Random induced subgraphs of Cayley graphs induced by transpositions

In this paper we study random induced subgraphs of Cayley graphs of the symmetric group induced by an arbitrary minimal generating set of transpositions. A random induced subgraph of this Cayley graph is obtained by selecting permutations with independent probability, $\lambda_n$. Our main result is that for any minimal generating set of transpositions, for probabilities $\lambda_n=\frac{1+\epsilon_n}{n-1}$ where $n^{-{1/3}+\delta}\le \epsilon_n0$, a random induced subgraph has a.s. a unique largest component of size $\wp(\epsilon_n)\frac{1+\epsilon_n}{n-1}n!$, where $\wp(\epsilon_n)$ is the survival probability of a specific branching process.Comment: 18 pages, 1 figur

Effect of vegetation on sound fields in idealised urban open spaces

Noise pollution is a major environmental problem within the EU and during the last years vegetation was examined for its benefits in increasing health and well-being of citizens from different viewpoints, including noise control and soundscape enhancement. This work focuses on numerical simulations to investigate the effectiveness of vegetation for controlling sound fields, especially in terms of the abatement of traffic noise. Two idealised urban squares were studied, one rectangular and one octagonal. Three plant types, climbing plants (ivy), living green walls with soil substrate and plants in pots (nephrolepis exaltata), were used in this investigation, based on their measured properties in laboratory, and four aspects of the use of vegetation were evaluated: effect of the amount of vegetation, effect of changing in the scattering coefficient of vegetation, effect of vegetation in different receiver positions and effect of vegetation on different groups of receivers. Parametric studies on the determination of a line source and on the definition of sound power levels referred to traffic noise were also developed. Three simulation tools were used, namely CATT-Acoustic®, CRR (Combined Ray-tracing and Radiosity) and Odeon