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C00002 00002		For hyperbolic differential equations, one gives initial
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	For hyperbolic differential equations, one gives initial
conditions, e.g. the function and a derivative on a space-like
surface.  For elliptic differential equations one gives boundary
conditions, e.g. the function OR a normal derivative on the boundary
of the region.  This is also true of the finite
difference schemes.  If one attempts to give initial conditions for
an elliptic system, the computations can be done but are grossly
unstable, i.e. the further in time one goes, the more sensitive they
are to the initial conditions.

	The differential equations of steady compressible gas flow are elliptic
for subsonic velocities and hyperbolic for supersonic velocities.  In this
case the independent variable is the downstream distance rather than
time since we are talking about steady flow.  In the case of a body
moving slightly slower than the speed of sound the flow is elliptic
over almost all of space, but there can be a region where the air
has to flow over the body faster than sound, where the equations
are hyperbolic.

	Now for the fantasy.  Imagine the time variable to be a space
variable in a space time, and suppose we are in a region of supersonic
flow where the differential equations are hyperbolic, but where the
main flow is elliptic.  Going forward in time for us corresponds
to going downstream.  From the point of view of an observer (God)
outside of space-time, our history corresponds to a variation over
space.  What happens when the transition to elliptic flow takes
place?

	Perhaps there is a shock wave at that transition, but
this isn't what is most interesting.  What is interesting is that
a transition occurs from the future being determined by the past,
to conditions at any point being determined by boundary conditions at
plus and minus infinity.  From the point of view of an observer, as
the transition approached, the world would become more and more
unstable.  Tiny causes would have giant effects, purposeful actions
would encounter more and more noise, and eventually purposeful action
and life itself would become impossible.

	The speculation is that our space-time is a small region of
hyperbolicity in a general elliptic situation.  So gather we rosebuds
while we may.