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Singularities of the divergence of continuous vector fields and uniform Hausdorff estimates
We prove that every closed set which is not sigma-finite with respect to the
Hausdorff measure H^{N-1} carries singularities of continuous vector fields in
the Euclidean space R^N for the divergence operator. We also show that finite
measures which do not charge sets of sigma-finite Hausdorff measure H^{N-1} can
be written as an L^1 perturbation of the divergence of a continuous vector
field. The main tool is a property of approximation of measures in terms of the
Hausdorff content
Structure of Divergence-Free Lie Algebras
One of the four well-known series of simple Lie algebras of Cartan type is
the series of Lie algebras of Special type, which are divergence-free Lie
algebras associated with polynomial algebras and the operators of taking
partial derivatives, connected with volume-preserving diffeomorphisms. In this
paper, we determine the structure space of the divergence-free Lie algebras
associated with pairs of a commutative associative algebra with an identity
element and its finite-dimensional commutative locally-finite derivation
subalgebra such that the commutative associative algebra is derivation-simple
with respect to the derivation subalgebra.Comment: 36 pages; Latex fil
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