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August 24, 2004, 11:00 am, Elvey Auditorium
A preliminary
study is made of the new phenomenology encountered when classical fluids are
confined to enclosures that are of the order of the particle size
in all but one spatial dimension. Self-diffusion is taken as the indicator
of this phenomenology. The anomalous diffusion occurring in strictly one-dimensional
flow is first reviewed, and then its extension to the single-file regime in
which particles cannot pass each other. When the system enters the parametric
regime in which particle exchange is first possible, a rapid transition to
the characteristics of normal diffusion takes place, which is organized by
the concept of “hopping time”.
August
24, 2004, 3:00 pm, Elvey Auditorium
A number of natural and man-made
activities can be cast in the form of various one-person games, and many
of these appear
as sequences of transitions
without memory, or Markov chains. It has been observed, initially with
surprise, that losing “games” can often be combined by selection,
or even randomly, to result in winning games. Here, we present the analysis
of such questions in concise mathematical form (exemplified by one nearly
trivial case and one which has received a fair amount of prior study),
showing that two wrongs can indeed make a right – but also that two
rights can make a wrong!
August 25, 2004, 11:00 am, Elvey Auditorium
We study a random walk with nearest neighbor transitions on a one-dimensional
lattice. The walk starts at the origin, as does a dividing line which moves
with constant speed gamma, but the outward transition probabilities p_A and
p_B differ on the right- and left- hand sides of the dividing line. This
problem is solved formally by taking advantage of the analytical properties
in the complex plane of an added variable generating function, and it is
found that (p_A, p_B) space decomposes into four regions of distinct qualitative
properties. The asymptotic probability of the walk being to the right of
the moving boundary is obtained explicitly in three of the four regions.
However, analysis in the fourth region is a sensitive function of the denominator
of the rational fraction gamma, and encounters some surprises. Applications
of random walk problems to sequential clinical trials will be mentioned.
August 25, 2004, 3:00 pm, Elvey Auditorium
We focus on several biologically
relevant situations in which small populations play a significant qualitative
role, and take some first steps to incorporate
such situations in the continuous dynamics format that has been so elegantly
developed in the past. We first describe a small number of model systems in
which the influence of small populations is evident. The we analyze in detail
a toy model, exactly solvable, that suggests a path towards the attainment
of our goal, and follow this by a formal vehicle for doing so. Application
to model systems, and comparison with numerical solutions, indicates the potential
utility of this approach.
Arctic Region Supercomputing Center
PO Box 756020, Fairbanks, AK 99775 | voice: 907-450-8600 | email:
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