Nonholonomic motion planning: steering using sinusoids

Methods for steering systems with nonholonomic constraints between arbitrary configurations are investigated. Suboptimal trajectories are derived for systems that are not in canonical form. Systems in which it takes more than one level of bracketing to achieve controllability are considered. The trajectories use sinusoids at integrally related frequencies to achieve motion at a given bracketing level. A class of systems that can be steered using sinusoids (claimed systems) is defined. Conditions under which a class of two-input systems can be converted into this form are given

Secular Effects of Tidal Damping in Compact Planetary Systems

We describe the long-term evolution of compact systems of terrestrial planets, using a set of simulations that match the statistical properties of the observed exoplanet distribution. The evolution is driven by tidal dissipation in the planetary interiors, but the systems evolve as a whole due to secular gravitational interactions. We find that, for Earth-like dissipation levels, planetary orbits can be circularised out to periods of order 100 days, an order of magnitude larger than is possible for single planets. The resulting distribution of eccentricities is a qualitative match to that inferred from transit timing variations, with a minority of non-zero eccentricities maintained by particular secular configurations. The coupling of the tidal and secular processes enhance the inward migration of the innermost planets in these systems, and can drive them to short orbital periods. Resonant interactions of both the mean motion and secular variety are observed, although the interactions are not strong enough to drive systemic instability in most cases. However, we demonstrate that these systems can easily be driven unstable if coupled to giant planets on longer period orbits.Comment: 17 pages, 22 figures, 2 tables, submitted to Monthly Notices of the Royal Astronomical Societ

The Impact of Early Positive Results on a Mathematics and Science Partnership: The Experience of the Institute for Chemistry Literacy Through Computational Science

After one year of implementation, the Institute for Chemistry Literacy through Computational Science, an NSF Mathematics and Science Partnership Institute Project led by the University of Illinois at Urbana-Champaign’s Department of Chemistry, College of Medicine, and National Center for Supercomputing Applications, experienced statistically significant gains in chemistry content knowledge among students of the rural high school teachers participating in its intensive, year-round professional development course, compared to a control group. The project utilizes a two-cohort, delayed-treatment, random control trial, quasi-experimental research design with the second cohort entering treatment one year following the first. The three-year treatment includes intensive two-week summer institutes, occasional school year workshops and year-round, on-line collaborative lesson development, resource sharing, and expert support. The means of student pre-test scores for Cohort I (η=963) and Cohort II (η=862) teachers were not significantly different. The mean gain (difference between pre-test and post-test scores) after seven months in the classroom for Cohort I was 9.8 percentage points, compared to 6.7 percentage points for Cohort II. This statistically significant difference (p\u3c.001) represented an effect size of .25 standard deviation units, and indicated unusually early confirmation of treatment effects. When post-tests were compared, Cohort I students scored significantly higher than Cohort II and supported the gain score differences. The impact of these results on treatment and research plans is discussed. concentrating on the effect of lessening rural teachers’ isolation and increasing access to tools to facilitate learning