Jake Wildstrom's Research

My publications:

I have a number of ongoing research projects. This is a partial list of problems I'm currently engaged in. If you're interested in learning more about any of them, write to me.

Realtime resource relocation

Service providers on communication, transportation, and any other sort of network may relocate in response to changing conditions. For instance, a mobile radio transmitter may be better served by moving physically closer to its clients, or a software construct might provide faster and less expensive service by being reinstalled on a new system as its users' locations shift. A significant question is when such relocations become expedient, given that there are costs associated with both relocation and long-distance service.

The mathematical approach to this problem under consideration is been to consider a finite set of possible client and server locations and costs associated with both relocation and remote service. Then, given an initial server location and a queue of client-requests to serve, there is clearly a selection of relocations to minimize the overall cost of providing service. This can be determined through (highly inefficient) linear programming. Moving beyond the question of perfect optimization with flawless knowledge of the entire request sequence, two fundamental questions (which raise ancillary questions with regard to network architecture and business practice) arise:

Useful References:

Permutations of k-tuples with distinct sums

It was shown by Marshall Hall that, given an abelian group of order $n$ and (not necessarily distinct) elements a1,..., an thereof, there is a permutation b1,..., bn of elements of the group such that each ai+bi is distinct if and only if the ai add up to zero.

A question which is somewhat similar to this field of inquiry is as follows: if we are given k elements a1,..., ak of Zn, are we guaranteed that there is a permutation σ of 1,...,k so that the sums aii are all distinct modulo n, and if so how many such σ are there? Such a selection is clearly impossible if k>n. Hall's result shows such a selection of σ is not necessarily possible if k=n, but that it always is if k=n-1.

Kézdy and Snevily have conejctured that such a σ always exists if k<n; and furthermore, that the number of such σ is nonstrictly increasing in as a function of n, and nonstrictily increasing as a function of k for k<n.

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