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[{"key": "dc.contributor.advisor", "value": "Juutinen, Petri", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Huttunen, Lauri", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2017-08-15T19:53:38Z", "language": "", "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2017-08-15T19:53:38Z", "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:1719463", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/55104", "language": "", "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on luoda perusta topologiselle asteelle, ja todistaa siihen liittyvi\u00e4 tuloksia. Topologinen aste m\u00e4\u00e4ritell\u00e4\u00e4n aluksi jatkuvasti derivoituville funktioille jossakin kyseisen funktion kuvapisteess\u00e4. N\u00e4m\u00e4 ovat useasti moniulotteisia funktioita, joiden m\u00e4\u00e4rittelyjoukko ja kuvapisteiden joukko ovat samassa ulottuvuudessa.\r\nTopologinen aste tarkastelee funktion kuvapisteen alkukuvien ymp\u00e4rist\u00f6n kuvautumista derivaattamatriisin determinantin avulla. Mik\u00e4li Jacobin determinantti saa positiivisen arvon, lis\u00e4t\u00e4\u00e4n topologiseen asteeseen kokonaisluku yksi. Jos taas derivaattamatriisin arvo on negatiivinen kyseisen kuvapisteen alkukuvassa, topologisesta asteesta v\u00e4hennet\u00e4\u00e4n luku yksi. Topologinen aste on siis funktio, joka laskee yhteen kuvapisteen alkukuvia, jossa derivaatan merkki m\u00e4\u00e4r\u00e4\u00e4 summattavan luvun.\r\nKun on luotu perustaa jatkuvasti derivoituvien funktioiden asteteorialle, m\u00e4\u00e4ritelm\u00e4\u00e4 laajennetaan my\u00f6s jatkuville funktioille. Topologinen aste jatkuvalle funktiolle m\u00e4\u00e4ritell\u00e4\u00e4n jatkuvasti derivoituvan funktion avulla, joka on l\u00e4hell\u00e4 alkuper\u00e4ist\u00e4 funktiota kaikissa pisteiss\u00e4.\r\nNiiss\u00e4 pisteiss\u00e4, joissa funktion Jacobin determinantti saa arvon nolla, topologista astetta ei pystyt\u00e4 my\u00f6sk\u00e4\u00e4n suoraan m\u00e4\u00e4rittelem\u00e4\u00e4n. T\u00e4m\u00e4 voidaan kiert\u00e4\u00e4 muutaman tuloksen avulla. Topologista astetta ei kuitenkaan koskaan m\u00e4\u00e4ritell\u00e4 m\u00e4\u00e4rittelyjoukon reunan kuvapisteiss\u00e4, sill\u00e4 funktion k\u00e4ytt\u00e4ytyminen joukon reunalla voi olla arvaamatonta. T\u00e4m\u00e4 m\u00e4\u00e4rittelyjoukon reunan kuvajoukko on siis k\u00e4yt\u00e4nn\u00f6ss\u00e4 jatkuville funktioille aina v\u00e4hint\u00e4\u00e4n osa kuvajoukon reunasta.\r\nTutkielman lopuksi k\u00e4yd\u00e4\u00e4n l\u00e4pi my\u00f6s muutamia lauseita, esimerkiksi Brouwerin kiintopistelause ja Jordanin erotuslause, joiden todistamisessa topologista astetta voidaan k\u00e4ytt\u00e4\u00e4 hyo\u00f6dyksi.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Lauri Huttunen (lajohutt) on 2017-08-15 19:53:38.018935. 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 2017-08-15T19:53:38Z\r\nNo. of bitstreams: 2\r\nURN:NBN:fi:jyu-201708153494.pdf: 801293 bytes, checksum: 9d7429a8904cca9af3fdb5319651eb48 (MD5)\r\nlicense.html: 4774 bytes, checksum: e6974d9a80abb508b191b741f7eccbcb (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2017-08-15T19:53:38Z (GMT). No. of bitstreams: 2\r\nURN:NBN:fi:jyu-201708153494.pdf: 801293 bytes, checksum: 9d7429a8904cca9af3fdb5319651eb48 (MD5)\r\nlicense.html: 4774 bytes, checksum: e6974d9a80abb508b191b741f7eccbcb (MD5)\r\n Previous issue date: 2017", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (43 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": "Brouwerin kiintopistelause", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Jordanin erotuslause", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Topologinen aste", "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-201708153494", "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": "2017-08-15T19:53:38Z", "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": "topologia", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "Jacobin matriisit", "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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