Literatur

Ohne Zusammenfassung     

Spectral methods with sparse matrices

Summary Spectral methods employ global polynomials for approximation. Hence they give very accurate approximations for smooth solutions. Unfortunately, for Dirichlet problems the matrices involved are dense and have condition numbers growing asO(N ⁴) for polynomials of degree N in each variable. We propose a new spectral method for the Helmholtz equation with a symmetric and sparse matrix whose condition number grows only asO(N ²). Certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system. Numerical results are presented which show that we have probably found the most effective solver for spectral systems.     

On the solution of singular linear systems of algebraic equations by semiiterative methods

Summary For a square matrixT ^n,n , where (I–T) is possibly singular, we investigate the solution of the linear fixed point problemx=T x+c by applying semiiterative methods (SIM's) to the basic iterationx ₀ ⁿ ,x _k T c _k–1+c(k1). Such problems arise if one splits the coefficient matrix of a linear systemA x=b of algebraic equations according toA=M–N (M nonsingular) which leads tox=M ^–1 N x+M ^–1 bT x+c. Even ifx=T x+c is consistent there are cases where the basic iteration fails to converge, namely ifT possesses eigenvalues 1 with ||1, or if =1 is an eigenvalue ofT with nonlinear elementary divisors. In these cases — and also ifx=T x+c is incompatible — we derive necessary and sufficient conditions implying that a SIM tends to a vector which can be described in terms of the Drazin inverse of (I–T). We further give conditions under which is a solution or a least squares solution of (I–T)x=c.Research supported in part by the Alexander von Humboldt-Stiftung     

N-(17-Phosphonooxylinolenoyl)glutamine and N-(17-phosphonooxylinoleoyl)glutamine from insect gut: the first backbone-phosphorylated fatty acid derivatives in nature

N-(17-Phosphonooxylinolenoyl)glutamine (1) and N-(17-phosphonooxylinoleoyl)glutamine (2) were isolated from the regurgitate of Spodoptora exigua and identified as the first natural alkyl chain-phosphorylated fatty acid derivatives. The compounds were characterized by HPLC-MS/MS and the assigned structures confirmed by synthesis via a dissymmetric bis-Wittig approach as the key reaction. Rearing of the larvae on a diet enriched with inorganic phosphate increased the amount of the phosphorylated N-acyl glutamines in the regurgitate.     

Impact of electron-electron interaction on the electron energy distribution function in molecular plasmas under rf field action

Calculations of the electron energy distribution and of relevant macroscopic quantities of collision-dominated, weakly ionized plasmas under rf field action have been performed with increasing degrees of ionization, and the impact of the electron-electron interaction on these quantities was determined. The investigations were performed for the gas plasmas in CO and H₂ as representatives of molecular plasmas The energy distribution and macroscopic quantities are obtained by solving the nonstationary Bolizmann equation for a given rf field and degree of ionization taking into accoung and additional Fokker-Planck term besides the collision integrals for the elastic and the main inelastic collision processes. In these molecular plasmas a remarkable impact of the electron-electron interaction connected with increasing Maxwellization is observed for degrees of ionization greater than 10.