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[{"key": "dc.contributor.advisor", "value": "Koskela, Pekka", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.advisor", "value": "Lu\u010di\u0107, Danka", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Leirimaa, Elisa", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2023-09-15T05:12:02Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2023-09-15T05:12:02Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2023", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/89115", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa k\u00e4sitell\u00e4\u00e4n Cesarin konstruktiota, joka on yksi tilan t\u00e4ytt\u00e4vist\u00e4 k\u00e4yrist\u00e4. Tilan t\u00e4ytt\u00e4v\u00e4n k\u00e4yr\u00e4n m\u00e4\u00e4ritelm\u00e4 edellytt\u00e4\u00e4, ett\u00e4 funktio kuvaa v\u00e4lin [0;1] koko neli\u00f6ksi [0;1]x[0;1]. T\u00e4llaisen funktion kuvajoukkoa kutsutaan tilan t\u00e4ytt\u00e4v\u00e4ksi k\u00e4yr\u00e4ksi. \n\nTilan t\u00e4ytt\u00e4vien k\u00e4yrien historian voidaan katsoa alkaneen vuodesta 1878, jolloin George Cantor osoitti, ett\u00e4 v\u00e4lin [0;1] voi kuvata bijektiivisesti neli\u00f6ksi [0;1]x[0;1]. Seuraavana vuonna E. Netto kuitenkin osoitti, ett\u00e4 t\u00e4llainen kuvaus ei voi olla jatkuva. T\u00e4st\u00e4 jatkui vastaavien k\u00e4yrien ominaisuuksien tutkiminen. Vuonna 1890 G. Peano osoitti, ett\u00e4 l\u00f6ytyy surjektiivinen jatkuva kuvaus v\u00e4lilt\u00e4 [0;1], jonka kuva on koko neli\u00f6 [0;1]x[0;1]. T\u00e4m\u00e4 on ensimm\u00e4inen tilan t\u00e4ytt\u00e4v\u00e4 k\u00e4yr\u00e4, joka tunnetaan my\u00f6s nimell\u00e4 Peanon k\u00e4yr\u00e4.\n\nCesarin konstruktio aloitetaan valitsemalla yksikk\u00f6pallon sis\u00e4lt\u00e4 nelj\u00e4 pistett\u00e4. N\u00e4iden pisteiden ymp\u00e4rille valitaan pienempi ja suurempi s\u00e4de. T\u00e4m\u00e4n j\u00e4lkeen m\u00e4\u00e4ritell\u00e4\u00e4n funktio, jonka arvo riippuu siit\u00e4, sijaitseeko piste pienemm\u00e4n s\u00e4teen sis\u00e4ll\u00e4, s\u00e4teiden v\u00e4liss\u00e4 vai suuremman s\u00e4teen ulkopuolella. Iteroimalla vastaavasti saadaan funktio, joka kuvaa yksikk\u00f6pallon neli\u00f6ksi [0;1]x[0;1]. Tarkemmin sanottuna jopa hyvin pieni osajoukko yksikk\u00f6pallosta kuvautuu t\u00e4ksi neli\u00f6ksi. \n\nT\u00e4ss\u00e4 tutkielmassa konstruoidaan Cesarin konstruktiota l\u00e4heisesti muistuttava funktio jakamalla v\u00e4li [0;1] nelj\u00e4\u00e4n osav\u00e4liin. Jokaisesta osav\u00e4list\u00e4 valitaan keskelt\u00e4 pienempi v\u00e4li, joka j\u00e4lleen jaetaan nelj\u00e4\u00e4n osav\u00e4liin. N\u00e4in muodostuu sis\u00e4kk\u00e4isi\u00e4 v\u00e4lej\u00e4, joiden pituus l\u00e4hestyy nollaa. Konstruktiossa m\u00e4\u00e4ritell\u00e4\u00e4n funktio, joka kuvaa jokaisella iteraatiokierroksella tihenev\u00e4n kuvan maalijoukkoon [-1;1]x[-1;1] t\u00e4ytt\u00e4en lopulta koko neli\u00f6n. Skaalaamalla ja siirt\u00e4m\u00e4ll\u00e4 kuvajoukoksi saadaan neli\u00f6 [0;1]x[0;1].\n\nKonstruktion iteraatiokierrosten osav\u00e4lien leikkaus osoittautuu nollamittaiseksi. T\u00e4m\u00e4 nollamittainen joukko kuvautuu positiivisen mittaiseksi. Lis\u00e4ksi osoitetaan, ett\u00e4 t\u00e4m\u00e4n nollamittaisen joukon Hausdorffin dimensio saadaan mielivaltaisen pieneksi, ja ett\u00e4 konstruktiossa m\u00e4\u00e4ritelty funktio on jatkuva.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2023-09-15T05:12:02Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2023-09-15T05:12:02Z (GMT). No. of bitstreams: 0\n Previous issue date: 2023", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "35", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": null, "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "Cesarin konstruktio ja sen ominaisuuksia", "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-202309155136", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "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.copyright", "value": "\u00a9 The Author(s)", "language": null, "element": "rights", "qualifier": "copyright", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "restrictedAccess", "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.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.rights.accessrights", "value": "The author has not given permission to make the work publicly available electronically. Therefore the material can be read only at the archival workstation at Jyv\u00e4skyl\u00e4 University Library (https://kirjasto.jyu.fi/collections/archival-workstation).", "language": "en", "element": "rights", "qualifier": "accessrights", "schema": "dc"}, {"key": "dc.rights.accessrights", "value": "Tekij\u00e4 ei ole antanut lupaa avoimeen julkaisuun, joten aineisto on luettavissa vain Jyv\u00e4skyl\u00e4n yliopiston kirjaston arkistoty\u00f6semalta. Ks. https://kirjasto.jyu.fi/kokoelmat/arkistotyoasema..", "language": "fi", "element": "rights", "qualifier": "accessrights", "schema": "dc"}]
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