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?

Tetration Forum > Tetration and Related Topics > Mathematical and General Discussion > 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?

09/02/2019, 03:40 PM

I am not sure I get your problem correctly.

Take the function as to be iterated, with, say .

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 . Then repeat it with a third plane, again 10 cm above, marking .

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 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 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

Take the function as to be iterated, with, say .

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 . Then repeat it with a third plane, again 10 cm above, marking .

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 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 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

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

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

Tetration Forum > Tetration and Related Topics > Mathematical and General Discussion > Thoughts on hyper-operations of rational but non-integer orders?

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