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.     

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.     

Singlet oxygen photooxygenation of furans : Isolation and reactions of (4+2)-cycloaddition products (unsaturated sec.-ozonides)

Tetraphenylporphin-photosensitized oxygenations of furan (19), 2-methylfuran (26), 2-ethylfuran (39), furfurylalcohol (24), 2-acetylfuran (40), 2-methoxyfuran (42), 2,5-dimethylfuran (30), furfural (25) and 5-methylfurfural (41) in non-polar aprotic solvents yield the corresponding monomeric unsaturated secondary ozonides due to a (4+2)-cycloaddition of singlet oxygen to these furans. With the exception of the ozonide derived from 25, the ozonides were isolated and characterized (¹H- and ¹³C-NMR spectra, etc.). In non-polar aprotic solvents, the ozonides derived from 19, 26 and 39 undergo thermal rearrangements to the corresponding cis-diepoxides and epoxylactones. Ozonide 31, derived from 30, however, dimerizes, only above about 60° is a cis-diepoxide formed from either 31 or its dimer. Rose bengal-photosensitized oxygenations of the furans in alcohols (MeOH, EtOH, i-PrOH) also produce the corresponding ozonides as the primary products of (4+2)-cycloadditions of singlet oxygen to these furans. However, reactions of the alcohols with the ozonides are too fast to allow the ozonides to be isolated. Instead, the same products are obtained as are isolated from reactions carried out by dissolving the ozonides in the alcohols. Depending on the structure of the ozonide, three pathways are available to ozonide/alcohol (ROH) interactions:(1) addition of ROH to yield alkoxy hydroperoxides; one out of several possible isomers is formed in a completely stereoselective and regiospecific reaction; (2) elimination of a bridgehead proton by ROH as a base, as observed with the ozonide derived from 19 to give hydroxy butenolide (78) in yields between 20 and 60%, and (3) ROH-attack on a carbonyl side-chain under elimination of the corresponding alkyl ester, as observed with furfural photooxygenation which yielded hydroxy butenolide (78) in high yields (95%). Interaction of ozonide 31 with tert.-butyl alcohol (t-BuOH) yields quantitatively cis-3-oxo-1-butenylacetate (81) by a Baeyer-Villiger-type rearrangement with vinyl group migration Hydrogenbonding between the alcohol and the peroxy group of the ozonides assist the heterolysis of the C—O bonds in the ozonides; the most stabilized cation develops. Front-attack of ROH on this cation explains the stereoselectivity as well as the regiospecificity of the alkoxy hydroperoxide formation; with a bulky alcohol like t-BuOH, ROH-attack on the cation is sterically hindered thus allowing a rearrangement to occur. 1,3-Dipolar cycloaddition of p-nitrophenyl azide to ozonide 31 proceeds stereoselectively to one of the isomers 87a/87b. Finally, kinetic results of furan photooxygenation in methanol show the following order of furan-reactivity towards singlet oxygen: 30 > 42 > 26 > 19 > 41 > 25, with absolute rate constants ranging from 1.8 × 10⁸ (with 30) to 8.4 × 1O⁴ M^-1P^-1 (with 25).     

A Hilbert space approach to maximum entropy reconstruction

We examine here the problem of reconstructing an X-ray attenuation function from measurements of its integrals. The approach that is taken is to maximize the difference of the entropy and the residual error in meeting the measurements. The solution of this optimization problem is constrained by requiring that the solution lie in a certain weakly compact subset of L², to be determined by physical information. We show that the constrained optimization problem is well-posed: there exists a unique solution (even when the measured data are inconsistent) and the solution depends continuously on the measurements. In the course of proving this, we show that the entropy functional is continuous on L². We further demonstrate that the solution of the optimization problem for a special case, must be piecewise constant.