160 research outputs found
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
Fresh look at randomly branched polymers
We develop a new, dynamical field theory of isotropic randomly branched
polymers, and we use this model in conjunction with the renormalization group
(RG) to study several prominent problems in the physics of these polymers. Our
model provides an alternative vantage point to understand the swollen phase via
dimensional reduction. We reveal a hidden Becchi-Rouet-Stora (BRS) symmetry of
the model that describes the collapse (-)transition to compact
polymer-conformations, and calculate the critical exponents to 2-loop order. It
turns out that the long-standing 1-loop results for these exponents are not
entirely correct. A runaway of the RG flow indicates that the so-called
-transition could be a fluctuation induced first order
transition.Comment: 4 page
Another case of discrepancies when evaluating power theories using real data
Abstract: Most non-sinusoidal power theories have been developed from theoretical techniques, as well as relying on hypothetical and experimental networks to highlight the advantages of each. The drawback of the power theories is that no single one has been universally accepted as a benchmark for other developments. This paper will however show this weakness by means of evaluating two power theories in the time domain, using real recorded data
Detection of human papillomavirus DNA with in situ hybridisation in oral squamous carcinoma in a rural black population
No Abstract
Collapsing lattice animals and lattice trees in two dimensions
We present high statistics simulations of weighted lattice bond animals and
lattice trees on the square lattice, with fugacities for each non-bonded
contact and for each bond between two neighbouring monomers. The simulations
are performed using a newly developed sequential sampling method with
resampling, very similar to the pruned-enriched Rosenbluth method (PERM) used
for linear chain polymers. We determine with high precision the line of second
order transitions from an extended to a collapsed phase in the resulting
2-dimensional phase diagram. This line includes critical bond percolation as a
multicritical point, and we verify that this point divides the line into two
different universality classes. One of them corresponds to the collapse driven
by contacts and includes the collapse of (weakly embeddable) trees, but the
other is {\it not yet} bond driven and does not contain the Derrida-Herrmann
model as special point. Instead it ends at a multicritical point where a
transition line between two collapsed phases (one bond-driven and the other
contact-driven) sparks off. The Derrida-Herrmann model is representative for
the bond driven collapse, which then forms the fourth universality class on the
transition line (collapsing trees, critical percolation, intermediate regime,
and Derrida-Herrmann). We obtain very precise estimates for all critical
exponents for collapsing trees. It is already harder to estimate the critical
exponents for the intermediate regime. Finally, it is very difficult to obtain
with our method good estimates of the critical parameters of the
Derrida-Herrmann universality class. As regards the bond-driven to
contact-driven transition in the collapsed phase, we have some evidence for its
existence and rough location, but no precise estimates of critical exponents.Comment: 11 pages, 16 figures, 1 tabl
Polar decompositions of quaternion matrices in indefinite inner product spaces
Polar decompositions of quaternion matrices with respect to a given
indefinite inner product are studied. Necessary and sufficient conditions for
the existence of an -polar decomposition are found. In the process an
equivalent to Witt's theorem on extending -isometries to -unitary
matrices is given for quaternion matrices
Simulations of lattice animals and trees
The scaling behaviour of randomly branched polymers in a good solvent is
studied in two to nine dimensions, using as microscopic models lattice animals
and lattice trees on simple hypercubic lattices. As a stochastic sampling
method we use a biased sequential sampling algorithm with re-sampling, similar
to the pruned-enriched Rosenbluth method (PERM) used extensively for linear
polymers. Essentially we start simulating percolation clusters (either site or
bond), re-weigh them according to the animal (tree) ensemble, and prune or
branch the further growth according to a heuristic fitness function. In
contrast to previous applications of PERM, this fitness function is {\it not}
the weight with which the actual configuration would contribute to the
partition sum, but is closely related to it. We obtain high statistics of
animals with up to several thousand sites in all dimension 2 <= d <= 9. In
addition to the partition sum (number of different animals) we estimate
gyration radii and numbers of perimeter sites. In all dimensions we verify the
Parisi-Sourlas prediction, and we verify all exactly known critical exponents
in dimensions 2, 3, 4, and >= 8. In addition, we present the hitherto most
precise estimates for growth constants in d >= 3. For clusters with one site
attached to an attractive surface, we verify the superuniversality of the
cross-over exponent at the adsorption transition predicted by Janssen and
Lyssy. Finally, we discuss the collapse of animals and trees, arguing that our
present version of the algorithm is also efficient for some of the models
studied in this context, but showing that it is {\it not} very efficient for
the `classical' model for collapsing animals.Comment: 17 pages RevTeX, 29 figures include
New results for virial coefficients of hard spheres in D dimensions
We present new results for the virial coefficients B_k with k <= 10 for hard
spheres in dimensions D=2,...,8.Comment: 10 pages, 5 figures, to appear in conference proceedings of STATPHYS
2004 in Pramana - Journal of Physic
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