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 short compilation:fractional iteration-> eigendecomposition Gottfried Ultimate Fellow Posts: 765 Threads: 119 Joined: Aug 2007 01/21/2008, 06:45 PM (This post was last modified: 01/21/2008, 08:17 PM by Gottfried.) I was considering remark of Dan Asimov, which Henryk cited in the thread "Bummer". Using my conversion-formula from T-tetration to U-tetration, which involves the fixpoints of the tetration-base b and application of the powerseries formula in my iteration-article, which I've announced here, I tend to conclude, that Asimov is right - because I get different powerseries, if I take different fixpoints, and where the related terms do not cancel pairwise. Let b = t1^(1/t1) = t2^(1/t2) , u1=log(t1) and u2 = log(t2). Then with the example b=sqrt(2), t1=2, u=u1=log(2), t2=4, u2=2*u and the conversion Tb°h(x) = (Ut°h(x/t-1)+1)*t I get for fixpoints t1 and t2 (and x=1 which is the assumption for tetration) the equation in question (U2°h(-1/2)+1)*2 =?= (U4°h(-3/4)+1)*4 or rewritten U2°h(-1/2) =?= U4°h(-3/4) + 1 The powerseries in u1 and u1^h resp u2 and u2^h seem so different, that one may possibly derive an argument from that, although I have not yet a convincing idea. Also possibly any general property of function-theory may be applicable then (for instance U2 is convergent and U4 is divergent, if I have it right). See a better formatted short statement of this speculation here [update] In a second view,and recalling the discussion and computations that we (and also myself) already had about the topic, the question is, why then is the difference so small? U_2 provides a convergent, U_4 a divergent series: the differences should be huge, if there are some, but: they are small. Hmmm... Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Messages In This Thread short compilation:fractional iteration-> eigendecomposition - by Gottfried - 01/19/2008, 09:37 AM RE: short compilation:fractional iteration-> eigendecomposition - by andydude - 01/19/2008, 09:50 AM RE: short compilation:fractional iteration-> eigendecomposition - by Gottfried - 01/19/2008, 11:17 AM RE: short compilation:fractional iteration-> eigendecomposition - by bo198214 - 01/19/2008, 01:20 PM RE: short compilation:fractional iteration-> eigendecomposition - by Gottfried - 01/19/2008, 02:29 PM RE: short compilation:fractional iteration-> eigendecomposition - by Gottfried - 01/20/2008, 02:32 PM RE: short compilation:fractional iteration-> eigendecomposition - by Gottfried - 01/20/2008, 07:50 AM RE: short compilation:fractional iteration-> eigendecomposition - by Gottfried - 01/21/2008, 06:45 PM RE: short compilation:fractional iteration-> eigendecomposition - by Gottfried - 01/23/2008, 07:35 AM

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