Dahlberg's theorem in higher co-dimension

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension $n?1$ in ${\mathbb{R}}^n$, and later this result has been extended to more general non-tangentially accessible domains and beyond.In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph Γ of dimension d in ${\mathbb{R}}^n$, $d<n?1$, with a small Lipschitz constant. We construct a linear degenerate elliptic operator L such that the corresponding harmonic measure ${\omega }_L$ is absolutely continuous with respect to the Hausdorff measure on Γ. More generally, we provide sufficient conditions on the matrix of coefficients of L which guarantee the mutual absolute continuity of ${\omega }_L$ and the Hausdorff measure.     

On the reactions of methyl radicals with nitrilotris(methylenephosphonic-acid) complexes in aqueous solutions

Abstract

The reactions of methyl radicals with M^II-(ntp)(H₂O)₂ complexes were studied. The formation of an unstable intermediate (ntp)(H₂O)M^III-CH₃ is observed. This reaction is an equilibrium process, i.e. the M^III-C bond decomposes via homolysis. The (ntp)(H₂O)M^III-CH₃ complexes isomerize to a more stable form. The results compared to those obtained in analogous reactions of the M^II-(nta)(H₂O)₂ complexes were shown to be similar with one exception – for the nta complexes no isomerization process is observed.     

PSEUDOHALOGEN CHEMISTRY. PART IV. HETEROLYTIC ADDITION OF THIOCYANOGEN TO ALKENES

Abstract

Thiocyanogen reacts slowly with alkenes, in the presence of a radical inhibitor in benzene or acetic acid in the dark at 25°, to yield α,β-dithiocyanates, α-isothiocyanates-β-thiocyanates and, in acetic acid, α-acetoxy-β-thiocyanates in varying proportions. The additions to alkyl alkenes are trans-stereospecific, and, in the case of the α-isothiocyanato-β-thiocyanates, non-regiospecific. The additions to aryl alkenes are trans-stereoselective and regiospecific, yielding the Markownikov-orientated α-isothiocyanato-β-thiocyanates. A heterolytic mechanism involving a two-step, kinetically controlled addition, with the formation of a cyano-sulfonium ion intermediate, e.g., 35, in the case of alkyl alkenes and an open carbonium ion, e.g., 36, in the case of aryl alkenes, is suggested. The diothiocyanate: isothiocyanato-thiocyanate ratios are discussed in terms of kinetic and steric control of reaction.