35,256 research outputs found
On the study of jamming percolation
We investigate kinetically constrained models of glassy transitions, and
determine which model characteristics are crucial in allowing a rigorous proof
that such models have discontinuous transitions with faster than power law
diverging length and time scales. The models we investigate have constraints
similar to that of the knights model, introduced by Toninelli, Biroli, and
Fisher (TBF), but differing neighbor relations. We find that such knights-like
models, otherwise known as models of jamming percolation, need a ``No Parallel
Crossing'' rule for the TBF proof of a glassy transition to be valid.
Furthermore, most knight-like models fail a ``No Perpendicular Crossing''
requirement, and thus need modification to be made rigorous. We also show how
the ``No Parallel Crossing'' requirement can be used to evaluate the provable
glassiness of other correlated percolation models, by looking at models with
more stable directions than the knights model. Finally, we show that the TBF
proof does not generalize in any straightforward fashion for three-dimensional
versions of the knights-like models.Comment: 13 pages, 18 figures; Spiral model does satisfy property
Holography in General Space-times
We provide a background-independent formulation of the holographic principle.
It permits the construction of embedded hypersurfaces (screens) on which the
entire bulk information can be stored at a density of no more than one bit per
Planck area. Screens are constructed explicitly for AdS, Minkowski, and de
Sitter spaces with and without black holes, and for cosmological solutions. The
properties of screens provide clues about the character of a manifestly
holographic theory.Comment: 30 pages, 8 figures. v2: references adde
Visualizing curved spacetime
I present a way to visualize the concept of curved spacetime. The result is a
curved surface with local coordinate systems (Minkowski Systems) living on it,
giving the local directions of space and time. Relative to these systems,
special relativity holds. The method can be used to visualize gravitational
time dilation, the horizon of black holes, and cosmological models. The idea
underlying the illustrations is first to specify a field of timelike
four-velocities. Then, at every point, one performs a coordinate transformation
to a local Minkowski system comoving with the given four-velocity. In the local
system, the sign of the spatial part of the metric is flipped to create a new
metric of Euclidean signature. The new positive definite metric, called the
absolute metric, can be covariantly related to the original Lorentzian metric.
For the special case of a 2-dimensional original metric, the absolute metric
may be embedded in 3-dimensional Euclidean space as a curved surface.Comment: 15 pages, 20 figure
New perspectives in Arakelov geometry
In this survey, written for the proceedings of the VII meeting of the CNTA
held in May 2002 in Montreal, we describe how Connes' theory of spectral
triples provides a unified view, via noncommutative geometry, of the
archimedean and the totally split degenerate fibers of an arithmetic surface.Comment: 20 pages, 10pt LaTeX, 2 eps figures (v3: some changes for the final
version
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