TY - THES A1 - Hannes, Sebastian T1 - Boundary Value Problems for the Lorentzian Dirac Operator N2 - The index theorem for elliptic operators on a closed Riemannian manifold by Atiyah and Singer has many applications in analysis, geometry and topology, but it is not suitable for a generalization to a Lorentzian setting. In the case where a boundary is present Atiyah, Patodi and Singer provide an index theorem for compact Riemannian manifolds by introducing non-local boundary conditions obtained via the spectral decomposition of an induced boundary operator, so called APS boundary conditions. Bär and Strohmaier prove a Lorentzian version of this index theorem for the Dirac operator on a manifold with boundary by utilizing results from APS and the characterization of the spectral flow by Phillips. In their case the Lorentzian manifold is assumed to be globally hyperbolic and spatially compact, and the induced boundary operator is given by the Riemannian Dirac operator on a spacelike Cauchy hypersurface. Their results show that imposing APS boundary conditions for these boundary operator will yield a Fredholm operator with a smooth kernel and its index can be calculated by a formula similar to the Riemannian case. Back in the Riemannian setting, Bär and Ballmann provide an analysis of the most general kind of boundary conditions that can be imposed on a first order elliptic differential operator that will still yield regularity for solutions as well as Fredholm property for the resulting operator. These boundary conditions can be thought of as deformations to the graph of a suitable operator mapping APS boundary conditions to their orthogonal complement. This thesis aims at applying the boundary conditions found by Bär and Ballmann to a Lorentzian setting to understand more general types of boundary conditions for the Dirac operator, conserving Fredholm property as well as providing regularity results and relative index formulas for the resulting operators. As it turns out, there are some differences in applying these graph-type boundary conditions to the Lorentzian Dirac operator when compared to the Riemannian setting. It will be shown that in contrast to the Riemannian case, going from a Fredholm boundary condition to its orthogonal complement works out fine in the Lorentzian setting. On the other hand, in order to deduce Fredholm property and regularity of solutions for graph-type boundary conditions, additional assumptions for the deformation maps need to be made. The thesis is organized as follows. In chapter 1 basic facts about Lorentzian and Riemannian spin manifolds, their spinor bundles and the Dirac operator are listed. These will serve as a foundation to define the setting and prove the results of later chapters. Chapter 2 defines the general notion of boundary conditions for the Dirac operator used in this thesis and introduces the APS boundary conditions as well as their graph type deformations. Also the role of the wave evolution operator in finding Fredholm boundary conditions is analyzed and these boundary conditions are connected to notion of Fredholm pairs in a given Hilbert space. Chapter 3 focuses on the principal symbol calculation of the wave evolution operator and the results are used to proof Fredholm property as well as regularity of solutions for suitable graph-type boundary conditions. Also sufficient conditions are derived for (pseudo-)local boundary conditions imposed on the Dirac operator to yield a Fredholm operator with a smooth solution space. In the last chapter 4, a few examples of boundary conditions are calculated applying the results of previous chapters. Restricting to special geometries and/or boundary conditions, results can be obtained that are not covered by the more general statements, and it is shown that so-called transmission conditions behave very differently than in the Riemannian setting. N2 - Der Indexsatz für elliptische Operatoren auf geschlossenen Riemannschen Mannigfaltigkeiten von Atiyah und Singer hat zahlreiche Anwendungen in Analysis, Geometrie und Topologie, ist aber ungeeignet für eine Verallgemeinerung auf Lorentz-Mannigfaltigkeiten. Durch die Einführung nicht-lokaler Randbedingungen, gewonnen aus der Spektralzerlegung eines induzierten Randoperators, beweisen Atiyah, Patodi und Singer (APS) einen Indexsatz für den Fall kompakter Riemannscher Mannigfaltigkeiten mit Rand. Aufbauend auf diesem Resultat und mit Hilfe der Charakterisierung des Spektralflusses durch Philipps gelangen Bär und Strohmaier zu einem Indexsatz für den Dirac-Operator auf global hyperbolischen Lorentz-Mannigfaltigkeiten mit kompakten und raumartigen Cauchy-Hyperflächen. Ihr Ergebnis zeigt unter anderem, dass der Dirac Operator auf solchen Mannigfaltigkeiten und unter APS Randbedingungen ein Fredholm-Operator mit glattem Kern ist und das sein Index sich aus einer zum Riemannschen Fall analogen Formel berechnen lässt. Zurück im Riemannschen Setup zeigen Bär und Ballmann eine allgemeine Charakterisierung von Randbedingungen für elliptische Differentialoperatoren erster Ordnung die sowohl die Regularität von Lösungen, als auch Fredholm-Eigenschaft des resultierenden Operators garantieren. Die dort entwickelten Randbedingungen können als Deformation auf den Graphen einer geeigneten Abbildung der APS-Randbedingung auf ihr orthogonales Komplement verstanden werden. Die vorliegende Arbeit hat das Ziel die von Bär und Ballmann beschriebenen Randbedingungen auf den Dirac-Operator von global hyperbolischen Lorentz-Mannigfaltigkeiten zu übertragen um eine allgemeinere Klasse von Randbedingungen zu finden unter denen der resultierende Dirac-Operator Fredholm ist und einen glatten Lösungsraum hat. Weiterhin wird analysiert wie sich derartige Deformation von APS-Randbedingungen auf den Index solcher Operatoren auswirken und wie dieser aus den bekannten Resultaten für den APS-Index berechnet werden kann. Es wird unter anderem gezeigt, dass im Gegensatz zum Riemannschen Fall beim Übergang von Randbedingungen zu ihrem orthogonalen Komplement die Fredholm-Eigenschaft des Operators erhalten bleibt. Andererseits sind zusätzliche Annahme nötig um die Regularität von Lösungen, sowie die Fredholm-Eigenschaft für Graph-Deformationen im Fall von Lorentz-Mannigfaltigkeiten zu erhalten. Die Arbeit ist dabei wie folgt aufgebaut. In Kapitel 1 werden grundlegende Fakten zu Lorentzschen und Riemannschen Spin-Mannigfaltigkeiten, ihren Spinor-Bündeln und Dirac-Operatoren zusammengetragen. Diese Informationen dienen als Ausgangspunkt zur Definition und Analyse von Randbedingungen in späteren Kapiteln der Arbeit. Kapitel 2 definiert allgemein den Begriff der Randbedingung wie er in dieser Arbeit verwendet wird und führt zudem den sogenannten ''wave-evolution-Operator'' ein, der eine wichtige Rolle im Finden und Analysieren von Fredholm-Randbedingungen für den Dirac-Operator spielen wird. Zuletzt wird der Zusammenhang zwischen Fredholm-Paaren eines Hilbert-Raumes und Fredholm-Randbedingungen für den Dirac-Operator erklärt. Kapitel 3 beschäftigt sich mit der Berechnung des Hauptsymbols des wave-evolution-Operators und die dort erzielten Resultate werden verwendet um Fredholm-Eigenschaft, sowie Regularität von Lösungen für geeignete Deformationen von APS-Randbedingungen zu beweisen. Weiterhin werden hinreichende Bedingungen für (pseudo-)lokale Randbedingungen abgeleitet, die Fredholm-Eigenschaft und Regularität für den resultierenden Dirac-Operator garantieren. Kapitel 4 zeigt, aufbauend auf den Ergebnissen der Kapitel 1-3, einige Beispiele von lokalen und nicht-lokalen Randbedingungen für den Dirac-Operator. Unter gewissen Einschränkungen an die Geometrie der zugrunde liegenden Mannigfaltigkeit bzw. den gestellten Randbedingungen können Ergebnisse erzielt werden die in den allgemeineren Resultaten der vorangehenden Kapitel nicht enthalten sind. Zuletzt werden sogenannte Transmission-Bedingungen analysiert und die Unterschiede dieser Randbedingungen zum Riemannschen Fall aufgezeigt. T2 - Randwertprobleme für den Lorentschen Diracoperator KW - Dirac Operator KW - Boundary Value Problems KW - Lorentzian Geometry KW - Randwertprobleme KW - Diracoperator KW - Lorentzgeometrie Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-548391 ER - TY - JOUR A1 - Lau, Stephan A1 - Kubiak, Thomas A1 - Burchert, Sebastian A1 - Goering, Mark A1 - Oberlaender, Nils A1 - von Mauschwitz, Hannes A1 - von Sass, Sarah A1 - Selle, Mareen A1 - Hiemisch, Anette T1 - Disentangling the effects of optimism and attributions on feelings of success JF - Personality and individual differences : an international journal of research into the structure and development of personality, and the causation of individual differences N2 - Two experiments examined the effects of dispositional optimism and attributions on feelings of success in a performance setting. In Experiment 1, participants successfully solved three cognitive tasks and attributed the success either internally (i.e., to themselves) or externally (i.e., to a teammate). We found no effect of optimism, but a significant effect of the attribution: Internal attribution predicted an increase in feelings of success. In Experiment 2, we replicated the design and adopted an extreme groups approach in order to include the extremes of the optimism dimension. Only optimism affected feelings of success in this sample: Pessimistic participants showed higher increases in feelings of success than optimistic participants. We conclude that optimism, if disentangled from attribution, may have an effect on affect, with pessimism showing potential affective benefits. However, this association may be concealed if samples with a restricted range of the optimism dimension are studied. KW - Optimism KW - Performance setting KW - Attribution KW - Success KW - Affect Y1 - 2014 U6 - https://doi.org/10.1016/j.paid.2013.08.030 SN - 0191-8869 VL - 56 SP - 78 EP - 82 PB - Elsevier CY - Oxford ER - TY - JOUR A1 - Heimann, Sebastian A1 - Vasyura-Bathke, Hannes A1 - Sudhaus, Henriette A1 - Isken, Marius Paul A1 - Kriegerowski, Marius A1 - Steinberg, Andreas A1 - Dahm, Torsten T1 - A Python framework for efficient use of pre-computed Green's functions in seismological and other physical forward and inverse source problems JF - Solid earth N2 - The computation of such synthetic GFs is computationally and operationally demanding. As a consequence, the onthe-fly recalculation of synthetic GFs in each iteration of an optimisation is time-consuming and impractical. Therefore, the pre-calculation and efficient storage of synthetic GFs on a dense grid of source to receiver combinations enables the efficient lookup and utilisation of GFs in time-critical scenarios. We present a Python-based framework and toolkit - Pyrocko-GF - that enables the pre-calculation of synthetic GF stores, which are independent of their numerical calculation method and GF transfer function. The framework aids in the creation of such GF stores by interfacing a suite of established numerical forward modelling codes in seismology (computational back ends). So far, interfaces to back ends for layered Earth model cases have been provided; however, the architecture of Pyrocko-GF is designed to cover back ends for other geometries (e.g. full 3-D heterogeneous media) and other physical quantities (e.g. gravity, pressure, tilt). Therefore, Pyrocko-GF defines an extensible GF storage format suitable for a wide range of GF types, especially handling elasticity and wave propagation problems. The framework assists with visualisations, quality control, and the exchange of GF stores, which is supported through an online platform that provides many pre-calculated GF stores for local, regional, and global studies. The Pyrocko-GF toolkit comes with a well-documented application programming interface (API) for the Python programming language to efficiently facilitate forward modelling of geophysical processes, e.g. synthetic waveforms or static displacements for a wide range of source models. Y1 - 2019 U6 - https://doi.org/10.5194/se-10-1921-2019 SN - 1869-9510 SN - 1869-9529 VL - 10 IS - 6 SP - 1921 EP - 1935 PB - Copernicus CY - Göttingen ER - TY - JOUR A1 - Isken, Marius Paul A1 - Vasyura-Bathke, Hannes A1 - Dahm, Torsten A1 - Heimann, Sebastian T1 - De-noising distributed acoustic sensing data using an adaptive frequency-wavenumber filter JF - Geophysical journal international N2 - Data recorded by distributed acoustic sensing (DAS) along an optical fibre sample the spatial and temporal properties of seismic wavefields at high spatial density. Often leading to massive amount of data when collected for seismic monitoring along many kilometre long cables. The spatially coherent signals from weak seismic arrivals within the data are often obscured by incoherent noise. We present a flexible and computationally efficient filtering technique, which makes use of the dense spatial and temporal sampling of the data and that can handle the large amount of data. The presented adaptive frequency-wavenumber filter suppresses the incoherent seismic noise while amplifying the coherent wavefield. We analyse the response of the filter in time and spectral domain, and we demonstrate its performance on a noisy data set that was recorded in a vertical borehole observatory showing active and passive seismic phase arrivals. Lastly, we present a performant open-source software implementation enabling real-time filtering of large DAS data sets. KW - Fourier analysis KW - Image processing KW - Time-series analysis KW - Seismic noise KW - Distributed acoustic sensing Y1 - 2022 U6 - https://doi.org/10.1093/gji/ggac229 SN - 0956-540X SN - 1365-246X VL - 231 IS - 2 SP - 944 EP - 949 PB - Oxford University Press CY - Oxford ER -