Differential galois groups and representation of quivers for seismic models with constant hessian of square of slowness
The trajectory of energy is modeled by the solution of the Eikonal equation, which can be solved by solving a Hamiltonian system. This system is amenable of treatment from the point of view of the theory of differential algebra. In particular, by Morales-Ramis theory, it is possible to analyze integ...
- Autores:
-
Acosta-Humánez, Primitivo
Giraldo, Hernán
Piedrahita, Carlos
- Tipo de recurso:
- Fecha de publicación:
- 2017
- Institución:
- Universidad Simón Bolívar
- Repositorio:
- Repositorio Digital USB
- Idioma:
- eng
- OAI Identifier:
- oai:bonga.unisimon.edu.co:20.500.12442/1896
- Acceso en línea:
- http://hdl.handle.net/20.500.12442/1896
- Palabra clave:
- Differential Galois theory
Eikonal equation
Hamilton equation
Helmholtz equation
High frequency approximation
Morales-Ramis theory
Ray theory
Representations of quivers
- Rights
- License
- Licencia de Creative Commons Reconocimiento-NoComercial-CompartirIgual 4.0 Internacional
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| dc.title.eng.fl_str_mv |
Differential galois groups and representation of quivers for seismic models with constant hessian of square of slowness |
| title |
Differential galois groups and representation of quivers for seismic models with constant hessian of square of slowness |
| spellingShingle |
Differential galois groups and representation of quivers for seismic models with constant hessian of square of slowness Differential Galois theory Eikonal equation Hamilton equation Helmholtz equation High frequency approximation Morales-Ramis theory Ray theory Representations of quivers |
| title_short |
Differential galois groups and representation of quivers for seismic models with constant hessian of square of slowness |
| title_full |
Differential galois groups and representation of quivers for seismic models with constant hessian of square of slowness |
| title_fullStr |
Differential galois groups and representation of quivers for seismic models with constant hessian of square of slowness |
| title_full_unstemmed |
Differential galois groups and representation of quivers for seismic models with constant hessian of square of slowness |
| title_sort |
Differential galois groups and representation of quivers for seismic models with constant hessian of square of slowness |
| dc.creator.fl_str_mv |
Acosta-Humánez, Primitivo Giraldo, Hernán Piedrahita, Carlos |
| dc.contributor.author.none.fl_str_mv |
Acosta-Humánez, Primitivo Giraldo, Hernán Piedrahita, Carlos |
| dc.subject.eng.fl_str_mv |
Differential Galois theory Eikonal equation Hamilton equation Helmholtz equation High frequency approximation Morales-Ramis theory Ray theory Representations of quivers |
| topic |
Differential Galois theory Eikonal equation Hamilton equation Helmholtz equation High frequency approximation Morales-Ramis theory Ray theory Representations of quivers |
| description |
The trajectory of energy is modeled by the solution of the Eikonal equation, which can be solved by solving a Hamiltonian system. This system is amenable of treatment from the point of view of the theory of differential algebra. In particular, by Morales-Ramis theory, it is possible to analyze integrable Hamiltonian systems through the abelian structure of their variational equations. In this paper, we obtain the abelian differential Galois groups and the representation of the quiver, that allow us to obtain such abelian differential Galois groups, for some seismic models with constant Hessian of square of slowness, proposed in [20], which are equivalent to linear Hamiltonian systems with three uncoupled harmonic oscillators. |
| publishDate |
2017 |
| dc.date.issued.none.fl_str_mv |
2017 |
| dc.date.accessioned.none.fl_str_mv |
2018-03-21T22:41:49Z |
| dc.date.available.none.fl_str_mv |
2018-03-21T22:41:49Z |
| dc.type.eng.fl_str_mv |
article |
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http://purl.org/coar/resource_type/c_6501 |
| dc.identifier.issn.none.fl_str_mv |
09720871 |
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http://hdl.handle.net/20.500.12442/1896 |
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09720871 |
| url |
http://hdl.handle.net/20.500.12442/1896 |
| dc.language.iso.eng.fl_str_mv |
eng |
| language |
eng |
| dc.rights.coar.fl_str_mv |
http://purl.org/coar/access_right/c_16ec |
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Licencia de Creative Commons Reconocimiento-NoComercial-CompartirIgual 4.0 Internacional |
| rights_invalid_str_mv |
Licencia de Creative Commons Reconocimiento-NoComercial-CompartirIgual 4.0 Internacional http://purl.org/coar/access_right/c_16ec |
| dc.publisher.eng.fl_str_mv |
Pushpa Publishing House |
| dc.source.eng.fl_str_mv |
Far East Journal of Mathematical Sciences (FJMS) Vol. 102, No. 3 (2017) |
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Universidad Simón Bolívar |
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http://www.pphmj.com/index.php?act=show_login&msg=Please%20first%20login! |
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Licencia de Creative Commons Reconocimiento-NoComercial-CompartirIgual 4.0 Internacionalhttp://purl.org/coar/access_right/c_16ecAcosta-Humánez, Primitivo943741ee-b9b5-4867-b148-541f0701eaea-1Giraldo, Hernán4888c746-3086-4f87-8e90-5a7a6b26b3f5-1Piedrahita, Carlosa49ee4c6-8efe-4c38-b98f-fa414a648151-12018-03-21T22:41:49Z2018-03-21T22:41:49Z201709720871http://hdl.handle.net/20.500.12442/1896The trajectory of energy is modeled by the solution of the Eikonal equation, which can be solved by solving a Hamiltonian system. This system is amenable of treatment from the point of view of the theory of differential algebra. In particular, by Morales-Ramis theory, it is possible to analyze integrable Hamiltonian systems through the abelian structure of their variational equations. In this paper, we obtain the abelian differential Galois groups and the representation of the quiver, that allow us to obtain such abelian differential Galois groups, for some seismic models with constant Hessian of square of slowness, proposed in [20], which are equivalent to linear Hamiltonian systems with three uncoupled harmonic oscillators.engPushpa Publishing HouseFar East Journal of Mathematical Sciences (FJMS)Vol. 102, No. 3 (2017)http://www.pphmj.com/index.php?act=show_login&msg=Please%20first%20login!Differential Galois theoryEikonal equationHamilton equationHelmholtz equationHigh frequency approximationMorales-Ramis theoryRay theoryRepresentations of quiversDifferential galois groups and representation of quivers for seismic models with constant hessian of square of slownessarticlehttp://purl.org/coar/resource_type/c_6501P. Acosta-Humánez, M. Álvarez-Ramírez, D. Blázquez-Sanz and J. Delgado, Non-integrability criterium for normal variational equations around an integrable subsystem and an example: the Wilberforce spring-pendulum, Discrete and Continuous Dynamical Systems - Series A (DCDS-A) 33(1) (2013), 965-986.P. Acosta-Humánez, M. Álvarez-Ramírez and J. Delgado, Non-integrability of some few body problems in two degrees of freedom, Qual. Theory Dyn. Syst. 8(2) (2009), 209-239.P. Acosta-Humánez and D. Blázquez-Sanz, Hamiltonian systems and variational equations with polynomial coefficients, Dynam. Systems Appl. 5(1) (2008), 6-10.P. Acosta-Humánez and D. Blázquez-Sanz, Non-integrability of some hamiltonians with rational potentials, Discrete Contin. Dyn. Syst. Series B 10(2&3) (2008), 265-293.P. Acosta-Humánez, D. Blázquez-Sanz and C. Vargas-Contreras, On hamiltonian potentials with quartic polynomial normal variational equations, Nonlinear Studies. The International Journal 16 (2009), 299-313.P. Acosta-Humánez, J. J. Morales-Ruiz and J.-A. Weil, Galoisian approach to integrability of Schrödinger equation, Rep. Math. Phys. 67(3) (2011), 305-374.P. Acosta-Humánez and Ch. Pantazi, Darboux integrals for Schrödinger planar vector fields via Darboux transformations, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 8(043) (2012), 1-26.P. Acosta- Humánez and E. Suazo, Liouvillian propagators, Riccati equation and differential Galois theory, J. Phys. A: Math. Theor. 46(45) (2013), 455203, 17 pp.P. Acosta-Humánez and E. Suazo, Liouvillian propagators and degenerate parametric amplification with time-dependent pump amplitude and phase, Analysis, Modelling, Optimization, and Numerical Techniques, Springer Proceedings in Mathematics & Statistics 121(1) (2015), 295-307.P. B. Acosta-Humánez, Galoisian approach to supersymmetric quantum mechanics. Ph.D. Thesis, Universitat Politècnica de Catalunya. Available in ArXiv: 0906.3532, 2009.P. B. Acosta-Humánez. Nonautonomous Hamiltonian systems and Morales-Ramis theory I. The case x = f (x, t), SIAM J. Appl. Dyn. Syst. 8(1) (2009), 279-297.P. B. Acosta-Humánez, Galoisian Approach to Supersymmetric Quantum Mechanics. The Integrability Analysis of the Schrödinger Equation by Means of Differential Galois Theory, VDM Verlag, Dr Muller, Berlin, 2010.P. B. Acosta-Humánez, S. I. Kryuchkov, E. Suazo and S. K. Suslov, Degenerate parametric amplification of squeezed photons: explicit solutions, statistics, means and variances, J. Nonlinear Optic. Phys. Mat. 24(2) (2015), 1550021, 27 pp.P. B. Acosta-Humánez, J. T. Lázaro, J. Morales-Ruiz and Ch. Pantazi, On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory, Discrete and Continuous Dynamical Systems - Series A (DCDS-A) 35(5) (2015), 1767-1800.V. Arnold, Mathematical methods of classical mechanics, Graduate Texts in Mathematics, Vol. 60, 2nd ed., Springer Verlag, New York, USA, 1989.I. Assem, D. D. Simson and A. Skowroński, Elements of the representation theory of associative algebras, London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 2006.M. Auslander, I. Reiten and S. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, Vol. 36, Cambridge University Press, 1995.N. Bleistein, Mathematical Methods for Wave Phenomena, Academic Press, 1986.N. Bleistein and R. Handelsman, Asymptotic Expansion of Integrals, 2nd ed., Dover, New York, USA, 1984.V. Cerveny, Seismic Ray Theory, Cambridge University Press, Cambridge, UK, 2001.J. de la Peña, Tame Algebras and Derived Categories, 1st ed., XV Escola de Álgebra. UFRGS, Brasil, Canela-RS, 1998.L. Evans, Partial differential equations, Graduate Studies in Mathematics, Vol. 19, 2nd ed., American Mathematical Society, Providence, Rhode Island, USA, 2010.J. Fritz, Partial differential equations, Applied Mathematical Sciences, Vol. 1, 4th ed., Springer Verlag, New York, USA, 1982.P. Gabriel, Unzerlegbare dartellungen I, Manuscripta Math. 6 (1972), 71-103.P. Gabriel, Auslander-Reiten sequences and representation-finite algebras, Proc. ICRA II (Ottawa, Canada 1979), Lecture Notes in Math. 831, Springer-Verlag, 1980, pp. 1-71.W. H. Gustafson, The history of algebras and their representations, Proc. ICRA III (Puebla, Mexico 1980), Lecture Notes in Math. 944, Springer-Verlag, 1982, pp. 1-28.M. Herzberger, Modern Geometrical Optics, John Wiley and Sons (Interscience), New York, 1958.I. Kaplansky, An Introduction to Differential Algebra, Hermann, 1957.T. Kimura, On Riemann’s equations which are solvable by quadratures, Funkcialaj Ekvacioj 12(1) (1969), 269-281.J. Kovacic, An algorithm for solving second order linear homogeneous differential equations, J. Symbolic Computation 2(1) (1986), 3-43.A. Magid, Lectures on differential Galois theory, University Lecture Series, American Mathematical Society, Providence, RI, 1994.J. Martinet and J. P. Ramis, Theorie de Galois differentielle et resommation, Computer Algebra and Differential Equations 193(1) (1989), 117-214.J. Morales-Ruiz, Differential Galois Theory and Non-integrability of Hamiltonian Systems, Progress in Mathematics, Birkhäuser, Basel, 1999.J. J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrability of hamiltonian systems I, Methods Appl. Anal. 8(1) (2001), 33-96.J. J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrability of hamiltonian systems II, Methods Appl. Anal. 8(1) (2001), 97-112.J. J. Morales-Ruiz and J.-P. Ramis, Integrability of dynamical systems through differential Galois theory: a practical guide, Differential algebra, complex analysis and orthogonal polynomials, Contemp. Math., Amer. Math. Soc. 509(1) (2010), 143-220.J. Rauch, Hyperbolic partial differential in geometrical optics, Graduate Studies in Mathematics, Vol. 133, 1st ed., American Mathematical Society, Providence, Rhode Island, USA, 2012.I. Reiten, An introduction to the representation of Artin algebras, Bull London Math. Soc. 17 (1985), 209-223.C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, Vol. 1099, Springer-Verlag, 1984.J. Schleicher, M. Tygel and P. Hubral, Seismic true-amplitude imaging, SEG Geophysical Developments, Vol. 12, 1st ed., Society of Exploration Geophysics, Tulsa, OK, USA, 2007.M. F. Singer, An outline of differential Galois theory, Computer Algebra and Differential Equations 121(1) (1989), 3-58.M. van der Put and M. Singer, Galois theory in linear differential equations, Graduate Text in Mathematics, Springer Verlag, New York, 2003.M. Zworski, Semiclassical analysis, Graduate Studies in Mathematics, Vol. 138, 1st ed., American Mathematical Society, Providence, Rhode Island, USA, 2012.LICENSElicense.txtlicense.txttext/plain; charset=utf-81748https://bonga.unisimon.edu.co/bitstreams/c1f4c2b0-3542-463d-8be7-160b9d663a70/download8a4605be74aa9ea9d79846c1fba20a33MD5220.500.12442/1896oai:bonga.unisimon.edu.co:20.500.12442/18962019-04-11 21:51:32.652metadata.onlyhttps://bonga.unisimon.edu.coDSpace UniSimonbibliotecas@biteca.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 |