Fractals

Probability, Analysis and Mathematical Physics on Fractals 2018

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Each year we are looking for a group of undergraduate students to work on Probability, Analysis and Mathematical Physics on Fractals. The aim of the projects will be exploration of differential equations and various operators on fractal domains. Students in the project are supposed to have the usual background in linear algebra and differential equations. Knowledge of Matlab, Mathematica, other computer algebra systems, or programming, as well as proof writing, mathematical analysis, and probability may be helpful but is not required. Previous undergraduate work includes published papers on the eigenmodes (vibration modes) of the Laplacian (2nd derivative) of functions that live on Sierpinski gasket type fractals, and the electrical resistance of fractal networks, as well as work on Laplacians on projective limit spaces. The exact choice of the topics to study will depend on the students’ background and interests. Besides being interesting, taking part in a research project like this may be very useful in the future (for instance, when applying to graduate schools).

Luke Rogers, Gamal Mograby, Sasha Teplyaev, Patricia Alonso-Ruiz

Gradients on Higher Dimensional Sierpinski Gaskets

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Group Members

Luke Brown,  Giovanni E Ferrer Suarez, Karuna Sangam.

Supervisors

Gamal Mograby, Dan Kelleher, Luke Rogers, Sasha Teplyaev.

Overview

Laplacians have been well studied on post-critically finite (PCF) fractals. However, less is known about gradients on such fractals. Building on work by Teplyaev, we generalize results regarding the existence and continuity of the gradient on the standard Sierpinski Gasket to higher dimensional Sierpinski Gaskets. In particular, we find that, for functions with a continuous Laplacian, the gradient must be defined almost everywhere, and specify a set of points for which it is defined. Furthermore, we provide a counterexample on higher-dimensional Sierpinski gaskets where the Laplacian is continuous but the gradient is not defined everywhere. We conjecture that Hölder continuity of the Laplacian is a condition strong enough to guarantee that the gradient exists at each point.

Publication: arXiv:1908.10539  Fractals Vol. 28, No. 06, 2050108 (2020)

doi.org/10.1142/S0218348X2050108X

Presentation

Poster

Spectral Analysis on Graphs Related to the Basilica Julia Set

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Group Members

Courtney George, Samantha Jarvis.

Supervisors

Dan Kelleher, Luke Rogers, Sasha Teplyaev.

Overview

We analyze the spectra of a sequence of graphs constructed from the Schreier graphs of the Basilica group.  Our analysis differs from earlier work of Grigorchuk and Zuk in that it is based on a macroscopic decomposition of the graphs. This method gives precise information about the multiplicities of eigenvalues and, consequently, good information about the spectral measures of large graphs. It also permits a proof of the existence of gaps in the spectrum of limiting graphs.

Publication: arXiv:1908.10505

Spectral properties of graphs associated to the Basilica group

Antoni Brzoska, Courtney George, Samantha Jarvis, Luke G. Rogers, Alexander Teplyaev

Presentation

Poster

Spectrum of the Magnetic Laplacian on the Diamond Fractal

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Group MembersIMG_5606

Stephen Loew, Madeline Hansalik, Aubrey Coffey

Supervisors

Luke Rogers, Antoni Brzoska

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: Journal of Physics A: Mathematical and Theoretical, Volume 50, Number 32

arXiv:1704.01609

Presentation

Magnetic Spectral Decimation

Poster

Power Dissipation in Fractal AC Circuits

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Publication

Paper

Joe P Chen8,1, Luke G Rogers9,2, Loren Anderson3, Ulysses Andrews2, Antoni Brzoska2, Aubrey Coffey4, Hannah Davis3, Lee Fisher5, Madeline Hansalik6, Stephen Loew7Show full author list

Published 14 July 2017 • © 2017 IOP Publishing Ltd
Journal of Physics A: Mathematical and Theoretical, Volume 50, Number 32 Citation Joe P Chen et al 2017 J. Phys. A: Math. Theor. 50 325205

arXiv.org

Project

Project Information

Power Dissipation in Fractal AC Circuits

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Group Members

Loren Anderson, Aubrey Coffey, Hannah Davis, Lee Fisher, Madeline Hansalik, Stephen LoewIMG_5534

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

Fractal AC Circuits

Poster