Zur Spektralinvarianz von Algebren von Pseudodifferentialoperatoren in derL p -Theorie

Zusammenfassung Die Hörmander-Klassen _1, ⁰ (0<1) von Pseudodifferentialoperatoren sind -Algebren. Insbesondere ist die Inverse eines inL(L ^p ( ⁿ )) invertierbaren Pseudodifferentialoperators der Klasse _1, ⁰ selbst wieder ein Pseudodifferential-operator derselben Klasse.Der Beweis beruht auf den von R. Beals, R. R. Coifman, Y. Meyer und H. O. Cordes entwickelten Methoden zur Charakterisierung von Algebren von Pseudodifferentialoperatoren.     

Asymmetric Diels-Alder reactions with 5-menthyloxy-2-(5H)-furanone

A new class of chiral dienophiles, 5-alkoxy-2(5H)-furanones, has been developed. Both enantiomers of 5-menthyloxy-2(5H)-ftiranone are readily available in enantiomerically pure form, starting from furfural and d- or l-menthol. Excellent diastereoselectivities (d.e. β99%) are obtained in thermal Diels-Alder reactions with several cyclic and acyclic dienes. The use of silyl dienol ethers has resulted in new routes to enantiomerically pure cyclohexanones in a highly regioselective manner.     

Etorphine‐Related Ferrocenyl‐Substituted Morphinan Alkaloids

The two diastereoisomeric ferrocenyl‐substituted orvinols 2 and 3 were prepared. The modified alkaloids are still able to interact with opioid receptors (see Table). The ferrocene moiety allows highly selective and sensitive electrochemical detection. The X‐ray crystal structure of the major isomer 2 was determined. The combination of a metallocene and a morphinan alkaloid holds promise for useful antitumor activity.     

The average a posteriori error of numerical methods

Summary The definition of the average error of numerical methods (by example of a quadrature formula to approximateS(f)= f d on a function classF) is difficult, because on many important setsF there is no natural probability measure in the sense of an equidistribution. We define the average a posteriori error of an approximation by an averaging process over the set of possible information, which is used by (in the example of a quadrature formula,N(F)={(f(a ₁), ...,f/fF} is the set of posible information). This approach has the practical advantage that the averaging process is related only to finite dimensional sets and uses only the usual Lebesgue measure. As an application of the theory I consider the numerical integration of functions of the classF={f:[0,1]/f(x)–f(y)||x–y|}. For arbitrary (fixed) knotsa _i we determine the optimal coefficientsc _i for the approximation and compute the resulting average error. The latter is minimal for the knots . (It is well known that the maximal error is minimal for the knotsa _i .) Then the adaptive methods for the same problem and methods for seeking the maximum of a Lipschitz function are considered. While adaptive methods are not better when considering the maximal error (this is valid for our examples as well as for many others) this is in general not the case with the average error.     

Pyrolysis studies of aromatic polyesters

The products of pyrolysis, at 400°C in vacuo (1 Pa), of poly(ethyleneterephthalate) (PET) and poly(butyleneterephthalate) (PBT) were studied. The products were identified by gas chromatography/mass spectrometry technique. The highly volatile products of PET contained acetaldehyde, benzene, toluene, styrene and ethylbenzene and in the case of PBT, butadiene, tetrahydrofuran, vinylcyclohexene and ethylbenzene were the major products. A qualitative analysis of the products of low volatility revealed that the main components were benzoic acid, terephthalic acid, monovinyl esters of terephthalic acid and higher oligomers in the case of PET, whereas benzoic acid, monobutenyl esters of terephthalic acid and higher oligomers were the main products from PBT. The results obtained were in good agreement with those obtained from pyrolysis experiments, carried out directly in the mass spectrometer. Mechanisms to explain the occurrence of the different products are proposed.     

One-dimensional magnetism and crystal structure ofcatena-di-bromobis(3,5-dimethylpyridine)copper(II)

Summary Crystals ofcatena-di--bromobis(3,5-dimethylpyridine)copper(II) are monoclinic, space group P2₁/a. The unit cell constants area=13.900(2),b=14.416(2),c=4.097(1) Å,=93.49(2)°, V=819.4 ų and Z=2. The structure was determined from powder data using a Guinier-Johansson focussing powder camera. The structure was solved using a simplex method for function minimization to a conventional R-value of 0.13.The structure consists of infinite linear chains parallel toc in which the copper coordination is distorted elongated octahedral. Cu-Br distances were found to be 2.449(7) and 3.286(7) Å, whereas the Cu-N bond length is 2.02(2) Å. All distances are in the range usually observed for this type of compounds.The antiferromagnetic superexchange interactions between adjacent Cu^II ions (J = –21 cm^–1) has been compared with those observed in structural similar CuBr₂L₂ compounds. The differences in observed J-values are discussed briefly, in relation to the structural variations. It appears that very small changes in structural parameters strongly affect the magnetic exchange.