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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

+ Z2 invariant under the

+ Z2 invariant under the action of ~N on Epq. 218 Modulo M the prenormal form given by Proposition 4.1 is:- 3 j=l where Ilij = oqj + 13ijw 1 + 7ijw2. This is infinitesimally stable if and only if A = det 0 711 0~22 0 0 0 0t12 (X22 ~21 ~312 ~22 721 712 722 In particular, for stability we must have 0t12 # 0 or 0~22 ~ 0. Some algebraic manipulation shows that the coefficients in the coordinate changes :- x 2 ~ (al0 + allW 1 + al2w2)x l + (a20 + a21w I + a22w2)x 2 wj ~ bjlw 1 + bj2w 2 j-- 1,2 ~,xl, Yl, Y2, Y3 ~[0 can bechosento set:- ot 1 tx 2 unchanged, I: :] I: :] Eo tX = if0~12= 0, 0~22# 0, A ~ 0, and otherwise affect only terms in M. This completes the proof of (iii). I: [: :1 (iv) This is clear from Corollary 4.4 and the proof above. O. if ctl2 ~ 0, A # 0, and

R~P'ERF~CES [AI [ADI [AGVI [BPW] [DI 219 Adams, J.F., Lectures on Lie Groups, Benjamin, New York, 1969. Armbruster, D. and Dangelmayr, G., Singularities in phonon focusing, Z. Phys. B52 (1983), 87-94. Arnold, V.I., Gusein-Zade, S.M. and Varchenko, A.N., Singularities of Differentiable Maps, Vol. I, Birkhauser, Boston, 1985. Bruce, J.W., du Plessis, A.A. and Wall, C.T.C., Determinacy and unipotency, Invent. Math. 88, 521-554. Damon, J.N., The unfolding and determinacy theorems for subgroups of .A, and 2(, Memoirs A.M.S. 306 (1984). [El Ericksen, J.L., Some phase transitions in crystals, Arch. Rational Mech. Anal. 73 (1980), 99-124. [GI Gaffney, T., New methods in the classification theory of bifurcation problems, in MultiparameterBifurcation Theory (eds. M. Golubitsky and J. Guckenheimer), Contemp. Math. 56, Amer. Math. Sot., Providence, R.I., 1986. [Y] Janeczko, S., On G-versal Lagrangian submanifolds, Bull. Polish Acad. So. 31 (1983), 183-190. [JKII Janeczko, S. and Kowalczyk, A., Equivariant singularities of Lagrangian manifolds and uniaxial ferromagnet, SIAM J. Appl. Math. 47 (1987), 1342- 1360. [YK2] Janeczko, S. and Kowalczyk, A., Classification of generic 3-dimensional Lagrangian singularities with (z2)l-symmetries, preprint, (1988). Janeczko, S. and Roberts, R.M., Classification of symmetric caustics II : caustic equivalence, in preparation. [Ma] Mather, J.N., Stability of C ~ mappings IV: classification of stable germs by R-algebras, Publ. Math. IHES 37 (1970), 223-248. [Mo] Montaldi, J.A., Bounded caustics in time reversible systems, this volume. [hq-Iw] Northrop, G.A., Hebboul, S.E. and Wolfe, LP., Lattice dynamics from phonon imaging, Phys. Rev. Lett. 55 (1985), 95-98. IN] Nye, J.F., The catastrophe optics of liquid drop lenses, Proe. R. Soc. London A403 (1986), 27-50. [P] Poenaru, V., Singularitts C "~ en prtsence de symttrie, Lecture Notes in Math. 192, Springer, Berlin, Heidelberg, New York, 1976. [gll Roberts, R.M., On the genericity of some properties of equivariant map germs, J. London Math. Soc. 32 (1985), 177-192. [R21 Roberts, R.M., Characterisations of finitely determined equivariant map germs, Math. Ann. 275 (1986), 583-597. tSl Slodowy, P., Einige Bemerkungen zur Enffaltung symmetrischer Functionen, Math. Z. 158 (1978), 157-170. [Wl Wolfe, J.P., Ballistic heat pulses in crystals, Phys. Today 33 (1980), no.12, 44-50.

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