| Mathematical Sciences / Activity Information | |
| Title: | ADM Seminar | |
| Location: | M-102 Martin | |
| Schedule: | Wednesday, April 09, 2008 3:30 PM to 4:30 PM | |
| Notes: | Algebra and Discrete Math Seminar Paul H. Edelman Vanderbilt University The Inverse Banzhaf Problem TIME&PLACE: Wednesday, April 9, 3:30pm @ Martin M102 REFRESHMENTS: 3:00 pm, Martin O112 ABSTRACT: Let \Delta be a simplicial complex on the ground set {1,2,...,n}=[n]. For each i in [n] let B(\Delta,i) be the number of subsets A in \Delta, which do not contain i, such that A+i is not in \Delta. That is, B(\Delta,i) is the number of times adding i to a set in \Delta produces a set not in \Delta. B(\Delta)=(B(\Delta,1), B(\Delta,2),..., B(\Delta,n)) is called the Banzhaf index of \Delta, and after normalizing it to get a sum of 1 we have the NB(\Delta), the normalized Banzhaf index. The normalized Banzhaf index has been used to measure power in voting games as well as in Boolean circuit theory. In this talk I will consider the inverse problem. Given a nonnegative vector v, of sum 1, when can we find a simplicial complex \Delta so that NB(\Delta) is a good approximation to v? There is virtually nothing known on this subject. I will present some preliminary results that show that one can never well-approximate vectors on the boundary of the standard simplex. Coordinators: Elena Dimitrova [edimit@clemson.edu] and H. Xue [huixue@clemson.edu] |
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