This thesis presents the theory of affine algebraic sets defined over an arbitrary field K. We define basic concepts such as the Zariski topology, coordinate ring of functions, regular functions, and dimension. We are interested in the relationship between the geometry of an affine algebraic set over a field K and its geometry induced by the algebraic closure of K. Various versions of Hilbert-Nullstellensatz are presented, introducing a new variant over finite fields. Examples are provided throughout the paper and a question on the dimension of irreducible affine algebraic sets is formulated.

Range arithmetic is a way of calculating with variables that hold ranges of real values. This ability to manage uncertainty during computation has many applications. Examples in graphics include rendering and surface modeling, and there are more general applications like global optimization and solving systems of nonlinear equations. This thesis focuses on affine arithmetic, one kind of range arithmetic. The main drawbacks of affine arithmetic are that it taxes processors with heavy use of floating point arithmetic and uses expensive sparse vectors to represent noise symbols. Stream processors like graphics processing units (GPUs) excel at intense computation, since they were originally designed for high throughput media applications. Heavy control flow and irregular data structures pose problems though, so the conventional implementation of affine arithmetic with dynamically managed sparse vectors runs slowly at best. The goal of this thesis is to map affine arithmetic efficiently onto GPUs by turning sparse vectors into shorter dense vectors at compile time using static analysis. In addition, we look at how to improve efficiency further during the static analysis using unique symbol condensation. We demonstrate our implementation and performance of the condensation on several graphics applications.

The multiple sequence alignment (MSA) is a fundamental technique of molecular biology. Biological sequences are aligned with each other vertically in order to show the similarities and differences among them. Due to its importance, many algorithms have been proposed. With dynamic programming, finding the optimal alignment for a pair of sequences can be done in O(n2) time, where n is the length of the two strings. Unfortunately, for the general optimization problem of aligning k sequences of length n , O(nk) time is required. In this thesis, we shall first propose an efficient group alignment method to perform the alignment between two groups of sequences. Then we shall propose a clustering method to build the tree topology for merging. The clustering method is based on the concept that the two sequences having the longest distance should be split into two clusters. By our experiments, both the alignment quality and required time of our algorithm are better than those of NJ (neighbor joining) algorithm and Clustal W algorithm.

No description available.

Reduction theory for systems with symmetry deals with the problem of understanding dynamics on a manifold with an action of a Lie group. In geometric mechanics, this problem can be formulated in the Lagrangian, Hamiltonian or affine connection frameworks. While the Lagrangian and Hamiltonian formulations have been well developed, the results obtained in these setups are based on variational principles and symplectic geometry. These methods cannot be used directly in the affine connection formulation unless additional structure is available. In this thesis, a manifold with an arbitrary affine connection is considered, and the geodesic spray associated with the connection is studied in the presence of a Lie group action. In particular, results are obtained that provide insight into the structure of the reduced dynamics associated with the given invariant affine connection. The geometry of the frame bundle of the given manifold is used to provide an intrinsic description of the geodesic spray. A fundamental relationship between the geodesic spray, the tangent lift and the vertical lift of the symmetric product is obtained, which provides a key to understanding reduction in this formulation. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2007-07-24 01:00:05.635

Intermittent power sources such as wind and solar are increasingly penetrating electrical grids, mainly motivated by global warming concerns and government policies. These intermittent and non-dispatchable sources of power affect the operation and control of the power system because of the uncertainties associated with their output power. Depending on the penetration level of intermittent sources of power, the electric grid may experience considerable changes in power flows and synchronizing torques associated with system stability, because of the variability of the power injections, among several other factors. Thus, adequate and efficient techniques are required to properly analyze the system stability under such uncertainties. A variety of methods are available in the literature to perform power flow, transient, and voltage stability analyses considering uncertainties associated with electrical parameters. Some of these methods are computationally inefficient and require assumptions regarding the probability density functions (pdfs) of the uncertain variables that may be unrealistic in some cases. Thus, this thesis proposes computationally efficient Affine Arithmetic (AA)-based approaches for voltage and transient stability assessment of power systems, considering uncertainties associated with power injections due to intermittent sources of power. In the proposed AA-based methods, the estimation of the output power of the intermittent sources and their associated uncertainty are modeled as intervals, without any need for assumptions regarding pdfs. This is a more desirable characteristic when dealing with intermittent sources of power, since the pdfs of the output power depends on the planning horizon and prediction method, among several other factors. The proposed AA-based approaches take into account the correlations among variables, thus avoiding error explosions attributed to other self-validated techniques such as Interval Arithmetic (IA).

Thesis (Ph.D.)--McMaster University, 2005. / Supervisor: Jean-Pierre Gabardo. Includes bibliographical references (leaves 138-145).

Thesis (Ph. D.)--University of Hong Kong, 2008. / Includes bibliographical references (leaf 31-35) Also available in print.

Thesis (Ph. D.)--University of Chicago, Graduate School of Business. / Includes bibliographical references. Also available on the Internet.

The segmentation of medical images is a difficult task due to the inhomogeneous intensity variations that occurs during digital image acquisition, the complicated shape of the object, and the medical expert’s lack of semantic knowledge. Automated segmentation algorithms work well for some medical images, but no algorithm has been general enough to work for all medical images. In practice, most of the time the segmentation results are corrected by the experts before the actual use. In this work, we are motivated to determine how to make use of manually segmented data in automatic segmentation. The key idea is to transfer the ground truth segmentation from the database of train images to a given test image. The ground truth segmentation of MR images is done by experts. The process includes a hierarchical image decomposition approach that performs the shape matching of test images at several levels, starting with the image as a whole (i.e. level 0) and then going through a pyramid decomposition (i.e. level 1, level 2, etc.) with the database of the train images and the given test image. The goal of pyramid decomposition is to find the section of the training image that best matches a section of the test image of a different level. After that, a re-composition approach is taken to place the best matched sections of the training image to the original test image space. Finally, the ground truth segmentation is transferred from the best training images to their corresponding location in the test image. We have tested our method on a hip joint MR image database and the experiment shows successful results on level 0, level 1 and level 2 re-compositions. Results improve with deeper level decompositions, which supports our hypotheses.