Zur Reziprozität quadratischer Charaktersummen in algebraischen Zahlkörpern

Ohne Zusammenfassung     

The decomposition numbers of the hecke algebra of typeF 4

LetW be the finite Coxeter group of typeF ₄, andH _r (q) be the associated Hecke algebra, with parameter a prime powerq, defined over a valuation ringR in a large enough extension field ofQ, with residue class field of characteristicr. In this paper, ther-modular decomposition numbers ofH _R (q) are determined for allq andr such thatr does not divideq. The methods of the proofs involve the study of the generic Hecke algebra of typeF ₄ over the ringA = ℤ[u ^1/2,u ^-1/2] of Laurent polynomials in an indeterminateu ^1/2 and its specializations onto the ring of integers in various cyclotomic number fields. Substancial use of computers and computer program systems (GAP, MAPLE, Meat-Axe) has been made.     

Strukturänderungen bei der Reduktion von 9-(Diphenylmethyliden)fluoren mit Natrium-Metall zu Schichten (R2O)Na⊕-verknüpfter Kohlenwasserstoff-Dianionen

Structural Changes on Reduction of 9-(Diphenylmethylidene)fluorene by Sodium Metal to Sheets of (R₂O)Na⊕- Connected Hydrocarbon Dianions The single-crystal structure of 9-(diphenylmethylidene)fluorene (monoclinic, space group P2₁/n, Z = 4) proves a C?C bond length of 135 pm and a twisting of the two Ph rings out of the methylidene-fluorene plane by 70° and 83°. Its reduction in Et₂O solution by a Na mirror under Ar yields dark green crystals with a metallic luster. Their structure, determined in a N₂ flow cooled to 223 K (monoclinic, space group C2/c, Z = 4), reveals sheets of hydrocarbon dianions, connected by sodium(diethylether) links. The Na centers are sixfold coordinated as half-sandwiches to the fluorene benzo rings and threefold allylic to the Ph rings. In the dianion of 9-(diphenyl-methylidene)fluorene, the central C?C bond is stretched by 13 pm to 148 pm length, and the two molecular cyanine-like halves are twisted by 51° relative to each other. Both the contact-ion multiple structures as well as the structural changes on twofold reduction are discussed by way of comparison with literature examples and based on semiempirical calculations.     

Simulation of dispersive Fourier interferograms of gases in the visible wave number range

Dispersive two-beam Fourier interferograms of N₂-gas in the visible wave number range are calculated numerically for nonabsorbing as well as absorbing gas and compared with experiment. Some problems of inverse Fourier transformation of dispersive interferograms are also discussed.     

Bis-1,3-diol type liquid crystals from pentaerythritol [1]

Several esters and ethers of pentaerythritol have been prepared. Four of them, containing each a 1,4-trans-disubstituted cyclo-hexane unit and three hydroxyl functions, emerge as examples of a novel type of thermotropic, intermolecularly hydrogen bridge-supported multiol mesogens exhibiting the smectic A phase. A new case of a heteroatom effect preventing mesophase formation by causing stable intramolecular hydrogen bonding situations will also be discussed.