There is a class of multivariate functions that have some kind of “univariate character” except that they “combine effects”. A+B is king of them. Some of his subordinates are:

1. $\frac{1}{1/A+1/B}$
2. $\sqrt{A^2+B^2}$
3. $e^{\ln{A}+\ln{B}}$

These expressions are all commonly used. Number 3 is so common that it has it’s own shorthand: $A \times B$.

Let’s define a family of operators

$+_f$

where $f$ is a monotonic univariate function that takes us to the domain in which the summing happens, and $f^{-1}$ brings us back. So by definition…

$A +_f B=f^{-1}(f(A)+f(B))$

Those three expressions listed in the beginning then become:

1. $A+_{-1}B$        (  because $(A^{-1}+B^{-1})^\frac{1}{-1}$   )
2. $A+_{2}B$          (  because $(A^2+B^2)^\frac{1}{2}$   )
3. $A+_{\ln}B$         (  because $\ln^{-1}(\ln{A} + \ln{B})$   )

We can combine, for example, parallel resistors like this:

$ResistorA +_{-1} ResistorB +_{-1} ResistorC$

and independent Gaussian deviates like this:

$ErrorA +_2 ErrorB + _2 ErrorC$

Short-hand could be useful in itself, but what properties/identities might these operators permit? It isn’t hard to verify that the entire family inherits the associative and commutative properties of it’s inner champion $+$. This symmetry allows us to construct multivariate expressions in the exact same way as it works for $+$:

$B +_f (C +_f A) = A +_f B +_f C$

In calculus we have the product rule to help us with $d[A \cdot B] = dA \cdot B + A \cdot dB$. The generalization for $+_f$ turns out to be

$d[A +_f B] =$ $\frac{f'(A)dA+f'(B)dB}{f'(A +_f B)}$

and again, this generalizes straightforwardly to higher dimensions.

The proof isn’t difficult, but for now let’s see how the product rule emerges as a special case of this rule:

$d[A \cdot B] = d[A +_{\ln} B] =$ $\frac{\ln'(A)dA + \ln'(B)dB}{\ln'(A +_{\ln} B)}$ $= \frac{dA/A+dB/B}{\frac{1}{AB}}$ $= B \cdot dA + A \cdot dB$

Might this family yield more identities that allow it to share in the abilities of its special cases ($+$ and $\times$) to reduce complex expressions to simple insights?

Giving these expressions full multivariate-function status just seems like overkill. The area and the diagonal length of a (b&w) photograph are simpler things than the photograph itself, and deserve a simpler abstraction.

TODO: Check more identities.

TODO: Note on nomograms. Each of $+_{-1}$, $+_2$, $+_{ln}$ has a simple visualization using perpendicular axes.

### One Response to “Addition in Other Domains”

• Jisang Yoo says:

This reminds me of the “under” operator from J which I heard from an article by James Hague: http://prog21.dadgum.com/121.html

For example, the three plus operators could be called “Sum under reciprocal”, “Sum under square”, “Sum under log”