Dearest Administrator and Dear Friends !
Concerning the behaviour of the tetrational function y = b # x, in the domain of base b where b >= e^(1/e), i.e. Eta (;->), as described according to the various models and approximations discussed in this Forum, I think that it would be interesting to start putting together an overall general picture. In fact, in this domain, y = b # x is (or it seems to be) a one-value always increasing smooth "function". The "general behaviour" of this "function" seems to show first derivatives y' with one minimum and, accordingly, second derivatives y" with one zero point, corresponding to one inflection point of y.
Moreover, it can be shown that, for bases b "near to" (and greater than) e^(1/e), the "critical path" can be chosen to be approximatly linear (see the attached notes). In particular, for exactly b = Eta = e^(1/e), this approximation can be made with any required precision.
In this respect, I should like to draw your attention to the fact that, in all the most effective implementations, approximations or simulations of y = b # x (b-tetra-x) presented in the Forum, its second derivative (y") shows one zero (y" = 0) for values of x strongly depending from base b.
In particular, I qualitatively detected that we have y" = 0:
- for b = e^(1/e) = 1.44466.., when x -> +oo
- for b = 1.47 when x -> 31 (about)
- for b = 2 at x = 0
- for b = e at x = - 0.5 (about?)
- for b > 10 when x -> - 1
The attached notes are only a provocation and were prepared for inviting the Participants to a deeper analysis of these interesting aspects of the problem (if this had not already been done).
To be more precise (and ... serious), I should like to see with your collaboration if we can obtain the "exact" coordinates of the following points, concerning y = e # x (e-tetra-x). In fact, looking at the y = e # x function, as described by Andrew, we can see that:
- y" = 0 for x = - 0.5 (perhaps ..., to be verified); or:
- y" = 0 for x = - 0.4446678.. (...who knows!)
- y'(-1) = y'(0) (this can be demonstrated);
- y'(-1) = y'(0) > 0 (this needs a demonstration);
- y'(-0.5) < 0 (idem, as before).
The same observations on the "andydude's" plots show that, concerning y = 2 # x (2-tetra-x), we might have:
- y" = 0 for x = 0 (perhaps again ..., to be verified).
Can somebody confirm or infirm (!) these observations/conjectures of mine? They seem to be superficial observations, but they are not. In fact, they may suggest some very interesting new research strategies.
Thank you for your interest.
GFR
Concerning the behaviour of the tetrational function y = b # x, in the domain of base b where b >= e^(1/e), i.e. Eta (;->), as described according to the various models and approximations discussed in this Forum, I think that it would be interesting to start putting together an overall general picture. In fact, in this domain, y = b # x is (or it seems to be) a one-value always increasing smooth "function". The "general behaviour" of this "function" seems to show first derivatives y' with one minimum and, accordingly, second derivatives y" with one zero point, corresponding to one inflection point of y.
Moreover, it can be shown that, for bases b "near to" (and greater than) e^(1/e), the "critical path" can be chosen to be approximatly linear (see the attached notes). In particular, for exactly b = Eta = e^(1/e), this approximation can be made with any required precision.
In this respect, I should like to draw your attention to the fact that, in all the most effective implementations, approximations or simulations of y = b # x (b-tetra-x) presented in the Forum, its second derivative (y") shows one zero (y" = 0) for values of x strongly depending from base b.
In particular, I qualitatively detected that we have y" = 0:
- for b = e^(1/e) = 1.44466.., when x -> +oo
- for b = 1.47 when x -> 31 (about)
- for b = 2 at x = 0
- for b = e at x = - 0.5 (about?)
- for b > 10 when x -> - 1
The attached notes are only a provocation and were prepared for inviting the Participants to a deeper analysis of these interesting aspects of the problem (if this had not already been done).
To be more precise (and ... serious), I should like to see with your collaboration if we can obtain the "exact" coordinates of the following points, concerning y = e # x (e-tetra-x). In fact, looking at the y = e # x function, as described by Andrew, we can see that:
- y" = 0 for x = - 0.5 (perhaps ..., to be verified); or:
- y" = 0 for x = - 0.4446678.. (...who knows!)
- y'(-1) = y'(0) (this can be demonstrated);
- y'(-1) = y'(0) > 0 (this needs a demonstration);
- y'(-0.5) < 0 (idem, as before).
The same observations on the "andydude's" plots show that, concerning y = 2 # x (2-tetra-x), we might have:
- y" = 0 for x = 0 (perhaps again ..., to be verified).
Can somebody confirm or infirm (!) these observations/conjectures of mine? They seem to be superficial observations, but they are not. In fact, they may suggest some very interesting new research strategies.
Thank you for your interest.
GFR
