31,434 research outputs found
On theories of random variables
We study theories of spaces of random variables: first, we consider random
variables with values in the interval , then with values in an arbitrary
metric structure, generalising Keisler's randomisation of classical structures.
We prove preservation and non-preservation results for model theoretic
properties under this construction: i) The randomisation of a stable structure
is stable. ii) The randomisation of a simple unstable structure is not simple.
We also prove that in the randomised structure, every type is a Lascar type
Continuous and discrete models of cooperation in complex bacterial colonies
We study the effect of discreteness on various models for patterning in
bacterial colonies. In a bacterial colony with branching pattern, there are
discrete entities - bacteria - which are only two orders of magnitude smaller
than the elements of the macroscopic pattern. We present two types of models.
The first is the Communicating Walkers model, a hybrid model composed of both
continuous fields and discrete entities - walkers, which are coarse-graining of
the bacteria. Models of the second type are systems of reaction diffusion
equations, where the branching of the pattern is due to non-constant diffusion
coefficient of the bacterial field. The diffusion coefficient represents the
effect of self-generated lubrication fluid on the bacterial movement. We
implement the discreteness of the biological system by introducing a cutoff in
the growth term at low bacterial densities. We demonstrate that the cutoff does
not improve the models in any way. Its only effect is to decrease the effective
surface tension of the front, making it more sensitive to anisotropy. We
compare the models by introducing food chemotaxis and repulsive chemotactic
signaling into the models. We find that the growth dynamics of the
Communication Walkers model and the growth dynamics of the Non-Linear diffusion
model are affected in the same manner. From such similarities and from the
insensitivity of the Communication Walkers model to implicit anisotropy we
conclude that the increased discreteness, introduced be the coarse-graining of
the walkers, is small enough to be neglected.Comment: 16 pages, 10 figures in 13 gif files, to be published in proceeding
of CMDS
Estimating graph parameters with random walks
An algorithm observes the trajectories of random walks over an unknown graph
, starting from the same vertex , as well as the degrees along the
trajectories. For all finite connected graphs, one can estimate the number of
edges up to a bounded factor in
steps, where
is the relaxation time of the lazy random walk on and
is the minimum degree in . Alternatively, can be estimated in
, where is
the number of vertices and is the uniform mixing time on
. The number of vertices can then be estimated up to a bounded factor in
an additional steps. Our
algorithms are based on counting the number of intersections of random walk
paths , i.e. the number of pairs such that . This
improves on previous estimates which only consider collisions (i.e., times
with ). We also show that the complexity of our algorithms is optimal,
even when restricting to graphs with a prescribed relaxation time. Finally, we
show that, given either or the mixing time of , we can compute the
"other parameter" with a self-stopping algorithm
On the spectrum of the transfer operators of a one-parameter family with intermittency transition
We study the transfer operators for a family depending
on the parameter , which interpolates between the tent map and the
Farey map. In particular, considering the action of the transfer operator on a
suitable Hilbert space, we can define a family of infinite matrices associated
to the operators and study their spectrum by numerical methods.Comment: 6 pages, 3 figure
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