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 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 inputoutput data. Identifiable models are models such that the unknown parameters can be determined to have a finite number of values given inputoutput 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:
 20180309

 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:
 20180302

 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 reuse, and suggestions for areas of improvement and investment that could facilitate broader curation of, access to, and reuse of research data in the fu…
 Date Created:
 20170831

 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:
 20170706

 Creator(s):
 Jonathan Hauenstein
 Description:
The Kuramoto model describes synchronization behavior among coupled oscillators and enjoys successful application in a wide variety of fields. Many of these applications seek phasecoherent solutions, i.e., equilibria of the model. Historically, research has focused on situations where the number of oscillators, n, is extremely large and can be treated as being infinite. More recently, however, applications have arisen in areas such as electrical engineering with more modest values of n. For…
 Date Created:
 20170407

 Creator(s):
 Jonathan Hauenstein
 Description:
StewartGough platforms are mechanisms which consist of two rigid objects, a base and a platform, connected by six legs via spherical joints. For fixed leg lengths, a generic StewartGough platform is rigid with 40 assembly configurations (over the complex numbers) while exceptional StewartGough platforms have infinitely many assembly configurations and thus have selfmotion. We define a family of exceptional StewartGough platforms called Segredependent StewartGough platforms which aris…
 Date Created:
 20170117

 Creator(s):
 Jonathan Hauenstein
 Description:
We define tensors, most of which correspond with cubic polynomials, which have the same exponent w as the matrix multiplication tensor. In particular, we study the symmetrized matrix multiplication tensor defined on an nxn matrix A by trace(A^3). The use of polynomials enables the introduction of additional techniques from algebraic geometry in the study of the matrix multiplication exponent w.
 Date Created:
 20170107

 Author(s):
 Daniel Bates, Andrew Sommese, Jonathan Hauenstein
 Abstract:
Numerical algebraic geometry is the area devoted to the solution and manipulation of polynomial systems by numerical methods, which are mainly based on continuation. Due to the extreme intrinsic parallelism of continuation, polynomial systems may be successfully dealt with that are much larger than is possible with other methods. Singular solutions require special numerical methods called endgames, and the endgames currently used do not take advantage of parallelism. This article gives an ove…

 Author(s):
 Charles Wampler II, Daniel Bates, Andrew Sommese, Jonathan Hauenstein
 Abstract:
Dedicated to our collaborator, mentor, and friend, Andrew Sommese, by Bates, Hauenstein, and Wampler on the occasion of his sixtieth birthday.
When numerically tracking implicitlydefined paths, such as is required for homotopy continuation methods, efficiency and reliability are enhanced by using adaptive stepsize and adaptive multiprecision methods. Both efficiency and reliability can be further improved by adapting precision and stepsize simultaneously. This paper presents a strategy fo…

12
Article
 Author(s):
 Charles Wampler, Andrew Sommese, Jonathan Hauenstein
 Abstract:
Though numerical methods to find all the isolated solutions of nonlinear systems of multivariate polynomials go back 30 years, it is only over the last decade that numerical methods have been devised for the computation and manipulation of algebraic sets coming from polynomial systems over the complex numbers. Collectively, these algorithms and the underlying theory have come to be known as numerical algebraic geometry. Several software packages are capable of carrying out some of the operati…