Topological problems of the theory of wave propagation

Russian Mathematical Surveys

Volumn 51, Issue 1, 1996, Pages 1-47

  Arnold, V I ^a  

^a NONE

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EID: 0030494625     PISSN: 00360279     EISSN: 14684829     Source Type: Journal    
DOI: 10.1070/RM1996v051n01ABEH002734     Document Type: Article

References (88)

1. H. Weyl, "Invariants", Duke Math. J. 5 (1939), 500.
2. R. Feynman, Surely you're joking, Mr. Feynman!, Bantam Books, Toronto-New York 1986, pp. 191-198.
3. C. G. J. Jacobi, Vorlesungen über Dynamik, G. Reiner, Berlin 1884.
4. V. M. Zakalyukin, "Small deformations of Lagrangian collapse", Preprint, The University of Liverpool 1995.
5. V. I. Arnold, "Sur les propriétés des projections Lagrangiennes en géométrie symplectique des caustiques", Cahiers de Mathématiques de la Decision, Vol. 9320, CEREMADE, 1993, pp. 1-9.
6. S. L. Tabachnikov, "Around four vertices", Uspekhi Mat. Nauk 45:1 (1990), 229-230; English transl. in Russian Math. Surveys 45:1 (1990).
7. S. L. Tabachnikov, "Around four vertices", Uspekhi Mat. Nauk 45:1 (1990), 229-230; English transl. in Russian Math. Surveys 45:1 (1990).
8. A. Agrachev, "Methods of control theory in nonholonomic geometry", Proc. ICM Zürich, Birkhäuser 1995, pp. 1-11.
9. V. I. Arnold, "On topological properties of Legendrian projections in the contact geometry of wave fronts", Algebra i Analiz 6:3 (1994), 1-16; English transl. in St. Petersburg Math. J. 6 (1995).
10. V. I. Arnold, "On topological properties of Legendrian projections in the contact geometry of wave fronts", Algebra i Analiz 6:3 (1994), 1-16; English transl. in St. Petersburg Math. J. 6 (1995).
11. V. I. Arnold, "Plane curves, their invariants, perestroikas and classifications", in: Singularities and bifurcations (V. I. Arnold, ed.), Adv. Soviet Math., 21, AMS, Providence 1994, pp. 33-91.
12. V. I. Arnold, Topological invariants of plane curves and caustics, University Lecture Series, 5, AMS, Providence 1994.
13. V. I. Arnold, "Invariants and perestroikas of fronts on the plane", Trudy Mat. Inst. Steklov. 209 (1995), 14-64; English transl. in Proc. Steklov. Inst. Mat. 209 (1996).
14. V. I. Arnold, "Invariants and perestroikas of fronts on the plane", Trudy Mat. Inst. Steklov. 209 (1995), 14-64; English transl. in Proc. Steklov. Inst. Mat. 209 (1996).
15. V. I. Arnold, "The geometry of spherical curves and the algebra of quaternions", Uspekhi Mat. Nauk 50:1 (1995), 3-68; English transl. in Russian Math. Surveys 50:1 (1995).
16. V. I. Arnold, "The geometry of spherical curves and the algebra of quaternions", Uspekhi Mat. Nauk 50:1 (1995), 3-68; English transl. in Russian Math. Surveys 50:1 (1995).
17. F. Aicardi, "Tree-like curves", in: Singularities and bifurcations (V. I. Arnold, ed.), Adv. Soviet Math., 21, AMS, Providence 1994, pp. 1-31.
18. O. Viro, "First degree invariants of generic curves on surfaces", Preprint U.U.D.M. Report 1994: 21. ISSN 1101-3591, Dept. of Math., Uppsala University 1994
19. S. Tabachnikov, "Invariants of smooth triple point free plane curves", Preprint U of A-R-77, 1994.
20. A. Shumakovich, "Explicit formulae for the strangeness of a plane curve", Algebra i Analiz 7:3 (1995), 165-199; English transl. in St. Petersburg Math. J. 7 (1996).
21. A. Shumakovich, "Explicit formulae for the strangeness of a plane curve", Algebra i Analiz 7:3 (1995), 165-199; English transl. in St. Petersburg Math. J. 7 (1996).
22. M. Polyak, "Invariants of plane curves and fronts via Gauss diagrams", Preprint, Max Planck Inst. für Math., Bonn 1994.
23. M. Polyak, "On the Bennequin invariant of Legendrian curves and its quantization", Preprint, Max Planck Inst. für Math., Bonn 1995; C.R. Acad. Sci. Paris Sér. I 322 (1996), 77-82.
24. M. Polyak, "On the Bennequin invariant of Legendrian curves and its quantization", Preprint, Max Planck Inst. für Math., Bonn 1995; C.R. Acad. Sci. Paris Sér. I 322 (1996), 77-82.
25. G. Mikhalkin and M. Polyak, "Whitney formula in higher dimensions", Preprint, Max Planck Inst. für Math., Bonn 1995.
26. F. Aicardi, "Topological invariants of knots and framed knots in the solid torus", C.R. Acad. Sci. Paris 321 (1995), 81-86.
27. F. Aicardi, "Topological invariants of legendrian curves", C.R. Acad. Sci. Paris 321 (1995), 199-204.
28. X.-S. Lin and Z. Wang, "Integral geometry of plane curves and knot invariants", J. Diff. Geom. (1995) (to appear).
29. E. Ferrand, "On the Bennequin invariant and the geometry of wave fronts", Preprint, Ecole Polytechnique Palaiseau, 1995.
30. V. I. Arnold, "The Vassiliev theory of discriminants and knots", First Europ. Congr. Math. Paris, 1992; Progress in Mathematics, vol. 119, Birkhäuser, Basel 1994, pp. 1-29.
31. V. I. Arnold, "The Vassiliev theory of discriminants and knots", First Europ. Congr. Math. Paris, 1992; Progress in Mathematics, vol. 119, Birkhäuser, Basel 1994, pp. 1-29.
32. A. Hurwitz, "Über die Fourierschen Konstanten integrierbarer Funktionen", Math. Ann. 57 (1903), 425-446.
33. O. Kellogg, "The oscillation of functions on an orthogonal set", Amer. J. Math. 38 (1916), 1-5.
34. V. I. Arnold, "On the number of flattening points on space curves", Sinai's Moscow seminar on dynamical systems, 11-22 AMS Translations (2) 171.
35. V. I. Arnold, "Sur une propriété topologique des applications globalement canoniques de la mécanique classique", C.R. Acad. Sci. Paris Sec. A 261 (1965), 3719-3722.
36. V. Ovsienko and S. Tabachnikov, "Sturm theory, Ghys' theorem on zeroes of the Schwarzian derivative and flattening of Legendrian curves", Preprint CPT-CNRS. Luminy-Marseille 1995; Selecta Math., 1995 (to appear).
37. V. I. Arnold, "Remarks on the extatic points of plane curves", Cahiers de Mathematiques de la Decision, vol. 9529, CEREMADE (URA CNRS 749), 1995, pp. 1-16; V. I. Arnold, I. M. Gelfand, and M. S. Smirnov (eds.), Geometry and singularity theory, Birkhäuser, Basel 1995.
38. V. I. Arnold, "Remarks on the extatic points of plane curves", Cahiers de Mathematiques de la Decision, vol. 9529, CEREMADE (URA CNRS 749), 1995, pp. 1-16; V. I. Arnold, I. M. Gelfand, and M. S. Smirnov (eds.), Geometry and singularity theory, Birkhäuser, Basel 1995.
39. V. I. Arnold, "Commentary on "A geometrical theorem" of H. Poincaré", The collected works of H. Poincaré, Vol. 2, Nauka, Moscow 1972, pp. 987-989. (Russian)
40. C. Conley and E. Zehnder, "The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnold", Invent. Math. 73 (1983), 33-49.
41. M. Chaperon, "Quelque questions de géométrie symplectique [d'après, entre autres, Poincaré, Arnold, Conley et Zehnder]", Astérisque 105-106 (1983), 231-249.
42. F. Laudenbach and J.-C. Sikorav, "Persistence d'intersection avec la section nulle au course d'une isotopie hamiltonienne dans un fibré cotangent", Invent. Math. 82 (1985), 349-357.
43. Yu. V. Chekanov, "Legendrian Morse theory", Uspekhi Mat. Nauk 42:4 (1987), 139-141.
44. A. Floer, "Morse theory and Lagrangian intersections", J. Diff. Geom. 28 (1988), 513-547.
45. A. Floer, "Proof of the Arnold conjecture and generalizations to certain Kähler manifolds", Duke Math. J. 53 (1986), 1-32.
46. A. Floer, "Symplectic fixed points and holomorphic spheres", Comm. Math. Phys. 120 (1989), 576-611.
47. A. Floer and H. Hofer, "Symplectic homology. I", Math. Z. 215 (1994), 37-88.
48. M. Gromov, "Pseudo-holomorphic curves in symplectic manifolds", Invent. Math. 82 (1985), 307-347.
49. K. Ono, "On the Arnold conjecture for weakly monotone symplectic manifolds", Preprint, 1994.
50. A. Floer, "An instanton invariant for 3-manifolds", Comm. Math. Phys. 118 (1988), 215-240.
51. M. Atiyah, "New invariants of 3-and 4-dimensional manifolds", Proc. Sympos. Pure Math., 48, AMS, Providence 1988, pp. 285-289.
52. V. I. Arnold, "First steps in symplectic topology", Uspekhi Mat. Nauk 41:6 (1986), 3-18; English transl. in Russian Math. Surveys 41:6 (1986).
53. V. I. Arnold, "First steps in symplectic topology", Uspekhi Mat. Nauk 41:6 (1986), 3-18; English transl. in Russian Math. Surveys 41:6 (1986).
54. Yu. V. Chekanov, "Critical points of quasifunctions and generating families of Legendrian manifolds", Funktsional. Anal. i Prilozhen. 29 (1995) (to appear); English transl. in Functional Anal. Appl. 29 (1995).
55. Yu. V. Chekanov, "Critical points of quasifunctions and generating families of Legendrian manifolds", Funktsional. Anal. i Prilozhen. 29 (1995) (to appear); English transl. in Functional Anal. Appl. 29 (1995).
56. M. Chaperon, "On generating families", Floer Memorial Volume, Birkhäuser, Basel 1995, pp. 283-296.
57. A. B. Givental, "Nonlinear generalization of the Maslov index", in: Theory of singularities and its applications (V. I. Arnold, ed.); Adv. Soviet Math. 1, 1990, pp. 71-103.
58. A. B. Givental, "Nonlinear generalization of the Maslov index", in: Theory of singularities and its applications (V. I. Arnold, ed.); Adv. Soviet Math. 1, 1990, pp. 71-103.
59. M. Kazarian, "Nonlinear version of Arnold's theorem on flattening points", C.R. Acad. Sci. Paris (1996) (to appear).
60. n", Bull. Amer. Math. Soc. 25 (1991), 75-80.
61. S. Anisov, "Projective convex curves", in: Geometry and singularity theory (V. I. Arnold, I. M. Gelfand, M. S. Smirnov, eds.), Birkhäuser, Basel-Boston 1995.
62. 4, hyperbolicity and projective topology", Sibirsk. Mat. Zh. 29:4 (1988), 36-47; English trans. in Siberian Math. J. 29:4 (1988).
63. 4, hyperbolicity and projective topology", Sibirsk. Mat. Zh. 29:4 (1988), 36-47; English trans. in Siberian Math. J. 29:4 (1988).
64. R. Uribe, "On the vertices of convex space curves", C.R. Acad. Sci. Paris (1995) (to appear).
65. V. D. Sedykh, "A connection between Lagrangian and Legendrian singularities under stereographic projection", Mat. Sb. 185:12 (1994), 123-130; English transl. in Russian Acad. Sci. Sb. Math. 83 (1996).
66. V. D. Sedykh, "A connection between Lagrangian and Legendrian singularities under stereographic projection", Mat. Sb. 185:12 (1994), 123-130; English transl. in Russian Acad. Sci. Sb. Math. 83 (1996).
67. V. Ovsienko and B. Khesin, "Symplectic leaves of the Gelfand-Dikii brackets and homotopy classes of nondegenerate curves", Funktsional. Anal. i Prilozhen. 24:1 (1990), 38-47; English transl. in Functional Anal. Appl. 24 (1990).
68. V. Ovsienko and B. Khesin, "Symplectic leaves of the Gelfand-Dikii brackets and homotopy classes of nondegenerate curves", Funktsional. Anal. i Prilozhen. 24:1 (1990), 38-47; English transl. in Functional Anal. Appl. 24 (1990).
69. V. I. Arnold, "Bernoulli-Euler updown numbers associated with function singularities, their combinatorics and arithmetics", Duke Math. J. 63 (1991), 537-555.
70. V. I. Arnold, "Critical points of functions and the classification of caustics", Uspekhi Mat. Nauk 29:2 (1974), 243-244.
71. E. Looijenga, "The complement of the bifurcation variety of a simple singularity", Invent. Math. 23 (1974), 105-116.
72. S. Zdravkovska, "The topological classification of polynomial maps", Uspekhi Mat. Nauk 25:4 (1970), 179-180.
73. R. Thorn, "L'équivalence d'une fonction différentiable et d'un polynome", Topology 3 (1965), 297-307.
74. C. Davis, "Properties of maps of certain Chebyshev systems", Dokl. Akad. Nauk 175 (1958), 280-283.
75. C. Davis, "Extrema of a polynomial", Amer. Math. Monthly 64 (1957), 679-680.
76. V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of differentiable maps, Vol. 2, Nauka, Moscow 1984, pp. 64-66; English transl., Birkhäuser, Boston 1988, pp. 85-88.
77. V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade, Singularities of differentiable maps, Vol. 2, Nauka, Moscow 1984, pp. 64-66; English transl., Birkhäuser, Boston 1988, pp. 85-88.
78. A. Glutsyuk, "An analogue of Cayley's theorem for cyclic symmetric connected graphs", Uspekhi Mat. Nauk 48:2 (1993), 173-174; English transl. in Russian Math. Surveys 48:2 (1993).
79. A. Glutsyuk, "An analogue of Cayley's theorem for cyclic symmetric connected graphs", Uspekhi Mat. Nauk 48:2 (1993), 173-174; English transl. in Russian Math. Surveys 48:2 (1993).
80. L. Guieu, E. Mourre, and V. Yu. Ovsienko, "Theorem on six vertices of a plane curve via Sturm theory", Preprint CPT, Luminy-Marseille 1995.
81. R. Courant and D. Hubert, Methods of mathematical physics, Vol. 1, Ch. 6, Interscience, New York 1953.
82. B. Segre, The non-singular cubic surfaces, Clarendon Press, Oxford 1942.
83. S. M. Gusein-Zade and S. M. Natanson, "Arf invariant and Arnold invariant", in: Geometry and singularity theory (V. I. Arnold, I. M. Gelfand, M. S. Smirnov, eds.), Birkhäuser, Basel-Boston 1995.
84. V. V. Goryunov, "Geometry of bifurcation diagrams of simple projections onto the line", Funktsional. Anal. i Prilozhen. 15:2 (1981), 1-8; English transl. in Functional Anal. Appl. 15 (1981).
85. V. V. Goryunov, "Geometry of bifurcation diagrams of simple projections onto the line", Funktsional. Anal. i Prilozhen. 15:2 (1981), 1-8; English transl. in Functional Anal. Appl. 15 (1981).
86. μ singularities", in: Singularities and bifurcations (V. I. Arnold, ed.), Adv. Soviet Math., 21, AMS, Providence 1994, pp. 163-188.
87. V. I. Arnold, "The topological classification of complex trigonometric polynomials and the combinatorics of graphs with the same number of vertices and edges", Funktsional. Anal. i Prilozhen. 30:1 (1996); English transl. in Functional Anal. Appl. 30 (1996).
88. V. I. Arnold, "The topological classification of complex trigonometric polynomials and the combinatorics of graphs with the same number of vertices and edges", Funktsional. Anal. i Prilozhen. 30:1 (1996); English transl. in Functional Anal. Appl. 30 (1996).