Restrictions of Steiner bundles and divisors on the Hilbert scheme of points in the plane

The Hilbert scheme of n points in the projective plane parameterizes degree n zero-dimensional subschemes of the projective plane. We examine the dual cones of effective divisors and moving curves on the Hilbert scheme. By studying interpolation, restriction, and stability properties of certain vector bundles on the plane we fully determine these cones for just over three fourths of all values of n. A general Steiner bundle on P^N is a vector bundle admitting a linear resolution of a particular form. We complete the classification of slopes of semistable Steiner bundles on P^N by showing every admissible slope is realized by a bundle which restricts to a semistable bundle on a rational curve. The proof involves a basic question about multiplication of polynomials on P^1 which is interesting in its own right.Comment: Comments welcome; 27 pages, 1 figur

Hydrodynamics of the continuously filtering slurry reactor. Influence of load of solids and particle size distribution.

Internal filtration in slurry bubble columns offers a possible solution to the filtration problems related to this reactor type. The applicability of the concept has already been demonstrated at full-scale for waste water treatment. Theoretical description of internal filtration is lacking, however.\ud \ud In previous work it was derived on basis of two different models, that in three-phase filtration solids concentration in the vicinity of the filter should uniquely depend on a single dimensionless number, i.e. \ud \ud \ud \ud \ud The validity of the so-called cake build-up number was demonstrated by determining filter cake resistance as an indirect measure of solids concentration for a system in batch operation with respect to solids.\ud \ud In the present work the force balance model is extended to include the effect of particle size distribution. It is demonstrated that particle polydispersity can explain trends observed upon variation of load of solids, which cannot be understood assuming the particles to be monodispersed. Based on a simplified version of the model including particle polydispersity, a novel diameter characteristic to filtration behaviour of polydisperse solids is derived. Using this novel diameter in the cake build-up number, experimental results for four different size distributions can be reasonably well described by a single master curve