On the optimality of a characterization theorem for the gamma function using the multiplication formula

Summary The following Artin type characterization of : _{+ +} is proved: Assume thatf: _{+ +} satisfies the Gauss multiplication formula for some fixedp 2,f is absolutely continuous on [l/p, 1 + ] for some > 0 and lim _{x 0} xf(x) = 1. Thenf(x) = (x) forx > 0.The optimality of this result is checked by means of counterexamples. For instance, it is shown that the result is no longer true, if f is absolutely continuous is replaced by f is continuous and of finite variation.     

Geometric H/D isotope effects and cooperativity of the hydrogen bonds in porphycene.

We investigate the primary, secondary, and vicinal hydrogen/deuterium (H/D) isotope effects on the geometry of the two intramolecular hydrogen bonds in porphycene. Multidimensional potential energy surfaces describing the anharmonic motion in the vicinity of the trans isomer are calculated for the different symmetric (HH/DD) and asymmetric (HD) isotopomers. From the solution of the nuclear Schr?dinger equation the ground-state wavefunction is obtained, which is further used to determine the quantum corrections to the classical equilibrium geometries of the hydrogen bonds and thus the geometric isotope effects. In particular, it is found that the hydrogen bonds are cooperative, that is, both expand simultaneously even in the case of an asymmetric isotopic substitution. The theoretical predictions compare favorably with NMR chemical-shift data.     

Linear-Reactor-IR.-Matrix and Microwave Spectroscopy of the System O3/NO2/(Z)-2-Butene

Investigation of the formation of complex reaction products in the gas-phase system O₃/NO₂/(Z)-2-butene by combination of linear reactors with IR. matrix and microwave Stark Spectroscopy is reported. Besides the polyatomic products observed earlier in the gas-phase ozonolysis of (Z)-2-butene, the following products were identified; N₂O₅, HNO₃, HNO₄, CH₃NO₂, CH₃ONO, CH₃COONO₂ and CH₃COO₂NO₂ (peroxyacetyl nitrate, PAN). Matrix IR. spectra of N₂O₅, HNO₃. CH₃COONO, CH₃COONO₂ required for reference purposes are presented. It is shown that PAN-formation occurs already in the absence of light. A reaction scheme is proposed for explanation of the observed complex NO_x-containing products, which assumes methyldioxirane as a central intermediate. Particular reaction steps of the scheme will be discussed, including thermochemical estimates of reaction enthalpies.