# Fractals

# Spectrum of the Magnetic Laplacian on the Diamond Fractal

## Group Members

Stephen Loew, Madeline Hansalik, Aubrey Coffey

## Supervisors

## Overview

The diamond fractal is a fractal that is obtained in the following manner. Start with a graph with two vertices and an edge and replace the edge with two new vertices connected to our original vertices to obtain a diamond shaped graph. The diamond fractal is defined to be the limiting object after continuing with the edge replacement indefinitely. In the project, the spectrum of magnetic Laplacian operators on graph approximations to the diamond fractal was computed.

Given a level n approximation to the fractal with known magnetic field strengths through cells and holes, it is possible to determine the net magnetic field through the cells and holes of the preceding graph approximations. The spectral similarity relation between the operators on successive graph approximations was worked out, with the corresponding spectral decimation polynomial depending on the magnetic field strengths. A poster and talk on this work was presented at the REU Mini-Symposium at UConn.

## Publication

# arXiv:1704.01609

## Presentation

# Power Dissipation in Fractal AC Circuits

# Publication

# Power Dissipation in Fractal AC Circuits

# Power Dissipation in Fractal AC Circuits

## Group Members

Loren Anderson, Aubrey Coffey, Hannah Davis, Lee Fisher, Madeline Hansalik, Stephen Loew

## Supervisors

Alexander Teplyaev, Luke Rogers, Antoni Brzoska, Ulysses Andrews

## Overview

In this project, Feynman’s analysis of an infinite ladder circuit is extended to fractal circuits, in particular, a Sierpinski Ladder circuit and two variants of a Hanoi circuit. These circuits are constructed by taking the limit of graph approximations with inductors and capacitors placed along edges in a well-defined manner. Inductors, capacitors and resistors all contribute to the impedance within a circuit; but whereas a resistor imparts a real impedance, inductors and capacitors impart a purely imaginary impedance.

For each circuit, the following was accomplished. First, the net impedance between certain boundary points of the circuit was computed. Second, the filter conditions for each circuit were found. A circuit becomes a filter when the net impedance between boundary vertices has a positive real part! Third, it was proved that these impedances can be obtained by placing a small positive resistance epsilon on each edge of the graph approximations, finding the limiting impedance between the boundary vertices, and then taking epsilon to zero. Finally, the construction of harmonic functions on these circuits was outlined.

## Publication

# arXiv:1605.03890

## Presentation

# Spectrum of the Magnetic Laplacian on the Diamond Fractal

# Wave Propagation through a Fractal Medium

# Group Members

## Supervisors

Alexander Teplyaev, Luke Rogers, Ulysses Andrews

## Overview

We consider the wave equation on the unit interval with fractal measure, and use two numerical models to study wave speed and propagation distance. The first approach uses a Fourier series of eigenfunctions of the fractal Laplacian, while the second uses a Markov chain to model the transmission and reflection of classical waves on an approximation of the fractal. These models have complementary advantages and limitations, and we conjecture that they approximate the same fractal wave.

## Presentation

# Magnetic Laplacians of Locally Exact Forms on the Sierpinski Gasket

# Gaps in the Spectrum of the Laplacian on 3N-Gaskets

## Publication

D. Kelleher, N. Gupta, M. Margenot, J. Marsh, W. Oakley, A. Teplyaev, ** Gaps in the spectrum of the Laplacian on 3N-Gaskets**. Communications on Pure and Applied Analysis (CPAA) Pages: 2509 – 2533, Volume 14, Issue 6, November 2015