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

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

276 by six vector

276 by six vector fields. In fact we only need the 1-jets of these vector fields at (co, O, 0, 0), and only 4 of the generators have non-zero 1-jets. We record these 1-jets here. jlv 3 : i o) jl vl = u v ' i jl v2 = 0 w 0 4c0 3c0 1v/0 lv4 (4c0 3c0 1)0 0 ' 0 " 0 0 Remark These six vector fields generate the module 0~" of analytic vector fields tangent to V (or rather the real part of the complex discriminant) but not necessaxily the module Ov of smooth vector fields tangent to V. However, all we need is that these vector fields are contained in the module of smooth vector fields tangent to V. PItOOF OF THEOREM 7: Let S denote the germ ofc, u, v, w-spmce at ((c0,4"),0,0,0), with (co 2, ={=) ~ (1, T), (3/4, T). Let V C S be the germ of the discriminant of ~ defined in (2). Suppose g : (R3,0) --, S is transverse to the c-axis. Coordinates in R a can be chosen so that g takes the form (p,q,r) H (co + h(p,q,r),p,q,r), where h E m3. We wish to show that g is K:v-equivalent to the map, g~(p,q,~)=(co,p,q,~). We use the unipotency results of Bruce, du Plessis and Wall, [5]. Consider the submodule Ov,1 COv defined by, Or,1 = m~sOs n Ov + Cs.{v2, va, v,i}. The group this defines is a jet-unipotent subgroup of K: in the sense of [5], since the 1-jet part of ®v,1 consists only of strictly upper triangular matrices. Let ~ be the subgroup of K: generated by Or,1 and m3203 (the latter bcing a submodnle of the module of vector fields on the source). This is also jet-unipotent. With this G we use the notation of [5, Proposition (4.1)]. Let A be the module of smooth map germs g : (R3,0) ---} 5', with g(0) = (co,0,0,0), and let M= {g e A[g(p,q,r) = (h(p,q,r),O,O,O),h e ma} +m~.A. First note that the map gl is 2-K:v-determined by [6], since, TJCv,~.gl = tgl(03) + g~Ov = (m3, C~, C3, C3).

277 Let L = £(J20) and m E M. Working modulo m33 we have, L.j2(gl + m) + m3.M = t(gz + m)(m~6)3) + (gl + m)*Ov,1 + m3.M = (h,, O, Z, O), O, O, Z)} + (gl + m)'{jlv2,jlv3,Jlv4} + rn3.M = (0,0,1,0), (0,0,0,1)} + (gz + m)*{jlv2,jzv3,jlv4} + m3.M = M, provided (4Co 2 :F 3)(Co 2 q: 1) # 0. Thus, by Nakayama's lemma, ]5.j2(gz 4- m) = M for all m E M, and the result follows. [] References [1] A.M. Ozorio de Almeida, J.H. Hannay. Geometry of two dimensional tori in phase space: projections, sections and the Wigner function. Annals of Phys. 138 (1982), 115-154. [2] V.I. Arnold. Mathematical methods of classical mechanics. Springer, New York etc., 1978. [3] V.I. Arnold, S.M. Gussein-Zade, A.N. Varchenko. Singularities of differentiable maps, Volume I. Birkhauser, Boston etc., 1985. [4] Th. Br6cker, L. Lander. Differentiable Germs and Catastrophes. L.M.S. Lecture Note Series 17, C.U.P., Cambridge, 1975. [5] J.W. Bruce, A.A. du Plessis, C.T.C. Wall. Determinacy and unipotency. Invent. math. 88 (1987), 521-554. [6] J. Damon. Deformations of sections of singularities and Gorenstein surface singulari- ties, Am. J. Math. 109 (1987), 695-722. [7] J.B. Delos. Catastrophes and stable caustics in bound states of Hamiltonian systems. J. Chem. Phys. 86 (1987), 425--439. [8] S. Janeczko, I~.M. Roberts. Classification of symmetric caustics I: Symplectic equiv- alence. These proceedings. [9] S. Janeczko, tt.M. Roberts. Classification of symmetric canstics II: Caustic equiva- lence. In preparation. [10] D.W. Noid, R.A. Marcus, Semiclassical calculation of bound states in a multidimen- sional system for nearly 1:1 degenerate systems. J. Chem. Phys. 67 (1977), 559-567. Mathematics Institute University of Warwick Coventry CV4 7AL U.K.

  • Page 1 and 2:

    Lecture Notes in Mathematics Editor

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    Editors Mark Roberts Ian Stewart Ma

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    Co~e~s P.J. Aston, Scaling laws and

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    Scaling Laws and Bifurcation P.J.As

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    where = It follows immediately from

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    esults for unitary representations

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    and T are orthogonal representation

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    which is also a subgroup of F. Clea

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    Proof 11 The equivalent result that

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    13 such that the Equlvariant Branch

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    ]5 since h is orthogonal. As b, e,

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    17 and from Lemma 4.2, hj : X ~k --

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    ]9 in a bifurcating branch of solut

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    21 Duncan, K. and Eilbeck, 3. C. (1

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    23 the symmetry arises naturally fr

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    25 the traction problem in nonlinea

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    27 First we dispose of the case k =

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    29 The symbols 0 ..... 4 indicate r

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    31 In terms of the radially project

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    S I --U 33 A Figure 4 .. • . ".

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    35 k z 4 I For k s 6 the ideas abov

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    37 [GG] Golubitsky, M. and Guillemi

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    39 small equivariant perturbations

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    41 It may happen that an obstructio

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    , 43 0(3) {yo} {xO'Yo, Y2+Y-2} D 2

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    45 stability requires order 5 (Golu

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    3.1. Phase portrait in FixfD2~.Z ~

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    49 (11) kl=Z,l+- "-c+ c2~2 2 anothe

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    51 When c=0, the eigenvalue in the

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    53 Remark 2. This heteroclinic cycl

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    55 Armbruster et al. [1988]. This w

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    57 Notes: 1) each picture shows pro

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    ~< o( 59 i o( o(" ~X1r IX~£~ Figur

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    ¢* 1 61 T ........... "--7 y¢, --

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    Boundary Conditions as Symmetry Con

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    65 We illustrate this point in the

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    U T(u) 67 m=2 -~ ................ 0

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    69 be the reduced bifurcation equat

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    71 3 The Couette-Taylor Experiment

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    73 00 u(x,y) = v(x,y) = ~-x (x,y) =

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    75 (dS) F of S at F has an eigenval

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    77 experimental geometry. 'Upper bo

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    79 M.G.M. Gomes [1989]. Steady-stat

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    81 To describe the results, we supp

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    83 The author's work on these quest

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    85 certain elements of G may interc

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    87 isolated singularity at 0. We sa

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    in particular, Zp~ = 1 + ~ iii) Vp

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    91 Similarly we let X(G) denote the

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    93 F I Fix(G') x ~ : Fix(G') x $t -

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    Example 4.5: 95 Let G = Z/mE act on

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    97 Thus, we see that X q = 1 + 4V.

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    99 Q(F1) = Cx,~/I(F1) = Cx,~/I(F 0)

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    Corollary 7 101 For i) we know by t

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    103 generator for m A, the module o

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    105 D3 On the number of branches fo

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    On a Codimension-four Bifurcation O

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    109 Setting x = u3 and introducing

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    111 not matter. We choose b > 0. Th

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    113 SLot x SLs Figure 2: Phase port

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    SLs(H} H.(SL s) SNs I SN~ (~6) 115

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    iii 1 SLo/" SNo ,y X 117 i 11 21 .

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    119 extent the dynamics is influenc

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    121 [12] J. Guckenhelmer, SIAM J. M

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    123 produces a continuous function.

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    is generated by 125 V(A~) = {f : f(

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    and the abstract integral equation

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    129 (iii). X® and X+ are finite di

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    131 Definition 6.1 Let E and F be B

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    133 6.3 Contractions on embedded Ba

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    135 Theorem 6.13 (Center Manifold)

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    137 The AIE (6.14) is equivalent to

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    and 139 (q,p) = foh d-"~)p(-r) (7.8

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    [Cha71] [Die87] [Dui76] [DvG84] [Ha

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    143 Much is already known about the

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    145 We refer to ,g = {Su[y E a} as

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    147 ttemark 3.1 It follows by our m

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    149 Definition 4.1 Let 11 be a clos

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    151 Remark 4.3 It follows from Theo

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    (a) f(=) = ~(=)x, au x E GIH. (b) f

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    155 Since ] and ~ are smooth, so ar

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    157 of X at all points z E a. Neces

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    159 In this section we wish to desc

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    161 A straightforward application o

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    8.1 Poincard maps 163 We review the

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    165 Proposition 8.3 Let P. be a rel

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    On the Bifurcations of Subharmonics

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    169 and so we give in a fourth part

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    171 of KerL, the elements of B are

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    173 From the fixed-point subspaces

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    175 and the following curve is the

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    q even q=3 a g 177 s~llx-ag odd Fig

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    q=3 S 179 s=lk-aa q ~ - - q>5 ~ a ~

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    3.2 dimB--1 181 We are now in the s

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    183 Puting them back into (14) and

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    4.2 The recognition problem 185 Thi

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    case I II 1V V VI VII IX case I U 1

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    qo r0 =0 =O #o ~o Pa. ~o 1=° top-c

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    qo =0 #o References .p~ =0 191 ~o ~

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    Classification of Symmetric Caustic

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    195 internal variables p and extern

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    197 In the classification of Lagran

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    Remark 1.3 More generally, the orga

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    (W*~V,0) (v,o) (where n2 is the nat

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    G $~G-versal unfolding of F(.,0) in

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    Z (V,U) as an E q module. It follow

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    3 FINITE DETERMINACY 207 Good deter

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    209 Definition 3.3 G Let ~ be any g

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    211 where q%(x,y) = Z Vbc(X'y)xc an

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    Theorem 4.5 213 (i) If r >_. s then

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    215 (4) In [JR] we show that the ca

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    217 In terms of the invariants the

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    R~P'ERF~CES [AI [ADI [AGVI [BPW] [D

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    221 instrument in terms of composed

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    ~p'_-sin0' ~0' sin0' 3p -sin0 30 si

  • Page 231 and 232: 225 perfect gas with V = volume, S
  • Page 233 and 234: 227 according to co and all maximal
  • Page 235 and 236: a) a submersion p : X -~ Y, 229 b)
  • Page 237 and 238: 231 Then the reduced symplectic spa
  • Page 239 and 240: 233 Definition The phase space of a
  • Page 241 and 242: where m ~t-- 1 +,~2 = a9 '2 - 2 9'
  • Page 243 and 244: 237 The Billiard Map as an Optical
  • Page 245 and 246: exists a local generating Morse fam
  • Page 247 and 248: Proof q 241 aperture aperture apert
  • Page 249 and 250: A1 :FlCO = -7 i2" A3 : F3f/) = - ¼
  • Page 251 and 252: 245 Now taking an inflection point
  • Page 253 and 254: 247 1 5 2 3 q2)+kl(ql~.2+q2)2 H 4 :
  • Page 255 and 256: 249 A2(k+l) singularities by specif
  • Page 257 and 258: 251 Assume that the surfaces have t
  • Page 259 and 260: 253 -- tsin @ )+(t-'+~in~>t_~os2q>
  • Page 261 and 262: 255 Poston, T. and Stewart, I. [197
  • Page 263 and 264: 257 0ii) A(),)-A 0 + B(X) is a hoto
  • Page 265 and 266: 259 their lists. The correct lists
  • Page 267 and 268: 261 denote the generator of the Lie
  • Page 269 and 270: 263 4 ÷ D6 d O(3), O(2)-, O- O(2)-
  • Page 271 and 272: References 265 Chossat, P. [1Q70]:
  • Page 273 and 274: 267 In this paper I consider invari
  • Page 275 and 276: 269 be the set of critical points i
  • Page 277 and 278: 271 Figure 1: Two trajectories in a
  • Page 279 and 280: 273 Now we return to generating fam
  • Page 281: ~LLL I 275 ~::'.'~'.C': • ,',': :
  • Page 285 and 286: §1. Introduction 279 In this paper
  • Page 287 and 288: where 281 Im(ei~R0) - 0 , (l.4)b M
  • Page 289 and 290: 283 b 0 to avoid negative suffices.
  • Page 291 and 292: and where 285 r r r i i i r i Ro ~
  • Page 293 and 294: with R 0 as before in (2.8), but no
  • Page 295 and 296: is always real. such that is real,
  • Page 297 and 298: 291 I. The fixed point (2.9) remain
  • Page 299 and 300: References 293 [i] R. W. Lucky (196
  • Page 301 and 302: 295 -- ba+~+~t h For small values o
  • Page 303 and 304: h =0 XX ~0 (h~ 1, h~L 2) H = (0, O)
  • Page 305 and 306: 299 4 Description of the proof of t
  • Page 307 and 308: Versal Deformations of Infinitesima
  • Page 309 and 310: 303 form. (2) By dropping the sympl
  • Page 311 and 312: 305 Jij q + L~j=l Ji]~j )' we have
  • Page 313 and 314: 307 For (0) n, n--even,~= 1, set Ix
  • Page 315 and 316: D% I -% ,,~ Fig. 1 309 Each oblique
  • Page 317 and 318: 311 I -- T/ Fig. 2 7~-form Now, def
  • Page 319 and 320: 313 b'st = r (-1)s-t[; s t' case(c)
  • Page 321 and 322: H i [X22"FILXl 2] ~ [(2x2x4-x32) -F
  • Page 323 and 324: H 2Re[+2AZlZ2+Z2Z2 +#zlT" l ] 2E--R
  • Page 325 and 326: 319 unfolding H(g) of a Hamiltonian
  • Page 327 and 328: ADDRESSES OF CONTRIBUTORS D.Armbrus
Reading grade 6 2.A.5.b - mdk12
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