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[{"key": "dc.contributor.advisor", "value": "N\u00e4kki, Raimo", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.advisor", "value": "Parkkonen, Jouni", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Kym\u00e4l\u00e4inen, Petri", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2014-11-16T19:59:26Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2014-11-16T19:59:26Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2014", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1451530", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/44666", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa esitell\u00e4\u00e4n viisi erilaista Riemannin monistoa, jotka toimivat hyperbolisen geometrian analyyttisin\u00e4 malleina. Geometria voidaan karkeasti jakaa kahteen eri tapaukseen, euklidiseen ja ep\u00e4euklidiseen. Euklidisessa geometriassa p\u00e4tee Eukleideen geometrian viides aksiooma, paralleeliaksiooma. N\u00e4in ei kuitenkaan ole laita hyperbolisessa geometriassa, joka luokitellaan ep\u00e4euklidiseksi geometriaksi. Analyyttisell\u00e4 geometrialla taas tarkoitetaan koordinaatistoon sidottua geometriaa. T\u00e4ss\u00e4 tapauksessa n\u00e4m\u00e4 geometrian mallit ovat topologisia 2-ulotteisia pintoja euklidisessa avaruudessa. \n\nLis\u00e4ksi tutkielman malleissa hy\u00f6dynnet\u00e4\u00e4n n\u00e4ille pinnoille m\u00e4\u00e4ritelty\u00e4 sile\u00e4\u00e4 differentiaalirakennetta, jolloin voidaan k\u00e4ytt\u00e4\u00e4 mallista nimityst\u00e4 sile\u00e4 monisto. Riemannin monistossa on m\u00e4\u00e4ritelty 2-kovariantti tensorikentt\u00e4, Riemannin metriikka, jonka avulla voidaan selvitt\u00e4\u00e4 geodeesi eli sile\u00e4 polku, jonka kuva monistolla on kahden pisteen v\u00e4linen lyhin reitti eli geometrinen jana. My\u00f6s sellaiset geometriset oliot kuin suora ja kulma m\u00e4\u00e4ritell\u00e4\u00e4n Riemannin metriikan avulla. \n\nTutkielmassa johdetaan keino geodeesien laskemiseksi geodeettisen differentiaaliyht\u00e4l\u00f6ryhm\u00e4n avulla. T\u00e4m\u00e4n yht\u00e4l\u00f6ryhm\u00e4n ratkaisuna hahmotellaan suorat yhdess\u00e4 hyperbolisen geometrian mallissa, Poincar\u00e9n puolitasossa. Lopuksi tutkielmassa esitell\u00e4\u00e4n metriikan siirto, joka on keino muodostaa alkuper\u00e4isen mallin kaltaisia uusia hyperbolisen geometrian malleja sopivien diffeomorfismien avulla. T\u00e4t\u00e4 menetelm\u00e4\u00e4 k\u00e4ytet\u00e4\u00e4n nelj\u00e4n muun Riemannin moniston muodostamiseen.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Petri Kym\u00e4l\u00e4inen (peoskyma) on 2014-11-16 19:59:25.461957. 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@noreply.fi) on 2014-11-16T19:59:26Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201411163268.pdf: 3418823 bytes, checksum: 33bdb384b7143d8b1bce10ed2797c5f7 (MD5)\nlicense.html: 4812 bytes, checksum: 9b26b48d25f17f9fcb31e9d3831f1bd1 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2014-11-16T19:59:26Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201411163268.pdf: 3418823 bytes, checksum: 33bdb384b7143d8b1bce10ed2797c5f7 (MD5)\nlicense.html: 4812 bytes, checksum: 9b26b48d25f17f9fcb31e9d3831f1bd1 (MD5)\n Previous issue date: 2014", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (27 sivua)", "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": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "monisto", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "geodeesi", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Hyperbolisen geometrian analyyttisi\u00e4 malleja", "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-201411163268", "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": "2014-11-16T19:59:27Z", "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": "geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "metriikka", "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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