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[{"key": "dc.contributor.advisor", "value": "Pankka, Pekka", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Moisala, Terhi", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2016-03-30T07:25:33Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2016-03-30T07:25:33Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2016", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1525140", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/49211", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Todistamme t\u00e4ss\u00e4 ty\u00f6ss\u00e4 Poincar\u00e9n dualiteetin Euklidisten avaruuksien avoimille\njoukoille. Annamme lyhyen johdatuksen differentiaaligeometriaan ja m\u00e4\u00e4rittelemme de Rham -kohomologian k\u00e4sitteen. Itse Poincar\u00e9n dualiteetin todistuksen aloitamme muutamalla aputuloksella. N\u00e4yt\u00e4mme ensin, ett\u00e4 Poincar\u00e9n\ndualiteetti p\u00e4tee joukoille, jotka ovat diffeomorfisia avaruuteen R^n . Todis-\ntamme sitten Poincar\u00e9n dualiteetin avointen joukkojen yhdisteille erin\u00e4isten\nlis\u00e4oletusten vallitessa. T\u00e4t\u00e4 varten esittelemme Mayer\u2013Vietoris jonon de\nRham -kohomologialle. Lopulta n\u00e4yt\u00e4mme Poincar\u00e9n dualiteetin mielivaltaiselle\navoimelle joukolle k\u00e4ytten Whitney-jakoa. Annamme my\u00f6s havainnollistavan\nesimerkin Poincar\u00e9n dualiteetista punkteeratussa tasossa.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "In this thesis we prove the Poincar\u00e9 duality for open sets in Euclidean spaces.\nWe start with a brief introduction to differential geometry and introduce then\nthe de Rham cohomology. The actual proof begins with some auxiliary results.\nWe prove first the Poincar\u00e9 duality for sets that are diffeomorphic to R^n . We\nthen introduce the Mayer\u2013Vietoris sequence for de Rham cohomology and show\nthat the Poincar\u00e9 duality holds for unions of open sets with some additional\nassumptions. Finally we prove the Poincar\u00e9 duality for an arbitrary open set\nusing the Whitney decomposition. We give also an illustrative example of the\nPoincar\u00e9 duality in the punctured plane.", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Terhi Moisala (tekamois) on 2016-03-30 07:25:33.185313. Form: Pro gradu -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija.group@korppi.jyu.fi) on 2016-03-30T07:25:33Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201603301961.pdf: 492500 bytes, checksum: 09b61b718ba01105ffadc26e44cd1358 (MD5)\nlicense.html: 4807 bytes, checksum: 815b03f0c0d9889fd35c227000e9a2a6 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2016-03-30T07:25:33Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201603301961.pdf: 492500 bytes, checksum: 09b61b718ba01105ffadc26e44cd1358 (MD5)\nlicense.html: 4807 bytes, checksum: 815b03f0c0d9889fd35c227000e9a2a6 (MD5)\n Previous issue date: 2016", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (55 s.)", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "eng", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "Poincare\u0301 duality for open sets in Euclidean spaces", "language": null, "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-201603301961", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.date.updated", "value": "2016-03-30T07:25:34Z", "language": null, "element": "date", "qualifier": "updated", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": null, "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": "fi", "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": null, "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "differentiaaligeometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "euklidinen geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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