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[{"key": "dc.contributor.advisor", "value": "Le Donne, Enrico", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Kivioja, Ville", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2015-11-14T09:36:18Z", "language": "", "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2015-11-14T09:36:18Z", "language": "", "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2015", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1504035", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/47675", "language": "", "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Ta\u0308ssa\u0308 tyo\u0308ssa\u0308 esiteta\u0308a\u0308n uusi tulos koskien isometrioiden sa\u0308a\u0308nno\u0308llisyytta\u0308 nilpotenttien yhtena\u0308isten metristen Lien ryhmien va\u0308lilla\u0308. Termilla\u0308 metrinen Lien ryhma\u0308 tarkoitamme Lien ryhma\u0308a\u0308, joka on varustettu eta\u0308isyysfunktiolla siten, etta\u0308 ryhma\u0308n (vasen) siirtokuvaus on isometria, ja eta\u0308isyysfunktio indusoi topologian, joka Lien ryhma\u0308lla\u0308 on monistona alun perin olemassa. Todistamme, etta\u0308 isometriat ta\u0308ssa\u0308 tilanteessa ovat va\u0308ltta\u0308ma\u0308tta\u0308 affiinikuvauksia: jokainen isometria voidaan esitta\u0308a\u0308 yhdistettyna\u0308 kuvauksena siirrosta ja isomorfismista. Ta\u0308ma\u0308n seurauksena kaksi isometrista ryhma\u0308a\u0308 ovat va\u0308ltta\u0308ma\u0308tta\u0308 isomorfiset.\r\nKlassisesti isometrioiden lineaariaffiinisuus on tunnettu Euklidisessa avaruudessa, mutta myo\u0308hemmin vastaava yleistetty tulos on todistettu reaalisissa normiavaruuksissa (Mazur\u2013Ulam lause) ja nilpotenteissa yhtena\u0308isissa\u0308 Riemannilaisissa Lien ryhmissa\u0308 (E.N. Wilson). Viime vuosina tulos on onnistuttu todistamaan myo\u0308s subRiemannilaisissa ja subFinslerila\u0308isissa\u0308 Carnot\u2019n ryhmissa\u0308. Metrinen Lien ryhma\u0308 on na\u0308ita\u0308 kaikkia yleisempi avaruus, lukuun ottamatta a\u0308a\u0308reto\u0308nulotteisia normiavaruuksia.\r\nTodistus perustuu Montgomery\u2013Zippinin lokaalisti kompaktien ryhmien teoriasta johdettaviin isometrioiden sa\u0308a\u0308nno\u0308llisyysominaisuuksiin seka\u0308 mainitun Wilsonin tuloksen ka\u0308ytto\u0308o\u0308n.\r\nToteamme lopuksi, etta\u0308 niin yhtena\u0308isyys kuin nilpotenttiuskin ovat va\u0308ltta\u0308ma\u0308tto\u0308mia\u0308 oletuksia siina\u0308 mielessa\u0308, etta\u0308 voimme esitta\u0308a\u0308 vastaesimerkit kummasta tahansa na\u0308ista\u0308 oletuksista luovuttaessa.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "We show that any isometry between two connected nilpotent metric Lie groups can be expressed as a composition of a translation and an isomorphism, i.e. isometries have an affine decomposition. By the term metric Lie group we mean a Lie group with a left-invariant distance that induces the topology of the manifold. It also follows that two isometric groups are isomorphic in this setting. Classically isometries are known to have the affine decomposition in the setting of Euclidean space and more generally in normed vector spaces over reals (Mazur-Ulam theorem) and nilpotent connected Riemannian Lie groups (E.N. Wilson). The result is also recently proved in subRiemannian (even subFinsler) Carnot groups. Metric Lie groups are more general spaces than these, excluding the infinite dimensional normed spaces. Our proof is based on the theory of locally compact groups of Montgomery\u2013Zippin and on the usage of the above mentioned result by Wilson. In a sense our result is a maximal generalization: After proving the result we construct counterexamples to the result in the cases where either nilpotency or connectedness is dropped from the list of assumptions.", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Ville Kivioja (vikakivi) on 2015-11-14 09:36:16.745894. Form: Pro gradu -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija.group@korppi.jyu.fi) on 2015-11-14T09:36:18Z\r\nNo. of bitstreams: 2\r\nURN:NBN:fi:jyu-201511143657.pdf: 670032 bytes, checksum: 53a96d87f44310e7cf9c3dcd6ddae9e2 (MD5)\r\nlicense.html: 4815 bytes, checksum: c7513ec6da417afe4fa3bd035e0d3dd7 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2015-11-14T09:36:18Z (GMT). No. of bitstreams: 2\r\nURN:NBN:fi:jyu-201511143657.pdf: 670032 bytes, checksum: 53a96d87f44310e7cf9c3dcd6ddae9e2 (MD5)\r\nlicense.html: 4815 bytes, checksum: c7513ec6da417afe4fa3bd035e0d3dd7 (MD5)\r\n Previous issue date: 2015", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (55 sivua)", "language": null, "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": "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.subject.other", "value": "isometria", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "nilpotentit ryhm\u00e4t", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "Lien ryhm\u00e4t", "language": null, "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Affine decomposition of isometries in nilpotent Lie groups", "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-201511143657", "language": null, "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": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "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": "dc.date.updated", "value": "2015-11-14T09:36:19Z", "language": "", "element": "date", "qualifier": "updated", "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": "fi", "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": null, "element": "subject", "qualifier": "oppiainekoodi", "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"}]
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