The combinatorics of the bar resolution in group cohomology

We study a combinatorially defined double complex structure on the ordered chains of any simplicial complex. Its columns are related to the cell complex K_n whose face poset is isomorphic to the subword ordering on words without repetition from an alphabet of size n. This complex is shellable and as an application we give a representation theoretic interpretation for derangement numbers and a related symmetric function considered by Désarménien and Wachs [11].

We analyze the two spectral sequences arising from the double complex in the case of the bar resolution for a group. This spectral sequence converges to the cohomology of the group and provides a method for computing group cohomology in terms of the cohomology of subgroups. Its behavior is influenced by the complex of oriented chains of the simplicial complex of finite subsets of the group, and we examine the Ext class of this complex.     

A quasiconformal dynamic property of the disk

It is known that every conformai embedding of the disk into the extended complex plane possesses a r²-quasiconformal extension across eachr-level line (the r²-property of a domain). We show here that this is a characteristic property of the disk: any simply connected domain which is not a disk does not admit ther ²-property. Supported by the RiP-program of the Volkswagen-Stiftung in the Mathematisches Forschungsinstitut Oberwolfach.     

Über Verbindungen mit Urotropin-Struktur, 37. Mitt.: Eine neue Methode zur Herstellung von Hydroxy-ketonen und Diketonen sowie ihre Anwendung auf die Synthese des 2,4-Dioxa-adamantan-Ringsystems

Zusammenfassung Es wird eine neue Methode zur Herstellung von Hydroxyketonen und Diketonen beschrieben, die im ersten Schritt der Reaktionsfolge die Umsetzung von Hydroxycarbonsäureestern mit Phenylsulfonalkylmagnesiumbromid zu -Sulfonylketonen erfordert. Diese werden dann im zweiten Schritt mit Al-Amalgam zu den Hydroxyketonen oder den Halbketalen der Diketone entschwefelt. Mit Hilfe dieser Reaktionsfolge gelang auch, ausgehend von 3,5-Dihydroxy-phenylessigsäureester, die Synthese des bisher noch unbekannten Ringsystems des 2,4-Dioxa-adamantans.Herrn Prof. Dr. habil.Friedrich Asinger zum 60. Geburtstag in freundschaftlicher Verbundenheit zugeeignet36. Mitt.:H. Stetter, J. Gärtner undP. Tacke, Chem. Ber.99, 1435 (1966).     

A vanadium-51 NMR study of the binding of vanadate and peroxovanadate to proteins

51V quadrupolar central transition NMR spectra of buffered (pH 7.6-8.0) solutions of bovine apo-transferrin (Tf) and bovine prostatic acid phosphatase (Pp) treated with vanadate show normal features (chemical shifts between -515 and -542 ppm) corresponding to the complexation of VO2+ to the Tf binding site and the Pp active centre, respectively. Addition of H2O2 leads to the temporary formation of complexed VO(O2)+ (delta approximately -595). Vanadate-dependent bromoperoxidase from the alga Ascophyllum nodosum exhibits an unusually high shielding both for the native (delta = -931) and the peroxo form (delta = -1135) of the enzyme. A resonance at -471 ppm is traced back to an inactive form with oxovanadium(V) in a trigonal-bipyramidal array.     

A combined theoretical and experimental study of efficient and fast titanocene-catalyzed 3-exo cyclizations

The mechanism of titanocene mediated 3-exo cyclizations was investigated by a combined theoretical and experimental study. A gradient corrected density functional theory (DFT) method has been scaled against titanocene dichloride, the parent butenyl radical, and in bond dissociation energy (BDE) calculations. The BP86 method using density fitting, and a basis set of triple-zeta quality emerged as a highly reliable tool for studying titanocene mediated radical reactions. The computational results revealed important kinetic and thermodynamic features of cyclopropane formation. Surprisingly, the beta-titanoxy radicals, the first intermediates of our investigations, were demonstrated to possess essentially the same thermodynamic stabilization as the corresponding alkyl radicals by comparison of the calculated BDEs. In contrast to suggestions for samarium mediated reactions, the cyclization was shown to be thermodynamically favorable in agreement with earlier kinetic studies. It was established that stereoselectivity of the cyclization is governed by the stability of the intermediates and thus the trans disubstituted products are formed preferentially. The observed ratios of products are in good to excellent agreement with the DFT results. By a combination of computational and experimental results, it was also shown that for the completion of the overall cyclopropane formation the efficiency of the trapping of the cyclopropylcarbinyl radicals is decisive.     

Free arrangements and rhombic tilings

Let Z be a centrally symmetric polygon with integer side lengths. We answer the following two questions:

1. When is the associated discriminantal hyperplane arrangementfree in the sense of Saito and Terao?
2. When areall of the tilings of Z by unit rhombicoherent in the sense of Billera and Sturmfels?

Surprisingly, the answers to these two questions are very similar. Furthermore, by means of an old result of MacMahon on plane partitions and some new results of Elnitsky on rhombic tilings, the answer to the first question helps to answer the second. These results then also give rise to some interesting geometric corollaries. Consideration of the discriminantal arrangements for some particular octagons leads to a previously announced counterexample to the conjecture by Saito [ER2] that the complexified complement of a real free arrangement is aK (π, 1) space.