Addition in Other Domains
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:
These expressions are all commonly used. Number 3 is so common that it has it’s own shorthand: .
Let’s define a family of operators
where is a monotonic univariate function that takes us to the domain in which the summing happens, and
brings us back. So by definition…
Those three expressions listed in the beginning then become:
( because
)
( because
)
( because
)
We can combine, for example, parallel resistors like this:
and independent Gaussian deviates like this:
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
:
In calculus we have the product rule to help us with . The generalization for
turns out to be
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:
Might this family yield more identities that allow it to share in the abilities of its special cases ( and
) 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 ,
,
has a simple visualization using perpendicular axes.
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”