02/03/2012, 08:07 PM
Hello Dmitri -
But that was not true - but the difference to the expected value of zero was systematically distorted which shows a sinusoidal curve. I expect, that that curve has some sinusoidal function and that this function might be remotely related to that integral which we need if we do Ramanujan-summation.
I have described this more precisely at
http://go.helms-net.de/math/tetdocs/Tetr...roblem.pdf
Some more introductory remarks to the concept of iteration-series are in the introductory remarks at my tetration-homepage at http://go.helms-net.de/math/tetdocs
So it has the known deficients of the Carleman-matrix-approach:
a) there is nothing known yet which would make the Carlemanmatrix-approach a unique preferable solution,
b) the solution for the fractional iterates is dependent on the fixpoint which was chosen to center the power series around, and
c) we get complex-valued power series for real valued fractional heights.
Because of that unsolved problems I've put my studies on low energy and I hoped, that your solution would come out as *the* general (and generally accepted) method for the tetration. I'd really like to see that this would happen!
It's very nice to hear from you now!
Gottfried
(02/03/2012, 06:19 PM)Kouznetsov Wrote:My conjecture, based on standard matrix-identities which I assumed could be extended to the case of infinite size, was the following. I constructed the doubly-infinite series of powertowers of increasing/decreasing height, so for the index/height h of -infinity to +infinity. By the matrix-identities (involving the von-Neumann-series of the according Carlemanmatrix and its inverse) I expected that the sum of that doubly-infinite series was always zero.(02/03/2012, 04:47 PM)Gottfried Wrote: ..But I did not find yet an appropriate expression for this in the case of the iteration-series of powertowers/tetration. ..?
But that was not true - but the difference to the expected value of zero was systematically distorted which shows a sinusoidal curve. I expect, that that curve has some sinusoidal function and that this function might be remotely related to that integral which we need if we do Ramanujan-summation.
I have described this more precisely at
http://go.helms-net.de/math/tetdocs/Tetr...roblem.pdf
Some more introductory remarks to the concept of iteration-series are in the introductory remarks at my tetration-homepage at http://go.helms-net.de/math/tetdocs
Quote:Does your method allow to evaluate tetration faster (or more precise) than my one?No, this is just the (re-)discovery of the concept of "Carleman-matrix" which I did not know when I came across the problem of tetration and my general matrix-approach to some number-theoretic problems.
So it has the known deficients of the Carleman-matrix-approach:
a) there is nothing known yet which would make the Carlemanmatrix-approach a unique preferable solution,
b) the solution for the fractional iterates is dependent on the fixpoint which was chosen to center the power series around, and
c) we get complex-valued power series for real valued fractional heights.
Because of that unsolved problems I've put my studies on low energy and I hoped, that your solution would come out as *the* general (and generally accepted) method for the tetration. I'd really like to see that this would happen!
Quote:There are already two wikis about tetration,Yes I know them; however their ambition seem to be very high and I have only undergraduate courses in mathematics in computer-science in the 70ties, and my number-theory-knowledge is amateurishly compiled single-topic-material. So I felt I could not contribute on that formal/rigorous level of definition and knowledge. But perhaps there is something I can help with...
http://tori.ils.uec.ac.jp/TORI/index.php/Tetration
http://math.eretrandre.org/hyperops_wiki
Quote:I tried to create an account at the second one and failed. I see you are successful.. Did you do it by yourself or Henryk had created your account?Well, that was Henryk - now that you call for him I'm getting aware that we've heard little or nothing of him lastly...
I should greatly appreciate your comments about http://tori.ils.uec.ac.jp/TORI
It's very nice to hear from you now!
Gottfried
Gottfried Helms, Kassel
