Hi -
I'm not really happy to start a new thread here on a subject, which we had discussed several times and in some depth. But I am not able to follow the discussion of Henryk and Dmitri, that's too elaborate maths currently.
So my *basic* fiddlings/reinventing the wheels(?) may be inappropriate in the recent uniqueness-threads - but well, one moderator may move this post to an appropriate place...
Ok, so much as a foreword.
(I'm reproducing a msg to sci.math here, which I sent today)
I'm still trying to learn more about this uniqueness thing. This time I've considered to look at the problem via the function graphs.
For instance, for the function f(x) = exp(x) the function f°0.5(x) should have a curve somehow between the f°0(x) = x and the f°1(x) = exp(x) lines/curves.
I thought about creating the trajectory for f°0.5(x) depending on one initial setting, say f°0.5(1) = 1 .
The trajectories of iterates of f°(h+0.5)(x) give then more or less smooth curves (when interpolated). From the graphs it is obvious, that there is one initial setting which gives the smoothest result.
Images for various initial settings (-1,-0.7,-0.37,-0.2,0)
To have better graphs I used f(x) = exp(1/2 x)
See images below.
The best "smoothness", on one hand, seems simply that, where the derivatives are monotonuous.
But the "smoothness" can also be expressed by the sum (or integral?) of the area of the rectangles which occur, when the trajectories f°h(x) and f°h(f°0.5(x)) are drawn and the vertical lines at x=h and x=h+1 and the horizontal lines at the according y-values are taken as borders of rectangles (thin green and blue lines)
The worst initial settings give rectangles all of area zero, so I assume, the best guess (smoothest graph) gives the maximal sum of areas, so the "unique best definition of f°0.5(x)" is then that, which gives the highest area/integral (or something related to this)
Seems to be a useful/meaningful uniqueness-criterion?
From this would then also occur the idea to use the maximizing criterion on some integral-formula for that criterion (and possibly for the general case).
Gottfried
![[Image: attachment.php?aid=439]](http://math.eretrandre.org/tetrationforum/attachment.php?aid=439)
I'm not really happy to start a new thread here on a subject, which we had discussed several times and in some depth. But I am not able to follow the discussion of Henryk and Dmitri, that's too elaborate maths currently.
So my *basic* fiddlings/reinventing the wheels(?) may be inappropriate in the recent uniqueness-threads - but well, one moderator may move this post to an appropriate place...
Ok, so much as a foreword.
(I'm reproducing a msg to sci.math here, which I sent today)
I'm still trying to learn more about this uniqueness thing. This time I've considered to look at the problem via the function graphs.
For instance, for the function f(x) = exp(x) the function f°0.5(x) should have a curve somehow between the f°0(x) = x and the f°1(x) = exp(x) lines/curves.
I thought about creating the trajectory for f°0.5(x) depending on one initial setting, say f°0.5(1) = 1 .
The trajectories of iterates of f°(h+0.5)(x) give then more or less smooth curves (when interpolated). From the graphs it is obvious, that there is one initial setting which gives the smoothest result.
Images for various initial settings (-1,-0.7,-0.37,-0.2,0)
To have better graphs I used f(x) = exp(1/2 x)
See images below.
The best "smoothness", on one hand, seems simply that, where the derivatives are monotonuous.
But the "smoothness" can also be expressed by the sum (or integral?) of the area of the rectangles which occur, when the trajectories f°h(x) and f°h(f°0.5(x)) are drawn and the vertical lines at x=h and x=h+1 and the horizontal lines at the according y-values are taken as borders of rectangles (thin green and blue lines)
The worst initial settings give rectangles all of area zero, so I assume, the best guess (smoothest graph) gives the maximal sum of areas, so the "unique best definition of f°0.5(x)" is then that, which gives the highest area/integral (or something related to this)
Seems to be a useful/meaningful uniqueness-criterion?
From this would then also occur the idea to use the maximizing criterion on some integral-formula for that criterion (and possibly for the general case).
Gottfried
Gottfried Helms, Kassel