1,303 research outputs found
BFACF-style algorithms for polygons in the body-centered and face-centered cubic lattices
In this paper the elementary moves of the BFACF-algorithm for lattice
polygons are generalised to elementary moves of BFACF-style algorithms for
lattice polygons in the body-centred (BCC) and face-centred (FCC) cubic
lattices. We prove that the ergodicity classes of these new elementary moves
coincide with the knot types of unrooted polygons in the BCC and FCC lattices
and so expand a similar result for the cubic lattice. Implementations of these
algorithms for knotted polygons using the GAS algorithm produce estimates of
the minimal length of knotted polygons in the BCC and FCC lattices
Lattice Knots in a Slab
In this paper the number and lengths of minimal length lattice knots confined
to slabs of width , is determined. Our data on minimal length verify the
results by Sharein et.al. (2011) for the similar problem, expect in a single
case, where an improvement is found. From our data we construct two models of
grafted knotted ring polymers squeezed between hard walls, or by an external
force. In each model, we determine the entropic forces arising when the lattice
polygon is squeezed by externally applied forces. The profile of forces and
compressibility of several knot types are presented and compared, and in
addition, the total work done on the lattice knots when it is squeezed to a
minimal state is determined
Minimal knotted polygons in cubic lattices
An implementation of BFACF-style algorithms on knotted polygons in the simple
cubic, face centered cubic and body centered cubic lattice is used to estimate
the statistics and writhe of minimal length knotted polygons in each of the
lattices. Data are collected and analysed on minimal length knotted polygons,
their entropy, and their lattice curvature and writhe
The Compressibility of Minimal Lattice Knots
The (isothermic) compressibility of lattice knots can be examined as a model
of the effects of topology and geometry on the compressibility of ring
polymers. In this paper, the compressibility of minimal length lattice knots in
the simple cubic, face centered cubic and body centered cubic lattices are
determined. Our results show that the compressibility is generally not
monotonic, but in some cases increases with pressure. Differences of the
compressibility for different knot types show that topology is a factor
determining the compressibility of a lattice knot, and differences between the
three lattices show that compressibility is also a function of geometry.Comment: Submitted to J. Stat. Mec
A simple model of a vesicle drop in a confined geometry
We present the exact solution of a two-dimensional directed walk model of a
drop, or half vesicle, confined between two walls, and attached to one wall.
This model is also a generalisation of a polymer model of steric stabilisation
recently investigated. We explore the competition between a sticky potential on
the two walls and the effect of a pressure-like term in the system. We show
that a negative pressure ensures the drop/polymer is unaffected by confinement
when the walls are a macroscopic distance apart
Adsorbed self-avoiding walks subject to a force
We consider a self-avoiding walk model of polymer adsorption where the
adsorbed polymer can be desorbed by the application of a force. In this paper
the force is applied normal to the surface at the last vertex of the walk. We
prove that the appropriate limiting free energy exists where there is an
applied force and a surface potential term, and prove that this free energy is
convex in appropriate variables. We then derive an expression for the limiting
free energy in terms of the free energy without a force and the free energy
with no surface interaction. Finally we show that there is a phase boundary
between the adsorbed phase and the desorbed phase in the presence of a force,
prove some qualitative properties of this boundary and derive bounds on the
location of the boundary
Partially directed paths in a wedge
The enumeration of lattice paths in wedges poses unique mathematical
challenges. These models are not translationally invariant, and the absence of
this symmetry complicates both the derivation of a functional recurrence for
the generating function, and solving for it. In this paper we consider a model
of partially directed walks from the origin in the square lattice confined to
both a symmetric wedge defined by , and an asymmetric wedge defined
by the lines and Y=0, where is an integer. We prove that the
growth constant for all these models is equal to , independent of
the angle of the wedge. We derive functional recursions for both models, and
obtain explicit expressions for the generating functions when . From these
we find asymptotic formulas for the number of partially directed paths of
length in a wedge when .
The functional recurrences are solved by a variation of the kernel method,
which we call the ``iterated kernel method''. This method appears to be similar
to the obstinate kernel method used by Bousquet-Melou. This method requires us
to consider iterated compositions of the roots of the kernel. These
compositions turn out to be surprisingly tractable, and we are able to find
simple explicit expressions for them. However, in spite of this, the generating
functions turn out to be similar in form to Jacobi -functions, and have
natural boundaries on the unit circle.Comment: 26 pages, 5 figures. Submitted to JCT
On trivial words in finitely presented groups
We propose a numerical method for studying the cogrowth of finitely presented
groups. To validate our numerical results we compare them against the
corresponding data from groups whose cogrowth series are known exactly.
Further, we add to the set of such groups by finding the cogrowth series for
Baumslag-Solitar groups and prove
that their cogrowth rates are algebraic numbers.Comment: This article has been rewritten as two separate papers, with improved
exposition. The new papers are arXiv:1309.4184 and arXiv:1312.572
- …