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 Dmitrii Kouznetsov's Tetration Extension bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 05/18/2008, 05:09 PM (This post was last modified: 05/18/2008, 05:11 PM by bo198214.) Kouznetsov Wrote:bo198214 Wrote: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.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?Yes I mean that paper. In (3.2) there is not yet the logarithm but in the following formulas. Its just the 1 in the denominator $1+ip-z$ of the second summand that has to be changed into a -1. There are also some minor corrections for the description of the path on page 6 A.,C.: instead of just writing $t=-iA$ or $t=iA$ you have to add the corresponding 1's, as you mention before $\Re(t)=\pm 1$. But that didnt injure the readability. Quote: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? Unfortunately I am not the deep into the matter and also not familiar with complex function visualization. Hm, does that mean that I can not just take the normally cutted $0\dots -\infty$ logarithm but have to use something else? I dint get you algorithm to work, it does always diverge. Though I use not Laguerre Gauss integration but simple equidistant integration this shouldnt stop the convergence. Can you send me a sample of your code? Quote:Yes, I assume $F(z^*)=F(z)^*$, but it follows from the analyticity and the assumption that $F(z)$ is real at z>-2. Oh I wasnt aware of this. Kouznetsov Fellow Posts: 151 Threads: 9 Joined: Apr 2008 05/21/2008, 12:20 AM Kouznetsov Wrote:bo198214 Wrote: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.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 $L$ and $\overline{L}$) if you would put $F(\overline{z})=\overline{F(z)}$ into your assumption, which is quite reasonable and which you are also using in (4.7).Yes, I assume $F(z^*)=F(z)^*$, but it follows from the analyticity and the assumption that $F(z)$ is real at z>-2. bo198214 Wrote:In formula (3.4),(4.4) either the $A$ in ${\mathcal K}_A$ should be omitted, or it should be appended whenever you use ${\mathcal K}$Yes, 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. Dear Bo: 1. I prepare the updated version with correction of the misprints you have indicated. May I specify your real name in the Acknowledgement? 2. I have described behavior of tertation at the $1<\ln(b)<1/\rm e$ at http://en.citizendium.org/wiki/User:Dmit...base._Base Could you plot Fig.3 ? Would you like to take the text and write the rest? bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 05/21/2008, 06:22 AM (This post was last modified: 05/21/2008, 06:22 AM by bo198214.) Kouznetsov Wrote:Dear Bo: 1. I prepare the updated version with correction of the misprints you have indicated. May I specify your real name in the Acknowledgement?Its no secret: Henryk Trappmann. Quote:2. I have described behavior of tertation at the $1<\ln(b)<1/\rm e$ at http://en.citizendium.org/wiki/User:Dmit...base._Base Could you plot Fig.3 ? Would you like to take the text and write the rest? What is citizendium? Is it a competition to wikipedia? If you mean to put tetration by regular iteration there, yes I can do. But allow me some time. However its your user page so I dont know about permissions. Kouznetsov Fellow Posts: 151 Threads: 9 Joined: Apr 2008 05/21/2008, 11:18 AM ... What is citizendium? Is it a competition to wikipedia? Henrik, sorry, I was sure that the page is available from anywhere. I attach two most important files: map of the complex tetration at base sqrt(2) and graphic of the asymptotic parameters of tetration versus base. I should move it to some readable and writable place. How about wikisource? Attached Files Image(s)         bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 05/22/2008, 07:12 PM Kouznetsov Wrote:Henrik, sorry, I was sure that the page is available from anywhere. Nono this was a misunderstanding, I can read your page. However I can not probably write on it. I was just curious what citizendium is. It states something like the "world's most trusted encyclopedia". So is there some peer reviewing going on? Quote:I should move it to some readable and writable place. How about wikisource? Seems more to be about historic texts isnt it? Kouznetsov Fellow Posts: 151 Threads: 9 Joined: Apr 2008 05/22/2008, 10:43 PM bo198214 Wrote:Kouznetsov Wrote:Henrik, sorry, I was sure that the page is available from anywhere. Nono this was a misunderstanding, I can read your page. However I can not probably write on it. I was just curious what citizendium is. It states something like the "world's most trusted encyclopedia". So is there some peer reviewing going on? Quote:I should move it to some readable and writable place. How about wikisource? Seems more to be about historic texts isnt it?1. Yes, it is. I agree. Could you register as citizendium? It seems to be the only wiki that allows original researches. 2. There are two regular superexponentials at base $b$ such that $1. I have plotted the only one, $F$ such that $F(0)=1$. At $b=\sqrt{2}$, for example, $\lim_{x \rightarrow \infty} F_{\sqrt{2}}(x+{\rm i}y)=2$; $\lim_{x \rightarrow -\infty} F_{\sqrt{2}}(x+{\rm i}y)=4$. There is another one, $G$ that grows up along the real axis faster than any exponential and aproaches its limiting values in the opposite direction. I am writing source for its evaluation. 3. Then we have covered the ranges $1 and $b>\exp(1/\rm e)$; and I think about cases $b=\exp(1/\rm e})$ and $b<1$. I suggest that you use the same idea: first, find the asymptotics and periodicity (if any); then recover the analytic function with these properties. Could you calculate some pictures (similar to those I have posted) for these cases? andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 05/22/2008, 10:59 PM Also, formula (1. uses "x" where the function takes "z", have you fixed this yet? Andrew Robbins Kouznetsov Fellow Posts: 151 Threads: 9 Joined: Apr 2008 05/22/2008, 11:36 PM (This post was last modified: 05/23/2008, 02:40 AM by Kouznetsov.) andydude Wrote:Also, formula (1. uses "x" where the function takes "z", have you fixed this yet? Andrew Robbins 1. Yes. Thank you, Andrew! 2. Arthur Knoebel tried sending to me a report about my paper on tetration, but it was returned. (as I understand, Arthur uses the non-electronic mail) I asked him to send it again. If anybody works with A.Knoebel, the corrections could be boosted with scanning and posting or emailing his notes. I am very interested to read the crytics. 3. I updated that I could at http://www.ils.uec.ac.jp/~dima/PAPERS/2008analuxp71.pdf bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 05/23/2008, 06:21 AM (This post was last modified: 11/20/2008, 09:26 AM by bo198214.) Kouznetsov Wrote:Could you register as citizendium? It seems to be the only wiki that allows original researches. Hm, yes I will do. However I am not convinced about the concept of Citizendium. I mean if you work at an expert level probably publishing in peer reviewed journals is a more desirable and established way to ensure quality. And if you work on an amateur level then you can put it into Wikipedia. If I had to design and start a project like Citizendium then I just make a wikipedia with the additional feature of digitally signing articles and members. So if someone puts his signature below (a version) of an article then he states by best knowledge that the contents is thorough. And if he puts his signature below a person then he states by best knowing that the person is reliable. Based on whom signs whom and what you can compute a ranking of trustworthiness for articles and persons. Ok, but if someone wants to continue this discussion then please start a new thread in "General discussions and questions". Quote:2. There are two regular superexponentials at base $b$ such that $1. What do you mean by regular? I would deprecate the term regular superexponential except in the sense of "the superexponential retrieved by regular iteration": $\text{sexp}_b(x)=\exp_b^{\circ x}(1)$. Otherwise we impose major confusion on this forum. I think your use of "regular" can be replaced by "analytic", or something similar? Quote:I have plotted the only one, $F$ such that $F(0)=1$. At $b=\sqrt{2}$, for example, $\lim_{x \rightarrow \infty} F_{\sqrt{2}}(x+{\rm i}y)=2$; $\lim_{x \rightarrow -\infty} F_{\sqrt{2}}(x+{\rm i}y)=4$. This contradicts $F(z^\ast)=F(z)^\ast$? How does it come anyway that you now can compute tetration also for bases $? Quote:There is another one, $G$ that grows up along the real axis faster than any exponential and aproaches its limiting values in the opposite direction. But if we use base $\sqrt{2}$ it can not grow to infinity, it has to be limited by $2$ on the real axis. Or what exactly do you mean? Quote:I am writing source for its evaluation. Did you read my previous post? I asked you to send me some code of your computations. Quote:3. Then we have covered the ranges $1 and $b>\exp(1/\rm e)$; and I think about cases $b=\exp(1/\rm e})$ and $b<1$. Yes, we would have covered them in one way. However if you read through this forum we have established several other (at least 3) ways to compute analytic tetration. And we even dont know yet which are equal or which are equal to yours. Quote: I suggest that you use the same idea: first, find the asymptotics and periodicity (if any); then recover the analytic function with these properties. Could you calculate some pictures (similar to those I have posted) for these cases? As I said: for the case of tetration by regular iteration, which is applicable in the range $e^{-e}, I can (and did) compute pictures. Apropos pictures: I think you have to explain something more about your second posted picture. Kouznetsov Fellow Posts: 151 Threads: 9 Joined: Apr 2008 05/23/2008, 08:48 AM Dear Bo. I am glad to see your interest. I cut your post for pieces and try to answer one by one. bo198214 Wrote:Quote:2. There are two regular superexponentials at base $b$ such that $1. What do you mean by regular? I would deprecate the term regular superexponential except in the sense of "the superexponential retrieved by regular iteration": $\text{sexp}_b(x)=\exp_b^{\circ x}(1)$. Otherwise we impose major confusion on this forum. I think your use of "regular" can be replaced by "analytic", or something similar?Yes. Sorry about slang. Regular means that $F(z)$ has no singularities at $\Re(z)>-2$. Quote:Quote:I have plotted the only one, $F$ such that $F(0)=1$. At $b=\sqrt{2}$, for example, $\lim_{x \rightarrow \infty} F_{\sqrt{2}}(x+{\rm i}y)=2$; $\lim_{x \rightarrow -\infty} F_{\sqrt{2}}(x+{\rm i}y)=4$. This contradicts $F(z^\ast)=F(z)^\ast$? No. $F_{\sqrt{2}}(x+{\rm i}y)$ approaches the limiting values at any real $y$. (in spite the cutlines!) Quote:How does it come anyway that you now can compute tetration also for bases $? Yes, I can. You can too. I tried to post the code as "source" together with the picture, but it was not accepted... I shall post it in different way. Quote:Quote:There is another one, $G$ that grows up along the real axis faster than any exponential and aproaches its limiting values in the opposite direction. But if we use base $\sqrt{2}$ it can not grow to infinity, it has to be limited by $2$ on the real axis. Or what exactly do you mean? Yes, it is limited, you can see, it approaches 2 at the picture. But if we withdraw the equation $F(0)=1$ and replace it with requirement $F(z) = 2 + \mathcal O (\exp(Qz))$ at $\Re(z)\rightarrow -\infty$, then the "second" tetration can be plotted, let us call it $G$; and it should be so beautiful, as tetration $F$ for $b=2$ and $b=\rm e$. It is that I expect, I did not yet plot it. Quote:Quote:I am writing source for its evaluation. Did you read my previous post? I asked you to send me some code of your computations. Oops... I see, I used to read it "by diagonal". I should stop to write new codes and share that I already have. I know that usually the "self-made" codes dislike to run at another computer, so, let us do it with few steps, reproducing the figures one by one. Let us begin with very simple code. Please, reproduce first Figure 1 from the source posted at http://en.citizendium.org/wiki/Image:Exa...nLog01.png please, tell me how does it run at your computer and send me (or post) the resulting picture. It is supposed to be identical with the eps file I got at my computer. I begin with so simple picture because its source is in "one piece" and does not require any input files. Quote:Quote:3. Then we have covered the ranges $1 and $b>\exp(1/\rm e)$; and I think about cases $b=\exp(1/\rm e})$ and $b<1$. Yes, we would have covered them in one way. However if you read through this forum we have established several other (at least 3) ways to compute analytic tetration. And we even dont know yet which are equal or which are equal to yours. Yes, but they seem to be singular.. What example would you suggest to begin with? "Equal to mine" is tetration that has no singularities at the right hand side of the complex halfplane. I did not see any map of real and imaginary parts, nor those of modulus and phase. Tell me if I am wrong. Quote:Quote:I suggest that you use the same idea: first, find the asymptotics and periodicity (if any); then recover the analytic function with these properties. Could you calculate some pictures (similar to those I have posted) for these cases? As I said: for the case of tetration by regular iteration, which is applicable in the range $e^{-e}, I can (and did) compute pictures. Ah! very good! Will you plot the real and imaginary parts in the complex plane? Then any difference (even exponentially-small) leads to singularities which are easy to see. Quote:Apropos pictures: I think you have to explain something more about your second posted picture. Yes. But let us begin with Fig.1. I expect, is trivial and you can easy reproduce it. Please, confirm, that you can do it namely with my code; if we need any adjustment, it is important to reveal this with simplest possible case. While I prepare the description of Figure 2 and post its source. « Next Oldest | Next Newest »

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