Exploring student difficulties with observation location

Throughout introductory physics, students create and interpret free body diagrams in which multiple forces act on an object, typically at a single location (the object's center of mass). The situation increases in difficulty when multiple objects are involved, and further when electric and magnetic fields are present. In the latter, sources of the fields are often identified as a set of electric charges or current-carrying wires, and students are asked to determine the electric or magnetic field at a separate location defined as the observation location. Previous research suggests students struggle with accounting for how a measurement or calculation depends on the observation location. We present preliminary results from a studio-style, algebra-based, introductory electricity and magnetism course showing the prevalence of correct and incorrect responses to questions about observation location by analyzing student written work involving vector addition of fields

Post-Swift Gamma-ray Burst Science and Capabilities Needed to EXIST

The exhilerating results from Swift in its first year of operations have opened a new era of exploration of the high energy universe. The surge to higher redshifts of the Gamma-ray bursts now imaged with increased sensitivity establishes them as viable cosmic probes of the early universe. Wide-field coded aperture imaging with solid-state pixel detectors (Cd-Zn-Te) has been also established as the optimum approach for GRB discovery and location as well as to conduct sensitive full-sky hard X-ray sky surveys. I outline the current and future major science questions likely to dominate the post-Swift era for GRBs and several related disciplines and the mission requirements to tackle these. The EXIST mission, under study for NASA's Black Hole Finder Probe (BHFP) in the Beyond Einstein Program, could achieve these objectives as the Next Generation GRB Mission with `ultimate' sensitivity and wide-field survey capability. Analysis tools for processing Swift/BAT slew data are under development at CfA and will both test EXIST scanning imaging and provide new data on GRBs and transients.Comment: Invited review to appear in the Proceedings of the 16th Annual October Astrophysics Conference in Maryland, "Gamma Ray Bursts in the Swift Era", eds. S. Holt, N. Gehrels and J. Nousek; 11 pages, 5 figure

Principal Component Analysis for Functional Data on Riemannian Manifolds and Spheres

Functional data analysis on nonlinear manifolds has drawn recent interest. Sphere-valued functional data, which are encountered for example as movement trajectories on the surface of the earth, are an important special case. We consider an intrinsic principal component analysis for smooth Riemannian manifold-valued functional data and study its asymptotic properties. Riemannian functional principal component analysis (RFPCA) is carried out by first mapping the manifold-valued data through Riemannian logarithm maps to tangent spaces around the time-varying Fr\'echet mean function, and then performing a classical multivariate functional principal component analysis on the linear tangent spaces. Representations of the Riemannian manifold-valued functions and the eigenfunctions on the original manifold are then obtained with exponential maps. The tangent-space approximation through functional principal component analysis is shown to be well-behaved in terms of controlling the residual variation if the Riemannian manifold has nonnegative curvature. Specifically, we derive a central limit theorem for the mean function, as well as root-$n$ uniform convergence rates for other model components, including the covariance function, eigenfunctions, and functional principal component scores. Our applications include a novel framework for the analysis of longitudinal compositional data, achieved by mapping longitudinal compositional data to trajectories on the sphere, illustrated with longitudinal fruit fly behavior patterns. RFPCA is shown to be superior in terms of trajectory recovery in comparison to an unrestricted functional principal component analysis in applications and simulations and is also found to produce principal component scores that are better predictors for classification compared to traditional functional functional principal component scores

Single-Trial Phase Precession in the Hippocampus

During the crossing of the place field of a pyramidal cell in the rat hippocampus, the firing phase of the cell decreases with respect to the local theta rhythm. This phase precession is usually studied on the basis of data in which many place field traversals are pooled together. Here we study properties of phase precession in single trials. We found that single-trial and pooled-trial phase precession were different with respect to phase-position correlation, phase-time correlation, and phase range. Whereas pooled-trial phase precession may span 360°, the most frequent single-trial phase range was only ∼180°. In pooled trials, the correlation between phase and position (r = −0.58) was stronger than the correlation between phase and time (r = −0.27), whereas in single trials these correlations (r = −0.61 for both) were not significantly different. Next, we demonstrated that phase precession exhibited a large trial-to-trial variability. Overall, only a small fraction of the trial-to-trial variability in measures of phase precession (e.g., slope or offset) could be explained by other single-trial properties (such as running speed or firing rate), whereas the larger part of the variability remains to be explained. Finally, we found that surrogate single trials, created by randomly drawing spikes from the pooled data, are not equivalent to experimental single trials: pooling over trials therefore changes basic measures of phase precession. These findings indicate that single trials may be better suited for encoding temporally structured events than is suggested by the pooled data