Group extensions with special properties

Given a group G and a G-module A, we show how to determine up to isomorphism the extensions E of A by G so that A embeds as smallest non-trivial term of the derived series or of the lower central series into E.Comment: 13 pages, 2 figure

The number of nilpotent semigroups of degree 3

A semigroup is \emph{nilpotent} of degree 3 if it has a zero, every product of 3 elements equals the zero, and some product of 2 elements is non-zero. It is part of the folklore of semigroup theory that almost all finite semigroups are nilpotent of degree 3. We give formulae for the number of nilpotent semigroups of degree 3 with $n\in\N$ elements up to equality, isomorphism, and isomorphism or anti-isomorphism. Likewise, we give formulae for the number of nilpotent commutative semigroups with $n$ elements up to equality and up to isomorphism

Coclass theory for nilpotent semigroups via their associated algebras

Coclass theory has been a highly successful approach towards the investigation and classification of finite nilpotent groups. Here we suggest a similar approach for finite nilpotent semigroups. This differs from the group theory setting in that we additionally use certain algebras associated to the considered semigroups. We propose a series of conjectures on our suggested approach. If these become theorems, then this would reduce the classification of nilpotent semigroups of a fixed coclass to a finite calculation. Our conjectures are supported by the classification of nilpotent semigroups of coclass 0 and 1. Computational experiments suggest that the conjectures also hold for the nilpotent semigroups of coclass 2 and 3.Comment: 13 pages, 2 figure

Finite nilpotent semigroups of small coclass

The parameter coclass has been used successfully in the study of nilpotent algebraic objects of different kinds. In this paper a definition of coclass for nilpotent semigroups is introduced and semigroups of coclass 0, 1, and 2 are classified. Presentations for all such semigroups and formulae for their numbers are obtained. The classification is provided up to isomorphism as well as up to isomorphism or anti-isomorphism. Commutative and self-dual semigroups are identified within the classification.Comment: 11 page

Exploring the Moduli Space of (0,2) Strings

We use an exactly solvable (0,2) supersymmetric conformal field theory with gauge group SO(10) to investigate the superpotential of the corresponding classical string vacuum. We provide evidence that the rational point lies in the Landau-Ginzburg phase of the linear sigma-model and calculate exactly all three- and four-point functions of the gauge singlets. These couplings already put severe constraints on the possible flat directions of the superpotential. Finally, we contemplate about the flat direction related to Kahler deformations of the underlying linear sigma-model.Comment: 19 pages, plain TeX, 2 postscript figures, epsf include

Die Rolle von Fullerenen bei der Photodegradation organischer Solarzellen

This dissertation constitutes a detailed study of the degradation behavior of organic photovoltaics under different ambient conditions, with particular focus on the role of the fullerene in all processes. We find that fullerenes do not only influence the solar cell lifetime by stabilizing or destabilizing the pi-conjugated polymer against photo-oxidation, but they are also able to cause direct device degradation by undergoing photo-dimerization under inert atmosphere. In the first part of this thesis, we investigate the degradation of organic solar cells under photo-oxidative conditions. Firstly, a correlation between fullerene electron affinity and device degradation is established for a series of eight different fullerenes. A fullerene LUMO level of -3.75 eV is found to yield the highest stability, while the degradation is accelerated for both higher and lower electron affinities. Secondly, we study the photo-oxidation behavior of three different polymers, namely poly(3-hexylthiophene) (P3HT), poly[2,6-(4,4-bis(2-ethylhexyl)-4H-cyclopenta[2,1-b;3,4-b′]dithiophene)-alt-4,7-(2,1,3-benzothiadiazole)] (C-PCPDTBT), and poly[2,6-(4,4-bis(2-ethylhexyl)dithieno[3,2-b:2′,3′-d]silole)-alt-4,7-(2,1,3-benzothiadiazole)] (Si-PCPDTBT), in neat polymer films and in blends with the fullerene [6,6]-phenyl-C61-butyric acid methyl ester (PC60BM). While P3HT is strongly stabilized by PC60BM, Si-PCPDTBT and C-PCPDTBT are hardly stabilized or even destabilized, respectively. Photo-induced absorption spectroscopy reveals that the enhanced degradation of C-PCPDTBT correlates with the population of the polymer triplet state via the polymer:fullerene charge-transfer state. Thirdly, we compare PC60BM with two other small molecules, namely 2,7 dinitrofluorenone (DNF) and nickel(II)bisdibutyldithiocarbamate (Ni(dtc)2), in order to obtain a mechanistic understanding of the stabilization process. It is evidenced that quenching of the polymer singlet-excited states does not lead to a significant stabilization. Finally, we perform calculations to quantify the stabilization effect by optical screening, and demonstrate it to play only a minor role as well. An extensive discussion about the different possible stabilizing and destabilizing effects of the fullerene completes the first part, concluding that radical scavenging is the main stabilizing mechanism, whereas sensitization of superoxide radicals is the predominant destabilizing mechanism. The second part of this work deals with the photo-degradation of organic photovoltaics under inert atmosphere. The exclusion of oxygen leads to considerably larger lifetimes of organic solar cells, since the π-conjugated polymer is found to be chemically stable under these conditions. Nevertheless, in the case of some polymer:fullerene combinations, device performance losses even emerge from mere exposure to light. For example, encapsulated solar cells based on PC60BM and the polymer poly[4,4' bis(2-ethylhexyl)dithieno[3,2-b:2',3'-d]silole)-2,6-diyl-alt-[2,5-bis(3-tetradecylthiophen 2-yl)thiazole[5,4-d]thiazole)-1,8-diyl] (PDTSTzTz) show substantial performance losses upon illumination (“burn-in effect”). Our electrical, spectroscopic, and analytic measurements on this model system provide evidence that photo-dimerization of PC60BM, i.e., a [2+2] cycloaddition between two adjacent fullerenes, is responsible for the observed degradation. Simulation of the electrical device parameters reveals that this dimerization process results in a significant reduction of the charge carrier mobility. BisPC60BM, the bis-substituted analog of PC60BM, is shown to be resistant towards light exposure, which in turn enables the manufacture of photostable PDTSTzTz:bisPC60BM solar cells. In addition, detailed investigations on the photo-degradation behavior of PC60BM and other fullerenes in both neat films and blend films with polymers are performed under different conditions. Within a series of eleven different fullerenes, multi-substituted fullerenes as well as C70-based fullerenes are observed not to undergo dimerization, in contrast to all mono-substituted C60-fullerenes. Dimerization is found to be suppressed in the presence of oxygen and is independent of the temperature. At temperatures above 100 °C, however, both the dimerization of PC60BM and the associated photovoltaic device performance loss become reversible. Moreover, the photo-transformation yield of PC60BM in thin films is shown to be independent of the photon energy and exhibits first-order dependence on the incident photon flux density. The extent of PC60BM dimerization in blend films with polymers depends strongly on the type of polymer. The dimerization yield is demonstrated to mainly depend on the morphology of the bulk-heterojunction, that is to say: the bigger the fullerene domains, the more dimerization occurs. Finally, the formation of different photogenerated species and their kinetics in PC60BM films and blends with polymer is studied by ultrafast transient absorption spectroscopy. By this means, we establish a mechanistic understanding of the photo-induced fullerene dimerization in the solid state. In films of PC60BM, fullerene triplet-excited states, initiating species of the dimerization process, are not formed via intersystem crossing, but are shown to be formed upon charge recombination of fullerene anions and cations, which are already formed on a sub-picosecond scale.Diese Dissertation stellt eine detaillierte Studie des Degradationsverhaltens organischer Photovoltaik unter verschiedenen Umgebungsbedingungen, mit besonderem Fokus auf der Rolle des Fullerens in allen Prozessen, dar. Fullerene beeinflussen die Lebensdauer organischer Solarzellen nicht nur durch Stabilisierung oder Destabilisierung des pi-konjugierten Polymers gegen Photooxidation, sondern können durch Photodimerisierung unter inerter Atmosphäre auch selbst direkt eine Degradation hervorrufen. Im ersten Teil der Arbeit wird die Degradation von organischen Solarzellen unter photooxidativen Bedingungen untersucht. Zunächst stellen wir für eine Reihe von acht verschiedenen Fullerenen einen Zusammenhang zwischen der Elektronenaffinität des Fullerens und der Solarzellendegradation her. Ein Fulleren-LUMO-Level von -3,75 eV erzielt die höchste Stabilität, während die Degradation sowohl durch höhere als auch durch niedrigere Elektronenaffinitäten beschleunigt wird. Des Weiteren untersuchen wir das Photooxidationsverhalten drei verschiedener Polymere, Poly(3-hexylthiophen) (P3HT), Poly[2,6-(4,4-bis(2-ethylhexyl)-4H-cyclopenta[2,1-b;3,4-b′]dithiophen)-alt-4,7-(2,1,3-benzothiadiazol)] (C-PCPDTBT) und Poly[2,6-(4,4-bis(2-ethylhexyl)dithieno[3,2-b:2′,3′-d]silol)-alt-4,7-(2,1,3-benzothiadiazol)] (Si-PCPDTBT), in reinen Polymerfilmen und im Gemisch mit dem Fulleren [6,6]-Phenyl-C61-buttersäuremethylester (PC60BM). Während P3HT von PC60BM stark stabilisiert wird, wird Si-PCPDTBT kaum stabilisiert und C-PCPDTBT sogar destabilisiert. Photoinduzierte Absorptionsmessungen zeigen, dass die gesteigerte Degradation von C-PCPDTBT mit der Population des Polymer-Triplett-Zustandes über den Charge-Transfer-Zustand von Polymer und Fulleren korreliert. Darüber hinaus vergleichen wir PC60BM mit zwei anderen niedermolekularen Verbindungen, 2,7-Dinitrofluorenon (DNF) und Nickel(II)bisdibutyldithiocarbamat (Ni(dtc)2), um ein mechanistisches Verständnis des Stabilisierungsprozesses zu erlangen. Es wird aufgezeigt, dass das Löschen von angeregten Polymer-Singulett-Zuständen zu keiner signifikanten Stabilisierung führt. Zudem liefern Berechnungen das Ergebnis, dass auch die Stabilisierung durch optische Abschirmung eine untergeordnete Rolle spielt. Eine ausführliche Diskussion über die verschiedenen möglichen Stabilisierungs- und Destabilisierungseffekte des Fullerens rundet den ersten Teil ab und kommt zu dem Schluss, dass das Einfangen von Radikalen den wichtigsten Stabilisierungsmechanismus darstellt, wohingegen die Sensibilisierung von Superoxidradikalen der dominierende Destabilisierungsmechanismus ist. Der zweite Teil dieser Arbeit behandelt die Photodegradation organischer Photovoltaik unter inerter Atmosphäre. Der Ausschluss von Sauerstoff führt zu einer beträchtlich längeren Lebensdauer organischer Solarzellen, da das π-konjugierte Polymer unter diesen Bedingungen chemisch stabil ist. Dennoch führt bei einigen Polymer:Fulleren-Kombinationen sogar die bloße Lichtexposition zu Zelldegradation. So zeigen zum Beispiel verkapselte Solarzellen basierend auf PC60BM und dem Polymer Poly[4,4' bis(2-ethylhexyl)dithieno[3,2-b:2',3'-d]silol)-2,6-diyl-alt-[2,5-bis(3-tetradecylthiophen 2-yl)thiazol[5,4-d]thiazol)-1,8-diyl] (PDTSTzTz) unter Beleuchtung erhebliche Leistungsverluste („burn-in-Effekt“). Unsere elektrischen, spektroskopischen und analytischen Messungen an diesem Modellsystem liefern den Beweis, dass Photodimerisierung von PC60BM, d. h. eine [2+2]-Cycloaddition zwischen zwei benachbarten Fullerenen, verantwortlich für die beobachtete Degradation ist. Eine Simulation der elektrischen Zellparameter lässt erkennen, dass dieser Dimerisierungsprozess eine verringerte Ladungsträgermobilität zur Folge hat. BisPC60BM, das bis-substituierte Analog zu PC60BM, ist hingegen resistent gegenüber Bestrahlung, was die Herstellung lichtstabiler PDTSTzTz:bisPC60BM-Solarzellen ermöglicht. Des Weiteren werden detaillierte Untersuchungen des Photodegradationsverhaltens von PC60BM und anderen Fullerenen in Filmen mit und ohne Polymer unter verschiedenen Bedingungen durchgeführt. In einer Reihe von elf verschiedenen Fullerenen weisen sowohl mehrfach substituierte als auch C70-basierte Fullerene im Gegensatz zu allen einfach substituierten C60-Fullerenen Dimerisierung auf. Die Dimerisierung ist temperaturunabhängig und in Gegenwart von Sauerstoff unterdrückt. Bei Temperaturen über 100 °C wird jedoch sowohl die Dimerisierung von PC60BM als auch der damit verbundene Leistungsverlust der Solarzelle reversibel. Darüber hinaus zeigen wir, dass die Ausbeute der Photodimerisierung von PC60BM in dünnen Schichten wellenlängenunabhängig ist und linear von der eingestrahlten Lichtintensität abhängt. Das Ausmaß an PC60BM-Dimerisierung in Mischfilmen mit Polymeren hängt stark vom Polymertyp ab. Die Dimerisierungsausbeute wird in erster Linie durch die Morphologie in der Schicht festgelegt. Je größer die Fullerendomänen, desto ausgeprägter ist die Dimerisierung. Schließlich wird mittels transienter Femtosekunden-Absorptionsspektroskopie die Bildung verschiedener photogenerierter Spezies und deren Kinetik in PC60BM-Filmen mit und ohne Polymer ermittelt. Auf diese Weise wird ein mechanistisches Verständnis der lichtinduzierten Dimerisierung von Fullerenen geschaffen. In PC60BM-Filmen werden angeregte Fulleren-Triplett-Zustände, initiierende Spezies des Dimerisierungsprozesses, nicht über Intersystem Crossing gebildet, sondern entstehen bei der Ladungsrekombination von Fullerenanionen und -kationen, welche bereits auf einer sub-Pikosekundenskala gebildet werden

(0,2) Mirror Symmetry

We generalize the previously established (0,2) triality of exactly solvable models, Landau-Ginzburg theories and Calabi-Yau manifolds to a number of different classes of (0,2) compactifications derived from (2,2) vacua. For the resulting models we show that the known (2,2) mirror constructions induce mirror symmetry in the (0,2) context.Comment: plain TeX, harvmac incl., 38(b)/22(l) pp., 3 PS figs., epsf incl., references adde

More on N=1 Self-Dualities and Exceptional Gauge Groups

Starting from a generalization of a recent result on self-duality we systematically analyze self-dual models. We find a criterion to judge whether a given model is self-dual or not. With this tool we construct some new self-dual pairs, focussing on examples with exceptional gauge groups.Comment: 10 pages, LaTeX2e, using utarticle.cls (included

Spacetime Subsystem Symmetries

One characteristic feature of many fractonic lattice models, and a defining property of the exotic field theories developed to describe them, are subsystem symmetries including a conservation of not just net electric charge but also electric dipole moments or charges living on submanifolds. So far all such theories were based on internal subsystem symmetries. In this work we generalize the notion of subsystem symmetries to system with subsystem spacetime symmetries with locally conserved energies.Comment: 7 page