Talk:Superfunction

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Superfunction evaluated *at point t*

I seem to not to become familiar with the concept of Superfunctions. But I understand now at least one reason. If we say "S(z) = f(f(f(...f(t))), where f is evaluated z times" then we obscure, that we evaluate f initially *at the point t*. That means, for each *t* we have another superfunction! So a) we should notice that in the article and b) should change the notation, to something like "S_t(z) = f(f(f(...f(t))), where f is evaluated z times beginning at t".

We'll see then, that the range of the "superfunction at some t" is limited to intervals around t between fixpoints and that "S_t(z)" cannot exceed that intervals by change of the parameter z (as long as z is real, there is one option for complex z to step to a neighboured interval). An example is "f(t)=2^t-1 ", whose superfunction -if evaluated beginning at any value "0<t<1" - cannot exceed this range.

--Gotti 07:00, 24 November 2011 (UTC)

Of course the independent variable t is everywhere implicit. You might use the notation S(z;t), but I don't see why... The subject, minus eccentric voodoo notation, is covered in a conventional two-variable language, in Iterated function, orbit, Schröder's equation, and such less amateurish articles. It is a glorious subject. Note the two variables, n and x in the definition of the family (orbit) of Chebyshev_polynomials#Trigonometric_definition: there are two variables, but no need of introducing weird nonstandard terms such as "superfunction". Cuzkatzimhut (talk) 11:36, 24 November 2011 (UTC)

Come to think of it, you might have the time, patience, and gumption, to close down this page and merge its contents, in mainstream language, with those of orbit, or Iterated function, functional composition, or such... It is a bit of work to do it nicely. I tried to help a little, but it is still a mess when it gets to the examples and the use of Abel's equation. I supplanted x for t, since t is normally the evolution/recursion variable, what this article calls z. Sadly, however, x is the variable which normally goes complex, as in Kneser's seminal work... So I let it stand in Abel's equation which can only result in grief and confusion.... I fear the article needs more time than I have, to be salvaged ...

Cuzkatzimhut (talk) 22:38, 25 November 2011 (UTC)

The problem is, though, that none of those really address the specific topic of "fractional iteration" -- i.e. how can we interpolate a discrete iterated function to something continuous? Perhaps it would be better to spin off a new article called "fractional iteration" or "continuous iteration" or something similar. This page would then become a redirect to it. mike4ty4 (talk) 23:27, 26 November 2011 (UTC)

I'm not sure what you mean... Schröder's equation already answered all these questions in Schroeder's inspired 1870 method, and for 140 years no users have been let down, no? I fear fewer articles, not more are the answer... So far, the only ones in the overlapping stack which are not borderline crank are Iterated function, Schröder's equation, and, by way of illustration, functional square root. Why couldn't the sound 30% of this article jam into section 7 of the first? Just a thought... However, I am reluctant to argue it at length, given the huge amount of work involved, which I could hardly contribute to... Cuzkatzimhut (talk) 00:46, 27 November 2011 (UTC)

The Schröder's equation is good while the transfer function has as least one real fixed point. But if there are no fixed points, then it is difficult to formulate the problem of non-integer iterates in terms of the Schroeder function. 130.153.147.215 (talk) 04:44, 28 March 2013 (UTC)

Should article Superfunction be merged with article Flow???

That is the question that led me to this discussion page.

I say, Yes indeed, they should be merged. But only after the enormous number of errors, incongruities, and infelicitous attempts at exposition be fixed in both articles.

For one thing, Superfunctions refers many times to the [function with some property], in many cases where there is no uniqueness, although the author seems unaware of or unconcerned with this issue.

For another, Flows discusses one-parameter semigroups of operators on Hilbert space, which is sufficiently unrelated to the ordinary definition of flow (a group action of the real numbers on a manifold) that it absolutely does not belong in the same article.

Finally, all of these articles have many statements where the domain and/or codomain of some function remain unmentioned and hence a total mystery to anyone who comprehends mathematical notation but who is trying to learn the subject of the article. This happens time and again, and I seriously wish that people would learn how to write expository essays before tampering with a Wikipedia article.

And post-finally, the "celebrated Bernoulli flow" is by no means the first flow that comes to mind among most people who are familiar with flows. It does not deserve mention in the Flows article, since it requires a lot more explanation than an example should require.Daqu (talk) 20:26, 19 April 2013 (UTC)

The article outlines, in parochial idiosyncratic terms, iteration orbits in the complex domain, as covered quite adequately in the main article, iterated function whose traffic stats dominate this one by 3 orders of magnitude. If one had the energy to clarify it and condense it to a sectionlet late in that article, that could not be a bad thing. But, starting with the daffy name, this one here does not rise above stub status.Cuzkatzimhut (talk) 21:07, 9 December 2013 (UTC)