On meaningful transformation equations

  • Jean-Claude FalmagneEmail author
  • Christopher W. Doble
Open Access


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


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

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


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

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

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

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



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


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

Mathematics Subject Classification




  1. 1.
    Aczél, J.: Lectures on Functional Equations and Their Applications. Academic Press, New York (1966). (Paperback edition, Dover, (2006)) zbMATHGoogle Scholar
  2. 2.
    Falmagne, J.-Cl.: Deriving meaningful scientific laws from abstract, “gedanken” type, axioms: five examples. Aequ. Math. 89, 393–435 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Falmagne, J.-Cl., Doble, C.W.: On Meaningful Scientific Laws. Springer, Berlin (2015)Google Scholar
  4. 4.
    Falmagne, J-Cl., Doble, C.W.: Meaningfulness as a “principle of theory construction”. J. Math. Psychol. 75, 59–67 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hosszú, M.: Note on commutable mappings. Publ. Math. Debrecen 9, 105–106 (1962a)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Hosszú, M.: Néhány lineáris függvényegyenletről. Mat. Lapok 13, 202 (1962b)Google Scholar
  7. 7.
    Hosszú, M.: Algebrai rendszereken értelmezett függvényegyenletek, i. algebrai módszerek a függvényegyenletek elméletében. Magyar Tud. Acad. Mat. Fiz. Oszt. Közl 12, 303–315 (1962)Google Scholar
  8. 8.
    Maksa, G.: CM solutions of some functional equations of associative type. Ann. Univ. Sci. Budapest. Sect. Comp 24, 125–132 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Vincze, E.: Eine allgemeinere Methode in der Theorie der Funktionalgleichungen I. Publ. Math. Debrecen 9, 149–163 (1962)MathSciNetGoogle Scholar

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© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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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