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[{"key": "dc.contributor.advisor", "value": "Juutinen, Petri", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Kangas, Aleksi", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2022-03-23T12:56:08Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2022-03-23T12:56:08Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2022", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/80317", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoitus on auttaa lukijaa ymm\u00e4rt\u00e4m\u00e4\u00e4n matematiikkaa dynaamisten systeemien ja kaaoksen takana. Kirjoitelman toivotaan her\u00e4tt\u00e4v\u00e4n lukijassa kiinnostusta matemaattisia ilmi\u00f6it\u00e4 kohtaan, sek\u00e4 kannustavan empiiriseen matematiikan tutkimiseen. Tutkielmassa esitell\u00e4\u00e4n diskreettien dynaamisten systeemien toimintaperiaate, tutustutaan kaaokseen, sek\u00e4 n\u00e4iden sovelluksena esitell\u00e4\u00e4n Mandelbrotin joukko ja sen ominaisuuksia. Tutkielmassa k\u00e4sitteit\u00e4 esitell\u00e4\u00e4n p\u00e4\u00e4osin sanallisesti selitt\u00e4en menem\u00e4tt\u00e4 liian syv\u00e4llisiin matemaattisiin m\u00e4\u00e4ritelmiin. Lukijan oletetaan hallitsevan lukion pitk\u00e4n matematiikan oppim\u00e4\u00e4r\u00e4\u00e4 vastaavat taidot. Tekstin ymm\u00e4rt\u00e4misen kannalta lukijan oletetaan hallitsevan algebran ja analyyttisen geometrian lis\u00e4ksi funktioiden ja yht\u00e4l\u00f6iden, raja-arvon ja derivaatan, lukujonojen, numeeristen menetelmien, sek\u00e4 matemaattisen v\u00e4itelauseen todistamisen perustaidot.\n\nDynaamisella systeemill\u00e4 tarkoitetaan j\u00e4rjestelm\u00e4\u00e4, joka riippuu ajasta. J\u00e4rjestelm\u00e4 voi olla hyvin yksinkertainen, kuten bakteerin levi\u00e4minen, tai todella monimutkainen, kuten p\u00f6rssiosakkeen hinta. Ensimm\u00e4isess\u00e4 luvussa tutustutaan miten systeemi k\u00e4ytt\u00e4ytyy, kun sen taustalla on jokin yksinkertaisen funktio. Luvun lopussa perehdyt\u00e4\u00e4n, miten dynaamista systeemi\u00e4 voidaan hy\u00f6dynt\u00e4\u00e4 populaation mallintamisessa ja samalla johdetaan populaation mallinnuksessa k\u00e4ytetyn logistisen funktion lauseke, jolla ilmenee olevan mielenkiintoisia ominaisuuksia.\n\nMatematiikassa kaaoksella tarkoitetaan ilmi\u00f6t\u00e4, jossa tarkasti m\u00e4\u00e4ritelty systeemi alkaa k\u00e4ytt\u00e4yty\u00e4 pitk\u00e4ll\u00e4 aikav\u00e4lill\u00e4 t\u00e4ysin arvaamattomasti. Arkikieless\u00e4 kaaoksella tarkoitetaan usein sekasortoa ja totaalista j\u00e4rjestyksen puuttumista. Niin ristiriitaiselta kuin se kuulostaakin, matematiikassa kaaoksella tarkoitetaan t\u00e4ysin p\u00e4invastaista, t\u00e4ydellisesti m\u00e4\u00e4ritelty\u00e4 systeemi\u00e4 niiden arvmaattomasta k\u00e4ytt\u00e4ytymisest\u00e4 huolimatta. Toisessa luvussa perehdyt\u00e4\u00e4n logistisen funktion avulla ratajoukon kaoottiseen k\u00e4ytt\u00e4ytymiseen. Luvussa m\u00e4\u00e4ritell\u00e4\u00e4n kaoottisuus ja tarkastellaan hieman tarkemmin mik\u00e4 tekee systeemist\u00e4 kaoottisen. Luvun lopussa tutkitaan my\u00f6s, miten satunnaisuus eroaa kaoottisuudesta.\n\nViimeisess\u00e4 luvussa keskityt\u00e4\u00e4n matemaattisemmasta n\u00e4k\u00f6kulmasta iteroinnin avulla aikaan saatavaan Mandelbrotin joukkoon. T\u00e4t\u00e4 varten luvussa esitell\u00e4\u00e4n kompleksiluvut, sek\u00e4 muutamia muita kaksiuloitteisen avaruuden relevantteja k\u00e4sitteit\u00e4. Mandelbrotin joukon ominaisuuksien esitt\u00e4misen ja todistamisen yhteydess\u00e4 lukijaa haastetaan hieman syv\u00e4llisemp\u00e4\u00e4n matemaattiseen ajatteluun. Lis\u00e4ksi tarkastellaan, miten tietokoneen avulla voidaan luoda h\u00e4mm\u00e4stytt\u00e4v\u00e4n upeita kuvia Mandelbrotin joukosta ja miten kaaos liittyy Mandelbrotin joukkoon.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2022-03-23T12:56:07Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2022-03-23T12:56:08Z (GMT). No. of bitstreams: 0\n Previous issue date: 2022", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "68", "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": "diskreetti dynaaminen systeemi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "logistinen funktio", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Mandelbrotin joukko", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Johdatus diskreetteihin dynaamisiin systeemeihin ja kaaokseen", "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-202203232017", "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": "kaaos", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "iterointi", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "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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