Hypercentral unit groups and the hyperbolicity of a modular group algebra

We classify groups G such that the unit group U-1 (ZG) is hypercentral. In the second part, we classify groups G whose modular group algebra has hyperbolic unit groups U-1 (KG).

Symplectic Representation of a Braid Group on 3-Sheeted Covers of the Riemann Sphere

We define Picard cycles on each smooth three-sheeted Galois cover C of the Riemann sphere. The moduli space of all these algebraic curves is a nice Shimura surface, namely a symmetric quotient of the projective plane uniformized by the complex two-dimensional unit ball. We show that all Picard cycles on C form a simple orbit of the Picard modular group of Eisenstein numbers. The proof uses a special surface classification in connection with the uniformization of a classical Picard-Fuchs system. It yields an explicit symplectic representation of the braid groups (coloured or not) of four strings.

Isomorphism of Commutative Modular Group Algebras

∗ The work was supported by the National Fund “Scientific researches” and by the Ministry of Education and Science in Bulgaria under contract MM 70/91.

Complex Hyperbolic Surfaces of Abelian Type

2000 Mathematics Subject Classification: 11G15, 11G18, 14H52, 14J25, 32L07.

Transformation formula for exponential sums involving fourier coefficients of modular forms

In 1984 Jutila [5] obtained a transformation formula for certain exponential sums involving the Fourier coefficients of a holomorphic cusp form for the full modular group SL(2, Z). With the help of the transformation formula he obtained good estimates for the distance between consecutive zeros on the critical line of the Dirichlet series associated with the cusp form and for the order of the Dirichlet series on the critical line, [7]. In this paper we follow Jutila to obtain a transformation formula for exponential sums involving the Fourier coefficients of either holomorphic cusp forms or certain Maass forms for congruence subgroups of SL(2, Z) and prove similar estimates for the corresponding Dirichlet series.

Sylow P-Subgroups of Abelian Group Rings

2000 Mathematics Subject Classification: Primary 20C07, 20K10, 20K20, 20K21; Secondary 16U60, 16S34.

Über die Linearität der Teichmüllerschen Modulgruppe des Torus mit zwei Punktierungen

Über die Liniarität der Teichmüllerschen Modulgruppe des Torus mit zwei Punktierungen. In meiner Arbeit beschäftige ich mich mit Darstellungen der Teichmüllerschen Modulgruppe des Torus mit zwei Punktierungen. Mein Ansatz hierbei ist, die Teichmüllersche Modulgruppe in eine p-adische Liegruppe einzubetten. Sei nun F die von zwei Elementen erzeugte freie Gruppe und Aut(F) die Automorphismengruppe von F. Inhalt des ersten Kapitels ist es nun zu zeigen, daß folgende Aussagen äquivalent sind: - Die Teichmüllersche Modulgruppe des Torus mit zwei Punktierungen ist linear, - Aut(F)ist linear, - F besitzt eine p-Kongruenzstruktur, deren Folgen- glieder von Aut(F) festgehalten werden, also charak- teristisch sind. Im zweiten Kapitel wird unter anderem gezeigt, daß es eine Einbettung einer Untergruppe endlichen Indexes der Aut(F) in die Automorphismengruppe einer einfachen p-adischen Liegruppe gibt. Bisher ist unbekannt, ob die Buraudarstellung treu ist.In dieser Arbeit wird ein unendliches, lineares Gleichungssystem, dessen Lösungen gerade die Koeffizienten der Wörter des Kernes der Buraudarstellung sind, vorgestellt.Im dritten Kapitel wird mit den Methoden des 1.Kapitels gezeigt, daß der Torus mit zwei Punktierungen genau dann linear ist, wenn die Teichmüllersche Modulgruppe der Sphäre mit 5 Punktierungen es auch ist. Bekanntlich ist die 4. Braidgruppe linear. Nun ist aber die 4. Braidgruppe letztlich die Teichmüllersche Modulgruppe der abgeschlossenen Kreisscheibe mit 5 Punktierungen. Wenn man nun deren Randpunkte miteinander identifiziert und anschließend wegläßt, erhält man die 5-fach punktiereSphäre.Mit der eben beschriebenen Abbildung kann man zeigen, daß die Teichmüllersche Modulgruppe der fünffach punktierten Sphäre linear ist.

The M-theory origin of global properties of gauge theories

We show that global properties of gauge groups can be understood as geometric properties in M-theory. Different wrappings of a system of N M5-branes on a torus reduce to four-dimensional theories with AN−1 gauge algebra and different unitary groups. The classical properties of the wrappings determine the global properties of the gauge theories without the need to impose any quantum conditions. We count the inequivalent wrappings as they fall into orbits of the modular group of the torus, which correspond to the S-duality orbits of the gauge theories.

On realizability of p-groups as Galois groups

2000 Mathematics Subject Classification: 12F12, 15A66.

Nadel's Subschemes of Fano Manifolds X with a Picard Group Pic(X) Isomorphic to Z

∗The author supported by Contract NSFR MM 402/1994.

Macromolecular engineering via RAFT chemistry : from sequential to modular design

Sequential Design Molecular Weight Range Functional Monomers: Possibilities, Limits, and Challenges Block Copolymers: Combinations, Block Lengths, and Purities Modular Design End-Group Chemistry Ligation Protocols Conclusions

Associated schools of construction of southern Africa infrastructure design and delivery workshop - Report. CIB task group 91 infrastructure

This workshop comprised a diverse group of African construction experts, ranging far wider than RSA. Each of the attendees had attended the annual ASOCSA conference and was additionally provided with a short workshop pre-brief. The aim was to develop a view of their 15-20 year vision of construction improvement in RSA and the steps necessary to get there. These included sociological, structural, technical and process changes. Whilst some suggestions are significantly challenging, none are impossible, given sufficient collaboration between government, industry, academia and NGOs. The highest priority projects (more properly, programmes) were identified and further explored. These are: 1. Information Hub (‘Open Africa’). Aim – to utilise emerging trends in Open Data to provide a force for African unity. 2. Workforce Development. Aim – to rebuild a competent, skilled construction industry for RSA projects and for export. 3. Modular DIY Building. Aim – to accelerate the development of sustainable, cost-efficient and desirable housing for African economic immigrants and others living in makeshift and slum dwellings. Open Data is a maturing theme in different cities and governments around the world and the workshop attendees were very keen to seize such a possibility to assist in developing an environment where Africans can share information and foster collaboration. It is likely that NGOs might be keen to follow up such an initiative. There are significant developments taking place around the world in the construction sector currently, with comparatively large savings being made for taxpayers (20% plus in the UK). Not all of these changes would be easy to transplant to RSA (even more so to much of the rest of Africa). Workforce development was a keen plea amongst the attendees, who seemed concerned that expertise has leaked away and is not being replaced with sufficient intensity. It is possible today to develop modular buildings in such a way that even unskilled residents can assist in their construction, and even their appropriate design. These buildings can be sited nearly autonomously from infrastructures, thus relieving the tensions on cities and townships, whilst providing humane accommodation for the economically disadvantaged. Development of suitable solutions could either be conducted with other similarly stressed countries or developed in-country and the expertise exported. Finally, it should be pointed out that this was very much a first step. Any opportunity to collaborate from an Australian, QUT or CIB perspective would be welcomed, whilst acknowledging that the leading roles belong to RSA, CSIR, NRF, ASOCSA and the University of KwaZulu-Natal.

Modular design of synthetic protein mimics. Crystal structure of two seven-residue helical peptide segments linked by .epsilon.-aminocaproic acid

Two seven-residue helical segments, Val-Ala-Leu-Aib-Val-Ala-Leu, were linked synthetically with an epsilon-aminocaproic acid (Acp) linker with the intention of making a stable antiparallel helix-helix motif. The crystal structure of the linked peptide Boc-Val-Ala-Leu-Aib-Val-Ala-Leu-Acp-Val-Ala-Leu-Aib-Val-Ala-Leu-OMe (1) shows the two helices displaced laterally from each other by the linker, but the linker has not folded the molecule into a close-packed antiparallel conformation. Two strong intermolecular NH...O = C hydrogen bonds are formed between the top of the lower helix of one molecule and the bottom of the upper helix in a laterally adjacent molecule to give the appearance of an extended single helix. The composite peptide with Boc and OMe end groups, C76H137N15O18.H2O, crystallize in space group P2(1) with a = 8.802 (1) angstrom, b = 20.409 (4) angstrom, c = 26.315 (3) angstrom, and beta = 90.72 (1)degrees; overall agreement R = 7.86% for 5030 observed reflections (\F(o)\ > 3-sigma(F)); resolution = 0.93 angstrom. Limited evidence for a more compact conformation in solution consistent with an antiparallel helix arrangement is obtained by comparison of the HPLC retention times and CD spectra of peptide 1 with well-characterized continuous helices of similar length and sequence.

L-infinity norms of holomorphic modular forms in the case of compact quotient

We prove a sub-convex estimate for the sup-norm of L-2-normalized holomorphic modular forms of weight k on the upper half plane, with respect to the unit group of a quaternion division algebra over Q. More precisely we show that when the L-2 norm of an eigenfunction f is one, parallel to f parallel to(infinity) <<(epsilon) k(1/2-1/33+epsilon) for any epsilon > 0 and for all k sufficiently large.