Covering algorithms, continuum percolation and the geometry of wireless networks

Continuum percolation models in which each point of a two-dimensional Poisson point process is the centre of a disc of given (or random) radius r, have been extensively studied. In this paper, we consider the generalization in which a deterministic algorithm (given the points of the point process) places the discs on the plane, in such a way that each disc covers at least one point of the point process and that each point is covered by at least one disc. This gives a model for wireless communication networks, which was the original motivation to study this class of problems. We look at the percolation properties of this generalized model, showing that an unbounded connected component of discs does not exist, almost surely, for small values of the density lambda of the Poisson point process, for any covering algorithm. In general, it turns out not to be true that unbounded connected components arise when lambda is taken sufficiently high. However, we identify some large families of covering algorithms, for which such an unbounded component does arise for large values of lambda. We show how a simple scaling operation can change the percolation properties of the model, leading to the almost sure existence of an unbounded connected component for large values of lambda, for any covering algorithm. Finally, we show that a large class of covering algorithms, which arise in many practical applications, can get arbitrarily close to achieving a minimal density of covering discs. We also construct an algorithm that achieves this minimal density

Driving sandpiles to criticality and beyond

A popular theory of self-organized criticality relates driven dissipative systems to systems with conservation. This theory predicts that the stationary density of the abelian sandpile model equals the threshold density of the fixed-energy sandpile. We refute this prediction for a wide variety of underlying graphs, including the square grid. Driven dissipative sandpiles continue to evolve even after reaching criticality. This result casts doubt on the validity of using fixed-energy sandpiles to explore the critical behavior of the abelian sandpile model at stationarity.Comment: v4 adds referenc

Scaling relation for determining the critical threshold for continuum percolation of overlapping discs of two sizes

We study continuum percolation of overlapping circular discs of two sizes. We propose a phenomenological scaling equation for the increase in the effective size of the larger discs due to the presence of the smaller discs. The critical percolation threshold as a function of the ratio of sizes of discs, for different values of the relative areal densities of two discs, can be described in terms of a scaling function of only one variable. The recent accurate Monte Carlo estimates of critical threshold by Quintanilla and Ziff [Phys. Rev. E, 76 051115 (2007)] are in very good agreement with the proposed scaling relation.Comment: 4 pages, 3 figure

Catch them ... if you can

An important part of forensic science is dedicated to the evaluation of physical traces left at the crime scene like fingerprints, bullets, toolmarks etc. These traces are compared with traces from a suspect. The evaluation of physical traces can be interpreted as the comparison of two noisy signals. We introduce an evaluation of the matching of two noisy signals at diverse scales and localisations in space. In a multi-resolution way a "probability" of matching is computed. Furthermore, a description is given to evaluate the complexity of a shoemark. A likelihood ratio approach is used for comparing two shoemark traces

Black Dialect in Children\u27s Books

Black non-Standard English is different in grammar (syntax) from Standard English. The advent of the 60\u27s produced authors who explored the full possibilities of language to deal with their themes. The increased use of dialect by black authors, particularly children\u27s authors, was a sign that the nature of the black experience as they wanted to convey it did not have to rely on traditional forms, and literary devices; that they could treat familiar, realistic ideas and situations using a familiar dialect and relate that idea more effectively

Limiting shapes for deterministic centrally seeded growth models

We study the rotor router model and two deterministic sandpile models. For the rotor router model in $\mathbb{Z}^d$, Levine and Peres proved that the limiting shape of the growth cluster is a sphere. For the other two models, only bounds in dimension 2 are known. A unified approach for these models with a new parameter $h$ (the initial number of particles at each site), allows to prove a number of new limiting shape results in any dimension $d \geq 1$. For the rotor router model, the limiting shape is a sphere for all values of $h$. For one of the sandpile models, and $h=2d-2$ (the maximal value), the limiting shape is a cube. For both sandpile models, the limiting shape is a sphere in the limit $h \to -\infty$. Finally, we prove that the rotor router shape contains a diamond.Comment: 18 pages, 3 figures, some errors corrected and more explanation added, to appear in Journal of Statistical Physic

Strict inequalities of critical values in continuum percolation

We consider the supercritical finite-range random connection model where the points $x,y$ of a homogeneous planar Poisson process are connected with probability $f(|y-x|)$ for a given $f$. Performing percolation on the resulting graph, we show that the critical probabilities for site and bond percolation satisfy the strict inequality $p_c^{\rm site} > p_c^{\rm bond}$. We also show that reducing the connection function $f$ strictly increases the critical Poisson intensity. Finally, we deduce that performing a spreading transformation on $f$ (thereby allowing connections over greater distances but with lower probabilities, leaving average degrees unchanged) {\em strictly} reduces the critical Poisson intensity. This is of practical relevance, indicating that in many real networks it is in principle possible to exploit the presence of spread-out, long range connections, to achieve connectivity at a strictly lower density value.Comment: 38 pages, 8 figure

Clarifying European terminology in plastics recycling

The increasing activities in plastics recycling have led to a sprawl of terminology describing different technologies and technology categorizations. This creates not only linguistic confusion but also makes it difficult for regulators, investors, corporate leaders and other stakeholders to fully understand the relationship between different technologies, potentially leading to suboptimal decisions on policy, investment, or collaboration. To bring clarity to this topic, this manuscript provides an overview of (i) the different circular pathways for plastics, with a focus on recycling, (ii) the most common categorization of recycling technologies, (iii) what is considered ‘recycling’ by the European Commission and (iv) some alternative terms used in grey and academic literature to describe recycling technologies