[Regular tetration] [Iteration series] norming fixpointdependencies  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: [Regular tetration] [Iteration series] norming fixpointdependencies (/showthread.php?tid=483) Pages:
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[Regular tetration] [Iteration series] norming fixpointdependencies  Gottfried  07/28/2010 If we do regular tetration having a "nice" base 1<b<eta we have two real fixpoints, fp0,fp1. So, for base b=sqrt(2) we have fp0=2 and fp1=4 . Iterations on the real axis constitute at a first sight three different segments Code: . seg1 seg2 seg3 For some x2 in seg2 we can iterate with arbitrary heights to some y2, but again are confined to y2 in seg2. For some x3 in seg3 we can iterate with arbitrary negative height to some y3 and in principle also to arbitrary positive heigt, but practically encounter numerical overflow very soon. The powerseries for regular tetration can be developed around fp0 or fp1. Let's call that tet0 and tet1 for shortness. Then if we look at the schroederfunction of tet0 for all x1 we get negative values and for all x2 we get positive values. Thus we can map the set of seg1 to that of seg2 by negating the schroedervalue. This means for instance, that x1 = 1 gets mapped to x2 = 2.467914... and because the heightlimits of seg2 are infinity and +infinity we could use that value x2 to define a norm for that segment, so in seg2 the value x2=2.467914 could be said has (real) height 0 by definition. We have a similar problem with seg3: here too we have infinity at both heightlimits. But we can repeat the normingprocess, now using the schroedervalues of tet1. We compute the schroedervalue of x2 using tet1 and compute x3 by negating that value. However, without further measures we get infinity here. If we reduce the height of x2 by 2, then we get x3 = 417.234406762 So we have the segments with the normed heights Code: . seg1 seg2 seg3 Unfortunately this has two asymmetries: the tet0 and tet1 have somehow opposite sign; but more inconvenient is, that we cannot have the same heightnorm. What we can do is to shift left and use x1=0 as reference. We get then Code: . x1 = 0 x2 = 2.606584 (x3=417.2344) and still x3 computed by x2 seems to become infinite. We may reduce again x2 by height 1 to get the usable x3value of 417.2344... We cannot reduce x1 by one more height, but my proposal here is to use x1 = b^^1.5 as reference value. Then we have, for base b=sqrt(2) the referencevalues for height 1.5 Code: . x1 = 1.33729937324 x2 = 2.68345013524 x3 = 3465302.30778 The inversion of sign of the schroederfunctionvalue is essentially the iteration with an imaginary height. For notation I introduce now u0 = ln(fp0) and u1 = ln(fp1) If we have, for some x, the schroedervalue s, then the schroedervalue of the h'th iterate of x is s*u^h and the negation of sign can be achieved by supplying the according complex value in h. Using the different fixpoints and different u0 and u1 we can state this norming more explicitely Code: . x1 = tet0(1, 1.5) What is now interesting is, whether the observed wobbling of the tetrates in seg2 using the different fixpoints changes in some interesting way. I remember that the shifting of the height by a halfunit made some significant change in the wobbling when I considered the infinite alternating iteration series (tetraseries) in one of my older msgs, I'll have a look at it soon. Gottfried RE: [Regular tetration] norming fixpointdependencies  Gottfried  07/28/2010 I have added a plot of the principle of norming. For the plot I took my old norming where I identify height h=0 in the middle segment by Code: . [attachment=725] Gottfried RE: [Regular tetration] norming fixpointdependencies  bo198214  07/29/2010 (07/28/2010, 09:49 PM)Gottfried Wrote: I have added a plot of the principle of norming. Ah, that makes your idea more accessible. RE: [Regular tetration] norming fixpointdependencies  Gottfried  07/29/2010 Another picture which shows the wobble of different values for the regular heightfunction when fp0 or fp1 is used. Example: base b=sqrt(2). Then the value "normzero" is the map of 1 into the segment seg1 between 2 and 4 using fp0powerseries and has referenceheight 0 in that segment. It is that value of 2.46... = tet0(1,Pi*I/ln(fp0)) To have computations numerically nearer at the fixpointvalue 2, I increase its height (tetrate it using tet0) by 13.5. Then I generate a set of xcoordinates in small steps in the heightinterval hgh0(x)= 13... 17. Now I determine the heights of these xcoordinates using the hgh1()function which employs the second fixpoint fp1. Then the heightvalues using hgh0(x) and hgh1(x) differ periodically by small differences of about 1e25. This is the basic idea of the curves in the plot. But we find, that the norming process has more implications. If we connect the tet0 and tet1functions using a common x at a fractional iterate from "normzero", then the differencecurve becomes asymmetric. Examples: if we use the connectionvalue at tet0(normzero,+0.25) all differences are positive, if at tet0(normzero,+0.5) we have nearly the same curve as with tet0(normzero,0) itself, and if we connect tet0 and tet1 at tet0(normzero,+0.75) all differences become negative. So the selection of the connectionpoint for the norming is an important aspect. However, the matter is not yet satisfactorily solved: still we have a small (but seemingly constant ~ 2e26) difference of the curves for connectionpoint tet0(normzero,+0) and tet0(normzero,+0.5). So the wobbling is not exact the same even at halfinteger steps of the connectionpoint. Gottfried [attachment=726] RE: [Regular tetration] norming fixpointdependencies  Gottfried  08/29/2010 Just came across an older subject and thought it would fit into this "norming"thread. As older fellows here may remember, nearly my first contact with tetration was the question of alternating iterationseries for which I worked out some interesting heuristics. (see [1] and [2]) [update] I should explain, that for convenient asciinotation of the tetration I "misuse" here the common notation. With z^^h I mean in the context of a given fixed base b, the value of z^^h := [/update] Using base b=sqrt(2) we have the realvalued interval 2..4 for which we may find iteration heights from inf to +inf if we start at some value z, say z=3, in this interval. Because in both direction the values of z^^h are finite we can compute a value for the alternating series of that values. So using Pari/Gp we can compute f(z) = sumalt(h=0,(1)^h*iter(z,h)) + sumalt(h=0,(1)^h*iter(z,h) )  z to evaluate the alternating iterationseries with center at the chosen z. It is clear that this series is periodic for z in the interval z..z^^2 . But what's interesting is, that in general the f(z) is "small" and even we find f(z)=0 Because this is a remarkable result (and matches, for instance, the analogue problem when applied to a doublyinfinite geometric series by analytic continuation) this value z (where f(z)=0) introduces itself gently as candidate for a normvalue, at which the height is defined to be zero or at least an integer. Here is a picture of the sinusoidal curve f(z) when z is moved from z to z^^2 beginning at some arbitrary value z0: [attachment=760] We see that astonishing approximation to a sinecurve, where the amplitude should be normed. Actually the deviance from the sinecurve is of the order of 1e3 : I mean, if the heightparameter of this curve is compared with the abscissa of the sinecurve after the two curves are matched (for instance by binary search of the same yvalues). I'm not experienced with Fourieranalysis, but I think, it would be profitable to try to describe the f(z)function by a fourierdecomposition. Analoguously this could be done for the other bases 1<b<eta. Gottfried [update]: obviously this provides also a "fixpointindependent" definition for the real fractional tetration: just match the values of the sincurve with that of f(z) and define the height h for the representation of the z according to the found abszissa of the sine (though this provides only approximation). [/update] (both articles are *very* freshmanlike and need being improved...) [1] Short article of magazinetype [2] longer version RE: [Regular tetration] norming fixpointdependencies  tommy1729  08/30/2010 let me recap A is carleman of z. B is carleman of base^z. carleman f(z) = A/(1+B) but why is 1+B invertible for bases > eta ?? RE: [Regular tetration] norming fixpointdependencies  Gottfried  08/30/2010 (08/30/2010, 09:32 AM)tommy1729 Wrote: let me recap Let M=(I+B), W = M^1; ß some eigenvalue of B ß>1 , µ =ß+1 eigenvalue of M w = 1/(1+ß) the according eigenvalue of W
No proof yet RE: [Regular tetration] norming fixpointdependencies  tommy1729  08/30/2010 ok that makes some sense. now explain me what you are doing in this thread and WHY ? and WHY is f(z) periodic ??? how to compute its period ?? this reminds me of my thread " 452 pi " where i conjectured a period. maybe related ?? regards tommy1729 RE: [Regular tetration] norming fixpointdependencies  Gottfried  08/30/2010 (08/30/2010, 11:08 AM)tommy1729 Wrote: ok that makes some sense.Using base b= sqrt(2)
Quote:and WHY is f(z) periodic ??? Code: ´ f(z) and f(y)=f(z^^2) are equal => f(z) is periodic in terms of the heightparameter Quote:how to compute its period ?? f(z) = f(z^^2) where z^^2 = b^b^z ==> periodlength is delta_h = 2 It is more difficult to find the amplitude RE: [Regular tetration] norming fixpointdependencies  tommy1729  08/30/2010 ahh i think .... f(z) is NOT periodic , ONLY in the interval [z,z^^2] IF f(z) = 0. meaning that not only f(z) = f(z^^2) = 0 but also f(z+(z^^2z)/2) = = f(z/2 + z^^2 /2) = f(z) = 0. hmm. f(z) = f(z^^2). for all z. f(z) =  f(z^z) right ? but if f(z) = f(z^^2) = 0 then  f(z^z) must be 0 but z^z is not in the middle of z and z^^2. i.e. z<> z^z => z^z =/= (z + z^^2)/2. or ... is that the equation to solve for f(z) = 0 ?? man your f(z) is weird ! 