{"id":{"repo_id":"aachen","oai_identifier":"oai:publications.rwth-aachen.de:59441"},"canonical_url":"https://search.dev.ndltd.org/etd/aachen/oai:publications.rwth-aachen.de:59441","repository":{"repo_id":"aachen","name":"RWTH Aachen University","base_url":"https://publications.rwth-aachen.de/oai2d"},"display":{"title":"Functional spaces : a direct approach","abstract":"In this thesis I define the notion of a functional tensor space, and derive the specific form of the Lie derivative (w.r.t. generalized vector fields) on the constructed spaces, giving them a Lie module structure over the Lie algebra of generalized vector fields. I also construct the Euler complex in a recursive manner using the celebrated Cartan formula, and write down the first three operators explicitly. I call the third one, the Takens operator. Then, in the context of functional spaces, I prove and use the fact, that a Hamiltonian structure of an evolution equation is invariant under the flow of the equation, to derive all Hamiltonian structures (up to a certain order) of KdV equation and the Boussinesq equation. These are well known examples for nonlinear completely integrable evolution equations.","abstract_html":"In this thesis I define the notion of a functional tensor space, and derive the specific form of the Lie derivative (w.r.t. generalized vector fields) on the constructed spaces, giving them a Lie module structure over the Lie algebra of generalized vector fields. I also construct the Euler complex in a recursive manner using the celebrated Cartan formula, and write down the first three operators explicitly. I call the third one, the Takens operator. Then, in the context of functional spaces, I prove and use the fact, that a Hamiltonian structure of an evolution equation is invariant under the flow of the equation, to derive all Hamiltonian structures (up to a certain order) of KdV equation and the Boussinesq equation. These are well known examples for nonlinear completely integrable evolution equations.","abstract_has_math":false,"creators":["Barakat, Mohamed"],"institution":"Publikationsserver der RWTH Aachen University","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Plesken, Wilhelm"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2001,"date_issued":"2001","date_published":"2001","updated_at":"2026-07-30T19:42:39Z","subjects":["info:eu-repo/classification/ddc/510","Mathematik","Funktionalraum","Lie-Ableitung","Vektorfeld"],"languages":["eng"],"rights":["info:eu-repo/semantics/openAccess"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-121226%22"],"render_values":[{"text":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-121226%22","href":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-121226%22","code":true}]}]},"links":{"outbound_url":"https://publications.rwth-aachen.de/record/59441","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Plesken, Wilhelm"]},{"key":"dc:creator","label":"Author","values":["Barakat, Mohamed"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:coverage","label":"Dc Coverage","values":["DE"]},{"key":"dc:date","label":"Dc Date","values":["2001"]},{"key":"dc:publisher","label":"Institution","values":["Publikationsserver der RWTH Aachen University"]},{"key":"dc:relation","label":"Dc Relation","values":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-2803"]},{"key":"dc:type","label":"Dc Type","values":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["info:eu-repo/classification/ddc/510","Mathematik","Funktionalraum","Lie-Ableitung","Vektorfeld"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["info:eu-repo/semantics/openAccess"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/record/59441","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-121226%22"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["In this thesis I define the notion of a functional tensor space, and derive the specific form of the Lie derivative (w.r.t. generalized vector fields) on the constructed spaces, giving them a Lie module structure over the Lie algebra of generalized vector fields. I also construct the Euler complex in a recursive manner using the celebrated Cartan formula, and write down the first three operators explicitly. I call the third one, the Takens operator. Then, in the context of functional spaces, I prove and use the fact, that a Hamiltonian structure of an evolution equation is invariant under the flow of the equation, to derive all Hamiltonian structures (up to a certain order) of KdV equation and the Boussinesq equation. These are well known examples for nonlinear completely integrable evolution equations."]},{"key":"dc:source","label":"Dc Source","values":["Aachen : Publikationsserver der RWTH Aachen University 71 S. (2001). = Aachen, Techn. Hochsch., Diss., 2001"]},{"key":"dc:title","label":"Title","values":["Functional spaces : a direct approach"]}]}],"canonical_facts":{"dc:contributor":["Plesken, Wilhelm"],"dc:coverage":["DE"],"dc:creator":["Barakat, Mohamed"],"dc:date":["2001"],"dc:description":["In this thesis I define the notion of a functional tensor space, and derive the specific form of the Lie derivative (w.r.t. generalized vector fields) on the constructed spaces, giving them a Lie module structure over the Lie algebra of generalized vector fields. I also construct the Euler complex in a recursive manner using the celebrated Cartan formula, and write down the first three operators explicitly. I call the third one, the Takens operator. Then, in the context of functional spaces, I prove and use the fact, that a Hamiltonian structure of an evolution equation is invariant under the flow of the equation, to derive all Hamiltonian structures (up to a certain order) of KdV equation and the Boussinesq equation. These are well known examples for nonlinear completely integrable evolution equations."],"dc:identifier":["https://publications.rwth-aachen.de/record/59441","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-121226%22"],"dc:language":["eng"],"dc:publisher":["Publikationsserver der RWTH Aachen University"],"dc:relation":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-2803"],"dc:rights":["info:eu-repo/semantics/openAccess"],"dc:source":["Aachen : Publikationsserver der RWTH Aachen University 71 S. (2001). = Aachen, Techn. Hochsch., Diss., 2001"],"dc:subject":["info:eu-repo/classification/ddc/510","Mathematik","Funktionalraum","Lie-Ableitung","Vektorfeld"],"dc:title":["Functional spaces : a direct approach"],"dc:type":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]},"updated_at":"2026-07-30T19:42:39Z"}