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[{"key": "dc.contributor.advisor", "value": "Ilmavirta, Joonas", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Manu, Miika", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2024-11-26T06:25:35Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2024-11-26T06:25:35Z", "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/123456789/98622", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa tarkastellaan sek\u00e4 geodeettisen s\u00e4demuunnoksen ett\u00e4 valons\u00e4demuunnoksen injektiivisyytt\u00e4. S\u00e4demuunnokset ovat integraalioperaattoreita, jotka muuttavat moniston \\(M\\) kuvauksen \\(f\\) funktioksi \\(If\\) moniston \\(M\\) geodeesien joukkoon siten, ett\u00e4 \\(If(\\gamma)\\) on kuvauksen \\(f\\) integraali geodeesin \\(\\gamma\\) yli. S\u00e4demuunnokset ovat lineaarisia, jolloin injektiivisyys voidaan tarkistaa osoittamalla, ett\u00e4 \\(\\ker I=\\{0\\}\\).\n\nTutkielman yhten\u00e4 p\u00e4\u00e4tuloksena todistetaan geodeettisen s\u00e4demuunnoksen injektiivisyys yksinkertaisella Riemannin monistolla kompaktisti kannetuille funktioille. Todistus tapahtuu muodostamalla funktiosta \\(f\\in\\ker I\\) pallonkuorikimpun osittaisdifferentiaaliyht\u00e4l\u00f6, jolla osoitetaan olevan yksik\u00e4sitteinen ratkaisu, josta seuraa, ett\u00e4 \\(f=0\\). Osittaisdifferentiaaliyht\u00e4l\u00f6n ratkaisu tapahtuu hy\u00f6dynt\u00e4m\u00e4ll\u00e4 Pestovin identiteettin\u00e4 tunnettua energiaidentiteetti\u00e4.\n\nTutkielman viimeisess\u00e4 osassa osoitetaan, ett\u00e4 Lorentzin tulomonistolla \\([0,T]\\times M\\), miss\u00e4 \\(M\\) on Riemannin monisto, valons\u00e4demuunnoksen injektiivisyyden tarkastelu voidaan palauttaa Riemannin monistolle kompaktisti kannettujen funktioiden ja 1-muotojen tapauksessa. Erityisesti valons\u00e4demuunnos on injektiivinen, jos geodeettinen s\u00e4demuunnos on injektiivinen Riemannin monistolla. Palauttaminen tapahtuu k\u00e4ytt\u00e4m\u00e4ll\u00e4 Fourier-muunnosta aikakoordinaatin suhteen sek\u00e4 hy\u00f6dynt\u00e4m\u00e4ll\u00e4 Paley-Wienerin lausetta, jonka mukaan kompaktisti kannetun funktion Fourier-muunnos on reaalianalyyttinen.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2024-11-26T06:25:35Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2024-11-26T06:25:35Z (GMT). No. of bitstreams: 0\n Previous issue date: 2024", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "54", "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": "valons\u00e4demuunnos", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "geodeettinen s\u00e4demuunnos", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "inversio-ongelmat", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Geodeettisen s\u00e4demuunnoksen ja valons\u00e4demuunnoksen injektiivisyys", "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-202411267447", "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": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "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": "differentiaaligeometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "monistot", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "geometria", "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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