Interpolation of Morrey-Campanato and related smoothness spaces

We study the interpolation of Morrey-Campanato spaces and some smoothness spaces based on Morrey spaces, e. g., Besov-type and Triebel-Lizorkin-type spaces. Various interpolation methods, including the complex method, the ±-method and the Peetre-Gagliardo method, are studied in such a framework. Special emphasis is given to the quasi-Banach case and to the interpolation property.     

1,2- and 1,3-diphosphetanes from alkylidenephosphines

Abstract

Alkyl- or arylbis(trimethylsilyl)phosphines as well as tris(trimethylsilyl)phosphine and the corresponding arsines react with acyl chlorides to give [1-(trimethylsiloxy)alkylidene]phosphines 1 and -arsines 2; most of their 2,2-dimethylpropylidene derivatives are thermally stable at room temperature. With the same class of phosphines as starting compounds and carbon disulfide [bis(trimethylsilylsulfano)methylidene]phosphines 3 are formed, whereas [(dialkylamino)methylidene]-4 and [diarylmethylidene]phosphines 5 or the corresponding arsines 6 and 7 can be obtained from acyl amides or ketones.¹     

Enzymatic Conversion of Flavonoids using Bacterial Chalcone Isomerase and Enoate Reductase

Flavonoids are a large group of plant secondary metabolites with a variety of biological properties and are therefore of interest to many scientists, as they can lead to industrially interesting intermediates. The anaerobic gut bacterium Eubacterium ramulus can catabolize flavonoids, but until now, the pathway has not been experimentally confirmed. In the present work, a chalcone isomerase (CHI) and an enoate reductase (ERED) could be identified through whole genome sequencing and gene motif search. These two enzymes were successfully cloned and expressed in Escherichia coli in their active form, even under aerobic conditions. The catabolic pathway of E. ramulus was confirmed by biotransformations of flavanones into dihydrochalcones. The engineered E. coli strain that expresses both enzymes was used for the conversion of several flavanones, underlining the applicability of this biocatalytic cascade reaction.     

A -normal, not densely normal Tychonoff space

We give an example of a -normal space which is not densely normal.

     

Synthese arylsubstituierter 2-Pyrimidinamine,Dihydro-2-pyrimidinamine und Pyrimido[1,2-a]pyrimidine durch Reaktion von Guanidin mit Chalkonen

4- and 4-methoxy-, 4-hydroxy- and 2-hydroxychalcone4 e,k,f andg, respectively, are transformed by action of guanidine in benzene to yield 4-(4-methoxyphenyl)-6-phenyl-2-pyrimidinamine (6 e=6 k) and hydroxyphenylpyrimidinamines6 f andg, respectively. In contrast, 4- and 4-chlor-, 4-brom- and 4-phenylchalcone4 h,l–n resp. react with guanidine under analogous conditions to give 4,6-diaryl-1,4-dihydro-2-pyrimidinamines5 h,l–n. The bases5 h,l–n also tend to aromatize, but they can be stabilized by transformation into salts5 h,l–n · HCl with hydrochloric acid. Heating of5 n inDMF under atmospheric oxygen yields 4-(4-biphenyl)-6-phenyl-2-pyrimidinamine (6 n). Action of guanidine on 4-nitrochalcone4 o in chloroform affords 2-amino-4-nitrophenyl-6-phenyl-tetrahydro-4-pyrimidinol8 o, which is transformed by hydrochloric acid into 6-nitrophenyl-dihydropyrimidinamine-hydrochloride5 o · HCl. Treating of the latter with sodiummethylat in methanol yields the very stable dihydropyrimidinamine5 o. Action of guanidine on 4-chlor- and 4-bromchalcone4 h andm respectively (in addition to5 h andm) yields 2,4,6,8-tetraaryl-1,4-dihydro-2H-pyrimido[1,2-a]pyrimidines7 h andm, respectively.

Herrn Prof. Dr.Erwin Schauenstein mit den besten Wünschen zum 65. Geburtstag gewidmet.     

Optical metrics and birefringence of anisotropic media

The material tensor of linear response in electrodynamics is constructed out of products of two symmetric second rank tensor fields which in the approximation of geometrical optics and for uniaxial symmetry reduce to “optical” metrics, describing the phenomenon of birefringence. This representation is interpreted in the context of an underlying internal geometrical structure according to which the symmetric tensor fields are vectorial elements of an associated two-dimensional space.