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From: Osher Doctorow on 15 May 2010 20:04
From Osher Doctorow

With t and u independent time dimensions, then the equation discussed
in recent posts:

1) y = y(t, u) = y(0)exp(kt) - y(0)exp(ku), k constant > 0

has some interesting properties. If both t and u increase in our
local part of the Universe at least, then for 0 < = u < = t, we have
the following:

2) For t, u increasing and 0 < = u < = t, y in (1) can assume any
values from 0 to infinity by the continuity of the exponential
function. For example, if t = u, then y = 0, while if u << t (is much
less than t), then y --> infinity under a wide variety of conditions.

3) y can refer to the "radius" or "principal radius" or "scale factor"
of the Universe, so that (1) is its expansion under a wide variety of
conditions.

4) The equation (1) is derived from Dt(y) + Du(y) = ky, which it
satisfies by substitution of (1) into the latter equation.

Osher Doctorow
From: Osher Doctorow on 15 May 2010 20:15
From Osher Doctorow

Notice that if we define:

1) y1 = exp(kt), y2 = exp(ku)

then we have:

2) Dt(y1) = kexp(kt) = ky1, Dtt(y1) = k2(exp(kt)), both > 0 for k > 0
3) Similarly for Dt(y2), Dtt(y2)

Thus, for y as above in terms of y1 and y2:

4) y = y(0)[y1 + y2]

under a wide variety of conditions, y is accelerating. Depending on
the size of k and on how close u is to t, this acceleration can go
from near 0 to infinity. We might even have k variable and depending
on some other quantity.

Osher Doctorow
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