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 Notations and Opinions andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 04/08/2008, 09:04 PM OK, I think we need to look at all the possibilities before we go too far into determining notation. We seem to be stuck on two things that are very closely related: auxiliary hyper-operations and iterated hyper-operations, which are practically the same things, but from two different viewpoints (an example difference is that iterated hyper-3 is auxiliary hyper-4). Since the term "auxiliary" is new and "iterated" is old and venerable, it is more appropriate to call them "iterated" hyper-operations, although either term could suffice. A comparison between the notation I used and the notation that GFR used: $\mathtt{b[N]\^h(x)} = \mathtt{b[N]x}$ however, since GFR's notation requires angle-brackets around the 'y', it prevents it from being used with slash-notation, especially for some inverse hyperops. To illustrate the difficulties, I will use GFR's instead. $ \begin{tabular}{c|l|l|l} \text{inv} & \text{my} & \text{GFR's} & \text{name} \\ \hline {z =} & \mathtt{b[N]\^h(x)} & \mathtt{b[N]x} & \text{iterated hyper-operations} \\ b = & \mathtt{z/[N]\^h(x)} & \mathtt{z/[N]x} & \text{auxiliary hyper-roots} \\ h = & \mathtt{b[N]\^{\backslash}z(x)} & \mathtt{b[N]< >x{\backslash}z}? & \text{auxiliary hyper-logarithms} \\ x = & \mathtt{b[N]\^(-h)(z)} & \mathtt{b[N]<-h>z} & \text{negatively iterated hyper-operations} \end{tabular}$ Fortunately, however, we do not need a notation for auxiliary hyper-logarithms, because: $h = \mathtt{b[N]\^{\backslash}z(x)} = \left({}^N_b\begin{tabular}{|c} z \\\hline\end{tabular}\right) - \left({}^N_b\begin{tabular}{|c} x \\\hline\end{tabular}\right)$ So if neccessary, this can be written $h = \mathtt{b[N]{\backslash}z - b[N]{\backslash}x}$ which means we really don't need either my notation, nor GFR's notation for auxiliary hyper-logarithms. Also, as you can see, we also don't need a notation for the auxiliary inverse, because these can be represented by negatively iterated hyper-operations. What this means is that we only need a notation for auxiliary hyper-roots. Andrew Robbins « Next Oldest | Next Newest »

 Messages In This Thread Notations and Opinions - by andydude - 01/21/2008, 02:08 AM RE: Notations and Opinions - by Gottfried - 01/21/2008, 06:48 AM RE: Notations and Opinions - by GFR - 01/21/2008, 10:23 PM RE: Notations and Opinions - by andydude - 01/22/2008, 05:04 AM RE: Notations and Opinions - by Ivars - 01/22/2008, 08:27 AM RE: Notations and Opinions - by andydude - 01/22/2008, 07:04 PM RE: Notations and Opinions - by Ivars - 01/22/2008, 09:56 PM RE: Notations and Opinions - by GFR - 01/22/2008, 10:12 AM RE: Notations and Opinions - by Ivars - 01/22/2008, 12:57 PM RE: Notations and Opinions - by GFR - 01/22/2008, 05:47 PM My Notation - by James Knight - 03/25/2008, 07:58 PM Notation needed - by bo198214 - 03/26/2008, 02:26 PM RE: Notations and Opinions - by GFR - 03/30/2008, 12:51 AM RE: Notations and Opinions - by andydude - 03/30/2008, 05:12 AM RE: Notations and Opinions - by GFR - 04/04/2008, 01:20 PM RE: Notations and Opinions - by bo198214 - 04/04/2008, 01:24 PM RE: Notations and Opinions - by GFR - 04/04/2008, 09:53 PM RE: Notations and Opinions - by GFR - 04/05/2008, 08:26 AM RE: Notations and Opinions - by GFR - 04/08/2008, 10:52 AM RE: Notations and Opinions - by GFR - 04/08/2008, 03:31 PM RE: Notations and Opinions - by bo198214 - 04/08/2008, 04:22 PM RE: Notations and Opinions - by GFR - 04/09/2008, 05:10 PM RE: Notations and Opinions - by andydude - 04/08/2008, 09:04 PM RE: Notations and Opinions - by bo198214 - 04/09/2008, 11:30 AM RE: Notations and Opinions - by GFR - 04/10/2008, 09:15 PM

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