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[{"key": "dc.contributor.advisor", "value": "Parviainen, Mikko", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Uusitalo, Susanna", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2023-10-26T06:04:42Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2023-10-26T06:04:42Z", "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/123456789/90669", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman ensimm\u00e4isess\u00e4 kokonaisuudessa tarkoituksena on tutustua harmonisiin funktioihin ja niiden ominaisuuksiin. Niist\u00e4 keskeinen on harmonisen funktion keskiarvoperiaate. Toisessa kokonaisuudessa tarkastellaan todenn\u00e4k\u00f6isyysteoriaa ja harmonisen funktion yhteytt\u00e4 pallossa tapahtuvaan satunnaisk\u00e4velyyn. Tutkielman alussa tutustutaan Laplacen yht\u00e4l\u00f6\u00f6n, jonka avulla harmoniset funktiot m\u00e4\u00e4ritell\u00e4\u00e4n. Erityisesti tarkastellaan sen fysikaalista tulkintaa ja johdetaan perusratkaisu. Seuraavaksi esitell\u00e4\u00e4n kolme harmonisen funktion ominaisuutta. N\u00e4ist\u00e4 harmonisen funktion keskiarvoperiaate kertoo harmonisen funktion saavan pallon keskipisteess\u00e4 yht\u00e4 suuren arvon sek\u00e4 pallon pinnalla olevien arvojen keskiarvon ett\u00e4 pallon sis\u00e4ll\u00e4 olevien arvojen keskiarvon kanssa. Keskiarvoperiaatteesta seuraa, ett\u00e4 harmoniset funktiot ovat s\u00e4\u00e4nn\u00f6llisi\u00e4 eli \u00e4\u00e4rett\u00f6m\u00e4n monta kertaa jatkuvasti differentioituvia. Lis\u00e4ksi todistetaan s\u00e4\u00e4nn\u00f6llisyystuloksen paranneltu versio, joka kertoo harmonisen funktion olevan analyyttinen eli jonkin pisteen ymp\u00e4rill\u00e4 esitett\u00e4viss\u00e4 suppenevana potenssisarjana. Harmonisten funktioiden ominaisuuksien j\u00e4lkeen tutkielmassa perehdyt\u00e4\u00e4n todenn\u00e4k\u00f6isyysteoriaan ja erityisesti martingaaliteoriaan. Ensin tutustutaan stokastiikan esitietoihin ja m\u00e4\u00e4ritell\u00e4\u00e4n martingaali ja pys\u00e4ytysaika. Niiden j\u00e4lkeen todistetaan optionaalisen pys\u00e4ytt\u00e4misen lause, jonka mukaan martingaalin odotusarvot ovat yht\u00e4 suuret aloitus- ja pys\u00e4ytyshetkell\u00e4. Tutkielman lopuksi esitell\u00e4\u00e4n harmonisen funktion keskiarvoperiaatetta muistuttava dynaamisen ohjelmoinnin periaate, jolle etsit\u00e4\u00e4n ratkaisufunktio siten, ett\u00e4 se saavuttaa annetun reuna-arvofunktion. Lis\u00e4ksi n\u00e4ytet\u00e4\u00e4n l\u00f6ydetyn ratkaisufunktion olevan pallossa tapahtuvan satunnaisk\u00e4velyn odotusarvo.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2023-10-26T06:04:42Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2023-10-26T06:04:42Z (GMT). No. of bitstreams: 0\n Previous issue date: 2023", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "49", "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.subject.other", "value": "Laplacen yht\u00e4l\u00f6", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "harmoniset funktiot", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "martingaaliteoria", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "satunnaisk\u00e4vely", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Harmoniset funktiot ja satunnaisk\u00e4vely", "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-202310266698", "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": "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.copyright", "value": "\u00a9 The Author(s)", "language": null, "element": "rights", "qualifier": "copyright", "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": "funktiot", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "todenn\u00e4k\u00f6isyyslaskenta", "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"}]
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