Applied and Computational Mathematics and Statistics
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 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
 Record Visibility:
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14
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…
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15
Article
 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…
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16
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…
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17
Article
 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 predictorcorrector 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:
 20160831
 Record Visibility:
 Public

 Author(s):
 Daniel Bates, Charles Wampler, Andrew Sommese, Jonathan Hauenstein
 Abstract:
This book is a guide to concepts and practice in numerical algebraic geometry—the solution of systems of polynomial equations by numerical methods. Through numerous examples, the authors show how to apply the wellreceived and widely used opensource Bertini software package to compute solutions, including a detailed manual on syntax and usage options. The authors also maintain a complementary web page where readers can find supplementary materials and Bertini input files.
 Date Created:
 20160223
 Record Visibility:
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20
Software

 Author(s):
 Andrew Sommese, Azat M. Gainutdinov, Wenrui Hao, Rafael I. Nepomechie
 Abstract:
We consider the sl(2)qinvariant open spin½ XXZ quantum spin chain of finite length N. For the case that q is a root of unity, we propose a formula for the number of admissible solutions of the Bethe ansatz equations in terms of dimensions of irreducible representations of the TemperleyLieb algebra; and a formula for the degeneracies of the transfer matrix eigenvalues in terms of dimensions of tilting sl(2)qmodules. These formulas include corrections that appear if two or more tilting …
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 Author(s):
 Parker Ladwig, Andrew Sommese
 Abstract:
“Supplying accurate CPU [costperserial use] information to faculty and appropriate marketing of the alternate modes of delivery … become the key to achieving an optimal costefficient serials collection in an academic library.” (Marisa Scigliano, “Serial Use in a Small Academic Library: Determining Costeffectiveness,” Serials Review 26 (2000): 43–52.)
A model is presented for adjusting use statistics using a journal’s ISI Journal Citation Reports cited halflife.The goal is to improve t…
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