Percolation of randomly distributed growing clusters

We investigate the problem of growing clusters, which is modeled by two dimensional disks and three dimensional droplets. In this model we place a number of seeds on random locations on a lattice with an initial occupation probability, $p$. The seeds simultaneously grow with a constant velocity to form clusters. When two or more clusters eventually touch each other they immediately stop their growth. The probability that such a system will result in a percolating cluster depends on the density of the initially distributed seeds and the dimensionality of the system. For very low initial values of $p$ we find a power law behavior for several properties that we investigate, namely for the size of the largest and second largest cluster, for the probability for a site to belong to the finally formed spanning cluster, and for the mean radius of the finally formed droplets. We report the values of the corresponding scaling exponents. Finally, we show that for very low initial concentration of seeds the final coverage takes a constant value which depends on the system dimensionality.Comment: 5 pages, 7 figure

Percolation of randomly distributed growing clusters: Finite Size Scaling and Critical Exponents

We study the percolation properties of the growing clusters model. In this model, a number of seeds placed on random locations on a lattice are allowed to grow with a constant velocity to form clusters. When two or more clusters eventually touch each other they immediately stop their growth. The model exhibits a discontinuous transition for very low values of the seed concentration $p$ and a second, non-trivial continuous phase transition for intermediate $p$ values. Here we study in detail this continuous transition that separates a phase of finite clusters from a phase characterized by the presence of a giant component. Using finite size scaling and large scale Monte Carlo simulations we determine the value of the percolation threshold where the giant component first appears, and the critical exponents that characterize the transition. We find that the transition belongs to a different universality class from the standard percolation transition.Comment: 5 two-column pages, 6 figure

Some discussions of D. Fearnhead and D. Prangle's Read Paper "Constructing summary statistics for approximate Bayesian computation: semi-automatic approximate Bayesian computation"

This report is a collection of comments on the Read Paper of Fearnhead and Prangle (2011), to appear in the Journal of the Royal Statistical Society Series B, along with a reply from the authors.Comment: 10 page

‘The International Teacher Leadership project,’ a case of international action research.

Copyright CARNThe paper arises from the International Teacher Leadership project, a research and development project involving researchers and practitioners in 14 European countries. The paper provides a conceptual exploration of the idea of teacher leadership and its role in educational reform, central to which is the idea that teachers, regardless of their level of power and organisational position, can engage in the leadership of enquiry-based development activity aimed at influencing their colleagues and embedding improved practices in their schools. The paper provides an outline of the project’s methodology which builds on that used in the Carpe Vitam Leadership for Learning project (Frost, 2008a). It is a form of collaborative action research which is highly developmental and discursive. It seeks to identify principles, strategies and tools that can be applied in a range of cultural settings. The paper includes a thematic analysis of the cultural contexts and policy environments of the participating countries in order to identify the obstacles to teacher leadership and to inform the nature of the support strategies employed

Fabrication of gradient hydrogels using a thermophoretic approach in microfluidics

The extracellular matrix presents spatially varying physical cues that can influence cell behavior in many processes. Physical gradients within hydrogels that mimic the heterogenous mechanical microenvironment are useful to study the impact of these cues on cellular responses. Therefore, simple and reliable techniques to create such gradient hydrogels are highly desirable. This work demonstrates the fabrication of stiffness gradient Gellan gum (GG) hydrogels by applying a temperature gradient across a microchannel containing hydrogel precursor solution. Thermophoretic migration of components within the precursor solution generates a concentration gradient that mirrors the temperature gradient profile, which translates into mechanical gradients upon crosslinking. Using this technique, GG hydrogels with stiffness gradients ranging from 20 to 90 kPa over 600 µm are created, covering the elastic moduli typical of moderately hard to hard tissues. MC3T3 osteoblast cells are then cultured on these gradient substrates, which exhibit preferential migration and enhanced osteogenic potential toward the stiffest region on the gradient. Overall, the thermophoretic approach provides a non-toxic and effective method to create hydrogels with defined mechanical gradients at the micron scale suitable for in vitro biological studies and potentially tissue engineering applications

Fabrication of gradient hydrogels using a thermophoretic approach in microfluidics

The extracellular matrix presents spatially varying physical cues that can influence cell behavior in many processes. Physical gradients within hydrogels that mimic the heterogenous mechanical microenvironment are useful to study the impact of these cues on cellular responses. Therefore, simple and reliable techniques to create such gradient hydrogels are highly desirable. This work demonstrates the fabrication of stiffness gradient Gellan gum (GG) hydrogels by applying a temperature gradient across a microchannel containing hydrogel precursor solution. Thermophoretic migration of components within the precursor solution generates a concentration gradient that mirrors the temperature gradient profile, which translates into mechanical gradients upon crosslinking. Using this technique, GG hydrogels with stiffness gradients ranging from 20 to 90 kPa over 600 µm are created, covering the elastic moduli typical of moderately hard to hard tissues. MC3T3 osteoblast cells are then cultured on these gradient substrates, which exhibit preferential migration and enhanced osteogenic potential toward the stiffest region on the gradient. Overall, the thermophoretic approach provides a non-toxic and effective method to create hydrogels with defined mechanical gradients at the micron scale suitable for in vitro biological studies and potentially tissue engineering applications

Two-dimensional SIR epidemics with long range infection

We extend a recent study of susceptible-infected-removed epidemic processes with long range infection (referred to as I in the following) from 1-dimensional lattices to lattices in two dimensions. As in I we use hashing to simulate very large lattices for which finite size effects can be neglected, in spite of the assumed power law $p({\bf x})\sim |{\bf x}|^{-\sigma-2}$ for the probability that a site can infect another site a distance vector ${\bf x}$ apart. As in I we present detailed results for the critical case, for the supercritical case with $\sigma = 2$, and for the supercritical case with $0< \sigma < 2$. For the latter we verify the stretched exponential growth of the infected cluster with time predicted by M. Biskup. For $\sigma=2$ we find generic power laws with $\sigma-$dependent exponents in the supercritical phase, but no Kosterlitz-Thouless (KT) like critical point as in 1-d. Instead of diverging exponentially with the distance from the critical point, the correlation length increases with an inverse power, as in an ordinary critical point. Finally we study the dependence of the critical exponents on $\sigma$ in the regime $0<\sigma <2$, and compare with field theoretic predictions. In particular we discuss in detail whether the critical behavior for $\sigma$ slightly less than 2 is in the short range universality class, as conjectured recently by F. Linder {\it et al.}. As in I we also consider a modified version of the model where only some of the contacts are long range, the others being between nearest neighbors. If the number of the latter reaches the percolation threshold, the critical behavior is changed but the supercritical behavior stays qualitatively the same.Comment: 14 pages, including 29 figure

Some discussions of D. Fearnhead and D. Prangle's Read Paper "Constructing summary statistics for approximate Bayesian computation: semi-automatic approximate Bayesian computation"

This report is a collection of comments on the Read Paper of Fearnhead and Prangle (2011), to appear in the Journal of the Royal Statistical Society Series B, along with a reply from the authors