arXiv subjects

Matthew Begue

Publications and source records attributed to Matthew Begue.

3 recordsLinked to original sources

arXiv2012

Harmonic functions and the spectrum of the Laplacian on the Sierpinski carpet

Kusuoka and Zhou have defined the Laplacian on the Sierpinski carpet using average values of functions on small cells and the graph structure of cell adjacency. We have implemented an algorithm that uses their method to approximate solutions to boundary value problems. As a result we have a wealth of data concerning harmonic functions with prescribed boundary values, and eigenfunctions of the Laplacian with either Neumann or Dirichlet boundary conditions. We will present some of this data and discuss some ideas for defining normal derivatives on the boundary of the carpet.

math-ph↗

arXiv2011

Random walks on barycentric subdivisions and the Strichartz hexacarpet

We investigate the relation between simple random walks on repeated barycentric subdivisions of a triangle and a self-similar fractal, Strichartz hexacarpet, which we introduce. We explore a graph approximation to the hexacarpet in order to establish a graph isomorphism between the hexacarpet approximations and Barycentric subdivisions of the triangle, and discuss various numerical calculations performed on the these graphs. We prove that equilateral barycentric subdivisions converge to a self-similar geodesic metric space of dimension log(6)/log(2), or about 2.58. Our numerical experiments give evidence to a conjecture that the simple random walks on the equilateral barycentric subdivisions converge to a continuous diffusion process on the Strichartz hexacarpet corresponding to a different spectral dimension (estimated numerically to be about 1.74).

math.MG↗

arXiv2010

Spectrum and Heat Kernel Asymptotics on General Laakso Spaces

We introduce a method of constructing a general Laakso space while calculating the spectrum and multiplicities of the Laplacian operator on it. Using this information, we found the leading term of the trace of the heat kernel of a Laakso space and its spectral dimension.

math.CA↗