Looking a bit deeper at the sigma-function the above plot looked a bit strange to me, when I thought in terms of "height of powertowers": because at x>=t we should have infinite heights, but where is the infinity here? But this resolves (and possibly there is nothing new here for the sigma-experienced reader...)
Let me restate the use of standard variable-names:
Also I use the fixpoint-shift x/t-1 instead of x-t, but that doesn't matter besides the same scaling in the results.
For the example I use here as in the previous msg
From the functional description it should be, that the function sigma decomposes its parameter x into a power of u; additionally it provides some constant c for sigma(1).
If x is seen as a powertower of height h to base b, such that \( x = 1 \lbrace b \rbrace h \) then formally
\( \hspace{24} \sigma(x ) = \sigma(1 \lbrace b \rbrace h) = u^h * \sigma(1) \)
where sigma(1) is a constant (call it "c") where c=-0.327639416789 (If the fixpoint-shift "x-t" is done then c1=-0.540186075580 = t*c)
So the sigma-function decomposes x, replaces b by u and replaces height by exponent. Thus we can get the height-parameter h simply by \( \log(\sigma(x)/c)/ \log(u) \)
Obviously, for x increasing to \( (1 \lbrace b \rbrace \infty) ((= t)) \) we get
\( \hspace{24} \lim_{x->t} \sigma(x) = \sigma(1 \lbrace b \rbrace \infty) \)
\( \hspace{24} =\lim_{h->\infty} u^h * c = 0 \)
and for all u with -1 < u < 1 this converges to zero, so in the plot we have the crossing with y-axis at x = t ((~ 1.6487))
The nice feature is, that the height-parameter does not occur explicitely in the sigma-function; otherwise we were lost in the case where x>t; this would require then a more-than-infinite height.
If we test one x>t we see, that sigma(x) gets positive, so sigma(x)/c is negative and the height h becomes complex due to log of a negative value. But the plot seems to be smooth in the vicinity of x=t; so if |u|<1 there seems to be a compensation of small u against the change of infinite real height to complex heights - I'll look at it later...
For bases b, where u>1 we need no infinite real heights nor complex heights to arrive at an arbitrary high real x; here all heights should be real.
So far for this msg.
(the data for the plot were computed in steps of h where h=0,1,2,...,32 and x=1 {b} 0,1 {b} 1, 1 {b} 2,...)
Let me restate the use of standard variable-names:
Code:
base = b,
t = h(b) such that b=t^(1/t) ,
u = log(t)
h = height of the powertowerAlso I use the fixpoint-shift x/t-1 instead of x-t, but that doesn't matter besides the same scaling in the results.
For the example I use here as in the previous msg
Code:
u=0.5, t=exp(0.5)~1.64872127070, b~1.35427374603From the functional description it should be, that the function sigma decomposes its parameter x into a power of u; additionally it provides some constant c for sigma(1).
If x is seen as a powertower of height h to base b, such that \( x = 1 \lbrace b \rbrace h \) then formally
\( \hspace{24} \sigma(x ) = \sigma(1 \lbrace b \rbrace h) = u^h * \sigma(1) \)
where sigma(1) is a constant (call it "c") where c=-0.327639416789 (If the fixpoint-shift "x-t" is done then c1=-0.540186075580 = t*c)
So the sigma-function decomposes x, replaces b by u and replaces height by exponent. Thus we can get the height-parameter h simply by \( \log(\sigma(x)/c)/ \log(u) \)
Obviously, for x increasing to \( (1 \lbrace b \rbrace \infty) ((= t)) \) we get
\( \hspace{24} \lim_{x->t} \sigma(x) = \sigma(1 \lbrace b \rbrace \infty) \)
\( \hspace{24} =\lim_{h->\infty} u^h * c = 0 \)
and for all u with -1 < u < 1 this converges to zero, so in the plot we have the crossing with y-axis at x = t ((~ 1.6487))
The nice feature is, that the height-parameter does not occur explicitely in the sigma-function; otherwise we were lost in the case where x>t; this would require then a more-than-infinite height.
If we test one x>t we see, that sigma(x) gets positive, so sigma(x)/c is negative and the height h becomes complex due to log of a negative value. But the plot seems to be smooth in the vicinity of x=t; so if |u|<1 there seems to be a compensation of small u against the change of infinite real height to complex heights - I'll look at it later...
For bases b, where u>1 we need no infinite real heights nor complex heights to arrive at an arbitrary high real x; here all heights should be real.
So far for this msg.
(the data for the plot were computed in steps of h where h=0,1,2,...,32 and x=1 {b} 0,1 {b} 1, 1 {b} 2,...)
Gottfried Helms, Kassel