Math and the Mouse: Explorations of Mathematics and Science in Walt Disney World

Math and the Mouse is an intensive, collaborative, project-driven, study away course that runs during the three-week May Experience term at Furman University and has many of the attributes of a course-based undergraduate research experience in mathematics. We take twelve students to Orlando, Florida to study the behind-the-scenes mathematics employed to make Walt Disney World operate efficiently. Students learn techniques of mathematical modeling (mostly resource allocation, logistics, and scheduling models), statistical analysis (mostly probability, clustering, data collection, and hypothesis testing), and ow management (queuing theory and some beginning ow dynamics) in an applied setting. Through planned course modules, collaborative activities, conversations with guest speakers, and three group projects, one of which is of the students\u27 choosing, this academic experience provides an engaged learning experience that shows how material from eleven academic courses comes together in connection with real-world applications

Epicyclic orbits in a viscous fluid about a precessing rod: Theory and experiments at the micro- and macro-scales

We present experimental observations and quantified theoretical predictions of the nanoscale hydrodynamics induced by nanorod precession emulating primary cilia motion in developing embryos. We observe phenomena including micron size particles which exhibit epicyclic orbits with coherent fluctuations distinguishable from comparable amplitude thermal noise. Quantifying the mixing and transport physics of such motions on small scales is critical to understanding fundamental biological processes such as extracellular redistribution of nutrients. We present experiments designed to quantify the trajectories of these particles, which are seen to consist of slow orbits about the rod, with secondary epicycles quasicommensurate with the precession rate. A first-principles theory is developed to predict trajectories in such time-varying flows. The theory is further tested using a dynamically similar macroscale experiment to remove thermal noise effects. The excellent agreement between our theory and experiments confirms that the continuum hypothesis applies all the way to the scales of such submicron biological motions

Regularized singularities and spectral deferred correction methods: a mathematical study of numerically modeling Stokes fluid flow

Regularized Stokeslets and spectral deferred correction methods are used to model variations of a rigid body precessing in Stokes flow. Numerical solutions are compared to exact and asymptotic closed form solutions for a spheroid precessing about its center. This provides the opportunity to perform careful error analysis and identify different numerical errors in regard to the motion of both slender and non-slender precessing spheroids. The error has components relating to quadrature, asymptotics, regularization, and time integration. Often, the quadrature error and time integration error are small with respect to the other error contributions, all of which are discussed. The motion of both slender and non-slender spheroids is studied to find the parameter and boundary condition choices that minimize velocity error. A system of regularized image singularities is developed to create a no-slip plane that mimics the effect of a nearby wall in the experiment setup. A temporal integration strategy based on spectral deferred correction (SDC) methods using an explicit treatment with different time steps for different components of the physical system is discussed. Multi-explicit SDC (MESDC) methods provide an increase in efficiency for stiff problems by allowing non-stiff parts of the physical setup to use a larger time step requiring fewer expensive computations. The numerical methods are used to study experimental fluid dynamics phenomena relating to precessing rods that are not described by an exact closed form solution. This work has biological motivations resulting from the study of pulmonary cilia in conjunction with cystic fibrosis research as well as the motion of primary nodal cilia in developing embryos whose motion plays a critical role in developing left-right asymmetry in mammals