Here's my code, and its results. That perimeter sure looks close to tau. -tom At 67108864 pts 3.99990034103394 area 2.6665184797472 perim 6.28303082552975 At 134217728 pts 3.9999551102519 area 2.66652574548337 perim 6.28309803073879 At 268435456 pts 3.99998798593879 area 2.66669666495159 perim 6.28329936437473 On Wed, Aug 7, 2019 at 9:47 PM Brad Klee <bradklee@gmail.com> wrote:
Andy,
In this case the function doesn't have poles, so you are probably right. I still think it's weird that a complete integral can be taken on an incomplete domain, so I'm willing to worry more.
Another "problem" is that, after more thought, it seems plausible that the configuration space R would need an odd volume element to match Jim Propp's preferred standard. I think it's still "okay" to start with Cartesian, but would not be surprised if this led to statistical diversity.
Also, nice reading of Keith's algorithm, I was confused by the part about "carving", but now I can see from your perspective that it actually sounds right. Is Keith's algorithm on Mathworld? Maybe we could get EW to add a few more equations?
Please don't blame Fred for listening to me. We are all trying to have fun and learn more, and it doesn't hurt to double or triple check for quality assurance!
Cheers --Brad
On Wed, Aug 7, 2019 at 11:28 PM Andy Latto <andy.latto@pobox.com> wrote:
If what we're interested in is the integral of a function with respect to a particular measure, there is nothing tricky about ignoring measure zero sets. If a set has measure 0, the integral is unchanged by integrating a function that has different, arbitrary, values on that set.
Andy andy.latto@pobox.com
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