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Rational operators (a {t} b); a,b > e solved
#11
(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:
aºb
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


therefore:


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

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