63 research outputs found
On certain families of planar patterns and fractals
This survey article is dedicated to some families of fractals that were
introduced and studied during the last decade, more precisely, families of
Sierpi\'nski carpets: limit net sets, generalised Sierpi\'nski carpets and
labyrinth fractals. We give a unifying approach of these fractals and several
of their topological and geometrical properties, by using the framework of
planar patterns.Comment: survey article, 10 pages, 7 figure
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
Derivations and Dirichlet forms on fractals
We study derivations and Fredholm modules on metric spaces with a local
regular conservative Dirichlet form. In particular, on finitely ramified
fractals, we show that there is a non-trivial Fredholm module if and only if
the fractal is not a tree (i.e. not simply connected). This result relates
Fredholm modules and topology, and refines and improves known results on p.c.f.
fractals. We also discuss weakly summable Fredholm modules and the Dixmier
trace in the cases of some finitely and infinitely ramified fractals (including
non-self-similar fractals) if the so-called spectral dimension is less than 2.
In the finitely ramified self-similar case we relate the p-summability question
with estimates of the Lyapunov exponents for harmonic functions and the
behavior of the pressure function.Comment: to appear in the Journal of Functional Analysis 201
Smooth bumps, a Borel theorem and partitions of smooth functions on p.c.f. fractals
We provide two methods for constructing smooth bump functions and for
smoothly cutting off smooth functions on fractals, one using a probabilistic
approach and sub-Gaussian estimates for the heat operator, and the other using
the analytic theory for p.c.f. fractals and a fixed point argument. The heat
semigroup (probabilistic) method is applicable to a more general class of
metric measure spaces with Laplacian, including certain infinitely ramified
fractals, however the cut off technique involves some loss in smoothness. From
the analytic approach we establish a Borel theorem for p.c.f. fractals, showing
that to any prescribed jet at a junction point there is a smooth function with
that jet. As a consequence we prove that on p.c.f. fractals smooth functions
may be cut off with no loss of smoothness, and thus can be smoothly decomposed
subordinate to an open cover. The latter result provides a replacement for
classical partition of unity arguments in the p.c.f. fractal setting.Comment: 26 pages. May differ slightly from published (refereed) versio
Contributions to computational phylogenetics and algorithmic self-assembly
This dissertation addresses some of the algorithmic and combinatorial problems at the interface between biology and computation.
In particular, it focuses on problems in both computational phylogenetics, an area of study in which computation is used to better understand evolutionary relationships, and algorithmic self-assembly, an area of study in which biological processes are used to perform computation.
The first set of results investigate inferring phylogenetic trees from multi-state character data. We give a novel characterization of when a set of three-state characters has a perfect phylogeny and make progress on a long-standing conjecture regarding the compatibility of multi-state characters.
The next set of results investigate inferring phylogenetic supertrees from collections of smaller input trees when the input trees do not fully agree on the relative positions of the taxa. Two approaches to dealing with such conflicting input trees are considered. The first is to contract a set of edges in the input trees so that the resulting trees have an agreement supertree. The second is to remove a set of taxa from the input trees so that the resulting trees have an agreement supertree. We give fixed-parameter tractable algorithms for both approaches.
We then turn to the algorithmic self-assembly of fractal structures from DNA tiles and investigate approximating the Sierpinski triangle and the Sierpinski carpet with strict self-assembly. We prove tight bounds on approximating the Sierpinski triangle and exhibit a class of fractals that are generalizations of the Sierpinski carpet that can approximately self-assemble.
We conclude by discussing some ideas for further research
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