In this thesis we address the analytic and numerical aspects of isospectral flows. Such flows occur in mathematical physics and numerical linear algebra. Their main structural feature is to retain the eigenvalues in the solution space. We explore the solution of Isospectral flows and their stochastic counterpart using explicit generalisation of Magnus expansion. \par In the first part of the thesis we expand the solution of Bloch--Iserles equations, the matrix ordinary differential system of the form $ X'=[N,X^{2}],\ \ t\geq0, \ \ X(0)=X_0\in \textrm{Sym}(n),\ N\in \mathfrak{so}(n), $ where $\textrm{Sym}(n)$ denotes the space of real $n\times n$ symmetric matrices and $\mathfrak{so}(n)$ denotes the Lie algebra of real $n\times n$ skew-symmetric matrices. This system is endowed with Poisson structure and is integrable. Various important properties of the flow are discussed. The flow is solved using explicit Magnus expansion and the terms of expansion are represented as binary rooted trees deducing an explicit formalism to construct the trees recursively. Unlike classical numerical methods, e.g.\ Runge--Kutta and multistep methods, Magnus expansion respects the isospectrality of the system, and the shorthand of binary rooted trees reduces the computational cost of the exponentially growing terms. The desired structure of the solution (also with large time steps) has been displayed. \par Having seen the promising results in the first part of the thesis, the technique has been extended to the generalised double bracket flow $ X^{'}=[[N,X]+M,X], \ \ t\geq0, \ \ X(0)=X_0\in \textrm{Sym}(n),$ where $N\in \textrm{diag}(n)$ and $M\in \mathfrak{so}(n)$, which is also a form of an Isospectral flow. In the second part of the thesis we define the generalised double bracket flow and discuss its dynamics. It is noted that $N=0$ reduces it to an integrable flow, while for $M=0$ it results in a gradient flow. We analyse the flow for various non-zero values of $N$ and $M$ by assigning different weights and observe Hopf bifurcation in the system. The discretisation is done using Magnus series and the expansion terms have been portrayed using binary rooted trees. Although this matrix system appears more complex and leads to the tri-colour leaves; it has been possible to formulate the explicit recursive rule. The desired structure of the solution is obtained that leaves the eigenvalues invariant in the solution space.

Le réseau de Toeplitz est un système hamiltonien dont la structure de Poisson est connue. Dans cette thèse, nous donnons l'origine de cette structure de Poisson et nous en déduisons des équations de Lax associées au réseau de Toeplitz. Nous construisons tout d'abord une sous-variété de Poisson Hn de GLn(C), ce dernier étant vu comme un groupe de Lie-Poisson réel ou complexe dont la structure de Poisson provient d'un R-crochet quadratique sur gln(C) pour une R-matrice fixée. L'existence d'hamiltoniens associés au réseau de Toeplitz pour la structure de Poisson sur Hn ainsi que les propriétés du R-crochet quadratique permettent alors d'expliciter des équations de Lax du système. On en déduit alors l'intégrabilité au sens de Liouville du réseau de Toeplitz. Dans le point de vue réel, nous pouvons ensuite construire une sous-variété de Poisson Han du groupe Un qui est lui-même une sous-variété de Poisson-Dirac de GLR n(C). Nous construisons alors un hamiltonien, pour la structure de Poisson induite sur Han, correspondant à un autre système déduit du réseau de Toeplitz : le réseau de Schur modifié. Grâce aux propriétés des sous-variétés de Poisson-Dirac, nous explicitons une équation de Lax pour ce nouveau système et nous en déduisons une équation de Lax pour le réseau de Schur. On en déduit également l'intégrabilité au sens de Liouville du réseau de Schur modifié. / The Toeplitz lattice is a Hamiltonian system whose Poisson structure is known. In this thesis, we reveil the origins of this Poisson structure and we derive from it the associated Lax equations for this lattice. We first construct a Poisson subvariety Hn of GLn(C), which we view as a real or complex Poisson-Lie group whose Poisson structure comes from a quadratic R-bracket on gln(C) for a fixed R-matrix. The existence of Hamiltonians, associated to the Toeplitz lattice for the Poisson structure on Hn, combined with the properties of the quadratic R-bracket allow us to give explicit formulas for the Lax equation. Then, we derive from it the integrability in the sense of Liouville of the Toeplitz lattice. When we view the lattice as being defined over R, we can construct a Poisson subvariety Han of Un which is itself a Poisson-Dirac subvariety of GLR n(C). We then construct a Hamiltonian for the Poisson structure induced on Han, corresponding to another system which derives from the Toeplitz lattice : the modified Schur lattice. Thanks to the properties of Poisson-Dirac subvarieties, we give an explicit Lax equation for the new system and derive from it a Lax equation for the Schur lattice. We also deduce the integrability in the sense of Liouville of the modified Schur lattice.

L'objet de cette thèse est l'étude de l'espace de modules des G_2-fibrés principaux sur une courbe complexe projective connexe lisse, où G_2 désigne le groupe de Lie exceptionnel de plus petit rang. Le groupe G_2 est tout d'abord présenté comme le groupe des automorphismes de l'algèbre complexe des octaves de Cayley. D'autres définitions sont ensuite proposées. Les différentes réductions et extensions que peut admettre un G_2-fibré principal sont étudiées ainsi que la relation entre la stabilité d'un G_2-fibré principal et celle de son fibré vectoriel associé. L'espace de modules des G_2-fibrés principaux semistables est analysé. Nous obtenons notamment une caractérisation de son lieu lisse, une décomposition explicite de son lieu singulier en trois composantes connexes et une analyse de l'espace de Verlinde de niveau 1 pour le groupe G_2. / This thesis studies the moduli space of principal G_2-bundles over a smooth connected projective curve, where G_2 is the exceptional Lie group of smallest rank. The group G_2 is first introduced as the group of automorphisms of the complex algebra of the Cayley numbers. Other equivalent definitions are also proposed. We study the reductions and extensions that a principal G_2_bundle can admit, as well as the link between a principal G_2-bundle and its associated vector bundle in relation to the notion of (semi)stability. The moduli space of semistable principal G_2-bundles is analysed. We notably obtain a characterisation of its smooth locus, with an explicit decomposition of its singular locus into three connected componants. We also give an analysis of the Verlinde space of G_2 at level 1.

A pair (V, G) is called geometric structure, where V is a vector space and G is a subgroup GL(V), which is a set of transmission matrices. In this thesis we classify structures, which are based on properties of quaternions. Geometric structures based on quaternions are called triple structures. Triple structures are four structures with similar properties as quaternions. Quaternions are generated from real numbers and three complex units. We write quaternions in this shape a+bi+cj+dk.

Unter der Voraussetzung, dass q keine Einheitswurzel ist und dass die Differentiale duij der Fundamentalmatrix den Linksmodul der 1-Formen erzeugen, werden die bikovarianten Differentialkalküle auf den Quantengruppen SLq(N), Oq(N) und Spq(N) klassifiziert. Es wird gezeigt, dass es auf den Quantengruppen SLq(N), N ≥ 3, abgesehen von eindimensionalen Kalkülen und endlich vielen Werten von q genau 2N bikovariante Differentialkalküle gibt. Diese Kalküle haben die Dimension N². Für die Quantengruppen Oq(N) und Spq(N), N ≥ 3, gibt es unter den genannten Voraussetzungen bis auf endlich viele Werte von q genau zwei bikovariante Differentialkalküle der Dimension N². Die Bimodulstruktur der Kalküle sowie die zugeordneten ad-invarianten Rechtsideale werden explizit angegeben. Für die Quantengruppen SLq(N), N ≥ 3, wird gezeigt, dass es, sofern q keine Einheitwurzel ist, genau 2N² + 2N bikovariante Bimoduln vom Typ (u^c u; f) gibt. / If q is not a root of unity and under the assumption that the differentials duij of the fundamental matrix (uij) generate the left module of 1-forms, all bicovariant differential calculi on quantum groups SLq(N), Oq(N) and Spq(N) are classified. It is shown that on quantum groups SLq(N), N ≥ 3, except of 1-dimensional calculi and finitely many values of q, thre are exactly 2N bicovariant differential calculi. The space of invariant forms has dimension N². For quantum groups Oq(N) and Spq(N), N ≥ 3, under the same assumptions and up to finitely many values of q, there are exactly two bicovariant differential calculi of dimension N². The bimodule structure of the calculi as well as the corresponding ad-invariant right ideals are explicitely described. For quantum groups SLq(N), N ≥ 3, there are exactly 2N² + 2N bicovariant bimodules of type (u^c u; f) provided q is not a root of unity.

Multiple target tracking (MTT) is the process of localizing targets in an environment using sensors that perceive the environment. MTT has many applications such as wildlife monitoring, air traffic monitoring, and surveillance. These applications motivate further research in the different challenging aspects of MTT. One of these challenges that we will focus on in this dissertation is constructing a high fidelity target model. A common approach to target modeling is to use linear models or other simplified models that do not properly describe the target's pose (position and orientation), motion, and uncertainty. These simplified models are typically used because they are easy to implement and computationally efficient. A more accurate approach that improves tracking performance is to define the target model using a geometric representation of the target's natural configuration manifold. In essence, this geometric approach seeks to define a target model that can express every pose and motion of the target while preserving geometric properties such as distances and angles. We restrict our discussion of MTT to objects that move in physical space and can be modeled as a rigid body. This restriction allows us to construct generic geometric target models defined on Lie groups. Since not every Lie group has additional structure that permits vector space arithmetic like Euclidean space, many components of MTT such as data association, track initialization, track propagation and updating, track association and fusing, etc, must be adapted to work with Lie groups. The main contribution of this dissertation is the presentation of a novel MTT algorithm that implements the different MTT components to work with target models defined on Lie groups. We call this new algorithm, Geometric Multiple Target Tracking (G-MTT). This dissertation also serves as a guide on how other MTT algorithms can be modified to work with geometric target models. As part of the presentation there are various experimental results that strengthen the argument that a geometric approach to target modeling improves tracking performance.

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.

Made available in DSpace on 2015-05-15T11:45:59Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 323962 bytes, checksum: b1f72af0670744659eabe72c7c444dc3 (MD5) Previous issue date: 2008-08-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The Weierstrass representation of minimal surfaces in R3 and its generalization to Rn shows is a very useful tool in the study of minimal surfaces in these spaces. In this work we want to describe a type Weierstrass representation for immersions simply connected in the group of Heisenberg H3. Using applications harmonics is possible obtain a formula for general representation, type Weierstrass for minimal immersions in manifolds Riemannian simply connected general, is that, useful of point view theoretical, however it is very difficult find solutions explicit. The dimention 3 and the structure of group Lie of the group of Heisenberg H3 allow a description Geometric simple and we can get some classic examples. / A representação deWeierstrass para superfícies mínimas em R3 e sua generalização a Rn mostra-se uma ferramenta muito útil no estudo de superfícies mínimas nestes espaços. Neste trabalho pretendemos descrever uma representação tipo Weierstrass para imersões simplesmente conexas no grupo de Heisenberg H3. Usando aplicações harmónicas é possível obter uma fórmula de representação geral, tipo Weierstrass, para imersões mínimas simplesmente conexas em variedades Riemannianas gerais, isto é útil do ponto de vista teórico, entretanto é muito difícil encontrar soluções explicitas. A dimensão 3 e a estrutura de grupo de Lie do grupo de Heisenberg H3 permitem uma descrição geométrica simples e podemos obter alguns exemplos clássicos.

This work deals with the orientation of the camera in real-time with a single camera. Offline methods are described and used as a reference for comparison of a real-time metods. Metods work in real-time Monocular SLAM and PTAM methods are there described and compared. Further, paper shows hints of advanced methods whereas future work is possible.

Cette thèse est consacrée à une classe de groupes, appelés groupes de Kac-Moody, qui généralise de façon naturelle les groupes de Lie semi-simples, ou plus précisément, les groupes algébriques réductifs, dans un contexte infini-dimensionnel. On s'intéresse plus particulièrement au problème d'isomorphismes pour ces groupes, en vue d'obtenir un analogue infini-dimensionnel de la célèbre théorie des homomorphismes 'abstraits' de groupes algébriques simples, due à Armand Borel et Jacques Tits.<p><p>Le problème d'isomorphismes qu'on étudie s'avère être un cas particulier d'un problème plus général, qui consiste à caractériser les homomorphismes de groupes algébriques vers les groupes de Kac-Moody, dont l'image est bornée. Ce problème peut à son tour s'énoncer comme un problème de rigidité pour les actions de groupes algébriques sur les immeubles, via l'action naturelle d'un groupe de Kac-Moody sur une paire d'immeubles jumelés. Les résultats partiels, relatifs à ce problème de rigidité, que nous obtenons, nous permettent d'apporter une solution complète au problème d'isomorphismes pour les groupes de Kac-Moody déployés.<p>En particulier, on obtient un résultat de dévissage pour les automorphismes de ces objets. Celui-ci fournit à son tour une description complète de la structure du groupe d'automorphismes d'un groupe de Kac-Moody déployé sur un corps de caractéristique~$0$.<p><p>Nos arguments permettent également de traiter de façon analogue certaines formes anisotropes de groupes de Kac-Moody complexes, appelées formes unitaires. On montre en particulier que la topologie Hausdorff naturelle que portent ces formes est un invariant de leur structure de groupe abstrait. Ceci généralise un résultat bien connu de H. Freudenthal pour les groupes de Lie compacts.<p><p>Enfin, l'on s'intéresse aux homomorphismes de groupes de Kac-Moody à image fini-dimensionnelle, et l'on démontre la non-existence de tels homomorphismes à noyau central, lorsque le domaine est un groupe de Kac-Moody de type indéfini sur un corps infini. Ceci réduit un problème ouvert, dit problème de linéarité pour les groupes de Kac-Moody, au cas de corps de base finis. / Doctorat en sciences, Spécialisation mathématiques / info:eu-repo/semantics/nonPublished