Zur Theorie der harmonischen Differentialformen

Generalizing harmonic differential forms (rot =0, div =0 in M, M being a smooth riemannian manifold with boundary) of first and second kind (=0 and *=0 on M resp.) within the framework of Hilbert space notation, it is possible to extend the meaning of the boundary conditions to non-smooth boundaries. It turns out that in this case the classical result is still valid for certain open subregions G of M: The dimension of the space of harmonic differential forms of second kind is given by the q-th Betti number of G; *-duality leads to the respective result for harmonic differential forms of first kind.     

Gewöhnliche lineare Differentialgleichungen zweiter Ordnung mit Distributionskoeffizient

Zusammenfassung Läßt man in der gew. lin. Dgl. zweiter Ordnung einen Distributionskoeffizient zu, der Ableitung einerL _loc ² -Funktion ist, so existieren eindeutig festlegbare, stetige Lösungen derart, daß die in der Dgl. auftretenden Produkte definiert werden können. Mit einer Regularisierung des Distr.-Koeffizienten können die Lösungen durchC -Funktionen approximiert werden. Es wird der Sturmsche Vergleichssatz verallgemeinert. Hieraus und aus den Eigenschaften der mit zwei Basislösungen gebildeten Lösungskurve folgt die Gültigkeit des Ljapunovschen Satzes über die Eigenwertverteilung der verallgemeinerten Hillschen Dgl. mit positivem Distr.-Koeffizient, falls eine zusätzliche Bedingung erfüllt ist. Die vorliegende Arbeit enthält einige Ergebnisse meiner Dissertation [6] (weitere Resultate und Details finden sich dort).     

Fibrous Networks with Incorporated Macrocycles: A Chiral Stimuli‐Responsive Supramolecular Supergelator and Its Application to Biocatalysis in Organic Media

A new and versatile, crown ether appended, chiral supergelator has been designed and synthesized based on the bis‐urea motif. The introduction of a stereogenic center improved its gelation ability significantly relative to its achiral analogue. This low‐molecular‐weight gelator forms supramolecular gels in a variety of organic solvents. It is sensitive to multiple chemical stimuli and the sol–gel phase transitions can be reversibly triggered by host–guest interactions. The gel can be used to trap enzymes and release them on demand by chemical stimuli. It stabilizes the microparticles in Pickering emulsions so that enzyme‐catalyzed organic reactions can take place in the polar phase inside the microparticles, the organic reactants diffusing through the biphasic interface from the surrounding organic phase. Because of the higher interface area between the organic and polar phases, enzyme activity is enhanced in comparison with simple biphasic systems.     

Synthesis and Rearrangement of P‐Nitroxyl‐Substituted PIII and PV Phosphanes: A Combined Experimental and Theoretical Case Study

Low‐temperature generation of P‐nitroxyl phosphane 2 (Ph₂POTEMP), which was obtained by the reaction of Ph₂PH ( 1 ) with two equivalents of TEMPO, is presented. Upon warming, phosphane 2 decomposed to give P‐nitroxyl phosphane P‐oxide 3 (Ph₂P(O)OTEMP) as one of the final products. This facile synthetic protocol also enabled access to P‐sulfide and P‐borane derivatives 7 and 13 , respectively, by using Ph₂P(S)H ( 6 ) or Ph₂P(BH₃)H ( 11 ) and TEMPO. Phosphane sulfide 7 revealed a rearrangement to phosphane oxide 8 (Ph₂P(O)STEMP) in CDCl₃ at ambient temperature, whereas in THF, thermal decomposition of sulfide 7 yielded salt 10 ([TEMP‐H₂][Ph₂P(S)O]). As well as EPR and detailed NMR kinetic studies, indepth theoretical studies provided an insight into the reaction pathways and spin‐density distributions of the reactive intermediates.     

Pd@SnO2 and SnO2@Pd Core@Shell Nanocomposite Sensors

Pd@SnO₂ and SnO₂@Pd core@shell nanocomposites are prepared via a microemulsion approach. Both nanocomposites exhibit high‐surface, porous matrices of SnO₂ shells (>150 m² g^?1) with very small SnO₂ crystallites (<10 nm) and palladium (Pd) nanoparticles (<10 nm) that are uniformly distributed in the porous SnO₂ matrix. Although similar by first sight, Pd@SnO₂ and SnO₂@Pd are significantly different in view of their structure with Pd inside or outside the SnO₂ shell and in view of their sensor performance. As SMOX‐based sensors (SMOX: semiconducting metal oxide), both nanocomposites show a very good sensor performance for the detection of CO and H₂. Especially, the Pd@SnO₂ core@shell nanocomposite is unique and shows a fast response time (τ₉₀ < 30 s) and a very good response at low temperature (<250 °C), especially under humid‐air conditions. Extraordinarily high sensor signals are observed when exposing the Pd@SnO₂ nanocomposite to CO in humid air. Under these conditions, even commercial sensors (Figaro TGS 2442, Applied Sensor MLC, E2V MICS 5521) are outperformed.