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 Functional super-iteration and hierarchy of functional hyper-iterations bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 05/03/2009, 08:13 PM (This post was last modified: 05/03/2009, 08:39 PM by bo198214.) (05/03/2009, 07:21 PM)Tetratophile Wrote: For the function x -> x+1, you simply hyper-n-iterate the function f x0+1 times at x=x0, (eg. at x=2 you iterate 3 times, and at x=3.5 you iterate 4.5 times etc.) So I don't think that a non-constant function in the second argument would behave differently from a constant function in this respect. If we take the equations as recursive definitions then $\operatorname{It}$ is only defined for functions in the second argument that are constant. Thatswhy I proposed the different version. Quote:Also $I_0^m f = f$ would not be true, because at n=0, $I^m_0 f = f \operatorname{It}_0 m = f(m), \mbox{ not }f.$ Ok, I also made another mistake in the previous posting, so here again the recursive definition $(f \operatorname{It}_n g)(x) := (I_n^{g(x)} f)(x)$. $(I_0^m f)(x) := f(m)$ $I_n^1 f :=f$ and $I_{n}^{m+1} f := f \operatorname{It}_{n-1} (I_n^m f).$ From that follows $(f \operatorname{It}_0 g)(x) = (I_0^{g(x)} f)(x) = f(g(x))$ $I_1^m f = f^{\circ m}$ $(f \operatorname{It}_1 g)(x) = (I_1^{g(x)} f)(x) = f^{\circ g(x)}(x)$ $I_2^m f = {^{\circ m} f}$ $(f \operatorname{It}_2 g)(x) = {^{\circ g(x)} f}(x)$ .... « Next Oldest | Next Newest »

 Messages In This Thread Functional super-iteration and hierarchy of functional hyper-iterations - by Base-Acid Tetration - 05/02/2009, 02:23 AM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/02/2009, 07:48 AM RE: Functional super-iteration and hierarchy of functional composition-based operations - by andydude - 05/02/2009, 08:32 AM RE: Functional super-iteration and hierarchy of functional composition-based operations - by andydude - 05/02/2009, 09:23 AM RE: Functional super-iteration and hierarchy of functional composition-based operations - by tommy1729 - 05/02/2009, 12:22 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/02/2009, 01:14 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by andydude - 05/02/2009, 06:38 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/02/2009, 06:14 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/02/2009, 07:27 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/03/2009, 01:08 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/03/2009, 03:25 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/03/2009, 07:21 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/03/2009, 08:13 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/03/2009, 09:45 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/03/2009, 10:27 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/04/2009, 01:06 AM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/04/2009, 07:56 AM RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/04/2009, 08:04 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/04/2009, 08:50 PM RE: Functional super-iteration and hierarchy of functional hyper-iterations - by Base-Acid Tetration - 05/04/2009, 08:57 PM RE: Functional super-iteration and hierarchy of functional hyper-iterations - by bo198214 - 05/04/2009, 09:01 PM RE: Functional super-iteration and hierarchy of functional hyper-iterations - by Base-Acid Tetration - 05/04/2009, 09:06 PM RE: Functional super-iteration and hierarchy of functional hyper-iterations - by bo198214 - 05/04/2009, 09:12 PM RE: Functional super-iteration and hierarchy of functional hyper-iterations - by Base-Acid Tetration - 05/12/2009, 02:29 AM RE: Functional super-iteration and hierarchy of functional hyper-iterations - by bo198214 - 05/12/2009, 07:11 AM

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