Developing Mathematics Enrichment Workshops for Middle School Students: Philosophy and Sample Workshops

This paper describes our approach to organizing enrichment activities using advanced mathematics topics for diverse audiences of middle school students. We discuss our philosophy and approaches for the structure of these workshops, and then provide sample schedules and resource materials. The workshops cover activities on the following topics: Graphing Calculators; The Chaos Game; Statistical Sampling; CT Scans–the reconstruction problem; The Platonic and Archimedean solids; The Shape of Space; Symmetry; The Binary Number System and the game of NIM; Graph Theory: Proof by Counterexample

Statics and Dynamics of Yukawa Cluster Crystals on Ordered Substrates

We examine the statics and dynamics of particles with repulsive Yukawa interactions in the presence of a two-dimensional triangular substrate for fillings of up to twelve particles per potential minimum. We term the ordered states Yukawa cluster crystals and show that they are distinct from the colloidal molecular crystal states found at low fillings. As a function of substrate and interaction strength at fixed particle density we find a series of novel crystalline states that we characterize using the structure factor. For fillings greater than four, shell and ring structures form at each potential minimum and can exhibit sample-wide orientational order. A disordered state can appear between ordered states as the substrate strength varies. Under an external drive, the onsets of different orderings produce clear changes in the critical depinning force, including a peak effect phenomenon that has generally only previously been observed in systems with random substrates. We also find a rich variety of dynamic ordering transitions that can be observed via changes in the structure factor and features in the velocity-force curves. The dynamical states encompass a variety of moving structures including one-dimensional stripes, smectic ordering, polycrystalline states, triangular lattices, and symmetry locking states. Despite the complexity of the system, we identify several generic features of the dynamical phase transitions which we map out in a series of phase diagrams. Our results have implications for the structure and depinning of colloids on periodic substrates, vortices in superconductors and Bose-Einstein condensates, Wigner crystals, and dusty plasmas.Comment: 14 pages, 17 postscript figure

Ordering and Melting in Colloidal Molecular Crystal Mixtures

We show in simulations that a rich variety of novel orderings such as pinwheel and star states can be realized for colloidal molecular crystal mixtures at rational ratios of the number of colloids to the number of minima from an underlying periodic substrate. These states can have multi-step melting transitions and also show coexistence in which one species disorders while the other species remains orientationally disordered. For other mixtures, only partially ordered or frustrated states form.Comment: 4 pages, 4 postscript figure

Fecal near Infrared Reflectance Spectroscopy (NIRS) and the Nutrition Balance Analyzer (NUTBAL) Case Study in South Dakota

The objective of this study was to compare fecal near infrared reflectance spectroscopy (NIRS) and the nutrition balance analyzer (NUTBAL) results with diet samples and cattle performance to determine if fecal NIRS and NUTBAL can accurately predict forage quality and cattle performance in South Dakota

Colloidal Dynamics on Disordered Substrates

Using Langevin simulations we examine driven colloids interacting with quenched disorder. For weak substrates the colloids form an ordered state and depin elastically. For increasing substrate strength we find a sharp crossover to inhomogeneous depinning and a substantial increase in the depinning force, analogous to the peak effect in superconductors. The velocity versus driving force curve shows criticality at depinning, with a change in scaling exponent occuring at the order to disorder crossover. Upon application of a sudden pulse of driving force, pronounced transients appear in the disordered regime which are due to the formation of long-lived colloidal flow channels.Comment: 4 pages, 4 postscript figure

An Optical Study of BG Geminorum: An Ellipsoidal Binary with an Unseen Primar Star

We describe optical photometric and spectroscopic observations of the bright variable BG Geminorum. Optical photometry shows a pronounced ellipsoidal variation of the K0 I secondary, with amplitudes of ~0.5 mag at VRI and a period of 91.645 days. A deep primary eclipse is visible for wavelengths < 4400A; a shallower secondary eclipse is present at longer wavelengths. Eclipse timings and the radial velocity curve of the K0 secondary star indicate an interacting binary where a lobe-filling secondary, M_2 ~ 0.5 Msun, transfers material into a extended disk around a massive primary, M_1 ~ 4.5 Msun. The primary star is either an early B-type star or a black hole. If it did contain a black hole, BG Gem would be the longest period black hole binary known by a factor of 10, as well as the only eclipsing black hole binary system.Comment: 27 pages, includes 8 figures and 5 tables, accepted to A