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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:The Expected Total Curvature of Random Polygons -
Cantarella\, J (University of Georgia)
DTSTART;TZID=Europe/London:20121206T094000
DTEND;TZID=Europe/London:20121206T100000
UID:TALK41882AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/41882
DESCRIPTION:We consider the expected value for the total curva
ture of a random closed polygon. Numerical experim
ents have suggested that as the number of edges be
comes large\, the difference between the expected
total curvature of a random closed polygon and a r
andom open polygon with the same number of turning
angles approaches a positive constant. We show th
at 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 edgelengt
h distribution. \n\nWe then consider the symmetric
measure on closed polygons of fixed total length
constructed by Cantarella\, Deguchi\, and Shonkwil
er. For this measure\, the expected total curvatur
e of a closed n-gon is asymptotic to n pi/2 + pi/4
by our first result. With a more careful analysis
\, we are able to prove that the exact expected va
lue 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 hept
agons in 3-space are unknotted. \n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
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