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[{"key": "dc.contributor.advisor", "value": "Salo, Mikko", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Niemi, Tuomas", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2025-01-28T06:25:55Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2025-01-28T06:25:55Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2025", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/99794", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa tarkastellaan geodeesista s\u00e4demuunnosta ja siihen liittyvi\u00e4 integraalimuunnoksia. Geodeesinen s\u00e4demuunnos on Riemannin geometrian vastine klassiselle r\u00f6ntgens\u00e4demuunnokselle, jota hy\u00f6dynnet\u00e4\u00e4n esimerkiksi l\u00e4\u00e4ketieteellisess\u00e4 kuvantamisessa. Erona t\u00e4h\u00e4n klassiseen s\u00e4demuunnokseen, geodeesisessa s\u00e4demuunnoksessa integroidaan annettua funktiota\nkaikkien Riemannin moniston M maksimaalisten geodeesien yli. Mielekk\u00e4it\u00e4\nkysymyksi\u00e4 geodeesiselle s\u00e4demuunnokselle ovat esimerkiksi sen injektiivisyys ja jatkuvuus. Injektiivisyytt\u00e4 koskevissa kysymyksiss\u00e4 halutaan siis selvitt\u00e4\u00e4 voidaanko annettu funktio selvitt\u00e4\u00e4 yksik\u00e4sitteisesti sen integraaleista kaikkien geodeesien yli. Jatkuvuuskysymyksess\u00e4 puolestaan tarkastellaan\novatko s\u00e4demuunnoksella kuvatut funktiot l\u00e4hell\u00e4 toisiaan, jos alkuper\u00e4iset\nfunktiot ovat. Vastaus t\u00e4h\u00e4n kysymykseen riippuu oleellisesti tarkasteltavasta funktioavaruudesta, ja siit\u00e4 miss\u00e4 mieless\u00e4 funktioiden halutaan olevan\nl\u00e4hell\u00e4 toisiaan.\nTutkielmassa k\u00e4sitell\u00e4\u00e4n aluksi yksinkertaisempaa tapusta geodeesisesta\ns\u00e4demuunnoksesta, jossa tarkasteltavana on tason yksikk\u00f6kiekko ja Riemannin metriikka on euklidinen metriikka kerrottuna jollain radiaalisella funktiolla c(r). Luvussa osoitetaan, ett\u00e4 jos geodeesit toteuttavat niin kutsutun\nHerglotz-ehdon, niin funktio c voidaan selvitt\u00e4\u00e4, kun tunnetaan maksimaalisten geodeesien pituudet. T\u00e4m\u00e4 johtaa luonnollisella tavalla tarkastelemaan\nAbelin muunnokseksi kutsuttua integraalimuunnosta, jolle osoitetaan joitakin kuvausominaisuuksia seuraavassa luvussa.\nViimeisess\u00e4 luvussa syvennyt\u00e4\u00e4n geodeesiseen s\u00e4demuunnokseen yleisemmin ja osoitetaan tutkielman p\u00e4\u00e4tulos: geodeesinen s\u00e4demuunnos on injektiivinen kompaktisti kannetuille sileille funktioille kaksiulotteisilla yksinkertaisilla monistoilla.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2025-01-28T06:25:54Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2025-01-28T06:25:55Z (GMT). No. of bitstreams: 0\n Previous issue date: 2025", "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.title", "value": "Geodeesinen s\u00e4demuunnos", "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-202501281557", "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": "geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "matematiikka", "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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