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Previous processing of the Lunar Prospector magnetometer (LP-MAG) data has yielded ∼40% coverage of the Moon. Here, new mapping of the low-altitude LP-MAG data is reported with the goal of producing the first global vector map of the lunar crustal magnetic field. By considering all data regardless of the external plasma environment and using less restrictive editing criteria, 2360 partial and complete passes have been identified that can be used to investigate the lunar crustal magnetic anomalies. The cleanest global coverage is provided using 329 low-altitude nightside and terminator passes. An inverse power method has been used to continue the final mapping data to constant altitude. Using the 329 optimal passes, global maps of the lunar crustal magnetic field are constructed at 30 and 40 km. Consistent with previous studies: (1) the largest concentrations of anomalies are mapped antipodal to the Crisium, Serenitatis, Imbrium, and Orientale basins and (2) isolated anomalies at Reiner Gamma, Rima Sirsalis, Descartes, and Airy are mapped. Anomalies previously unmapped by the LP-MAG experiment include (1) isolated anomalies near the craters Abel and Hartwig, (2) weak magnetization within the Nectarian-aged Crisium and Moscoviense basins, and (3) a relatively weak anomaly in an area dominated by crater chains associated with the formation of Nectaris. Future work with the new low-altitude data set is discussed and will include determining whether the lunar anomalies are capable of deflecting the solar wind and investigating directions of magnetization to evaluate a possible former core dynamo.

The Support Vector Machine (SVM) classification method has recently gained popularity due to the ease of implementing non-linear separating surfaces. SVM is an optimization problem with the two competing goals, minimizing misclassification on training data and maximizing a margin defined by the normal vector of a learned separating surface. We develop and implement new SVM models based on previously conceived SVM with L_1-Norm regularization with ramp loss error terms. The goal being a new SVM model that is both robust to outliers due to ramp loss, while also easy to implement in open source and off the shelf mathematical programming solvers and relatively efficient in finding solutions due to the mixed linear-integer form of the model. To show the effectiveness of the models we compare results of ramp loss SVM with L_1-Norm and L_2-Norm regularization on human organ microbial data and simulated data sets with outliers.

The purpose of this thesis is to study methods by which conformal vector fields on pseudo-Riemannian manifolds can be simplified. A vector field on a manifold M with metric g is conformal if its local flows preserve the metric g up to a scaling and unlike Killing vector fields, which preserve g exactly, it cannot in general be linearised in a neighbourhood of any given point. The difference is that a Killing vector field is affine, that is it preserves a connection on the manifold. In this case the connection is the canonical (Levi-Civita) connection associated with g, but affine vector fields with respect to any connection are linearisable. The task is to find new connections with respect to which the set of conformal vector fields, or some subset of them, are affine. Suppose that we have a manifold M with a pseudo-Riemannian conformal structure and an orthogonal splitting of the tangent bundle. We construct, for a natural choice of torsion, a unique connection in the principal bundle of frames adapted to the splitting. Moreover this connection is preserved by any transformations which preserve the splitting of the tangent bundle. Thus any conformal vector field which preserves the splitting is affine. The splitting can be chosen to reflect the tangent to the orbits of a subalgebra of conformal vector fields, the principal null directions of the Weyl tensor or the flow of a perfect fluid. We also give a study of conformal vector fields in three-dimensional Lorentzian manifolds. An equivalent of the Cotton-York tensor is used to investigate the behaviour of these vector fields at their fixed points in the same spirit as the Weyl tensor is used in four dimensions.

A Dissertation Submitted of the Faculty of Science, University of the Witwatersrand, Johannesburg in Partial Fulfilment of the Requirements for the Degree of Master of Science / Let 4 be the class of pairs of loc ..My onvex spaces (X,V) “h ‘ch are such that every closed graph linear ,pp, 1 from X into V is continuous. It B is any class of locally . ivex l.ausdortf spaces. let & w . (X . (X.Y) e 4 for ,11 Y E B). " ‘his expository dissertation, * (B) is investigated, firstly i r arbitrary B . secondly when B is the class of C,-complete paces and thirdly whon B is a class of locally convex webbed s- .ces

by Cheung Kam Ching Leo. / Thesis (M.Ph.)--Chinese University of Hong Kong, 1988. / Bibliography: leaves 98-99.

Shea Yuen Cheuk. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 91-94). / Abstracts in English and Chinese. / Introduction --- p.5 / Chapter 1 --- Prelimaries --- p.9 / Chapter 1.1 --- Basic Notations --- p.9 / Chapter 1.2 --- Multiresolution Analysis --- p.12 / Chapter 1.3 --- Orthonormal Wavelets --- p.17 / Chapter 1.4 --- Theory of Frames --- p.24 / Chapter 2 --- Construction of Orthonormal Wavelets --- p.33 / Chapter 2.1 --- Compactly Supported Smooth Orthonormal Wavelet in R --- p.33 / Chapter 2.2 --- Compactly Supported Smooth Orthonormal Wavelet in R2 --- p.40 / Chapter 3 --- Wavelet Frames --- p.51 / Chapter 3.1 --- Basic Properties --- p.51 / Chapter 3.2 --- Dual Wavelet Frame --- p.56 / Chapter 3.3 --- Canonical Dual Frame --- p.66 / Chapter 3.4 --- Oversampling --- p.69 / Chapter 4 --- MRA-Based Wavelet Frames --- p.74 / Chapter 4.1 --- Definitions --- p.74 / Chapter 4.2 --- Tight Frames Constructed by MRA --- p.77 / Chapter 4.3 --- Approximation Order and Vanishing Moments for Wavelet Frames --- p.82 / Chapter 4.4 --- Construction of MRA-Based Wavelet Frames --- p.85 / Bibliography --- p.91

The regular representation of the symmetric group Sn is a vector space of dimension n! with many interesting invariant subspaces. The projections of a vector onto these subspaces may be computed by first considering projections onto certain basis elements in the subspace and then recombining later. If all of these projections are kept, it creates an explosion in the size of the data, making it difficult to store and work with. This is a study of techniques to compress this computed data such that it is of the same dimmension as the original vector.

VecTor© is a suite of computer programs developed for nonlinear finite element analysis of reinforced concrete. A graphics-based pre-processor (FormWorks) was developed for 2D concrete membrane structures, greatly contributing to the software’s utility and success. However, modeling other types of structures is a time consuming process, requiring manual definition of the finite element mesh, loads and analysis parameters in standard text files. A user-friendly pre-processor is required for the entire suite of programs if they are to be of greater use to design engineers. The purpose of this study is to develop an updated version of FormWorks, FormWorks 3.5, which is more user-friendly and compatible with the improvements made in VecTor2 over the past ten years. In addition, an extended version of FormWorks, FormWorks-Plus, will be created for the remaining VecTor programs with a wide range of viewing features, and facilities for specification of node coordinates, elements, loads and material properties.