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?
Thoughts on hyper-operations of rational but non-integer orders?
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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?
I am not sure I get your problem correctly.
Take the function 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 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 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 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
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 (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\
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 \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\) |
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