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[{"key": "dc.contributor.advisor", "value": "Rajala, Tapio", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Kankaanp\u00e4\u00e4, Saana Kaarina", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2020-02-18T07:39:00Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2020-02-18T07:39:00Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2020", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/67854", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4 tutkielma k\u00e4sittelee fraktaaligeometriaa sek\u00e4 tarkemmin fraktaalilaatoituksia tasossa. Tutkielman p\u00e4\u00e4asiallisena tarkoituksena on luoda pohja fraktaalilaatoitusten muodostamiselle. T\u00e4m\u00e4 tehd\u00e4\u00e4n m\u00e4\u00e4ritelem\u00e4ll\u00e4 ensin Hausdorffin et\u00e4isyys sek\u00e4 Banachin kiintopistelause. Jotta n\u00e4iden m\u00e4\u00e4ritt\u00e4minen voidaan tehd\u00e4, tutustutaan my\u00f6s metrisiin avaruuksiin sek\u00e4 niiden ominaisuuksiin.\n\nTutkielman t\u00e4rke\u00e4n\u00e4 osana ovat iteroidut funktioj\u00e4rjestelm\u00e4t (IFS) sek\u00e4 niiden attraktorit. IFS on fraktaalien rakentamiseksi luotu j\u00e4rjestelm\u00e4, jonka avulla fraktaalien mallintaminen on mahdollista. Tutkielmassa todistetaan Banachin\nkiintopistelausetta k\u00e4ytt\u00e4en, ett\u00e4 t\u00e4ydellisess\u00e4 metrisess\u00e4 avaruudessa maaritetyll\u00e4\nIFS:ll\u00e4 on olemassa yksik\u00e4sitteinen attraktori.\n\nT\u00e4ss\u00e4 tutkielmassa fraktaaleja tutkitaan kaksiulotteisessa avaruudessa. Fraktaaliesimerkkej\u00e4 k\u00e4yd\u00e4\u00e4n l\u00e4pi ensin nelj\u00e4 kappaletta. N\u00e4m\u00e4 ovat Cantorin joukko, Kochin k\u00e4yr\u00e4, Sierpinskin kolmio sek\u00e4 Mandelbrotin joukko. N\u00e4ist\u00e4 kolme ensimm\u00e4ist\u00e4 ovat IFS:n antamia. Annamme n\u00e4ille esimerkeille funktioj\u00e4rjestelm\u00e4t, joilla fraktaalit on helppo iteroiden muodostaa.\n\nItse fraktaalilaatoituksiin tutustutaan neli\u00f6aatoitusten pohjalta. Fraktaalilaatoitusten todetaan koostuvan itsesimilaarisista fraktaalilaatoista eli jokainen yksitt\u00e4inen laatta on toisensa kopio. Laattojen muodostamisessa k\u00e4ytet\u00e4\u00e4n iteroitua funktioj\u00e4rjestelm\u00e4\u00e4, jossa kutistussuhteet ovat kaikilla funktioilla samat ja saatu laatoitus riippuu vain siirtovektoreiden valinnasta. Tutkielma sis\u00e4lt\u00e4\u00e4 esimerkkej\u00e4, joiden avulla huomataan, mitk\u00e4 ovat j\u00e4rkevi\u00e4 valintoja siirtovektoreille ja mitk\u00e4 eiv\u00e4t. Tarkkoja ehtoja fraktaalilaatoitusten muodostamiselle ei kuitenkaan anneta.\n\nTutkielman alussa tutustutaan my\u00f6s hieman fraktaalien historiaan. Kerrotaan\nesimerkiksi ensimm\u00e4isist\u00e4 l\u00f6ydetyist\u00e4 fraktaaleista sek\u00e4 siit\u00e4, milloin fraktaaleja\nalettiin mallintamaan matemaattisesti.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2020-02-18T07:39:00Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2020-02-18T07:39:00Z (GMT). No. of bitstreams: 0\n Previous issue date: 2020", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "29", "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": "fraktaalilaatoitus", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Fraktaalilaatoitukset", "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-202002182093", "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": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "fraktaalit", "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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