The perimeter of large planar Voronoi cells: a double-stranded random walk

Let $p\_n$ be the probability for a planar Poisson-Voronoi cell to have exactly $n$ sides. We construct the asymptotic expansion of $\log p\_n$ up to terms that vanish as $n\to\infty$. We show that {\it two independent biased random walks} executed by the polar angle determine the trajectory of the cell perimeter. We find the limit distribution of (i) the angle between two successive vertex vectors, and (ii) the one between two successive perimeter segments. We obtain the probability law for the perimeter's long wavelength deviations from circularity. We prove Lewis' law and show that it has coefficient 1/4.Comment: Slightly extended version; journal reference adde

Topological correlations in soap froths

Correlation in two-dimensional soap froth is analysed with an effective potential for the first time. Cells with equal number of sides repel (with linear correlation) while cells with different number of sides attract (with NON-bilinear) for nearest neighbours, which cannot be explained by the maximum entropy argument. Also, the analysis indicates that froth is correlated up to the third shell neighbours at least, contradicting the conventional ideas that froth is not strongly correlated.Comment: 10 Pages LaTeX, 6 Postscript figure

From one cell to the whole froth: a dynamical map

We investigate two and three-dimensional shell-structured-inflatable froths, which can be constructed by a recursion procedure adding successive layers of cells around a germ cell. We prove that any froth can be reduced into a system of concentric shells. There is only a restricted set of local configurations for which the recursive inflation transformation is not applicable. These configurations are inclusions between successive layers and can be treated as vertices and edges decorations of a shell-structure-inflatable skeleton. The recursion procedure is described by a logistic map, which provides a natural classification into Euclidean, hyperbolic and elliptic froths. Froths tiling manifolds with different curvature can be classified simply by distinguishing between those with a bounded or unbounded number of elements per shell, without any a-priori knowledge on their curvature. A new result, associated with maximal orientational entropy, is obtained on topological properties of natural cellular systems. The topological characteristics of all experimentally known tetrahedrally close-packed structures are retrieved.Comment: 20 Pages Tex, 11 Postscript figures, 1 Postscript tabl

On Random Bubble Lattices

We study random bubble lattices which can be produced by processes such as first order phase transitions, and derive characteristics that are important for understanding the percolation of distinct varieties of bubbles. The results are relevant to the formation of topological defects as they show that infinite domain walls and strings will be produced during appropriate first order transitions, and that the most suitable regular lattice to study defect formation in three dimensions is a face centered cubic lattice. Another application of our work is to the distribution of voids in the large-scale structure of the universe. We argue that the present universe is more akin to a system undergoing a first-order phase transition than to one that is crystallizing, as is implicit in the Voronoi foam description. Based on the picture of a bubbly universe, we predict a mean coordination number for the voids of 13.4. The mean coordination number may also be used as a tool to distinguish between different scenarios for structure formation.Comment: several modifications including new abstract, comparison with froth models, asymptotics of coordination number distribution, further discussion of biased defects, and relevance to large-scale structur

The self-assessment INTERMED predicts healthcare and social costs of orthopaedic trauma patients with persistent impairments.

To use the self-assessment INTERMED questionnaire to determine the relationship between biopsychosocial complexity and healthcare and social costs of patients after orthopaedic trauma. Secondary prospective analysis based on the validation study cohort of the self-assessment INTERMED questionnaire. Inpatients orthopaedic rehabilitation with vocational aspects. In total, 136 patients with chronic pain and impairments were included in this study: mean (SD) age, 42.6 (10.7) years; 116 men, with moderate pain intensity (51/100); suffering from upper (n = 55), lower-limb (n = 51) or spine (n = 30) pain after orthopaedic trauma; with minor or moderate injury severity (severe injury for 25). Biopsychosocial complexity, assessed with the self-assessment INTERMED questionnaire, and other confounding variables collected prospectively during rehabilitation. Outcome measures (healthcare costs, loss of wage costs and time for fitness-to-work) were collected through insurance files after case settlements. Linear multiple regression models adjusted for age, gender, pain, trauma severity, education and employment contract were performed to measure the influence of biopsychosocial complexity on the three outcome variables. High-cost patients were older (+3.6 years) and more anxious (9.0 vs 7.3 points at HADS-A), came later to rehabilitation (+105 days), and showed higher biopsychosocial complexity (+3.2 points). After adjustment, biopsychosocial complexity was significantly associated with healthcare (ß = 0.02; P = 0.003; exp <sup>ß</sup> = 1.02) and social costs (ß = 0.03; P = 0.006, exp <sup>ß</sup> = 1.03) and duration before fitness-to-work (ß = 0.04; P < 0.001, exp <sup>ß</sup> = 1.04). Biopsychosocial complexity assessed with the self-assessment INTERMED questionnaire is associated with higher healthcare and social costs

Asymptotic statistics of the n-sided planar Poisson-Voronoi cell. I. Exact results

We achieve a detailed understanding of the $n$-sided planar Poisson-Voronoi cell in the limit of large $n$. Let ${p}\_n$ be the probability for a cell to have $n$ sides. We construct the asymptotic expansion of $\log {p}\_n$ up to terms that vanish as $n\to\infty$. We obtain the statistics of the lengths of the perimeter segments and of the angles between adjoining segments: to leading order as $n\to\infty$, and after appropriate scaling, these become independent random variables whose laws we determine; and to next order in $1/n$ they have nontrivial long range correlations whose expressions we provide. The $n$-sided cell tends towards a circle of radius (n/4\pi\lambda)^{\half}, where $\lambda$ is the cell density; hence Lewis' law for the average area $A\_n$ of the $n$-sided cell behaves as $A\_n \simeq cn/\lambda$ with $c=1/4$. For $n\to\infty$ the cell perimeter, expressed as a function $R(\phi)$ of the polar angle $\phi$, satisfies $d^2 R/d\phi^2 = F(\phi)$, where $F$ is known Gaussian noise; we deduce from it the probability law for the perimeter's long wavelength deviations from circularity. Many other quantities related to the asymptotic cell shape become accessible to calculation.Comment: 54 pages, 3 figure

Analysis of signalling pathways using continuous time Markov chains

We describe a quantitative modelling and analysis approach for signal transduction networks. We illustrate the approach with an example, the RKIP inhibited ERK pathway [CSK+03]. Our models are high level descriptions of continuous time Markov chains: proteins are modelled by synchronous processes and reactions by transitions. Concentrations are modelled by discrete, abstract quantities. The main advantage of our approach is that using a (continuous time) stochastic logic and the PRISM model checker, we can perform quantitative analysis such as what is the probability that if a concentration reaches a certain level, it will remain at that level thereafter? or how does varying a given reaction rate affect that probability? We also perform standard simulations and compare our results with a traditional ordinary differential equation model. An interesting result is that for the example pathway, only a small number of discrete data values is required to render the simulations practically indistinguishable