Robert P. Gilbert is an applied analyst, who began studying function theoretic methods applied to partial differential equations. He and Jerry Hile originated the method of generalized hyperanalytic function theory. Professor Gilbert and James Buchanan applied this methodology to elastic materials. Further applications of the function theoretic method were made by Robert Gilbert, Steve Xu and James Buchanan to marine acoustics and inverse problems. From this point on Dr. Gilbert's research centered on bio-medical research, in particular the study of the rigidity of human bone and the disease osteoporosis. This research was twin pronged, part being focussed on cell-biology, with James Buchanan, and the other with Alex Panchenko on the homogenization of the micro-differential equations involved with bone mechanics. Further investigations were made in collaboration with Yvonne Ou and Michael Shoushani. Professor Gilbert became a full professor at Indiana University, Bloomington at the age of 34. At 43 he became the Unidell Foundation Chair of Mathematics at the University of Delaware. He has published over 300 articles in professional journals and conference proceedings. He is the Founding Editor of two mathematics journals Complex Variables and Applicable Analysis. M. Yvonne Ou is an applied mathematician specializing in inverse problems, computational mathematics and rational approximations. Prior to joining the faculty of the Department of Mathematical Sciences, where she is currently a tenured associate professor, she was a research scientist member of the Computational Mathematics Group in the Oak Ridge Laboratory. This is her first book. Michael Shoushani is an applied mathematician at Western Connecticut State University. His research interests are in transmission and inverse problems specifically related to poro-elasticity, partial differential equations, and numerical methods for partial differential equations. Professor Shoushani received his BA in mathematics from Western Connecticut State University and his MS and PhD both in Applied Mathematics from the University of Delaware.