Let E and F be Hilbert C*-modules over a C*-algebra A. New classes of (possibly unbounded) operators t: E->F are introduced and investigated - first of all graph regular operators. Instead of the density of the domain D(t) we only assume that t is essentially defined, that is, D(t) has an trivial ortogonal complement. Then t has a well-defined adjoint. We call an essentially defined operator t graph regular if its graph G(t) is orthogonally complemented and orthogonally closed if G(t) coincides with its biorthogonal complement. A theory of these operators and related concepts is developed: polar decomposition, functional calculus. Various characterizations of graph regular operators are given: (a, a_*, b)-transform and bounded transform. A number of examples of graph regular operators are presented (on commutative C*-algebras, a fraction algebra related to the Weyl algebra, Toeplitz algebra, C*-algebra of the Heisenberg group). A new characterization of operators affiliated to a C*-algebra in terms of resolvents is given as well as a Kato-Rellich theorem for affiliated operators. The association relation is introduced and studied as a counter part of graph regularity for concrete C*-algebras.:Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Sightings 1. Unitary *-module spaces Algebraic essence of adjointability on Hilbert C*-modules . . . . . 13 a) Operators on Hilbert C*-modules - Notions. . . . . . . . . . . . . . 13 b) Essential submodules and adjointability . . . . . . . . . . . . . . . . 15 c) From Hilbert C*-modules to unitary *-module spaces . . . . . . 16 2. Operators on unitary *-module spaces Basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3. Graph regularity Pragmatism between weak and (strong) regularity . . . . . . . . . 27 a) Types of regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 b) The case C(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 c) Graph regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Transition. Orthogonal complementability and topology Back to Hilbert C*-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Graph regular operators on Hilbert C*-modules 4. Commutative case: Operators on C_0(X) Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Interjection. Unboundedness and graph regularity . . . . . . . . . . 55 5. Relation to adjointable operators Sources of graph regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6. Concrete C*-algebras Association relation and affiliation relation . . . . . . . . . . . . . . . . 61 7. Examples Graph regular operators that are not regular . . . . . . . . . . . . . 67 a) Position and momentum operators as graph regular operators on a fraction algebra related to the Weyl algebra . . 67 b) A graph regular but not regular operator on the group C*-algebra of the Heisenberg group . . . . . . . . . . . . . . . 69 c) Unbounded Toeplitz operators . . . . . . . . . . . . . . . . . . . . . . . 70 8. Bounded transform The canonical regular operator associated to a graph regular operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 9. Absolute value and polar decomposition . . . . . . . . . . . . . . . 79 10. Functional calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 11. Special matrices of C*-algebras Counter examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Abstract and open questions . . . . . . . . . . . . . . . . . . . . . . . . . 89 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Dank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Erklärung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Die Positionsbestimmung von Verkehrsteilnehmern basiert häufig auf der Auswertung von Signalen globaler Navigationssatellitensysteme (GNSS). Dabei werden die Laufzeiten der Signale und darauf beruhend die Entfernungen zwischen den Satelliten und dem Verkehrsteilnehmer ermittelt. Die Positionsbestimmung erfolgt dann nach dem Prinzip der Trilateration. Bei hohen Anforderungen an die Genauigkeit der Position wird hierfür insbesondere die Phase des Trägersignals ausgewertet. Eine besondere Herausforderung stellt dabei die Mehrwegeausbreitung der Signale dar. Hervorgerufen wird diese durch Objekte, wie Bäume, Gebäude oder Fahrzeuge in der Umgebung des Verkehrsteilnehmers. Durch die Mehrwegeausbreitung werden die Laufzeiten der Signale und somit die Position fehlerhaft bestimmt. Es ist daher von großer Bedeutung, die mehrwegebedingten Fehleranteile zu detektieren und sie zu reduzieren. In dieser Arbeit wird dafür der Zusammenhang zwischen dem Trägerphasenmessfehler und der Signalqualität genutzt. Durch Anwendung einer im Rahmen dieser Arbeit entwickelten adaptierten Hilbert-Huang-Transformation auf die aus dem Signalqualitätsparameter des Signal-zu-Rauschleistungsdichte-Verhältnis abgeleiteten Signalamplituden können Mehrwegesignale detektiert und der durch sie verursachte Trägerphasenmessfehler berechnet werden. Anhand der Auswertung eines Experimentaldatensatzes sowie Daten von GNSS-Referenzstationen des SAPOS-Netzes kann der Erfolg des Einsatzes der adaptierten Hilbert-Huang-Transformation nachgewiesen werden.

We compute the Hilbert-Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen-Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals.

Indiana University-Purdue University Indianapolis (IUPUI) / We study the one-parameter family of determinants $det(I-\gamma K_{PII}),\gamma\in\mathbb{R}$ of an integrable Fredholm operator $K_{PII}$ acting on the interval $(-s,s)$ whose kernel is constructed out of the $\Psi$-function associated with the Hastings-McLeod solution of the second Painlev\'e equation. In case $\gamma=1$, this Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the Unitary Ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann-Hilbert method, we evaluate the large $s$-asymptotics of $\det(I-\gamma K_)$ for all values of the real parameter $\gamma$.

<p>Multi-linear operators play an important role in analysis due to their multiple connections with and applications to other mathematical areas such as ergodic theory, elliptic regularity, and other problems in partial differential equations.</p> <p>Within the area of multi-linear operators, powerful methods were developed originating from the problem of the almost everywhere convergence of Fourier series. Indeed, in their work, Carleson and Fefferman lay the foundation of time-frequency analysis. By further refining their methods, M. Lacey and C. Thiele proved the boundedness of the classical bilinear Hilbert transform for a suitable range of Hölder indices.</p> <p>In this thesis, we consider the general boundedness properties of the bilinear Hilbert transform and the sub-bilinear maximal function along a suitable family of curves.</p> <p>In the first part of our work, we present a short proof of the maximal boundedness range for the sub-bilinear maximal function along non-flat curves, giving a unified treatment of both the singular and the maximal operators.</p> <p>In the second part, we discuss the boundedness of these operators along hybrid curves. This work aims to present a unified perspective that treats the case obtained by joining the zero-curvature features of the operators along flat curves with the non-zero curvature features along non-flat curves.</p>

[pt] Foram desenvolvidas fontes ópticas para obtenção de imagens por tomografia de coerência óptica com alta resolução. Dois tipos de abordagens foram realizados, uma com um laser contínuo sintonizável, que neste trabalho foi instrumentado com marcadores de frequência óptica, outra com uma fonte óptica pulsada de banda larga. Mediante um processo de calibricação desenvolvido neste trabalho, a fonte contínua forneceu resoluções de 8 μm e alcances até 0,5 mm, enquanto que a fonte pulsada forneceu resoluções de 3 μm e alcances de 300 μm. A fonte pulsada permitiu ainda a obtenção de imagens em tempo real com capacidade de captura de movimento do objeto. / [en] Optical sources to obtain images through high resolution optical coherence tomography were developed. Two approaches were taken, one with a continuously tunable external cavity laser which, in this work, was modified to produce optical frequency markers, the other with an ultra-wideband pulsed source. Using a calibration process we developed in this work the continuously tunable source continued to achieve resolutions 8 μm and ranges of 0.5 mm, while the pulsed source achieved resolutions of 3.3 μm and ranges of 300 μm. The pulsed source has the capacity to capture real time images.

Doctor of Philosophy / Department of Mathematics / Alexander G. Ramm / A new method, the Dynamical Systems Method (DSM), justified recently, is applied to solving ill-conditioned linear algebraic system (ICLAS). The DSM gives a new approach to solving a wide class of ill-posed problems. In Chapter 1 a new iterative scheme for solving ICLAS is proposed. This iterative scheme is based on the DSM solution. An a posteriori stopping rules for the proposed method is justified. We also gives an a posteriori stopping rule for a modified iterative scheme developed in A.G.Ramm, JMAA,330 (2007),1338-1346, and proves convergence of the solution obtained by the iterative scheme. In Chapter 2 we give a convergence analysis of the following iterative scheme: u[subscript]n[superscript]delta=q u[subscript](n-1)[superscript]delta+(1-q)T[subscript](a[subscript]n)[superscript](-1) K[superscript]*f[subscript]delta, u[subscript]0[superscript]delta=0, where T:=K[superscript]* K, T[subscript]a :=T+aI, q in the interval (0,1),\quad a[subscript]n := alpha[subscript]0 q[superscript]n, alpha_0>0, with finite-dimensional approximations of T and K[superscript]* for solving stably Fredholm integral equations of the first kind with noisy data. In Chapter 3 a new method for inverting the Laplace transform from the real axis is formulated. This method is based on a quadrature formula. We assume that the unknown function f(t) is continuous with (known) compact support. An adaptive iterative method and an adaptive stopping rule, which yield the convergence of the approximate solution to f(t), are proposed in this chapter.

Quantum systems with finite Hilbert space where position x and momentum p take values in Z(d) (integers modulo d) are considered. Symplectic tranformations S(2ξ,Z(p)) in ξ-partite finite quantum systems are studied and constructed explicitly. Examples of applying such simple method is given for the case of bi-partite and tri-partite systems. The quantum correlations between the sub-systems after applying these transformations are discussed and quantified using various methods. An extended phase-space x-p-X-P where X, P ε Z(d) are position increment and momentum increment, is introduced. In this phase space the extended Wigner and Weyl functions are defined and their marginal properties are studied. The fourth order interference in the extended phase space is studied and verified using the extended Wigner function. It is seen that for both pure and mixed states the fourth order interference can be obtained.

L'océan et l'atmosphère sont des milieux stratifiés, supports d'ondes internes de gravité. Ces ondes peuvent se propager sur de très grandes distances, transportant de l'énergie loin de leurs sources et jouant ainsi un rôle clé dans le bilan énergétique global des écoulements géophysiques. Notre approche expérimentale modélise de façon idéalisée les topographies océaniques et la stratification du milieu afin de comprendre les mécanismes fondamentaux mis en jeu en présence d'ondes internes. Cette démarche a été couplée au développement de techniques d'analyse des données telles que la démodulation complexe. Différents types d'ondes internes ont été utilisées. Pour cela, un point clé a été la caractérisation et le contrôle d'une nouvelle source d'ondes. Notre étude combinant expériences, simulations numériques et prédictions analytiques permet la génération contrôlée de structures spatiales variées (faisceaux localisés, modes verticaux occupant toute la hauteur du uide, ondes planes). L'étude du problème classique de la réflexion d'une onde plane sur un plan incliné a permis de vérifier l'absence d'onde retour pour des pentes proches de la criticalité, tout en mettant en évidence que ce cas d'école est encore un sujet ouvert. Nous avons également étudié la conversion d'un mode vertical modélisant la marée interne lors de l'interaction avec une topographie de grande échelle. Cette conversion est régie par des mécanismes linéaires et non-linéaires. La présence de pentes sous-critiques inuence la conversion du mode incident en de plus petites échelles spatiales, tandis que la hauteur de la topographie joue un rôle notable dans l'amplitude des modes transmis ainsi que l'intensité de processus non-linéaires générant des ondes harmoniques et un courant moyen. Nous avons réalisé la première mise en évidence expérimentale de la génération locale de trains d'ondes non-linéaires (solitons) lors de la réflexion d'un faisceau d'ondes internes intense et localisé au niveau d'une pycnocline. Ce phénomène dépend de la stratification considérée, qui caractérise la nature de la pycnocline. Enfin, une étude de la dynamique couplée entre un bateau évoluant à force constante et le fluide stratifié environnant a été menée. Le régime particulier des dépend d'un unique nombre sans dimension, le nombre de Froude associé à l'onde la plus rapide de la stratification sur laquelle le bateau évolue. Nous avons généralisé le cas classique à deux couches à tout type de stratification constituée d'une couche homogène en surface.

This thesis aims to contribute to the area of kernel methods, which are a class of machine learning methods known for their wide applicability and state-of-the-art performance, but which suffer from high training and evaluation complexity. The work in this thesis utilizes the notion of reduced-set models to alleviate the training and testing complexities of these methods in a unified manner. In the first part of the thesis, we use recent results in kernel smoothing and integral-operator learning to design a generic strategy to speed up various kernel methods. In Chapter 3, we present a method to speed up kernel PCA (KPCA), which is one of the fundamental kernel methods for manifold learning, by using reduced-set density estimates (RSDE) of the data. The proposed method induces an integral operator that is an approximation of the ideal integral operator associated to KPCA. It is shown that the error between the ideal and approximate integral operators is related to the error between the ideal and approximate kernel density estimates of the data. In Chapter 4, we derive similar approximation algorithms for Gaussian process regression, diffusion maps, and kernel embeddings of conditional distributions. In the second part of the thesis, we use reduced-set models for kernel methods to tackle online learning in model-reference adaptive control (MRAC). In Chapter 5, we relate the properties of the feature spaces induced by Mercer kernels to make a connection between persistency-of-excitation and the budgeted placement of kernels to minimize tracking and modeling error. In Chapter 6, we use a Gaussian process (GP) formulation of the modeling error to accommodate a larger class of errors, and design a reduced-set algorithm to learn a GP model of the modeling error. Proofs of stability for all the algorithms are presented, and simulation results on a challenging control problem validate the methods.