We study the reduction of certain integral models of Shimura varieties of PEL type with Iwahori level structure. On these spaces we have the Kottwitz-Rapoport and the p-rank stratification. We show that the p-rank is constant on a KR stratum, generalizing a result of Ngô and Genestier. We prove an abstract, uniform formula for the p-rank on a KR stratum. In the symplectic and in the unitary case we derive explicit formulas for its value. We apply these formulas to the question of the density of the ordinary locus and to the question of the dimension of the p-rank 0 locus.
Laser beam welding is a contact free fusion technique which has gained importance during the last years due to rising need for automatization in industrial processes. However, a well‐known problem is the formation of solidification cracks in the rear region of the melting pool. In order to investigate the formation of solidification cracks which is highly dependent on the chemical composition, the welding speed, weld beam intensity and the resulting temperature gradient in the material, the modeling of the laser beam as a heat source plays a prominent role. In order to investigate heat sources in laser beam welding, this contribution compares two methods, the Goldak model and a second model that mimics the region of the melt pool with the idea of an isothermal region. Suitable boundary value problems to show the differences of the methods are demonstrated.
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