# On meaningful transformation equations

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

## 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

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

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

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

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

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

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

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

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

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

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

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

03A10

## References

1. 1.
Aczél, J.: Lectures on Functional Equations and Their Applications. Academic Press, New York (1966). (Paperback edition, Dover, (2006))
2. 2.
Falmagne, J.-Cl.: Deriving meaningful scientific laws from abstract, “gedanken” type, axioms: five examples. Aequ. Math. 89, 393–435 (2015)
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)
5. 5.
Hosszú, M.: Note on commutable mappings. Publ. Math. Debrecen 9, 105–106 (1962a)
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)
9. 9.
Vincze, E.: Eine allgemeinere Methode in der Theorie der Funktionalgleichungen I. Publ. Math. Debrecen 9, 149–163 (1962)