The divergence theorem for unbounded vector fields

In the context of Lebesgue integration, we derive the divergence theorem for unbounded vector fields that can have singularities at every point of a compact set whose Minkowski content of codimension greater than two is finite. The resulting integration by parts theorem is applied to removable sets of holomorphic and harmonic functions.

     

Finite element approximation of vector fields given by curl and divergence

In this paper a finite element approximation scheme for the system curl is considered. The use of pointwise approximation of the boundary condition leads to a nonconforming method. The error estimate is proved and numerically tested.     

Like-linearizations of vector fields

We characterize the n-dimensional vector fields (with or without null linear parts) which can be transformed, under conjugation or orbital equivalence, into their quasi-homogeneous parts of minimum degree and, therefore, have the same dynamics. We give several examples of nilpotent and degenerate systems.     

Vector fields on smooth threefolds vanishing on complete intersections

A result of J. Wahl shows that the existence of a vector field vanishing on an ample divisor of a projective normal variety X implies that X is a cone over this divisor. If X is smooth, X will be isomorphic to the n-dimensional projective space. This paper is a first attempt to generalize Wahl's theorem to higher codimensions: Given a complex smooth projective threefold X and a vector field on X vanishing on an irreducible and reduced curve which is the scheme theoretic intersection of two ample divisors, X is isomorphic to the 3-dimensional projective space or the 3-dimensional quadric. Received: 24 April 2001     

Two vanishing theorems for holomorphic vector bundles of mixed sign

This article was processed by the author using the Springer-Verlag T_EX QPMZGHB macro package 1991.     

Gradient approximation of vector fields

In this paper, we consider an approximation problem which arose in optimal control theory. We seek conditions on a compact subset K of Euclidean n-space such that every continuous vector field on K may be uniformly approximated (on K) by vector fields with strictly positive integrating factors. We prove that such approximation is possible for all K in a particular subclass of the compact sets with topological dimension less than or equal to 1.