> On Fri, 1 Nov 1996 09:22:59 -0600 (CST) zaimoni@ksu.edu writes:
>
> >Local maxima: points where the gradient of the evaluation function for
> >
> >relative survival is 0, and all local orthonormal frames have the
> >survival function decreasing on all basis vectors. [This is a fancy
> >way
> >of saying 'locally optimal solutions that Natural selection ends up
> >at'.]
> >
> >If the mutations don't go far enough, they will wipe before finding
> >any
> >nearby [improved] local maximum.
>
>
> Kenneth, could you say something more about 'ortholnormal frames'?
> That's a tool not in my cognitive tool box.
>
> Take care. -KMO
"Orthonormal frames":
This applies to the domain of the function.
First of all, I had a fit of overgeneralization and subconsciously
decided to write as if the survival function was defined on something
that "up close" looked like Euclidean space, instead of IS Euclidean
space. The technical term for the former is "manifold".
[4-D oversimplifcation: I'm allowing the function to be defined on
Einsteinian space-time, not just Newtonian space-time. Newtonian is
flat, i.e. IS euclidean space [4-d]. Einsteinian is curved; gravity is a
virtual force resulting from this curvature.]
[2-D oversimplification: I'd like to allow surfaces of spheres as
domains, not just planes.]
At any given point that isn't a corner/edge/otherwise sharply bent,
we can, "at sufficiently small scales", neglect the difference between the
tangent space to a manifold and its manifold. [curve: tangent line;
surface: tangent plane; and so on. The tangent space must have the same
dimension as what it is tangent to.]
We proceed to coordinatize the tangent space under a given
metric, using as many axes as it has dimensions [things get nasty for
infinite-dimensional stuff, so let's keep it finite], s.t. the axes are
all at right angles to each other. For each axis, we choose a vector of
length 1 parallel to the axis. This set of vectors is an orthonormal
frame at the point I constructed the tangent space for. Since it didn't
come from coordinates, it is not necessarily a workable coordinate system
at range.
The Lorentz frames I mentioned earlier are simply orthonormal frames
in General Relativity, for the Lorentz metric, with the time axis forced
to be time-like.
//////////////////////////////////////////////////////////////////////////
/ Towards the conversion of data into information....
/
/ Kenneth Boyd
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