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On meaningful transformation equations

  • Jean-Claude Falmagne
  • Christopher W. Doble
Open Access
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Abstract

The meaningfulness condition, applied to scientific or geometric laws, requires that the mathematical form of an equation does not change when we change the units of its ratio scale variables. Suitably formalized, this condition considerably limits the possible form of a law. In this paper, we give five new examples of such restricted representations. We use the meaningfulness condition on the five transformation equations displayed in the left column of the table below, in which x, y, z, u and v are real numbers and F, K, G, H, M and N are real valued functions. We show that under relatively weak general conditions (such as continuity, symmetry, monotonicity, homogeneity), each transformation equation must have, up to some real constants, the representation in the right column.

Transformation equations

Representations

\({\begin{matrix}F(F(x,y),z)= F(x,K(y,z))\end{matrix}}\)

\({\begin{matrix}F(x,y)= x\,\text {e}^{\frac{x^\theta }{\kappa }}\\ K(y,z)= \left( y^{\frac{1}{\theta }} +z^{\frac{1}{\theta }} \right) ^\theta \end{matrix}}\)

\({\begin{matrix}F(G(x,y),z)=F(G(x,z),y)\end{matrix}}\)

\({\begin{matrix}F(x,y) = \phi \,x\,y^\gamma \end{matrix}}\)

\({\begin{matrix}F(G(x,y),z)= G(F(x,z),F(y,z))\end{matrix}}\)

\({\begin{matrix}F(x,z)= \phi xz^\gamma \\ G(x,y)=(x^\theta + y^\theta )^{\frac{1}{\theta }}\end{matrix}}\)

\({\begin{matrix}F(G(x,y),z)= H(x,K(y,z))\end{matrix}}\)

\({\begin{matrix}F(G(x,y),z)\end{matrix}}\)

 

   \({\begin{matrix}= \phi xy z^\gamma = H(x,K(y,z))\end{matrix}}\)

\({\begin{matrix}F(G(x,y),H(u,v))=K(M(x,u),N(y,v))\end{matrix}}\)

\({\begin{matrix} F(G(x,y),H(u,v)) &{}=&{} (x^\theta +y^\theta +u^\theta +v^\theta )^{\frac{1}{\theta }}\\ &{}=&{}K(M(x,u),N(y,v))\end{matrix}}\)

Mathematics Subject Classification

03A10 

Notes

References

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Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.University of CaliforniaIrvineUSA
  2. 2.McGraw-Hill EducationNew YorkUSA

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