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 Notations and Opinions andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 03/30/2008, 09:24 AM Gottfried Wrote:... y = x{4,b}h for iterated exponentiation beginning at x: b^...^b^b^x (and from earlier discussions) y = x{3,b}h for iterated multiplication beginning at x: x*b^h y = x{2,b}h for iterated addition beginning at x: x+b*h for my needs for the time being, the "height"-function h = hgh(x,b) if x = 1 {4,b} h = b [4] h // related to the tetrational notation ... I personally think that the Arrow-Iteration-Section notations discussed in my first post cover most of these use cases, but U-tetration is different enough to require a special notation. Here are my recommendations: $ \begin{tabular}{l|c|c} \text{ASCII} & \text{Arrow-Iteration-Section} & \text{Special} \\ \hline \mathtt{x\^\^y(a)}\text{ or }\mathtt{(x\^)\^y(a)} & (x {\uparrow})^y(a) & \exp^y_x(a) \\ \mathtt{x\^\^{\backslash}z(a)} & (y \mapsto (x {\uparrow})^y(a))^{-1}(z) & \text{slog}_x(z) - \text{slog}_x(a) \\ \mathtt{z/\^\^y(a)} & (x \mapsto (x {\uparrow})^y(a))^{-1}(z) & \ \\ \hline \mathtt{x\^-\^y(a)}\text{ or }\mathtt{(x\^-)\^y(a)} & (t \mapsto x^t - 1)^y(a) & DE^y_x(a) \\ \mathtt{x\^-\^{\backslash}z(a)} & (y \mapsto (t \mapsto x^t - 1)^y(a))^{-1}(z) & \ \\ \mathtt{z/\^-\^y(a)} & (x \mapsto (t \mapsto x^t - 1)^y(a))^{-1}(z) & \ \\ \end{tabular}$ but I've seen other notations elsewhere. The one I've seen used the most is x^^y@a, although I had also used yxa in the past. Also, GFR uses x\$y*a or something like that, which I find confusing. Thats all about iter-exp. Starting from scratch using Arrow-Iteration-Section notation, we find that the natural expression in ASCII is (x^)^y(a) which could be shortened to x^^y(a) which means the corresponding notation for iterated decremented exponentials is (x^-)^y(a) which could be shortened to x^-^y(a), what do you think? About iter-dec-exp/U-tetration, this would mean that your "height" function is h = hgh(x, b, a) = b^-^\x(a) and h = hgh(x, b) = b^-^\x which I would've called the "super-decremented-logarithm" or something. We might even go so far as to use similar notations for superroot and superlog, so srt_n = (/^^n) and slog_b = (b^^\). While I'm at it, I might as well summarize the other suggestions (based on BO's): $ \begin{tabular}{l|c|c} \text{ASCII} & \text{Arrow-Iteration-Section} & \text{Special} \\ \hline \mathtt{x\^\^y} & x {\uparrow}{\uparrow} y & {}^{y}{x} \\ \mathtt{x\^\^{\backslash}z} & (x {\uparrow}{\uparrow})^{-1}(z) & \text{slog}_x(z) \\ \mathtt{z/\^\^y} & ({\uparrow}{\uparrow} y)^{-1}(z) & \ \\ \hline \mathtt{x[n]y} & x {\uparrow}^{n-2} y & x \begin{tabular}{|c|}\hline n \\\hline\end{tabular} y \\ \mathtt{x[n]{\backslash}z} & (x {\uparrow}^{n-2})^{-1}(z) & {}^{n}_{x}\begin{tabular}{|c}z \\\hline\end{tabular} \\ \mathtt{z/[n]y} & ({\uparrow}^{n-2} y)^{-1}(z) & {}_{n}^{y}\begin{tabular}{|c}\hline z \\\end{tabular} \\ \hline \mathtt{x[n]\^y(a)} & (x {\uparrow}^{n-2})^y(a) & \ \\ \mathtt{x[n]\^{\backslash}z(a)} & (y \mapsto (x {\uparrow}^{n-2})^y(a))^{-1}(z) & \ \\ \mathtt{z/[n]\^y(a)} & (x \mapsto (x {\uparrow}^{n-2})^y(a))^{-1}(z) & \ \end{tabular}$ I must say, the slash notation is by far the most expressive tetration notation I've ever seen. It allows full expression of practically anything I can think of that is hyperop/tetration related. As you can see, it covers many topics that do not have a specialized notation yet. 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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