The challenge of density functional theory is the useful approximation of the exchange - correlation energy. This energy can be approximated with the local electron density and the gradient of the density. Many different generalized gradient approximations (GGA) have been made recently and there is controversy over the best overall functional. Recent Monte Carlo simulations give evidence that the Laplacian of the density might be a better starting place than the gradient to correct the local density approximation. We have tested several Laplacian based GGA models for exchange for small atoms. We use known constraints on the exchange energy used in current GGA’s. In many models unphysical oscillations occur in the potential when using the Laplacian, and understanding and eliminating them is part of the focus of this research. We also find that mixing gradient and Laplacian seems to give a better result than only using one or the other. / Department of Physics and Astronomy

Electroencephalography (EEG) is a valuable tool for clinical and cognitive applications. EEG allows for measuring and imaging of scalp potentials emitted by brain activity and allows researchers to draw conclusions about underlying brain activity and function. However EEG is limited by poor spatial resolution due to various factors. One reason is the fact that EEG electrodes are separated from current sources in the brain by cerebrospinal fluid (CSF), the skull, and the scalp. Unfortunately the conductivities of these tissues are not yet well known which limits the spatial resolution of EEG.Based on prior research, spatial resolution of the EEG can be improved via use of various mathematical techniques that provide increased accuracy of the representation of scalp potentials. One such method is the surface Laplacian. It has been shown to be a direct approach to improving EEG spatial resolution. Yet this approach depends on a geometric head model and much work has been done on assuming the human head to be spherical.In this project, we will develop a mathematical model for ellipsoidal head geometry based on surface Laplacian calculations by Law [1]. The ellipsoidal head model is more realistic to the human head shape and can therefore improve accuracy of the EEG imaging calculations. We will construct a computational program that utilizes the ellipsoidal head geometry in hopes to provide a more accurate representation of data fits compared to the spherical head models. Also, we will demonstrate that the spline surface Laplacian calculations do indeed increase the spatial resolution thereby affording a greater impact to the clinical and cognitive study community involving EEG. / Department of Physics and Astronomy

Thesis (Ph. D.)--University of Missouri-Columbia, 1997. / Typescript. Vita. Includes bibliographical references (leaves 76-79). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 1997. / Typescript. Vita. Includes bibliographical references (leaves 76-79). Also available on the Internet.

Thesis (Ph. D.)--University of California, Riverside, 2009. / Includes abstract. Includes bibliographical references (leaves 92-93). Issued in print and online. Available via ProQuest Digital Dissertations.

Thesis (Ph. D.)--University of Washington, 2004. / Vita. Includes bibliographical references (p. 102-107).

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, August 1999. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--University of Washington, 1999. / Vita. Includes bibliographical references (leaves 42-43).

Thesis (Ph. D.)--Massachusetts Institute of Technology and Woods Hole Oceanographic Institution, 1980. / Vita. Photocopy of typescript. Bibliography: p. 196-201.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Meteorology, 1980. / Supervised by Carl Wunsch. Vita. Includes bibliographical references (leaves 196-201).