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[{"key": "dc.contributor.advisor", "value": "Eriksson-Bique, Sylvester", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Mattila, Vera", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2024-12-19T06:52:02Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2024-12-19T06:52:02Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2024", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/99078", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman aiheena on P\u00f3lyan lause. Se on stokastinen tulos, jonka mukaan symmetrinen satunnaisk\u00e4vely yksi- ja kaksiulotteisessa hilassa on palautuva, mutta kolme- ja ylempiulotteisessa hilassa poistuva. T\u00e4m\u00e4 tarkoittaa, ett\u00e4 kun yksi- tai kaksiulotteisessa hilassa l\u00e4hdet\u00e4\u00e4n mist\u00e4 tahansa pisteest\u00e4 liikkeelle niin, ett\u00e4 jokaisen suunnan valitsemisen todenn\u00e4k\u00f6isyys on sama, p\u00e4\u00e4dyt\u00e4\u00e4n melkein varmasti joskus takaisin alkupisteeseen, kun taas kolme- ja ylempiulotteisessa hilassa ei v\u00e4ltt\u00e4m\u00e4tt\u00e4 koskaan palata takaisin. Tutkielman tavoitteena on todistaa P\u00f3lyan lause. Se todistetaan tekem\u00e4ll\u00e4 hiloista s\u00e4hk\u00f6verkkoja ja k\u00e4ytt\u00e4m\u00e4ll\u00e4 tuloksia, jotka satunnaisk\u00e4velylle todistetaan fysikaalista intuitiota k\u00e4ytt\u00e4en. Tutkielma aloitetaan verkkoteorialla, sill\u00e4 hilat ovat verkkoja. M\u00e4\u00e4ritell\u00e4\u00e4n verkko sek\u00e4 annetaan m\u00e4\u00e4ritelmi\u00e4 verkkojen ominaisuuksille. Yksi n\u00e4ist\u00e4 ominaisuuksista on verkon k\u00e4vely, jota tarvitaan satunnaisk\u00e4velyn m\u00e4\u00e4rittelyyn. T\u00e4m\u00e4n j\u00e4lkeen esitell\u00e4\u00e4n Markovin ketjut, sill\u00e4 satunnaisk\u00e4vely on tietyt ehdot t\u00e4ytt\u00e4v\u00e4 Markovin ketju. Annetaan m\u00e4\u00e4ritelm\u00e4 Markovin ketjulle, ja esitell\u00e4\u00e4n Markovin ketjujen ominaisuuksia. Kerrotaan esimerkiksi, milloin Markovin ketju on palautuva ja milloin poistuva, mik\u00e4 on t\u00e4rke\u00e4\u00e4 P\u00f3lyan lausetta varten. T\u00e4m\u00e4n j\u00e4lkeen m\u00e4\u00e4ritell\u00e4\u00e4n satunnaisk\u00e4vely Markovin ketjujen ja verkkojen avulla. Seuraavaksi k\u00e4sitell\u00e4\u00e4n s\u00e4hk\u00f6verkkoja. Aluksi m\u00e4\u00e4ritell\u00e4\u00e4n s\u00e4hk\u00f6verkko, mink\u00e4 j\u00e4lkeen k\u00e4sitell\u00e4\u00e4n s\u00e4hk\u00f6verkkojen virtauksia ja potentiaaleja. Lis\u00e4ksi annetaan virtauksen energialle m\u00e4\u00e4ritelm\u00e4 sek\u00e4 esitell\u00e4\u00e4n efektiivinen resistanssi. T\u00e4m\u00e4n j\u00e4lkeen todistetaan Markovin ketjun palautuvuuteen ja poistuvuuteen liittyv\u00e4 tulos efektiivisen resistanssin sek\u00e4 virtauksien ja energian avulla. Kyseinen tulos on yksi tutkielman t\u00e4rkeimmist\u00e4, koska sit\u00e4 k\u00e4ytet\u00e4\u00e4n P\u00f3lyan lauseen todistuksessa. Tutkielman lopuksi muotoillaan P\u00f3lyan lause ja todistetaan se.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2024-12-19T06:52:02Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2024-12-19T06:52:02Z (GMT). No. of bitstreams: 0\n Previous issue date: 2024", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "56", "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": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "P\u00f3lyan lause", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "satunnaisk\u00e4vely", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "stokastiikka", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "P\u00f3lyan lause", "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-202412197889", "language": null, "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": "Stokastiikka ja todenn\u00e4k\u00f6isyysteoria", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Stochastics and Probability", "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": "s\u00e4hk\u00f6verkot", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "verkkoteoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "Markovin ketjut", "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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