Polar Multiplicities and Euler obstruction of the stable types in weighted homogeneous map germs from C-n to C-3 n >= 3

In this article we show that for corank 1, quasi-homogeneous and finitely determined map germs f : (C-n, 0)-> (C-3, 0), n >= 3 one can obtain formulae for the polar multiplicities defined on the following stable types of f, f(Delta(f) and f(Sigma(n-2,1)(f), in terms of the weights and degrees of f. As a consequence we show how to compute the Euler obstruction of such stable types, also in terms of the weights and degrees of f.

Manganese(III) porphyrins: Catalytic activity and intermediate studies in homogeneous systems

We studied how solvent, stirring method, PhIO/MnP molar ratio, presence of water and axial ligand affect the catalytic activities of Mn(TPP)Cl, Mn(MNPP)Cl, Mn(TDCPP)Cl and Mn(TFPP)Cl in the oxidation of cyclohexane by PhIO. A study of the catalytic intermediates in the reaction between Mn(TPP)Cl or Mn(TDCPP)Cl and PhIO was also carried out by UV-Vis and EPR spectroscopies. The reaction of Mn(TPP)Cl with PhIO showed the formation of a mixture of species Mn-IV(OP+ and Mn-V(O)P as intermediates, which were confirmed by the deconvolution of the UV-Vis spectra. Addition of imidazole as cocatalyst favoured the formation of the intermediate species Mn-V(O)P, evidenced by the UV-Vis band at 408 nm. The corresponding EPR spectra gave evidence that in the presence of imidazole, Mn-IV(OP+ species are formed only in very low amounts. For Mn(TDCPP)Cl the dominating intermediate species is Mn-IV(OP+. Addition of imidazole to halogen-substituted MnP systems does not result in increase of the C-ol yields because very stable bis-imidazole-MnP complexes are formed. Anchoring of such MnP on imidazole propyl gel (IPG) results in better catalytic activity because in this case, the catalyst is mono-coordinated to the support and imidazole favours the formation of the intermediate species Mn-V(O)P.