DEPARTMENT OF MATHEMATICS, STATISTICS, AND PHYSICS ###
Applied Mathematics

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Inverse Problems

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Acoustical Holography

Inverse problems are a major area of both theoretical and computational research in the department. The theoretical analysis of such problems requires a solid knowedge of such areas of mathematics as partial differential equations and Fourier analysis. Aircraft acoustics and crack detection are just two applications investigated by research groups in the department.

Nearfield Acoustical Holography is the inverse problem in acoustics of computing the normal velocities on the boundary of a region from measurements of the acoustical pressure on an interior surface. Methods for solving this problem are important for the reduction of noise levels in airplane cabins. A group of several faculty and graduate students have developed such methods with support from the National Science Foundation as well as a local aircraft company. The figure shows the reconstruction of normal velocity in comparison with the original normal velocity distribution.

Mathematical Physics

Various geometries are used in modern physics. In addition to the classic Euclidean geometry, other geometries have been used; first Bolyai-Lobachevskian, then Riemannian, then Kleinian, and more recently, various combinations of the last two. The graphic shows one of these: a geodesic surface in the Heisenberg group with an indefinite metric of signature (+--).

The curves spiraling out and up from (0,0,0) are geodesics, and the transverse circles are curves of constant geodesic parameter *t*. From outside in, we have spacelike, null, and timelike geodesic surfaces, cut away to reveal their interiors. Thus less than one full period of each of these infinitely periodic surfaces is visible.

(Parker)