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 Functional super-iteration and hierarchy of functional hyper-iterations Base-Acid Tetration Fellow Posts: 94 Threads: 15 Joined: Apr 2009 05/03/2009, 07:21 PM (This post was last modified: 05/03/2009, 07:23 PM by Base-Acid Tetration.) (05/03/2009, 03:25 PM)bo198214 Wrote: (05/03/2009, 01:08 PM)Tetratophile Wrote: $\operatorname{It}_k: \lbrace f: \mathbb{R} \rightarrow \mathbb{R} \rbrace \times \lbrace f:\mathbb{R} \rightarrow \mathbb{R} \rbrace \rightarrow \lbrace f:\mathbb{R} \rightarrow \mathbb{R} \rbrace,$ where It_0 is function composition (g It_0 f = g(f)). All of the operators has the property $f \operatorname{It}_n x+1 = f \operatorname{It}_{n-1} (f \operatorname{It}_n x).$ It is a bit sloppy and misunderstandable to write $x+1$ or $x$ in the second argument. As one would associate the function $x\mapsto x+1$, but actually you mean the constant function: $f \operatorname{It}_n (x\mapsto m+1) = f \operatorname{It}_{n-1} (f \operatorname{It}_n (x\mapsto m)).$ To make it a definition that makes at least on the natural numbers sense, we need to add the induction start: $f \operatorname{It}_n (x\mapsto 1) = f$ The question then however is how do we know what $\operatorname{It}_n$ for a non-constant function in the second argument is? I think to properly define the hierarchy we need an additional operator $I_n: (\mathbb{R}\to\mathbb{R})\times \mathbb{R}\to(\mathbb{R}\to\mathbb{R})$, $(f,m)\mapsto I_n^m f$ such that $(f \operatorname{It}_n g)(x) = (I_n^{f(x)} g)(x)$. with $I_0^m f = f$ $I_n^1 f =f$ and $I_{n}^{m+1} f = f \operatorname{It}_{n-1} (I_n^m f).$ My brain is flickering, I hope I got everything right. 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. Also $I_0^m f = f$ would not be true, because at n=0, $I^m_0 f = f \operatorname{It}_0 m = f(m), \mbox{ not }f.$ Use of the $I$ operator would make the notation more compact, however « 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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