Beyond + and - tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 05/28/2014, 10:44 PM (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. Notice this is just the generalized distributive property with r = -1 : http://math.eretrandre.org/tetrationforu...hp?tid=520 And also the box derivative is known to me. Reminds me of the " q-derivative ". regards tommy1729 « Next Oldest | Next Newest »