Dipartimento di Scienze di Base e Applicate per l'Ingegneria

Alexander Teplyaev

University of Connecticut

title:
Average power dissipation in AC fractal networks

abstract:
Energy forms on graphs and on fractals can be interpreted in terms of
electric linear networks by assuming that current flows between nodes
(vertices) connected by resistors (edges). A resistor is called a
dissipative element because there is a loss of energy when an
alternating current runs through it. To the contrary, no loss is
caused when the current flows through a non-dissipative element such
an inductor or a capacitor.

In the 60s Feynman described an infinite passive AC linear network
(the infinite ladder), whose nodes were connected by inductors and
capacitors, that would lead to actual power dissipation at some
frequencies. Based on this idea, we study the concept of power
dissipation on graphs and fractals associated to passive linear
networks with non-dissipative components.

We present in detail the so-called Sierpinski ladder fractal network
and construct the power dissipation measure associated with continuous
potentials in this network and prove it to be continuous as well as
singular with respect to an appropriate Hausdorff measure defined on
the fractal dust related to the network.

The talk will be based on recent results of Patricia Alonso-Ruiz,
and earlier preprints
arXiv:1507.05682
arXiv:1605.03890