985 research outputs found
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
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
Knotting probabilities after a local strand passage in unknotted self-avoiding polygons
We investigate the knotting probability after a local strand passage is
performed in an unknotted self-avoiding polygon on the simple cubic lattice. We
assume that two polygon segments have already been brought close together for
the purpose of performing a strand passage, and model this using Theta-SAPs,
polygons that contain the pattern Theta at a fixed location. It is proved that
the number of n-edge Theta-SAPs grows exponentially (with n) at the same rate
as the total number of n-edge unknotted self-avoiding polygons, and that the
same holds for subsets of n-edge Theta-SAPs that yield a specific
after-strand-passage knot-type. Thus the probability of a given
after-strand-passage knot-type does not grow (or decay) exponentially with n,
and we conjecture that instead it approaches a knot-type dependent amplitude
ratio lying strictly between 0 and 1. This is supported by critical exponent
estimates obtained from a new maximum likelihood method for Theta-SAPs that are
generated by a composite (aka multiple) Markov Chain Monte Carlo BFACF
algorithm. We also give strong numerical evidence that the after-strand-passage
knotting probability depends on the local structure around the strand passage
site. Considering both the local structure and the crossing-sign at the strand
passage site, we observe that the more "compact" the local structure, the less
likely the after-strand-passage polygon is to be knotted. This trend is
consistent with results from other strand-passage models, however, we are the
first to note the influence of the crossing-sign information. Two measures of
"compactness" are used: the size of a smallest polygon that contains the
structure and the structure's "opening" angle. The opening angle definition is
consistent with one that is measurable from single molecule DNA experiments.Comment: 31 pages, 12 figures, submitted to Journal of Physics
Punctured polygons and polyominoes on the square lattice
We use the finite lattice method to count the number of punctured staircase
and self-avoiding polygons with up to three holes on the square lattice. New or
radically extended series have been derived for both the perimeter and area
generating functions. We show that the critical point is unchanged by a finite
number of punctures, and that the critical exponent increases by a fixed amount
for each puncture. The increase is 1.5 per puncture when enumerating by
perimeter and 1.0 when enumerating by area. A refined estimate of the
connective constant for polygons by area is given. A similar set of results is
obtained for finitely punctured polyominoes. The exponent increase is proved to
be 1.0 per puncture for polyominoes.Comment: 36 pages, 11 figure
Confinement of knotted polymers in a slit
We investigate the effect of knot type on the properties of a ring polymer
confined to a slit. For relatively wide slits, the more complex the knot, the
more the force exerted by the polymer on the walls is decreased compared to an
unknotted polymer of the same length. For more narrow slits the opposite is
true. The crossover between these two regimes is, to first order, at smaller
slit width for more complex knots. However, knot topology can affect these
trends in subtle ways. Besides the force exerted by the polymers, we also study
other quantities such as the monomer-density distribution across the slit and
the anisotropic radius of gyration.Comment: 9 pages, 6 figures, submitted for publicatio
Fasting plasma glucose and risk factor assessment: Comparing sensitivity and specificity in identifying gestational diabetes in urban black African women
Background. Identifying women with gestational diabetes mellitus (GDM) allows interventions to improve perinatal outcomes. A fasting plasma glucose (FPG) level ≥5.1 mmol/L is 100% specific for a diagnosis of GDM. The International Association of Diabetes and Pregnancy Study Groups acknowledges that FPG <4.5 mmol/L is associated with a low probability of GDM.Objectives. The validity of selective screening based on the presence of risk factors was compared with the universal application of FPG ≥4.5 mmol/L to identify women with GDM. FPG ≥4.5 mmol/L or the presence of one or more risk factors was assumed to indicate an intermediate to high risk of GDM and therefore the need for an oral glucose tolerance test (OGTT).Methods. Consecutive black South African (SA) women were recruited to a 2-hour 75 g OGTT at 24 - 28 weeks’ gestation in an urban community health clinic. Of 969 women recruited, 666 underwent an OGTT, and of these 589 were eligible for analysis. The glucose oxidase laboratory method was used to measure plasma glucose concentrations. The World Health Organization GDM diagnostic criteria were applied. All participants underwent a risk factor assessment. The χ2 test was used to determine associations between risk factors and a positive diagnosis of GDM. The sensitivity and specificity of a positive diagnosis of GDM were calculated for FPG ≥4.5 mmol/L, FPG ≥5.1 mmol/L, and the presence of one or more risk factors.Results. The prevalence of overt diabetes mellitus and GDM was 0.5% and 7.0%, respectively. Risk factor-based selective screening indicated that 204/589 (34.6%) of participants needed an OGTT, but 18/41 (43.9%) of positive GDM diagnoses were missed. Universal screening using the FPG threshold of ≥4.5 mmol/L indicated that 152/589 (25.8%) of participants needed an OGTT, and 1/41 (2.4%) of positive diagnoses were missed. An FPG of ≥5.1 mmol/L identified 36/41 (87.8%) of GDM-positive participants. The sensitivity and specificity of the presence of one or more risk factors were 56% and 67%, respectively. The sensitivity and specificity of FPG ≥4.5 mmol/L were 98% and 80%, respectively.Conclusions. Universal screening using FPG ≥4.5 mmol/L had greater sensitivity and specificity in identifying GDM-affected women and required fewer women to undergo a resource-intensive diagnostic OGTT than risk factor-based selective screening. A universal screening strategy using FPG ≥4.5 mmol/L may be more efficient and cost-effective than risk factor-based selective screening for GDM in black SA women.
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
Knot localization in adsorbing polymer rings
We study by Monte Carlo simulations a model of knotted polymer ring adsorbing
onto an impenetrable, attractive wall. The polymer is described by a
self-avoiding polygon (SAP) on the cubic lattice. We find that the adsorption
transition temperature, the crossover exponent and the metric exponent
, are the same as in the model where the topology of the ring is
unrestricted. By measuring the average length of the knotted portion of the
ring we are able to show that adsorbed knots are localized. This knot
localization transition is triggered by the adsorption transition but is
accompanied by a less sharp variation of the exponent related to the degree of
localization. Indeed, for a whole interval below the adsorption transition, one
can not exclude a contiuous variation with temperature of this exponent. Deep
into the adsorbed phase we are able to verify that knot localization is strong
and well described in terms of the flat knot model.Comment: 27 pages, 10 figures. Submitter to Phys. Rev.
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