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:

  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


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”

Leave a Reply

Your email address will not be published. Required fields are marked *

Cheap NFL Jerseys Cheap NFL Jerseys goedkope air max online Basket Air Jordan 11 Christian Louboutin Outlet Cheap Jerseys Wholesale Jerseys Michael Kors Outlet Online Ray Ban Sunglasses Jerseys Wholesale Coach Outlet Store michael jordan air jordan 11 Coach Outlet Cheap NFL Jerseys Kate Spade Outlet Sale air max 90 pas cher junior Wholesale Jerseys Jerseys Cheap nba jerseys vegas Jerseys Wholesale China