Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Generalized recursive operators
#33
bo198214 Wrote:May I ask if anyone has an idea how much this result depends on the actually chosen tetration extension to real numbers? I mean the b[n]-k=-k+1 for 0kn-3 does not depend on the extension.

Surely. Just as is the fixed point obtained by iterating , so is the fixed point obtained by iterating which also corresponds to the fixed point of . This works well when the fixed point is an attracting fixed point, but poorly for a repelling fixed point.

For tetration has 2 fixed points, and for (does ?) tetration has only the lower fixed point. Since the lower fixed point of tetration falls between two integers is does depend on which extension is used.

To summarize, one thing we can know for sure regardless of which extension is used, is that even hyper-exponentials follow: and odd hyper-exponentials follow: . All odd hyper-exponentials should be real-valued over all reals. All even hyper-logarithms should be real-valued over all reals. All hyper-exponentials above 3 map [-1,0] -> [0,1] and all hyper-logarithms above 3 map [0,1] -> [-1,0]. Another common property is that the range of odd hyper-exponentials is bounded below, just as the domain of even hyper-logarithms is bounded below.

Since these findings are all using negative hyper-exponents, then they are essentially specifying the number of times to iterate the appropriate hyper-(N-1)-logarithm. All odd hyper-exponentials are real-valued over all reals BECAUSE the hyper-(N-1)-logarithm is also real-valued over all reals. One conclusion we can draw from the statements above is that the domain of even hyper-exponentials is bounded below, because the domain of odd hyper-(N-1)-logarithms is bounded below. Accordingly, the range of odd hyper-exponentials is bounded below, because the range of even hyper-(N-1)-logarithms is bounded below.

I hope that made sense. I will try and make this more formal and more clear in a further post.

Andrew Robbins
Reply


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

Possibly Related Threads...
Thread Author Replies Views Last Post
  Where is the proof of a generalized integral for integer heights? Chenjesu 2 477 03/03/2019, 08:55 AM
Last Post: Chenjesu
  Hyper operators in computability theory JmsNxn 5 3,159 02/15/2017, 10:07 PM
Last Post: MphLee
  Recursive formula generating bounded hyper-operators JmsNxn 0 1,262 01/17/2017, 05:10 AM
Last Post: JmsNxn
  Rational operators (a {t} b); a,b > e solved JmsNxn 30 33,859 09/02/2016, 02:11 AM
Last Post: tommy1729
  holomorphic binary operators over naturals; generalized hyper operators JmsNxn 15 14,671 08/22/2016, 12:19 AM
Last Post: JmsNxn
  The bounded analytic semiHyper-operators JmsNxn 2 3,139 05/27/2016, 04:03 AM
Last Post: JmsNxn
  Bounded Analytic Hyper operators JmsNxn 25 17,153 04/01/2015, 06:09 PM
Last Post: MphLee
  Generalized arithmetic operator hixidom 16 11,725 06/11/2014, 05:10 PM
Last Post: hixidom
  Incredible reduction for Hyper operators JmsNxn 0 2,043 02/13/2014, 06:20 PM
Last Post: JmsNxn
  Generalized Bieberbach conjectures ? tommy1729 0 1,528 08/12/2013, 08:11 PM
Last Post: tommy1729



Users browsing this thread: 1 Guest(s)