Der Rang der Lösungen von Y2=X3±p3 über 2e

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Über Gruppen mit einer definierenden Relation

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Richardson's iteration for nonsymmetric matrices

To solve the linear N×N system (1) Ax=a for any nonsingular matrix A, Richardson's iteration (2) x_j+1=x_j-α_j(Ax_j-a), j=1,2,…,n, which is applied in a cyclic manner with cycle length n is investigated, where the α_j are free parameters. The objective is to minimize the error |x_n+1-x|, where x is the solution of (1). If the spectrum of A is known to lie in a compact set S, one is led to the Chebyshev-type approximation problem (3) min_p-1∈Vnmax_z∈S|p(z)|, where V_n is the linear span of z,z²,…,zⁿ. If p solves (3), then the reciprocals of the zeros of p are optimal iteration parameters α_j. It is shown that for a real problem (1) the iteration (2) can be carried out with real arithmetic alone, even when there are complex α_j. The stationary case n=1 is solved completely, i.e., for all compact sets S the problem (3) is solved explicitly. As a consequence, the converging stationary iteration processes can be characterized. For arbitrary closed disks S the problem (3) can be solved for all n∈$\text{N}$, and a simple proof is provided. The lemniscates associated with S are introduced. They appear as an important tool for studying the stability of the iteration (2).     

On the effect of off-shell wavefunctions onK andL shell charge transfer in fast,asymmetric collisions

Within the semiclassical strong potential Born approximation (SPB) we have calculated the electron capture from theK andL shell of a heavy target atom by protons using two different peaking approximations. Both total capture cross sections and impact parameter distributions are compared with the impulse approximation (IA) and in the case of transfer from the ArK-shell also with an exact evaluation of the SPB and with experimental data. While for theK shell, all theories give a similar impact-parameter dependence, there is a substantial difference between IA and SPB results for the 2s subshell.     

Das Verhalten von Lösungen semilinearer elliptischer Systeme an Ecken eines Gebietes

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