Title: Metric and Ultrametric Spaces of Resistances
Speaker: Vladimir Gurvich, RUTCOR, Rutgers University
Date: Wednesday, February 9, 2011 12:10pm
Location: Graduate Student Lounge, 7th Floor, Hill Center, Rutgers University, Busch Campus, Piscataway, NJ
Given an electrical circuit each edge e of which is an isotropic conductor with a monomial conductivity function ye* = yer / µes, where, ye is the potential difference and ye* current in e, while µe is the resistance of e; furthermore, r and s are two strictly positive real parameters common for all edges. In particular, r = s = 1 is the standard Ohm law; r = 1/2 is typical for hydraulics or gas dynamics.
For every two nodes a, b the effective resistance µa,b is well-defined and for every three nodes a,b,c the inequality µa,bs/r = µa,cs/r + µc,bs/r holds. It implies the standard metric inequality whenever s = r and ultramentric one when s(t)/r(t) ? 8, as t ? 8.
One can get several metric and ultrametric spaces playing with parameters r and s. In particular, the effective Ohm resistance, the length of a shortest path, the inverse width of a bottleneck path, and the inverse capacity (maximum flow per unit time) between any pair of terminals.
Graduate Student Combinatorics Seminars