Conformal capacities and conformally invariant functions on Riemannian manifolds

This paper develops the theory of conformal invariants initiated inJ. Differential Geom 8 (1973), 487–510 for a Riemannian manifoldM with dimensionn2. We construct and study four conformally invariant functions M, M, M, M resp. depending on 4, 3 or 2 points onM, defined as extremal capacities for condensers associated with those points. These functions have similarities with the classical invariants onS ⁿ ,R ⁿ orH ⁿ . Their properties, and especially their continuity, are efficient tools for solving some problems of conformal geometry in the large.     

Synthesis, Structural Characterization, and Oxidation-Reduction Behavior of the gamma-Isomer of the Dodecatungstosilicate Anion

The reaction of tungstate ions with the gamma-10-tungstosilicate in mixed water-ethanol (v/v) yields the corresponding isomer of the 12-tungstosilicate isolated as its tetrabutylammonium salt. It was characterized by means of (183)W NMR, infrared, Raman, and UV spectroscopies and was identified with the isomer resulting from the Keggin structure (alpha-isomer) by rotation of two tritungstic groups by pi/3. Cyclic voltammetry shows that the reducibility increases in the sequence alpha-beta-gamma as the number of rotated tritungstic groups. Whereas the gamma-12-tungstosilicate anion is unstable in pure aqueous solution whatever the pH value, it is stable in mixed aqueous-organic or pure organic solvents, even at boiling temperatures. In contrast, the two- and four-electron-reduced blue species are stable in aqueous media but not the one-electron-reduced blue. Study of the (183)W NMR spectrum of the two-electron-reduced species showed that the two spin-paired electrons are delocalized on all the W atoms on the NMR time scale but have a larger residency time on one group of four atoms.