Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Rational operators (a {t} b); a,b > e solved
(06/06/2011, 11:59 AM)tommy1729 Wrote: what is the idea or intention behind working with base eta ?

First I noticed that at base 2 humps would appear at around 0.5 and 1.5, and if we increased the base, the humps would get sharper and sharper and taller and taller. So I figured, there must be a base value where the humps disappear altogether. My first guess was eta, since everyone talks about how closely related it is to tetration, but when I made a graph of there were still visible humps at 0.5 and 1.5. Disappointed, and intrigued by the cheta function, I tried using the upper superfunction of eta, and voila, the values converged miraculously.

So it was more just a shot in the dark as opposed to a real mathematical deduction as to why we used base eta.

(06/06/2011, 05:16 PM)nuninho1980 Wrote:
(06/06/2011, 04:39 AM)sheldonison Wrote: I wonder what it means that fatb(3,-1,4)=5.429897..?
no, it isn't correct, sorry.
fatb(2,-1,2) = 2º2=4 -> 2+2=4

fatb(3,-1,3) = 3º3=5 -> 3+2=5

fatb(4,-1,3) = 4º3=5 - it's correct. because that's here down:
if a>b then a+1
if b>a then b+1
if a=b then a+2 or b+2

no, this isn't zeration, this a different operator designed to preserve the ring structure of operators [t-1] and [t].

(06/06/2011, 09:23 AM)bo198214 Wrote: Well not on the whole complex plane, but on the real axis, wouldnt that be nice?
I anyway wonder whether thats possible at all.
As is not even differentiable at 1, I wonder whether f is. Did you compare the derivations from left and right?
Yes it would be nice to have it analytic on the real axis, I think it should be potentially analytic for at least (-oo, 1). I'm just not sure about the convergence radius.

No I haven't compared the derivations, I'll do that though. It's hard for me to think up tests I can do with only highschool math under my belt Tongue

(06/06/2011, 09:23 AM)bo198214 Wrote: This is also in sync with the convention for quasigroups, i.e. groups with non-associative operation but with left- and right-inverse. There the left- and right inverse are written as / and \.

hmm, I really understand where you're coming from with using a standard notation instead of the triangles, it saves a lot of confusion, but I think that for the same reason the gamma function is written instead of , logarithmic semi-operators should be written using their own notation--just to be clear this is only one extension of hyper operators. I know that simply stating this is the natural extension of hyper operators will step on a lot of people's toes.

Also, I have something very interesting to report!

My conjecture can be proved for .

and if anyone knows their deltation, like I just explained


The proof can actually be made even simpler,
consider, :

therefore since:

, and since there's a fixpoint at e,

This proves a beautiful connection between logarithmic semi operators and the cheta function.

And also gives me a new beautiful identity:

Messages In This Thread
RE: Rational operators (a {t} b); a,b > e solved - by JmsNxn - 06/06/2011, 05:44 PM

Possibly Related Threads…
Thread Author Replies Views Last Post
  The modified Bennet Operators, and their Abel functions JmsNxn 6 243 07/22/2022, 12:55 AM
Last Post: JmsNxn
  The \(\varphi\) method of semi operators, the first half of my research JmsNxn 13 681 07/17/2022, 05:42 AM
Last Post: JmsNxn
  Thoughts on hyper-operations of rational but non-integer orders? VSO 4 4,734 06/30/2022, 11:41 PM
Last Post: MphLee
  The bounded analytic semiHyper-operators JmsNxn 4 7,733 06/29/2022, 11:46 PM
Last Post: JmsNxn
  Holomorphic semi operators, using the beta method JmsNxn 71 5,325 06/13/2022, 08:33 PM
Last Post: JmsNxn
  [MSE-SOLVED] Subfunction is functorial!!!! MphLee 14 5,919 06/06/2021, 11:16 PM
Last Post: JmsNxn
  Hyper operators in computability theory JmsNxn 5 10,832 02/15/2017, 10:07 PM
Last Post: MphLee
  Recursive formula generating bounded hyper-operators JmsNxn 0 3,708 01/17/2017, 05:10 AM
Last Post: JmsNxn
  holomorphic binary operators over naturals; generalized hyper operators JmsNxn 15 31,170 08/22/2016, 12:19 AM
Last Post: JmsNxn
  Bounded Analytic Hyper operators JmsNxn 25 43,422 04/01/2015, 06:09 PM
Last Post: MphLee

Users browsing this thread: 6 Guest(s)