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[{"key": "dc.contributor.advisor", "value": "Rajala, Tapio", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Laitinen, Pietari", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2019-12-18T05:14:35Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2019-12-18T05:14:35Z", "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/123456789/66889", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 matematiikan sivuainetutkielmassa tarkastellaan rajoitettuja konvekseja joukkoja tasossa sek\u00e4 suoran leikkauspituuksia annettujen joukkojen kanssa. \nTutkielmassa esitet\u00e4\u00e4n injektiivinen suoran esitys tason kaikille suorille kierto ja siirto parametrin avulla. T\u00e4m\u00e4n esityksen avulla tarkastellaan ortogonaaliprojektioita ja n\u00e4iden mitallisuutta. \nErityisesti tasossa tarkastelun kohteena on rajoitetun konveksin joukon reuna. Tutkielmassa osoitetaan, ett\u00e4 rajoitetun konveksin joukon reuna on umpinainen Jordan k\u00e4yr\u00e4, jos konveksin joukon sis\u00e4joukko on ep\u00e4tyhj\u00e4. T\u00e4m\u00e4 todistetaan konveksien joukkojen ominaisuuksien ja puolisuorien avulla, jonka seurauksena saadaan ett\u00e4 rajoitetut konveksit joukot ovat Lebesgue-mitallisia. \nTutkielman p\u00e4\u00e4lauseena on osoittaa, ett\u00e4 parametrien joukolla, jolla mainitun injektiivisen suoran esityksen suorat leikkaavat annettua rajoitettua konveksia joukkoa on sellainen Lebesguen mitta, joka vastaa kertoimella 1 tai 1/2 kyseisen konveksin joukon reunan 1-Hausdorffin mittaa. Tulos voidaan yleist\u00e4\u00e4 kahteen tapaukseen, joista toisessa konveksin joukon sis\u00e4joukko on tyhj\u00e4 ja toisessa ep\u00e4tyhj\u00e4. Ep\u00e4tyhj\u00e4 tapaus on kiinnostavampi sill\u00e4 t\u00e4ll\u00f6in p\u00e4tee yht\u00e4suuruus kertoimella 1. Tyhj\u00e4ss\u00e4 tapauksessa konveksi joukko vastaa janaa tai tyhj\u00e4\u00e4 joukkoa, jolloin saadaan yht\u00e4suuruus kertoimella 1/2. \nTyhj\u00e4n joukon tapaus osoitetaan janan ominaisuuksien avulla. Ep\u00e4tyhj\u00e4\u00e4 tapausta varten hy\u00f6dynnet\u00e4\u00e4n niin tuloksia mittateoriasta kuin konveksianalyysist\u00e4 ja suoristuvista poluista.\nLis\u00e4ksi osoitetaan, ett\u00e4 tasossa saadaan Lebesgue ja Borel-mitallisia suoran leikkauspituusfunktioita tutkielmassa esitettyjen suoran esitysten avulla. N\u00e4m\u00e4 osoitetaan Fubinin Lauseen ja muuttujanvaihdon avulla.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2019-12-18T05:14:34Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2019-12-18T05:14:35Z (GMT). No. of bitstreams: 0\n Previous issue date: 2019", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "225", "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.title", "value": "Konvekseja joukkoja ja suoran leikkauksia tasossa", "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-201912185365", "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.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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