Again this year, two of our undergraduates have been involved in original research in mathematics. Last summer, in an NSF-funded project, Jason Aubrey worked with Professor Carl Cowen on a problem arising in the design of robotic arms with more joints than necessary for full motion. Such robots are used in environments that make repair impossible, and the extra joints are used to compensate for broken ones. Engineers use a standard for fault tolerance involving the singular value decomposition of a Jacobian matrix; Jason investigated the fault tolerance of a robotic arm immediately after a joint breaks.
Dan Crosby, a freshman from Fort Wayne, worked with Professor Cowen on a project entitled "Redundant matrices for linear transformations." While one ordinarily represents a linear transformation with a matrix that comes from a basis, it is possible to get a "redundant" matrix for the transformation using a spanning set instead of a basis. Crosby analyzed the way in which the resulting matrix still reflects properties of the transformation.
Both Aubrey and Crosby presented talks on their work at a regional meeting of the American Mathematical Society in March. They plan further research this summer: Jason Aubrey will be at the National Security Agency and Dan Crosby will investigate linear fractional maps on n-dimensional balls in a project here at Purdue.