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SUMMARY:The Expected Total Curvature of Random Polygons - Cantarella\, J (
 University of Georgia)
DTSTART:20121206T094000Z
DTEND:20121206T100000Z
UID:TALK41882@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:We consider the expected value for the total curvature of a ra
 ndom closed polygon. Numerical experiments have suggested that as the numb
 er of edges becomes large\, the difference between the expected total curv
 ature of a random closed polygon and a random open polygon with the same n
 umber of turning angles approaches a positive constant. We show that this 
 is true for a natural class of probability measures on polygons\, and give
  a formula for the constant in terms of the moments of the edgelength dist
 ribution. \n\nWe then consider the symmetric measure on closed polygons of
  fixed total length constructed by Cantarella\, Deguchi\, and Shonkwiler. 
 For this measure\, the expected total curvature of a closed n-gon is asymp
 totic to n pi/2 + pi/4 by our first result. With a more careful analysis\,
  we are able to prove that the exact expected value of total curvature is 
 n pi/2 + (2n/2n-3) pi/4. As a consequence\, we show that at least 1/3 of f
 ixed-length hexagons and 1/11 of fixed-length heptagons in 3-space are unk
 notted. \n
LOCATION:Seminar Room 1\, Newton Institute
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