4,221 research outputs found
Fire Containment in Planar Graphs
In a graph , a fire starts at some vertex. At every time step,
firefighters can protect up to vertices, and then the fire spreads to all
unprotected neighbours. The -surviving rate of is the
expectation of the proportion of vertices that can be saved from the fire, if
the starting vertex of the fire is chosen uniformly at random. For a given
class of graphs \cG we are interested in the minimum value such that
for some constant and all G\in\cG i.e.,
such that linearly many vertices are expected to be saved in every graph from
\cG).
In this note, we prove that for planar graphs this minimum value is at most
4, and that it is precisely 2 for triangle-free planar graphs.Comment: 15 pages, one reference adde
Decoupling Limits in M-Theory
Limits of a system of N Dn-branes in which the bulk and string degrees of
freedom decouple to leave a `matter' theory are investigated and, for n>4,
either give a free theory or require taking . The decoupled
matter theory is described at low energies by the limit of n+1
dimensional \sym, and at high energies by a free type II string theory in a
curved space-time. Metastable bound states of D6-branes with mass and
D0-branes with mass are shown to have an energy proportional to
and decouple, whereas in matrix theory they only decouple in
the large N limit.Comment: 23 Pages, Tex, Phyzzx Macro. Minor correction
Robust spatial memory maps encoded in networks with transient connections
The spiking activity of principal cells in mammalian hippocampus encodes an
internalized neuronal representation of the ambient space---a cognitive map.
Once learned, such a map enables the animal to navigate a given environment for
a long period. However, the neuronal substrate that produces this map remains
transient: the synaptic connections in the hippocampus and in the downstream
neuronal networks never cease to form and to deteriorate at a rapid rate. How
can the brain maintain a robust, reliable representation of space using a
network that constantly changes its architecture? Here, we demonstrate, using
novel Algebraic Topology techniques, that cognitive map's stability is a
generic, emergent phenomenon. The model allows evaluating the effect produced
by specific physiological parameters, e.g., the distribution of connections'
decay times, on the properties of the cognitive map as a whole. It also points
out that spatial memory deterioration caused by weakening or excessive loss of
the synaptic connections may be compensated by simulating the neuronal
activity. Lastly, the model explicates functional importance of the
complementary learning systems for processing spatial information at different
levels of spatiotemporal granularity, by establishing three complementary
timescales at which spatial information unfolds. Thus, the model provides a
principal insight into how can the brain develop a reliable representation of
the world, learn and retain memories despite complex plasticity of the
underlying networks and allows studying how instabilities and memory
deterioration mechanisms may affect learning process.Comment: 24 pages, 10 figures, 4 supplementary figure
Competition and cooperation:aspects of dynamics in sandpiles
In this article, we review some of our approaches to granular dynamics, now
well known to consist of both fast and slow relaxational processes. In the
first case, grains typically compete with each other, while in the second, they
cooperate. A typical result of {\it cooperation} is the formation of stable
bridges, signatures of spatiotemporal inhomogeneities; we review their
geometrical characteristics and compare theoretical results with those of
independent simulations. {\it Cooperative} excitations due to local density
fluctuations are also responsible for relaxation at the angle of repose; the
{\it competition} between these fluctuations and external driving forces, can,
on the other hand, result in a (rare) collapse of the sandpile to the
horizontal. Both these features are present in a theory reviewed here. An arena
where the effects of cooperation versus competition are felt most keenly is
granular compaction; we review here a random graph model, where three-spin
interactions are used to model compaction under tapping. The compaction curve
shows distinct regions where 'fast' and 'slow' dynamics apply, separated by
what we have called the {\it single-particle relaxation threshold}. In the
final section of this paper, we explore the effect of shape -- jagged vs.
regular -- on the compaction of packings near their jamming limit. One of our
major results is an entropic landscape that, while microscopically rough,
manifests {\it Edwards' flatness} at a macroscopic level. Another major result
is that of surface intermittency under low-intensity shaking.Comment: 36 pages, 23 figures, minor correction
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