Upper limit of bandwidth improvement by optical equalization of graded-index fibres

Optical equalization was suggested as a means for increasing the bandwidth of optical fibre communication links when non-optimal graded-index fibre profiles are available. In this paper it is shown that by proper choice of the length as well as of the index of the compensating fibre one can significantly increase the bandwidth capability of the link. An upper limit of this improvement is derived, which is much larger than that obtained with alternating fibres of equal lengths.     

Über eine scharfe Schranke in der Regularitätstheorie elliptischer Systeme

Ohne Zusammenfassung     

Families of sets with locally bounded width

A family of sets ? islocally k- wide if and only if the width (as a poset ordered by inclusion) of ? is at mostk for everyx. The directed covering graph of a locally 1-wide family of sets is a forest of rooted trees. It is shown that if ? is a locallyk-wide family of subsets of {1,...,n}, then |?|≤(2k) ^k?1 n. The proof involves a counting argument based on families of closed sets associated with theSperner closures in the filters of ?. The Sperner closure ofU in ? is the intersection of the members of the greatest Sperner antichain of ? _U = {V ∈ ?|V ?U}. This closure operation is related to a generalization of maximality in posets.