A Newton method for a class of quasi-variational inequalities

A variant of the Newton method for nonsmooth equations is applied to solve numerically quasivariational inequalities with monotone operators. For this purpose, we investigate the semismoothness of a certain locally Lipschitz operator coming from the quasi-variational inequality, and analyse the generalized Jacobian of this operator to ensure local convergence of the method. A simplified variant of this approach, applicable to implicit complementarity problems, is also studied. Small test examples have been computed.This work has been supported in parts by a grant from the German Scientific Foundation and by a grant from the Czech Academy of Sciences.     

An iterative two-step algorithm for linear complementarity problems

Summary. We propose an algorithm for the numerical solution of large-scale symmetric positive-definite linear complementarity problems. Each step of the algorithm combines an application of the successive overrelaxation method with projection (to determine an approximation of the optimal active set) with the preconditioned conjugate gradient method (to solve the reduced residual systems of linear equations). Convergence of the iterates to the solution is proved. In the experimental part we compare the efficiency of the algorithm with several other methods. As test example we consider the obstacle problem with different obstacles. For problems of dimension up to 24\,000 variables, the algorithm finds the solution in less then 7 iterations, where each iteration requires about 10 matrix-vector multiplications. Received July 14, 1993 / Revised version received February 1994     

über die Streuung relativistischer Elektronen im Coulombfeld

An approximative method for summing the partial wave series for the scattering amplitude of the relativistic scattering of electrons by an atomic nucleus is investigated in the simplest case of a point charge nucleus for electron energies not less 10 MeV. In the course of the computationΒ=1 is assumed (Β=v/c,v andc denote the velocity of the electron and of light, respectively). The Sommerfeld-Watson-transformation is the starting point. The method consists in setting up an asymptotic expansion in 1/sin² (θ/2) (θ is the scattering angle) of the amount coming from the Regge-poles. Two further amounts, the dependence of which on the scattering angle especially for easy nuclei is little, are computed numerically without simplification, which should be possible. The method is developed from a special way of summing the partial wave series for the non-relativistic Rutherford scattering, which may be done analytically.Mott's scattering formula is verified and finally the applicability of the method is examined for summing the relativistic partial wave series with the exact S-Matrix. The results are as exact as the expension of the cross section to order (Z/137)⁵ (Z is the nuclear charge). The method should in principle be applicable for an extended nucleus, but in this case there remain still several problems.     

Through-bond 13C-13C correlation at the natural abundance level: refining dynamic regions in the crystal structure of vitamin-D3 with solid-state NMR

Recently, we presented a novel nuclear magnetic resonance experiment for establishing through-bond connectivity in disordered solids using scalar coupling-driven correlation. This method, a variant of the popular double-quantum-filtered correlation spectroscopy experiment in liquids, is robust under fast magic-angle-spinning conditions and in the presence of dynamics. Here, we show that this new experiment, the UC2QF COSY, can be extended to 13C natural abundance correlation in moderately sized molecules, allowing the assignment of the 54 peaks of the solid-state NMR spectrum of microcrystalline vitamin-D3. In this case, comparison between the assigned peaks and ab initio calculations of the chemical shifts based on the crystal coordinates permits a refinement of the average structure in dynamic regions reported as disordered in the crystal structure.     

Studies in mass spectrometry. III—formation of a common intermediate in the primary fragmentation process of 1-phenylcyclobutene and of 2-phenyl-1,3-butadiene

The mass spectra of the title compounds show peaks corresponding to the primary fragmentation of H, CH₃ and (H₂ + C₂H₂). Kinetic study suggests that the molecular ion of 1-phenylcyclobutene and 2-phenyl-1,3-butadiene loses H and CH₃ through formation of a common intermediate. This is confirmed by appearance potential and ionisation potential measurements, fragmentation of deuterium labelled compounds and the corrected Z values.