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Generalized recursive operators
#24
andydude Wrote:I just found the asymptotes of pentation, hexation, heptation, octation, and beyond! And they're fascinating:

...

I suppose you could see this from the integer versions of these operators, but I think the continuous (or if not continuous, mostly real-valued) versions make it easier to see.

First, Andrew, these are really fascinating findings.

andydude Wrote:
for all

meaning, in the limit, all hyper-operators return to the successor operation, like the circle of life... Smile

I dare a proof by induction which only needs the integer operations.

Proposition:
If we have a sequence of operations [n] on the natural numbers (>0) that satisfy b[n+1]1=b, b[n+1](x+1)=b[n](b[n+1]x) for n 1, then we can extend the domain of the right operand of [n] to integer k with k 3-n and the only way to do so still satisfying the above conditions and injectivity of the functions f(x)=b[n]x is by b[n](-k)=-k+1 for 0 k n-3.

Proof:
We prove by induction over k that b[n](-k)=-k+1 for all n k+3.
Induction Start k=0:
b[n]1=b=b[n+1]1=b[n](b[n+1]0), for n2,
by injectivity follows
1=b[n+1]0 for n+13=0+3

Induction Step k=k+1:
by induction assumption for nk+3 :
b[n](-k)=-k+1=b[n+1](-k)=b[n](b[n+1]-(k+1))
by injectivity:
-k = b[n+1]-(k+1)
which is the induction assertion:
-(k+1)+1=b[n+1]-(k+1) for n+1k+1+3
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Messages In This Thread
Generalized recursive operators - by Whiteknox - 11/23/2007, 06:42 AM
RE: Generalized recursive operators - by bo198214 - 11/23/2007, 08:41 AM
RE: Generalized recursive operators - by andydude - 11/25/2007, 01:02 AM
RE: Generalized recursive operators - by andydude - 11/29/2007, 04:45 AM
RE: Generalized recursive operators - by andydude - 11/29/2007, 05:55 AM
RE: Generalized recursive operators - by andydude - 11/29/2007, 06:20 AM
RE: Generalized recursive operators - by andydude - 11/30/2007, 06:12 PM
RE: Generalized recursive operators - by andydude - 11/30/2007, 09:18 PM
RE: Generalized recursive operators - by bo198214 - 03/07/2008, 06:58 PM
RE: Generalized recursive operators - by Ivars - 02/02/2008, 10:11 PM
RE: Generalized recursive operators - by Ivars - 02/03/2008, 10:41 AM
RE: Generalized recursive operators - by andydude - 02/11/2008, 09:47 PM
RE: Generalized recursive operators - by Ivars - 02/14/2008, 06:05 PM
RE: Generalized recursive operators - by GFR - 02/03/2008, 04:12 PM
RE: Generalized recursive operators - by Ivars - 02/03/2008, 08:48 PM
RE: Generalized recursive operators - by GFR - 02/06/2008, 02:44 PM
RE: Generalized recursive operators - by Ivars - 02/06/2008, 02:56 PM
RE: Generalized recursive operators - by Ivars - 02/06/2008, 03:43 PM
RE: Generalized recursive operators - by GFR - 03/10/2008, 09:53 PM
RE: Generalized recursive operators - by GFR - 03/11/2008, 10:24 AM
RE: Generalized recursive operators - by bo198214 - 03/11/2008, 10:53 AM
RE: Generalized recursive operators - by GFR - 03/12/2008, 12:13 AM
RE: Generalized recursive operators - by GFR - 03/13/2008, 06:41 PM
RE: Generalized recursive operators - by Stan - 04/04/2011, 11:52 PM

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