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[{"key": "dc.contributor.author", "value": "Pitk\u00e4nen, Heikki", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2012-02-25T12:39:12Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2012-02-25T12:39:12Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2012", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1194728", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/37447", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n pro gradu -tutkielman tarkoituksena on esitell\u00e4 Diofantoksen approksimoinnin tuloksia ja antaa todistus Rothin lauseelle. \nDiofantoksen approksimoinnissa ollaan kiinnostuneita siit\u00e4, kuinka hyvin irrationaalilukuja voidaan arvioida rationaaliluvuilla.\nN\u00e4iden rationaalilukuarvioiden m\u00e4\u00e4r\u00e4n perusteella voidaan antaa riitt\u00e4v\u00e4 ja v\u00e4ltt\u00e4v\u00e4 ehto luvun irrationaalisuudesta. Osoittautuu, ett\u00e4 ainoastaan irrationaaliluvuilla on \u00e4\u00e4ret\u00f6n m\u00e4\u00e4r\u00e4 ''hyvi\u00e4'' arvioita. T\u00e4m\u00e4n ehdon riitt\u00e4vyys ja v\u00e4ltt\u00e4vyys todistetaan ja lis\u00e4ksi esitell\u00e4\u00e4n tehokas menetelm\u00e4 n\u00e4iden arvioiden laskemiseksi ketjumurtolukujen avulla.\n\nKun on todettu, ett\u00e4 n\u00e4it\u00e4 hyvi\u00e4 arvioita on olemassa ja niit\u00e4 voidaan laskea, voidaan kysy\u00e4, olisiko niit\u00e4 mahdollista l\u00f6yt\u00e4\u00e4 viel\u00e4 tehokkaammin. T\u00e4t\u00e4 ongelmaa l\u00e4hestyy Liouvillen lause, joka antaa rajan algebrallisten lukujen rationaaliarvioiden hyvyydelle ja mahdollistaa transkendenttilukujen konstruoinnin. Lopulta ratkaisun ongelmaan antaa Rothin lause, jonka nojalla vastaus on kielteinen algebrallisten lukujen kohdalla. Tutkielman toisen puoliskon p\u00e4\u00e4paino on Rothin lauseen todistuksessa.\n\nLis\u00e4ksi tutkielmassa esitell\u00e4\u00e4n muutama Rothin lauseen sovellus ja tutustutaan siihen, kuinka lausetta voitaisiin parantaa.", "language": null, "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Heikki Pitk\u00e4nen (hejupitk) on 2012-02-25 12:39:10.063677. Form: Pro gradu -lomake (1 tekij\u00e4) (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake-1-tekijae). JyX data:", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija@noreply.fi) on 2012-02-25T12:39:11Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201202251299.pdf: 530973 bytes, checksum: 4105bd956f50c96609468e77fb3c5389 (MD5)\nlicense.html: 4953 bytes, checksum: 3005d65ce07a43142769c3a221db4dc0 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2012-02-25T12:39:12Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201202251299.pdf: 530973 bytes, checksum: 4105bd956f50c96609468e77fb3c5389 (MD5)\nlicense.html: 4953 bytes, checksum: 3005d65ce07a43142769c3a221db4dc0 (MD5)\n Previous issue date: 2012", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "47 sivua", "language": null, "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": "Diofantos", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Rothin lause", "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-201202251299", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.dcmitype", "value": "Text", "language": "en", "element": "type", "qualifier": "dcmitype", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu", "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": "2012-02-25T12:39:12Z", "language": null, "element": "date", "qualifier": "updated", "schema": "dc"}, {"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": "lukuteoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "approksimointi", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "yht\u00e4l\u00f6t", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "murtoluvut", "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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