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[{"key": "dc.contributor.advisor", "value": "Parkkonen, Jouni", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Toivonen, Jaakko", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2024-07-01T10:52:44Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2024-07-01T10:52:44Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2024", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/96247", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa k\u00e4sitell\u00e4\u00e4n toisen asteen imagin\u00e4\u00e4risten lukukuntien luokkaluvun yhteytt\u00e4 kunnan kokonaislukurenkaan viritt\u00e4m\u00e4n hyperbolisen avaruuden isometrioiden ryhm\u00e4n PSL2(OK) eli Bianchin ryhm\u00e4n muodostamaan perusalueeseen. Tutkielma perustuu p\u00e4\u00e4osin J\u00fcrgen Elstrodtin, Fritz Grunewaldin ja Jens Mennicken kirjaan Groups Acting on Hyperbolic Spaces; Harmonic Analysis and Number Theory kappaleeseen 7. [9] Tutkielma esittelee kappaleen sis\u00e4llon todistaen sen tulokset l\u00e4hdeteosta perusteellisemmin. Lis\u00e4ksi tutkielmassa esitet\u00e4\u00e4n vaadittavat algebralliset ja geometriset esitiedot. Keskeisen\u00e4 ty\u00f6kaluna t\u00e4ss\u00e4 tutkielmassa k\u00e4ytet\u00e4\u00e4n ideaalien teoriaa. Tutkielmassa esitell\u00e4\u00e4n keskeisten m\u00e4\u00e4ritelmien lis\u00e4ksi merkitt\u00e4vi\u00e4 algebrallisia tuloksia kuten Kiinalainen j\u00e4\u00e4nn\u00f6slause ja Minkowskin lause. Algebrallisia tuloksia ei v\u00e4ltt\u00e4m\u00e4tt\u00e4 esitell\u00e4 yleisimm\u00e4ss\u00e4 mahdollisessa muodossa todistusten selkeyden vuoksi. Tutkielmassa esitell\u00e4\u00e4n lis\u00e4ksi kolmiulotteisen hyperbolisen avaruuden m\u00e4\u00e4ritelm\u00e4 ja joitain sen geometrisi\u00e4 ominaisuuksia. Keskeisen\u00e4 m\u00e4\u00e4ritelm\u00e4n\u00e4 on 2 \u00d7 2-matriisien erityinen lineaarinen ryhm\u00e4, jonka todetaan toimivan isometrioilla hyperbolisessa avaruudessa. Erityisen lineaarisen ryhm\u00e4n aliryhm\u00e4ll\u00e4, jonka kertoimet ovat toisen asteen imagin\u00e4\u00e4risen lukukunnan kokonaislukuja, todistetaan olevan algebrallisesti merkitt\u00e4vi\u00e4 ominaisuuksia. Tutkielman kaksi viimeist\u00e4 kappaletta osoittavat yhteyden luokkaluvun ja Bianchin ryhm\u00e4n hyperbolisen perusalueen v\u00e4lill\u00e4. Lukukunnan perusalueen konstruktio esitell\u00e4\u00e4n ensin ja t\u00e4m\u00e4n j\u00e4lkeen joukon todistetaan olevan perusalue. Perusalueen konstruktiosta p\u00e4\u00e4tell\u00e4\u00e4n joitain joukon geometrisi\u00e4 ominaisuuksia. Lopuksi esitell\u00e4\u00e4n ja visualisoidaan kuntien Q(i), Q( \u221a \u22123) ja Q\u221a \u22125 perusalueet sek\u00e4 huomioidaan perusalueen ja kompleksitason leikkauspisteiden yhteys kunnan luokkalukuun.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2024-07-01T10:52:44Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2024-07-01T10:52:44Z (GMT). No. of bitstreams: 0\n Previous issue date: 2024", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "62", "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": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "Bianchin ryhm\u00e4", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "luokkaluku", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "lukukunnat", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Toisen asteen imagin\u00e4\u00e4risten lukukuntien perusalueet hyperbolisessa avaruudessa", "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-202407015082", "language": null, "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": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "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.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": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "algebra", "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": "lukuteoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "ep\u00e4euklidinen 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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