| fullrecord |
[{"key": "dc.contributor.advisor", "value": "Koskela, Pekka", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Porkola, Jussi", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2022-04-06T05:15:08Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2022-04-06T05:15:08Z", "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/123456789/80499", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on todistaa Brouwerin kiintopistelause tason suljetussa yksikk\u00f6pallossa. Brouwerin kiintopistelauseen mukaan jokaisella jatkuvalla funktiolla tason suljetulta yksikk\u00f6pallolta itselleen on kiintopiste. Johdannossa k\u00e4yd\u00e4\u00e4n l\u00e4pi Brouwerin el\u00e4m\u00e4kertaa. Keskityt\u00e4\u00e4n el\u00e4m\u00e4kerran kuvailussa erityisesti Brouwerin akateemiseen uraan. Lis\u00e4ksi johdannossa kuvaillaan lyhyesti tutkielman kappaleiden aiheita.\nToisessa luvussa k\u00e4sitell\u00e4\u00e4n lyhyesti perusm\u00e4\u00e4ritelmi\u00e4 ja merkint\u00f6j\u00e4. Kolmannessa luvussa k\u00e4sitell\u00e4\u00e4n tutkielmassa tarvittavia esitietoja topologisista avaruuksista. M\u00e4\u00e4ritell\u00e4\u00e4n topologisten avaruuksien perusk\u00e4sitteit\u00e4, kuten jatkuvuus, topologian kanta, yhten\u00e4isyys ja kompaktius sek\u00e4 todistetaan n\u00e4ihin liittyvi\u00e4 tuloksia. Nelj\u00e4nness\u00e4 luvussa l\u00e4hdet\u00e4\u00e4n tarkastelemaan topologisia avaruuksia abstraktin algebran k\u00e4sitteiden, kuten ekvivalenssiluokkien kautta. Luvun alussa m\u00e4\u00e4ritell\u00e4\u00e4n homotopia ja polkuhomotopia, jotka osoittautuvat ekvivalenssirelaatioiksi. Polkuhomotopia on jatkuva kuvaus, joka muuttaa topologisen avaruuden polun toiseksi poluksi niin, ett\u00e4 polkujen yhteiset alku ja -p\u00e4\u00e4tepisteet pysyv\u00e4t muunnoksessa muuttumattomina. Luvussa m\u00e4\u00e4ritell\u00e4\u00e4n my\u00f6s polkuhomotopian ekvivalenssirelaation ekvivalenssiluokkien v\u00e4linen laskutoimitus ja n\u00e4hd\u00e4\u00e4n, ett\u00e4 t\u00e4ll\u00e4 laskutoimituksella on hyvin samanlaisia ominaisuuksia kuin mit\u00e4 aksioomia ryhm\u00e4ll\u00e4 on. Viidennen luvun alussa m\u00e4\u00e4ritell\u00e4\u00e4n topologisen avaruuden perusryhm\u00e4. Avaruuden perusryhm\u00e4\u00e4 varten kiinnitet\u00e4\u00e4n avaruudesta piste, jota kutsutaan avaruuden kantapisteeksi. Lis\u00e4ksi m\u00e4\u00e4ritell\u00e4\u00e4n silmukan olevan sellainen polku, jolla on sama alku- ja p\u00e4\u00e4tepiste. T\u00e4ll\u00f6in avaruuden perusryhm\u00e4 on kantapisteess\u00e4 olevien silmukoiden polkuluokkien joukko varustettuna polkuluokkien v\u00e4lisell\u00e4 laskutoimituksella. T\u00e4m\u00e4n j\u00e4lkeen luvussa k\u00e4sitell\u00e4\u00e4n perusryhm\u00e4n riippuvuutta kantapisteest\u00e4 ja m\u00e4\u00e4ritell\u00e4\u00e4n kuvaus, jonka avulla perusryhm\u00e4 voidaan kuvata toiseksi perusryhm\u00e4ksi. M\u00e4\u00e4ritell\u00e4\u00e4n my\u00f6s kuvauksen indusoima homomorfismi. Luvun lopussa k\u00e4sitell\u00e4\u00e4n viel\u00e4 lyhyesti peiteavaruuksia, kuvausten nostoja sek\u00e4 ympyr\u00e4n perusryhm\u00e4\u00e4. Kuudennessa luvussa m\u00e4\u00e4ritell\u00e4\u00e4n aluksi retraktio. Retraktio on sellainen jatkuva kuvaus topologiselta avaruudelta saman topologisen avaruuden osajoukkoon, jonka rajoittuma t\u00e4h\u00e4n osajoukkoon on sama kuin identtinen kuvaus. Sitten todistetaan joitakin tuloksia liittyen retraktioihin. Luvun lopussa p\u00e4\u00e4st\u00e4\u00e4n todistamaan tutkielman p\u00e4\u00e4tulos, eli Brouwerin kiintopistelause tason suljetussa yksikk\u00f6pallossa. Seitsem\u00e4nness\u00e4 ja tutkielman viimeisess\u00e4 luvussa k\u00e4sitell\u00e4\u00e4n kiintopistelauseiden historiaa sek\u00e4 esitell\u00e4\u00e4n Banachin, Schauderin ja Browderin kiintopistelauseet. Luvun lopussa esitell\u00e4\u00e4n viel\u00e4 kiintopistelauseiden sovelluksia niin teoreettisen kuin sovelletun matematiikan osa-alueilta.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2022-04-06T05:15:08Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2022-04-06T05:15:08Z (GMT). No. of bitstreams: 0\n Previous issue date: 2022", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "38", "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": "kiintopistelause", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "algebrallinen topologia", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "homotopia", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "perusryhm\u00e4", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Brouwerin kiintopistelause", "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-202204062181", "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": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "topologia", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "abstrakti algebra", "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"}]
|