Talk:Ultra exponential function

message: For the discussion that led to the notice that has been put on this article see here: Wikipedia talk:WikiProject Mathematics#"Ultra power" etc.. 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)

message: 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)
replies: text: 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)
replies: text: 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)

text: 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)

text: 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)
thread_title:  definition 

message: 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)
thread_title:  Explanation 

message: 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)
thread_title: Nothing personal

message: 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)
replies: text: 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)
replies: text:  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)
thread_title:  again definition. 

message: 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)
replies: text: 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)
replies: text: 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.  — Arthur Rubin |  (talk) 05:21, 16 January 2008 (UTC)
thread_title:  General tidy up 

message: I don't suppose anyone else has noted that
:\operatorname{uxp}_a(x)=x+1  for arbitrary a if -1 < x < 0? — Arthur Rubin |  (talk) 01:32, 18 January 2008 (UTC)
replies: text: 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.  — Arthur Rubin |  (talk) 01:57, 18 January 2008 (UTC)
thread_title:  Value on (-1,0) 

message: 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)
replies: text: 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)
replies: text: 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)

text: 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)

text: 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)
replies: text: Sounds like a good idea. 70.101.145.204 (talk) 08:26, 4 February 2008 (UTC)
thread_title:  This is nothing new.