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Re: Isn't it the limit?



Date: Sat, 29 Mar 1997 10:20:34 -0500 (EST)
From: "Donald E. Simanek" <dsimanek@eagle.lhup.edu>

There is a related problem with the mathematical definition of these
limits in two or more dimensions, in which triangulations (simplicial
approximations) produce problems when the ratio of volume and longest
edge of any component triangles becomes unbounded (either above or
away from zero, depending on which way the ratio is set up). Then the
phenomenon noted by H. Whitney appears, in which (for example) a
finite cylinder can seem to be assigned any positive number as its
area! Using "rectanglular" elements instead of triangles usually
manages to avoid this problem, as the implied triangulation almost
always satisfies the bounded criterion. Thus Riemann sums do give
areas as desired.
The problem is quite deep, unfortunately, and belongs to one of
those parts of math noted for its abstruseness: geometric measure
theory. (Whitney's examples are in his book of that name; no longer
the standard, but far more accessible than Federer's.) While the
example can be explained to math undergrads, I'm not sure how well
it is understood; most of our grad students have a hard time with
the underlying concepts. John Barrett and I wrote a paper using it
a few years ago. Since then we have seen numerous examples of its
inadvertant appearance in simplicial approximations used in
"discrete gravity" (e.g., Regge calculus). In each case limits went
very badly awry, and the authors were left wondering, "What happened
to our intuition?"
Not exactly the problem you asked about, but closely related.

---------------------------------------------
Phil Parker pparker@twsuvm.uc.twsu.edu
Random quote for this second:
Only God can make random selections.