New Monte Carlo method for planar Poisson-Voronoi cells

By a new Monte Carlo algorithm we evaluate the sidedness probability p_n of a planar Poisson-Voronoi cell in the range 3 \leq n \leq 1600. The algorithm is developed on the basis of earlier theoretical work; it exploits, in particular, the known asymptotic behavior of p_n as n\to\infty. Our p_n values all have between four and six significant digits. Accurate n dependent averages, second moments, and variances are obtained for the cell area and the cell perimeter. The numerical large n behavior of these quantities is analyzed in terms of asymptotic power series in 1/n. Snapshots are shown of typical occurrences of extremely rare events implicating cells of up to n=1600 sides embedded in an ordinary Poisson-Voronoi diagram. We reveal and discuss the characteristic features of such many-sided cells and their immediate environment. Their relevance for observable properties is stressed.Comment: 35 pages including 10 figures and 4 table

Travelling-wave analysis of a model describing tissue degradation by bacteria

We study travelling-wave solutions for a reaction-diffusion system arising as a model for host-tissue degradation by bacteria. This system consists of a parabolic equation coupled with an ordinary differential equation. For large values of the `degradation-rate parameter' solutions are well approximated by solutions of a Stefan-like free boundary problem, for which travelling-wave solutions can be found explicitly. Our aim is to prove the existence of travelling waves for all sufficiently large wave-speeds for the original reaction-diffusion system and to determine the minimal speed. We prove that for all sufficiently large degradation rates the minimal speed is identical to the minimal speed of the limit problem. In particular, in this parameter range, nonlinear selection of the minimal speed occurs.Comment: 15 pages, 3 figure

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

Some Open Points in Nonextensive Statistical Mechanics

We present and discuss a list of some interesting points that are currently open in nonextensive statistical mechanics. Their analytical, numerical, experimental or observational advancement would naturally be very welcome.Comment: 30 pages including 6 figures. Invited paper to appear in the International Journal of Bifurcation and Chao

Blind Spots: Domestic Entrepreneurship and Private-sector Development in South-Sudan

Policy discourse on private sector development in fragile states has started attributing great importance to domestic entrepreneurship. This chapter follows Dutch initiatives to support entrepreneurs in South Sudan between 2009 and 2015. Despite the rhetoric, support for entrepreneurs did not materialise. Donor representatives refer to a lack of small and medium-sized enterprises and entrepreneurial skills as prime reasons. Our research reveals that in practice, the apolitical interventionist rationale characterising donor support in South Sudan conflicts with the politicised nature of the private sector. In fact, interventions side-line domestic entrepreneurship, reinforcing an image of a ‘missing middle’ and diverting attention to international firms as ‘capacity builders’

Heuristic theory for many-faced d-dimensional Poisson-Voronoi cells

We consider the d-dimensional Poisson-Voronoi tessellation and investigate the applicability of heuristic methods developed recently for two dimensions. Let p_n(d) be the probability that a cell have n neighbors (be `n-faced') and m_n(d) the average facedness of a cell adjacent to an n-faced cell. We obtain the leading order terms of the asymptotic large-n expansions for p_n(d) and m_n(3). It appears that, just as in dimension two, the Poisson-Voronoi tessellation violates Aboav's `linear law' also in dimension three. A confrontation of this statement with existing Monte Carlo work remains inconclusive. However, simulations upgraded to the level of present-day computer capacity will in principle be able to confirm (or invalidate) our theory.Comment: 17 pages, 6 figure