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Uspekhi Mat. Nauk, 1988, Volume 43, Issue 3(261), Pages 125–160 (Mi umn1892)  

This article is cited in 36 scientific papers (total in 36 papers)

Wavefronts and reflection groups

O. P. Shcherbak

Abstract: Typical singularities of wave fronts and ray systems passing by smooth obstacle in $3$-space are described in the article. These singularities turn out to be connected with noncristallographic Coxeter groups $I_2(5)$, $H_3$, $H_4$. Proofs are based on the detail investigation of the discriminants of these groups by their inclusion into cristallographic ones $A_4$, $D_6$, $E_8$ correspondently. Besides, there is given a geometrical description of some singularities of bicaustics in collisionless flows of particles. It is based on inclusions of Coxeter groups $A_1^\mu$, $D_\mu$, $D_4$ into $B_\mu$, $G_\mu$, $F_4$ as normal subgroups. The article contains a wide table matherial on neutral stratification of discriminants of reflection groups.
32 refs.

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English version:
Russian Mathematical Surveys, 1988, 43:3, 149–194

Bibliographic databases:

UDC: 514.8
MSC: 20F55, 51F15
Received: 31.03.1987

Citation: O. P. Shcherbak, “Wavefronts and reflection groups”, Uspekhi Mat. Nauk, 43:3(261) (1988), 125–160; Russian Math. Surveys, 43:3 (1988), 149–194

Citation in format AMSBIB
\by O.~P.~Shcherbak
\paper Wavefronts and reflection groups
\jour Uspekhi Mat. Nauk
\yr 1988
\vol 43
\issue 3(261)
\pages 125--160
\jour Russian Math. Surveys
\yr 1988
\vol 43
\issue 3
\pages 149--194

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    1. I. A. Bogaevsky, “Singularities of projective hypersurfaces”, Funct. Anal. Appl., 24:2 (1990), 97–103  mathnet  crossref  mathscinet  zmath  isi
    2. V. M. Zakalyukin, R. M. Roberts, “On stable singular Lagrangian varieties”, Funct. Anal. Appl., 26:3 (1992), 174–178  mathnet  crossref  mathscinet  zmath  isi
    3. R V Moody, J Patera, J Phys A Math Gen, 26:12 (1993), 2829  crossref  mathscinet  zmath  adsnasa  isi
    4. G. G. Ilyuta, “The A'Campo Theorem on the Discriminant”, Funct. Anal. Appl., 28:2 (1994), 85–91  mathnet  crossref  mathscinet  zmath  isi
    5. C S Lam, J Patera, R T Sharp, J Phys A Math Gen, 29:23 (1996), 7705  crossref  mathscinet  zmath  adsnasa  isi
    6. J. -B. Zuber, “Graphs and reflection groups”, Comm Math Phys, 179:2 (1996), 265  crossref  mathscinet  zmath  adsnasa  isi
    7. O. M. Myasnichenko, “Direct Product of Unfurled Swallowtails in the Problem of Going around an Obstacle”, Funct. Anal. Appl., 30:4 (1996), 230–236  mathnet  crossref  crossref  mathscinet  zmath  isi
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