Cotangent functors of curve singularities

Let R be a complete analytic k-algebra of dimension one, where k is a field of characteristic 0. For a Noetherian normalization , the cotangent functors T_i(R/A, R) are of finite length. It is shown that if R is smoothable, then for all j, where denotes the complementary module. This result is used to give estimates for the length of the torsion of the module of differentials _R/k.     

Connectedness properties of polynomial maps between affine spaces

Let f:A^mAⁿ be a polynomial map between affine spaces. We give some sufficient conditions for the connectedness of the difference kernel of f and relate this to the Jacobian Conjecture.     

Mixed finite element approximation of the vector potential

Summary We consider a mixed finite element approximation of the three dimensional vector potential, which plays an important rôle in the simulation of perfect fluids and in the calculation of rotational corrections to transonic potential flows. The central point of our approach is a saddlepoint formulation of the essential boundary conditions. In particular, this avoids the wellknown Babuka paradox when approximating smooth domains by polyhedrons. Using piecewise linear/piecewise constant elements for the vector potential/the boundary terms, we obtain optimal error estimates under minimal regularity assumptions for the solution of the continuous problem.