Theses and Dissertations  Mathematics
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Grassmann, Flag, and Schubert varieties in applications
This dissertation develops mathematical tools for signal processing and pattern recognition tasks where data with the same identity is assumed to vary linearly. We build on the growing canon of techniques for analyzing and ... 
Avoiding singularities during homotopy continuation
In numerical algebraic geometry, the goal is to find solutions to a polynomial system F(x1,x2,...xn). This is done through a process called homotopy continuation. During this process, it is possible to encounter areas of ... 
General modelbased decomposition framework for polarimetric SAR images
Polarimetric synthetic aperture radars emit a signal and measure the magnitude, phase, and polarization of the return. Polarimetric decompositions are used to extract physically meaningful attributes of the scatterers. Of ... 
Mathematical modeling of groundwater anomaly detection
Public concerns about groundwater quality have increased in recent years due to the massive exploitation of shale gas through hydraulic fracturing which raises the risk of groundwater contamination. Groundwater monitoring ... 
An integrated mathematics/science activity for secondary students : development, implementation, and student feedback
Mathematics teachers are often challenged by their students to give reasoning for why learning mathematics is necessary. An approach to address this question is to show students the value in learning mathematics by ... 
On the formulation and uses of SVDbased generalized curvatures
In this dissertation we consider the problem of computing generalized curvature values from noisy, discrete data and applications of the provided algorithms. We first establish a connection between the FrenetSerret Frame, ... 
A constrained optimization model for partitioning students into cooperative learning groups
The problem of the constrained partitioning of a set using quantitative relationships amongst the elements is considered. An approach based on constrained integer programming is proposed that permits a group objective ... 
Computational advancements in the Dbar reconstruction method for 2D electrical impedance tomography
We study the problem of reconstructing 2D conductivities from boundary voltage and current density measurements, also known as the electrical impedance tomography (EIT) problem, using the Dbar inversion method, based on ... 
Counting ArtinSchreier curves over finite fields
Several authors have considered the weighted sum of various types of curves of a certain genus g over a finite field k := Fq of characteristic p where p is a prime and q = pm for some positive integer m. These include ... 
Algorithms for Feature Selection and Pattern Recognition on Grassmann Manifolds
This dissertation presents three distinct applicationdriven research projects united by ideas and topics from geometric data analysis, optimization, computational topology, and machine learning. We first consider ... 
Electromechanical and curvaturedriven molecular flows for lipid membranes
Lipid membranes play a crucial role in sustaining life, appearing ubiquitously in biology. Gaining a quantitative understanding of the flows of lipid membranes is critical to understanding how living systems operate. ... 
Algorithms in Numerical Algebraic Geometry and Applications
The topics in this dissertation, while independent, are unified under the field of numerical algebraic geometry. With ties to some of the oldest areas in mathematics, numerical algebraic geometry is relatively young as a ... 
Preconditioning Polynomial Systems Using Macaulay Dual Spaces
Polynomial systems arise in many applications across a diverse landscape of subjects. Solving these systems has been an area of intense research for many years. Methods for solving these systems numerically fit into the ... 
Mathematical Models for HIV1 Viral Capsid Structure and Assembly
HIV1 (human immunodeﬁciency virus type 1) is a retrovirus that causes the acquired immunodeﬁciency syndrome (AIDS). This infectious disease has high mortality rates, encouraging HIV1 to receive extensive research interest ... 
Abstract Hyperovals, Partial Geometries, and Transitive Hyperovals
A hyperoval is a (q+2) arc of a projective plane π, of order q with q even. Let G denote the collineation group of π containing a hyperoval Ω. We say that Ω is transitive if for any pair of points x, y is an element of ... 
Modeling Local Pattern Formation on Membrane Surfaces using Nonlocal Interactions
The cell membrane is of utmost importance in the transportation of nutrients and signals to the cell which are needed for survival. The magnitude of this is the inspiration for our study of the lipid bilayer which forms ... 
Object and action detection methods using MOSSE filters
In this thesis we explore the application of the Minimum Output Sum of Squared Error (MOSSE) filter to object detection in images as well as action detection in video. We exploit the properties of the Fourier transform ... 
Sparse multivariate analyses via ℓ1regularized optimization problems solved with Bregman Iterative Techniques
In this dissertation we propose Split Bregman algorithms for several multivariate analytic techniques for dimensionality reduction and feature selection including Sparse Principal Components Analysis, Bisparse Singular ... 
Mean variants on matrix manifolds
The geometrically elegant Stiefel and Grassmann manifolds have become organizational tools for data applications, such as illumination spaces for faces in digital photography. Modern data analysis involves increasingly ... 
Homotopy continuation methods, intrinsic localized modes, and cooperative robotic workspaces
This dissertation considers three topics that are united by the theme of application of geometric and nonlinear mechanics to practical problems. Firstly we consider the parallel implementation of numerical solution of ...