Hodge-Helmholtz Decompositions of Weighted Sobolev Spaces in Irregular Exterior Domains with Inhomogeneous and Anisotropic Media

We study in detail Hodge-Helmholtz decompositions in non-smooth exterior domains filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to weighted Sobolev spaces into irrotational and solenoidal forms. These decompositions are essential tools, for example, in electro-magnetic theory for exterior domains. In the appendix we translate our results to the classical framework of vector analysis.Comment: Key Words: Hodge-Helmholtz decompositions, Maxwell's equations, electro-magnetic theory, weighted Sobolev space

Three new species of Eupetersia Blüthgen, 1928 (Hymenoptera, Halictidae) from the Oriental Region

Three new species, Eupetersia (Nesoeupetersia) singaporensis sp. nov., collected in a mangrove swamp in Singapore, and Eupetersia (Nesoeupetersia) sabahensis sp. nov., collected in the mountains of Sabah, Borneo, and Eupetersia (Nesoeupetersia) yanegai sp. nov., collected in Thailand, are described. This genus is more diversified in the sub-Saharan region, including Madagascar. The only other Oriental species, E. nathani (Baker, 1974), was described from India and is diagnosed and re-illustrated here

On Maxwell's and Poincare's Constants

We prove that for bounded and convex domains in three dimensions, the Maxwell constants are bounded from below and above by Friedrichs' and Poincar\'e's constants. In other words, the second Maxwell eigenvalues lie between the square roots of the second Neumann-Laplace and the first Dirichlet-Laplace eigenvalue.Comment: Key Words: Maxwell's equations, Maxwell constant, second Maxwell eigenvalue, electro statics, magneto statics, Poincare's inequality, Friedrichs' inequality, Poincare's constant, Friedrichs' constan

On cubics and quartics through a canonical curve

We construct families of quartic and cubic hypersurfaces through a canonical curve, which are parametrized by an open subset in a Grassmannian and a Flag variety respectively. Using G. Kempf's cohomological obstruction theory, we show that these families cut out the canonical curve and that the quartics are birational (via a blowing-up of a linear subspace) to quadric bundles over the projective plane, whose Steinerian curve equals the canonical curve.Comment: 16 page