Modeling Surface Diagrams | The n-Category Café

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Existence and uniqueness of diffusions on finitely ramified self-similar fractals also by C. Sabot.

http://linkinghub.elsevier.com/retrieve/pii/S001295939789934X

In particular the latter contains (p609, i.e. p5 in the PDF):

“With each bond conductivity matrix J [circuit] we associate a positive bilinear form A on E x E by [the energy form]

The following result is well-known:

PROPOSITION 1.7. – The quadratic form A is a Dirichlet form on F …

The map J |-> A is bijective from the set of bond conductivity matrices [circuits] to the set of Dirichlet forms.”

I rather dislike these “This result is well known” appeals to folk theorems, but oh well, at least you’ve got another reference for your result!

]]>Alan Weinstein pointed me to this article, which you may enjoy too:

Christophe Sabot, Electrical networks, symplectic reductions, and application to the renormalization map of self-similar lattices, available as arXiv:math-ph/0304015.

It proposes a class of quadratic forms which it uses as a way to formalize electrical circuits made of resistors. It does a lot of interesting things with quadratic forms of this type. However, I didn’t see a proof that every circuit made of resistors gives a quadratic form of this type, or conversely. I should reread it and compare it to the reference you provide.

]]>http://www-an.acs.i.kyoto-u.ac.jp/~kigami/AOF.pdf

In it, the author shows that Dirichlet forms that I defined above correspond exactly with “Laplacians” which are the linear maps f from potentials to currents that you wrote about in week296, John.

The condition on Laplacians which corresponds to the Markov condition (QT <= Q) of Dirichlet forms is exactly Ben Tilly's intuition that the off-diagonal elements must be non-positive (he has the inverse sense so he writes non-negative).

He also demonstrates that every Dirichlet form, i.e. every Laplacian, comes from an electrical network.

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