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 Zh. Mat. Fiz. Anal. Geom.: Year: Volume: Issue: Page: Find

 Zh. Mat. Fiz. Anal. Geom., 2009, Volume 5, Number 1, Pages 12–24 (Mi jmag114)

Retroreflecting curves in nonstandard analysis

R. Almeidaa, V. Nevesa, A. Plakhovab

a Department of Mathematics, University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal
b Institute of Mathematical and Physical Sciences, University of Aberystwyth, Aberystwyth SY23 3BZ, Ceredigion, UK

Abstract: We present a direct construction of retroreecting curves by means of Nonstandard Analysis. We construct non self-intersecting curves which are of class $C^1$, except for a hyper-nite set of values, such that the probability of a particle being reected from the curve with the velocity opposite to the velocity of incidence, is innitely close to 1. The constructed curves are of two kinds: a curve innitely close to a straight line and a curve innitely close to the boundary of a bounded convex set. We shall see that the latter curve is a solution of the problem: nd the curve of maximum resistance innitely close to a given curve.

Key words and phrases: Nonstandard Analysis, retroreflectors, maximum resistance problems, reflection, billiards.

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Bibliographic databases:
MSC: 26E35, 49K30, 49Q10
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Citation: R. Almeida, V. Neves, A. Plakhov, “Retroreflecting curves in nonstandard analysis”, Zh. Mat. Fiz. Anal. Geom., 5:1 (2009), 12–24

Citation in format AMSBIB
\Bibitem{AlmNevPla09} \by R.~Almeida, V.~Neves, A.~Plakhov \paper Retroreflecting curves in nonstandard analysis \jour Zh. Mat. Fiz. Anal. Geom. \yr 2009 \vol 5 \issue 1 \pages 12--24 \mathnet{http://mi.mathnet.ru/jmag114} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2528397} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000269825400002} \elib{http://elibrary.ru/item.asp?id=12796277}