Avoided level crossings in the quantization of a mixed regular-chaotic system

The study of avoided level crossings in the spectra of quantum Hamiltonians whose classical counterparts exhibit mixed regular-chaotic dynamics reveals important information about the quantum signatures of such dynamics. Among other topics, one can study delocalization phenomena and exchange of structure in Husimi distributions and compare the quantum dynamics one observes to the dynamics of corresponding classical systems at the same parameter values

A Perturbative Analysis of Modulated Amplitude Waves in Bose-Einstein Condensates

We apply Lindstedt's method and multiple scale perturbation theory to analyze spatio-temporal structures in nonlinear Schr\"odinger equations and thereby study the dynamics of quasi-one-dimensional Bose-Einstein condensates with mean-field interactions. We determine the dependence of the intensity of modulated amplitude waves on their wave number. We also explore the band structure of Bose-Einstein condensates in detail using Hamiltonian perturbation theory and supporting numerical simulations.Comment: 24 pages, 20 figs (numbered to 9), 6 tables, to appear in Chao

Two-Particle Circular Billiards Versus Randomly Perturbed One-Particle Circular Billiards

We study a two-particle circular billiard containing two finite-size circular particles that collide elastically with the billiard boundary and with each other. Such a two-particle circular billiard provides a clean example of an "intermittent" system. This billiard system behaves chaotically, but the time scale on which chaos manifests can become arbitrarily long as the sizes of the confined particles become smaller. The finite-time dynamics of this system depends on the relative frequencies of (chaotic) particle-particle collisions versus (integrable) particle-boundary collisions, and investigating these dynamics is computationally intensive because of the long time scales involved. To help improve understanding of such two-particle dynamics, we compare the results of diagnostics used to measure chaotic dynamics for a two-particle circular billiard with those computed for two types of one-particle circular billiards in which a confined particle undergoes random perturbations. Importantly, such one-particle approximations are much less computationally demanding than the original two-particle system, and we expect them to yield reasonable estimates of the extent of chaotic behavior in the two-particle system when the sizes of confined particles are small. Our computations of recurrence-rate coefficients, finite-time Lyapunov exponents, and autocorrelation coefficients support this hypothesis and suggest that studying randomly perturbed one-particle billiards has the potential to yield insights into the aggregate properties of two-particle billiards, which are difficult to investigate directly without enormous computation times (especially when the sizes of the confined particles are small).Comment: 9 pages, 7 figures (some with multiple parts); published in Chao

Ground Control to Niels Bohr: Exploring Outer Space with Atomic Physics

We provided an introduction to transition state theory and the connections it provides between atomic and celestial physics. We include brief discussions of historical background, recent applications in space mission design, and current research efforts.Comment: 17 pages, 3 figures, expository article, submitted to the Notices of the American Mathematical Society; better figs available at http://www.math.gatech.edu/~mason

Yoga practice in the UK: a cross- sectional survey of motivation, health benefits and behaviours

Objectives Despite the popularity of yoga and evidence of its positive effects on physical and mental health, little is known about yoga practice in the UK. This study investigated the characteristics of people who practise yoga, reasons for initiating and maintaining practice, and perceived impact of yoga on health and well-being. Design, setting and participants A cross-sectional online anonymous survey distributed through UK-based yoga organisations, studios and events, through email invites and flyers. 2434 yoga practitioners completed the survey, including 903 yoga teachers: 87% were women, 91% white and 71% degree educated; mean age was 48.7 years. Main outcome measures Perceived impact of yoga on health conditions, health outcomes and injuries. Relationships between yoga practice and measures of health, lifestyle, stress and well-being. Results In comparison with national population norms, participants reported significantly higher well-being but also higher anxiety; lower perceived stress, body mass index and incidence of obesity, and higher rates of positive health behaviours. 47% reported changing their motivations to practise yoga, with general wellness and fitness key to initial uptake, and stress management and spirituality important to current practice. 16% of participants reported starting yoga to manage a physical or mental health condition. Respondents reported the value of yoga for a wide range of health conditions, most notably for musculoskeletal and mental health conditions. 20.7% reported at least one yoga-related injury over their lifetime. Controlling for demographic factors, frequency of yoga practice accounted for small but significant variance in health-related regression models (p<0.001). Conclusion The findings of this first detailed UK survey were consistent with surveys in other Western countries. Yoga was perceived to have a positive impact on physical and mental health conditions and was linked to positive health behaviours. Further investigation of yoga’s role in self-care could inform health-related challenges faced by many countries

Dynamical Systems on Networks: A Tutorial

We give a tutorial for the study of dynamical systems on networks. We focus especially on "simple" situations that are tractable analytically, because they can be very insightful and provide useful springboards for the study of more complicated scenarios. We briefly motivate why examining dynamical systems on networks is interesting and important, and we then give several fascinating examples and discuss some theoretical results. We also briefly discuss dynamical systems on dynamical (i.e., time-dependent) networks, overview software implementations, and give an outlook on the field.Comment: 39 pages, 1 figure, submitted, more examples and discussion than original version, some reorganization and also more pointers to interesting direction