Random approximation and the vertex index of convex bodies

We prove that there exists an absolute constant \({\alpha > 1}\) with the following property: if K is a convex body in \({{\mathbb R}^n}\) whose center of mass is at the origin, then a random subset \({X\subset K}\) of cardinality \({{\rm card}(X)=\lceil\alphan\rceil }\) satisfies with probability greater than \({1-e^{-c_1n}}\)

$$K\subseteq c_2n\, {\rm conv}(X),$$

where \({c_1, c_2 > 0}\) are absolute constants. As an application we show that the vertex index of any convex body K in \({{\mathbb R}^n}\) is bounded by \({c_3n^2}\), where \({c_3 > 0}\) is an absolute constant, thus extending an estimate of Bezdek and Litvak for the symmetric case.

     

Interfacial oxidation of ultrathin nickel and chromium films on yttria-stabilized zirconia

The substrate-induced oxidation upon prolonged annealing in UHV of ultrathin films of Ni and Cr vapor deposited on yttria-stabilized zirconia YSZ(100) was studied by X-ray photoelectron spectroscopy (XPS) to obtain information about the oxidation mechanism, determine the available quantity of reactive oxygen in YSZ, and investigate the thermal stability of the thin oxide films. Up to about 0.8 ML of Ni deposited at room temperature was oxidized to NiO at a constant rate at 650 K via the substrate, whereas at slightly higher coverage, the oxidation rate under identical conditions was drastically reduced. In contrast to Ni, up to 4.8 ML of Cr deposited at 275 K could be oxidized via the substrate to Cr2O3 upon extensive UHV annealing at increasing temperature up to 820 K, indicating a reactive oxygen content of at least 4 x 10(-6) with respect to the lattice oxygen in the YSZ specimen. The Cr2O3 decomposed to metallic Cr above about 800 K, whereas NiO was stable up to the maximum temperature of 875 K. These results indicate that the oxidation via the substrate is kinetically analogous to the gas-phase oxidation of bulk Ni and Cr. The reactive oxygen content of the single-crystal YSZ is larger than expected, and part of it is accommodated at the surface of the substrate. The thermal stability of the thin oxide films is determined by the oxygen exchange with YSZ and not by the respective bulk oxide thermodynamic decomposition temperature.     

Hydrogen-bonded networks based on manganese(II), nickel(II), copper(II) and zinc(II) complexes of N,N′-dimethylurea

A project related to the crystal engineering of hydrogen-bonded coordination complexes has been initiatied and some of our first results are presented here. The compounds [Mn(DMU)₆](ClO₄)₂ (1), [Ni(DMU)₆](ClO₄)₂ (2), [Cu(OClO₃)₂(DMU)₄] (3) and [Zn(DMU)₆](ClO₄)₂ (4) have all been prepared from the reaction of N,N-dimethylurea (DMU) and the appropriate hydrated metal perchlorate salt. Crystal structure determinations of the four compounds demonstrate the existence of [M(DMU)₆]²⁺ cations and ClO₄ ^– counterions in (1), (2) and (4), whereas in (3) monodentate coordination of the perchlorate groups leads to molecules. The [M(DMU)₆]²⁺ cations and ClO₄ ^– anions self-assemble to form a hydrogen-bonded one-dimensional (1D) architecture in (1) and different 2D hydrogen-bonded networks in (2) and (4). The hydrogen bonding functionalities on the molecules of (3) create a 2D structure. The complexes were also characterised by room-temperature effective magnetic moments and i.r. studies. The data are discussed in terms of the nature of bonding and the known structures.