Properties of the Collision Operators of the Electron Boltzmann Equation for a Weakly Ionized Plasma. I. Representation and General Properties

In the kinetic theory a great variety of physical systems is investigated by means of Boltzmann-like equations. This approach is used for neutral gases, neutron as well as radiation transport, plasmas etc. For many problems the knowledge of the properties of the collision operators is of great importance, especially if eigenvalue problems occur. The paper presents an investigation of the properties of the collision operators of the Boltzmann equation covering elastic, exciting and deexciting processes in a weakly ionized plasma. First, a short survey of the importance of eigenfunctions and eigenvalues in the kinetic theory of various systems is given. Then, properties of the outscattering operator as dependent on the course of the differential cross section are considered. Finally, for the inscattering operator such properties as selfadjointness and rotational invariance are investigated in detail. These considerations provide the basis for the proof of compactness and for first conclusions on the spectral properties of the collision operators in the second part of this paper.     

Fragmentation of the protonated stereoisomers of 5,7-Undecanediol and its bis(trimethylsilyl) ether under conditions of isobutane chemical ionization

The chemical ionization induced fragmentation behaviour of an open chain model compound, 5,7-undecanediol (meso and d,l stereoisomers), is studied and compared with that of previously studied alicyclic models, e.g. cyclohexanediols (cis and trans isomers). In close analogy to the latter, the major fragmentation processes are the one- and twofold loss of H₂O from the protonated molecules [MH]⁺. As borne out by appropriately deuterium labelled analogues, the first loss of H₂O is invariably a ‘clean’ heterolysis. The second loss is considerably more complex in that 1,3- and 1,4-elimination processes occur concurrently. While the 1,4-mode may be a direct elimination process in both types of compounds, the 1,3-mode is rearrangement induced in both. Whereas skeletal rearrangement (ring contraction) precedes this 1,3-elimination in the alicyclic models, hydrogen rearrangement (1,2-hydride shift) precedes it in the open chain model. No pronounced stereochemical effect is observed on the mechanistic course of the elimination processes, but is observed in their relative ease. The trimethylsilyl ethers of the stereoisomers of 5,7-undecanediol are also studied for analogous reactions, and are discussed in this context.     

A New Clathrate Featuring a Spiro-Type Ammonium Host. X-Ray Structure of the Inclusion Compound with Chloroform

A new onium salt host molecule 1 featuring a spiro-type ammonium backbone was synthesized and shown to form a crystalline inclusion compound with chloroform, 1 · chloroform (1:2), the crystal structure of which is reported. The host molecular interactions including C–H···Br, Cl···Cl and aromatic stacking contacts are typical of the packing structure.     

An extrapolation method with step size control for nonlinear Volterra integral equations

Summary In [6] it has been shown that the midpoint rule applied to second kind volterra integral equations possesses an asymptotic expansion in even powers of the stepsizeh. In this paper we describe an extrapolation method based on the midpoint rule, together with a mechanism of step size control.