Applied Mathematicsematics

Download American Mathematical Monthly, volume 117, number 3, March by Daniel J. Velleman PDF

By Daniel J. Velleman

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Extra info for American Mathematical Monthly, volume 117, number 3, March 2010

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4169/000298910X480072 220 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 Theorem 2 (T. Lyons [11]). A graph G is transient if and only if there exists a flow θ on G (from some vertex a) with θ > 0 and E (θ) < ∞. While Theorem 2 is very well known, the published proofs we are aware of require either a rather lengthy introduction to electrical networks, as in [4], or familiarity with infinite-dimensional Hilbert space, as in [11]; while these are valuable tools, we have tried here to give a short and elementary route to Theorem 2 by presenting only the relevant part of the electrical network theory.

Given any convex polyhedron Q together with an isomorphism β : (V Q , F Q , I Q ) ∼ = (V , F , I ) of incidence structures we obtain a realization of , by mapping each vertex of P to the position of the corresponding (under β) vertex of Q and mapping each face of P to the plane containing the corresponding face of Q. In particular, P itself gives a realization of , and when convenient we’ll also use the letter P for this realization. Conversely, while not every realization of comes from a convex polyhedron in this way, any realization of that’s sufficiently close to P in the natural topology for the space of realizations gives—for example, by taking the convex hull of the image of αV —a convex polyhedron whose underlying abstract polyhedron can be identified with .

Lyons [11]). A graph G is transient if and only if there exists a flow θ on G (from some vertex a) with θ > 0 and E (θ) < ∞. While Theorem 2 is very well known, the published proofs we are aware of require either a rather lengthy introduction to electrical networks, as in [4], or familiarity with infinite-dimensional Hilbert space, as in [11]; while these are valuable tools, we have tried here to give a short and elementary route to Theorem 2 by presenting only the relevant part of the electrical network theory.

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