Zig-zag chains and metric equivalences between ultrametric spaces

The zig-zag product

The expander constructions based on algebraic methods can give expanders that are both explicit (i.e. we can quickly construct the graph, or even obtain neighborhood information without constructing the entire graph, and Ramanujan, meaning that the spectral gap is essentially as large as possible. It also follows from this spectral bound that the edge expansion of Ramanujan graphs is essentiall...

Metric and ultrametric spaces of resistances a

Given an electrical circuit each edge e of which is an isotropic conductor with a monomial conductivity function y∗ e = y r e/μ s e. In this formula, ye is the potential difference and y∗ e 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 correspond to the standard Ohm low. In 1987, Gvish...

Metric and ultrametric spaces of resistances

Given an electrical circuit each edge e of which is an isotropic conductor with a monomial conductivity function y∗ e = y r e/μ s e. In this formula, ye is the potential difference and y ∗ e 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 law. In ...

Zig-zag Path to Understanding

OUT understanding of the fundamental physical limits of information handling has developed along a very convoluted path. Most of the initially plausible physical conjectures have turned out to be wrong. A participant’s personal view of these events is not a disciplined contribution to the history of science. I do, however, last my own mistakes along with those of others.

Reverse mathematics and Isbell's zig-zag theorem