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Theses and Dissertations

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Browsing Theses and Dissertations by Subject "algebraic"

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    The conjugacy extension problem
    (Colorado State University. Libraries, 2021) Afandi, Rebecca, author; Hulpke, Alexander, advisor; Achter, Jeff, committee member; Pries, Rachel, committee member; Rajopadhye, Sanjay, committee member
    In this dissertation, we consider R-conjugacy of integral matrices for various commutative rings R. An existence theorem of Guralnick states that integral matrices which are Zp-conjugate for every prime p are conjugate over some algebraic extension of Z. We refer to the problem of determining this algebraic extension as the conjugacy extension problem. We will describe our contributions to solving this problem. We discuss how a correspondence between Z-conjugacy classes of matrices and certain fractional ideal classes can be extended to the context of R-conjugacy for R an integral domain. In the case of integral matrices with a fixed irreducible characteristic polynomial, this theory allows us to implement an algorithm which tests for conjugacy of these matrices over the ring of integers of a specified number field. We also describe how class fields can be used to solve the conjugacy extension problem in some examples.
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    The numerical algebraic geometry approach to polynomial optimization
    (Colorado State University. Libraries, 2017) Davis, Brent R., author; Bates, Daniel J., advisor; Peterson, Chris, advisor; Kirby, Michael, committee member; Maciejewski, A. A., committee member
    Numerical algebraic geometry (NAG) consists of a collection of numerical algorithms, based on homotopy continuation, to approximate the solution sets of systems of polynomial equations arising from applications in science and engineering. This research focused on finding global solutions to constrained polynomial optimization problems of moderate size using NAG methods. The benefit of employing a NAG approach to nonlinear optimization problems is that every critical point of the objective function is obtained with probability-one. The NAG approach to global optimization aims to reduce computational complexity during path tracking by exploiting structure that arises from the corresponding polynomial systems. This thesis will consider applications to systems biology and life sciences where polynomials solve problems in model compatibility, model selection, and parameter estimation. Furthermore, these techniques produce mathematical models of large data sets on non-euclidean manifolds such as a disjoint union of Grassmannians. These methods will also play a role in analyzing the performance of existing local methods for solving polynomial optimization problems.