61 research outputs found
The combinatorics of the leading root of the partial theta function
Recently Alan Sokal studied the leading root of the partial theta
function , considered
as a formal power series. He proved that all the coefficients of
are positive integers. I give here an
explicit combinatorial interpretation of these coefficients. More precisely, I
show that enumerates rooted trees that are enriched by certain
polyominoes, weighted according to their total area.Comment: 15 pages, 7 figure
Winding angles for two-dimensional polymers with orientation dependent interactions
We study winding angles of oriented polymers with orientation-dependent
interaction in two dimensions. Using exact analytical calculations, computer
simulations, and phenomenological arguments, we succeed in finding the variance
of the winding angle for most of the phase diagram. Our results suggest that
the winding angle distribution is a universal quantity, and that the
--point is the point where the three phase boundaries between the
swollen, the normal collapsed, and the spiral collapsed phase meet. The
transition between the normal collapsed phase and the spiral phase is shown to
be continuous.Comment: 21 pages (incl 5 figures
Uniform asymptotics of area-weighted Dyck paths
Using the generalized method of steepest descents for the case of two
coalescing saddle points, we derive an asymptotic expression for the bivariate
generating function of Dyck paths, weighted according to their length and their
area in the limit of the area generating variable tending towards 1. The result
is valid uniformly for a range of the length generating variable, including the
tricritical point of the model.Comment: 14 pages, 5 figure
Dynamics of a single particle in a horizontally shaken box
We study the dynamics of a particle in a horizontally and periodically shaken
box as a function of the box parameters and the coefficient of restitution. For
certain parameter values, the particle becomes regularly chattered at one of
the walls, thereby loosing all its kinetic energy relative to that wall. The
number of container oscillations between two chattering events depends in a
fractal manner on the parameters of the system. In contrast to a vertically
vibrated particle, for which chattering is claimed to be the generic fate, the
horizontally shaken particle can become trapped on a periodic orbit and follow
the period-doubling route to chaos when the coefficient of restitution is
changed. We also discuss the case of a completely elastic particle, and the
influence of friction between the particle and the bottom of the container.Comment: 11 pages RevTex. Some postscript files have low resolution. We will
send the high-resolution files on reques
Pressure exerted by a vesicle on a surface
Several recent works have considered the pressure exerted on a wall by a
model polymer. We extend this consideration to vesicles attached to a wall, and
hence include osmotic pressure. We do this by considering a two-dimensional
directed model, namely that of area-weighted Dyck paths.
Not surprisingly, the pressure exerted by the vesicle on the wall depends on
the osmotic pressure inside, especially its sign. Here, we discuss the scaling
of this pressure in the different regimes, paying particular attention to the
crossover between positive and negative osmotic pressure. In our directed
model, there exists an underlying Airy function scaling form, from which we
extract the dependence of the bulk pressure on small osmotic pressures.Comment: 10 pages, 7 figure
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