**IMA Complex Systems Seminar**

**1:30, Thursday, April 8,
2004 **

Incomplete U-statistics, random permanents and beyond

Department of Mathematics

University of Louisville

Louisville, KY 40292 USA

The talk will present some recent results
from the general theory of the incomplete U-statistics of increasing order. In particular,
the concepts of the so-called "permanent " and "minimal"
designs will be developed. These
designs turn out to be asymptotically efficient as well as (under some
conditions on the kernel functions) optimal with respect to the minimal variance
criterion. Applications of this theory
to investigating asymptotic behavior of a random permanent function (for instance
for counting matchings in a bipartite random graph), general sub-sampling
methods, as well as and some problems in the protein sequences alignments will
be discussed.