  The Castelnuovo-Mumford regularity of a graded ring is an important invariant
in computational commutative algebra, and there is increasing interest in
multigraded generalizations. We study connections between two recent
definitions of multigraded regularity with a view towards a better
understanding of the multigraded Hilbert function of fat point schemes in
P^{n_1} x ... x P^{n_k}.
