Functional equations of polylogarithms in motivic cohomology

For an infinite field F, we study the integral relationship between the Bloch group B_2(F) and the higher Chow group CH^2(F,3) by proving some relations corresponding to the functional equations of the dilogarithm. As a second result, the groups involved in Suslin’s exact sequence 0 → Tor^1(F^× ,F^×)∼ → CH^2(F,3) → B_2(F) → 0 are identified with homology groups of the cycle complex Z^2(F,•) computing Bloch’s higher Chow groups. Using these results, we give explicit cycles in motivic cohomology generating the integral motivic cohomology groups of some specific number fields and determine whether a given cycle in the Chow group already lives in one of the other groups of Suslin’s sequence. In principle, this enables us to find a presentation of the codimension two Chow group of an arbitrary number field. Finally, we also prove some relations in the higher Chow groups of codimension three modulo 2-torsion coming from relations in the higher Bloch group B_3(F) modulo 2-torsion. Further, we can prove a series of relations in CH^ 3(Q(zeta_p),5) for a primitive pth root of unity zeta_p.

Supramolecular order and dynamics of functional materials studied by solid state NMR

The goal of this thesis was the investigation of the structure, conformation, supramolecular order and molecular dynamics of different classes of functional materials (phthalocyanine, perylene and hexa-peri-hexabenzocoronene derivatives and mixtures of those), all having planar aromatic cores modified with various types of alkyl chains. The planar aromatic systems are known to stack in the solid and the liquid-crystalline state due to p-p interactions forming columnar superstructures with high one-dimensional charge carrier mobility and potential application in photovoltaic devices. The different functionalities attached to the aromatic cores significantly influence the behavior of these systems allowing the experimentalists to modify the structures to fine-tune the desired thermotropic properties or charge carrier mobility. The aim of the presented studies was to understand the interplay between the driving forces causing self-assembly by relating the structural and dynamic information about the investigated systems. The supramolecular organization is investigated by applying 1H solid state NMR recoupling techniques. The results are related with DSC and X-ray scattering data. Detailed information about the site-specific molecular dynamics is gained by recording spinning sideband patterns using 1H-1H and 13C-1H solid state NMR recoupling techniques. The determined dipole-dipole coupling constants are then related with the coupling constants of the respective rigid pairs, thus providing local dynamic order parameters for the respective moieties. The investigations presented reveal that in the crystalline state the preferred arrangement in the columnar stack of discotic molecules modified with alkyl chains is tilted. This leads to characteristic differences in the 1H chemical shifts of otherwise chemically equivalent protons. Introducing branches and increasing the length of the alkyl chains results in lower mesophase transitions and disordered columnar stacks. In the liquid-crystalline state some of the discs lose the tilted orientation, others do not, but all start a rapid rotation about the columnar axis.

Finite-element discretization of 3D energy-transport equations for semiconductors

In this thesis a mathematical model was derived that describes the charge and energy transport in semiconductor devices like transistors. Moreover, numerical simulations of these physical processes are performed. In order to accomplish this, methods of theoretical physics, functional analysis, numerical mathematics and computer programming are applied. After an introduction to the status quo of semiconductor device simulation methods and a brief review of historical facts up to now, the attention is shifted to the construction of a model, which serves as the basis of the subsequent derivations in the thesis. Thereby the starting point is an important equation of the theory of dilute gases. From this equation the model equations are derived and specified by means of a series expansion method. This is done in a multi-stage derivation process, which is mainly taken from a scientific paper and which does not constitute the focus of this thesis. In the following phase we specify the mathematical setting and make precise the model assumptions. Thereby we make use of methods of functional analysis. Since the equations we deal with are coupled, we are concerned with a nonstandard problem. In contrary, the theory of scalar elliptic equations is established meanwhile. Subsequently, we are preoccupied with the numerical discretization of the equations. A special finite-element method is used for the discretization. This special approach has to be done in order to make the numerical results appropriate for practical application. By a series of transformations from the discrete model we derive a system of algebraic equations that are eligible for numerical evaluation. Using self-made computer programs we solve the equations to get approximate solutions. These programs are based on new and specialized iteration procedures that are developed and thoroughly tested within the frame of this research work. Due to their importance and their novel status, they are explained and demonstrated in detail. We compare these new iterations with a standard method that is complemented by a feature to fit in the current context. A further innovation is the computation of solutions in three-dimensional domains, which are still rare. Special attention is paid to applicability of the 3D simulation tools. The programs are designed to have justifiable working complexity. The simulation results of some models of contemporary semiconductor devices are shown and detailed comments on the results are given. Eventually, we make a prospect on future development and enhancements of the models and of the algorithms that we used.

Assessing the functional structure of molecular transporters by EPR spectroscopy

In this thesis, the self-assembled functional structure of a broad range of amphiphilic molecular transporters is studied. By employing paramagnetic probe molecules and ions, continuous-wave and pulse electron paramagnetic resonance spectroscopy reveal information about the local structure of these materials from the perspective of incorporated guest molecules. First, the transport function of human serum albumin for fatty acids is in the focus. As suggested by the crystal structure, the anchor points for the fatty acids are distributed asymmetrically in the protein. In contrast to the crystallographic findings, a remarkably symmetric entry point distribution of the fatty acid binding channels is found, which may facilitate the uptake and release of the guest molecules. Further, the metal binding of 1,2,3-triazole modified star-shaped cholic acid oligomers is studied. These biomimetic molecules are able to include and transport molecules in solvents of different polarity. A pre-arrangement of the triazole groups induces a strong chelate-like binding and close contact between guest molecule and metal ion. In absence of a preordering, each triazole moiety acts as a single entity and the binding affinity for metal ions is strongly decreased. Hydrogels based on N-isopropylacrylamide phase separate from water above a certain temperature. The macroscopic thermal collapse of these hydrogels is utilized as a tool for dynamic nuclear polarization. It is shown that a radical-free hyperpolarized solution can be achieved with a spin-labeled gel as separable matrix. On the nanoscale, these hydrogels form static heterogeneities in both structure and function. Collapsed regions protect the spin probes from a chemical decay while open, water-swollen regions act as catalytic centers. Similarly, thermoresponsive dendronized polymers form structural heterogeneities, which are, however, highly dynamic. At the critical temperature, they trigger the aggregation of the polymer into mesoglobules. The dehydration of these aggregates is a molecularly controlled non-equilibrium process that is facilitated by a hydrophobic dendritic core. Further, a slow heating rate results in a kinetically entrapped non-equilibrium state due to the formation of an impermeable dense polymeric layer at the periphery of the mesoglobule.

A novel functional renormalization group framework for gauge theories and gravity

In this thesis we develop further the functional renormalization group (RG) approach to quantum field theory (QFT) based on the effective average action (EAA) and on the exact flow equation that it satisfies. The EAA is a generalization of the standard effective action that interpolates smoothly between the bare action for krightarrowinfty and the standard effective action rnfor krightarrow0. In this way, the problem of performing the functional integral is converted into the problem of integrating the exact flow of the EAA from the UV to the IR. The EAA formalism deals naturally with several different aspects of a QFT. One aspect is related to the discovery of non-Gaussian fixed points of the RG flow that can be used to construct continuum limits. In particular, the EAA framework is a useful setting to search for Asymptotically Safe theories, i.e. theories valid up to arbitrarily high energies. A second aspect in which the EAA reveals its usefulness are non-perturbative calculations. In fact, the exact flow that it satisfies is a valuable starting point for devising new approximation schemes. In the first part of this thesis we review and extend the formalism, in particular we derive the exact RG flow equation for the EAA and the related hierarchy of coupled flow equations for the proper-vertices. We show how standard perturbation theory emerges as a particular way to iteratively solve the flow equation, if the starting point is the bare action. Next, we explore both technical and conceptual issues by means of three different applications of the formalism, to QED, to general non-linear sigma models (NLsigmaM) and to matter fields on curved spacetimes. In the main part of this thesis we construct the EAA for non-abelian gauge theories and for quantum Einstein gravity (QEG), using the background field method to implement the coarse-graining procedure in a gauge invariant way. We propose a new truncation scheme where the EAA is expanded in powers of the curvature or field strength. Crucial to the practical use of this expansion is the development of new techniques to manage functional traces such as the algorithm proposed in this thesis. This allows to project the flow of all terms in the EAA which are analytic in the fields. As an application we show how the low energy effective action for quantum gravity emerges as the result of integrating the RG flow. In any treatment of theories with local symmetries that introduces a reference scale, the question of preserving gauge invariance along the flow emerges as predominant. In the EAA framework this problem is dealt with the use of the background field formalism. This comes at the cost of enlarging the theory space where the EAA lives to the space of functionals of both fluctuation and background fields. In this thesis, we study how the identities dictated by the symmetries are modified by the introduction of the cutoff and we study so called bimetric truncations of the EAA that contain both fluctuation and background couplings. In particular, we confirm the existence of a non-Gaussian fixed point for QEG, that is at the heart of the Asymptotic Safety scenario in quantum gravity; in the enlarged bimetric theory space where the running of the cosmological constant and of Newton's constant is influenced by fluctuation couplings.

Synthetic routes toward functional block copolymers and bioconjugates via RAFT polymerization

Synthetic Routes toward Functional Block Copolymers and Bioconjugates via RAFT PolymerizationrnSynthesewege für funktionelle Blockcopolymere und Biohybride über RAFT PolymerisationrnDissertation von Dipl.-Chem. Kerstin T. WissrnIm Rahmen dieser Arbeit wurden effiziente Methoden für die Funktionalisierung beider Polymerkettenenden für Polymer- und Bioanbindung von Polymeren entwickelt, die mittels „Reversible Addition-Fragmentation Chain Transfer“ (RAFT) Polymerisation hergestellt wurden. Zu diesem Zweck wurde ein Dithioester-basiertes Kettentransferagens (CTA) mit einer Aktivestereinheit in der R-Gruppe (Pentafluorphenyl-4-phenylthiocarbonylthio-4-cyanovaleriansäureester, kurz PFP-CTA) synthetisiert und seine Anwendung als universelles Werkzeug für die Funktionalisierung der -Endgruppe demonstriert. Zum Einen wurde gezeigt, wie dieser PFP-CTA als Vorläufer für die Synthese anderer funktioneller CTAs durch einfache Aminolyse des Aktivesters genutzt werden kann und somit den synthetischen Aufwand, der üblicherweise mit der Entwicklung neuer CTAs verbunden ist, reduzieren kann. Zum Anderen konnte der PFP-CTA für die Synthese verschiedener Poly(methacrylate) mit enger Molekulargewichtsverteilung und wohl definierter reaktiver -Endgruppe verwendet werden. Dieses Kettenende konnte dann erfolgreich mit verschiedenen primären Aminen wie Propargylamin, 1-Azido-3-aminopropan und Ethylendiamin oder direkt mit den Amin-Endgruppen verschiedener Peptide umgesetzt werden.rnAus der Reaktion des PFP-CTAs mit Propargylamin wurde ein Alkin-CTA erhalten, der sich als effizientes Werkzeug für die RAFT Polymerisation verschiedener Methacrylate erwiesen hat. Der Einbau der Alkin-Funktion am -Kettenende wurde mittels 1H und 13C NMR Spektroskopie sowie MALDI TOF Massenspektroskopie bestätigt. Als Modelreaktion wurde die Kopplung eines solchen alkin-terminierten Poly(di(ethylenglykol)methylethermethacrylates) (PDEGMEMA) mit azid-terminiertem Poly(tert-butylmethacrylat), das mittels Umsetzung einer Aktivester-Endgruppe erhalten wurde, als kupferkatalysierte Azid-Alkin-Cycloaddition (CuAAC) durchgeführt. Die Aufarbeitung des resultierenden Diblockcopolymers durch Fällen ermöglichte die vollständige Abtrennung des Polymerblocks 1, der im Überschuss eingesetzt wurde. Darüber hinaus blieb nur ein sehr kleiner Anteil (< 2 Gew.-%) nicht umgesetzten Polymerblocks 2, was eine erfolgreiche Polymeranbindung und die Effizienz der Endgruppen-Funktionalisierung ausgehend von der Aktivester--Endgruppe belegt.rnDie direkte Reaktion von stimuli-responsiven Polymeren mit Pentafluorphenyl(PFP)ester-Endgruppen, namentlich PDEGMEMA und Poly(oligo(ethylenglykol)methylethermethacrylat), mit kollagen-ähnlichen Peptiden ergab wohl definierte Polymer-Peptid-Diblockcopolymere und Polymer-Peptid-Polymer-Triblockcopolymer unter nahezu quantitativer Umsetzung der Endgruppen. Alle Produkte konnten vollständig von nicht umgesetztem Überschuss des Homopolymers befreit werden. In Analogie zu natürlichem Kollagen und dem nicht funktionalisierten kollagen-ähnlichen Peptid bilden die PDEGMEMA-basierten, entschützten Hybridcopolymere Trimere mit kollagen-ähnlichen Triple-Helices in kalter wässriger Lösung, was mittels Zirkular-Dichroismus-Spektroskopie (CD) nachgewiesen werden konnte. Temperaturabhängige CD-Spektroskopie, Trübungsmessungen und dynamische Lichtstreuung deuteten darauf hin, dass sie bei höheren Temperaturen doppelt stimuli-responsive Überstrukturen bilden, die mindestens zwei konformative Übergänge beim Aufheizen durchlaufen. Einer dieser Übergänge wird durch den hydrophoben Kollaps des Polymerblocks induziert, der andere durch Entfalten der kollagen-ähnlichen Triple-Helices.rnAls Ausweitung dieser synthetischen Strategie wurde homotelecheles PDEGMEMA mit zwei PFP-Esterendgruppen dargestellt, wozu der PFP-CTA für die Funktionalisierung der -Endgruppe und die radikalische Substitution des Dithioesters durch Behandlung mit einem Überschuss eines funktionellen AIBN-Derivates für die Funktionalisierung der -Endgruppe ausgenutzt wurde. Die Umsetzung der beiden reaktiven Kettenenden mit dem N-Terminus eines Peptidblocks ergab ein Peptid-Polymer-Peptid Triblockcopolymer.rnSchließlich konnten die anorganisch-organischen Hybridmaterialien PMSSQ-Poly(2,2-diethoxyethylacrylat) (PMSSQ-PDEEA) und PMSSQ-Poly(1,3-dioxolan-2-ylmethylacrylat) (PMSSQ-PDMA) für die Herstellung robuster, peptid-reaktiver Oberflächen durch Spin Coaten und thermisch induziertes Vernetzen angewendet werden. Nach saurem Entschützen der Acetalgruppen in diesen Filmen konnten die resultierenden Aldehydgruppen durch einfaches Eintauchen in eine Lösung mit einer Auswahl von Aminen und Hydroxylaminen umgesetzt werden, wodurch die Oberflächenhydrophilie modifiziert werden konnte. Darüber hinaus konnten auf Basis der unterschiedlichen Stabilität der zwei hier verglichenen Acetalgruppen Entschützungsprotokolle für die exklusive Entschützung der Diethylacetale in PMSSQ-PDEEA und deren Umsetzung ohne Entschützung der zyklischen Ethylenacetale in PMSSQ-PDMA entwickelt werden, die die Herstellung multifunktioneller Oberflächenbeschichtungen z.B. für die Proteinimmobilisierung ermöglichen.