# Applied and Computational Mathematics and Statistics

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• 1

Dataset

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
• Author:
Martin P. Barron
Fang Liu, Steven Buechler, Jun Li
Degree Area:
Applied and Computational Mathematics and Statistics
Degree:
PhD
Defense Date:
2018-06-14
• 3

Dataset

Creator(s):
Jonathan Hauenstein, Alan Liddell, Yi Zhang
Description:

Standard interior point methods in semidefinite programming track a solution path for a homotopy defined by a system of polynomial equations. By viewing this in the context of numerical algebraic geometry, we are able to employ techniques to handle various cases which can arise. Adaptive precision path tracking techniques can help navigate around ill-conditioned areas. When the optimizer is singular with respect to the first-order optimality conditions, endgames can be used to efficiently and…

Date Created:
2018-04-10
• 4

Doctoral Dissertation

Author:
Claire McKay Bowen
Fang Liu
Degree Area:
Applied and Computational Mathematics and Statistics
Degree:
PhD
Defense Date:
2018-03-27
• 5

Dataset

Creator(s):
Jonathan Hauenstein
Description:

A common problem when analyzing models, such as a mathematical modeling of a biological process, is to determine if the unknown parameters of the model can be determined from given input-output data. Identifiable models are models such that the unknown parameters can be determined to have a finite number of values given input-output data. The total number of such values over the complex numbers is called the identifiability degree of the model. Unidentifiable models are models such that th…

Date Created:
2018-03-09
• 6

Dataset

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

Computational tools in numerical algebraic geometry can be used to numerically approximate solutions to a system of polynomial equations. If the system is well-constrained (i.e., square), Newton’s method is locally quadratically convergent near each nonsingular solution. In such cases, Smale’s alpha theory can be used to certify that a given point is in the quadratic convergence basin of some solution. This was extended to certifiably determine the reality of the corresponding sol…

Date Created:
2018-03-08
• 7

Dataset

Creator(s):
Margaret Regan, Jonathan Hauenstein
Description:

A common computational problem is to compute topological information about a real surface defined by a system of polynomial equations. Our software, called polyTop, leverages numerical algebraic geometry computations from Bertini and Bertini_real with topological computations in javaPlex to compute the Euler characteristic, genus, Betti numbers, and generators of the fundamental group of a real surface. Several examples are used to demonstrate this new software.

Date Created:
2018-03-02
• Author(s):
Michael Hildreth
Abstract:

This report is a direct result of consultation with the research communities funded by the Mathematical and Physical Sciences (MPS) Directorate at the National Science Foundation (NSF). The goal of this effort is to provide feedback to NSF on current best practices with regard to research data curation, discovery, access, preservation, and re-use, and suggestions for areas of improvement and investment that could facilitate broader curation of, access to, and re-use of research data in the fu…

Date Created:
2017-08-31
Resource Type
Report
• 9

Doctoral Dissertation

Author:
Francesco Pancaldi
Andrew Sommese, Mark Alber
Degree Area:
Applied and Computational Mathematics and Statistics
Degree:
PhD
Defense Date:
2017-06-20
• 10

Doctoral Dissertation

Author:
Alan Claude Liddell, Jr.
Jonathan Hauenstein, Alan Lindsay, Andrew Sommese
Degree Area:
Applied and Computational Mathematics and Statistics
Degree:
PhD
Defense Date:
2017-06-22
• Creator(s):
Margaret Regan, Jonathan Hauenstein
Description:

Three key aspects of applying homotopy continuation to parameterized systems of polynomial equations are investigated. First, for parameterized systems which are homogenized with solutions in projective space, we investigate options for selecting the affine patch where computations are performed. Second, for parameterized systems which are overdetermined, we investigate options for randomizing the system for improving the numerically stability of the computations. Finally, since one is typica…

Date Created:
2017-07-06
• 12

Doctoral Dissertation

Author:
Dong Lu