Toeplitz operators on finite and infinite dimensional spaces with associated psi *-Fréchet algebras

The present thesis is a contribution to the multi-variable theory of Bergman and Hardy Toeplitz operators on spaces of holomorphic functions over finite and infinite dimensional domains. In particular, we focus on certain spectral invariant Frechet operator algebras F closely related to the local symbol behavior of Toeplitz operators in F. We summarize results due to B. Gramsch et.al. on the construction of Psi_0- and Psi^*-algebras in operator algebras and corresponding scales of generalized Sobolev spaces using commutator methods, generalized Laplacians and strongly continuous group actions. In the case of the Segal-Bargmann space H^2(C^n,m) of Gaussian square integrable entire functions on C^n we determine a class of vector-fields Y(C^n) supported in complex cones K. Further, we require that for any finite subset V of Y(C^n) the Toeplitz projection P is a smooth element in the Psi_0-algebra constructed by commutator methods with respect to V. As a result we obtain Psi_0- and Psi^*-operator algebras F localized in cones K. It is an immediate consequence that F contains all Toeplitz operators T_f with a symbol f of certain regularity in an open neighborhood of K. There is a natural unitary group action on H^2(C^n,m) which is induced by weighted shifts and unitary groups on C^n. We examine the corresponding Psi^*-algebra A of smooth elements in Toeplitz-C^*-algebras. Among other results sufficient conditions on the symbol f for T_f to belong to A are given in terms of estimates on its Berezin-transform. Local aspects of the Szegö projection P_s on the Heisenbeg group and the corresponding Toeplitz operators T_f with symbol f are studied. In this connection we apply a result due to Nagel and Stein which states that for any strictly pseudo-convex domain U the projection P_s is a pseudodifferential operator of exotic type (1/2, 1/2). The second part of this thesis is devoted to the infinite dimensional theory of Bergman and Hardy spaces and the corresponding Toeplitz operators. We give a new proof of a result observed by Boland and Waelbroeck. Namely, that the space of all holomorphic functions H(U) on an open subset U of a DFN-space (dual Frechet nuclear space) is a FN-space (Frechet nuclear space) equipped with the compact open topology. Using the nuclearity of H(U) we obtain Cauchy-Weil-type integral formulas for closed subalgebras A in H_b(U), the space of all bounded holomorphic functions on U, where A separates points. Further, we prove the existence of Hardy spaces of holomorphic functions on U corresponding to the abstract Shilov boundary S_A of A and with respect to a suitable boundary measure on S_A. Finally, for a domain U in a DFN-space or a polish spaces we consider the symmetrizations m_s of measures m on U by suitable representations of a group G in the group of homeomorphisms on U. In particular,in the case where m leads to Bergman spaces of holomorphic functions on U, the group G is compact and the representation is continuous we show that m_s defines a Bergman space of holomorphic functions on U as well. This leads to unitary group representations of G on L^p- and Bergman spaces inducing operator algebras of smooth elements related to the symmetries of U.

Applications of remote sensing and geographic information systems in geomorphological studies: Safaga - El Quseir area, Red Sea coast, Egypt as an example

In den letzten drei Jahrzehnten sind Fernerkundung und GIS in den Geowissenschaften zunehmend wichtiger geworden, um die konventionellen Methoden von Datensammlung und zur Herstellung von Landkarten zu verbessern. Die vorliegende Arbeit befasst sich mit der Anwendung von Fernerkundung und geographischen Informationssystemen (GIS) für geomorphologische Untersuchungen. Durch die Kombination beider Techniken ist es vor allem möglich geworden, geomorphologische Formen im Überblick und dennoch detailliert zu erfassen. Als Grundlagen werden in dieser Arbeit topographische und geologische Karten, Satellitenbilder und Klimadaten benutzt. Die Arbeit besteht aus 6 Kapiteln. Das erste Kapitel gibt einen allgemeinen Überblick über den Untersuchungsraum. Dieser umfasst folgende morphologische Einheiten, klimatischen Verhältnisse, insbesondere die Ariditätsindizes der Küsten- und Gebirgslandschaft sowie das Siedlungsmuster beschrieben. Kapitel 2 befasst sich mit der regionalen Geologie und Stratigraphie des Untersuchungsraumes. Es wird versucht, die Hauptformationen mit Hilfe von ETM-Satellitenbildern zu identifizieren. Angewandt werden hierzu folgende Methoden: Colour Band Composite, Image Rationing und die sog. überwachte Klassifikation. Kapitel 3 enthält eine Beschreibung der strukturell bedingten Oberflächenformen, um die Wechselwirkung zwischen Tektonik und geomorphologischen Prozessen aufzuklären. Es geht es um die vielfältigen Methoden, zum Beispiel das sog. Image Processing, um die im Gebirgskörper vorhandenen Lineamente einwandfrei zu deuten. Spezielle Filtermethoden werden angewandt, um die wichtigsten Lineamente zu kartieren. Kapitel 4 stellt den Versuch dar, mit Hilfe von aufbereiteten SRTM-Satellitenbildern eine automatisierte Erfassung des Gewässernetzes. Es wird ausführlich diskutiert, inwieweit bei diesen Arbeitsschritten die Qualität kleinmaßstäbiger SRTM-Satellitenbilder mit großmaßstäbigen topographischen Karten vergleichbar ist. Weiterhin werden hydrologische Parameter über eine qualitative und quantitative Analyse des Abflussregimes einzelner Wadis erfasst. Der Ursprung von Entwässerungssystemen wird auf der Basis geomorphologischer und geologischer Befunde interpretiert. Kapitel 5 befasst sich mit der Abschätzung der Gefahr episodischer Wadifluten. Die Wahrscheinlichkeit ihres jährlichen Auftretens bzw. des Auftretens starker Fluten im Abstand mehrerer Jahre wird in einer historischen Betrachtung bis 1921 zurückverfolgt. Die Bedeutung von Regentiefs, die sich über dem Roten Meer entwickeln, und die für eine Abflussbildung in Frage kommen, wird mit Hilfe der IDW-Methode (Inverse Distance Weighted) untersucht. Betrachtet werden außerdem weitere, regenbringende Wetterlagen mit Hilfe von Meteosat Infrarotbildern. Genauer betrachtet wird die Periode 1990-1997, in der kräftige, Wadifluten auslösende Regenfälle auftraten. Flutereignisse und Fluthöhe werden anhand von hydrographischen Daten (Pegelmessungen) ermittelt. Auch die Landnutzung und Siedlungsstruktur im Einzugsgebiet eines Wadis wird berücksichtigt. In Kapitel 6 geht es um die unterschiedlichen Küstenformen auf der Westseite des Roten Meeres zum Beispiel die Erosionsformen, Aufbauformen, untergetauchte Formen. Im abschließenden Teil geht es um die Stratigraphie und zeitliche Zuordnung von submarinen Terrassen auf Korallenriffen sowie den Vergleich mit anderen solcher Terrassen an der ägyptischen Rotmeerküste westlich und östlich der Sinai-Halbinsel.

Ancient Greek harbours used as geo-archives for palaeotsunami research : case studies from Krane (Cefalonia), Lechaion (Gulf of Corinth) and Kyllini (Peloponnese)

Since historical times, coastal areas throughout the eastern Mediterranean are exposed to tsunami hazard. For many decades the knowledge about palaeotsunamis was solely based on historical accounts. However, results from timeline analyses reveal different characteristics affecting the quality of the dataset (i.e. distribution of data, temporal thinning backward of events, local periodization phenomena) that emphasize the fragmentary character of the historical data. As an increasing number of geo-scientific studies give convincing examples of well dated tsunami signatures not reported in catalogues, the non-existing record is a major problem to palaeotsunami research. While the compilation of historical data allows a first approach in the identification of areas vulnerable to tsunamis, it must not be regarded as reliable for hazard assessment. Considering the increasing economic significance of coastal regions (e.g. for mass tourism) and the constantly growing coastal population, our knowledge on the local, regional and supraregional tsunami hazard along Mediterranean coasts has to be improved. For setting up a reliable tsunami risk assessment and developing risk mitigation strategies, it is of major importance (i) to identify areas under risk and (ii) to estimate the intensity and frequency of potential events. This approach is most promising when based on the analysis of palaeotsunami research seeking to detect areas of high palaeotsunami hazard, to calculate recurrence intervals and to document palaeotsunami destructiveness in terms of wave run-up, inundation and long-term coastal change. Within the past few years, geo-scientific studies on palaeotsunami events provided convincing evidence that throughout the Mediterranean ancient harbours were subject to strong tsunami-related disturbance or destruction. Constructed to protect ships from storm and wave activity, harbours provide especially sheltered and quiescent environments and thus turned out to be valuable geo-archives for tsunamigenic high-energy impacts on coastal areas. Directly exposed to the Hellenic Trench and extensive local fault systems, coastal areas in the Ionian Sea and the Gulf of Corinth hold a considerably high risk for tsunami events, respectively.Geo-scientific and geoarcheaological studies carried out in the environs of the ancient harbours of Krane (Cefalonia Island), Lechaion (Corinth, Gulf of Corinth) and Kyllini (western Peloponnese) comprised on-shore and near-shore vibracoring and subsequent sedimentological, geochemical and microfossil analyses of the recovered sediments. Geophysical methods like electrical resistivity tomography and ground penetrating radar were applied in order to detect subsurface structures and to verify stratigraphical patterns derived from vibracores over long distances. The overall geochronological framework of each study area is based on radiocarbon dating of biogenic material and age determination of diagnostic ceramic fragments. Results presented within this study provide distinct evidence of multiple palaeotsunami landfalls for the investigated areas. Tsunami signatures encountered in the environs of Krane, Lechaion and Kyllini include (i) coarse-grained allochthonous marine sediments intersecting silt-dominated quiescent harbour deposits and/or shallow marine environments, (ii) disturbed microfaunal assemblages and/or (iii) distinct geochemical fingerprints as well as (iv) geo-archaeological destruction layers and (v) extensive units of beachrock-type calcarenitic tsunamites. For Krane, geochronological data yielded termini ad or post quem (maximum ages) for tsunami event generations dated to 4150 ± 60 cal BC, ~ 3200 ± 110 cal BC, ~ 650 ± 110 cal BC, and ~ 930 ± 40 cal AD, respectively. Results for Lechaion suggest that the harbour was hit by strong tsunami impacts in the 8th-6th century BC, the 1st-2nd century AD and in the 6th century AD. At Kyllini, the harbour site was affected by tsunami impact in between the late 7th and early 4th cent. BC and between the 4th and 6th cent. AD. In case of Lechaion and Kyllini, the final destruction of the harbour facilities also seems to be related to the tsunami impact. Comparing the tsunami signals obtained for each study areas with geo-scientific data from palaeotsunami events from other sites indicates that the investigated harbour sites represent excellent geo-archives for supra-regional mega-tsunamis.

Volatile organic compounds at the air-sea interface : gas exchange rates, oceanic emissions and the effect of ocean acidification

Oceans are key sources and sinks in the global budgets of significant atmospheric trace gases, termed Volatile Organic Compounds (VOCs). Despite their low concentrations, these species have an important role in the atmosphere, influencing ozone photochemistry and aerosol physics. Surprisingly, little work has been done on assessing their emissions or transport mechanisms and rates between ocean and atmosphere, all of which are important when modelling the atmosphere accurately.rnA new Needle Trap Device (NTD) - GC-MS method was developed for the effective sampling and analysis of VOCs in seawater. Good repeatability (RSDs <16 %), linearity (R2 = 0.96 - 0.99) and limits of detection in the range of pM were obtained for DMS, isoprene, benzene, toluene, p-xylene, (+)-α-pinene and (-)-α-pinene. Laboratory evaluation and subsequent field application indicated that the proposed method can be used successfully in place of the more usually applied extraction techniques (P&T, SPME) to extend the suite of species typically measured in the ocean and improve detection limits. rnDuring a mesocosm CO2 enrichment study, DMS, isoprene and α-pinene were identified and quantified in seawater samples, using the above mentioned method. Based on correlations with available biological datasets, the effects of ocean acidification as well as possible ocean biological sources were investigated for all examined compounds. Future ocean's acidity was shown to decrease oceanic DMS production, possibly impact isoprene emissions but not affect the production of α-pinene. rnIn a separate activity, ocean - atmosphere interactions were simulated in a large scale wind-wave canal facility, in order to investigate the gas exchange process and its controlling mechanisms. Air-water exchange rates of 14 chemical species (of which 11 VOCs) spanning a wide range of solubility (dimensionless solubility, α = 0:4 to 5470) and diffusivity (Schmidt number in water, Scw = 594 to 1194) were obtained under various turbulent (wind speed at ten meters height, u10 = 0:8 to 15ms-1) and surfactant modulated (two different sized Triton X-100 layers) surface conditions. Reliable and reproducible total gas transfer velocities were obtained and the derived values and trends were comparable to previous investigations. Through this study, a much better and more comprehensive understanding of the gas exchange process was accomplished. The role of friction velocity, uw* and mean square slope, σs2 in defining phenomena such as waves and wave breaking, near surface turbulence, bubbles and surface films was recognized as very significant. uw* was determined as the ideal turbulent parameter while σs2 described best the related surface conditions. A combination of both uw* and σs2 variables, was found to reproduce faithfully the air-water gas exchange process. rnA Total Transfer Velocity (TTV) model provided by a compilation of 14 tracers and a combination of both uw* and σs2 parameters, is proposed for the first time. Through the proposed TTV parameterization, a new physical perspective is presented which provides an accurate TTV for any tracer within the examined solubility range. rnThe development of such a comprehensive air-sea gas exchange parameterization represents a highly useful tool for regional and global models, providing accurate total transfer velocity estimations for any tracer and any sea-surface status, simplifying the calculation process and eliminating inevitable calculation uncertainty connected with the selection or combination of different parameterizations.rnrn

Polarization observables in virtual compton scattering

Virtual Compton Scattering (VCS) is an important reaction for understanding nucleon structure at low energies. By studying this process, the generalized polarizabilities of the nucleon can be measured. These observables are a generalization of the already known polarizabilities and will permit theoretical models to be challenged on a new level. More specifically, there exist six generalized polarizabilities and in order to disentangle them all, a double polarization experiment must be performed. Within this work, the VCS reaction p(e,e p)gamma was measured at MAMI using the A1 Collaboration three spectrometer setup with Q2=0.33 (GeV/c)2. Using the highly polarized MAMI beam and a recoil proton polarimeter, it was possible to measure both the VCS cross section and the double polarization observables. Already in 2000, the unpolarized VCS cross section was measured at MAMI. In this new experiment, we could confirm the old data and furthermore the double polarization observables were measured for the first time. The data were taken in five periods between 2005 and 2006. In this work, the data were analyzed to extract the cross section and the proton polarization. For the analysis, a maximum likelihood algorithm was developed together with the full simulation of all the analysis steps. The experiment is limited by the low statistics due mainly to the focal plane proton polarimeter efficiency. To overcome this problem, a new determination and parameterization of the carbon analyzing power was performed. The main result of the experiment is the extraction of a new combination of the generalized polarizabilities using the double polarization observables.

Pseudodifferential operators on Hilbert space riggings with associated psi *-algebras and generalized Hörmander classes

The present thesis is concerned with certain aspects of differential and pseudodifferential operators on infinite dimensional spaces. We aim to generalize classical operator theoretical concepts of pseudodifferential operators on finite dimensional spaces to the infinite dimensional case. At first we summarize some facts about the canonical Gaussian measures on infinite dimensional Hilbert space riggings. Considering the naturally unitary group actions in $L^2(H_-,gamma)$ given by weighted shifts and multiplication with $e^{iSkp{t}{cdot}_0}$ we obtain an unitary equivalence $F$ between them. In this sense $F$ can be considered as an abstract Fourier transform. We show that $F$ coincides with the Fourier-Wiener transform. Using the Fourier-Wiener transform we define pseudodifferential operators in Weyl- and Kohn-Nirenberg form on our Hilbert space rigging. In the case of this Gaussian measure $gamma$ we discuss several possible Laplacians, at first the Ornstein-Uhlenbeck operator and then pseudo-differential operators with negative definite symbol. In the second case, these operators are generators of $L^2_gamma$-sub-Markovian semi-groups and $L^2_gamma$-Dirichlet-forms. In 1992 Gramsch, Ueberberg and Wagner described a construction of generalized Hörmander classes by commutator methods. Following this concept and the classical finite dimensional description of $Psi_{ro,delta}^0$ ($0leqdeltaleqroleq 1$, $delta< 1$) in the $C^*$-algebra $L(L^2)$ by Beals and Cordes we construct in both cases generalized Hörmander classes, which are $Psi^*$-algebras. These classes act on a scale of Sobolev spaces, generated by our Laplacian. In the case of the Ornstein-Uhlenbeck operator, we prove that a large class of continuous pseudodifferential operators considered by Albeverio and Dalecky in 1998 is contained in our generalized Hörmander class. Furthermore, in the case of a Laplacian with negative definite symbol, we develop a symbolic calculus for our operators. We show some Fredholm-criteria for them and prove that these Fredholm-operators are hypoelliptic. Moreover, in the finite dimensional case, using the Gaussian-measure instead of the Lebesgue-measure the index of these Fredholm operators is still given by Fedosov's formula. Considering an infinite dimensional Heisenberg group rigging we discuss the connection of some representations of the Heisenberg group to pseudo-differential operators on infinite dimensional spaces. We use this connections to calculate the spectrum of pseudodifferential operators and to construct generalized Hörmander classes given by smooth elements which are spectrally invariant in $L^2(H_-,gamma)$. Finally, given a topological space $X$ with Borel measure $mu$, a locally compact group $G$ and a representation $B$ of $G$ in the group of all homeomorphisms of $X$, we construct a Borel measure $mu_s$ on $X$ which is invariant under $B(G)$.

Neural network activity in the neonatal acute slice, slice culture and cell culture

Information processing and storage in the brain may be presented by the oscillations and cell assemblies. Here we address the question of how individual neurons associate together to assemble neural networks and present spontaneous electrical activity. Therefore, we dissected the neonatal brain at three different levels: acute 1-mm thick brain slice, cultured organotypic 350-µm thick brain slice and dissociated neuronal cultures. The spatio-temporal properties of neural activity were investigated by using a 60-channel Micro-electrode arrays (MEA), and the cell assemblies were studied by using a template-matching method. We find local on-propagating as well as large- scale propagating spontaneous oscillatory activity in acute slices, spontaneous network activity characterized by synchronized burst discharges in organotypic cultured slices, and autonomous bursting behaviour in dissociated neuronal cultures. Furthermore, repetitive spike patterns emerge after one week of dissociated neuronal culture and dramatically increase their numbers as well as their complexity and occurrence in the second week. Our data indicate that neurons can self-organize themselves, assembly to a neural network, present spontaneous oscillations, and emerge spatio-temporal activation patterns. The spontaneous oscillations and repetitive spike patterns may serve fundamental functions for information processing and storage in the brain.