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A new formulation of the generalized Toda lattice equations and their fixed point analysis via the momentum map
Authors:
Anthony M. Bloch, Roger W. Brockett and Tudor S. Ratiu
Journal:
Bull. Amer. Math. Soc. 23 (1990), 477-485
MSC (1985):
Primary 34A05; Secondary 22E46
MathSciNet review:
1027895
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Additional Information
- 1.
M.
Adler, On a trace functional for formal pseudo differential
operators and the symplectic structure of the Korteweg-de\thinspace Vries
type equations, Invent. Math. 50 (1978/79),
no. 3, 219–248. MR 520927
(80i:58026), http://dx.doi.org/10.1007/BF01410079
- 2.
M.
F. Atiyah, Convexity and commuting Hamiltonians, Bull. London
Math. Soc. 14 (1982), no. 1, 1–15. MR 642416
(83e:53037), http://dx.doi.org/10.1112/blms/14.1.1
- 3.
Anthony
Bloch, Estimation, principal components and Hamiltonian
systems, Systems Control Lett. 6 (1985), no. 2,
103–108. MR
801020 (86k:93040), http://dx.doi.org/10.1016/0167-6911(85)90005-2
- 4.
A.
M. Bloch, Steepest descent, linear programming, and Hamiltonian
flows, Mathematical developments arising from linear programming
(Brunswick, ME, 1988), Contemp. Math., vol. 114, Amer. Math. Soc.,
Providence, RI, 1990, pp. 77–88. MR 1097866
(92d:58073), http://dx.doi.org/10.1090/conm/114/1097866
- 5.
A. M. Bloch, H. Flaschka, and T. Ratiu, A convexity theorem for isospectral sets of Jacobi matrices in a compact Lie algebra, Duke Math. J. (to appear).
- 6.
Anthony
M. Bloch, Roger
W. Brockett, and Tudor
S. Ratiu, Completely integrable gradient flows, Comm. Math.
Phys. 147 (1992), no. 1, 57–74. MR 1171760
(93f:58094)
- 7.
M.
R. Bremner, R.
V. Moody, and J.
Patera, Tables of dominant weight multiplicities for
representations of simple Lie algebras, Monographs and Textbooks in
Pure and Applied Mathematics, vol. 90, Marcel Dekker Inc., New York,
1985. MR
779462 (86f:17002)
- 8.
R.
W. Brockett, Least squares matching problems, Linear Algebra
Appl. 122/123/124 (1989), 761–777. MR 1020010
(90i:90116), http://dx.doi.org/10.1016/0024-3795(89)90675-7
- 9.
R. W. Brockett, Dynamical systems that sort lists and solve linear programming problems, Proc. 27th IEEE Conf. on Decision and Control, IEEE, New Jersey, 1988, pp. 799-803.
- 10.
C. I. Byrnes and J. C. Willems, Least squares estimation, linear programming and momentum, unpublished manuscript, 1984.
- 11.
Moody
T. Chu, The generalized Toda flow, the 𝑄𝑅 algorithm
and the center manifold theory, SIAM J. Algebraic Discrete Methods
5 (1984), no. 2, 187–201. MR 745438
(86g:58071), http://dx.doi.org/10.1137/0605020
- 12.
Michael
W. Davis, Some aspherical manifolds, Duke Math. J.
55 (1987), no. 1, 105–139. MR 883666
(88j:57044), http://dx.doi.org/10.1215/S0012-7094-87-05507-4
- 13.
P.
Deift, T.
Nanda, and C.
Tomei, Ordinary differential equations and the symmetric eigenvalue
problem, SIAM J. Numer. Anal. 20 (1983), no. 1,
1–22. MR
687364 (86k:58101), http://dx.doi.org/10.1137/0720001
- 14.
H.
Flaschka, The Toda lattice. I. Existence of integrals, Phys.
Rev. B (3) 9 (1974), 1924–1925. MR 0408647
(53 #12411)
- 15.
David
Fried, The cohomology of an isospectral
flow, Proc. Amer. Math. Soc.
98 (1986), no. 2,
363–368. MR
854048 (88b:58110), http://dx.doi.org/10.1090/S0002-9939-1986-0854048-6
- 16.
V.
Guillemin and S.
Sternberg, Convexity properties of the moment mapping, Invent.
Math. 67 (1982), no. 3, 491–513. MR 664117
(83m:58037), http://dx.doi.org/10.1007/BF01398933
- 17.
V.
Guillemin and S.
Sternberg, On the method of Symes for integrating systems of the
Toda type, Lett. Math. Phys. 7 (1983), no. 2,
113–115. MR
708432 (85k:58038), http://dx.doi.org/10.1007/BF00419928
- 18.
Alfred
Horn, Doubly stochastic matrices and the diagonal of a rotation
matrix, Amer. J. Math. 76 (1954), 620–630. MR 0063336
(16,105c)
- 19.
Bertram
Kostant, The principal three-dimensional subgroup and the Betti
numbers of a complex simple Lie group, Amer. J. Math.
81 (1959), 973–1032. MR 0114875
(22 #5693)
- 20.
Bertram
Kostant, On convexity, the Weyl group and the Iwasawa
decomposition, Ann. Sci. École Norm. Sup. (4)
6 (1973), 413–455 (1974). MR 0364552
(51 #806)
- 21.
Bertram
Kostant, The solution to a generalized Toda lattice and
representation theory, Adv. in Math. 34 (1979),
no. 3, 195–338. MR 550790
(82f:58045), http://dx.doi.org/10.1016/0001-8708(79)90057-4
- 22.
G.
L. Luke (ed.), Representation theory of Lie groups, London
Mathematical Society Lecture Note Series, vol. 34, Cambridge
University Press, Cambridge, 1979. MR 568880
(81j:22001)
- 23.
J. Lagarias, Monotonicity properties of the generalized Toda flow and QR flow, preprint, 1988.
- 24.
G.
L. Luke (ed.), Representation theory of Lie groups, London
Mathematical Society Lecture Note Series, vol. 34, Cambridge
University Press, Cambridge, 1979. MR 568880
(81j:22001)
- 25.
J. Moser, Finitely many mass points on the line under the influence of an exponential potential, Batelles Recontres, Springer Notes in Physics, Springer-Verlag, Berlin, 1974, pp. 417-497.
- 26.
W.
W. Symes, Hamiltonian group actions and integrable systems,
Phys. D 1 (1980), no. 4, 339–374. MR 601577
(83j:58063), http://dx.doi.org/10.1016/0167-2789(80)90017-2
- 27.
W.
W. Symes, Systems of Toda type, inverse spectral problems, and
representation theory, Invent. Math. 59 (1980),
no. 1, 13–51. MR 575079
(81g:58019), http://dx.doi.org/10.1007/BF01390312
- 28.
W.
W. Symes, The 𝑄𝑅 algorithm and scattering for the
finite nonperiodic Toda lattice, Phys. D 4 (1981/82),
no. 2, 275–280. MR 653781
(83h:58053), http://dx.doi.org/10.1016/0167-2789(82)90069-0
- 29.
Morikazu
Toda, Studies of a non-linear lattice, Phys. Rep.
18C (1975), no. 1, 1–123. MR 0489137
(58 #8610)
- 30.
Carlos
Tomei, The topology of isospectral manifolds of tridiagonal
matrices, Duke Math. J. 51 (1984), no. 4,
981–996. MR
771391 (86d:58091), http://dx.doi.org/10.1215/S0012-7094-84-05144-5
- 31.
Pierre
van Moerbeke, The spectrum of Jacobi matrices, Invent. Math.
37 (1976), no. 1, 45–81. MR 0650253
(58 #31226)
- 1.
- M. Adler, On a trace functional for pseudo-differential operators and the symplectic structure of Korteveg-Devries type equations,, Invent. Math. 50 (1979), 219-248. MR 520927
- 2.
- M.F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 16(1982), 1-15. MR 642416
- 3.
- A. M. Bloch, Estimation, principal components and Hamiltonian systems, Systems Control Lett., vol. 6, North-Holland, Amsterdam, 1985, pp. 103-108. MR 801020
- 4.
- A. M. Bloch, Steepest descent, linear programming and Hamiltonian Flows, Contemporary Math, (to appear). MR 1097866
- 5.
- A. M. Bloch, H. Flaschka, and T. Ratiu, A convexity theorem for isospectral sets of Jacobi matrices in a compact Lie algebra, Duke Math. J. (to appear).
- 6.
- A. M. Bloch, R. W. Brockett, and T. Ratiu, Completely integrable gradient flows (to appear). MR 1171760
- 7.
- M. R. Bremner, R. V. Moody, and J. Patera, Tables of dominant weight multiplicities for representations of simple Lie algebras, Marcel Dekker, New York, 1985. MR 779462
- 8.
- R. W. Brockett, Least squares matching problems,, Linear Algebra Appl. 122/123/124 (1989), 761-777. MR 1020010
- 9.
- R. W. Brockett, Dynamical systems that sort lists and solve linear programming problems, Proc. 27th IEEE Conf. on Decision and Control, IEEE, New Jersey, 1988, pp. 799-803.
- 10.
- C. I. Byrnes and J. C. Willems, Least squares estimation, linear programming and momentum, unpublished manuscript, 1984.
- 11.
- M. T. Chu, The generalized Toda lattice, the QR-algorithm and the centre manifold theory, SIAM J. Algebraic Discrete Methods 5 (1984), 187-201. MR 745438
- 12.
- M. W. Davis, Some aspherical manifolds, Duke Math. J. 5 (1987), 105-139. MR 883666
- 13.
- P. Deift, T. Nanda, and C. Tomei, Differential equations for the symmetric eigenvalue problem, SIAM J. Numer. Anal. 20 (1983), 1-22. MR 687364
- 14.
- H. Flaschka, The Toda lattice, Phys. Rev. B(3) 9 (1976), 1924-1925. MR 408647
- 15.
- D. Fried, The cohomology of an isospectral flow, Proc. Amer. Math. Soc. 98 (1986), 363-368. MR 854048
- 16.
- V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), 491-513. MR 664117
- 17.
- V. Guillemin and S. Sternberg, On the method of Symes for integrating systems of the Toda type, Lett. Math. Phys. 7 (1983), 113-115. MR 708432
- 18.
- A. Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, Amer. J. Math. 76 (1956), 620-630. MR 63336
- 19.
- B. Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973-1032. MR 114875
- 20.
- B. Kostant, On convexity, the Weyl group and the Iwasawa decomposition, Ann. Sci. École Norm. Sup. (4) 6 (1973), 413-455. MR 364552
- 21.
- B. Kostant, The solution to a generalized Toda lattice and representation theory, Adv. in Math. 34 (1979), 195-338. MR 550790
- 22.
- B. Kostant, Quantization and representation theory, in Representation theory of Lie groups, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, New York, vol. 34, 1979, pp. 91-150. MR 568880
- 23.
- J. Lagarias, Monotonicity properties of the generalized Toda flow and QR flow, preprint, 1988.
- 24.
- I. G. MacDonald, Algebraic structure of Lie groups, in Representation theory of Lie groups, London Math. Soc. Lecture Note Ser. Cambridge University Press, New York, vol. 34, 1979, pp. 91-150. MR 568880
- 25.
- J. Moser, Finitely many mass points on the line under the influence of an exponential potential, Batelles Recontres, Springer Notes in Physics, Springer-Verlag, Berlin, 1974, pp. 417-497.
- 26.
- W. W. Symes, Hamiltonian group actions and integrable systems, Phys. D 1 (1980), 339-376. MR 601577
- 27.
- W. W. Symes, Systems of Toda type, inverse spectral problems and representation theory, Invent. Math. 59 (1982), 13-51. MR 575079
- 28.
- W. W. Symes, The QR algorithm and scattering for the nonperiodic Toda lattice, Phys. D 4 (1982), 275-280. MR 653781
- 29.
- M. Toda, Studies of a non-linear lattice, Phys. Rep. 8 (1975), 1-125. MR 489137
- 30.
- C. Tomei, The topology of isospectral manifolds of diagonal matrices, Duke Math. J. 51 (1984), 981-996. MR 771391
- 31.
- P. van Moerbeke, The spectrum of Jacobi matrices, Invent. Math. 37 (1976), 45-81. MR 650253
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0273-0979-1990-15960-9
PII:
S 0273-0979(1990)15960-9
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