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 $L$, 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

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 $Y = \pm pX$, and an asymmetric wedge defined by the lines $Y= pX$ and Y=0, where $p > 0$ is an integer. We prove that the growth constant for all these models is equal to $1+\sqrt{2}$, independent of the angle of the wedge. We derive functional recursions for both models, and obtain explicit expressions for the generating functions when $p=1$. From these we find asymptotic formulas for the number of partially directed paths of length $n$ in a wedge when $p=1$. 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 $\theta$-functions, and have natural boundaries on the unit circle.Comment: 26 pages, 5 figures. Submitted to JCT

Determination of prey capture rates in the stony coral Galaxea fascicularis: a critical reconsideration of the clearance rate concept

In order to determine optimal feeding regimes for captive corals, prey capture by the scleractinian coral Galaxea fascicularis was determined by measuring clearance of prey items from the surrounding water. Colonies of G. fascicularis (sized between 200 and 400 polyps) were incubated in 1300 ml incubation chambers. Nauplii of the brine shrimp Artemia sp. were used as the prey item. A series of incubation experiments was conducted to determine the maximal capture per feeding event and per day. To determine maximal capture per feeding event, total uptake of nauplii after one hour was determined for different prey item availabilities ranging from 50 to 4000 nauplii per polyp. To determine maximal capture per day, the corals were subjected to four repetitive feeding events at three different prey item densities (50, 100 and 150 nauplii per polyp). Alongside these quantitative experiments, it was tested to what extent the feeding response of corals is triggered by chemical cues. One hour after food addition, extract of Artemia nauplii was added to the incubation chambers to test its effect on subsequent prey capture rates. In all experiments, prey capture was expressed as the number of nauplii consumed per coral polyp. Total capture of Artemia nauplii by G. fascicularis after a single feeding event increased linearly up till a prey item availability of 2000 nauplii per polyp. Maximal capture per feeding event was estimated at 1200 nauplii per polyp, which is higher than rates reported in previous studies. It became apparent that at high densities of Artemia nauplii, the clearance rate method does not discriminate between active capture and passive sedimentation. Repetitive feeding with 50 nauplii per polyp resulted in a constant total prey capture per feeding event. At a supply of 100 nauplii per polyp, total capture decreased after the first feeding event, and remained constant during the subsequent feeding events at a level comparable to the lower food availability. However, at a supply of 150 nauplii per polyp, total capture per event was higher throughout the entire four-hour incubation period, which obfuscates an accurate estimation of the maximal daily food uptake. In all incubations, a decrease in capture efficiency was observed within the course of the feeding event. In all repetitive feeding experiments, capture efficiency increased immediately upon addition of a new batch of food. This increase in efficiency was not caused by a priming effect of extract of Artemia. The inconsistencies in the data show that estimates of prey capture based on clearance rates should be interpreted with caution, because this method does not take into account potential dynamics of prey capture and release

Light intensity, photoperiod duration, daily light flux and coral growth of Galaxea fascicularis in an aquarium setting: a matter of photons?

Light is one of the most important abiotic factors influencing the (skeletal) growth of scleractinian corals. Light stimulates coral growth by the process of light-enhanced calcification, which is mediated by zooxanthellar photosynthesis. However, the quantity of light that is available for daily coral growth is not only determined by light intensity (i.e. irradiance), but also by photoperiod (i.e. the light duration time). Understanding and optimizing conditions for coral growth is essential for sustainable coral aquaculture. Therefore, in this study, the question was explored whether more light (i.e. more photons), presented either as irradiance or as light duration, would result in more growth. A series of nine genetically identical coral colonies of Galaxea fascicularis L. were cultured for a period of 18 weeks at different light duration times (8 hours 150 µE m-2 s-1:16 hours dark, 12 hours 150 µE m-2 s-1:12 hours dark, 16 hours 150 µE m-2 s-1:8 hours dark, 24 hours 150 µE m-2 s-1:0 hours dark) and different irradiance levels (8 hours 150 µE m-2 s-1:16 hours dark, 8 hours 225 µE m-2 s-1:16 hours dark and 8 hours 300 µE m-2 s-1:16 hours dark). Growth was determined every two weeks by measuring buoyant weight. Temperature, salinity and feeding levels were kept constant during the experiment. To detect possible acclimation of the corals to an increased light duration, rates of net photosynthesis and dark respiration were measured, hereby comparing coral colonies grown under an 8:16 hours light (150 µE m-2 s-1):dark cycle with corals grown under a 16:8 hours light (150 µE m-2 s-1):dark cycle. No increase in growth was detected with either increasing photoperiod or irradiance. Continuous lighting (24 hours 150 µE m-2 s-1:0 hours dark) resulted in immediate bleaching and the corals died after 14 weeks. Hourly photosynthetic rates were significantly reduced in the 16 hour light treatment compared to the 8 hour light treatment. As a result, daily net photosynthetic rates were not significantly different, which may explain the observed specific growth rates. Acclimation to photoperiod duration appeared neither to be mediated by changes in chlorophyll-a concentration nor zooxanthellae density. Based on the results of this study, we can conclude that the enhancing effect of light on coral growth is not only a matter of photons. Obviously, the availability of light was not limiting growth in these experiments and was probably in excess (i.e. stressful amounts). Other factors are discussed that play a role in determining growth rates and might explain our results