Volume 10, Issue 4

Strong solvability is proved in the Sobolev space W 2, p (Ω), 1 < p < ∞, for the regular oblique derivative problem assuming .

We give a short proof, in the case of representations over the complex numbers, of G.D. James characterization of the irreducible representations of the symmetric group as intersection of the kernels of suitable invariant operators. Following E. Bolker, P. Diaconis and S. Sternberg, those operators are interpreted as Radon transforms. Our proof is based on the harmonic analysis of the invariant Laplacian that describes the Bernoulli-Laplace diffusion model with many urns.

Goldman's and Mostow's works on bisectors in ℂ H n provide the motivation for this paper. By a linear submanifold of ℂ H n we mean a submanifold of the form exp( V ), where V is a subspace of T x ℂ H n for some x ∈ℂ H n . In this case, x is called an origin of L . Every bisector in ℂ H n is a linear submanifold. We obtain some results about linear submanifolds, bisectors, and their intersections, and generalize some results about bisectors to linear submanifolds. Among other things, we characterize minimal linear submanifolds, obtain sufficient conditions for two linear submanifolds to intersect in a nice set and for a geodesic to lie in a linear submanifold, and obtain equivalent conditions for two bisectors to intersect at constant angle.

Suppose that the real projective plane is contained as a Baer subplane in a compact 4-dimensional plane . Salzmann [15] has shown that if the group of projective collineations of extends to , then is isomorphic to the complex plane. This means that there are no compact 4-dimensional Hughes planes. If the group AGL 2 ℝ of offine collineations of extends to , then we call the extended group an offine Hughes group. Here, non-classical examples exist. We show that, up to duality, every action of AGL 2 ℝ on a compact 4-dimensional plane is an offine Hughes group, and we describe explicitly the possible actions on the point space of as transformation groups. In subsequent work (jointly with N. Knarr and H. Klein), this will be used to determine the possible planes , as well. This will complete the classification of 4-dimensional planes admitting a non-solvable group of dimension at least 6.

We define a sub-conformal structure on a contact distribution over a smooth manifold and find a complete set of local invariants. This structure is shown to be a generalization of CR structures and the sub-conformal invariants reduce to the CR invariants in that case. It includes a class of almost CR structures which arise naturally as hypersurfaces of almost complex manifolds. The main difficulty of our construction is that, contrary to the integrable CR case, the appropriate bundle of coframes where the invariants are defined is not a G -structure.

Since 1946 it has been an open question which compact Lie groups can act smoothly on some sphere with exactly one fixed point. In this paper we solve the problem completely for finite groups: these groups are exactly those which can act smoothly on some disk without fixed points, a class determined by R. Oliver. Our main tools are the Burnside ring and the Grothendieck-Witt ring (classical to some extent) and a form of equivariant surgery theory allowing middle-dimensional singular sets developed recently.