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Dimensional analysis using toric ideals: primitive invariants

Atherton, Mark A. and Bates, Ronald A. and Wynn, Henry P. (2014) Dimensional analysis using toric ideals: primitive invariants. PLoS ONE, 9 (12). e112827. ISSN 1932-6203

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Identification Number: 10.1371/journal.pone.0112827

Abstract

Classical dimensional analysis in its original form starts by expressing the units for derived quantities, such as force, in terms of power products of basic units etc. This suggests the use of toric ideal theory from algebraic geometry. Within this the Graver basis provides a unique primitive basis in a well-defined sense, which typically has more terms than the standard Buckingham approach. Some textbook examples are revisited and the full set of primitive invariants found. First, a worked example based on convection is introduced to recall the Buckingham method, but using computer algebra to obtain an integer matrix from the initial integer matrix holding the exponents for the derived quantities. The matrix defines the dimensionless variables. But, rather than this integer linear algebra approach it is shown how, by staying with the power product representation, the full set of invariants (dimensionless groups) is obtained directly from the toric ideal defined by . One candidate for the set of invariants is a simple basis of the toric ideal. This, although larger than the rank of , is typically not unique. However, the alternative Graver basis is unique and defines a maximal set of invariants, which are primitive in a simple sense. In addition to the running example four examples are taken from: a windmill, convection, electrodynamics and the hydrogen atom. The method reveals some named invariants. A selection of computer algebra packages is used to show the considerable ease with which both a simple basis and a Graver basis can be found.

Item Type: Article
Official URL: http://www.plosone.org/
Additional Information: © 2014 The Authors © CC BY 4.0
Subjects: H Social Sciences > HD Industries. Land use. Labor > HD61 Risk Management
Q Science > QA Mathematics
Sets: Research centres and groups > Decision Support and Risk Group (DSRG)
Date Deposited: 12 Jan 2015 13:37
Last Modified: 04 Feb 2015 14:37
Projects: 1-SST-U445, MUCM EP/D049993/1
Funders: Leverhulme Trust Emeritus Fellowship, United Kingdom EPSRC
URI: http://eprints.lse.ac.uk/id/eprint/60729

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