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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Deterministic approximations of random reflectors
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by Omer Angel, Krzysztof Burdzy and Scott Sheffield PDF
Trans. Amer. Math. Soc. 365 (2013), 6367-6383 Request permission

Abstract:

Within classical optics, one may add microscopic “roughness” to a macroscopically flat mirror so that parallel rays of a given angle are reflected at different outgoing angles. Taking the limit (as the roughness becomes increasingly microscopic) one obtains a flat surface that reflects randomly, i.e., the transition from incoming to outgoing ray is described by a probability kernel (whose form depends on the nature of the microscopic roughness).

We consider two-dimensional optics (a.k.a. billiards) and show that every random reflector on a line that satisfies a necessary measure-preservation condition (well established in the theory of billiards) can be approximated by deterministic reflectors in this way.

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Additional Information
  • Omer Angel
  • Affiliation: Department of Mathematics, University of British Columbia, 121 - 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2
  • MR Author ID: 667585
  • Email: angel@math.ubc.ca
  • Krzysztof Burdzy
  • Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
  • Email: burdzy@math.washington.edu
  • Scott Sheffield
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 2-180, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
  • Email: sheffield@math.mit.edu
  • Received by editor(s): March 4, 2012
  • Received by editor(s) in revised form: April 7, 2012, and April 9, 2012
  • Published electronically: June 3, 2013
  • Additional Notes: The first author’s research was supported in part by NSERC and by the Sloan Foundation. The second author’s research was supported in part by NSF Grant DMS-0906743 and by grant N N201 397137, MNiSW, Poland. The third author’s research was supported in part by NSF Grant DMS 0645585.
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 365 (2013), 6367-6383
  • MSC (2010): Primary 37D50
  • DOI: https://doi.org/10.1090/S0002-9947-2013-05851-5
  • MathSciNet review: 3105755