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[{"key": "dc.contributor.advisor", "value": "Nicolussi Golo, Sebastiano", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Pesonen, Sofia", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2022-01-31T06:50:49Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2022-01-31T06:50:49Z", "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/123456789/79562", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa todistetaan Marstrandin projektiolause k\u00e4ytt\u00e4en apuna potentiaaliteoriaa.\nProjektiolauseen mukaan 2-ulotteisen Borel joukon ortogonaaliprojektion Hausdorffin dimensio on luvun 1 ja kyseisen Borel joukon dimension minimi melkein kaikkiin eri suuntiin.\nIntuitiivisesti lause kertoo, ett\u00e4 joukon varjon dimensio on suurin mahdollinen.\n\nMarstrandin projektiolauseen todistamiseksi tutkielmassa rakennetaan teoria alkaen yleisen mittateorian perustuloksista.\nMittateorian pohjalta m\u00e4\u00e4ritell\u00e4\u00e4n Hausdorffin mitta, jonka avulla m\u00e4\u00e4ritell\u00e4\u00e4n joukon Hausdorffin dimensio.\nIntuitiivisesti Hausdorffin dimensio kuvaa joukon geometrista kokoa ja se on yksik\u00e4sitteinen jokaiselle joukolle.\nHausdorffin dimensio mahdollistaa monimutkaisten joukkojen, kuten fraktaalien, geometrian tutkimisen.\nLis\u00e4ksi esitell\u00e4\u00e4n dimensioihin liittyvi\u00e4 merkint\u00f6j\u00e4 ja tapoja arvioida joukon Hausdorffin dimension suuruutta.\nTutkielman lopussa esitell\u00e4\u00e4n algoritminen menetelm\u00e4, jonka avulla voidaan muodostaa esimerkkej\u00e4 fraktaaleista.\nLopuksi sovelletaan Marstrandin projektiolausetta erilaisiin joukkoihin.\n\nJohn Marstrand todisti projektiolauseen vuonna 1954.\nRobert Kaufman todsti tuloksen k\u00e4ytt\u00e4en potentiaaliteoriaa vuonna 1968.\nMy\u00f6hemmin Kenneth Falconer esitteli Kaufmania mukaillen potentiaaliteoriaan perustuvan todistuksen.\nT\u00e4ss\u00e4 tutkielmassa esitell\u00e4\u00e4n kyseinen todistus yksityiskohtaisemmin.\nMarstrandin projektiolause tuli tunnetuksi, kun Mandelbrot popularisoi fraktaalin k\u00e4sitteen 1970-luvulla.\nLause voidaan yleist\u00e4\u00e4 korkeampiin dimensioihin ja se on t\u00e4rke\u00e4 ty\u00f6kalu fraktaalien geometrian tarkastelussa.\nVaikka lause on tunnettu pitk\u00e4\u00e4n, siihen liittyy edelleen avoimia ongelmia.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "In this thesis we prove Marstrand's projection theorem using potential theoretical methods.\nProjection theorem claims that the Hausdorff dimension of the orthogonal projection of a Borel set in $\\mathbb{R}^2$ is the minimum between 1 and the dimension of the set for almost all angles.\nIntuitively, the theorem gives that the shadow of the set has the highest possible dimension.\nThis result was first proven by John Marstrand in 1954 and it became well known after Mandelbrot popularized the notion of fractal in the 1970s.\nMarstrand\u2019s theorem has generalizations to higher dimensions and it is an important tool to look into the geometry of fractals.\nAlthough the theorem has been known for long time, there are still open problems related to it.", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Paivi Vuorio (paelvuor@jyu.fi) on 2022-01-31T06:50:49Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2022-01-31T06:50:49Z (GMT). No. of bitstreams: 0\n Previous issue date: 2022", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "41", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "eng", "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": "Dimension of projection : Marstrand's theorem", "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-202201311329", "language": "", "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": "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": "mittateoria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "fraktaalit", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "ulottuvuus", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "mathematics", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "measure theory", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "fractals", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "dimension", "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.rights.accessrights", "value": "", "language": null, "element": "rights", "qualifier": "accessrights", "schema": "dc"}]
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