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  • Description(s):
    A movie showing the change in the number of stable steady-state solutions as a function of the cell-to-cell communication. Software code using Matlab and Bertini is also provided which was used to generate the frames of this movie.
    Creator(s):
    Jonathan Hauenstein
    Date Published:
    2016
    Record Visibility:
    Public
  • Creator(s):
    Jonathan Hauenstein
    Description:

    Stewart-Gough 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 Stewart-Gough platform is rigid with 40 assembly configurations (over the complex numbers) while exceptional Stewart-Gough platforms have infinitely many assembly configurations and thus have self-motion. We define a family of exceptional Stewart-Gough platforms called Segre-dependent Stewart-Gough platforms which aris…

    Date Created:
    2017-01-17
    Record Visibility:
    Public
  • 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:
    2017-01-07
    Record Visibility:
    Public
  • 16

    Article

    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…

    Record Visibility:
    Public
  • 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 implicitly-defined 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…

    Record Visibility:
    Public
  • 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…

    Record Visibility:
    Public
  • Author(s):
    Daniel Bates, Andrew Sommese, Jonathan Hauenstein
    Abstract:

    Path tracking is the fundamental computational tool in homotopy continuation and is therefore key in most algorithms in the emerging field of numerical algebraic geometry. Though the basic notions of predictor-corrector methods have been known for years, there is still much to be considered, particularly in the specialized algebraic setting of solving polynomial systems. In this article, the effects of the choice of predictor method on the performance of a tracker is analyzed, and details for…

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  • Creator(s):
    Jonathan Hauenstein
    Description:

    A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness sets via numerical elimination theory, we develop computational methods for computing ranks and border ranks of tensors along with decompositions. More generally, we present our approach using joins of any collection of irreducible and nondegenerate projecti…

    Date Created:
    2016-08-31
    Record Visibility:
    Public