ATON is a hopefully evolving classification theory. It aspires to unify knowledge around numbers and prefers naive methods. Some of the older posts are wrong but I'll keep them for the sake of continuity.

Sunday, March 26, 2006

Fundamental Theorem of Calculus

FTC states the existence of the integral;
Integral[f(x)dx,a,b] = F(b)-F(a)
An iteration of FTC gives Taylor's expansion and maps e^z to polynomials;
e^z = Sum[z^n/n!, n=0,...,Infinity]
The weighing factor 1/n! diminishes the higher polynomials and ensures absolute convergence.
It reminds the number of states of ESet;
SEn = n!(n-1)!(n-2)!...1!0!
ESet is derived from iterating a simple operation, like the e^z sum.
Then one would expect the Integral[] definition to be like the ESet operator E.
ESet accumulates its past and determined uniquely by its initial and final states, like Integral[].
ESet seems isomorphic to e^z.

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