Title : Automatic Geometric Theorem Proving: Sangaku From an Algebraic PerspectiveAbstract: During the Edo period (1603-1867 CE) Japan was almost completely closed off from the rest of the world and developed its own mathematical tradition called wasan. Part of this tradition was to hang tablets, known as sangaku, in the eaves of a shrine or temple as an act of devotion. The sangaku tablets were made of solid wood and each contained a colorful illustration of one or more results in Euclidean geometry. I have looked at one of these ancient geometric results and proven it using algebraic tools instead of the methods used in classical Euclidean geometry. (Faculty mentor: Dr. I. Kogan)
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Erin Beaton

Title: Offshore Airborne Wind Energy System Submersible Platform SimulationAbstract: Airborne Wind Energy (AWE) systems offer a controllable method for maximizing energy generation. As opposed to turbines on a standing tower, AWE systems are turbines suspended by a kite and tethered to a ground station. Researchers have analyzed buoyant platforms for offshore AWE systems. This project considers a platform with a linear actuator that can retract the kite, turbine, and docking station for the AWE system beneath the surface during storm conditions. The platform is subjected to tension forces caused by the anchors, a tension force caused by the airborne system, and gravity waves characterizable by the body of water. A numerical case study is performed on a dynamic model through the analysis of geometric and parameter variation of the platform. The results can be used to further design controllable AWE systems. (Faculty Mentor: Dr. C. Vermillion)
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Natalie Cody

Title: Population Model for the decline of Homalodisca vitripennis (Hemiptera: Cicadellidae) over a ten year periodAbstract: The glassy-winged sharpshooter, Homalodisca vitripennis (Germar), is an invasive pest which presents a major economic threat to grape industries in California, because it spreads a disease-causing bacterium, Xylella fastidiosa. In this note we develop a time and temperature dependent mathematical model to analyze aggregate population data for H. vitripennis from a 10-year study consisting of bi-weekly monitoring of H. vitripennis populations on unsprayed citrus, during which H. vitripennis decreased significantly. This model was fitted to the aggregate H. vitripennis time series data using iterative reweighted weighted least squares (IRWLS) with assumed probability distributions for certain parameter values. Results indicate that the H. vitripennis model fits the phenological and temperature data reasonably well, but the observed population decrease may possibly be attributed to factors other than the abiotic effect of temperature. A key factor responsible for this decline but not analyzed here could be biotic, for example, potentially parasitism of H. vitripennis eggs by Cosmocomoidea ashmeadi. A biological control program targeting H. vitripennis utilizing the mymarid egg parasitoid Cosmocomoidea (formerly Gonatocerus) ashmeadi (Girault) is described. (Faculty mentor: Dr. H.T. Banks)
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Ethan Dudley

Title: Dynamical Data Collection for Parameter EstimationAbstract: Mathematical models use parameters to represent physical phenomenon. Most of the time, some, if not all, of the parameters are unknown and require calibration. These parameters can be obtained by solving an inverse problem. However, budgetary constraints limit our ability to deploy sensors to collect the required data. Hence our motivating question is: where should we collect the data from so that the uncertainty associated with our estimates is as small as possible? Some sensor locations provide more information than others, but the best sensor locations at one time might not be the same at another time. To exemplify this kind of behavior we will consider a model problem involving a transient diffusive partial differential equation. We consider two different data collection strategies: a static/ stationary sensor configuration and a dynamic/ switching strategy. We use a novel optimization approach for the sensor placement strategy and compare the two data collection strategies in cost, relative error, and their ability to reconstruct the initial condition. (Faculty mentor: Dr. A. Alexanderian & A. Saibaba)