elevated total dispersion looking farther in ⇔ continuous collapse upon exposure of proclivity. easy
#Chebyshev #MarkovNumbers #Unicity Wow. So in my complete and utter misery realizing that I'm basically well... let's not go into my intelligence level at the moment, given that I literally derived Chebyshev polynomials in the past... I, baffled over an error, decided to see how something played out with like, actual numbers. Aaaand we have a new identity that is just, crazy powerful. I was /so/ right to go with b being a power of 2, because then you can just recursively bust down those U_b-1s, and U_n-bs if you had an odd n... what is the best way to optimize this thing?! This_is_awesome.
So it must be my technique. Nobody has ever derived this identity OR the Markov Number identity from a week and a half ago. This popping-out-the-triangular-coefficients on the right technique. In the first case, I did it because well, you just couldn't divide by the variable anymore. In the second case, I did it because I had a specific target depth I wanted to hit.
What makes it so powerful is that you are going from "of the second kind" on the left to "on the second kind" on the right, reducing to arbitrarily lower degrees, and never, EVER adding any coefficients whatsoever.
=) =) =)
elevated total dispersion looking farther in ⇔ continuous collapse upon exposure of proclivity. easy
2moWhy did I /need/ to do this, anyway? Well there is this ridiculous case that can arise where a Markov Number is way way waaaay out on a single branch and never in its life was re-introduced to the concept of "turning" or "picking up exponentiality." When the coefficients on the polynomials start to blow up fast enough, their pseudo-factorial growth supersedes the Markov Number operands exponential decay as you go across the polynomial, and there's no way to get a reasonable approximation of the value of the polynomial operating on the Markov Number. I now have a tool I can use to reduce the degree of the polynomial arbitrarily without ever encountering new coefficients ad absurdum then to reduce.