Applied and Computational Mathematics and Statistics

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  • Creator(s):
    Jonathan Hauenstein, Samantha Sherman
    Description:

    Synthesis problems for linkages in kinematics often yield large structured parameterized polynomial systems which generically have far fewer solutions than traditional upper bounds would suggest. This paper describes statistical models for estimating the generic number of solutions of such parameterized polynomial systems. The new approach extends previous work on success ratios of parameter homotopies to using monodromy loops as well as the addition of a trace test that provides a stopping…

    Date Created:
    2020-04-21
    Record Visibility:
    Public
  • Author:
    Christina Horr
    Advisory Committee:
    Fan Liu, Jun Li, Steven A. Buechler
    Degree Area:
    Applied and Computational Mathematics and Statistics
    Degree:
    Doctor of Philosophy
    Defense Date:
    2020-03-19
    Record Visibility:
    Public
  • Creator(s):
    Jonathan Hauenstein, Margaret Regan
    Description:

    Polynomials which arise via elimination can be difficult to compute explicitly. By using a pseudo-witness set, we develop an algorithm to explicitly compute the restriction of a polynomial to a given line. The resulting polynomial can then be used to evaluate the original polynomial and directional derivatives along the line at any point on the given line. Several examples are used to demonstrate this new algorithm including examples of computing the critical points of the discriminant locu…

    Date Created:
    2020-03-27
    Record Visibility:
    Public
  • Author:
    Yinan Li
    Advisory Committee:
    Fang Liu
    Degree Area:
    Applied and Computational Mathematics and Statistics
    Degree:
    Doctor of Philosophy
    Defense Date:
    2020-03-05
    Record Visibility:
    Public
  • Author:
    Cody Baker
    Advisory Committee:
    Robert J. Rosenbaum, Daniele Schiavazzi, Alexandra Jilkine
    Degree Area:
    Applied and Computational Mathematics and Statistics
    Degree:
    Doctor of Philosophy
    Defense Date:
    2020-02-26
    Record Visibility:
    Public
  • Creator(s):
    Jonathan Hauenstein, Martin Helmer
    Description:

    Alt’s problem, formulated in 1923, is to count the number of four-bar linkages whose coupler curve interpolates nine general points in the plane. This problem can be phrased as counting the number of solutions to a system of polynomial equations which was first solved numerically using homotopy continuation by Wampler, Morgan, and Sommese in 1992. Since there is still not a proof that all solutions were obtained, we consider upper bounds for Alt’s problem by counting the number of sol…

    Date Created:
    2020-03-03
    Record Visibility:
    Public
  • Creator(s):
    Jonathan Hauenstein
    Description:

    Many algorithms for determining properties of real algebraic or semi-algebraic sets rely upon the ability to compute smooth points. Existing methods to compute smooth points on semi-algebraic sets use symbolic quantifier elimination tools. In this paper, we present a simple algorithm based on computing the critical points of some well-chosen function that guarantees the computation of smooth points in each connected compact component of a real (semi)-algebraic set. Our technique is intuitive …

    Date Created:
    2020-01-18
    Record Visibility:
    Public
  • Author:
    Evercita Cuevas Eugenio
    Advisory Committee:
    Ick-Hoon Jin, Fang Liu, Lizhen Lin
    Degree Area:
    Applied and Computational Mathematics and Statistics
    Degree:
    Doctor of Philosophy
    Defense Date:
    2019-06-04
    Record Visibility:
    Public
  • Author:
    Kelsey DiPietro
    Advisory Committee:
    Alan E. Lindsay
    Degree Area:
    Applied and Computational Mathematics and Statistics
    Degree:
    Doctor of Philosophy
    Defense Date:
    2019-05-02
    Record Visibility:
    Public
  • Author:
    Ryan Pyle
    Advisory Committee:
    Robert J. Rosenbaum, Alan Lindsay, Lizhen Lin
    Degree Area:
    Applied and Computational Mathematics and Statistics
    Degree:
    Doctor of Philosophy
    Defense Date:
    2019-04-16
    Record Visibility:
    Public
  • Creator(s):
    Dan Bates, David Eklund, Jonathan Hauenstein, Chris Peterson
    Description:

    A fundamental problem in algebraic geometry is to decompose the solution set of a given polynomial system. A numerical description of this solution set is called a numerical irreducible decomposition and currently all standard algorithms use a sequence of homotopies forming a dimension-by-dimension approach. In this article, we pair a classical result to compute a smooth point on every irreducible component in every dimension using a single homotopy together with the theory of isosingular s…

    Date Created:
    2019-04-26
    Record Visibility:
    Public
  • Author:
    Kirsten Marie Kozlovsky
    Advisory Committee:
    Steven Schmid
    Degree Area:
    Aerospace and Mechanical Engineering
    Degree:
    Doctor of Philosophy
    Defense Date:
    2019-03-26
    Record Visibility:
    Public