Dmitrii Kouznetsov's Tetration Extension - Printable Version +- Tetration Forum ( https://math.eretrandre.org/tetrationforum)+-- Forum: Tetration and Related Topics ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1)+--- Forum: Mathematical and General Discussion ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3)+--- Thread: Dmitrii Kouznetsov's Tetration Extension ( /showthread.php?tid=214) |

Dmitrii Kouznetsov's Tetration Extension - andydude - 04/16/2008
It appears that there is a new extension related to complex heights. See Analytic solution of F(z+1)=exp(F(z)) in complex z-plane for more information. Andrew Robbins RE: Dmitrii Kouznetsov's Tetration Extension - bo198214 - 04/21/2008
andydude Wrote:See Analytic solution of F(z+1)=exp(F(z)) in complex z-plane for more information. About the uniqueness: It is well known that if we have a solution of the Abel equation then for any 1-periodic function also is a solution to the Abel equation. (Because ). So let be one solution of (*) with (**) and then is another solution of (*). Let us now consider (**). We know that and . As at least for x=0 also [edit] fixed some negligences. [/edit] RE: Dmitrii Kouznetsov's Tetration Extension - Kouznetsov - 04/22/2008
Hello, bo198214; you have cathced the important point! I see, in my paper, I have to write Theorem 0. There exist function , analytic in the whole complex plane except , satisfying (100) , (101) and (102) at any fixed and Theorem 1. There exist only one such function. I hope, the reviewer catches this point, and I already have the corresponding correction above. By the way, your deduction gives the hint, how to prove the Theorem 1. (However, we have to scale the argument of sin function.) How about the collaboration? RE: Dmitrii Kouznetsov's Tetration Extension - bo198214 - 04/22/2008
Kouznetsov Wrote:By the way, your deduction gives the hint, how to prove the Theorem 1. Perhaps. I am not convinced yet that it is true, but if it was, this would be great. We even know vice versa if we have two solutions and of the Abel equation then , meaning that is a 1-periodic function, so we know already that each other solution of the Abel equation must be of the form for some 1-periodic . To prove theorem 1 everything depends on the behaviour of those 1-periodic functions for . For uniqueness roughly the real value must be go to infinity for the imaginary argument going to infinity. Quote:(However, we have to scale the argument of sin function.) Corrected in the original post. Quote: How about the collaboration?That would be great. Quote:(P.S. Does the number of replies I should answer grow as the Ackermann function of time, or just exponentially?) I would guess its rather logarithmically (much questions at the start but then slowly ebbing away)! At least here is another question: Can you please compute values on the real axis for bases for example for ? I would like to compare your solution with the regular tetration developed at the lower real fixed point (which would be 2 in the case of ) of . @Andrew Can you post a comparison graph with your slog/sexp? The values are in Dmitrii's paper. How does the periodicity of your slog (or was it sexp?) at the imaginary axis compare with Dmitrii's limit (where is a fixed point of )? RE: Dmitrii Kouznetsov's Tetration Extension - Kouznetsov - 04/24/2008
bo198214 Wrote:andydude Wrote:See Analytic solution of F(z+1)=exp(F(z)) in complex z-plane for more information.About the uniqueness: It is well known that if we have a solution of the Abel equation then for any 1-periodic function also is a solution to the Abel equation. (Because ). bo198214 Wrote:Kouznetsov Wrote:By the way, your deduction gives the hint, how to prove the Theorem 1. Bo, I got your message about base b=e^(1/e) and b=sqrt(2). In these cases, the real part of quasiperiod is zero, and I cannot run my algorithm as is. I need to adopt it. It will take time. I do not think that b=sqrt(2) is of specific interest (just integer L(b)=4); we need to consider the general case. You may advance faster than I do. You may begin with the plot of the asymptotic period T(b) and analysis of its limiting behavior in vicinity of b=1 and b=e^(1/e). Please, provide the good approximation for (at least) the leading terms. P.S. you may also correct misprints in your post: invert the scaling factor for the argument of sin, and delete the expression with unmatched parenthesis. RE: Dmitrii Kouznetsov's Tetration Extension - bo198214 - 04/25/2008
Kouznetsov Wrote:Bo, I got your message about base b=e^(1/e) and b=sqrt(2). In these cases, the real part of quasiperiod is zero, and I cannot run my algorithm as is. I need to adopt it. It will take time. I do not think that b=sqrt(2) is of specific interest (just integer L(b)=4); we need to consider the general case.is indeed only of interest because it has a simple (integer) fixed point 2. So that is our standard reference (on this forum) base to compare two different methods of computing a tetration. As for example real regular iteration/tetration is no more possible for because there is no real fixed point. Quote:You may advance faster than I do. You may begin with the plot of the asymptotic period T(b) and analysis of its limiting behavior in vicinity of b=1 and b=e^(1/e). Please, provide the good approximation for (at least) the leading terms.yeah, I am also not that richly blessed with time. I will see, what I can do. Quote:P.S. you may also correct misprints in your post:blush (again!). RE: Dmitrii Kouznetsov's Tetration Extension - Kouznetsov - 04/26/2008
Bo, when you make the first step, please, plot the asymptotics in the complex plane, and we compare our results. It will be the second step. bo198214 Wrote: is indeed only of interest because it has a simple (integer) fixed point 2. So that is our standard reference (on this forum) base to compare two different methods of computing a tetration.At 1<b<e^(1/e), there are two real fixed points; each of them should correspond to the analytic tetration. Now I try to plot them both; then hope to provide the algorithm for the precize evaluation. Then I shall run it at b=sqrt(2). In such a way, my theorem is wrong at b < e^(1/e); and, perhaps, at b=e^(1/e); so, it should be reformulated: the equirement b > e^(1/e) should be included into the conditions of the Theorem. bo198214 Wrote:As for example real regular iteration/tetration is no more possible for because there is no real fixed point.I am not sure if I understand you well. At b=2 and b=e, tetration F(z) looks pretty regular (except ), and it is real at z>-2. Complex fixed points are easy to work with. RE: Dmitrii Kouznetsov's Tetration Extension - bo198214 - 04/26/2008
Kouznetsov Wrote:bo198214 Wrote:As for example real regular iteration/tetration is no more possible for because there is no real fixed point.I am not sure if I understand you well. At b=2 and b=e, tetration F(z) looks pretty regular (except ), and it is real at z>-2. Complex fixed points are easy to work with. Oh I use "regular" in the sense of "regular iteration" this is a well studied (mostly by Szekeres and Ecalle) way to compute arbitrary real or complex iterates of a function at a fixed point. There is only one solution for the iterates such that the fixed point still remains analytic or at least asymptotically analytic, this is called regular iteration. You will find the iterational formulas as well the formulas for the coefficients of the powerseries of regular iteration throughout the forum (keywords: hyperbolic and parabolic iteration). For tetration we have . So if we have a fixed point of then we can just consider to be the regular iteration, which gives us the regular tetration. In almost all cases the regular iteration at different fixed points give different solutions. As I now see those regular tetration (at the lower real fixed point, which is btw the only attracting fixed point of ) is cyclic along the imaginary axis: For regular iteration we have the iterational formula: where is the fixed point. We see that the regular tetration is periodic with so it can not have a limit for . RE: Dmitrii Kouznetsov's Tetration Extension - bo198214 - 05/17/2008
Dmitrii, I am currently programming your tetration extension. I just want to mention some misprints, that shouldnt go into your published paper. In formulas (3.2), (3.3), (3.4), (3.6), (4.4) you always omit the minus sign in front of the 1 below the log. Only in the computation formula 4.2 the minus sign is at the right place. There may be also some simplifications (avoidance of doublification with and ) if you would put into your assumption, which is quite reasonable and which you are also using in (4.7). In formula (3.4),(4.4) either the in should be omitted, or it should be appended whenever you use RE: Dmitrii Kouznetsov's Tetration Extension - Kouznetsov - 05/18/2008
bo198214 Wrote:Dmitrii, I am currently programming your tetration extension.Bo, I am glad to read from you! Do you mean paper at http://www.ils.uec.ac.jp/~dima/PAPERS/2008analuxp.pdf ? There is no log in formula (3.2), is there? Now I have doubts about (3.3); I remember I had to play with phases manually combinig the logarithms; both Mathematica and Maple failed to do it as I wanted. Can you reproduce Figure 2? Could you show it? bo198214 Wrote:There may be also some simplifications (avoidance of doublification with and ) if you would put into your assumption, which is quite reasonable and which you are also using in (4.7).Yes, I assume , but it follows from the analyticity and the assumption that is real at z>-2. bo198214 Wrote:In formula (3.4),(4.4) either the in should be omitted, or it should be appended whenever you useYes, your are right. K is always A-dependent. Better to trash this subscript. Thank you for your comment. I had corrected few misprints in the update http://www.ils.uec.ac.jp/~dima/PAPERS/2008analuxp64.pdf but not yet those you indicate; perhaps, I sould trace it again. |