[FoRK] FoRK Digest, Vol 108, Issue 19
Dave Long
dave.long at bluewin.ch
Fri Sep 21 05:19:40 PDT 2012
> ... and point is a subset of interval since points are just
> degenerate intervals.
Another way to look at it is that intervals are bounded by points
(resp. vertices) and because differentiation- and integration-like
operators are linear, we can easily get log factors (or, in some
cases, even achieve a streaming computation) by shuffling where (more
creatively, if) these operations occur in a computation. This is
useful in many situations, not the least of which is high-school
calculus*, in which we often reduce problems given over an interval
to calculations on the boundary. Further afield, one might even
argue that a financial report is an exercise in gathering data from
interiors to produce information on the boundaries: double-entry
bookkeeping is that field's answer to winged-edge meshes. That books
balance is a homotopic constraint, demonstrating that at some level,
Pacioli anticipated Poincaré by four centuries.
-Dave
* Spivak's _Calculus on Manifolds_ states: (4-13, p.104 -- under
asciification of notation)
> Finally, if c is a k-chain +/a[i]*c[i], we have
> <c|dw> = +/a[i]<c[i]|dw> = +/a[i]<bdy(c[i])|w> = <bdy(c)|w>
> Stokes' theorem shares three important attributes with many fully
> evolved major theorems:
> 1. It is trivial.
> 2. It is trivial because the terms appearing in it have been
> properly defined.
> 3. It has significant consequences.
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