Reconstructions of fronts and caustics depending on a parameter and versality of mappings. (English) Zbl 0554.58011
Translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 22, 56-93 (Russian) (1983; Zbl 0534.58014).
Citations:
Zbl 0534.58014References:
| [1] | V. I. Arnol’d, ?Normal forms of functions near degenerate critical points, the Weyl groups Ak, Dk, Ek, and Lagrangian singularities,? Funkts. Anal. Prilozhen.,6, No. 4, 3?25 (1972). |
| [2] | V. I. Arnol’d, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1974). |
| [3] | V. I. Arnol’d, ?Critical points of functions on a manifold with boundary, the simple Lie groups Bk, Ck, Fk, and singularities of evolutes,? Usp. Mat. Nauk,33, No. 5, 91?105 (1978). |
| [4] | V. I. Arnol’d, ?Indices of singular points of 1-forms on a manifold with boundary, displacement of invariants of groups generated by reflections, and singular projections of smooth surfaces,? Usp. Mat. Nauk,34, No. 2, 3?38 (1979). |
| [5] | V. I. Arnol’d, ?Spectral sequences for reducing functions to normal forms,? in: Problems of Mechanics and Mathematical Physics [in Russian], Nauka, Moscow (1976), pp. 7?20. |
| [6] | V. I. Arnol’d, ?On the theory of envelopes,? Usp. Mat. Nauk,31, No. 3, 172?173 (1976). |
| [7] | V. I. Arnol’d, ?Lagrange and Legendre cobordisms,? Funkts. Anal. Prilozhen.,14, No. 3, 1?13; No. 4, 8?17 (1980). · Zbl 0452.22014 · doi:10.1007/BF01078407 |
| [8] | V. I. Arnol’d, ?On the characteristic class contained in quantization conditions,? Punkts. Anal. Prilozhen.,1, No. 1, 1?14 (1967). |
| [9] | V. I. Arnol’d, ?Classification of unimodal critical points of functions,? Funkts. Anal. Prilozhen.,7, No. 3, 75?77 (1973). |
| [10] | V. I. Arnol’d, ?Classifications of bimodal critical points of functions,? Funkts. Anal. Prilozhen.,9, No. 1, 49?51 (1975). |
| [11] | V. I. Arnol’d, ?Rearrangements of singularities of potential flows in a collisionless medium and metamorphoses of caustics in three-dimensional space,? Tr. Sem. im. I. G. Petrovskii, Mosk. Gos. Univ., No. 8, 21?57 (1982). |
| [12] | V. I. Arnol’d, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of Differentiable Mappings [in Russian], Nauka, Moscow (1982). |
| [13] | G. R. Belitskii, ?Equivalence and normal forms of germs of smooth mappings,? Usp. Mat. Nauk,33, No. 1, 95?155 (1978). |
| [14] | A. N. Varchenko and A. B. Givental’, ?Mapping of periods and the intersection form,? Funkts. Anal. Prilozhen.,16, No. 2, 7?20 (1982). |
| [15] | V. A. Vasil’ev, ?Characteristic classes of Lagrange and Legendre manifolds dual to singularities of caustics and wave fronts,? Punkts. Anal. Prilozhen.,15, No. 3, 10?22 (1981). |
| [16] | E. P. Gomozov, ?A versality theorem for two-sided groups of changes of variables,? Funkts. Anal. Prilozhen.,9, No. 4, 69?70 (1975). · Zbl 0317.14013 · doi:10.1007/BF01078187 |
| [17] | E. P. Gomozov, ?Versal deformations of germs of diffeomorphisms of finite smoothness class,? Dokl. Akad. Nauk Ukr. SSR, A, No. 8, 679?681 (1975). · Zbl 0312.57022 |
| [18] | E. P. Gomozov, ?Finite determination of germs of diffeomorphisms relative to conjugation,? Dokl. Akad. Nauk Ukr. SSR, A, No. 9, 773?775 (1976). · Zbl 0333.57019 |
| [19] | V. M. Zakalyukin, ?On singularities of caustics and fronts,? Usp. Mat. Nauk,31, No. 3, 257?258 (1976). |
| [20] | V. M. Zakalyukin, ?A versality theorem,? Funkts. Anal. Prilozhen.,7, No. 2, 28?31 (1973). |
| [21] | V. M. Zakalyukin, ?On Lagrange and Legendre singularities,? Funkts. Anal. Prilozhen.,10, No. 1, 26?36 (1976). |
| [22] | V. M. Zakalyukin, ?Rearrangements of wave fronts depending on a parameter,? Funkts. Anal. Prilozhen.,10, No. 2, 69?70 (1976). · Zbl 0331.47009 · doi:10.1007/BF01075778 |
| [23] | V. M. Zakalyukin, ?Singularities of convex hulls of smooth manifolds,? Funkts. Anal. Prilozhen.,11, No. 3, 76?77 (1977). · Zbl 0376.60013 · doi:10.1007/BF01135546 |
| [24] | V. M. Zakalyukin, ?Legendre mappings in Hamiltonian systems,? in: Some Problems of Mechanics [in Russian], Moscow Aviation Inst. (1977), pp. 11?16. |
| [25] | Ya. B. Zel’dovich, ?Decomposition of homogeneous matter into parts under the influence of gravity,? Astrofizika,6, No. 2, 319?335 (1970). |
| [26] | V. V. Lychagin, ?On sufficient orbits of a group of contact diffeomorphisms,? Mat. Sb.,104, No. 2 (10), 248?270 (1977). · Zbl 0412.58019 |
| [27] | V. P. Maslov, Operator Methods [in Russian], Nauka, Moscow (1973). |
| [28] | V. D. Sedykh, ?Singularities of the convex hull of a curve in Rn,? Funkts. Anal. Prilozhen.,11, No. 1, 81?82 (1977). |
| [29] | A. N. Shoshitaishvili, ?On functions with isomorphic Jacobi ideals,? Funkts. Anal. Prilozhen.,10, No. 2, 57?62 (1976). |
| [30] | V. I. Arnol’d, ?Wave front evolution and equivariant Morse lemma,? Commun. Pure Appl. Math.,29, No. 6, 557?582 (1976). · Zbl 0343.58003 · doi:10.1002/cpa.3160290603 |
| [31] | W. Bruce, ?A stratification of space of cubic surfaces,? Math. Proc. Cambridge Philos. Soc.,87, No. 3, 427?441 (1980). · Zbl 0434.58009 · doi:10.1017/S0305004100056863 |
| [32] | M. Buchner, ?Stability of the cut-locus in dimensions less than or equal to 6. Appendix,? Inventiones Math.,43, No. 3, 199?231 (1977). · Zbl 0365.58010 · doi:10.1007/BF01390080 |
| [33] | E. Bierstone, ?General position of equivariant maps,? Trans. Am. Math. Soc.,234, No. 2, 447?466 (1977). · Zbl 0318.57044 · doi:10.1090/S0002-9947-1977-0464287-3 |
| [34] | J. G. Dubois and J. P. Dufour, ?La théorie des catastrophes. Caustiques de l’optic géometric,? Ann. Inst. Henri Poincare,A24, No. 8, 243?260 (1976). |
| [35] | J. G. Dubois and J. P. Dufour, ?La théorie des catastrophes. V. Transformées de Legendre et thermodynamique,? Preprint (1977). |
| [36] | J. P. Dufour, ?Sur la stabilité de diagrams d’applications differentiables, Ann. Sci. Ecole Norm Supér., A,10, No. 2, 153?174 (1977). · Zbl 0354.58011 · doi:10.24033/asens.1324 |
| [37] | J. P. Dufour, ?Une limite aux extensions due théorème de preparation,? C. R. Acad. Sci., Paris,A282, No. 4, A199-A202 (1976). · Zbl 0315.58011 |
| [38] | J. P. Dufour, ?Déploiements de cascades d’applications differentiables,? C. R. Acad. Sci., Paris, A,281, No. 1, 31?34 (1975). |
| [39] | J. P. Dufour, ?Bistabilité des fronces,? C. R. Acad. Sci., Paris,285, No. 4, A445?448 (1977). · Zbl 0362.58002 |
| [40] | J. J. Duistermaat, Fourier Integrals Operators, Univers Publishers Weekly, New York (1974). · Zbl 0285.35010 |
| [41] | J. J. Duistermaat, ?Oscillatory integrals, Lagrange immersions, and unfolding of singularities,? Commun. Pure Appl. Math.,27, No. 2, 207?281 (1974). · Zbl 0285.35010 · doi:10.1002/cpa.3160270205 |
| [42] | M. Field, ?Une version equivariant du théorème de la transversalité de Thom,? C. R. Acad. Sci., Paris, A,282, 379?381 (1976). |
| [43] | M. Golubitsky, ?Contact equivalence for Lagrangean submanifolds,? Adv. Math.,15, No. 3, 375?387 (1975). · Zbl 0295.57018 · doi:10.1016/0001-8708(75)90143-7 |
| [44] | M. Golubitsky and D. Schaeffer, ?A theory of imperfect bifurcations via singularity theory,? Commun. Pure Appl. Math.,32, No. 1, 21?98 (1979). · Zbl 0409.58007 · doi:10.1002/cpa.3160320103 |
| [45] | M. Golubitsky and D. Schaeffer, ?Imperfect bifurcations in the presence of symmetry,? Preprint (1978). · Zbl 0467.58019 |
| [46] | J. Guckenheimer, ?Caustics and nondegenerate Hamiltonians,? Topology,13, No. 2, 127?133 (1974). · Zbl 0291.58010 · doi:10.1016/0040-9383(74)90003-2 |
| [47] | J. Guckenheimer, ?Catastrophes and partial differential equations,? Ann. Inst. Fourier,23, No. 2, 31?59 (1973). · Zbl 0271.35006 · doi:10.5802/aif.455 |
| [48] | L. Hörmander, ?Fourier integral operators,? Acta Math.,127, Nos. 1?2, 79?183 (1971). · Zbl 0212.46601 · doi:10.1007/BF02392052 |
| [49] | K. Janich, ?Caustics and catastrophes,? Math. Ann.,209, No. 1, 161?180 (1974). · Zbl 0268.58005 · doi:10.1007/BF01351319 |
| [50] | W. Kucharz, ?Jets suffisants et functions de détermination finie,? C. R. Acad. Sci.,284, No. 15, A877-A879 (1977). · Zbl 0359.58005 |
| [51] | F. Latour, ?Stabilité de champs d’applications differentiables, Generalisation d’une théorème de J. Mather,? C. R. Acad. Sci.,268, No. 2, A1331-A1334 (1969). · Zbl 0184.48501 |
| [52] | J. A. Lees, ?On the classification of Lagrange immersions,? Duke Math. J.,43, No. 2, 217?224 (1976). · Zbl 0329.58006 · doi:10.1215/S0012-7094-76-04319-2 |
| [53] | J. A. Lees, ?Defining Lagrangian immersions by phase functions,? Preprint (1977). |
| [54] | J. Martinet, ?Déploiements versals des applications differentiables et classification des applications stables,? Lect. Notes Math.,535, 1?44 (1976). · doi:10.1007/BFb0080494 |
| [55] | J. Mather, ?Stability of C? mappings. V. Transversality, ? Adv. Math.,4, 301?335 (1970). · Zbl 0207.54303 · doi:10.1016/0001-8708(70)90028-9 |
| [56] | V. Poènaru, ?Versal unfoldings of G-invariant functions,? Bull. Am. Math. Soc.,82, No. 1, 86?88 (1976). · Zbl 0325.58005 · doi:10.1090/S0002-9904-1976-13970-5 |
| [57] | V. Poènaru, ?Singularities C? en présence de Symmétrie,? Lect. Notes Math., Springer-Verlag,510 (1976). |
| [58] | L. Solomon, ?Invariants of finite reflection groups,? Nagoya Math. J.,22, No. 1, 57?64 (1963). · Zbl 0117.27104 · doi:10.1017/S0027763000011028 |
| [59] | D. J. S. Trotman, ?A transversality property weaker than Whitney A-regularity,? Bull. London Math. Soc.,8, No. 3, 225?228 (1976). · Zbl 0344.57009 · doi:10.1112/blms/8.3.225 |
| [60] | G. Wassermann, ?Stability of unfoldings,? Lect. Notes Math.,393, IX (1974). |
| [61] | G. Wassermann, ?Stability of caustics,? Math. Ann.,216, No. 1, 43?50 (1975). · Zbl 0293.58008 · doi:10.1007/BF02547971 |
| [62] | G. Wassermann, ?(r, s)-Stable unfoldings and catastrophe theory,? Lect. Notes Math.,525, 253?262 (1976). · Zbl 0335.58009 · doi:10.1007/BFb0077852 |
| [63] | G. Wassermann, ?Stability of unfoldings in space and time,? Acta Math.,135, Nos. 1?2, 57?128 (1975). · Zbl 0315.58010 · doi:10.1007/BF02392016 |
| [64] | A. Weinstein, ?Symplectic manifolds and their Lagrangian submanifolds,? Adv. Math.,6, 329?346 (1971). · Zbl 0213.48203 · doi:10.1016/0001-8708(71)90020-X |
| [65] | A. Weinstein, ?Lagrangian submanifolds and Hamiltonian systems,? Ann. Math.,98, No. 3, 377?410 (1973). · Zbl 0271.58008 · doi:10.2307/1970911 |
| [66] | A. Weinstein, ?Lectures on symplectic manifolds,? Univ. of North Carolina (1976). |
| [67] | L. C. Wilson, ?Infinitely determined map germs,? Can. J. Math.,33, No. 3, 671?684 (1981). · Zbl 0476.58005 · doi:10.4153/CJM-1981-053-3 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
