[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.


* 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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