Talk:Ultra exponential function

For the discussion that led to the notice that has been put on this article see here: Wikipedia talk:WikiProject Mathematics. It would make sense for further discussion on this article to take place on this talk page. --Hans Adler (talk) 00:36, 13 January 2008 (UTC)

definition
Dear Ultra. I respect your extremistic ideas, and I want you to read the reference you suggest and provide the correct definition of the ultraexponential function. Can it be defined for real nombers? For complex numbers? It would be good if you provide also examples, when such a function appear as a solution of some physical problem, or some differential or integral equation,.. dima (talk) 03:34, 13 January 2008 (UTC)

P.S. Ultra, do you read the "discussion"? There is article Tetration which seems to be a little bit more developed.. Unfortunately, it has the same problems; there is no algorithmic definition of the operation for complex values of the arguments, and no examples. Even the generalization for the real values looks problematic and not well established. Please, provide the simple definition (take into account that the most of users are not mathematicians) and a reference to the implementaiton for the efficient evaluation, and the discussion of the terminology. I like the notation "uxp", but we need references. Please look how the article Gamma function is designed and try to do similar for uxp. In particular, the article (or at least this discussion) should be equipped with the following:

(Where are the singularities? Where are cuts if any? Where are zeros, if any? Where are minima of derivatives?)
 * 1. References what indicate that the termin ultraexponential is better (or more usual) than tetration.
 * 2. List of programming languages that support function uxp.
 * 3. References to appearence of uxp in applications.
 * 4. Graphics showing behavior of function uxp in the complex plane.

Without these, I cannot help you to defend the article aganist the deleters. Sorry. dima (talk) 02:33, 14 January 2008 (UTC)

P.P.S. Please, Do not delete the article while Ultra actively works on it. Let us look if Ultra makes it better than tetration. dima (talk) 02:33, 14 January 2008 (UTC)


 * So far he seems to be ignoring this talk page and even the messages on his talk page. That's not a good sign. Perhaps he will reply to email? --Hans Adler (talk) 02:54, 14 January 2008 (UTC)


 * Many of these things have already been worked out, but mostly under the umbrella subject of iterated exponentials, which can be written: $$\exp_a^t(x)$$ as noted on the tetration page. Ioannis Galidakis has several papers about series expansions of iterated exponentials in the base (a). Daniel Geisler has several pages about series expansions of iterated exponentials in the top exponent (x). Peter Walker has several papers about series expansions of iterated exponentials in the iterator (t). Iterated exponentials and tetration are well-developed functions. The problem is not that there is a lack of "nice" series expansions. The problem is that we have 3! AJRobbins (talk) 11:25, 19 January 2008 (UTC)


 * I wrote a paper about the applications of tetration, which is available from the Tetration Forum. Although it fails to mention all of the wonderful applications of the Lambert W-function (which infinite tetration is conjugate to), it covers a great deal of representation and combinatorial applications. AJRobbins (talk) 11:25, 19 January 2008 (UTC)


 * Also, if the definition of "uxp" given in this article is correct, then it is equivalent to the first extension given in Ioannis Galidakis' paper, which he calls the "fractional" method, and which I call the "linear" method, and which is equivalent to the "linear approximation" section that already exists here. If this is correct, then this article should be merged with tetration. Before this can be done, however, the claims made in this article should be clarified, and all errors should be corrected. AJRobbins (talk) 11:25, 19 January 2008 (UTC)

Explanation
I wrote the article, because I felt the subject is general and noticeable for some researcher who are working on special functions. Up to now ultra power have been defined for (possible) real numbers and there are five kinds of graphs for ultra exponential functions. For more information you can see the reference of the article. I have tried to improve it and I can not do more. I would appreciate if you do more improvements (if you feel the topic is useful and agrees to the wikipedia's rules). Otherwise you can delete it. —Preceding unsigned comment added by Ultra.Power (talk • contribs) 04:20, 14 January 2008 (UTC)

Nothing personal
Dear Ultra, I am glad that you appear at the Discussion page. Please do not take the crytics of the article as anything personal. Please type four tildas after your comments; the four tildes become your signature.

I tried fo format the article but I still do not understand even the definition. I read the article tetration; it is also poorly written. For example, why the differentiation is not continuous at integer values of argument? Why the linear approximation is required to plot the graphics? (and many other questions). Neither I am sasisfied with the literatude suggested. Notations are horrible, the deduction is not structured and so on. Both articles should be rewritten. Then we should consider to merge them.

Notation $$~^2 x~$$ is confusing. How should I read, for example, $$~a^2x~$$ is it $$~a(^2x)~$$ or $$~(a^2)x~$$? So, I would prefer if you defend the notation $$~{\rm uxp}(a,x)~$$.

Now the tetration article generalizes first for the case then first argument if complex. The uxp article (after to finish, I would move it to uxp, because the termin tetration is easy to confuse and Ultra exponential function is too long) could begin with generalization for the case of real and then complex values of the second argument. Then we should consider to merge them; but it should not be just deletion. Please keep doing. dima (talk) 06:02, 14 January 2008 (UTC)

again definition.
I would suggest the following definition.

Let $$~a~$$ be real number greater than unity. Then function $$~{\rm f}:(-2,+\infty)\rightarrow \mathbb{R}$$ is called ultraexponential on the base $$~a~$$, if it satisfies the conditions: (i) $$~{\rm f}(x)=a^{{\rm f}(x-1)} \mbox{for all} x>-1 ~,~ {\rm f}(0)=1$$ , (ii) $$~\rm f~$$ is differentiable on $$(-1,0)$$ and $$~\rm f'$$ is a nondecreasing or nonincreasing function on $$ (-1,0),$$ (iii) $$\lim_{x\rightarrow 0^+}{\rm f}'(x)=\ln(a) \lim_{x\rightarrow0^-} {\rm f}'(x),$$

Then we could define notation exp with subsctipt and superscript, like $$~\exp^{t}_a$$. It should be done before to formulate the Theorem. Then we should discuss the meaning of operation $$~[x]~$$; perhaps, even suggest a graphic of $$~t=[x]~$$ versus $$~x~$$ and so on. Most of readers of Wiki are not mathematicians. dima (talk) 11:09, 14 January 2008 (UTC)

P.S. The case $$~a=\rm e~$$ looks especially beautiful. I think, for this case, the ultraexponential function deserves a special name. dima (talk) 11:09, 14 January 2008 (UTC)


 * Sorry if I am being a bit direct, but we are not supposed to introduce new definitions on Wikipedia. We can make editorial choices among the definitions present in the literature, but even when we do that we have to consider WP:N. Your suggestions provide a good illustration why these policies actually make sense. Every widely published new definition further reduces the namespace that mathematicians can choose from when they need to name something that is actually important. For example, there may well be a special base for tetration that plays a role similar to e for exponentiation. And yes, e is the most likely candidate for this. But naming mathematical objects is the fun part, and must come after actually thinking about them. Otherwise you risk frivolous pollution of the namespace, and this harms mathematics as a science. (Obviously I am not talking about ad hoc definitions for private drafts or discussions with friends and colleagues here, but about wide dissemination.) --Hans Adler (talk) 11:33, 14 January 2008 (UTC)
 * ok, I understand about $$~a=\rm e~$$. How about to define exp with superscript before to use it? dima (talk) 11:46, 14 January 2008 (UTC)

Dubious
I've looked at that functional equation a long time ago (1975), in response to a query of Richard Feynmann, and it's not unique, at least for a=e. Aside from my personal views, and a clear misprint in the conditions, my interpretation can also be found in the literature of functional equations.

(The condition) $$\lim_{x\rightarrow 0^+}{\rm f}'(x)=\ln(a) \lim_{x\rightarrow0^-} {\rm f}'(x),$$

should clearly be

$$\lim_{x\rightarrow -1^+}{\rm f}'(x)=\ln(a) \lim_{x\rightarrow0^-} {\rm f}'(x),$$&mdash; Arthur Rubin | (talk) 13:38, 14 January 2008 (UTC)


 * Have you ever seen the reference of the article. If you look at it, then you can see the conditions and proof of the theorem (Theorem 2.1 in the paper "Ultra Power and Ultra Exponential Functions" ) Ultra.Power |  (talk) 01:12, 15 January 2008


 * Quite. I doubt the accuracy of the theorem, even if it (seems to) be published in a refereed journal, because it contradicts the research I and some others did in 1975, which wasn't published, because we didn't find the results interesting.  If we had found any hope of the possibility of uniqueness, we would have published. &mdash; Arthur Rubin |  (talk) 19:17, 15 January 2008 (UTC)

Well, Arthur, I have printed the paper, and the condition is in fact what Ultra.Power says it is, not your correction. It seems that the paper is not incorrect, but rather correct and pointless.

You have already identified the worst problem: The third condition requires incontinuity of f at all integer values. However, this problem disappears if we consider only the special case a=e. So let's do that.

We want a function f satisfying f(1)=e and f(n+1)=exp(f(n)), defined as widely as possible. We observe that n=0 and n=1 present no problems: f(0)=1 and f(-1)=0.

To extend f to all positive real numbers, it is clearly sufficient to define f on an interval of length 1, such as [0,1).

And we want f to be continuously differentiable. (The paper only requires f to be continuously differentiable on (0,1), or equivalently on (-1,0). This is obviously whacky, but we needn't worry about this because a=e.)

Next, for no obvious (or explained) reason we require that f ' is monotone on the interval [0,1]. From the functional equation f(x) = ln f(x+1) we get f '(x) = f '(x+1)/f(x+1). Hence f '(-1)=f '(0)/f(0)=f '(0), so f ' must be constant on [0,1]!

Not too surprisingly, the function we get in this way is patched together from smaller pieces in a relatively unnatural way: f(x) = exp[x](exp(x-[x])).

Thus what I learned from the paper is that the correct solution to the problem probably has a minimum of its derivative between -1 and 0. --Hans Adler (talk) 21:06, 15 January 2008 (UTC)

Note: About your dubious, I think it is useful if you try to solve the following problem:

Question. Let $$a>e$$. Is there any convex function on $$[-1,+\infty)$$ except that $$f=\mbox{uxp}_{a}$$ which $$f(0)=1$$ and satisfies the ultra exponential functional equation ($$f(x)=a^{f(x-1)}$$)? Ultra.Power |  (talk) 01:12, 17 January 2008.  —Preceding comment was added at 01:12, 17 January 2008 (UTC)


 * I'm afraid the answer is obviously yes. Let f be defined on {-1,0) with the following characteristics:
 * $$f(-1^+)=0$$
 * $$f(0^-)=1$$
 * f in convex (i.e., $$f'$$ is increasing) on [-1,0]
 * $$f'(0^-) \leq \left (\ln a \right )f'(-1^+)$$ (i.e., $$f'(0^-) \leq f'(0^+)$$)
 * Then the extension of f by the functional equation is convex on $$[-1,+\infty)$$.
 * For sufficiently small $$\epsilon>0$$, $$f(x)=(x+1)(1+\epsilon x)$$ satifies the equation.
 * If equality holds in the last equation, and f is differentiable on [-1,0], then f is C1 on $$(-2,+\infty)$$. With some effort, we can find such an f which is C∞ on $$(-2,+\infty)$$. &mdash; Arthur Rubin |  (talk) 01:36, 17 January 2008 (UTC)
 * In fact, this is what my paper is about, although the method is technically Cn. For updates on this front, please see the Tetration Forum as it represents the cumulative knowledge we have about tetration at this point. AJRobbins (talk) 10:59, 19 January 2008 (UTC)

Not only there is no any contradiction between the above example and the main theorem of "Ultra exponential function", but also Theorem 3.2 in the paper implies that $$\log_a(2+x)\leq (x+1)(1+\epsilon x)\leq 1+x $$, for the sufficiently small $$\epsilon>0$$ and every x in [-1,0].

What do you think about the case $$ a=e $$, in the above question? Ultra.Power (talk) 3:53, 17 January 2008.


 * Just that $$[-1,+\infty)$$ is arbitrary for the domain of convexity, since the natural domain of the function is $$(-2,+\infty)$$. Looking for C∞ or real-analytic or Cω solutions is more interesting.  &mdash; Arthur Rubin |  (talk) 08:21, 17 January 2008 (UTC)
 * I've removed the dubious tag, but will retag for importance unless some further interesting results are added, preferably from another paper. &mdash; Arthur Rubin |  (talk) 15:47, 17 January 2008 (UTC)


 * And I've confirmed C∞ above, based on a reply to my question on sci.math.research. &mdash; Arthur Rubin |  (talk) 20:15, 18 January 2008 (UTC)

General tidy up
I have attempted a general tidy up of the article, attemtping to fix grammar, remove use of first/second person, improve formatting, remove duplication etc. I still don't think that this extension of tetration merits its own article - everything here is either already in the tetration article or could be easily added to it. However, while we debate such issues, we can try to make this article readable in the meantime. Gandalf61 (talk) 14:24, 14 January 2008 (UTC)


 * I already merged the small piece, namely, the definition. But the Ultra exponential function uxp should also exist. The tetration deals with various upgrades of operation of increment. The uxp deal with very specific part of this problem, namely, solution of the equation $$~\exp(L~{\rm f}(x))={\rm f}(x+1))~$$ for $$~x \ge 0~$$; and the proof that $$~ \rm f ~$$ cannot be analytic function at $$~L\ne 1$$. I hope, Ultra will supply such a proof for wiki; this proof is not trivial and it will not fit the format of just a subsection. Therefore, uxp shoud be independent article. dima (talk) 03:48, 16 January 2008 (UTC)


 * Wrong, as can easily be seen from the paper. Only with that odd condition (which amounts to f' is constant on (-1,0) ) is there any claim it cannot be analytic.  &mdash; Arthur Rubin |  (talk) 05:21, 16 January 2008 (UTC)

Value on (-1,0)
I don't suppose anyone else has noted that
 * $$\operatorname{uxp}_a(x)=x+1 $$ for arbitrary a if -1 < x < 0? &mdash; Arthur Rubin | (talk) 01:32, 18 January 2008 (UTC)

This is gotten from the uniqueness conditions of the theorem.Ultra.Power (talk) 1:55, 18 January 2008.


 * I would rather have thought that the uniqueness condition was derived from the choice of function, but we'd have to ask the author of the paper about that. &mdash; Arthur Rubin |  (talk) 01:57, 18 January 2008 (UTC)

This is nothing new.
I've just noticed that this is nothing all that new: The extension to real numbers shown is identical to the "linear approximation" method on the page "Tetration", and is just the special case for b = e. Because of that, I'd suggest scrapping this page and merging any new encyclopedic material back into said page on tetration, perhaps mentioning in the "linear approximation" section that for b = e said method is first-differentiable with respect to the tower, but not second- or higher. Pretty much all this page does is introduce some "original research" notation and nomenclature, which isn't suitable for Wikipedia, and what non-original material is here is already covered elsewhere (namely, the aforementioned tetration page). Do you agree? mike4ty4 (talk) 09:32, 29 January 2008 (UTC)
 * It's probably not original research in the strict technical sense of Wikipedia because it was published in a refereed journal. Apart from that I agree. I think that this article gives undue weight (also not in the technical WP sense) to one of the most obvious and at the same time clearly not "natural" interpolations of tetration. I am afraid I am not sure there is anything here that is worth mentioning under tetration, so to me it looks more like this article should just be deleted. --Hans Adler (talk) 11:57, 29 January 2008 (UTC)
 * I agree with Mike4ty4 and Hans. I would vote for delete if this article were listed on AfD. Gandalf61 (talk) 12:12, 29 January 2008 (UTC)
 * Really, much of the information here is already mentioned at the tetration page, like I said this simply seems to introduce some non-notable and non-advantageous nomenclature/terminology. 70.101.145.204 (talk) 08:26, 4 February 2008 (UTC)
 * Actually, now I think it's better not to delete this page but to turn it into a redirect to tetration, to preserve the history. The discussions here are quite interesting after all. --Hans Adler (talk) 23:09, 29 January 2008 (UTC)
 * Sounds like a good idea. 70.101.145.204 (talk) 08:26, 4 February 2008 (UTC)

5th and 6th hyperoperations
I posted the following at Talk:Hyperoperation:
 * There's a second substantial cite for pentation, besides Goodstein (1947): K.K. Nambiar, 1995, Ackermann functions and transfinite ordinals, Appl. Math. Lett. 8 (6), 5153. That should justify a redirect. I don't see a case for a separate article.  As an aside, there's a fun attempt to generalise teration, pentation and hexation to continuous functions, but its unpublished so doesn't count for notability: Markus Müller, Reihenalgebra &mdash; What comes beyond exponentiation?&mdash; Charles Stewart (talk) 11:39, 1 July 2009 (UTC)

Unpublished=uncited, but maybe there is some value to Müller's observation for the content at this article.

Is there will to merge this article with hyperoperation? &mdash; Charles Stewart (talk) 09:06, 2 July 2009 (UTC)


 * This comment seems irrelevant to this article, only to Pentation (and Talk:Hyperoperation). There seems consensus that this article should be merged to Tetration.  — Arthur Rubin  (talk) 12:49, 2 July 2009 (UTC)


 * I was thinking that it was possible that this article should be extended to cover continuous generalisations of hyperoperations, rather than be merged with tetration. &mdash; Charles Stewart (talk) 13:26, 2 July 2009 (UTC)


 * Sorry to have misunderstood you. I disagree, as this article is on a particular continuous generalization of tetration.  It might make sense to split off the real and complex "heights" section of tetration to a separate article, and add the attempts at continuous pentation to that article.  — Arthur Rubin  (talk) 13:39, 2 July 2009 (UTC)


 * I agree with Arthur. There are more polite ways of saying it, but this article is about a completely and utterly non-notable stupid generalisation of tetration. I don't remember the details, but it was something obvious like the first or second partial derivative by the parameter doing a jump at every integer. Something that makes it implausible that this definition serves any purpose, combined with the fact that there is no indication that it does serve a purpose. Look at the earlier discussions for the details. The reason the article still exists is because nobody has done the work of "merging" it yet, i.e. mentioning this function at tetration. I would do it now, but I am currently on dial-up. --Hans Adler 20:06, 2 July 2009 (UTC)


 * I am happy to defer to you on the dismerits of this work, since I have no appetite to read it myself, and I had independently though how uninteresting the observation is that one can use splines to turn these number-theoretic functions into curves. I am simply speculating about where worthwhile content of this sort should go. Do have a look at the Müller paper, it is not at all like what you say of Hooshman's paper. &mdash; Charles Stewart (talk) 08:50, 3 July 2009 (UTC)


 * I'm afraid Muller's paper is only helpful for lower hyper operations; as I see it, what he has done that we don't already have seems to be:
 * If $$\bigodot$$ is defined for reals (greater than 1), then
 * $$ x \downarrow \bigodot 2 = x \uparrow \bigodot 2$$
 * $$ x \downarrow \bigodot n$$ and $$x \uparrow \bigodot n$$ are defined.
 * $$ n \downarrow \downarrow \bigodot x$$ may be defined for all real x.
 * (I may have missed something.) Unfortunately, that only applies for lower operations, not for upper operations.  — Arthur Rubin  (talk) 20:25, 3 July 2009 (UTC)