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...

Full description

Autores:
Tipo de recurso:
Fecha de publicación:
2017
Institución:
Universidad de Medellín
Repositorio:
Repositorio UDEM
Idioma:
eng
OAI Identifier:
oai:repository.udem.edu.co:11407/4276
Acceso en línea:
http://hdl.handle.net/11407/4276
Palabra clave:
Differential Galois theory
Eikonal equation
Hamilton equation
Helmholtz equation
High frequency approximation
Morales-Ramis theory
Ray theory
Representations of quivers
Rights
License
http://purl.org/coar/access_right/c_16ec
id REPOUDEM2_cf29375a29d472b1700c43ceba27266e
oai_identifier_str oai:repository.udem.edu.co:11407/4276
network_acronym_str REPOUDEM2
network_name_str Repositorio UDEM
repository_id_str
dc.title.spa.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.contributor.affiliation.spa.fl_str_mv Acosta-Humánez, P., School of Basic and Biomedical Sciences, Universidad Simón Bolívar, Barranquilla, Colombia
Giraldo, H., Institute of Mathematics, Universidad de Antioquia, Medellín, Colombia
Piedrahita, C., Department of Basic Sciences, Universidad de Medellín, Medellín, Colombia
dc.subject.keyword.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. © 2017 Pushpa Publishing House, Allahabad, India.
publishDate 2017
dc.date.accessioned.none.fl_str_mv 2017-12-19T19:36:44Z
dc.date.available.none.fl_str_mv 2017-12-19T19:36:44Z
dc.date.created.none.fl_str_mv 2017
dc.type.eng.fl_str_mv Article
dc.type.coarversion.fl_str_mv http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.coar.fl_str_mv http://purl.org/coar/resource_type/c_6501
http://purl.org/coar/resource_type/c_2df8fbb1
dc.type.driver.none.fl_str_mv info:eu-repo/semantics/article
dc.identifier.issn.none.fl_str_mv 9720871
dc.identifier.uri.none.fl_str_mv http://hdl.handle.net/11407/4276
dc.identifier.doi.none.fl_str_mv 10.17654/MS102030599
dc.identifier.reponame.spa.fl_str_mv reponame:Repositorio Institucional Universidad de Medellín
dc.identifier.instname.spa.fl_str_mv instname:Universidad de Medellín
identifier_str_mv 9720871
10.17654/MS102030599
reponame:Repositorio Institucional Universidad de Medellín
instname:Universidad de Medellín
url http://hdl.handle.net/11407/4276
dc.language.iso.none.fl_str_mv eng
language eng
dc.relation.isversionof.spa.fl_str_mv https://www.scopus.com/inward/record.uri?eid=2-s2.0-85027419050&doi=10.17654%2fMS102030599&partnerID=40&md5=cbf516e16e7dfa97fbf4463dc175d5b6
dc.relation.ispartofes.spa.fl_str_mv Far East Journal of Mathematical Sciences
Far East Journal of Mathematical Sciences Volume 102, Issue 3, August 2017, Pages 599-623
dc.relation.references.spa.fl_str_mv Acosta-Humánez, P., & BlAzquez-Sanz, D. (2008). Non-integrability of some hamiltonians with rational potentials. Discrete and Continuous Dynamical Systems - Series B, 10(2-3), 265-293.
Acosta-Humánez, P., & Blázquez-Sanz, D. (2008). Hamiltonian system and variational equations with polynomial coefficients. Dynamic Systems and Applications, Dynamic, Atlanta, GA, 5, 6-10.
Acosta-Humánez, P., & Suazo, E. (2013). Liouvillian propagators, riccati equation and differential galois theory. Journal of Physics A: Mathematical and Theoretical, 46(45) doi:10.1088/1751-8113/46/45/455203
Acosta-Humanez, P. B. (2010). Galoisian Approach to Supersymmetric Quantum Mechanics.the Integrability Analysis of the Schrodinger Equation by Means of Diérential Galois Theory.
Acosta-Humánez, P. B. (2009). Galoisian Approach to Supersymmetric Quantum Mechanics.
Acosta-Humánez, P. B. (2009). Nonautonomous hamiltonian systems and morales-ramis theory I. the case ẍ = f(x, t). SIAM Journal on Applied Dynamical Systems, 8(1), 279-297. doi:10.1137/080730329
Acosta-Humánez, P. B., Alvarez-Ramírez, M., Blázquez-Sanz, D., & Delgado, J. (2013). 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, 33(3), 965-986. doi:10.3934/dcds.2013.33.965
Acosta-Humánez, P. B., Álvarez-Ramírez, M., & Delgado, J. (2009). Non-integrability of some few body problems in two degrees of freedom. Qualitative Theory of Dynamical Systems, 8(2), 209-239. doi:10.1007/s12346-010-0008-7
Acosta-Humanez, P. B., Blazquez-Sanz, D., & Contreras, C. V. (2009). On hamiltonian potentials with quartic polynomial normal variational equations. Nonlinear Stud.Int.J., 16, 299-314.
Acosta-Humánez, P. B., Kryuchkov, S. I., Suazo, E., & Suslov, S. K. (2015). Degenerate parametric amplification of squeezed photons: Explicit solutions, statistics, means and variances. Journal of Nonlinear Optical Physics and Materials, 24(2) doi:10.1142/S0218863515500216
Acosta-Humánez, P. B., Lázaro, J. T., Morales-Ruiz, J. J., & Pantazi, C. (2015). On the integrability of polynomial vector fields in the plane by means of picard-vessiot theory. Discrete and Continuous Dynamical Systems- Series A, 35(5), 1767-1800. doi:10.3934/dcds.2015.35.1767
Acosta-Humánez, P. B., Morales-Ruiz, J. J., & Weil, J. -. (2011). Galoisian approach to integrability of schrödinger equation. Reports on Mathematical Physics, 67(3), 305-374. doi:10.1016/S0034-4877(11)60019-0
Acosta-Humánez, P. B., & Pantazi, C. (2012). Darboux integrals for schrödinger planar vector fields via darboux transformations. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 8 doi:10.3842/SIGMA.2012.043
Acosta-Humánez, P. B., & Suazo, E. (2015). Liouvillian propagators and degenerate parametric amplification with time-dependent pump amplitude and phase. Paper presented at the Springer Proceedings in Mathematics and Statistics, , 121 295-307. doi:10.1007/978-3-319-12583-1_21
Arnold, V. I. (1978). Mathematical methods of classical mechanics. Graduate Texts in Mathematics, 60.
Assem, I., Simson, D., & Skowronski, A. (2006). Elements of the Representation Theory of Associative Algebras.
Auslander, M., Reiten, I., & Smalø, S. O. (1995). "Representation theory of artin algebras". Representation Theory of Artin Algebras.
Bleistein, N. (1984). Mathematical Methods for Wave Phenomena.
Bleistein, N., & Handelsman, R. A. (1986). Asymptotic Expansions of Integrals.
Červený, V. (2001). Seismic Ray Theory.
De La Peña, J. (1998). Tame Algebras and Derived Categories.
Evans, L. (2010). Partial Differential Equations, Graduate Studies in Mathematics, 19.
Fritz, J. (1982). Partial differential equations. Applied Mathematical Sciences, 1.
Gabriel, P. (1972). Manuscripta Mathematica, 6(1), 71-103. doi:10.1007/BF01298413
Gabriel, P. (1980). Auslander-reiten sequences and representation-finite algebras. Lecture Notes in Mathematics, 831, 1-71.
Gustafson, W. H. (1982). The history of algebras and their representations. Lecture Notes in Math., 944, 1-28.
Herzberger, M. (1958). Modern Geometrical Optics.
Kaplansky, I. (1957). An Introduction to Differential Algebra.
Kimura, T. (1969). On riemann's equations which are solvable by quadratures. Funkcial.Ekvac., 12, 269-281.
Kovacic, J. J. (1986). An algorithm for solving second order linear homogeneous differential equations. Journal of Symbolic Computation, 2(1), 3-43. doi:10.1016/S0747-7171(86)80010-4
Magid, A. (1994). Lectures on differential galois theory, university lecture series. American Mathematical Society, Providence, RI.
Martinet, J., & Ramis, J. P. (1989). Théorie de galois différentielle et resommation. Computer Algebra and Differential Equations, 117-214.
Morales-Ruiz, J. J. (1999). Differential Galois Theory and Non-Integrability of Hamiltonian Systems.
Morales-Ruiz, J. J., & Ramis, J. P. (2001). Galoisian obstructions to integrability of hamiltonian systems. Methods Appl.Anal., 8(1), 33-95.
Morales-Ruiz, J. J., & Ramis, J. P. (2001). Galoisian obstructions to integrability of hamiltonian systems. Methods Appl.Anal., 8(1), 33-95.
Morales-Ruiz, J. J., & Ramis, J. -. (2010). Integrability of dynamical systems through differential galois theory: A practical guide. Differential Algebra, Complex Analysis and Orthogonal Polynomials, 509, 143-220.
Rauch, J. (2012). Hyperbolic Partial Differential in Geometrical Optics, Graduate Studies in Mathematics, 133.
Reiten, I. (1985). An introduction to the representation theory of artin algebras. Bulletin of the London Mathematical Society, 17(3), 209-233. doi:10.1112/blms/17.3.209
Ringel, C. M. (1984). Tame algebras and integral quadratic forms. Tame Algebras and Integral Quadratic Forms.
Schleicher, J., Tygel, M., & Hubral, P. (2007). Seismic True-Amplitude Imaging, SEG Geophysical Developments, 12.
Singer, M. F. (1990). An outline of differential galois theory. Computer Algebra and Differential Equations , 3-57.
Van Der Put, M., & Singer, M. (2003). Galois Theory in Linear Differential Equations, Graduate Text in Mathematics.
Zworski, M. (2012). Semiclassical Analysis, Graduate Studies in Mathematics, 138
dc.rights.coar.fl_str_mv http://purl.org/coar/access_right/c_16ec
rights_invalid_str_mv http://purl.org/coar/access_right/c_16ec
dc.publisher.spa.fl_str_mv Pushpa Publishing House
dc.publisher.faculty.spa.fl_str_mv Facultad de Ciencias Básicas
dc.source.spa.fl_str_mv Scopus
institution Universidad de Medellín
repository.name.fl_str_mv Repositorio Institucional Universidad de Medellin
repository.mail.fl_str_mv repositorio@udem.edu.co
_version_ 1814159100790964224
spelling 2017-12-19T19:36:44Z2017-12-19T19:36:44Z20179720871http://hdl.handle.net/11407/427610.17654/MS102030599reponame:Repositorio Institucional Universidad de Medellíninstname:Universidad de MedellínThe 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. © 2017 Pushpa Publishing House, Allahabad, India.engPushpa Publishing HouseFacultad de Ciencias Básicashttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85027419050&doi=10.17654%2fMS102030599&partnerID=40&md5=cbf516e16e7dfa97fbf4463dc175d5b6Far East Journal of Mathematical SciencesFar East Journal of Mathematical Sciences Volume 102, Issue 3, August 2017, Pages 599-623Acosta-Humánez, P., & BlAzquez-Sanz, D. (2008). Non-integrability of some hamiltonians with rational potentials. Discrete and Continuous Dynamical Systems - Series B, 10(2-3), 265-293.Acosta-Humánez, P., & Blázquez-Sanz, D. (2008). Hamiltonian system and variational equations with polynomial coefficients. Dynamic Systems and Applications, Dynamic, Atlanta, GA, 5, 6-10.Acosta-Humánez, P., & Suazo, E. (2013). Liouvillian propagators, riccati equation and differential galois theory. Journal of Physics A: Mathematical and Theoretical, 46(45) doi:10.1088/1751-8113/46/45/455203Acosta-Humanez, P. B. (2010). Galoisian Approach to Supersymmetric Quantum Mechanics.the Integrability Analysis of the Schrodinger Equation by Means of Diérential Galois Theory.Acosta-Humánez, P. B. (2009). Galoisian Approach to Supersymmetric Quantum Mechanics.Acosta-Humánez, P. B. (2009). Nonautonomous hamiltonian systems and morales-ramis theory I. the case ẍ = f(x, t). SIAM Journal on Applied Dynamical Systems, 8(1), 279-297. doi:10.1137/080730329Acosta-Humánez, P. B., Alvarez-Ramírez, M., Blázquez-Sanz, D., & Delgado, J. (2013). 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, 33(3), 965-986. doi:10.3934/dcds.2013.33.965Acosta-Humánez, P. B., Álvarez-Ramírez, M., & Delgado, J. (2009). Non-integrability of some few body problems in two degrees of freedom. Qualitative Theory of Dynamical Systems, 8(2), 209-239. doi:10.1007/s12346-010-0008-7Acosta-Humanez, P. B., Blazquez-Sanz, D., & Contreras, C. V. (2009). On hamiltonian potentials with quartic polynomial normal variational equations. Nonlinear Stud.Int.J., 16, 299-314.Acosta-Humánez, P. B., Kryuchkov, S. I., Suazo, E., & Suslov, S. K. (2015). Degenerate parametric amplification of squeezed photons: Explicit solutions, statistics, means and variances. Journal of Nonlinear Optical Physics and Materials, 24(2) doi:10.1142/S0218863515500216Acosta-Humánez, P. B., Lázaro, J. T., Morales-Ruiz, J. J., & Pantazi, C. (2015). On the integrability of polynomial vector fields in the plane by means of picard-vessiot theory. Discrete and Continuous Dynamical Systems- Series A, 35(5), 1767-1800. doi:10.3934/dcds.2015.35.1767Acosta-Humánez, P. B., Morales-Ruiz, J. J., & Weil, J. -. (2011). Galoisian approach to integrability of schrödinger equation. Reports on Mathematical Physics, 67(3), 305-374. doi:10.1016/S0034-4877(11)60019-0Acosta-Humánez, P. B., & Pantazi, C. (2012). Darboux integrals for schrödinger planar vector fields via darboux transformations. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 8 doi:10.3842/SIGMA.2012.043Acosta-Humánez, P. B., & Suazo, E. (2015). Liouvillian propagators and degenerate parametric amplification with time-dependent pump amplitude and phase. Paper presented at the Springer Proceedings in Mathematics and Statistics, , 121 295-307. doi:10.1007/978-3-319-12583-1_21Arnold, V. I. (1978). Mathematical methods of classical mechanics. Graduate Texts in Mathematics, 60.Assem, I., Simson, D., & Skowronski, A. (2006). Elements of the Representation Theory of Associative Algebras.Auslander, M., Reiten, I., & Smalø, S. O. (1995). "Representation theory of artin algebras". Representation Theory of Artin Algebras.Bleistein, N. (1984). Mathematical Methods for Wave Phenomena.Bleistein, N., & Handelsman, R. A. (1986). Asymptotic Expansions of Integrals.Červený, V. (2001). Seismic Ray Theory.De La Peña, J. (1998). Tame Algebras and Derived Categories.Evans, L. (2010). Partial Differential Equations, Graduate Studies in Mathematics, 19.Fritz, J. (1982). Partial differential equations. Applied Mathematical Sciences, 1.Gabriel, P. (1972). Manuscripta Mathematica, 6(1), 71-103. doi:10.1007/BF01298413Gabriel, P. (1980). Auslander-reiten sequences and representation-finite algebras. Lecture Notes in Mathematics, 831, 1-71.Gustafson, W. H. (1982). The history of algebras and their representations. Lecture Notes in Math., 944, 1-28.Herzberger, M. (1958). Modern Geometrical Optics.Kaplansky, I. (1957). An Introduction to Differential Algebra.Kimura, T. (1969). On riemann's equations which are solvable by quadratures. Funkcial.Ekvac., 12, 269-281.Kovacic, J. J. (1986). An algorithm for solving second order linear homogeneous differential equations. Journal of Symbolic Computation, 2(1), 3-43. doi:10.1016/S0747-7171(86)80010-4Magid, A. (1994). Lectures on differential galois theory, university lecture series. American Mathematical Society, Providence, RI.Martinet, J., & Ramis, J. P. (1989). Théorie de galois différentielle et resommation. Computer Algebra and Differential Equations, 117-214.Morales-Ruiz, J. J. (1999). Differential Galois Theory and Non-Integrability of Hamiltonian Systems.Morales-Ruiz, J. J., & Ramis, J. P. (2001). Galoisian obstructions to integrability of hamiltonian systems. Methods Appl.Anal., 8(1), 33-95.Morales-Ruiz, J. J., & Ramis, J. P. (2001). Galoisian obstructions to integrability of hamiltonian systems. Methods Appl.Anal., 8(1), 33-95.Morales-Ruiz, J. J., & Ramis, J. -. (2010). Integrability of dynamical systems through differential galois theory: A practical guide. Differential Algebra, Complex Analysis and Orthogonal Polynomials, 509, 143-220.Rauch, J. (2012). Hyperbolic Partial Differential in Geometrical Optics, Graduate Studies in Mathematics, 133.Reiten, I. (1985). An introduction to the representation theory of artin algebras. Bulletin of the London Mathematical Society, 17(3), 209-233. doi:10.1112/blms/17.3.209Ringel, C. M. (1984). Tame algebras and integral quadratic forms. Tame Algebras and Integral Quadratic Forms.Schleicher, J., Tygel, M., & Hubral, P. (2007). Seismic True-Amplitude Imaging, SEG Geophysical Developments, 12.Singer, M. F. (1990). An outline of differential galois theory. Computer Algebra and Differential Equations , 3-57.Van Der Put, M., & Singer, M. (2003). Galois Theory in Linear Differential Equations, Graduate Text in Mathematics.Zworski, M. (2012). Semiclassical Analysis, Graduate Studies in Mathematics, 138ScopusDifferential galois groups and representation of quivers for seismic models with constant hessian of square of slownessArticleinfo:eu-repo/semantics/articlehttp://purl.org/coar/version/c_970fb48d4fbd8a85http://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1Acosta-Humánez, P., School of Basic and Biomedical Sciences, Universidad Simón Bolívar, Barranquilla, ColombiaGiraldo, H., Institute of Mathematics, Universidad de Antioquia, Medellín, ColombiaPiedrahita, C., Department of Basic Sciences, Universidad de Medellín, Medellín, ColombiaAcosta-Humánez P.Giraldo H.Piedrahita C.School of Basic and Biomedical Sciences, Universidad Simón Bolívar, Barranquilla, ColombiaInstitute of Mathematics, Universidad de Antioquia, Medellín, ColombiaDepartment of Basic Sciences, Universidad de Medellín, Medellín, ColombiaDifferential Galois theoryEikonal equationHamilton equationHelmholtz equationHigh frequency approximationMorales-Ramis theoryRay theoryRepresentations of quiversThe 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. © 2017 Pushpa Publishing House, Allahabad, India.http://purl.org/coar/access_right/c_16ec11407/4276oai:repository.udem.edu.co:11407/42762020-05-27 15:40:07.644Repositorio Institucional Universidad de Medellinrepositorio@udem.edu.co

Publicaciones similares