On meaningful transformation equations

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 equationsRepresentations{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}}