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[{"key": "dc.contributor.advisor", "value": "Rajala, Tapio", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "M\u00e4enp\u00e4\u00e4, Vilma", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2022-05-03T05:41:11Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2022-05-03T05:41:11Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2022", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/80856", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 pro gradu -tutkielmassa tutustutaan lumihiutaleupotuksiin. P\u00e4\u00e4tuloksena todistetaan Assouadin upotuslause, mik\u00e4 osoittaa lumihiutaleupotusten olemassaolon. Esimerkkin\u00e4 lumihiutaleupotuksesta k\u00e4sitell\u00e4\u00e4n von Kochin lumihiutaletta.\n\nTutkielman alussa m\u00e4\u00e4ritell\u00e4\u00e4n keskeiset k\u00e4sitteet, joita ovat muun muassa metriset avaruudet, bi-Lipschitz-kuvaus, t\u00e4ydellisyys sek\u00e4 kompaktius. Lis\u00e4ksi todistetaan\ntuloksia, joita tarvitaan my\u00f6hemmin tutkielman muiden lemmojen ja lauseiden todistuksissa.\n\nToisessa luvussa m\u00e4\u00e4ritell\u00e4\u00e4n ensin metrisen avaruuden tuplaavuus, lumihiutalemetriikka ja metrisen avaruuden lumihiutaleversio. Lis\u00e4ksi osoitetaan lumihiutaleversion olevan metrinen avaruus. T\u00e4m\u00e4n j\u00e4lkeen todistetaan Assouadin upotuslause:\nOlkoon (X, d) tuplaava metrinen avaruus. T\u00e4ll\u00f6in sen jokainen lumihiutaleversio\n(X, d^\u03b1) voidaan bi-Lipschitz upottaa johonkin Euklidiseen avaruuteen R^N .\n\nTutkielman kolmannessa luvussa k\u00e4sitell\u00e4\u00e4n von Kochin lumihiutalek\u00e4yr\u00e4\u00e4. Jotta\nk\u00e4yr\u00e4 voidaan antaa iteroidun funktioj\u00e4rjestelm\u00e4n kiintopisteen\u00e4, m\u00e4\u00e4ritell\u00e4\u00e4n ensin kutistavat kuvaukset ja todistetaan Banachin kiintopistelause. Lis\u00e4ksi m\u00e4\u00e4ritell\u00e4\u00e4n Hausdorff-et\u00e4isyys kompakteille ep\u00e4tyhjille joukoille ja keskeisen\u00e4 tuloksena osoitetaan, ett\u00e4 jokaisella iteroidulla funktioj\u00e4rjestelm\u00e4ll\u00e4 on olemassa yksik\u00e4sitteinen kiintopiste. Lopuksi osoitetaan, ett\u00e4 von Kochin lumihiutalek\u00e4yr\u00e4 on v\u00e4lin [0, 1] lumihiutaleupotus.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2022-05-03T05:41:11Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2022-05-03T05:41:11Z (GMT). No. of bitstreams: 0\n Previous issue date: 2022", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "44", "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": "bi-Lipschitz upotus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Assouadin upotuslause", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "lumihiutalemetriikka", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "lumihiutaleupotukset", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "von Kochin lumihiutalek\u00e4yr\u00e4", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Lumihiutaleupotukset", "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-202205032520", "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": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in 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": "geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "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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