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.

Novel simulation methods for Coulomb and hydrodynamic interactions

This thesis presents new methods to simulate systems with hydrodynamic and electrostatic interactions. Part 1 is devoted to computer simulations of Brownian particles with hydrodynamic interactions. The main influence of the solvent on the dynamics of Brownian particles is that it mediates hydrodynamic interactions. In the method, this is simulated by numerical solution of the Navier--Stokes equation on a lattice. To this end, the Lattice--Boltzmann method is used, namely its D3Q19 version. This model is capable to simulate compressible flow. It gives us the advantage to treat dense systems, in particular away from thermal equilibrium. The Lattice--Boltzmann equation is coupled to the particles via a friction force. In addition to this force, acting on {it point} particles, we construct another coupling force, which comes from the pressure tensor. The coupling is purely local, i.~e. the algorithm scales linearly with the total number of particles. In order to be able to map the physical properties of the Lattice--Boltzmann fluid onto a Molecular Dynamics (MD) fluid, the case of an almost incompressible flow is considered. The Fluctuation--Dissipation theorem for the hybrid coupling is analyzed, and a geometric interpretation of the friction coefficient in terms of a Stokes radius is given. Part 2 is devoted to the simulation of charged particles. We present a novel method for obtaining Coulomb interactions as the potential of mean force between charges which are dynamically coupled to a local electromagnetic field. This algorithm scales linearly, too. We focus on the Molecular Dynamics version of the method and show that it is intimately related to the Car--Parrinello approach, while being equivalent to solving Maxwell's equations with freely adjustable speed of light. The Lagrangian formulation of the coupled particles--fields system is derived. The quasi--Hamiltonian dynamics of the system is studied in great detail. For implementation on the computer, the equations of motion are discretized with respect to both space and time. The discretization of the electromagnetic fields on a lattice, as well as the interpolation of the particle charges on the lattice is given. The algorithm is as local as possible: Only nearest neighbors sites of the lattice are interacting with a charged particle. Unphysical self--energies arise as a result of the lattice interpolation of charges, and are corrected by a subtraction scheme based on the exact lattice Green's function. The method allows easy parallelization using standard domain decomposition. Some benchmarking results of the algorithm are presented and discussed.

Use of computer algebra in the calculation of Feynman diagrams

The increasing precision of current and future experiments in high-energy physics requires a likewise increase in the accuracy of the calculation of theoretical predictions, in order to find evidence for possible deviations of the generally accepted Standard Model of elementary particles and interactions. Calculating the experimentally measurable cross sections of scattering and decay processes to a higher accuracy directly translates into including higher order radiative corrections in the calculation. The large number of particles and interactions in the full Standard Model results in an exponentially growing number of Feynman diagrams contributing to any given process in higher orders. Additionally, the appearance of multiple independent mass scales makes even the calculation of single diagrams non-trivial. For over two decades now, the only way to cope with these issues has been to rely on the assistance of computers. The aim of the xloops project is to provide the necessary tools to automate the calculation procedures as far as possible, including the generation of the contributing diagrams and the evaluation of the resulting Feynman integrals. The latter is based on the techniques developed in Mainz for solving one- and two-loop diagrams in a general and systematic way using parallel/orthogonal space methods. These techniques involve a considerable amount of symbolic computations. During the development of xloops it was found that conventional computer algebra systems were not a suitable implementation environment. For this reason, a new system called GiNaC has been created, which allows the development of large-scale symbolic applications in an object-oriented fashion within the C++ programming language. This system, which is now also in use for other projects besides xloops, is the main focus of this thesis. The implementation of GiNaC as a C++ library sets it apart from other algebraic systems. Our results prove that a highly efficient symbolic manipulator can be designed in an object-oriented way, and that having a very fine granularity of objects is also feasible. The xloops-related parts of this work consist of a new implementation, based on GiNaC, of functions for calculating one-loop Feynman integrals that already existed in the original xloops program, as well as the addition of supplementary modules belonging to the interface between the library of integral functions and the diagram generator.

The swelling behaviour of polyelectrolyte networks

Being basic ingredients of numerous daily-life products with significant industrial importance as well as basic building blocks for biomaterials, charged hydrogels continue to pose a series of unanswered challenges for scientists even after decades of practical applications and intensive research efforts. Despite a rather simple internal structure it is mainly the unique combination of short- and long-range forces which render scientific investigations of their characteristic properties to be quite difficult. Hence early on computer simulations were used to link analytical theory and empirical experiments, bridging the gap between the simplifying assumptions of the models and the complexity of real world measurements. Due to the immense numerical effort, even for high performance supercomputers, system sizes and time scales were rather restricted until recently, whereas it only now has become possible to also simulate a network of charged macromolecules. This is the topic of the presented thesis which investigates one of the fundamental and at the same time highly fascinating phenomenon of polymer research: The swelling behaviour of polyelectrolyte networks. For this an extensible simulation package for the research on soft matter systems, ESPResSo for short, was created which puts a particular emphasis on mesoscopic bead-spring-models of complex systems. Highly efficient algorithms and a consistent parallelization reduced the necessary computation time for solving equations of motion even in case of long-ranged electrostatics and large number of particles, allowing to tackle even expensive calculations and applications. Nevertheless, the program has a modular and simple structure, enabling a continuous process of adding new potentials, interactions, degrees of freedom, ensembles, and integrators, while staying easily accessible for newcomers due to a Tcl-script steering level controlling the C-implemented simulation core. Numerous analysis routines provide means to investigate system properties and observables on-the-fly. Even though analytical theories agreed on the modeling of networks in the past years, our numerical MD-simulations show that even in case of simple model systems fundamental theoretical assumptions no longer apply except for a small parameter regime, prohibiting correct predictions of observables. Applying a "microscopic" analysis of the isolated contributions of individual system components, one of the particular strengths of computer simulations, it was then possible to describe the behaviour of charged polymer networks at swelling equilibrium in good solvent and close to the Theta-point by introducing appropriate model modifications. This became possible by enhancing known simple scaling arguments with components deemed crucial in our detailed study, through which a generalized model could be constructed. Herewith an agreement of the final system volume of swollen polyelectrolyte gels with results of computer simulations could be shown successfully over the entire investigated range of parameters, for different network sizes, charge fractions, and interaction strengths. In addition, the "cell under tension" was presented as a self-regulating approach for predicting the amount of swelling based on the used system parameters only. Without the need for measured observables as input, minimizing the free energy alone already allows to determine the the equilibrium behaviour. In poor solvent the shape of the network chains changes considerably, as now their hydrophobicity counteracts the repulsion of like-wise charged monomers and pursues collapsing the polyelectrolytes. Depending on the chosen parameters a fragile balance emerges, giving rise to fascinating geometrical structures such as the so-called pear-necklaces. This behaviour, known from single chain polyelectrolytes under similar environmental conditions and also theoretically predicted, could be detected for the first time for networks as well. An analysis of the total structure factors confirmed first evidences for the existence of such structures found in experimental results.

Das neue digitale ‚Shakespeare-Bildarchiv Oppel-Hammerschmidt‘ an der Universitätsbibliothek Mainz : Reden und Beiträge anlässlich seiner öffentlichen Vorstellung am 17. November 2008 an der Johannes Gutenberg-Universität Mainz

On the basis of illustrations of Shakespeare's Hamlet, the new digital 'Oppel-Hammerschmidt Shakespeare Illustration Archive' at the Mainz University Library - together with a lavishly-constructed and multiply-linked Web interface version - was presented to the public on 17 November 2008. This e-book, edited by Andreas Anderhub and Hildegard Hammerschmidt-Hummel, contains the speeches and presentations given on the occasion of the opening ceremony of the electronic archive. The collection of the new archive, published here for the first time, holds about 3,500 images and is part of the only Shakespeare illustration archive in the world. The Shakespeare Illustration Archive was founded in 1946 by the internationally acclaimed Shakespeare and Goethe scholar, Prof. Horst Oppel. This part of the archive was donated to the Mainz University Library on condition that its holdings be digitalised and made available to the public. The collection has been named 'The Oppel-Hammerschmidt Shakespeare Illustration Archive' in accordance with the terms of the Agreement of Donation of 9, 15, and 16 September 2005, and honouring the 16 March 1988 Delegation of Authority and Declaration of Intent by Frau Ingeborg Oppel, Prof. Oppel's widow and legal assignee. Vice-President Prof. Jürgen Oldenstein opened the proceedings by noting that 2008 had been a good year for international Shakespeare scholarship. For, in London, the site of the 'Theatre' in Shoreditch, where Shakespeare's company performed, had been unearthed, and in Mainz the Shakespeare Archive had gone online with thousands of illustrations. The Dean of the Faculty of Philosophy and Philology, Prof. Mechthild Dreyer, who mentioned that she herself had long been successfully employing interdisciplinary research methods, took particular pleasure in the transdisciplinary approach to research resolutely pursued by Prof. Hammerschmidt-Hummel. Prof. Clemens Zintzen (Cologne), former President of the Mainz Academy of Literature and Sciences, recalled highlights from the more than sixty-year-long history of the Shakespeare Illustration Archive. Prof. Kurt Otten (Heidelberg and Cambridge) drew an impressive portrait of Horst Oppel's personality as an academic and praised his influential books on Goethe and Shakespeare. He pointed out that Oppel's Shakespeare Illustration Archive, the basis for many a dissertation, had enjoyed great popularity around the world. Prof. Otten also delineated the academic career of Prof. Hammerschmidt-Hummel and her new findings regarding Shakespeare's time, life and work. Prof. Rüdiger Ahrens OBE (Würzburg) drew attention to Prof. Hammerschmidt-Hummel's research results, directly or indirectly arising out of her work on the Shakespeare Illustration Archive. This research had centred on proving the authenticity of four visual representations of Shakespeare (the Chandos and Flower portraits, the Davenant bust and the Darmstadt Shakespeare death mask); solving the mystery around Shakespeare's 'Dark Lady'; and establishing the dramatist's Catholic religion. Prof. Hammerschmidt-Hummel reported on her 'Shakespeare Illustration' project, describing the nature, dimensions and significance of the Archive's pictorial material, which relates to all of Shakespeare's plays and stretches over five centuries. She explained that the digital 'Oppel-Hammerschmidt Illustration Archive' was an addition to the three-volume edition she had compiled, authored and edited for publication in 2003. Unlike the print version, however, the digital collection had only been partly editorially prepared. It represented source material and a basis for further work. Hammerschmidt-Hummel expressed her thanks to the Head of the Central University Library, Dr Andreas Anderhub, for his untiring commitment. After the initial donation had been made, he had entered enthusiastically into setting up the necessary contacts, getting all the work underway, and clearing the legal hurdles. Hammerschmidt-Hummel was especially grateful to University of Mainz librarian Heike Geisel, who had worked for nearly five years to carry out the large-scale digitalization of a total of 8,800 items. Frau Geisel was also extremely resourceful in devising ways of making the collection yield even more, e.g. by classifying and cross-linking the data, assembling clusters of individual topics that lend themselves to research, and (in collaboration with the art historian Dr Klaus Weber) making the archive's index of artists compatible with the data-bank of artists held by the University of Mainz Institute of Art History. In addition, she compiled an extremely helpful 'users' guide' to the new digital collection. Frau Geisel had enjoyed invaluable support from Dr Annette Holzapfel-Pschorn, the leading academic in the Central IT Department at the University, who set up an intelligent, most impressive Web interface using the latest application technologies. Frau Geisel and Dr Holzapfel-Pschorn were highly praised for their convincing demonstration, using illustrations to Hamlet, of how to access this well-devised and exceptionally user-friendly Web version. For legal reasons, Prof. Hammerschmidt-Hummel pointed out, the collection could not be released for open access on the internet. The media - as Dr Anderhub stressed in his foreword - had shown great interest in the new digital collection of thousands of Shakespearean illustrations (cf. Benjamin Cor's TV feature in "Tagesthemen", 17 November 2008, presented by Tom Buhrow). The ‘Oppel-Hammerschmidt Shakespeare Illustration Archive’ should also meet with particular interest not only among academic specialists, but also among the performers of the arts and persons active in the cultural realm in general, as well as theatre and film directors, literary managers, teachers, and countless Shakespeare enthusiasts.