Generalized soluble groups of finite co-central rank

Eine Gruppe G hat endlichen Prüferrang (bzw. Ko-zentralrang) kleiner gleich r, wenn für jede endlich erzeugte Gruppe H gilt: H (bzw. H modulo seinem Zentrum) ist r-erzeugbar. In der vorliegenden Arbeit werden, soweit möglich, die bekannten Sätze über Gruppen von endlichem Prüferrang (kurz X-Gruppen), auf die wesentlich größere Klasse der Gruppen mit endlichem Ko-zentralrang (kurz R-Gruppen) verallgemeinert.Für lokal nilpotente R-Gruppen, welche torsionsfrei oder p-Gruppen sind, wird gezeigt, dass die Zentrumsfaktorgruppe eine X-Gruppe sein muss. Es folgt, dass Hyperzentralität und lokale Nilpotenz für R-Gruppen identische Bediungungen sind. Analog hierzu sind R-Gruppen genau dann lokal auflösbar, wenn sie hyperabelsch sind. Zentral für die Strukturtheorie hyperabelscher R-Gruppen ist die Tatsache, dass solche Gruppen eine aufsteigende Normalreihe abelscher X-Gruppen besitzen. Es wird eine Sylowtheorie für periodische hyperabelsche R-Gruppen entwickelt. Für torsionsfreie hyperabelsche R-Gruppen wird deren Auflösbarkeit bewiesen. Des weiteren sind lokal endliche R-Gruppen fast hyperabelsch. Für R-Gruppen fallen sehr große Gruppenklassen mit den fast hyperabelschen Gruppen zusammen. Hierzu wird der Begriff der Sektionsüberdeckung eingeführt und gezeigt, dass R-Gruppen mit fast hyperabelscher Sektionsüberdeckung fast hyperabelsch sind.

Nearrings and a construction of triply factorized groups

In this thesis a connection between triply factorised groups and nearrings is investigated. A group G is called triply factorised by its subgroups A, B, and M, if G = AM = BM = AB, where M is normal in G and the intersection of A and B with M is trivial. There is a well-known connection between triply factorised groups and radical rings. If the adjoint group of a radical ring operates on its additive group, the semidirect product of those two groups is triply factorised. On the other hand, if G = AM = BM = AB is a triply factorised group with abelian subgroups A, B, and M, G can be constructed from a suitable radical ring, if the intersection of A and B is trivial. In these triply factorised groups the normal subgroup M is always abelian. In this thesis the construction of triply factorised groups is generalised using nearrings instead of radical rings. Nearrings are a generalisation of rings in the sense that their additive groups need not be abelian and only one distributive law holds. Furthermore, it is shown that every triply factorised group G = AM = BM = AB can be constructed from a nearring if A and B intersect trivially. Moreover, the structure of nearrings is investigated in detail. Especially local nearrings are investigated, since they are important for the construction of triply factorised groups. Given an arbitrary p-group N, a method to construct a local nearring is presented, such that the triply factorised group constructed from this nearring contains N as a subgroup of the normal subgroup M. Finally all local nearrings with dihedral groups of units are classified. It turns out that these nearrings are always finite and their order does not exceed 16.

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.

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 $B(G)$.

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.

Hyperbranched polyether polyols as building blocks for complex macromolecular architectures

The present thesis deals with the development of new branched polymer architectures containing hyperbranched polyglycerol. Materials investigated include hyperbranched oligomers, hyperbranched polyglycerols containing functional initiator-cores at the focal point, well-defined linear-hyperbranched block copolymers and also negatively charged hyperbranched polyelectrolytes.rnHyperbranched oligoglycerols (DPn = 7 and 14) have been synthesized for the first time. The materials show narrow polydispersity (Mw/Mn ca. 1.45) and a very low content in cyclic homopolymers. 13C NMR evidences the dendritic structure of the oligomers and the DB could be calculated (44% and 52%). These new oligoglycerols were compared with the industrial products obtained by polycondensation which exhibit narrow polydispersity (Mw/Mn<1.3) butrnmultimodal distribution in SEC. Detailed 13C NMR and Maldi-ToF studies reveal the presence of branched units and cyclic compounds. In comparison, the hyperbranched oligoglycerols comprise a very low proportion of cyclic homopolymer which render them very interesting materials for biomedical applications for example.rnThe site isolation of the core moiety in dendritic structure offers intriguing potential with respect to peculiar electro-optical properties. Various initiator-cores (n-alkyl amines, UVabsorbing amines and benzophenone) for the ROMBP of glycidol have been tested. The bisglycidolized amine initiator-cores show the best control over the molecular weight and the molecular weight distribution. The photochemical analyses of the naphthalene containingrnhyperbranched polyglycerols show a slight red shift, a pronounced hypochromic effect (decrease of the intensity of the band) compared with the parent model compound and the formation of a relative compact structure. The benzophenone containing polymers adopt an open structure in polar solvents. The fluorescence measurements show a clear “dendritic effect” on the fluorescence intensities and the quantum yield of the encapsulated benzophenone.rnA convenient 3-step strategy has been developed for the preparation of well-defined amphiphilic, linear-hyperbranched block copolymers via hypergrafting. The procedure represents a combination of carbanionic polymerization with the alkoxide-based, controlled ring-opening multibranching polymerization of glycidol. Materials consisting of a polystyrene linear block and a hyperbranched polyglycerol block exhibit narrow polydispersity (1.01-1.02rnfor 5.4% to 27% wt. PG and 1.74 for 52% wt. PG) with a high grafting efficiency. The strategy was also extended to materials with a linear polyisoprene block.rnDetailed investigations of the solution properties of the block copolymers with linear polystyrene blocks show that block copolymer micelles are stabilized by the highly branched block. The morphology of the aggregates is depending on the solvent: in chloroform monodisperse spherical shape aggregates and in toluene ellipsoidal aggregates are formed. On graphite these aggregates show interesting features, giving promising potential applications with respect to the presence of a very dense, functional and stable hyperbranched block.rnThe bulk morphology of the linear-hyperbranched block copolymers has been investigated. The materials with a linear polyisoprene block only behave like complex liquids due to the low Tg and the disordered nature of both components. For the materials with polystyrene, only the sample with 27% wt. hyperbranched polyglycerol forms some domains showing lamellae.rnThe preparation of hyperbranched polyelectrolytes was achieved by post-modification of the hydroxyl groups via Michael addition of acrylonitrile, followed by hydrolysis. In aqueous solution materials form large aggregates with size depending on the pH value. After deposition on mica the structures observed by AFM show the coexistence of aggregates andrnunimers. For the low molecular weight sample (PG 520 g·mol-1) extended and highly ordered terrace structures were observed. Materials were also successfully employed for the fabrication of composite organic-inorganic multilayer thin films, using electrostatic layer-bylayer self-assembly coupled with chemical vapor deposition.

Context-dependent interpretation of the conjunction 'und' in different age-groups

In contrast to formal semantics, the conjunction and is nonsymmetrical in pragmatics. The events in Marc went to bed and fell asleep seem to have occurred chronologically although no explicit time reference is given. As the temporal interpretation appears to be weaker in Mia ate chocolate and drank milk, it seems that the kind and nature of events presented in a context influences the interpretation of the conjunction. This work focuses on contextual influences on the interpretation of the German conjunction und (‘and’). A variety of theoretic approaches are concerned with whether and contributes to the establishment of discourse coherence via pragmatic processes or whether the conjunction has complex semantic meaning. These approaches are discussed with respect to how they explain the temporal and additive interpretation of the conjunction and the role of context in the interpre-tation process. It turned out that most theoretic approaches do not consider the importance of different kinds of context in the interpretation process.rnIn experimental pragmatics there are currently only very few studies that investigate the inter-pretation of the conjunction. As there are no studies that investigate contextual influences on the interpretation of und systematically or investigate preschoolers interpretation of the con-junction, research questions such as How do (preschool) children interpret ‘und’? and Does the kind of events conjoined influence children’s and adults’ interpretation? are yet to be answered. Therefore, this dissertation systematically investigates how different types of context influence children’s interpretation of und. Three auditory comprehension studies were conducted in German. Of special interest was whether and how the order of events introduced in a context contributes to the temporal read-ing of the conjunction und. Results indicate that the interpretation of und is – at least in Ger-man – context-dependent: The conjunction is interpreted temporally more often when events that typical occur in a certain order are connected (typical contexts) compared to events with-out typical event order (neutral contexts). This suggests that the type of events conjoined in-fluences the interpretation process. Moreover, older children and adults interpret the conjunc-tion temporally more often than the younger cohorts if the conjoined events typically occur in a certain order. In neutral contexts, additive interpretations increase with age. 5-year-olds reject reversed order statements more often in typical contexts compared to neutral contexts. However, they have more difficulties with reversed order statements in typical contexts where they perform at chance level. This suggests that not only the type of event but also other age-dependent factors such as knowledge about scripts influence children’s performance. The type of event conjoined influences children’s and adults’ interpretation of the conjunction. There-fore, the influence of different event types and script knowledge on the interpretation process does not only have to be considered in future experimental studies on language acquisition and pragmatics but also in experimental pragmatics in general. In linguistic theories, context has to be given a central role and a commonly agreed definition of context that considers the consequences arising from different event types has to be agreed upon.

Periodic Higgs-de Rham flows and representations of algebraic fundamental groups

Let k := bar{F}_p for p > 2, W_n(k) := W(k)/p^n and X_n be a projective smooth W_n(k)-scheme which is W_{n+1}(k)-liftable. For all n > 1, we construct explicitly a functor, which we call the inverse Cartier functor, from a subcategory of Higgs bundles over X_n to a subcategory of flat Bundles over X_n. Then we introduce the notion of periodic Higgs-de Rham flows and show that a periodic Higgs-de Rham flow is equivalent to a Fontaine-Faltings module. Together with a p-adic analogue of Riemann-Hilbert correspondence established by Faltings, we obtain a coarse p-adic Simpson correspondence.