G-invariante Morse-funktionen

If f is a Morse function on a smooth manifold M there exists a homotopy equivalence from M to a CW complex X such that the critical points of f with index are in a one-one correspondence to the -cells of X. In the equivariant case, a similar result holds for a special type of invariant Morse functions. In this paper we prove the existence of such special invariant Morse functions on compact smooth G-manifolds. As a consequence, any compact smooth G-manifold is homotopy equivalent to a G-CW complex. Other applications deal with the Euler number of the fixed point set and Morse inequalities in equivariant homology theory.     

On the stereospecific conversion of proximally-oxygenated trisubstituted vinyltriphenylstannanes into stereodefined trisubstituted alkenes

[reaction: see text] Allylically oxygenated vinyl alpha-triphenylstannanes such as 22 can be readily converted into vinyl iodides and thereafter stereodefined trisubstituted alkenes with retention of configuration.     

4-Amino-imidazole durchThorpe-Cyclisierung

The alkylation of arylaminomethylenecyanamides1 or cyano-imidothiocarbamates2 with -halogen carbonyl compounds followed by base catalysed cyclization yields substituted 4-amino-imidazoles4. Imidazo[4,5-d]pyrimidones5, 6 and imidazo[4,5-b]pyridines7 can be obtained from4.     

Formulas for \bar \beta parameters and their limitations

The operator relationship p=[r, h] between linear momentum, position vector and Hamiltonian is the basis for an investigation of formulas for ¦ atomic orbitals is discussed. Formulas for parameters useful in extended Hückel methods are presented.

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Zusammenfassung Die Operatorengleichung p=[r, h] zwischen Impuls, Ortsvektor und Hamiltonoperator ist die Grundlage für eine Untersuchung von Formeln für -Parameter in der erweiterten Hückel[1]- und CNDO [2]-Methode. Das Eichproblem dieser Gleichung in Verbindung mit abgebrochenen Entwicklungen für Matrixelemente ¦P¦ über Atomfunktionen wird diskutiert. Nützliche Formeln für die erweiterte Hückel-Methode werden abgeleitet.

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Résumé La relation entre opérateurs moment linéaire, position et hamiltonien: p=[r, h] sert de base à une étude de formules pour les paramètres dans les méthodes de type Hückel étendu [1] et CNDO [2]. Le problème de jauge de cette équation est discuté par rapport au développement limité des éléments de matrice ¦P¦ dans une base d'orbitales atomiques. Présentation de formules pour les paramètres utiles dans les méthodes Hückel étendu.