On C*-algebras related to constrained representations of a free group

We consider representations of the free group $F_2$ on two generators such that the norm of the sum of the generators and their inverses is bounded by $\mu\in[0,4]$. These $\mu$-constrained representations determine a C*-algebra $A_{\mu}$ for each $\mu\in[0,4]$. We prove that these C*-algebras form a continuous bundle of C*-algebras over $[0,4]$ and calculate their K-groups.Comment: 9 page

The role of community readiness in the prevention of overweight and obesity in pre-adolescent girls

Overweight and obesity prevention efforts focused on pre-adolescent girls are justified because the health of future mothers can significantly impact the health of their future offspring. Aims and Objectives This thesis aimed to investigate the role of community readiness in the prevention of overweight and obesity in pre-adolescent girls within the Charnwood Borough (Leicestershire) community. Specifically the objectives of this research were to: 1) further the understanding of the concept of community , 2) identify, using focus groups, the key informants and other ecological influences acting on pre-adolescent girls health behaviours, 3) employ the Community Readiness Model (CRM), using key informant interviews, to assess the Charnwood Borough community s knowledge, awareness and readiness to adopt behaviours associated with the prevention of overweight and obesity in pre-adolescent girls, 4) evaluate the usefulness of the CRM for this task and 5) form recommendations for a tailored intervention. Methods This research used focus groups with pre-adolescent girls to identify the key informants who influence their lifestyle choices regarding dietary and physical activity behaviours. The study recruited 56 female participants aged between 6-11 years from 8 primary schools. In total 13 focus groups with between 2 to 7 participants per group were performed. To assess the level of community readiness, 33 key informants identified as informing the health behaviours of pre-adolescent girls were recruited. Semi-structured interviews followed an adapted version of the CRM. The model consists of six dimensions: community efforts; community knowledge of efforts; leadership; community climate; community knowledge of the issue and resources which are scored on a scale of 1 (no awareness) to 9 (high level of community readiness). The average of these scores is the community readiness stage. Findings In relation to the first objective, the definition of a community given by those residing in the Charnwood Borough community included: people and groups of people (e.g. school; religious and parents community groups); characteristics (i.e. geographical area; where people reside; buildings; institutions and the size of the community); sense of belonging and the attempt to improve the community for its members. By valuing the perspectives of pre-adolescent girls, a deepened understanding of the broad range of key informants (Dinner staff, Doctors and dentists, Head teachers, Girl Guide leaders, Government, Grandparents, Neighbours, Parents, Peer group, School cooks, Shop keepers, Siblings, Sports Coaches and Teachers) acting on their health behaviours was achieved (Objective 2). The community readiness score relating to healthy eating and drinking corresponded to the Preparation Stage (Objective 3), whereby influential community members have begun planning efforts by deciding what to do and who will do it and the community offers modest support of efforts promoting healthy dietary behaviours. The community readiness score for physical activity was higher corresponding to the Initiation Stage where there is enough information available to justify efforts and activities are underway to support healthy physical activity behaviours. Qualitatively, the key findings from applying the CRM are that: the effectiveness of community initiatives promoting healthy behaviours can be improved; there are varying levels of concern and awareness regarding unhealthy behaviours and child weight status in the community; social disparities are perceived to exist in community health behaviours; communication between schools and parents are seen as important for the success of school policies; parental attitudes and behaviours inform child behaviours; enjoyment is an important predictor of girls activity levels and aspects of the Charnwood Borough s environment do not support healthy behaviours. The CRM was shown to be an appropriate tool for assessing community readiness (Objective 4) providing the qualitative strengths of the model are emphasised. The community readiness score and the qualitative analysis informed the recommendations for an intervention in the Charnwood Borough (Objective 5). Key aspects of these recommendations include: raising the community s awareness of pre-adolescent girls health behaviours; ensuring all initiatives are accessible and appealing to low income families by working directly with these groups; fostering a greater sense of community through increased community cohesion and social interaction; placing social interaction and enjoyment at the centre of all activities, and attempting to make the healthier option the easier choice through modifications to the environment. Conclusions This thesis is the first to apply the Community Readiness Model to the prevention of overweight and obesity in pre-adolescent girls in the UK and has demonstrated the value of applying a community readiness assessment prior to the implementation of interventions

The Politicisation of Diversity Planning in a Global City: Lessons from London

This paper explores the politics of diversity plan ning in one of Europe’s most socially and economically divided and globally-oriented cities, London. The analysis draws on Latour’s writings on modes of politicisation to examine the processes an d practices that shape contemporary urban governance. It uses the example of diversity planni ng to examine the ‘what’ and ‘how’ of urban politics. It shows that on the one hand diversity i s represented in pragmatic, consensual, and celebratory terms. Under prevailing conditions of c ontemporary global capitalism, the ‘what’ of diversity has been politicised into an agenda for l abour market-building and the attraction of ‘talented’ individuals and foreign investment. Howe ver, at the same time this celebratory rhetoric represents part of a wider effort to deflect politi cal attention away from the socially and economically divisive impacts of global models of e conomic growth and physical development. There is little discussion of the ways in which pla nning frameworks, the ‘how’ of diversity policy, ar e helping to generate new separations in and beyond t he city. Moreover, despite claiming that policy is pragmatic and non-ideological, the paper shows h ow diversity narratives have become an integral part of broader political projects to orientate the city’s economy towards the needs of a relatively small cluster of powerful economic sectors. The pap er concludes with reflections on the recent impacts of the vote for Brexit and the election of an openly Muslim London Mayor. It also assesses the broader relevance of a Latourian framework for the analysis of contemporary urban politics

Stability of Influence Maximization

The present article serves as an erratum to our paper of the same title, which was presented and published in the KDD 2014 conference. In that article, we claimed falsely that the objective function defined in Section 1.4 is non-monotone submodular. We are deeply indebted to Debmalya Mandal, Jean Pouget-Abadie and Yaron Singer for bringing to our attention a counter-example to that claim. Subsequent to becoming aware of the counter-example, we have shown that the objective function is in fact NP-hard to approximate to within a factor of $O(n^{1-\epsilon})$ for any $\epsilon > 0$. In an attempt to fix the record, the present article combines the problem motivation, models, and experimental results sections from the original incorrect article with the new hardness result. We would like readers to only cite and use this version (which will remain an unpublished note) instead of the incorrect conference version.Comment: Erratum of Paper "Stability of Influence Maximization" which was presented and published in the KDD1

Sharpness of the phase transition and exponential decay of the subcritical cluster size for percolation on quasi-transitive graphs

We study homogeneous, independent percolation on general quasi-transitive graphs. We prove that in the disorder regime where all clusters are finite almost surely, in fact the expectation of the cluster size is finite. This extends a well-known theorem by Menshikov and Aizenman & Barsky to all quasi-transitive graphs. Moreover we deduce that in this disorder regime the cluster size distribution decays exponentially, extending a result of Aizenman & Newman. Our results apply to both edge and site percolation, as well as long range (edge) percolation. The proof is based on a modification of the Aizenman & Barsky method.Comment: Latex 2e; 25 pages (a4wide); small editorial corrections; one reference adde

Bond percolation on isoradial graphs: criticality and universality

In an investigation of percolation on isoradial graphs, we prove the criticality of canonical bond percolation on isoradial embeddings of planar graphs, thus extending celebrated earlier results for homogeneous and inhomogeneous square, triangular, and other lattices. This is achieved via the star-triangle transformation, by transporting the box-crossing property across the family of isoradial graphs. As a consequence, we obtain the universality of these models at the critical point, in the sense that the one-arm and 2j-alternating-arm critical exponents (and therefore also the connectivity and volume exponents) are constant across the family of such percolation processes. The isoradial graphs in question are those that satisfy certain weak conditions on their embedding and on their track system. This class of graphs includes, for example, isoradial embeddings of periodic graphs, and graphs derived from rhombic Penrose tilings.Comment: In v2: extended title, and small changes in the tex

Restoring site percolation on a damaged square lattice

We study how to restore site percolation on a damaged square lattice with nearest neighbor (N$^2$) interactions. Two strategies are suggested for a density $x$ of destroyed sites by a random attack at $p_c$. In the first one, a density $y$ of new sites are created with longer range interactions, either next nearest neighbor (N$^3$) or next next nearest neighbor (N$^4$). In the second one, new longer range interactions N$^3$ or N$^4$ are created for a fraction $v$ of the remaining $(p_c-x)$ sites in addition to their N$^2$ interactions. In both cases, the values of $y$ and $v$ are tuned in order to restore site percolation which then occurs at new percolation thresholds, respectively $\pi_3$, $\pi_4$, $\pi_{23}$ and $\pi_{24}$. Using Monte Carlo simulations the values of the pairs $\{y, \pi_3 \}$, $\{y, \pi_4\}$ and $\{v, \pi_{23}\}$, $\{v, \pi_{24}\}$ are calculated for the whole range $0\leq x \leq p_c(\text{N}^2)$. Our schemes are applicable to all regular lattices.Comment: 5 pages, revtex

Induced topological pressure for countable state Markov shifts

We introduce the notion of induced topological pressure for countable state Markov shifts with respect to a non-negative scaling function and an arbitrary subset of finite words. Firstly, the scaling function allows a direct access to important thermodynamical quantities, which are usually given only implicitly by certain identities involving the classically defined pressure. In this context we generalise Savchenko's definition of entropy for special flows to a corresponding notion of topological pressure and show that this new notion coincides with the induced pressure for a large class of H\"older continuous height functions not necessarily bounded away from zero. Secondly, the dependence on the subset of words gives rise to interesting new results connecting the Gurevi{\vc} and the classical pressure with exhausting principles for a large class of Markov shifts. In this context we consider dynamical group extentions to demonstrate that our new approach provides a useful tool to characterise amenability of the underlying group structure.Comment: 28 page

Partially asymmetric exclusion models with quenched disorder

We consider the one-dimensional partially asymmetric exclusion process with random hopping rates, in which a fraction of particles (or sites) have a preferential jumping direction against the global drift. In this case the accumulated distance traveled by the particles, x, scales with the time, t, as x ~ t^{1/z}, with a dynamical exponent z > 0. Using extreme value statistics and an asymptotically exact strong disorder renormalization group method we analytically calculate, z_{pt}, for particlewise (pt) disorder, which is argued to be related to the dynamical exponent for sitewise (st) disorder as z_{st}=z_{pt}/2. In the symmetric situation with zero mean drift the particle diffusion is ultra-slow, logarithmic in time.Comment: 4 pages, 3 figure

The Parallel Complexity of Growth Models

This paper investigates the parallel complexity of several non-equilibrium growth models. Invasion percolation, Eden growth, ballistic deposition and solid-on-solid growth are all seemingly highly sequential processes that yield self-similar or self-affine random clusters. Nonetheless, we present fast parallel randomized algorithms for generating these clusters. The running times of the algorithms scale as $O(\log^2 N)$, where $N$ is the system size, and the number of processors required scale as a polynomial in $N$. The algorithms are based on fast parallel procedures for finding minimum weight paths; they illuminate the close connection between growth models and self-avoiding paths in random environments. In addition to their potential practical value, our algorithms serve to classify these growth models as less complex than other growth models, such as diffusion-limited aggregation, for which fast parallel algorithms probably do not exist.Comment: 20 pages, latex, submitted to J. Stat. Phys., UNH-TR94-0