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From: OsherD on 23 Jan 2010 03:40
From Osher Doctorow

It is generally accepted in physical cosmology that the Universe went
through the radiation-dominated era, then the matter-dominated era,
then the dark matter-dominated era.

This makes the comparison of these different eras in cosmology, for
example in arXiv, very similar to Memory solutions or Memory
Differential-Difference or Delay Equations and Volterra Integral
Equations.

Readers can download free online the pioneering work on Differential-
Difference Equations, namely:

1) Bellman, Richard and Cooke, Kenneth L., "Differential-Difference
Equations," The Rand Corporation Santa Monica California USA, R-374-PR
Jan 1963, www.rc.rand.org, or http://www.rand.org/pubs/reports/2006/R374.pdf.

The book is full of examples as well as theory, and Wikipedia's online
articles on various topics related to them including Delay Equations
are also useful.

One surprising result is that Delay Differential Equations, or
Differential-Difference Equations, tend to have different equations on
different time intervals, although they may be connected at endpoints
of intervals (with however possible discontinuities in derivatives).
This makes them at least conceptually similar to different phases and
different eras of the Universe.

Bellman and Cooke also give theorems on the correspondence between the
differential-difference equations and the Volterra or Volterra-like
integral equations, although the terminology "Volterra" was not used
much by those authors as far as I can determine.

Another interesting topic is to follow the stochastic (probabilistic)
versions of the above types of equations in the research literature in
arXiv. With careful study, this reveals some surprising results
about independence, dependence, Markovian, and Non-Markovian
scenarios, as for example the French researchers Serge Cohen, Fabien
Panloup, arXiv: 0912.2889 v1 [math.PR] 15 Dec 2009, 20 pages, U. de
Toulouse France, "Existence of the stationary regime of a non-
Markovian stochastic differential equation," where they find that as
soon as Gaussian processes have dependent instead of independent
increments, the initial random conditions and the driving process of
stationary solutions change from independent to dependent!

Osher Doctorow
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