7,191 research outputs found
Term Rewriting Systems as Topological Dynamical Systems
Topological dynamics is, roughly, the study of phenomena related to iterations of continuous maps from a metric space to itself. We show how the rewrite relation in term rewriting gives rise to dynamical systems in two distinct, natural ways: (A) One in which any deterministic rewriting strategy induces a dynamical system on the set of finite and infinite terms endowed with the usual metric, and (B) one in which the unconstrained rewriting relation induces a dynamical system on sets of sets of terms, specifically the set of compact subsets of the set of finite and infinite terms endowed with the Hausdorff metric.
For both approaches, we give sufficient criteria for the induced systems to be well-defined dynamical systems and for (A) we demonstrate how the classic topological invariant called topological entropy turns out to be much less useful in the setting of term rewriting systems than in symbolic dynamics
Dynamical maximum entropy approach to flocking
Peer reviewedPublisher PD
The Dynamics of Hybrid Metabolic-Genetic Oscillators
The synthetic construction of intracellular circuits is frequently hindered
by a poor knowledge of appropriate kinetics and precise rate parameters. Here,
we use generalized modeling (GM) to study the dynamical behavior of topological
models of a family of hybrid metabolic-genetic circuits known as
"metabolators." Under mild assumptions on the kinetics, we use GM to
analytically prove that all explicit kinetic models which are topologically
analogous to one such circuit, the "core metabolator," cannot undergo Hopf
bifurcations. Then, we examine more detailed models of the metabolator.
Inspired by the experimental observation of a Hopf bifurcation in a
synthetically constructed circuit related to the core metabolator, we apply GM
to identify the critical components of the synthetically constructed
metabolator which must be reintroduced in order to recover the Hopf
bifurcation. Next, we study the dynamics of a re-wired version of the core
metabolator, dubbed the "reverse" metabolator, and show that it exhibits a
substantially richer set of dynamical behaviors, including both local and
global oscillations. Prompted by the observation of relaxation oscillations in
the reverse metabolator, we study the role that a separation of genetic and
metabolic time scales may play in its dynamics, and find that widely separated
time scales promote stability in the circuit. Our results illustrate a generic
pipeline for vetting the potential success of a potential circuit design,
simply by studying the dynamics of the corresponding generalized model
Topological Model for Domain Walls in (Super-)Yang-Mills Theories
We derive a topological action that describes the confining phase of
(Super-)Yang-Mills theories with gauge group , similar to the work
recently carried out by Seiberg and collaborators. It encodes all the
Aharonov-Bohm phases of the possible non-local operators and phases generated
by the intersection of flux tubes. Within this topological framework we show
that the worldvolume theory of domain walls contains a Chern-Simons term at
level also seen in string theory constructions. The discussion can also
illuminate dynamical differences of domain walls in the supersymmetric and
non-supersymmetric framework. Two further analogies, to string theory and the
fractional quantum Hall effect might lead to additional possibilities to
investigate the dynamics
Thermodynamic formalism for the Lorentz gas with open boundaries in dimensions
A Lorentz gas may be defined as a system of fixed dispersing scatterers, with
a single light particle moving among these and making specular collisions on
encounters with the scatterers. For a dilute Lorentz gas with open boundaries
in dimensions we relate the thermodynamic formalism to a random flight
problem. Using this representation we analytically calculate the central
quantity within this formalism, the topological pressure, as a function of
system size and a temperature-like parameter \ba. The topological pressure is
given as the sum of the topological pressure for the closed system and a
diffusion term with a \ba-dependent diffusion coefficient. From the
topological pressure we obtain the Kolmogorov-Sinai entropy on the repeller,
the topological entropy, and the partial information dimension.Comment: 7 pages, 5 figure
Interpolating relativistic and non-relativistic Nambu-Goldstone and Higgs modes
When a continuous symmetry is spontaneously broken in non-relativistic
theories, there appear Nambu-Goldstone (NG) modes, whose dispersion relations
are either linear (type-I) or quadratic (type-II). We give a general framework
to interpolate between relativistic and non-relativistic NG modes, revealing a
nature of type-I and II NG modes in non-relativistic theories. The
interpolating Lagrangians have the nonlinear Lorentz invariance which reduces
to the Galilei or Schrodinger invariance in the non-relativistic limit. We find
that type-I and type-II NG modes in the interpolating region are accompanied
with a Higgs mode and a chiral NG partner, respectively, both of which are
gapful. In the ultra-relativistic limit, a set of a type-I NG mode and its
Higgs partner remains, while a set of type-II NG mode and gapful NG partner
turns to a set of two type-I NG modes. In the non-relativistic limit, the both
types of accompanied gapful modes become infinitely massive, disappearing from
the spectrum. The examples contain a phonon in Bose-Einstein condensates, a
magnon in ferromagnets, and a Kelvon and dilaton-magnon localized around a
skyrmion line in ferromagnets.Comment: 20 pages, no figur
Edge-state instabilities of bosons in a topological band
In this work, we consider the dynamics of bosons in bands with non-trivial
topological structure. In particular, we focus on the case where bosons are
prepared in a higher-energy band and allowed to evolve. The Bogoliubov theory
about the initial state can have a dynamical instability, and we show that it
is possible to achieve the interesting situation where the topological edge
modes are unstable while all bulk modes are stable. Thus, after the initial
preparation, the edge modes will become rapidly populated. We illustrate this
with the Su-Schrieffer-Heeger model which can be realized with a double-well
optical lattice and is perhaps the simplest model with topological edge states.
This work provides a direct physical consequence of topological bands whose
properties are often not of immediate relevance for bosonic systems.Comment: 7 pages, 2 figure
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