Pseudodifferential analysis in Psi*-algebras on transmission spaces, infinite solving ideal chains and K-theory for conformally compact spaces

The present thesis is a contribution to the theory of algebras of pseudodifferential operators on singular settings. In particular, we focus on the $b$-calculus and the calculus on conformally compact spaces in the sense of Mazzeo and Melrose in connection with the notion of spectral invariant transmission operator algebras. We summarize results given by Gramsch et. al. on the construction of $Psi_0$-and $Psi*$-algebras and the corresponding scales of generalized Sobolev spaces using commutators of certain closed operators and derivations. In the case of a manifold with corners $Z$ we construct a $Psi*$-completion $A_b(Z,{}^bOmega^{1/2})$ of the algebra of zero order $b$-pseudodifferential operators $Psi_{b,cl}(Z, {}^bOmega^{1/2})$ in the corresponding $C*$-closure $B(Z,{}^bOmega^{12})hookrightarrow L(L^2(Z,{}^bOmega^{1/2}))$. The construction will also provide that localised to the (smooth) interior of Z the operators in the $A_b(Z, {}^bOmega^{1/2})$ can be represented as ordinary pseudodifferential operators. In connection with the notion of solvable $C*$-algebras - introduced by Dynin - we calculate the length of the $C*$-closure of $Psi_{b,cl}^0(F,{}^bOmega^{1/2},R^{E(F)})$ in $B(F,{}^bOmega^{1/2}),R^{E(F)})$ by localizing $B(Z, {}^bOmega^{1/2})$ along the boundary face $F$ using the (extended) indical familiy $I^B_{FZ}$. Moreover, we discuss how one can localise a certain solving ideal chain of $B(Z, {}^bOmega^{1/2})$ in neighbourhoods $U_p$ of arbitrary points $pin Z$. This localisation process will recover the singular structure of $U_p$; further, the induced length function $l_p$ is shown to be upper semi-continuous. We give construction methods for $Psi*$- and $C*$-algebras admitting only infinite long solving ideal chains. These algebras will first be realized as unconnected direct sums of (solvable) $C*$-algebras and then refined such that the resulting algebras have arcwise connected spaces of one dimensional representations. In addition, we recall the notion of transmission algebras on manifolds with corners $(Z_i)_{iin N}$ following an idea of Ali Mehmeti, Gramsch et. al. Thereby, we connect the underlying $C^infty$-function spaces using point evaluations in the smooth parts of the $Z_i$ and use generalized Laplacians to generate an appropriate scale of Sobolev spaces. Moreover, it is possible to associate generalized (solving) ideal chains to these algebras, such that to every $ninN$ there exists an ideal chain of length $n$ within the algebra. Finally, we discuss the $K$-theory for algebras of pseudodifferential operators on conformally compact manifolds $X$ and give an index theorem for these operators. In addition, we prove that the Dirac-operator associated to the metric of a conformally compact manifold $X$ is not a Fredholm operator.

Survival of the fittest: Herausforderungen an den Fremdenverkehr im Zeichen des global change

Der Globale Wandel ist im Begriff, den Tourismus zu verändern. Die Wechselwirkung von Tourismus und Klimawandel sind beidseitiger Art. Die vorliegende Arbeit zeigt Möglichkeiten der Adaption und einen wandelbaren Fremdenverkehr. Eine Übersicht der gängigen Tourismusmodelle stellt den Stand der Forschung dar. Der Fremdenverkehr ist durch drei Faktoren massiv geprägt: Die Nachfrage und Motivation, die Reisemittler und Veranstalter sowie das Destinationsangebot. Bei der Motivation wirken Motiv und Anreiz Motivationspsychologisch betrachtet auf die Reiseentscheidung deren Grundlage verarbeitete Informationen sind. Reisemittler und Veranstalter haben einen großen Einfluss auf Entscheidungsprozesse. Neue IuK Technologien haben deren Arbeit grundlegend verändert. Das Tourismusangebot wird stark durch die naturräumlichen Gegebenheiten sowie das politische System bestimmt. Überlebenswichtig für die Destination ist die evolutionstheoretisch etrachtete Fitnessmaximierung also Adaption und Wandel, um sich an geänderte Rahmenbedingungen anpassen zu können. Gerade im Bereich des Klimawandels müssen Maßnahmen ergriffen werden. Aber auch die Marktsättigung gerade in Verbindung mit der aktuellen Finanzkrise wirkt besonders schwer auf die Destination. Eine hohes Innovationsvermögen, Trendscanning und der Zusammenschluss in flexiblen Netzwerkclustern können einen Kundenmehrwert erzeugen. Die Fitnessmaximierung ist somit Überlebensziel der Destination und führt zur Kundenzufriedenheit die im Sättigungsmarkt alleinig Wachstum generieren kann.

Mechanical microcontacts of powder particles studied with the quartz crystal microbalance

Within this work, a particle-polymer surface system is studied with respect to the particle-surface interactions. The latter are governed by micromechanics and are an important aspect for a wide range of industrial applications. Here, a new methodology is developed for understanding the adhesion process and measure the relevant forces, based on the quartz crystal microbalance, QCM. rnThe potential of the QCM technique for studying particle-surface interactions and reflect the adhesion process is evaluated by carrying out experiments with a custom-made setup, consisting of the QCM with a 160 nm thick film of polystyrene (PS) spin-coated onto the quartz and of glass particles, of different diameters (5-20µm), deposited onto the polymer surface. Shifts in the QCM resonance frequency are monitored as a function of the oscillation amplitude. The induced frequency shifts of the 3rd overtone are found to decrease or increase, depending on the particle-surface coupling type and the applied oscillation (frequency and amplitude). For strong coupling the 3rd harmonic decreased, corresponding to an “added mass” on the quartz surface. However, positive frequency shifts are observed in some cases and are attributed to weak-coupling between particle and surface. Higher overtones, i.e. the 5th and 7th, were utilized in order to derive additional information about the interactions taking place. For small particles, the shift for specific overtones can increase after annealing, while for large particle diameters annealing causes a negative frequency shift. The lower overtones correspond to a generally strong-coupling regime with mainly negative frequency shifts observed, while the 7th appears to be sensitive to the contact break-down and the recorded shifts are positive.rnDuring oscillation, the motion of the particles and the induced frequency shift of the QCM are governed by a balance between inertial forces and contact forces. The adherence of the particles can be increased by annealing the PS film at 150°C, which led to the formation of a PS meniscus. For the interpretation, the Hertz, Johnson-Kendall-Roberts, Derjaguin-Müller-Toporov and the Mindlin theory of partial slip are considered. The Mindlin approach is utilized to describe partial slip. When partial slip takes place induced by an oscillating load, a part of the contact ruptures. This results in a decrease of the effective contact stiffness. Additionally, there are long-term memory effects due to the consolidation which along with the QCM vibrations induce a coupling increase. However, the latter can also break the contact, lead to detachment and even surface damage and deformation due to inertia. For strong coupling the particles appear to move with the vibrations and simply act as added effective mass leading to a decrease of the resonance frequency, in agreement with the Sauerbrey equation that is commonly used to calculate the added mass on a QCM). When the system enters the weak-coupling regime the particles are not able to follow the fast movement of the QCM surface. Hence, they effectively act as adding a “spring” with an additional coupling constant and increase the resonance frequency. The frequency shift, however, is not a unique function of the coupling constant. Furthermore, the critical oscillation amplitude is determined, above which particle detach. No movement is detected at much lower amplitudes, while for intermediate values, lateral particle displacement is observed. rnIn order to validate the QCM results and study the particle effects on the surface, atomic force microscopy, AFM, is additionally utilized, to image surfaces and measure surface forces. By studying the surface of the polymer film after excitation and particle removal, AFM imaging helped in detecting three different meniscus types for the contact area: the “full contact”, the “asymmetrical” and a third one including a “homocentric smaller meniscus”. The different meniscus forms result in varying bond intensity between particles and polymer film, which could explain the deviation between number of particles per surface area measured by imaging and the values provided by the QCM - frequency shift analysis. The asymmetric and the homocentric contact types are suggested to be responsible for the positive frequency shifts observed for all three measured overtones, i.e. for the weak-coupling regime, while the “full contact” type resulted in a negative frequency shift, by effectively contributing to the mass increase of the quartz..rnThe interplay between inertia and contact forces for the particle-surface system leads to strong- or weak-coupling, with the particle affecting in three mentioned ways the polymer surface. This is manifested in the frequency shifts of the QCM system harmonics which are used to differentiate between the two interaction types and reflect the overall state of adhesion for particles of different size.rn

Spectral theory of differential operators on finite metric graphs and on bounded domains

Die vorliegende Arbeit widmet sich der Spektraltheorie von Differentialoperatoren auf metrischen Graphen und von indefiniten Differentialoperatoren auf beschränkten Gebieten. Sie besteht aus zwei Teilen. Im Ersten werden endliche, nicht notwendigerweise kompakte, metrische Graphen und die Hilberträume von quadratintegrierbaren Funktionen auf diesen betrachtet. Alle quasi-m-akkretiven Laplaceoperatoren auf solchen Graphen werden charakterisiert, und Abschätzungen an die negativen Eigenwerte selbstadjungierter Laplaceoperatoren werden hergeleitet. Weiterhin wird die Wohlgestelltheit eines gemischten Diffusions- und Transportproblems auf kompakten Graphen durch die Anwendung von Halbgruppenmethoden untersucht. Eine Verallgemeinerung des indefiniten Operators $-tfrac{d}{dx}sgn(x)tfrac{d}{dx}$ von Intervallen auf metrische Graphen wird eingeführt. Die Spektral- und Streutheorie der selbstadjungierten Realisierungen wird detailliert besprochen. Im zweiten Teil der Arbeit werden Operatoren untersucht, die mit indefiniten Formen der Art $langlegrad v, A(cdot)grad urangle$ mit $u,vin H_0^1(Omega)subset L^2(Omega)$ und $OmegasubsetR^d$ beschränkt, assoziiert sind. Das Eigenwertverhalten entspricht in Dimension $d=1$ einer verallgemeinerten Weylschen Asymptotik und für $dgeq 2$ werden Abschätzungen an die Eigenwerte bewiesen. Die Frage, wann indefinite Formmethoden für Dimensionen $dgeq 2$ anwendbar sind, bleibt offen und wird diskutiert.

Measurement of the forward-backward asymmetry of Z boson decays in electron-positron pairs with the ATLAS detector in proton-proton collisions at root out s = 7TeV at the LHC

In this thesis the measurement of the effective weak mixing angle wma in proton-proton collisions is described. The results are extracted from the forward-backward asymmetry (AFB) in electron-positron final states at the ATLAS experiment at the LHC. The AFB is defined upon the distribution of the polar angle between the incoming quark and outgoing lepton. The signal process used in this study is the reaction pp to zgamma + X to ee + X taking a total integrated luminosity of 4.8\,fb^(-1) of data into account. The data was recorded at a proton-proton center-of-mass energy of sqrt(s)=7TeV. The weak mixing angle is a central parameter of the electroweak theory of the Standard Model (SM) and relates the neutral current interactions of electromagnetism and weak force. The higher order corrections on wma are related to other SM parameters like the mass of the Higgs boson.rnrnBecause of the symmetric initial state constellation of colliding protons, there is no favoured forward or backward direction in the experimental setup. The reference axis used in the definition of the polar angle is therefore chosen with respect to the longitudinal boost of the electron-positron final state. This leads to events with low absolute rapidity have a higher chance of being assigned to the opposite direction of the reference axis. This effect called dilution is reduced when events at higher rapidities are used. It can be studied including electrons and positrons in the forward regions of the ATLAS calorimeters. Electrons and positrons are further referred to as electrons. To include the electrons from the forward region, the energy calibration for the forward calorimeters had to be redone. This calibration is performed by inter-calibrating the forward electron energy scale using pairs of a central and a forward electron and the previously derived central electron energy calibration. The uncertainty is shown to be dominated by the systematic variations.rnrnThe extraction of wma is performed using chi^2 tests, comparing the measured distribution of AFB in data to a set of template distributions with varied values of wma. The templates are built in a forward folding technique using modified generator level samples and the official fully simulated signal sample with full detector simulation and particle reconstruction and identification. The analysis is performed in two different channels: pairs of central electrons or one central and one forward electron. The results of the two channels are in good agreement and are the first measurements of wma at the Z resonance using electron final states at proton-proton collisions at sqrt(s)=7TeV. The precision of the measurement is already systematically limited mostly by the uncertainties resulting from the knowledge of the parton distribution functions (PDF) and the systematic uncertainties of the energy calibration.rnrnThe extracted results of wma are combined and yield a value of wma_comb = 0.2288 +- 0.0004 (stat.) +- 0.0009 (syst.) = 0.2288 +- 0.0010 (tot.). The measurements are compared to the results of previous measurements at the Z boson resonance. The deviation with respect to the combined result provided by the LEP and SLC experiments is up to 2.7 standard deviations.

High-mass Drell-Yan cross-section and search for new phenomena in multi-electron/positron final states with the ATLAS detector

The Standard Model of particle physics is a very successful theory which describes nearly all known processes of particle physics very precisely. Nevertheless, there are several observations which cannot be explained within the existing theory. In this thesis, two analyses with high energy electrons and positrons using data of the ATLAS detector are presented. One, probing the Standard Model of particle physics and another searching for phenomena beyond the Standard Model.rnThe production of an electron-positron pair via the Drell-Yan process leads to a very clean signature in the detector with low background contributions. This allows for a very precise measurement of the cross-section and can be used as a precision test of perturbative quantum chromodynamics (pQCD) where this process has been calculated at next-to-next-to-leading order (NNLO). The invariant mass spectrum mee is sensitive to parton distribution functions (PFDs), in particular to the poorly known distribution of antiquarks at large momentum fraction (Bjoerken x). The measurementrnof the high-mass Drell-Yan cross-section in proton-proton collisions at a center-of-mass energy of sqrt(s) = 7 TeV is performed on a dataset collected with the ATLAS detector, corresponding to an integrated luminosity of 4.7 fb-1. The differential cross-section of pp -> Z/gamma + X -> e+e- + X is measured as a function of the invariant mass in the range 116 GeV < mee < 1500 GeV. The background is estimated using a data driven method and Monte Carlo simulations. The final cross-section is corrected for detector effects and different levels of final state radiation corrections. A comparison isrnmade to various event generators and to predictions of pQCD calculations at NNLO. A good agreement within the uncertainties between measured cross-sections and Standard Model predictions is observed.rnExamples of observed phenomena which can not be explained by the Standard Model are the amount of dark matter in the universe and neutrino oscillations. To explain these phenomena several extensions of the Standard Model are proposed, some of them leading to new processes with a high multiplicity of electrons and/or positrons in the final state. A model independent search in multi-object final states, with objects defined as electrons and positrons, is performed to search for these phenomenas. Therndataset collected at a center-of-mass energy of sqrt(s) = 8 TeV, corresponding to an integrated luminosity of 20.3 fb-1 is used. The events are separated in different categories using the object multiplicity. The data-driven background method, already used for the cross-section measurement was developed further for up to five objects to get an estimation of the number of events including fake contributions. Within the uncertainties the comparison between data and Standard Model predictions shows no significant deviations.