This is fucking beautiful! I've been working at this for three years now and you beat me to the punch! LOL!
Do you have a closed form expression for it yet? I have so many ideas involving rational operators.
And one thing that MUST be true is
x {1.5} 2 = x {0.5} x
If that's not true then the system doesn't meet requirements. But it wouldn't be hard to improvise a system since 0 <= q <= 1, {q} is defined.
Also, I wonder if we could try to implement the laws of logarithmic semi operators into this system:
q:log(x) = exp^[-q](x)
q:log(x {q} y) = q:log(x) + q:log(y)
q:log(x {1+q} y) = q:log(x) * y
And if we implement this we can feasibly solve for a new variation of tetration.
Since
x {0.5} y = arithmetic/geometric limiting algo
= -0.5:log(0.5:log(x) + 0.5:log(y))
= sexp(slog(sexp(slog(x)-0.5) + sexp(slog(y)-0.5)) + 0.5)
Holy jesus, yes!
And also, I was wondering what our identities are?, if S(q) is the identity function and x {q} S(q) = x, S(1) = 1, and S(0) = 0 obvi, but what is S(q)?
I found S(0.5) = 0.7019920407 for 2 {0.5} S(0.5) = 2. I used 1000 cycles so my numbers are probably more accurate.
However, sadly, S(0.5) for 3 {0.5} S(0.5) = 3 is a different number and therefore {q} has no identity. This is very sad indeed.
Do you have a closed form expression for it yet? I have so many ideas involving rational operators.
And one thing that MUST be true is
x {1.5} 2 = x {0.5} x
If that's not true then the system doesn't meet requirements. But it wouldn't be hard to improvise a system since 0 <= q <= 1, {q} is defined.
Also, I wonder if we could try to implement the laws of logarithmic semi operators into this system:
q:log(x) = exp^[-q](x)
q:log(x {q} y) = q:log(x) + q:log(y)
q:log(x {1+q} y) = q:log(x) * y
And if we implement this we can feasibly solve for a new variation of tetration.
Since
x {0.5} y = arithmetic/geometric limiting algo
= -0.5:log(0.5:log(x) + 0.5:log(y))
= sexp(slog(sexp(slog(x)-0.5) + sexp(slog(y)-0.5)) + 0.5)
Holy jesus, yes!
And also, I was wondering what our identities are?, if S(q) is the identity function and x {q} S(q) = x, S(1) = 1, and S(0) = 0 obvi, but what is S(q)?
I found S(0.5) = 0.7019920407 for 2 {0.5} S(0.5) = 2. I used 1000 cycles so my numbers are probably more accurate.
However, sadly, S(0.5) for 3 {0.5} S(0.5) = 3 is a different number and therefore {q} has no identity. This is very sad indeed.
