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[{"key": "dc.contributor.advisor", "value": "Rajala, Kai", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Tuominen, Eemu", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2019-04-05T06:27:35Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2019-04-05T06:27:35Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2019", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/63401", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Tutkielmassa l\u00e4hdet\u00e4\u00e4n liikkeelle metristen avaruuksien t\u00e4ydellist\u00e4misest\u00e4 t\u00e4ydellisiksi metrisiksi avaruuksiksi, miss\u00e4 Cauchy-jonoilla on merkitt\u00e4v\u00e4 rooli. T\u00e4ydellistymisprosessista k\u00e4yd\u00e4\u00e4n l\u00e4pi esimerkkin\u00e4 rationaalilukujen t\u00e4ydellist\u00e4minen reaaliluvuiksi euklidisella normilla. My\u00f6hemmin tutkitaan rationaalilukujen t\u00e4ydellist\u00e4mist\u00e4 toisenlaisella normilla, josta tulee hyvin erilainen kunta. Kappaleen 2 lopuksi k\u00e4sitell\u00e4\u00e4n lause, jonka avulla saadaan muodostettua t\u00e4ydellinen metrinen avaruus k\u00e4tev\u00e4sti tietyill\u00e4 ehdoilla.\nT\u00e4m\u00e4n j\u00e4lkeen tutkitaan kunnan normin erilaisia tuloksia ja m\u00e4\u00e4ritell\u00e4\u00e4n jonon ominaisuuksia. Lis\u00e4ksi perehdyt\u00e4\u00e4n vahvaan kolmioep\u00e4yht\u00e4l\u00f6\u00f6n, jonka toteuttavaa normia sanotaan ep\u00e4arkhimediseksi.\nSeuraavaksi on j\u00e4rkev\u00e4\u00e4 yhdist\u00e4\u00e4 kahden aiemman kappaleen tuloksia ja tutkia normilla varustettujen kuntien t\u00e4ydellist\u00e4mist\u00e4.\n\nN\u00e4iden tulosten nojalla Kappaleessa 5 voidaan siirty\u00e4 p\u00e4\u00e4asiaan eli p-adisiin lukuihin. P-adisia lukuja varten m\u00e4\u00e4ritell\u00e4\u00e4n p-adinen normi, joka riippuu alkuluvusta p. P-adisten lukujen kunta Q_p saadaan t\u00e4ydellist\u00e4misprosessilla rationaaliluvuista p-adisella normilla. P-adisten lukujen joukosta saadaan my\u00f6s toinen mielenkiintoinen joukko, p-adisten kokonaislukujen joukko Z_p. Kun ollaan p\u00e4\u00e4sty p-adisiin lukuihin, niin tutkitaan kuinka niiden aritmeettiset laskustoimitukset toimivat. Ne eroavat jossain m\u00e4\u00e4rin perinteisist\u00e4 reaalilukujen laskutoimituksista, mutta jossain mieless\u00e4 ne ovat jopa helpompia laskea.\nT\u00e4m\u00e4n j\u00e4lkeen tutkitaan rationaalilukujen ja p-adisten lukujen yhteytt\u00e4. Tarkoituksena on my\u00f6s selvitt\u00e4\u00e4, voidaanko p-adisen luvun laajennuksesta p\u00e4\u00e4tell\u00e4 mink\u00e4laista lukua se esitt\u00e4\u00e4.\nSeuraavaksi tutkitaan kongruenssiin liittyvi\u00e4 tuloksia avaruudessa Q_p. Keskeisin asia kongruenssiin liittyviss\u00e4 tuloksissa on Henselin Lemma, jolla voidaan ratkaista polynomin juuria, kun kertoimet ovat p-adisia lukuja. T\u00e4m\u00e4 on periaatteessa p-adinen veriso Newtonin menetelm\u00e4st\u00e4, jolla etsit\u00e4\u00e4n reaalikertoimisen polynomin juuria.\n\nKappaleessa 9 tutkitaan topologisia perusominaisuuksia. T\u00e4st\u00e4 p\u00e4\u00e4st\u00e4\u00e4n avaruuden Q_p palloihin, jotka k\u00e4ytt\u00e4ytyv\u00e4t hyvin eri lailla kuin avaruuden R pallot.\nSeuraavaksi tutkitaan Cantorin joukkkoa eli ``Cantorin kolmasosajoukkoa'' ja sen ominaisuuksia. Lis\u00e4ksi kappaleessa kerrataan metristen avaruuksien kuvauksiin liittyvi\u00e4 ominaisuuksia. N\u00e4iden avulla luodaan yhteyksi\u00e4 Cantorin joukkojen ja p-adisten kokonaislukujen joukon Z_p v\u00e4lille.\nViimeisess\u00e4 kappaleessa tutkitaan avaruuden Z_p euklidisia mallinnuksia. T\u00e4t\u00e4 varten muodostetaan kuvaus joukolta Z_p v\u00e4lille [0,1]. Joukkojen Z_p kuvauksilla eri alkuluvun p arvoilla on jotain yhteist\u00e4: ne ovat kaikki fraktaaleja.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2019-04-05T06:27:35Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2019-04-05T06:27:35Z (GMT). No. of bitstreams: 0\n Previous issue date: 2019", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "57", "language": "", "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": "p-adiset luvut", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Caychy-jono", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "jatkuvat kuvaukset", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "P-adiset luvut ja avaruuksien Z_p euklidiset mallinnukset", "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-201904052078", "language": "", "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": "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": "normit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "metriset avaruudet", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "reaaliluvut", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "rationaaliluvut", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "alkuluvut", "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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