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Metric and ultrametric spaces of resistances-ScienceDirect

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Discrete Applied Mathematics

Volume 158, Issue 14

, 28 July 2010, Pages 1496-1505

Discrete Applied Mathematics

Metric and ultrametric spaces of resistances

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Abstract

Consider an electrical circuit, each edge e of which is an isotropic conductor with a monomial conductivity function ye∗=yer/μes. In this formula, 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, the case r=s=1 corresponds to the standard Ohm’s law.

In 1987, Gvishiani and Gurvich [A.D. Gvishiani, V.A. Gurvich, Metric and ultrametric spaces of resistances, in: Communications of the Moscow Mathematical Society, Russian Math. Surveys 42 (6 (258)) (1987) 235–236] proved that, for every two nodes a,b of the circuit, 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 obviously implies the standard triangle inequality μa,b≤μa,c+μc,b whenever s≥r. For the case s=r=1, these results were rediscovered in the 1990s. Now, after 23 years, I venture to reproduce the proof of the original result for the following reasons:

It is more general than just the case r=s=1 and one can get several interesting metric and ultrametric spaces playing with parameters r and s. In particular, (i) the effective Ohm resistance, (ii) the length of a shortest path, (iii) the inverse width of a bottleneck path, and (iv) the inverse capacity (maximum flow per unit time) between any pair of terminals a and b provide four examples of the resistance distances μa,b that can be obtained from the above model by the following limit transitions: (i) r(t)=s(t)≡1, (ii) r(t)=s(t)→∞, (iii) r(t)≡1,s(t)→∞, and (iv) r(t)→0,s(t)≡1, as t→∞. In all four cases the limits μa,b=limt→∞μa,b(t) exist for all pairs a,b and the metric inequality μa,b≤μa,c+μc,b holds for all triplets a,b,c, since s(t)≥r(t) for any sufficiently large t. Moreover, the stronger ultrametric inequality μa,b≤max(μa,c,μc,b) holds for all triplets a,b,c in examples (iii) and (iv), since in these two cases s(t)/r(t)→∞, as t→∞.

Communications of the Moscow Math. Soc. in Russ. Math. Surveys were (and still are) strictly limited to two pages; the present paper is much more detailed.

Although a translation in English of the Russ. Math. Surveys is available, it is not free in the web and not that easy to find.

The last but not least: priority.

Keywords

Metric and ultrametric spaces
Shortest and bottleneck paths
Maximum flow
Ohm law
Joule–Lenz heat
Maxwell principle

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