equation on R2, involving the coordenate curves that generate these hypersurfaces. Such differential equation, in its turn, is associated with a vector field X : R22 &#8594; R2 on the plan. The study of the orbits space in this field is essential; after all, because of it, it was possible to translate the X orbits' behavior into information concerning the profile curves and, finally, reach the theorem. / Fundação de Amparo a Pesquisa do Estado de Alagoas / Esta dissertação está baseada no artigo de Jocelino Sato e Vicente de Souza Neto intitulado Complete and Stable O(p+1) x O(q+1) - Invariant Hypersurfaces with Zero Scalar Curvature in Euclidean Space Rp+q+2, publicado na revista Annals of Global Analysis and Geometry, volume 29, em 2006. O principal resultado desta dissertação é o Teorema de Classicação, que afirma o seguinte: Uma hipersuperfície Mp+q+1 que é invariante pela açãoao do grupo O(p + 1) x O(q + 1), p; q > 1, com curvatura escalar identicamente nula deve pertencer a uma das seguintes classes: (1) Cones com uma singularidade na origem de Rp+q+2; (2) Hipersuperfícies possuindo uma órbita de singularidades e assintotando ambos os cones C&#945; e C&#946;; (3) Hipersuperfícies regulares que assintotam o cone C&#945;; (4) Hipersuperfícies regulares que assintotam o cone C&#946;; (5) Hipersuperfícies regulares que assintotam ambos os cones C&#945; e C&#946;. A demonstração do teorema requer um estudo de uma equação diferencial ordinária envolvendo as coordenadas das curvas, no plano, que geram estas hipersuperfícies. Esta equação diferencial, por sua vez, está associada a um campo de vetores X : R2 &#8594; R2 no plano. O estudo do retrato de fase deste campo é fundamental. Através dele, foi possível traduzir o comportamento das trajetórias de X em informações com respeito às curvas geratrizes e desta maneira obter o teorema.