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Functional super-iteration and hierarchy of functional hyper-iterations
#16
(05/03/2009, 10:27 PM)bo198214 Wrote:
(05/03/2009, 09:45 PM)Tetratophile Wrote: If one defined the It operator for functions in the second argument the definition would not be a "recursive" definition?
then say that the functional equations are true for the It operator for all functions, but the equations are not "recursive". ok.
What do you mean by "true for all functions"? That x is an arbitrary function?

Yes. x can, in principle, be any function*; interpreting the "iterants" (the n in f [It_k] n) as constant functions rather than as simply numbers is what makes the hierarchy possible (since a function outputs numbers we can use as iterant). I should have written the iterant as g(x) to emphasize that.

The order of the operations is that It_k for bigger k are performed first, and, if the same operator is nested, perform the innermost one first; top down, inside out.

*as long as everything is defined: f(x), g(x), some real x, some natural n, f(x) It_(n+1) g(x), f(x) It_(n) [f(x) It_(n+1) g(x)-1]
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Messages In This Thread
RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/04/2009, 01:06 AM

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