1,873 research outputs found
\v{C}ech-Delaunay gradient flow and homology inference for self-maps
We call a continuous self-map that reveals itself through a discrete set of
point-value pairs a sampled dynamical system. Capturing the available
information with chain maps on Delaunay complexes, we use persistent homology
to quantify the evidence of recurrent behavior. We establish a sampling theorem
to recover the eigenspace of the endomorphism on homology induced by the
self-map. Using a combinatorial gradient flow arising from the discrete Morse
theory for \v{C}ech and Delaunay complexes, we construct a chain map to
transform the problem from the natural but expensive \v{C}ech complexes to the
computationally efficient Delaunay triangulations. The fast chain map algorithm
has applications beyond dynamical systems.Comment: 22 pages, 8 figure
Symmetry and the thermodynamics of currents in open quantum systems
Symmetry is a powerful concept in physics, and its recent application to
understand nonequilibrium behavior is providing deep insights and
groundbreaking exact results. Here we show how to harness symmetry to control
transport and statistics in open quantum systems. Such control is enabled by a
first-order-type dynamic phase transition in current statistics and the
associated coexistence of different transport channels (or nonequilibrium
steady states) classified by symmetry. Microreversibility then ensues, via the
Gallavotti-Cohen fluctuation theorem, a twin dynamic phase transition for rare
current fluctuations. Interestingly, the symmetry present in the initial state
is spontaneously broken at the fluctuating level, where the quantum system
selects the symmetry sector that maximally facilitates a given fluctuation. We
illustrate these results in a qubit network model motivated by the problem of
coherent energy harvesting in photosynthetic complexes, and introduce the
concept of a symmetry-controlled quantum thermal switch, suggesting
symmetry-based design strategies for quantum devices with controllable
transport properties.Comment: 12 pages, 6 figure
Limits of geometries
A geometric transition is a continuous path of geometric structures that
changes type, meaning that the model geometry, i.e. the homogeneous space on
which the structures are modeled, abruptly changes. In order to rigorously
study transitions, one must define a notion of geometric limit at the level of
homogeneous spaces, describing the basic process by which one homogeneous
geometry may transform into another. We develop a general framework to describe
transitions in the context that both geometries involved are represented as
sub-geometries of a larger ambient geometry. Specializing to the setting of
real projective geometry, we classify the geometric limits of any sub-geometry
whose structure group is a symmetric subgroup of the projective general linear
group. As an application, we classify all limits of three-dimensional
hyperbolic geometry inside of projective geometry, finding Euclidean, Nil, and
Sol geometry among the limits. We prove, however, that the other Thurston
geometries, in particular and
, do not embed in any limit of
hyperbolic geometry in this sense.Comment: 40 pages, 2 figures. new in v2: figure 2 added, minor edits to
Sections 1,2,
Quantum Isometry Group for Spectral Triples with Real Structure
Given a spectral triple of compact type with a real structure in the sense of
[Dabrowski L., J. Geom. Phys. 56 (2006), 86-107] (which is a modification of
Connes' original definition to accommodate examples coming from quantum group
theory) and references therein, we prove that there is always a universal
object in the category of compact quantum group acting by orientation
preserving isometries (in the sense of [Bhowmick J., Goswami D., J. Funct.
Anal. 257 (2009), 2530-2572]) and also preserving the real structure of the
spectral triple. This gives a natural definition of quantum isometry group in
the context of real spectral triples without fixing a choice of 'volume form'
as in [Bhowmick J., Goswami D., J. Funct. Anal. 257 (2009), 2530-2572]
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