161 research outputs found
Hydrodynamic limit for a zero-range process in the Sierpinski gasket
We prove that the hydrodynamic limit of a zero-range process evolving in
graphs approximating the Sierpinski gasket is given by a nonlinear heat
equation. We also prove existence and uniqueness of the hydrodynamic equation
by considering a finite-difference scheme.Comment: 24 pages, 1 figur
Nontangential limits and Fatou-type theorems on post-critically finite self-similar sets
In this paper we study the boundary limit properties of harmonic functions on
, the solutions to the Poisson equation where is a p.c.f. set
and its Laplacian given by a regular harmonic structure. In
particular, we prove the existence of nontangential limits of the corresponding
Poisson integrals, and the analogous results of the classical Fatou theorems
for bounded and nontangentially bounded harmonic functions.Comment: 22 page
Vibration Spectra of the -Tree Fractal
We introduce a family of post-critically finite fractal trees indexed by the
number of branches they possess. Then we produce a Laplacian operator on graph
approximations to these fractals and use spectral decimation to describe the
spectrum of the Laplacian on these trees. Lastly we consider the behavior of
the spectrum as the number of branches increases.Comment: 21 pages, 4 figure
A renormalisation approach to excitable reaction-diffusion waves in fractal media
Of fundamental importance to wave propagation in a wide range of physical phenomena is the structural geometry of the supporting medium. Recently, there have been several investigations on wave propagation in fractal media. We present here a renormalization approach to the study of reaction-diffusion (RD) wave propagation on finitely ramified fractal structures. In particular we will study a Rinzel-Keller (RK) type model, supporting travelling waves on a Sierpinski gasket (SG), lattice
Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals
We investigate the existence of the meromorphic extension of the spectral
zeta function of the Laplacian on self-similar fractals using the classical
results of Kigami and Lapidus (based on the renewal theory) and new results of
Hambly and Kajino based on the heat kernel estimates and other probabilistic
techniques. We also formulate conjectures which hold true in the examples that
have been analyzed in the existing literature
Spectral analysis on infinite Sierpinski fractafolds
A fractafold, a space that is locally modeled on a specified fractal, is the
fractal equivalent of a manifold. For compact fractafolds based on the
Sierpinski gasket, it was shown by the first author how to compute the discrete
spectrum of the Laplacian in terms of the spectrum of a finite graph Laplacian.
A similar problem was solved by the second author for the case of infinite
blowups of a Sierpinski gasket, where spectrum is pure point of infinite
multiplicity. Both works used the method of spectral decimations to obtain
explicit description of the eigenvalues and eigenfunctions. In this paper we
combine the ideas from these earlier works to obtain a description of the
spectral resolution of the Laplacian for noncompact fractafolds. Our main
abstract results enable us to obtain a completely explicit description of the
spectral resolution of the fractafold Laplacian. For some specific examples we
turn the spectral resolution into a "Plancherel formula". We also present such
a formula for the graph Laplacian on the 3-regular tree, which appears to be a
new result of independent interest. In the end we discuss periodic fractafolds
and fractal fields
Analysis of ultrasonic transducers with fractal architecture
Ultrasonic transducers composed of a periodic piezoelectric composite are generally accepted as the design of choice in many applications. Their architecture is normally very regular and this is due to manufacturing constraints rather than performance optimisation. Many of these manufacturing restrictions no longer hold due to new production methods such as computer controlled, laser cutting, and so there is now freedom to investigate new types of geometry. In this paper, the plane wave expansion model is utilised to investigate the behaviour of a transducer with a self-similar architecture. The Cantor set is utilised to design a 2-2 conguration, and a 1-3 conguration is investigated with a Sierpinski Carpet geometry
Fredholm Modules on P.C.F. Self-Similar Fractals and their Conformal Geometry
The aim of the present work is to show how, using the differential calculus
associated to Dirichlet forms, it is possible to construct Fredholm modules on
post critically finite fractals by regular harmonic structures. The modules are
d-summable, the summability exponent d coinciding with the spectral dimension
of the generalized laplacian operator associated with the regular harmonic
structures. The characteristic tools of the noncommutative infinitesimal
calculus allow to define a d-energy functional which is shown to be a
self-similar conformal invariant.Comment: 16 page
From self-similar groups to self-similar sets and spectra
The survey presents developments in the theory of self-similar groups leading
to applications to the study of fractal sets and graphs, and their associated
spectra
From non-symmetric particle systems to non-linear PDEs on fractals
We present new results and challenges in obtaining hydrodynamic limits for
non-symmetric (weakly asymmetric) particle systems (exclusion processes on
pre-fractal graphs) converging to a non-linear heat equation. We discuss a
joint density-current law of large numbers and a corresponding large deviations
principle.Comment: v2: 10 pages, 1 figure. To appear in the proceedings for the 2016
conference "Stochastic Partial Differential Equations & Related Fields" in
honor of Michael R\"ockner's 60th birthday, Bielefel
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