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.

Structuring of polymer surface by evaporation of sessile microdrops

In this thesis, we investigated the evaporation of sessile microdroplets on different solid substrates. Three major aspects were studied: the influence of surface hydrophilicity and heterogeneity on the evaporation dynamics for an insoluble solid substrate, the influence of external process parameters and intrinsic material properties on microstructuring of soluble polymer substrates and the influence of an increased area to volume ratio in a microfluidic capillary, when evaporation is hindered. In the first part, the evaporation dynamics of pure sessile water drops on smooth self-assembled monolayers (SAMs) of thiols or disulfides on gold on mica was studied. With increasing surface hydrophilicity the drop stayed pinned longer. Thus, the total evaporation time of a given initial drop volume was shorter, since the drop surface, through which the evaporation occurs, stays longer large. Usually, for a single drop the volume decreased linearly with t1.5, t being the evaporation time, for a diffusion-controlled evaporation process. However, when we measured the total evaporation time, ttot, for multiple droplets with different initial volumes, V0, we found a scaling of the form V0 = attotb. The more hydrophilic the substrate was, the more showed the scaling exponent a tendency to an increased value up to 1.6. This can be attributed to an increasing evaporation rate through a thin water layer in the vicinity of the drop. Under the assumption of a constant temperature at the substrate surface a cooling of the droplet and thus a decreased evaporation rate could be excluded as a reason for the different scaling exponent by simulations performed by F. Schönfeld at the IMM, Mainz. In contrast, for a hairy surface, made of dialkyldisulfide SAMs with different chain lengths and a 1:1 mixture of hydrophilic and hydrophobic end groups (hydroxy versus methyl group), the scaling exponent was found to be ~ 1.4. It increased to ~ 1.5 with increasing hydrophilicity. A reason for this observation can only be speculated: in the case of longer hydrophobic alkyl chains the formation of an air layer between substrate and surface might be favorable. Thus, the heat transport to the substrate might be reduced, leading to a stronger cooling and thus decreased evaporation rate. In the second part, the microstructuring of polystyrene surfaces by drops of toluene, a good solvent, was investigated. For this a novel deposition technique was developed, with which the drop can be deposited with a syringe. The polymer substrate is lying on a motorized table, which picks up the pendant drop by an upward motion until a liquid bridge is formed. A consecutive downward motion of the table after a variable delay, i.e. the contact time between drop and polymer, leads to the deposition of the droplet, which can evaporate. The resulting microstructure is investigated in dependence of the processes parameters, i.e. the approach and the retraction speed of the substrate and the delay between them, and in dependence of the intrinsic material properties, i.e. the molar mass and the type of the polymer/solvent system. The principal equivalence with the microstructuring by the ink-jet technique was demonstrated. For a high approach and retraction speed of 9 mm/s and no delay between them, a concave microtopology was observed. In agreement with the literature, this can be explained by a flow of solvent and the dissolved polymer to the rim of the pinned droplet, where polymer is accumulated. This effect is analogue to the well-known formation of ring-like stains after the evaporation of coffee drops (coffee-stain effect). With decreasing retraction speed down to 10 µm/s the resulting surface topology changes from concave to convex. This can be explained with the increasing dissolution of polymer into the solvent drop prior to the evaporation. If the polymer concentration is high enough, gelation occurs instead of a flow to the rim and the shape of the convex droplet is received. With increasing delay time from below 0 ms to 1s the depth of the concave microwells decreases from 4.6 µm to 3.2 µm. However, a convex surface topology could not be obtained, since for longer delay times the polymer sticks to the tip of the syringe. Thus, by changing the delay time a fine-tuning of the concave structure is accomplished, while by changing the retraction speed a principal change of the microtopolgy can be achieved. We attribute this to an additional flow inside the liquid bridge, which enhanced polymer dissolution. Even if the pendant drop is evaporating about 30 µm above the polymer surface without any contact (non-contact mode), concave structures were observed. Rim heights as high as 33 µm could be generated for exposure times of 20 min. The concave structure exclusively lay above the flat polymer surface outside the structure even after drying. This shows that toluene is taken up permanently. The increasing rim height, rh, with increasing exposure time to the solvent vapor obeys a diffusion law of rh = rh0  tn, with n in the range of 0.46 ~ 0.65. This hints at a non-Fickian swelling process. A detailed analysis showed that the rim height of the concave structure is modulated, unlike for the drop deposition. This is due to the local stress relaxation, which was initiated by the increasing toluene concentration in the extruded polymer surface. By altering the intrinsic material parameters i.e. the polymer molar mass and the polymer/solvent combination, several types of microstructures could be formed. With increasing molar mass from 20.9 kDa to 1.44 MDa the resulting microstructure changed from convex, to a structure with a dimple in the center, to concave, to finally an irregular structure. This observation can be explained if one assumes that the microstructuring is dominated by two opposing effects, a decreasing solubility with increasing polymer molar mass, but an increasing surface tension gradient leading to instabilities of Marangoni-type. Thus, a polymer with a low molar mass close or below the entanglement limit is subject to a high dissolution rate, which leads to fast gelation compared to the evaporation rate. This way a coffee-rim like effect is eliminated early and a convex structure results. For high molar masses the low dissolution rate and the low polymer diffusion might lead to increased surface tension gradients and a typical local pile-up of polymer is found. For intermediate polymer masses around 200 kDa, the dissolution and evaporation rate are comparable and the typical concave microtopology is found. This interpretation was supported by a quantitative estimation of the diffusion coefficient and the evaporation rate. For a different polymer/solvent system, polyethylmethacrylate (PEMA)/ethylacetate (EA), exclusively concave structures were found. Following the statements above this can be interpreted with a lower dissolution rate. At low molar masses the concentration of PEMA in EA most likely never reaches the gelation point. Thus, a concave instead of a convex structure occurs. At the end of this section, the optically properties of such microstructures for a potential application as microlenses are studied with laser scanning confocal microscopy. In the third part, the droplet was confined into a glass microcapillary to avoid evaporation. Since here, due to an increased area to volume ratio, the surface properties of the liquid and the solid walls became important, the influence of the surface hydrophilicity of the wall on the interfacial tension between two immiscible liquid slugs was investigated. For this a novel method for measuring the interfacial tension between the two liquids within the capillary was developed. This technique was demonstrated by measuring the interfacial tensions between slugs of pure water and standard solvents. For toluene, n-hexane and chloroform 36.2, 50.9 and 34.2 mN/m were measured at 20°C, which is in a good agreement with data from the literature. For a slug of hexane in contact with a slug of pure water containing ethanol in a concentration range between 0 and 70 (v/v %), a difference of up to 6 mN/m was found, when compared to commercial ring tensiometry. This discrepancy is still under debate.

Räumliche Verbreitung geogener Schwermetallgehalte in Böden des Taunus: Einfluss von periglazialer Deckschichtengenese

Während der Glazialphasen kam es in den europäischen Mittelgebirgen bedingt durch extensive solifluidale Massenbewegungen zur Bildung von Deckschichten. Diese Deckschichten repräsentieren eine Mischung verschiedener Substrate, wie anstehendes Ausgangsgestein, äolische Depositionen und lokale Erzgänge. Die räumliche Ausdehnung der Metallkontaminationen verursacht durch kleinräumige Erzgänge wird durch die periglaziale Solifluktion verstärkt. Das Ziel der vorliegenden Untersuchung war a) den Zusammenhang zwischen den Reliefeigenschaften und den Ausprägungen der solifluidalen Deckschichten und Böden aufzuklären, sowie b) mittels Spurenelementgehalte und Blei-Isotopen-Verhältnisse als Eingangsdaten für Mischungsmodelle die Beitrage der einzelnen Substrate zum Ausgangsmaterial der Bodenbildung zu identifizieren und quantifizieren und c) die räumliche Verteilung von Blei (Pb) in Deckschichten, die über Bleierzgänge gewandert sind, untersucht, die Transportweite des erzbürtigen Bleis berechnet und die kontrollierenden Faktoren der Transportweite bestimmt werden. Sechs Transekte im südöstlichen Rheinischen Schiefergebirge, einschließlich der durch periglaziale Solifluktion entwickelten Böden, wurden untersucht. Die bodenkundliche Geländeaufnahme erfolgte nach AG Boden (2005). O, A, B und C-Horizontproben wurden auf ihre Spurenelementgehalte und teilweise auf ihre 206Pb/207Pb-Isotopenverhältnisse analysiert. Die steuernden Faktoren der Verteilung und Eigenschaften periglazialer Deckschichten sind neben der Petrographie, Reliefeigenschaften wie Exposition, Hangneigung, Hangposition und Krümmung. Die Reliefanalyse zeigt geringmächtige Deckschichten in divergenten, konvexen Hangbereichen bei gleichzeitig hohem Skelettgehalt. In konvergent, konkaven Hangbereichen nimmt die Deckschichtenmächtigkeit deutlich zu, bei gleichzeitig zunehmendem Lösslehm- und abnehmendem Skelettgehalt. Abhängig von den Reliefeigenschaften und -positionen reichen die ausgeprägten Bodentypen von sauren Braunerden bis hin zu Pseudogley-Parabraunerden. Des Weiteren kommen holozäne Kolluvien in eher untypischen Reliefpositionen wie langgestreckten, kaum geneigten Hangbereichen oder Mittelhangbereichen vor. Außer für Pb bewegen sich die Spurenelementgehalte im Rahmen niedriger Hintergrundgehalte. Die Pb-Gehalte liegen zwischen 20-135 mg kg-1. Abnehmende Spurenelementgehalte und Isotopensignaturen (206Pb/207Pb-Isotopenverhältnisse) von Pb zeigen, dass nahezu kein Pb aus atmosphärischen Depositionen in die B-Horizonte verlagert wurde. Eine Hauptkomponentenanalyse (PCA) der Spurenelementgehalte hat vier Hauptsubstratquellen der untersuchten B-Horizonte identifiziert (Tonschiefer, Löss, Laacher-See-Tephra [LST] und lokale Pb-Erzgänge). Mittels 3-Komponenten-Mischungsmodell, das Tonschiefer, Löss und LST einschloss, konnten, bis auf 10 Ausreißer, die Spurenelementgehalte aller 120 B-Horizontproben erklärt werden. Der Massenbeitrag des Pb-Erzes zur Substratmischung liegt bei <0,1%. Die räumliche Pb-Verteilung zeigt Bereiche lokaler Pb-Gehaltsmaxima hangaufwärtiger Pb-Erzgänge. Mittels eines 206Pb/207Pb-Isotopenverhältnis-Mischungsmodells konnten 14 Bereiche erhöhter lokaler Pb-Gehaltsmaxima ausgewiesen werden, die 76-100% erzbürtigen Bleis enthalten. Mit Hilfe eines Geographischen Informationssystems wurden die Transportweiten des erzbürtigen Bleis mit 30 bis 110 m bestimmt. Die steuerenden Faktoren der Transportweite sind dabei die Schluffkonzentration und die Vertikalkrümmung. Diese Untersuchung zeigt, dass Reliefeigenschaften und Reliefposition einen entscheidenden Einfluss auf die Ausprägung der Deckschichten und Böden im europäischen Mittelgebirgsbereich haben. Mischungsmodelle in Kombination mit Spurenelementanalysen und Isotopenverhältnissen stellen ein wichtiges Werkzeug zur Bestimmung der Beiträge der einzelnen Glieder in Bodensubstratmischungen dar. Außerdem können lokale Bleierzgänge die natürlichen Pb-Gehalte in Böden, entwickelt in periglazialen Deckschichten der letzten Vereisungsphase (Würm), bis über 100 m Entfernung erhöhen.

Ein inverses Rückstreuproblem der elektrischen Impedanztomographie

Die vorliegende Arbeit untersucht das inverse Hindernisproblem der zweidimensionalen elektrischen Impedanztomographie (EIT) mit Rückstreudaten. Wir präsentieren und analysieren das mathematische Modell für Rückstreudaten, diskutieren das inverse Problem für einen einzelnen isolierenden oder perfekt leitenden Einschluss und stellen zwei Rekonstruktionsverfahren für das inverse Hindernisproblem mit Rückstreudaten vor. Ziel des inversen Hindernisproblems der EIT ist es, Inhomogenitäten (sogenannte Einschlüsse) der elektrischen Leitfähigkeit eines Körpers aus Strom-Spannungs-Messungen an der Körperoberfläche zu identifizieren. Für die Messung von Rückstreudaten ist dafür nur ein Paar aus an der Körperoberfläche nahe zueinander angebrachten Elektroden nötig, das zur Datenerfassung auf der Oberfläche entlang bewegt wird. Wir stellen ein mathematisches Modell für Rückstreudaten vor und zeigen, dass Rückstreudaten die Randwerte einer außerhalb der Einschlüsse holomorphen Funktion sind. Auf dieser Grundlage entwickeln wir das Konzept des konvexen Rückstreuträgers: Der konvexe Rückstreuträger ist eine Teilmenge der konvexen Hülle der Einschlüsse und kann daher zu deren Auffindung dienen. Wir stellen einen Algorithmus zur Berechnung des konvexen Rückstreuträgers vor und demonstrieren ihn an numerischen Beispielen. Ferner zeigen wir, dass ein einzelner isolierender Einschluss anhand seiner Rückstreudaten eindeutig identifizierbar ist. Der Beweis dazu beruht auf dem Riemann'schen Abbildungssatz für zweifach zusammenhängende Gebiete und dient als Grundlage für einen Rekonstruktionsalgorithmus, dessen Leistungsfähigkeit wir an verschiedenen Beispielen demonstrieren. Ein perfekt leitender Einschluss ist hingegen nicht immer aus seinen Rückstreudaten rekonstruierbar. Wir diskutieren, in welchen Fällen die eindeutige Identifizierung fehlschlägt und zeigen Beispiele für unterschiedliche perfekt leitende Einschlüsse mit gleichen Rückstreudaten.

Packing a trunk : a physically based approach with full motional freedom

Geometric packing problems may be formulated mathematically as constrained optimization problems. But finding a good solution is a challenging task. The more complicated the geometry of the container or the objects to be packed, the more complex the non-penetration constraints become. In this work we propose the use of a physics engine that simulates a system of colliding rigid bodies. It is a tool to resolve interpenetration conflicts and to optimize configurations locally. We develop an efficient and easy-to-implement physics engine that is specialized for collision detection and contact handling. In succession of the development of this engine a number of novel algorithms for distance calculation and intersection volume were designed and imple- mented, which are presented in this work. They are highly specialized to pro- vide fast responses for cuboids and triangles as input geometry whereas the concepts they are based on can easily be extended to other convex shapes. Especially noteworthy in this context is our ε-distance algorithm - a novel application that is not only very robust and fast but also compact in its im- plementation. Several state-of-the-art third party implementations are being presented and we show that our implementations beat them in runtime and robustness. The packing algorithm that lies on top of the physics engine is a Monte Carlo based approach implemented for packing cuboids into a container described by a triangle soup. We give an implementation for the SAE J1100 variant of the trunk packing problem. We compare this implementation to several established approaches and we show that it gives better results in faster time than these existing implementations.

Computer simulation methods to study interfacial tensions : from the Ising model to colloidal crystals

In condensed matter systems, the interfacial tension plays a central role for a multitude of phenomena. It is the driving force for nucleation processes, determines the shape and structure of crystalline structures and is important for industrial applications. Despite its importance, the interfacial tension is hard to determine in experiments and also in computer simulations. While for liquid-vapor interfacial tensions there exist sophisticated simulation methods to compute the interfacial tension, current methods for solid-liquid interfaces produce unsatisfactory results.rnrnAs a first approach to this topic, the influence of the interfacial tension on nuclei is studied within the three-dimensional Ising model. This model is well suited because despite its simplicity, one can learn much about nucleation of crystalline nuclei. Below the so-called roughening temperature, nuclei in the Ising model are not spherical anymore but become cubic because of the anisotropy of the interfacial tension. This is similar to crystalline nuclei, which are in general not spherical but more like a convex polyhedron with flat facets on the surface. In this context, the problem of distinguishing between the two bulk phases in the vicinity of the diffuse droplet surface is addressed. A new definition is found which correctly determines the volume of a droplet in a given configuration if compared to the volume predicted by simple macroscopic assumptions.rnrnTo compute the interfacial tension of solid-liquid interfaces, a new Monte Carlo method called ensemble switch method'' is presented which allows to compute the interfacial tension of liquid-vapor interfaces as well as solid-liquid interfaces with great accuracy. In the past, the dependence of the interfacial tension on the finite size and shape of the simulation box has often been neglected although there is a nontrivial dependence on the box dimensions. As a consequence, one needs to systematically increase the box size and extrapolate to infinite volume in order to accurately predict the interfacial tension. Therefore, a thorough finite-size scaling analysis is established in this thesis. Logarithmic corrections to the finite-size scaling are motivated and identified, which are of leading order and therefore must not be neglected. The astounding feature of these logarithmic corrections is that they do not depend at all on the model under consideration. Using the ensemble switch method, the validity of a finite-size scaling ansatz containing the aforementioned logarithmic corrections is carefully tested and confirmed. Combining the finite-size scaling theory with the ensemble switch method, the interfacial tension of several model systems, ranging from the Ising model to colloidal systems, is computed with great accuracy.