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[{"key": "dc.contributor.advisor", "value": "Parkkonen, Jouni", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Peltokangas, Aleksi", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2023-06-13T08:23:30Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2023-06-13T08:23:30Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2023", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/87683", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa k\u00e4sitell\u00e4\u00e4n Hamiltonin kvaternioita sek\u00e4 niiden yhteytt\u00e4 s\u00e4\u00e4nn\u00f6llisten monitahokkaiden symmetriaan. Tutkielman aluksi kerrataan muutamia jo tunnettuja m\u00e4\u00e4ritelmi\u00e4 ryhmist\u00e4, renkaista ja matriiseista. Lis\u00e4ksi esitell\u00e4\u00e4n aliryhm\u00e4kriteeri ja alirengastesti, joiden avulla on helppo tutkia ryhmien ja renkaiden osajoukkojen ominaisuuksia.\nT\u00e4m\u00e4n j\u00e4lkeen tutkielmassa perehdyt\u00e4\u00e4n hieman siihen, kuinka Sir William Rowan Hamilton p\u00e4\u00e4tyi l\u00f6yt\u00e4m\u00e4\u00e4n kvaterniot etsiess\u00e4\u00e4n matemaattista mallinnusta kolmiulotteiselle maailmalle. Luvussa my\u00f6s tarkastellaan kvaternioiden algebrallisia ominaisuuksia ja osoitetaan, ett\u00e4 kvaterniot muodostavat vinon kunnan.\nKolmannessa luvussa keskityt\u00e4\u00e4n kvaternioiden lineaarialgebrallisiin ominaisuuksiin, kuten sis\u00e4tuloon ja normiin. T\u00e4st\u00e4 p\u00e4\u00e4dyt\u00e4\u00e4nkin tulokseen, ett\u00e4 kvaterniot vastaavat reaaliavaruutta R<sup>4</sup> ja imaginaariset kvaterniot niiden osajoukkona puolestaan avaruutta R<sup>3</sup>. Luvun lopuksi osoitetaan er\u00e4s tutkielman t\u00e4rkeimpi\u00e4 tuloksia, ett\u00e4 jokainen joukon Im H erityinen ortogonaalikuvaus voidaan esitt\u00e4\u00e4 yksikk\u00f6kvaternion ja sen k\u00e4\u00e4nteiskvaternion avulla kuvauksena \u03a6<sub>a</sub>=aua<sup>(-1)</sup>. Lis\u00e4ksi n\u00e4m\u00e4 kuvaukset vastaavat kiertoja ja ne kuvaavat yksikk\u00f6kvaterniot ortogonaaliryhm\u00e4ksi SO(3).\nTutkielman lopuksi tutustutaan s\u00e4\u00e4nn\u00f6llisten kolmiulotteisten monitahokkaiden symmetriaryhmiin. T\u00e4m\u00e4n j\u00e4lkeen tutkitaan, kuinka kuvaukset \u03a6 kuvaavat binaarisia polyedri- eli monitahokasryhmi\u00e4. Tutkielman viimeisen\u00e4 tuloksena osoitetaan, ett\u00e4 kvaternioiden yksik\u00f6iden H<sup>\u00d7</sup> kaikki \u00e4\u00e4relliset aliryhm\u00e4t ovat \u00e4\u00e4relliset sykliset ryhm\u00e4t, binaariset diedriryhm\u00e4t sek\u00e4 binaariset monitahokasryhm\u00e4t.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2023-06-13T08:23:29Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2023-06-13T08:23:30Z (GMT). No. of bitstreams: 0\n Previous issue date: 2023", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "57", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": null, "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "kvaterniot", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "s\u00e4\u00e4nn\u00f6lliset monitahokkaat", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Kvaterniot ja geometria", "language": "", "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-202306133751", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "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": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "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.copyright", "value": "\u00a9 The Author(s)", "language": null, "element": "rights", "qualifier": "copyright", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "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": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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