## ARCTIC REGION SUPERCOMPUTING CENTER, COLLEGE OF NATURAL SCIENCE AND MATHEMATICS, INSTITUTE OF ARCTIC BIOLOGY are pleased to present a series of lectures Hosted by Tom Marr, President’s Professor in Bioinformatics

### Fluids Under Tight ConfinementJerome K. Percus, Courant Institute of Mathematical Sciences and Department of Physics, New York University

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August 24, 2004, 11:00 am, Elvey Auditorium
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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”.

# Can Two Wrongs Make A Right? Coin Tossing Games and Parrondo’s Paradox Ora E. Percus, Courant Institute of Mathematical Sciences, New York University

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August 24, 2004, 3:00 pm, Elvey Auditorium
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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!

# Piecewise Homogeneous Random Walk with a Moving Boundary Ora E. Percus, Courant Institute of Mathematical Sciences, New York University

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August 25, 2004, 11:00 am, Elvey Auditorium
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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.

# Small Population Effects in Stochastic Population Dynamics Jerome K. Percus, Courant Institute of Mathematical Sciences and Department of Physics, New York University

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August 25, 2004, 3:00 pm, Elvey Auditorium
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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.

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