(05/28/2014, 04:33 PM)JmsNxn Wrote: I used to think about trying to generalize arithmetic operations like negation and addition. I never came to three but I did come up with the following operator that I thought was very interesting:

\( x \box y = \ln(e^x + e^y) \) which is holo in \( x \) and \( y \) by doing a little calculus that I'm too lazy to do atm ^_^.

It has the cool property \( (x \box y)+z = (x+z) \box (y+z) \)

and \( x + y \box x = x+ (y \box 0) \)

Then we define a nice metric:

\( ||x|| = |e^{x}| < e^{|x|} \)

so that
\( ||x \box y|| \le ||x|| + ||y|| \)

Now we see we can start talking about calculus even because this operator is continuous.

I always liked the box derivative:

\( \frac{\box}{\box x} f(x) = \lim_{h \to -\infty} [f(x\box h) \box (f(x) + \pi i)] - h \)

\( \frac{\box}{\box x}[ f(x) \box g(x)] = [\frac{\box}{\box x} f(x)] \box [\frac{\box}{\box x} g(x)] \)

\( \frac{\box}{\box x} nx = \ln(n) + (n-1)x \)

and general box analysis ^_^

This may be a little off topic, this thread just reminded me of this.