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) |

RE: Dmitrii Kouznetsov's Tetration Extension - bo198214 - 05/18/2008
Kouznetsov Wrote: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 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 or you have to add the corresponding 1's, as you mention before . But that didnt injure the readability.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 ? 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. 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 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 , but it follows from the analyticity and the assumption that is real at z>-2.Oh I wasnt aware of this. RE: Dmitrii Kouznetsov's Tetration Extension - Kouznetsov - 05/21/2008
Kouznetsov Wrote:bo198214 Wrote:Dmitrii, I am currently programming your tetration extension.Bo, I am glad to read from you! 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 at http://en.citizendium.org/wiki/User:Dmitrii_Kouznetsov/Analytic_Tetration#Small_base._Base Could you plot Fig.3 ? Would you like to take the text and write the rest? RE: Dmitrii Kouznetsov's Tetration Extension - bo198214 - 05/21/2008
Kouznetsov Wrote:Dear Bo:Its no secret: Henryk Trappmann. Quote:2. I have described behavior of tertation at the 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. RE: Dmitrii Kouznetsov's Tetration Extension - Kouznetsov - 05/21/2008
... 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? RE: Dmitrii Kouznetsov's Tetration Extension - bo198214 - 05/22/2008
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. Seems more to be about historic texts isnt it? RE: Dmitrii Kouznetsov's Tetration Extension - Kouznetsov - 05/22/2008
bo198214 Wrote:1. Yes, it is. I agree. Could you register as citizendium? It seems to be the only wiki that allows original researches.Kouznetsov Wrote:Henrik, sorry, I was sure that the page is available from anywhere. 2. There are two regular superexponentials at base such that . I have plotted the only one, such that . At , for example, ; . There is another one, 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 and ; and I think about cases and . 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? RE: Dmitrii Kouznetsov's Tetration Extension - andydude - 05/22/2008
Also, formula (1. uses "x" where the function takes "z", have you fixed this yet? Andrew Robbins RE: Dmitrii Kouznetsov's Tetration Extension - Kouznetsov - 05/22/2008
andydude Wrote:Also, formula (1. uses "x" where the function takes "z", have you fixed this yet? 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 RE: Dmitrii Kouznetsov's Tetration Extension - bo198214 - 05/23/2008
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 such that . What do you mean by regular? I would deprecate the term regular superexponential except in the sense of "the superexponential retrieved by regular iteration": . 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, such that . This contradicts ? How does it come anyway that you now can compute tetration also for bases ? Quote:There is another one, 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 it can not grow to infinity, it has to be limited by 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 and ; and I think about cases and .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 , I can (and did) compute pictures. Apropos pictures: I think you have to explain something more about your second posted picture. RE: Dmitrii Kouznetsov's Tetration Extension - Kouznetsov - 05/23/2008
Dear Bo. I am glad to see your interest. I cut your post for pieces and try to answer one by one. bo198214 Wrote:Yes. Sorry about slang. Regular means that has no singularities at .Quote:2. There are two regular superexponentials at base such that .What do you mean by regular? I would deprecate the term regular superexponential except in the sense of "the superexponential retrieved by regular iteration": . Quote:No. approaches the limiting values at any real . (in spite the cutlines!)Quote:I have plotted the only one, such that .This contradicts ? 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:Yes, it is limited, you can see, it approaches 2 at the picture. But if we withdraw the equation and replace it with requirement at , then the "second" tetration can be plotted, let us call it ; and it should be so beautiful, as tetration for and .Quote:There is another one, 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 it can not grow to infinity, it has to be limited by on the real axis. Or what exactly do you mean? It is that I expect, I did not yet plot it. Quote: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:ExampleEquationLog01.pngQuote:I am writing source for its evaluation.Did you read my previous post? I asked you to send me some code of your computations. please, tell me how does it run at your computer and send me (or post) the resulting picture. It is supposed to be <b>identical</b> 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:Yes, but they seem to be singular.. What example would you suggest to begin with?Quote:3. Then we have covered the ranges and ; and I think about cases and .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. "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: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: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 , I can (and did) compute pictures. 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. |