Bessel sequences of exponentials on fractal measures

Jorgensen and Pedersen have proven that a certain fractal measure ν has no infinite set of complex exponentials which form an orthonormal set in L²(ν). We prove that any fractal measure μ obtained from an affine iterated function system possesses a sequence of complex exponentials which forms a Riesz basic sequence, or more generally a Bessel sequence, in L²(μ) such that the frequencies have positive Beurling dimension.     

Herstellung einiger verzweigter Paraffine im oligomeren Bereich

Synthesis of some branched paraffins of high molecular weight Synthesis and properties of two series of branched paraffins of high molecular weight are described. In the course of the preparation of one of the starting materials the Wolff-Kishner reduction of a sterically hindered ketone, unstable in basic medium, has been elaborated.     

Monofunctional (Dimethylamino)silane as Silylating Agent

The reaction of triorganyl(dimethylamino)silanes with surface-hydrated silicon dioxide has been studied. These silylating agents are easy to prepare from the corresponding chloro or bromosilanes with dimethylamine. The resulting products are thermally stable and relatively volatile. Reaction with surface-hydrated silicon-dioxide preparations at 150–250°C for 170 h yields a dense grafted layer. However, with (dimethylamino)silanes having strongly polar substituents, a retreatment of the surface-modified silica seems to be necessary in order to attain maximum coverage.