# Sobolev Algebra Counterexamples

by Thierry Coulhon, Luke G. Rogers

In the Euclidean setting the Sobolev spaces $$W^{α,p}\cap L^\infty$$ are algebras for the pointwise product when α>0 and p∈(1,∞). This property has recently been extended to a variety of geometric settings. We produce a class of fractal examples where it fails for a wide range of the indices α,p.

Arxiv prepint version

Published version in JFG

# Power dissipation in fractal AC circuits

by Joe P Chen, Luke G. Rogers, Loren Anderson, Ulysses Andrews, Antoni Brzoska, Aubrey Coffey, Hannah Davis, Lee Fisher, Madeline Hansalik, Stephen Loew and Alexander Teplyaev.

We extend Feynman’s analysis of an infinite ladder circuit to fractal circuits, providing examples in which fractal circuits constructed with purely imaginary impedances can have characteristic impedances with positive real part. Continue reading