| fullrecord |
[{"key": "dc.contributor.advisor", "value": "Parkkonen, Jouni", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Elomaa, Esko", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2025-03-18T06:03:12Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2025-03-18T06:03:12Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2025", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/100753", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman aiheita ovat Lien matriisiryhm\u00e4t, Lien algebrat sek\u00e4 Baker-Campbell-Hausdorffin kaava. Lien matriisiryhm\u00e4t ovat matriisien muodostamia joukkoja, joille on asetettu laskutoimitukseksi matriisien kertolasku, ja t\u00e4lle laskutoimitukselle p\u00e4tev\u00e4t ryhm\u00e4n ominaiset aksioomat. Lien matriisiryhmist\u00e4 n\u00e4ytet\u00e4\u00e4n esimerkkej\u00e4 ja niihin liittyvi\u00e4 topologisia ominaisuuksia. Lien algebrat ovat Lien matriisiryhmist\u00e4 nousevia \u00e4\u00e4rellisulotteisia avaruuksia, joille on asetettu ominainen kuvaus nimelt\u00e4 Lien sulkuoperaatio. Lien matriisiryhm\u00e4n Lien algebralle asetettu Lien sulkuoperaatio on kahden matriisin v\u00e4linen kommutaattori, joka on erityinen matriisien kertolaskun ep\u00e4vaihdannaisuuden\ntakia.\nLien teorian ohella esitet\u00e4\u00e4n matriisin eksponenttikuvaus sek\u00e4 matriisin logaritmi. Kaikille neli\u00f6matriiseille n\u00e4m\u00e4 ovat laskettavissa, ja esimerkkej\u00e4 n\u00e4ytet\u00e4\u00e4n matriisieksponentin laskemisesta erityisille matriiseille, kuten diagonalisoituville matriiseille. Matriisien eksponentti- ja logaritmikuvaus toimivat toistensa k\u00e4\u00e4nteiskuvauksina vain silloin, kun matriisit rajoitetaan yksikk\u00f6matriisin tietynkokoisen ymp\u00e4rist\u00f6n sis\u00e4lle.\nBaker-Campbell-Hausdorffin kaava on t\u00e4rke\u00e4 ja hy\u00f6dyllinen apuv\u00e4line matriisin eksponentin logaritmin selvitt\u00e4misess\u00e4. Esimerkkin\u00e4 katsotaan Heisenbergin ryhm\u00e4\u00e4, joka muotonsa vuoksi antaa helposti k\u00e4sinlaskettavan tapauksen. BCH-kaavan t\u00e4rkeys juontuu matriisien ep\u00e4vaihdannaisuudesta: jos kaksi matriisia olisivat vaihdannaisia, niin t\u00e4ll\u00f6in niiden v\u00e4linen kommutaattori olisi nollaa, ja matriisieksponentit olisivat my\u00f6s vaihdannaisia, jolloin tilanne on paljon yksinkertaisempi. BCH-kaavan hy\u00f6dyllisyys tuleekin juuri esille ep\u00e4vaihdannaisilla matriiseilla, joista mielenkiintoisimmat tapaukset saadaan tehty\u00e4.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2025-03-18T06:03:12Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2025-03-18T06:03:12Z (GMT). No. of bitstreams: 0\n Previous issue date: 2025", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "40", "language": null, "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": "Lien algebrat", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "matriisiryhm\u00e4t", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "matriisin eksponentti", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "matriisin logaritmi", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Lien matriisiryhm\u00e4t ja Lien algebrat", "language": null, "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-202503182568", "language": null, "element": "identifier", "qualifier": "urn", "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": null, "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": null, "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": "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": "fi", "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.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matriisit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "algebra", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matriisilaskenta", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "ryhm\u00e4teoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "Lien ryhm\u00e4t", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "ryhm\u00e4t", "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"}, {"key": "dc.description.accessibilityfeature", "value": "unknown accessibility", "language": "en", "element": "description", "qualifier": "accessibilityfeature", "schema": "dc"}, {"key": "dc.description.accessibilityfeature", "value": "ei tietoa saavutettavuudesta", "language": "fi", "element": "description", "qualifier": "accessibilityfeature", "schema": "dc"}]
|