38,047 research outputs found
Probabilistic Approach to Pattern Selection
The problem of pattern selection arises when the evolution equations have
many solutions, whereas observed patterns constitute a much more restricted
set. An approach is advanced for treating the problem of pattern selection by
defining the probability distribution of patterns. Then the most probable
pattern naturally corresponds to the largest probability weight. This approach
provides the ordering principle for the multiplicity of solutions explaining
why some of them are more preferable than other. The approach is applied to
solving the problem of turbulent photon filamentation in resonant media.Comment: LaTex, 22 page
A probabilistic approach to Zhang's sandpile model
The current literature on sandpile models mainly deals with the abelian
sandpile model (ASM) and its variants. We treat a less known - but equally
interesting - model, namely Zhang's sandpile. This model differs in two aspects
from the ASM. First, additions are not discrete, but random amounts with a
uniform distribution on an interval . Second, if a site topples - which
happens if the amount at that site is larger than a threshold value
(which is a model parameter), then it divides its entire content in equal
amounts among its neighbors. Zhang conjectured that in the infinite volume
limit, this model tends to behave like the ASM in the sense that the stationary
measure for the system in large volumes tends to be peaked narrowly around a
finite set. This belief is supported by simulations, but so far not by
analytical investigations.
We study the stationary distribution of this model in one dimension, for
several values of and . When there is only one site, exact computations
are possible. Our main result concerns the limit as the number of sites tends
to infinity, in the one-dimensional case. We find that the stationary
distribution, in the case , indeed tends to that of the ASM (up
to a scaling factor), in agreement with Zhang's conjecture. For the case ,
we provide strong evidence that the stationary expectation tends to
.Comment: 47 pages, 3 figure
A probabilistic approach to composite micromechanics
Probabilistic composite micromechanics methods are developed that simulate expected uncertainties in unidirectional fiber composite properties. These methods are in the form of computational procedures using Monte Carlo simulation. A graphite/epoxy unidirectional composite (ply) is studied to demonstrate fiber composite material properties at the micro level. Regression results are presented to show the relative correlation between predicted and response variables in the study
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