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 tetration bending uniqueness ? bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 06/07/2011, 11:19 AM (This post was last modified: 06/07/2011, 11:22 AM by bo198214.) (06/07/2011, 10:56 AM)mike3 Wrote: Hmm. This makes me wonder about the following conjecture: The "principal" analytic fractional iterates $\exp^t(x)$, $t \ge 0$ of the natural exponential (and perhaps any with $b > \eta$) are uniquely characterized by $\frac{d^n}{dx^n} \exp^t(x) > 0$ for all $x$, all $t \ge 0$ and all $n > 0$. You should put that in the open problems section. Really interesting question. And you can even put $n\ge 0$. bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 06/07/2011, 12:27 PM (06/07/2011, 10:56 AM)mike3 Wrote: Hmm. This makes me wonder about the following conjecture: The "principal" analytic fractional iterates $\exp^t(x)$, $t \ge 0$ of the natural exponential (and perhaps any with $b > \eta$) are uniquely characterized by $\frac{d^n}{dx^n} \exp^t(x) > 0$ for all $x$, all $t \ge 0$ and all $n > 0$. There is also an associated line of thought of Szekeres, but not with the fractional iterates but with the Abel function. He wonders about the alternating signs in the logarithm and shows that the principal/regular Abel function of $e^x-1$ is uniquely determined by the "totally monotonic"-criterion. tommy1729 Ultimate Fellow Posts: 1,372 Threads: 336 Joined: Feb 2009 06/07/2011, 09:02 PM im sorry i do not wish to be annoying. but this does not seem like a new idea. in fact , it seems mine : quoting from my own posts : thread tid 474 posted on 7/11/10 title : tommy's uniqueness conditions post nr 1 , thread started by me : " i partially already mentioned the first uniqueness condition before : d f^n / d x^n sexp(slog(x) + k) > 0 for all positive integer n and all positive real k. and probably ( i.e. if im not mistaken because of the local heat wave )this is true if and only if the following is true : ( i.e. i assume " equivalent to " ) d f^n / d k^n sexp(slog(x) + k) > 0 for all positive integer n and all positive real k. " thread tid 484 posted on 7/29/10 title : final uniqueness condition ... probably post nr 1 , thread started by me : " d f^2 / d x^2 sexp(slog(x) + k) > 0 for all real x and all real k with 0 < k < Q where Q is any nonzero positive real. assuming sexp resp slog to be C^2 of course , i 'believe' this condition implies analytic as well. is this equivalent to d f^2 / d x^2 sexp(x) > 0 for all positive real x ? i assume because of the substitution x = sexp(y) " if you combine those two posts , i find it pretty clear that the idea has occured to me first. notice that exp^[t](x) is equivalent to sexp(slog(x) + t) despite one of those threads and this one contains mistakes , that idea is clearly mine. ( i even did a search on this forum too see if anyone else was first and looked on sci.math , mathoverflow , google and some books ) i believe my (tommy's) 2sinh method satisfies these conditions and hence it is conjectured for bases > sqrt(e) [ thus including e like mike ] ( yes these posts were made after i posted the 2sinh method - which is also by me and in fact way older than this forum ( i found it in my teenage notes ) and i mentioned that too ) thus for bases > sqrt(e) => d f^n / d x^n sexp(slog(x) + k) > 0 for all positive integer n and all positive real k. d f^n / d k^n sexp(slog(x) + k) > 0 for all positive integer n and all positive real k. d f^2 / d x^2 sexp(slog(x) + k) > 0 for all real x and all real k with 0 < k < Q where Q is any nonzero positive real. are all ideas of me , somewhat sloppy ( i could have written positive real k e.g. ) but mine. so basicly my opinion is that this conjecture of mike is actually a rewording of some of my conjectures. regards tommy1729 mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 06/07/2011, 09:10 PM (06/07/2011, 11:19 AM)bo198214 Wrote: (06/07/2011, 10:56 AM)mike3 Wrote: Hmm. This makes me wonder about the following conjecture: The "principal" analytic fractional iterates $\exp^t(x)$, $t \ge 0$ of the natural exponential (and perhaps any with $b > \eta$) are uniquely characterized by $\frac{d^n}{dx^n} \exp^t(x) > 0$ for all $x$, all $t \ge 0$ and all $n > 0$. You should put that in the open problems section. Really interesting question. And you can even put $n\ge 0$. Um, no. So far, the "best" candidate for $\exp^{1/2}(x)$ shows a zero. So $n = 0$ (taking no derivative at all) may not be a good idea. tommy1729 Ultimate Fellow Posts: 1,372 Threads: 336 Joined: Feb 2009 06/08/2011, 12:34 PM bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 06/08/2011, 08:37 PM (06/07/2011, 09:02 PM)tommy1729 Wrote: so basicly my opinion is that this conjecture of mike is actually a rewording of some of my conjectures. Tommy, keep cool. I dont think Mike did steal that idea from you, but discovered it independently. And then I never saw people arguing about whom had a *conjecture* first. I rather think when someone else has the same conjecture then it could attract more investigation and in the end leading to a proof or disproof of the conjecture. Which is imho is the interesting part: the proof, or the answer to the question. I mean to ask questions/to utter conjectures is easy. To ask good (with respect to fertility in the field) questions is a bit more difficult. And really tough work is finding proofs/solving open problems. So if you still consider it utmost important who first asked a question/uttered a conjecture, I have no problems attributing it to you (on this forum). But much more I would like to attribute the first proof of that conjecture to you. tommy1729 Ultimate Fellow Posts: 1,372 Threads: 336 Joined: Feb 2009 06/09/2011, 12:26 PM im not claiming mike stole it. although i find it offense of members not admitting i was first. its not a genius idea , but it might be an important one , so my honor matters imho. right questions are almost as important as proofs. im happy you give me credit afterall. regards tommy1729 « Next Oldest | Next Newest »

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