Thoughts on hyper-operations of rational but non-integer orders? VSO Newbie Posts: 1 Threads: 1 Joined: Jun 2019 09/01/2019, 04:34 AM I can't seem to find the right angle to approach this concept intuitively. Does anyone have any ideas of how to consider hyper-operations in a way that isn't recursive, such as to accept non-integers? Gottfried Ultimate Fellow Posts: 900 Threads: 130 Joined: Aug 2007 09/02/2019, 03:40 PM (This post was last modified: 09/10/2019, 02:33 AM by Gottfried.) I am not sure I get your problem correctly.            Take the function $f: b^z$ as to be iterated, with, say $b=sqrt(2)$ .              Assume one plane on a math-paper and look for easiness only the lines and their crossings of the coordinate-system of integer complex numbers $z_0$ .               Now take another paper, position it 10 cm above and for every point of the crossings (and ideally also of the lines) mark the values of $z_1=b^{z_0}$. Then repeat it with a third plane, again 10 cm above, marking $z_2=b ^{b^{z_0}}$ . After that, try to connect the related points of the zero'th, the first and the second plane by a weak string, say a spaghetti or so. Surely except of the fixpoints in $z_0$ it shall be difficult to make a meaningful and smooth curve - and in principle it seems arbitrary, except at the fixpoints, where we simple stitch a straight stick through the iterates of the $z_0$ at the fixpoint.            Of course the spaghetti on the second level is then no more arbitrary but must be - point for point - be computed by one iteration. But the spaghatti in the first level follow that vertically orientated curve, where a fictive/imaginative plane of paper is at fractional heights and the fractional iterates would be the marks on the coordinate-papers at the "fractional (iteration) height".                                                      I'd liked to construct some physical example, showing alternative paths upwards between the fixed basic planes, with matrial curves made by an 3-D-printer, but I've not yet started to initialize the required data. But I think, that mind-model alone makes it possibly already sufficiently intuitive for you. A somewhat better illustration is in my answer at MSE, see https://math.stackexchange.com/a/451755/1714 Gottfried Helms, Kassel tommy1729 Ultimate Fellow Posts: 1,906 Threads: 409 Joined: Feb 2009 09/09/2019, 10:38 PM I think the OP refers to concepts like , what i called " semi- super " operators. Like the semisuper operator of the semisuper operator of f(x) is the super of f(x). This is extremely difficult. Do not confuse with the functional half-iterate of the superfunction. Regards  Tommy1729 Catullus Fellow Posts: 213 Threads: 47 Joined: Jun 2022 06/28/2022, 08:33 AM (This post was last modified: 06/28/2022, 08:34 AM by Catullus.) (09/01/2019, 04:34 AM)VSO Wrote: Does anyone have any ideas of how to consider hyper-operations in a way that isn't recursive, such as to accept non-integers?Although, each level of the Ackermann function is primitive resursive. The entire Ackermann function itself can not be de-recursed. Please remember to stay hydrated. ฅ(ﾐ⚈ ﻌ ⚈ﾐ)ฅ Sincerely: Catullus /ᐠ_ ꞈ _ᐟ\ MphLee Long Time Fellow Posts: 374 Threads: 30 Joined: May 2013 06/30/2022, 11:41 PM (06/28/2022, 08:33 AM)Catullus Wrote: (09/01/2019, 04:34 AM)VSO Wrote: Does anyone have any ideas of how to consider hyper-operations in a way that isn't recursive, such as to accept non-integers?Although, each level of the Ackermann function is primitive resursive. The entire Ackermann function itself can not be de-recursed. This comment doesn't address VSO's question. Ackermann function do not coincides with hyperoperations. Secondly, hyperoperations need not to be defined recursively. Third point: ackermann function, goodstein function (natural hyperoperations), if considered as functions over the natural numbers are recursive. Are defined recursively. It is not fully clear what do you mean by de-recursed. Anyways, if the question is if it's possible to give an analytical, non recursive, expression to the ackermann function/hyperoperations, the answer is yes. JmsNxn gave one expression for that. The real question is if that analytical representation has desired properties: does it satifies a functional equation? Is it smooth? Holomorphic? MSE MphLee Mother Law $(\sigma+1)0=\sigma (\sigma+1)$ S Law $\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$ « Next Oldest | Next Newest »

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