For a local ring (R, \M) of infinite residue field and positive depth, we
address the question raised by H. Dao on how to control the asymptotic
behaviour of the \M-full, full, and weakly \M-full properties of certain
ideals (such notions were first investigated by D. Rees and J. Watanabe), by
means of bounding appropriate numbers which express such behaviour. We
establish upper bounds, and in certain cases even formulas for such invariants.
The main tools used in our results are reduction numbers along with
Ratliff-Rush closure of ideals, and also the Castelnuovo-Mumford regularity of
the Rees algebra of \M.Comment: 11 pages. Submitted for publicatio