592 research outputs found
Stretching and folding versus cutting and shuffling: An illustrated perspective on mixing and deformations of continua
We compare and contrast two types of deformations inspired by mixing
applications -- one from the mixing of fluids (stretching and folding), the
other from the mixing of granular matter (cutting and shuffling). The
connection between mechanics and dynamical systems is discussed in the context
of the kinematics of deformation, emphasizing the equivalence between stretches
and Lyapunov exponents. The stretching and folding motion exemplified by the
baker's map is shown to give rise to a dynamical system with a positive
Lyapunov exponent, the hallmark of chaotic mixing. On the other hand, cutting
and shuffling does not stretch. When an interval exchange transformation is
used as the basis for cutting and shuffling, we establish that all of the map's
Lyapunov exponents are zero. Mixing, as quantified by the interfacial area per
unit volume, is shown to be exponentially fast when there is stretching and
folding, but linear when there is only cutting and shuffling. We also discuss
how a simple computational approach can discern stretching in discrete data.Comment: REVTeX 4.1, 9 pages, 3 figures; v2 corrects some misprints. The
following article appeared in the American Journal of Physics and may be
found at http://ajp.aapt.org/resource/1/ajpias/v79/i4/p359_s1 . Copyright
2011 American Association of Physics Teachers. This article may be downloaded
for personal use only. Any other use requires prior permission of the author
and the AAP
Does a billiard orbit determine its (polygonal) table?
We introduce a new equivalence relation on the set of all polygonal
billiards. We say that two billiards (or polygons) are order equivalent if each
of the billiards has an orbit whose footpoints are dense in the boundary and
the two sequences of footpoints of these orbits have the same combinatorial
order. We study this equivalence relation with additional regularity conditions
on the orbit
Lorenz, G\"{o}del and Penrose: New perspectives on determinism and causality in fundamental physics
Despite being known for his pioneering work on chaotic unpredictability, the
key discovery at the core of meteorologist Ed Lorenz's work is the link between
space-time calculus and state-space fractal geometry. Indeed, properties of
Lorenz's fractal invariant set relate space-time calculus to deep areas of
mathematics such as G\"{o}del's Incompleteness Theorem. These properties,
combined with some recent developments in theoretical and observational
cosmology, motivate what is referred to as the `cosmological invariant set
postulate': that the universe can be considered a deterministic dynamical
system evolving on a causal measure-zero fractal invariant set in its
state space. Symbolic representations of are constructed explicitly based
on permutation representations of quaternions. The resulting `invariant set
theory' provides some new perspectives on determinism and causality in
fundamental physics. For example, whilst the cosmological invariant set appears
to have a rich enough structure to allow a description of quantum probability,
its measure-zero character ensures it is sparse enough to prevent invariant set
theory being constrained by the Bell inequality (consistent with a partial
violation of the so-called measurement independence postulate). The primacy of
geometry as embodied in the proposed theory extends the principles underpinning
general relativity. As a result, the physical basis for contemporary programmes
which apply standard field quantisation to some putative gravitational
lagrangian is questioned. Consistent with Penrose's suggestion of a
deterministic but non-computable theory of fundamental physics, a
`gravitational theory of the quantum' is proposed based on the geometry of
, with potential observational consequences for the dark universe.Comment: This manuscript has been accepted for publication in Contemporary
Physics and is based on the author's 9th Dennis Sciama Lecture, given in
Oxford and Triest
The Dynamics of Twisted Tent Maps
This paper is a study of the dynamics of a new family of maps from the
complex plane to itself, which we call twisted tent maps. A twisted tent map is
a complex generalization of a real tent map. The action of this map can be
visualized as the complex scaling of the plane followed by folding the plane
once. Most of the time, scaling by a complex number will "twist" the plane,
hence the name. The "folding" both breaks analyticity (and even smoothness) and
leads to interesting dynamics ranging from easily understood and highly
geometric behavior to chaotic behavior and fractals.Comment: 87 pages. This is my Ph.D. thesis from IUPU
Tiling billiards and Dynnikov's helicoid
Here are two problems. First, understand the dynamics of a tiling billiard in
a cyclic quadrilateral periodic tiling. Second, describe the topology of
connected components of plane sections of a centrally symmetric subsurface of genus . In this note we show that these two
problems are related via a helicoidal construction proposed recently by Ivan
Dynnikov. The second problem is a particular case of a classical question
formulated by Sergei Novikov. The exploration of the relationship between a
large class of tiling billiards (periodic locally foldable tiling billiards)
and Novikov's problem in higher genus seems promising, as we show in the end of
this note.Comment: 18 pages, 5 figure
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