The residue formula and the Tolman-Weitsman theorem
  We give a simple direct proof (for the case of Hamiltonian circle actions with isolated fixed points) that Tolman and Weitsman's description of the kernel of the Kirwan map (in other words the sum of those equivariant cohomology classes vanishing on one side of a collection of hyperplanes) is equivalent to the characterization of this kernel given by the residue theorem of Jeffrey and Kirwan. 
Lisa C. Jeffrey