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[{"key": "dc.contributor.advisor", "value": "Juha Lehrb\u00e4ck", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Takkinen, Mikko", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2018-03-19T17:45:58Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2018-03-19T17:45:58Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2018", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1862460", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/57351", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on tarkastella isometrioita ja similariteettej\u00e4 tasossa\r\nkompleksilukujen avulla. T\u00e4ss\u00e4 kirjoitelmassa tarvitaan kompleksilukujen ominaisuuksista\r\nainakin kompleksiluvun k\u00e4\u00e4nteisluku ja argumenttien laskus\u00e4\u00e4nn\u00f6t. Lis\u00e4ksi\r\nisometrioiden tulosten todistamista helpottavat erilaiset kompleksilukujen esitystavat,\r\nkuten napakoordinaattiesitys ja imagin\u00e4\u00e4riyksik\u00f6n avulla esitt\u00e4minen. Isometriat\r\novat kuvauksia, jotka s\u00e4ilytt\u00e4v\u00e4t jokaisen pisteen v\u00e4lisen et\u00e4isyyden kuvauksen\r\naikana. Tason isometrioita ovat siirto, kierto, peilaus ja peilauksen ja siirron yhdisteen\u00e4\r\nsaatava liukupeilaus.\r\nEdell\u00e4 mainitut kaikki Gaussin tason isometriat voidaan jakaa kahteen luokkaan:\r\nsuoriin tai vastakkaisiin isometrioihin. Suoriin isometrioihin kuuluvat siirrot ja kierrot.\r\nVastakkaisiin kuuluvat peilaukset ja liukupeilaukset.\r\nIsometriat jaotellaan my\u00f6s tarkemmin viel\u00e4 kiintopisteiden avulla. Jos suoralla\r\nisometrialla on kiintopiste, se on t\u00e4m\u00e4n kiintopisteen suhteen kierto, ja jos sill\u00e4 ei\r\nole kiintopistett\u00e4, se on siirto. Vastakkaiset isometriat jaotellaan kiintopisteen suhteen\r\nliukupeilaukseen ja suoran suhteen peilaukseen. Suoran suhteen peilauksilla on\r\nkiintopiste ja liukupeilauksilla ei ole kiintopistett\u00e4. Lis\u00e4ksi siirto ja kierto voidaan\r\nilmoittaa kahden suoran suhteen peilauksen avulla. N\u00e4iden tietojen avulla saadaan,\r\nett\u00e4 jokainen isometria voidaan esitt\u00e4\u00e4 korkeintaan kolmen suoran suhteen peilauksen\r\navulla.\r\nSimilariteetit joko venytt\u00e4v\u00e4t tai kutistavat kuvia. Similariteetit jaetaan suoriin\r\nja vastakkaisiin similariteetteihin, kuten isometriatkin. Kummatkin kuvaukset s\u00e4ilytt\u00e4v\u00e4t\r\nkolmioiden kulmien suuruudet samana kuvautuessa. Lis\u00e4ksi n\u00e4ille kummallekin\r\nkuvaukselle voidaan muotoilla Hjelmslevin lause. Isometroille t\u00e4m\u00e4 tarkoittaa sit\u00e4,\r\nett\u00e4 kuvatun suoran kuvapisteiden ja alkuper\u00e4isen suoran pisteiden v\u00e4listen janojen\r\nkeskipisteet ovat joko samalla suoralla tai kulkevat yhden ainoan pisteen kautta.\r\nSimilariteeteille n\u00e4m\u00e4 eiv\u00e4t ole keskipisteit\u00e4, vaan janat jaetaan samassa suhteessa.", "language": "", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Mikko Takkinen (mipetakk) on 2018-03-19 17:45:57.903359. 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 2018-03-19T17:45:58Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201803191768.pdf: 792635 bytes, checksum: 76555bf29de9c14616a42af73a88f666 (MD5)\nlicense.html: 4810 bytes, checksum: e9554fb9413b9259823cee55f17aada7 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2018-03-19T17:45:58Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201803191768.pdf: 792635 bytes, checksum: 76555bf29de9c14616a42af73a88f666 (MD5)\nlicense.html: 4810 bytes, checksum: e9554fb9413b9259823cee55f17aada7 (MD5)\n Previous issue date: 2018", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (41 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": "isometria", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "similariteetti", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Hjelmslevin lause", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Tason isometriat ja similariteetit kompleksiluvuilla", "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-201803191768", "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": "2018-03-19T17:45:58Z", "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": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "kompleksiluvut", "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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