Concerning my "old" comment about "parametration" of the perimeter of the h/sup and h/inf zone, for 0 < b < e^(-e):
a) - Let us define the joint union of the two real branches of the Lambert Function (the product-logarithm) as: plog(z) = {W/-1 (z), W/0 (z)}, where W/-1 and W/0 are "Mathematica's" ProductLog[-1,z] and ProductLog[x], respectively.
b) - As we know, a formula for h(b), the height of the "infinite towers", is given by h = plog(-ln b)/(-ln b), inverse of b = h^(1/h), the selfroot function. We also know that this expression is not sufficient to describe what happens at 0 < b < e^(-e), where a zone suddenly appears, indicating the asymptotic max and min values of an oscillating y = b[4]x (real part of a complex function), for b < e^(-e) [my interpretation]. It is also clear that these formulas were obtained taking into consideration y = b[3]y, implying y = b[4]oo, according to the DL.
c) - Now, if we take into consideration:
y = b[3]y = b[3](b[3]y), we can write:
b[3]\y = y = b[3]y, i.e.:
log[/b] y = y = b^y, i.e.:
ln (y^y) = y*(ln b)*e^y*(ln b), i.e.:
y*(ln b) = plog(ln (y^y)), i.e.:
b = e^((plog(ln(y^y)))/y).
I think that this formula is very accurate and, perhaps, the correct one, to be used instead of the simple selfroot. It goes without saying that we need to invert it for obtaining y = h = h(b).
In fact, with plog(z) = W/0 (z), the standard Lambert function, we get the function shown in annex Fig. 1, covering the upper part of the "yellow zone" [b = b(y), with b vertical], for y < 1/e, and the remaining part described by the selfroot, for y > 1/e.
Using the W/-1 branch of the Lambert Function, for y < 1/e, we get the middle b = b(y), also given by the selfroot b = y^(1/y). For y > 1/e, we obtain the second branch of the perimeter of the "yellow zone" (see Fig. 2).
Fig. 3 shows the two plots obtained by b = e^((plog(ln(y^y)))/y), jointly superposed. We see again the continuous line described by the selfroot, together with the perimeter of the "yellow zone", with a maximum at y = 1/e, with b = e^(-e). The plot is extremely similar to what was obtained by Andydude. We also see that b = b(y) = b(h) has two values, for 0 < y < 1.
Fig 4 shows the comparison between my old (insufficient) approximations and my new (I hope correct) formulas, for b = b(y) or b = b(h).
Fig. 5, 6, 7 are plots of low iterated logs and exps (to the base b = 0.025 < e^(-e)), showing the crosspoints for even and odd iterations, as well as the fixpoints, whenever they exist [f(x)=x].
Now, what is needed is to "invert" this description (Fig3) for getting h = h(b). But tis, will probably come next time. Fig. 8 is a qualitative presentation of what I am expecting.
May Day, May Day, please help me!!!
GFR
GFR Wrote:Actually, I should like to propose the following strategy and formulas:andydude Wrote:About GFR's "yellow zone" (YZ) or Ivars' "phase transition region" I was very interested to see the formula \( x = (y^{-y} -1)\frac{e^{-e}}{e^{1/e}-1} \). I think it is a very good approximation, but I think it is not accurate..... you are right! My formula is wrong, because it is only a very approximated .... approximation. The only way, I think, is to take into consideration [/b]log x = b^x. I shall come back to that asap.
a) - Let us define the joint union of the two real branches of the Lambert Function (the product-logarithm) as: plog(z) = {W/-1 (z), W/0 (z)}, where W/-1 and W/0 are "Mathematica's" ProductLog[-1,z] and ProductLog[x], respectively.
b) - As we know, a formula for h(b), the height of the "infinite towers", is given by h = plog(-ln b)/(-ln b), inverse of b = h^(1/h), the selfroot function. We also know that this expression is not sufficient to describe what happens at 0 < b < e^(-e), where a zone suddenly appears, indicating the asymptotic max and min values of an oscillating y = b[4]x (real part of a complex function), for b < e^(-e) [my interpretation]. It is also clear that these formulas were obtained taking into consideration y = b[3]y, implying y = b[4]oo, according to the DL.
c) - Now, if we take into consideration:
y = b[3]y = b[3](b[3]y), we can write:
b[3]\y = y = b[3]y, i.e.:
log[/b] y = y = b^y, i.e.:
ln (y^y) = y*(ln b)*e^y*(ln b), i.e.:
y*(ln b) = plog(ln (y^y)), i.e.:
b = e^((plog(ln(y^y)))/y).
I think that this formula is very accurate and, perhaps, the correct one, to be used instead of the simple selfroot. It goes without saying that we need to invert it for obtaining y = h = h(b).
In fact, with plog(z) = W/0 (z), the standard Lambert function, we get the function shown in annex Fig. 1, covering the upper part of the "yellow zone" [b = b(y), with b vertical], for y < 1/e, and the remaining part described by the selfroot, for y > 1/e.
Using the W/-1 branch of the Lambert Function, for y < 1/e, we get the middle b = b(y), also given by the selfroot b = y^(1/y). For y > 1/e, we obtain the second branch of the perimeter of the "yellow zone" (see Fig. 2).
Fig. 3 shows the two plots obtained by b = e^((plog(ln(y^y)))/y), jointly superposed. We see again the continuous line described by the selfroot, together with the perimeter of the "yellow zone", with a maximum at y = 1/e, with b = e^(-e). The plot is extremely similar to what was obtained by Andydude. We also see that b = b(y) = b(h) has two values, for 0 < y < 1.
Fig 4 shows the comparison between my old (insufficient) approximations and my new (I hope correct) formulas, for b = b(y) or b = b(h).
Fig. 5, 6, 7 are plots of low iterated logs and exps (to the base b = 0.025 < e^(-e)), showing the crosspoints for even and odd iterations, as well as the fixpoints, whenever they exist [f(x)=x].
Now, what is needed is to "invert" this description (Fig3) for getting h = h(b). But tis, will probably come next time. Fig. 8 is a qualitative presentation of what I am expecting.
May Day, May Day, please help me!!!
GFR