integrable systems
On hamiltonian and integrable systems.
(continued here).
References are taken from:
A.M. Bloch; R.W. Brockett; T.S. Ratiu "A new formulation of the generalized
Toda lattice equations and their fixed point analysis via the momentum map"
Bull. Amer. Math. Soc. 23 (1990), 477-485.
http://www.ams.org/journals/bull/1990-23-02/S0273-0979-1990-15960-9/home.html
and
M.-A. Belabbas, R.W. Brockett "Input-output Hamiltonian Systems and
Neural Signal Processing" Proc. of the 45th IEEE Conf. on Decision & Control.
Manchester Grand Hyatt Hotel, San Diego, CA, USA, Dec.13-15, 2006
Some references:
Flaschka H. "The Toda lattice. II. Existence of integrals"
Phys. Rev. B 9, 1924-1925 (1974)
http://prb.aps.org/abstract/PRB/v9/i4/p1924_1
Kostant B. ”The Solution to a Generalized Toda Lattice
and Representation Theory” Adv. Math., 39, pp 195-338, 1979.
http://dx.doi.org/10.1016/0001-8708(79)90057-4
Kostant, B. "Quantization and representation theory" Lond.M.S. 1979
http://free-books.dontexist.com/search?req=Kostant&nametype=orig
Kostant, B. "The principal three-dimensional subgroup and the Betti numbers
of a complex simple Lie group." Amer. J. Math. 81 1959 973--1032.
http://www.jstor.org/stable/2372999?origin=crossref&cookieSet=1
Toda, M. "Studies of a non-linear lattice." Phys. Rep. 18C (1975), no. 1, 1--123.
http://dx.doi.org/10.1016/0370-1573(75)90018-6
Lax P.D. "Integrals of nonlinear equations of evolution and solitary waves"
Communications on pure and applied mathematics 1968; 21: 467 - 90
http://dx.doi.org/10.1002/cpa.3160210503
Atiyah, M. F. "Convexity and commuting Hamiltonians."
Bull. London Math. Soc. 14 (1982), no. 1, 1--15.
http://blms.oxfordjournals.org/cgi/reprint/14/1/1
http://dx.doi.org/10.1112/blms/14.1.1
Symes, W. W. "Hamiltonian group actions and integrable
systems" Phys. D 1 (1980), no. 4, 339--374.
http://dx.doi.org/10.1016/0167-2789(80)90017-2
Symes, W. W. "Systems of Toda type, inverse spectral problems,
and representation theory" Invent. Math. 59 (1980), no. 1, 13--51.
http://www.springerlink.com/content/q772766522867532/
R. W. Brockett "Least squares matching problems" Linear Algebra
and its Applications, Vol.122-124, (1989), Pp.: 761-777
http://linkinghub.elsevier.com/retrieve/pii/0024379589906757
R. W. Brockett "Dynamical systems that sort lists, diagonalize matrices,
and solve linear programming problems" Linear algebra appl. 146(1991), 79-91.
http://linkinghub.elsevier.com/retrieve/pii/002437959190021N
NN implications:
Cariani, P. ”Neural timing nets.” Neural Networks, 14(6-7), pp 737-753, 2001
http://homepage.mac.com/cariani/CarianiWebsite/CarianiNN2001.pdf
Cariani, P. ”Temporal codes and computations for sensory representation and
scene analysis,” in IEEE Transactions on Neural Networks, Vol.15, I.5, 1100-1111
http://www.cariani.com/
additional:
"Representation Theory of Lie Groups"
Series: London Mathematical Society Lecture Note Series (No. 34)
M. F. Atiyah, R. Bott, S. Helgason, D. Kazhdan, B. Kostant, G. Lustztig
Paperback (ISBN-13: 9780521226363 | ISBN-10: 0521226368) 1980
"Representation theory of Lie groups" Proc. of the SRC/LMS Research
Symposium Oxford, (1977). Edited by G. L. Luke. London Mathematical
Society Lecture Note Series, 34. Cambridge University Press,
Cambridge-New York, 1979. v+341 pp. ISBN: 0-521-22636-8
c.f. " Topics in Mathematical System Theory" by Kalman, Falb, Arbib,
part IV (representing the state of art in 1969).
Some links from the review of B.Kostant:
I.M.Gelfand, D.A.Kazhdan, "Representations of the group Gl(n, K) where K is a local
field," in “Lie Groups and Their Representations,” pp. 95-118, Wiley, New York, 1975.
http://mi.mathnet.ru/rus/faa/v6/i4/p73
0.I.Bogoyavlensky, "On perturbations of the periodic Toda lattice," Comm. Math.
Phys. 51 (1976), 201-209. http://projecteuclid.org/euclid.cmp/1103900387
c.f.: http://www.mathnet.ru/php/person.phtml?personid=8578
M.A.Olshanetsky, A.M.Perelomov, "Completely integrable Hamiltonian systems
connected with semi-simple Lie algebras," Invent. Math. 37 (1976), 93-108.
http://www.springerlink.com/content/l402502357h75344/
c.f.: http://mi.mathnet.ru/rus/faa/v11/i1/p75
M.A.Olshanetsky, A.M.Perelomov, "Explicit solution of the classical generalized
Toda models," (ITEP-157-1978), Invent. Math. 54 (1979), 261-269.
http://www.springerlink.com/content/w2203m01647ll485/
c.f.: http://mi.mathnet.ru/rus/tmf/v45/i1/p3
A.N.Leznov, M.V.Saveliev, "Representation of zero curvature for the system of nonlinear
partial differential equations $x_{\alpha,z\overline{z}}=\exp(kx)_\alpha$, and its integrability,"
Функц. анализ и его прил., 14:3 (1980), 87-88 http://mi.mathnet.ru/rus/faa/v14/i3/p87
additional:
И. М. Гельфанд, М. А. Наймарк "Унитарные представления классических групп"
Тр. МИАН СССР, 36, Изд-во АН СССР, М.-Л., 1950, 288 с.
Д. П. Желобенко, “Представления полупростых комплексных групп Ли”,
Итоги науки и техн. Сер. Мат. анал., 11, ВИНИТИ, М., 1973, 51-90
http://mi.mathnet.ru/rus/intm/v11/p51
T. Ceccherini-Silberstein, F. Scarabotti and F. Tolli
"Representation Theory of the Symmetric Groups"
Cambridge Univ. Press, 2010 http://gigapedia.com/items/434624/
books in russian:
Переломов А.М. "Интегрируемые системы классической механики и алгебры Ли." 2002. 238 с.
Лезнов А.Н., Савельев М.В. "Групповые методы интегрирования нелинейных динамических систем" 1985. 280 с.
Желобенко Д.П., Штерн А.Н. "Представления групп Ли" Наука, 1983. 190 с.
Желобенко Д.П. "Основные структуры и методы теории представлений" 2004. 488 с.
Желобенко Д.П. "Гармонический анализ на полупростых комплексных группах Ли" 1974. 240 с.
Гельфанд И.М., Граев М.И., Виленкин Н.Я. "Интегральная геометрия и связанные с ней вопросы
теории представлений" [Обобщенные функции. Вып.5] (Гос. изд. физ.-мат. лит., 1962, 335 стр.)
Гельфанд И.М., Минлос Р.А., Шапиро З.Я. "Представления группы
вращений и группы Лоренца, их применения." 1958. 368 с.
Хамфрис Дж. "Введение в теорию алгебр Ли и их представлений" 2003. 216 с.
( previously, on switched systems).
(continued here).
References are taken from:
A.M. Bloch; R.W. Brockett; T.S. Ratiu "A new formulation of the generalized
Toda lattice equations and their fixed point analysis via the momentum map"
Bull. Amer. Math. Soc. 23 (1990), 477-485.
http://www.ams.org/journals/bull/1990-23-02/S0273-0979-1990-15960-9/home.html
and
M.-A. Belabbas, R.W. Brockett "Input-output Hamiltonian Systems and
Neural Signal Processing" Proc. of the 45th IEEE Conf. on Decision & Control.
Manchester Grand Hyatt Hotel, San Diego, CA, USA, Dec.13-15, 2006
Some references:
Flaschka H. "The Toda lattice. II. Existence of integrals"
Phys. Rev. B 9, 1924-1925 (1974)
http://prb.aps.org/abstract/PRB/v9/i4/p1924_1
Kostant B. ”The Solution to a Generalized Toda Lattice
and Representation Theory” Adv. Math., 39, pp 195-338, 1979.
http://dx.doi.org/10.1016/0001-8708(79)90057-4
Kostant, B. "Quantization and representation theory" Lond.M.S. 1979
http://free-books.dontexist.com/search?req=Kostant&nametype=orig
Kostant, B. "The principal three-dimensional subgroup and the Betti numbers
of a complex simple Lie group." Amer. J. Math. 81 1959 973--1032.
http://www.jstor.org/stable/2372999?origin=crossref&cookieSet=1
Toda, M. "Studies of a non-linear lattice." Phys. Rep. 18C (1975), no. 1, 1--123.
http://dx.doi.org/10.1016/0370-1573(75)90018-6
Lax P.D. "Integrals of nonlinear equations of evolution and solitary waves"
Communications on pure and applied mathematics 1968; 21: 467 - 90
http://dx.doi.org/10.1002/cpa.3160210503
Atiyah, M. F. "Convexity and commuting Hamiltonians."
Bull. London Math. Soc. 14 (1982), no. 1, 1--15.
http://blms.oxfordjournals.org/cgi/reprint/14/1/1
http://dx.doi.org/10.1112/blms/14.1.1
Symes, W. W. "Hamiltonian group actions and integrable
systems" Phys. D 1 (1980), no. 4, 339--374.
http://dx.doi.org/10.1016/0167-2789(80)90017-2
Symes, W. W. "Systems of Toda type, inverse spectral problems,
and representation theory" Invent. Math. 59 (1980), no. 1, 13--51.
http://www.springerlink.com/content/q772766522867532/
R. W. Brockett "Least squares matching problems" Linear Algebra
and its Applications, Vol.122-124, (1989), Pp.: 761-777
http://linkinghub.elsevier.com/retrieve/pii/0024379589906757
R. W. Brockett "Dynamical systems that sort lists, diagonalize matrices,
and solve linear programming problems" Linear algebra appl. 146(1991), 79-91.
http://linkinghub.elsevier.com/retrieve/pii/002437959190021N
NN implications:
Cariani, P. ”Neural timing nets.” Neural Networks, 14(6-7), pp 737-753, 2001
http://homepage.mac.com/cariani/CarianiWebsite/CarianiNN2001.pdf
Cariani, P. ”Temporal codes and computations for sensory representation and
scene analysis,” in IEEE Transactions on Neural Networks, Vol.15, I.5, 1100-1111
http://www.cariani.com/
additional:
"Representation Theory of Lie Groups"
Series: London Mathematical Society Lecture Note Series (No. 34)
M. F. Atiyah, R. Bott, S. Helgason, D. Kazhdan, B. Kostant, G. Lustztig
Paperback (ISBN-13: 9780521226363 | ISBN-10: 0521226368) 1980
"Representation theory of Lie groups" Proc. of the SRC/LMS Research
Symposium Oxford, (1977). Edited by G. L. Luke. London Mathematical
Society Lecture Note Series, 34. Cambridge University Press,
Cambridge-New York, 1979. v+341 pp. ISBN: 0-521-22636-8
c.f. " Topics in Mathematical System Theory" by Kalman, Falb, Arbib,
part IV (representing the state of art in 1969).
Some links from the review of B.Kostant:
I.M.Gelfand, D.A.Kazhdan, "Representations of the group Gl(n, K) where K is a local
field," in “Lie Groups and Their Representations,” pp. 95-118, Wiley, New York, 1975.
http://mi.mathnet.ru/rus/faa/v6/i4/p73
0.I.Bogoyavlensky, "On perturbations of the periodic Toda lattice," Comm. Math.
Phys. 51 (1976), 201-209. http://projecteuclid.org/euclid.cmp/1103900387
c.f.: http://www.mathnet.ru/php/person.phtml?personid=8578
M.A.Olshanetsky, A.M.Perelomov, "Completely integrable Hamiltonian systems
connected with semi-simple Lie algebras," Invent. Math. 37 (1976), 93-108.
http://www.springerlink.com/content/l402502357h75344/
c.f.: http://mi.mathnet.ru/rus/faa/v11/i1/p75
M.A.Olshanetsky, A.M.Perelomov, "Explicit solution of the classical generalized
Toda models," (ITEP-157-1978), Invent. Math. 54 (1979), 261-269.
http://www.springerlink.com/content/w2203m01647ll485/
c.f.: http://mi.mathnet.ru/rus/tmf/v45/i1/p3
A.N.Leznov, M.V.Saveliev, "Representation of zero curvature for the system of nonlinear
partial differential equations $x_{\alpha,z\overline{z}}=\exp(kx)_\alpha$, and its integrability,"
Функц. анализ и его прил., 14:3 (1980), 87-88 http://mi.mathnet.ru/rus/faa/v14/i3/p87
additional:
И. М. Гельфанд, М. А. Наймарк "Унитарные представления классических групп"
Тр. МИАН СССР, 36, Изд-во АН СССР, М.-Л., 1950, 288 с.
Д. П. Желобенко, “Представления полупростых комплексных групп Ли”,
Итоги науки и техн. Сер. Мат. анал., 11, ВИНИТИ, М., 1973, 51-90
http://mi.mathnet.ru/rus/intm/v11/p51
T. Ceccherini-Silberstein, F. Scarabotti and F. Tolli
"Representation Theory of the Symmetric Groups"
Cambridge Univ. Press, 2010 http://gigapedia.com/items/434624/
books in russian:
Переломов А.М. "Интегрируемые системы классической механики и алгебры Ли." 2002. 238 с.
Лезнов А.Н., Савельев М.В. "Групповые методы интегрирования нелинейных динамических систем" 1985. 280 с.
Желобенко Д.П., Штерн А.Н. "Представления групп Ли" Наука, 1983. 190 с.
Желобенко Д.П. "Основные структуры и методы теории представлений" 2004. 488 с.
Желобенко Д.П. "Гармонический анализ на полупростых комплексных группах Ли" 1974. 240 с.
Гельфанд И.М., Граев М.И., Виленкин Н.Я. "Интегральная геометрия и связанные с ней вопросы
теории представлений" [Обобщенные функции. Вып.5] (Гос. изд. физ.-мат. лит., 1962, 335 стр.)
Гельфанд И.М., Минлос Р.А., Шапиро З.Я. "Представления группы
вращений и группы Лоренца, их применения." 1958. 368 с.
Хамфрис Дж. "Введение в теорию алгебр Ли и их представлений" 2003. 216 с.
( previously, on switched systems).