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[{"key": "dc.contributor.advisor", "value": "Juutinen, Petri", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Pienim\u00e4ki, Santtu", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2017-12-07T17:30:07Z", "language": "", "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2017-12-07T17:30:07Z", "language": "", "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2017", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1804989", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/56170", "language": "", "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Hyperreaaliluvut ovat reaalilukujen joukon laajennus, jossa on olemassa \u00e4\u00e4rett\u00f6m\u00e4n\r\npieni\u00e4 ja suuria lukuja. Hyperreaalilukuja k\u00e4ytet\u00e4\u00e4n differentiaali- ja integraalilaskennassa.\r\nMetodi on suosittu erityisesti fyysikoiden keskuudessa. Analyysin\r\nosa-aluetta, jossa hy\u00f6dynnet\u00e4\u00e4n hyperreaalilukuja, kutsutaan ep\u00e4standardiksi analyysiksi.\r\nEp\u00e4standardissa analyysiss\u00e4 k\u00e4ytet\u00e4\u00e4n analyysille ep\u00e4standardeja ty\u00f6kaluja,\r\njosta nimi juontuu. Hyperreaalilukujen edut verrattaessa reaalilukuihin\r\ntulevat esille ep\u00e4standardissa analyysiss\u00e4. Varsinkin fysiikassa hy\u00f6dynnet\u00e4\u00e4n yht\u00e4l\u00f6iden\r\ndifferentiaalimuotoja ja integroinnissa l\u00e4ht\u00f6kohtana pidet\u00e4\u00e4n infinitesimaalin\r\nvalintaa.\r\nTutkielmassa hyperreaaliluvut m\u00e4\u00e4ritell\u00e4\u00e4n l\u00e4htien kuudesta aksioomasta. Kaksi\r\nensimm\u00e4ist\u00e4 aksioomaa vastaavat reaalilukujen aksioomia. Kolmas aksiooma takaa\r\nyhden positiivisen infinitesimaalin olemassaolon. T\u00e4m\u00e4n lis\u00e4ksi tarvitaan viel\u00e4\r\naksiooman standardiosalle ja kaksi aksioomaa funktioille. Jokainen \u00e4\u00e4rellinen hyperreaaliluku\r\non mielivaltaisen l\u00e4hell\u00e4 yht\u00e4 reaalilukua. Hyperreaaliluvun standardiosa\r\non se reaaliluku, jota l\u00e4hell\u00e4 hyperreaaliluku on.\r\nEp\u00e4standardista analyysista ensimm\u00e4isen\u00e4 m\u00e4\u00e4ritell\u00e4\u00e4n jatkuvuus funktioilla hyperreaalilukujen\r\navulla. Hyperreaalifunktiot ovat funktioita, jotka ovat m\u00e4\u00e4rtitelty\r\nhyperreaaliluvuilla. Viides aksiooma takaa jokaiselle funktiolle f vastaavan hyperreaalifunktion\r\nf*, jota kutsutaan funktion f luonnolliseksi jatkoksi. Raja-arvo\r\nm\u00e4\u00e4ritell\u00e4\u00e4n hyperreaaliluvuille standardiosan avulla. Jatkuvuus m\u00e4\u00e4ritell\u00e4\u00e4n rajaarvon\r\navulla, mutta todistuksissa hy\u00f6dynnet\u00e4\u00e4n standardiosaa. Jatkuvuustuloksista\r\nBolzanon lauseen todistus on selke\u00e4sti lyhyempi hyperreaalilukuja hy\u00f6dynt\u00e4en.\r\nDerivaatta m\u00e4\u00e4ritell\u00e4\u00e4n tutkielmassa standardiosan avulla raja-arvon sijaan. M\u00e4\u00e4ritelm\u00e4n\r\navulla osoitetaan rationaalifunktioiden derivointitulokset. Lauseiden todistuksia\r\nverrataan standardin analyysin todistuksiin. Viimeisess\u00e4 kappaleessa\r\nm\u00e4\u00e4ritell\u00e4\u00e4n m\u00e4\u00e4r\u00e4tty integraali hyperreaalilukujen avulla. M\u00e4\u00e4r\u00e4tyn integraalin\r\nominaisuuksien lis\u00e4ksi osoitetaan, ett\u00e4 m\u00e4\u00e4r\u00e4tty integraali\r\nv\u00e4lill\u00e4 [a,b] funktiosta f on funktion f\r\npinta-ala funktio.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Santtu Pienim\u00e4ki (saelpien) on 2017-12-07 17:30:07.345484. 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 2017-12-07T17:30:07Z\r\nNo. of bitstreams: 2\r\nURN:NBN:fi:jyu-201712074539.pdf: 921069 bytes, checksum: 4b29a042f383d151972ab1a4b60a5862 (MD5)\r\nlicense.html: 4780 bytes, checksum: 9ee788338e5ab0c0229eaee49573aff2 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2017-12-07T17:30:07Z (GMT). No. of bitstreams: 2\r\nURN:NBN:fi:jyu-201712074539.pdf: 921069 bytes, checksum: 4b29a042f383d151972ab1a4b60a5862 (MD5)\r\nlicense.html: 4780 bytes, checksum: 9ee788338e5ab0c0229eaee49573aff2 (MD5)\r\n Previous issue date: 2017", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (55 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": "hyperreaaliluvut", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "ep\u00e4standardi analyysi", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Hyperreaaliluvut", "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-201712074539", "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": "2017-12-07T17:30:08Z", "language": "", "element": "date", "qualifier": "updated", "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": "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": "reaaliluvut", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "differentiaalilaskenta", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "integraalilaskenta", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "jatkuvuus", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "derivaatta", "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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