From Osher Doctorow
From the last Section, using k = 2 (other models could use k > 2), the
time-probability equation during Inflation is (taking logarithms of
both sides of the exponential-quadratic equation there):
1) t = log[(p-1)^2 + 1], 0 < = p < = 1, 0 < = t < = log(2)
As usual in this type of model, we assume that time t > = 0. The
probability p is Probable Influence/Causation (PI).
Readers shoul graph or sketch t = log(p) and t = log(p^2) = 2log(p) and
t = log(p - 1), although they may be in for a surprise since t in (1)
decreases with p in the indicated domain and range (t in [0, log(2)], p
in [0, 1]) unlike t = log(p).
Although y = log(x) in general for real variables x, y is an increasing
function of x, the restrictions on t and p in (1) make t in (1) a
decreasing function of p in the indicated domain/range. For example,
when p = 0, then t = log(2), while when p = 1, then t = log(1) = 0.
The derivative dt/dp helps also in sketching or graphing since it is
2(p-1)/[(p-1)^2 + 1] which is always negative except at p = 1 since p <
= 1.
From the original equations (1) to (15) of the last few postings, we
can also see directly that s and t are increasing functions of each
other and are decreasing functions of p.
Osher Doctorow
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