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Computation of Minimal Units Monomials

Daniel Lichtblau
Organization: Wolfram Research, Inc.

ACA 2007
Conference location

Oakland University, Rochester, MI

Revised version of a talk given at the Conference on Applications of Computer Algebra (ACA) 2007, Session on Algebraic and Numerical Computation for Engineering and Optimization Problems (July 2007, Oakland University, Rochester, MI)


In this talk I will consider the following problem. We are given a "units monomial", that is, a product of (possibly negative) integer powers of physical units, e.g. ((meters^2) volts )/(farads seconds^2). We might try to make sense of this by finding all equivalent monomials subject to a minimality condition. Good candidates for such a condition involve minimizing exponents. For example one might minimize the sum of absolute values of exponents, or minimize the larger of the sum of numerator and sum of denominator exponents. Given a set of algebraic relations between pairs of such monomials, we will readily set these up as problems in integer linear programming, and discuss various ways in which it might be solved via algebraic or numeric programming.

This arose in-house several months ago in the context of a web site currently under development at Wolfram Research. Hence the ability to tackle it with reasonable computational efficiency (i.e. in real time) is paramount.

Afterword acknowledgment: I thank session organizer Dmytro Chibisov and several attendees, in particular Jaime Villate and Georg Regensberger, whose questions following the talk made me rethink various aspects of this problem.

*Applied Mathematics > Optimization
*Mathematics > Algebra > Field and Ring Theory
*Mathematics > Algebra > Polynomials
*Mathematics > Discrete Mathematics > Combinatorics

minimal units, optimization, integer linear programming, Gröbner bases

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ACA2007_MinimalUnits.pdf (65.4 KB) - PDF Document
ACA2007_MinimalUnits.nb (131.1 KB) - Mathematica Notebook