On modules of finite upper rank

For a group and a prime , the upper -rank of is the supremum of the sectional -ranks of all finite quotients of . It is unknown whether, for a finitely generated group , these numbers can be finite but unbounded as ranges over all primes. The conjecture that this cannot happen if is soluble is reduced to an analogous `relative' conjecture about the upper -ranks of a `quasi-finitely-generated' module for a soluble minimax group . The main result establishes a special case of this relative conjecture, namely when the module is finitely generated and the minimax group is abelian-by-polycyclic. The proof depends on generalising results of Roseblade on group rings of polycyclic groups to group rings of soluble minimax groups. (If true in general, the above-stated conjecture would imply the truth of Lubotzky's `Gap Conjecture' for subgroup growth, in the case of soluble groups; the Gap Conjecture is known to be false for non-soluble groups.)     

Versuche zur Gewinnung einiger neuer Azofarbstoffe

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A comparison of preconditioners for incompressible Navier–Stokes solvers

We consider solution methods for large systems of linear equations that arise from the finite element discretization of the incompressible Navier–Stokes equations. These systems are of the so‐called saddle point type, which means that there is a large block of zeros on the main diagonal. To solve these types of systems efficiently, several block preconditioners have been published. These types of preconditioners require adaptation of standard finite element packages. The alternative is to apply a standard ILU preconditioner in combination with a suitable renumbering of unknowns. We introduce a reordering technique for the degrees of freedom that makes the application of ILU relatively fast. We compare the performance of this technique with some block preconditioners. The performance appears to depend on grid size, Reynolds number and quality of the mesh. For medium‐sized problems, which are of practical interest, we show that the reordering technique is competitive with the block preconditioners. Its simple implementation makes it worthwhile to implement it in the standard finite element method software. Copyright © 2007 John Wiley & Sons, Ltd.     

Zur Bestimmung der Titans?ure neben Eisen

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