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+--- Thread: Rank-Wise Approximations of Hyper-Operations (/showthread.php?tid=1394)
RE: Rank-Wise Approximations of Hyper-Operations - JmsNxn - 06/27/2022
That's a very deep question, Catullus. I honestly don't know the answer to it.
But to point out, when you use \(1 \le \alpha \le \eta\), you have a very fine growth in the hyper-operations. In that, they tend to \(\alpha\) for \(\Re(z) > 0\). So doing this in this interval is about the closest I ever got. Where, as I've written before, I'll write again.
\[
\vartheta(w,u) = \sum_{n=0}^\infty\sum_{k=0}^\infty \alpha \uparrow^{n+2} (k+1)\frac{u^nw^k}{n!k!}\\
\]
If we differentiate repeatedly in \(w\), we get:
\[
\frac{d^j}{dw^j} \vartheta(w,u) \approx e^w\sum_{n=0}^\infty \omega_{n+2} \frac{u^n}{n!}\\
\]
Where \(\omega_{n+2} = \alpha \uparrow^{n+2} \infty\), which are the fixed points of \(\alpha \uparrow^{n+1} z\). And if we differentiate repeatedly in \(u\), we get:
\[
\frac{d^l}{du^l} \vartheta(w,u) \approx \alpha e^{w}e^{u}\\
\]
Now as asymptotic limits, both of these things are differintegrable. Whereby:
\[
\frac{d^{s}}{du^{s}} \frac{d^{z}}{dw^z}\Big{|}_{u=0,w=0} \vartheta(w,u) = \alpha \uparrow^{s+2} z+1\\
\]
This will satisfy the functional equation, but only if you can turn this heuristic into a proof. Which, I was unable to do. But I was only off by a few lemmas.
Not sure what else you can do here... Your question is very fucking hard, don't expect an honest and easy answer to it.
Regards, James
RE: Rank-Wise Approximations of Hyper-Operations - tommy1729 - 06/28/2022
Maybe this is a dumb remark but ..
What if 2 fixpoints have the same Taylor polynomial when the fixpoints are recentered at 0 ?
So around fixpoint A we get a X + b X^2 + … ( recenter )
And the same polynomial around Fixpoint B.
The the fixpoint formulas would agree around the 2 fixpoints.
Another fixpoint in the middle might destroy the connection.
Otherwise I see little objection ?
Ofcourse you might need a periodic theta adjustment to get desirable properties such as real to real
( kneser for instance )
Or even to make them analytically connected.
But what is stopping us to unite 3 fixpoints if they also locally have the same Taylor ?
—-
Regards
Tommy1729
RE: Rank-Wise Approximations of Hyper-Operations - JmsNxn - 06/30/2022
Ah unfortunately that's not possible in the general sense, it's very restrictive.
Assume that:
\[
f(z+c) - c = \sum_{j=1}^\infty a_j z^j\\
\]
And
\[
f(z+b) - b = \sum_{j=1}^\infty a_j z^j\\
\]
Then:
\[
f(z) = f(z+b-c) -b+c\\
\]
Then if \(\mu = b-c\) we're restricted to functions such that:
\[
f(z+\mu) = f(z) + \mu\\
\]
Which implies that \(f(z) - z\) is \(\mu\)-periodic.
RE: Rank-Wise Approximations of Hyper-Operations - tommy1729 - 07/01/2022
(06/30/2022, 12:05 AM)JmsNxn Wrote: Ah unfortunately that's not possible in the general sense, it's very restrictive.
Assume that:
\[
f(z+c) - c = \sum_{j=1}^\infty a_j z^j\\
\]
And
\[
f(z+b) - b = \sum_{j=1}^\infty a_j z^j\\
\]
Then:
\[
f(z) = f(z+b-c) -b+c\\
\]
Then if \(\mu = b-c\) we're restricted to functions such that:
\[
f(z+\mu) = f(z) + \mu\\
\]
Which implies that \(f(z) - z\) is \(\mu\)-periodic.
wow this resonates so hard with the new thread i started before i read this !
regards
tommy1729
RE: Rank-Wise Approximations of Hyper-Operations - Catullus - 07/05/2022
If you used similar uniqueness criterion to the one I proposed for tetration‚ but for higher hyper-operations than for tetration‚ when in the complex plane would a circulated to the infinity converge?
RE: Rank-Wise Approximations of Hyper-Operations - JmsNxn - 07/05/2022
(07/05/2022, 01:49 AM)Catullus Wrote: If you used similar uniqueness criterion to the one I proposed for tetration‚ but for higher hyper-operations than for tetration‚ when in the complex plane would a circulated to the infinity converge?
I'd wager a conjecture is in order.
There exists a domain \(a \in \mathcal{D}\) such that:
\[
a \uparrow^\infty z\\
\]
is finite. I'd propose that solely (under any iteration protocol) this domain is \(\mathcal{D} \supseteq \mathcal{S}\) for the Shell-thron region \(\mathcal{S}\). This would be based on the fact that I know \(1 \le \alpha \le \eta\) converges for \(\Re(z) > 0\).
As we sort of move and iterate more exotic areas of \(a \in \mathcal{S}\), the domain \(\Re(z) > 0\) moves in some manner. I do not know how. But, bounded iterations ellicit bounded iterations. So since bounded iterations of the exponential exist in the shell-thron region, and these iterations ellicit bounded iterations themself. We can expect that the Shell thron region \(a \in \mathcal{S}\) always has a value \(z\) such that:
\[
a \uparrow^\infty z = a\\
\]
This is all I'm willing to talk on the matter. When the simple answer is I don't know, and I don't think anyone knows. I think this would be worthy of the greats if you could answer this question entirely.
RE: Rank-Wise Approximations of Hyper-Operations - Catullus - 07/05/2022
What about i circulated to the infinity?
Does that converge?
If so, then what does it converge to?
RE: Rank-Wise Approximations of Hyper-Operations - Catullus - 09/03/2022
What would happen if you used did something similar to the uniqueness criteria for tetration I proposed, but for defining non whole number rank functions in the Fast-growing hierarchy?
RE: Rank-Wise Approximations of Hyper-Operations - JmsNxn - 09/08/2022
(09/03/2022, 06:36 AM)Catullus Wrote: What would happen if you used did something similar to the uniqueness criteria for tetration I proposed, but for defining non whole number rank functions in the Fast-growing hierarchy?
Catullus, this question is a question very similarly asked by John H. Conway. Conway was huge on number puzzles/game theory/weird integer iteration stuff; but what gets overlooked is his work on Hyper Operators. It is where we get Conway Chained Arrows notation: \(a \to b \to n = a \uparrow^{n} b\). And where he describes a formal algebra, very well mind you, using this notation. John H. Conway also went on to develop surreal numbers. The idea of a surreal number system, is very similar to the Fast growing hierarchy. It is intended to act in a similar manner. The thing is, these things are not proven to be equivalent (As I remember, unless there's been an update).
So you can, in the surreal number system, theoretically take a surreal number \(\omega\) and write \(2 \to 2 \to \omega = 2\). Whether the \(\omega\) in surreal numbers equates to something like \(\aleph_0\)--in a meaningful manner, is still up for grabs. I highly suggest you study Conway in this direction. But, again, this has little to do with the calculus you'll find on this forum. This forum leans heavily into calculus; and what you are asking are foundational logic questions. Which, on similar forums you might find more interesting takes.
Regards