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 open problems survey andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 10/23/2009, 05:27 AM (This post was last modified: 10/25/2009, 07:49 AM by andydude.) Conjecture The following holds for regular tetration: $0 \,< \,({}^{y}x)({}^{-y}x) \,< \,1$ for all $1 < x < e^{1/e}$ and 0 < |y| < 1. The following holds for intuitive tetration: $0 \,< \,({}^{y}x)({}^{-y}x) \,< \,1$ for all $x > e^{1/e}$ and 0 < |y| < 1. Discussion This would be interesting in its own right, partly because it is symmetric, but also because it is useful in demonstrating the difference between reciprocal heights and super-roots. Another notable aspect of this set of bounds is that it is very extension-dependent. This does not hold for linear tetration. Appended are several graphs of the function $f(y) = ({}^{y}x)({}^{-y}x)$ for x = 1.001, 1.1, eta, e, 10. The first three (1.001, 1.1, eta) were calculated with regular tetration, and the last two (e, 10) were calculated with intuitive tetration. More discussion of this is here Attached Files   xty-times-xtny-base-1p001.pdf (Size: 6.39 KB / Downloads: 630)   xty-times-xtny-base-1p1.pdf (Size: 6.28 KB / Downloads: 627)   xty-times-xtny-base-eta.pdf (Size: 6.26 KB / Downloads: 610)   xty-times-xtny-base-e.pdf (Size: 6.2 KB / Downloads: 611)   xty-times-xtny-base-10.pdf (Size: 6.31 KB / Downloads: 613) « Next Oldest | Next Newest »

 Messages In This Thread open problems survey - by bo198214 - 05/17/2008, 10:03 AM eigenvalues of Carleman matrix for b^x, TPID 1 - by bo198214 - 05/17/2008, 10:23 AM Exponential Factorial, TPID 2 - by andydude - 05/26/2008, 03:24 PM eigenvalues of Carleman matrix for (x+s)^p-s, TPID 3 - by bo198214 - 06/29/2008, 12:36 PM Existence of bounded b^z TPID 4 - by bo198214 - 10/08/2008, 04:22 PM sqrt(2) tetrational is completely discontinuous TPID 5 - by bo198214 - 05/01/2009, 09:20 AM Limit of self-super-roots is e^1/e. TPID 6 - by andydude - 10/07/2009, 12:03 AM A conjecture on bounds. TPID 7 - by andydude - 10/23/2009, 05:27 AM Elementary superfunction of polynomial without real fixed points TPID 8 - by bo198214 - 04/25/2010, 10:53 AM Logarithm reciprocal TPID 9 - by bo198214 - 07/20/2010, 05:50 AM Convergence of Eulers and Etas. TPID 10 - by dantheman163 - 10/31/2010, 07:13 PM RE: open problems survey - by nuninho1980 - 10/31/2010, 09:50 PM Tommy's conjecture about Eulers and Etas. TPID 11. - by tommy1729 - 12/01/2010, 03:56 PM convergence of self-tetra-root polynomial interpolation. TPID 12 - by bo198214 - 05/31/2011, 04:54 PM convergence of self-root polynomial interpolation. TPID 13 - by bo198214 - 05/31/2011, 07:02 PM Tommy's conjecture about andrew slog method. TPID 14 - by tommy1729 - 06/01/2011, 06:23 PM Tommy's conjecture TPID 16 - by tommy1729 - 06/07/2014, 10:44 PM Error terms for fake function theory TPID 17 - by tommy1729 - 03/28/2015, 10:48 PM The third super-root TPID 18 - by andydude - 12/25/2015, 06:16 AM

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