Statistics of a Family of Piecewise Linear Maps

We study statistical properties of the truncated flat spot map $f_t(x)$. In particular, we investigate whether for large $n$, the deviations $\sum_{i=0}^{n-1} \left(f_t^i(x_0)-\frac 12\right)$ upon rescaling satisfy a $Q$-Gaussian distribution if $x_0$ and $t$ are both independently and uniformly distributed on the unit circle. This was motivated by the fact that if $f_t$ is the rotation by $t$, then it has been shown that in this case the rescaled deviations are distributed as a $Q$-Gaussian with $Q=2$ (a Cauchy distribution). This is the only case where a non-trivial (i.e. $Q\neq 1$) $Q$-Gaussian has been analytically established in a conservative dynamical system. In this note, however, we prove that for the family considered here, $\lim_n S_n/n$ converges to a random variable with a curious distribution which is clearly not a $Q$-Gaussian or any other standard smooth distribution

\u27Traffic-light\u27 nutrition labelling and \u27junk-food\u27 tax : a modelled comparison of cost-effectiveness for obesity prevention

Introduction: Cost-effectiveness analyses are important tools in efforts to prioritise interventions for obesity prevention.Modelling facilitates evaluation of multiple scenarios with varying assumptions. This study compares the cost-effectiveness ofconservative scenarios for two commonly proposed policy-based interventions: front-of-pack &lsquo;traffic-light&rsquo; nutrition labelling(traffic-light labelling) and a tax on unhealthy foods (&lsquo;junk-food&rsquo; tax).Methods: For traffic-light labelling, estimates of changes in energy intake were based on an assumed 10% shift in consumptiontowards healthier options in four food categories (breakfast cereals, pastries, sausages and preprepared meals) in 10% of adults. For the &lsquo;junk-food&rsquo; tax, price elasticities were used to estimate a change in energy intake in response to a 10% price increase in seven food categories (including soft drinks, confectionery and snack foods). Changes in population weight and body mass index by sex were then estimated based on these changes in population energy intake, along with subsequent impacts on disability-adjusted life years (DALYs). Associated resource use was measured and costed using pathway analysis, based on a health sector perspective (with some industry costs included). Costs and health outcomes were discounted at 3%. The cost-effectiveness of each intervention was modelled for the 2003 Australian adult population.Results: Both interventions resulted in reduced mean weight (traffic-light labelling: 1.3 kg (95% uncertainty interval (UI): 1.2;1.4); &lsquo;junk-food&rsquo; tax: 1.6 kg (95% UI: 1.5; 1.7)); and DALYs averted (traffic-light labelling: 45 100 (95% UI: 37 700; 60 100);&lsquo;junk-food&rsquo; tax: 559 000 (95% UI: 459 500; 676 000)). Cost outlays were AUD81 million (95% UI: 44.7; 108.0) for traffic-lightlabelling and AUD18 million (95% UI: 14.4; 21.6) for &lsquo;junk-food&rsquo; tax. Cost-effectiveness analysis showed both interventions were&lsquo;dominant&rsquo; (effective and cost-saving).Conclusion: Policy-based population-wide interventions such as traffic-light nutrition labelling and taxes on unhealthy foods arelikely to offer excellent &lsquo;value for money&rsquo; as obesity prevention measures.<br /

Scaffolding School Pupils’ Scientific Argumentation with Evidence-Based Dialogue Maps

This chapter reports pilot work investigating the potential of Evidence-based Dialogue Mapping to scaffold young teenagers’ scientific argumentation. Our research objective is to better understand pupils’ usage of dialogue maps created in Compendium to write scientific ex-planations. The participants were 20 pupils, 12-13 years old, in a summer science course for “gifted and talented” children in the UK. Through qualitative analysis of three case studies, we investigate the value of dialogue mapping as a mediating tool in the scientific reasoning process during a set of learning activities. These activities were published in an online learning envi-ronment to foster collaborative learning. Pupils mapped their discussions in pairs, shared maps via the online forum and in plenary discussions, and wrote essays based on their dialogue maps. This study draws on these multiple data sources: pupils’ maps in Compendium, writings in science and reflective comments about the uses of mapping for writing. Our analysis highlights the diversity of ways, both successful and unsuccessful, in which dialogue mapping was used by these young teenagers

Piecewise Linear Models for the Quasiperiodic Transition to Chaos

We formulate and study analytically and computationally two families of piecewise linear degree one circle maps. These families offer the rare advantage of being non-trivial but essentially solvable models for the phenomenon of mode-locking and the quasi-periodic transition to chaos. For instance, for these families, we obtain complete solutions to several questions still largely unanswered for families of smooth circle maps. Our main results describe (1) the sets of maps in these families having some prescribed rotation interval; (2) the boundaries between zero and positive topological entropy and between zero length and non-zero length rotation interval; and (3) the structure and bifurcations of the attractors in one of these families. We discuss the interpretation of these maps as low-order spline approximations to the classic ``sine-circle'' map and examine more generally the implications of our results for the case of smooth circle maps. We also mention a possible connection to recent experiments on models of a driven Josephson junction.Comment: 75 pages, plain TeX, 47 figures (available on request

State attachment variability across distressing situations in middle childhood

Contemporary research suggests that attachment has both a trait-like, stable component, and a state-like component that varies across contexts. In the current study, we assessed state attachment variability across comparably distressing situations in middle childhood. In two samples, children reported their expectations of maternal support in each situation. Additionally, we administered trait attachment and psychological well-being measures. Results indicated that, overall, children varied in their expectations across situations: more than half of the variance was explained by intra-individual differences across situations. Results revealed two components underlying variability: a Signal-and-Support component reflecting expectations of support-seeking and receiving, and a Back-on-Track component reflecting expectations of stress reduction and comfort. State attachment variability was related to individual differences in trait attachment: children who are more securely attached at the trait level, overall appear to vary less in their state attachment, likely due to their high mean state attachment scores across situations. When the mean state attachment scores are accounted for, more securely attached children seem to vary more, suggesting that their state attachment expectations are more sensitive to contextual fluctuations. Importantly, degree of state attachment variability explained psychological well-being over and above trait attachment