{"?xml":{"@version":"1.0"},"edm:RDF":{"@xmlns:dc":"http://purl.org/dc/elements/1.1/","@xmlns:edm":"http://www.europeana.eu/schemas/edm/","@xmlns:wgs84_pos":"http://www.w3.org/2003/01/geo/wgs84_pos","@xmlns:foaf":"http://xmlns.com/foaf/0.1/","@xmlns:rdaGr2":"http://rdvocab.info/ElementsGr2","@xmlns:oai":"http://www.openarchives.org/OAI/2.0/","@xmlns:owl":"http://www.w3.org/2002/07/owl#","@xmlns:rdf":"http://www.w3.org/1999/02/22-rdf-syntax-ns#","@xmlns:ore":"http://www.openarchives.org/ore/terms/","@xmlns:skos":"http://www.w3.org/2004/02/skos/core#","@xmlns:dcterms":"http://purl.org/dc/terms/","edm:WebResource":[{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:DOC-8MW6VJ50/b28eb6936f760--f-90bfc-92c505a4cf57b/PDF","dcterms:extent":"1197 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:DOC-8MW6VJ50/b26cba36-f5f8-4ff7-bce2-799569c00b50/TEXT","dcterms:extent":"0 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:DOC-8MW6VJ50/c0eeedd9-84dd-47c6-82a6-be10f87e5f09/WEB","dcterms:extent":"0 KB"}],"edm:ProvidedCHO":{"@rdf:about":"URN:NBN:SI:DOC-8MW6VJ50","dcterms:issued":"2022","dc:creator":"Lemež, Boštjan","dc:contributor":["Repovš, Dušan","Virk, Žiga"],"dc:format":{"@xml:lang":"sl","#text":"96 str., 30 cm"},"dc:identifier":["COBISSID:132132355","PID:https://repozitorij.uni-lj.si/IzpisGradiva.php?id=142912","URN:URN:NBN:SI:doc-8MW6VJ50"],"dc:language":"en","dc:publisher":{"@xml:lang":"sl","#text":"B. Lemež"},"dc:source":{"@xml:lang":"sl","#text":"visokošolska dela"},"dc:subject":[{"@xml:lang":"en","#text":"computational topology"},{"@xml:lang":"en","#text":"geodesic space"},{"@xml:lang":"sl","#text":"geodetski prostor"},{"@xml:lang":"sl","#text":"homotopska rekonstukcija"},{"@xml:lang":"en","#text":"homotopy reconstruction"},{"@xml:lang":"sl","#text":"izrek o živcu"},{"@xml:lang":"sl","#text":"matematika"},{"@xml:lang":"en","#text":"mathematics"},{"@xml:lang":"en","#text":"nerve theorem"},{"@xml:lang":"sl","#text":"računska topologija"},{"@xml:lang":"en","#text":"Riemannian manifolds"},{"@xml:lang":"sl","#text":"Riemannove mnogoterosti"},{"@xml:lang":"en","#text":"selective Rips complex"},{"@xml:lang":"sl","#text":"selektivni Ripsov kompleks"},{"@xml:lang":"en","#text":"Vietoris-Rips complex"},{"@xml:lang":"sl","#text":"Vietoris-Ripsov kompleks"}],"dc:title":{"@xml:lang":"sl","#text":"Persistent homology and geometry| doctoral thesis|"},"dc:description":[{"@xml:lang":"sl","#text":"In thesis we introduce a novel simplicial complex assigned to a decreasing sequence of scales, called the selective Rips complex. It is a generalization of the Vietoris-Rips complex, where a sequence of scales is used instead of a single scale. A simpler version of the selective Rips complexes was previously designed to detect more geometric features than their Rips counterparts. We study the properties of selective Rips complexes and generalize the theory of Vietoris-Rips complexes. We prove the Stability Theorem for selective Rips complexes. The main contributions of this dissertation are various reconstruction results up to the homotopy type using selective Rips complexes. First, we prove that the selective Rips complex of a closed Riemannian manifold ?$X$? is homotopy equivalent to ?$X$? for appropriately small scales. When restricted to Vietoris-Rips complexes, we provide a novel proof of the Hausmann's reconstruction result. Next, we present finite reconstruction results with selective Rips complexes and intrinsic Čech complexes. We prove that if a metric space ?$S$? is close enough to a closed Riemannian manifold ?$X$? in the Gromov-Hausdorff distance, the selective Rips complex (and also the intrinsic Čech complex) of ?$S$? is homotopy equivalent to ?$X$? for appropriately small scales. As a special case, we provide a novel proof of the Latschev's reconstruction result. Finally, we classify the one-dimensional persistence of geodesic spaces arising from selective Rips complexes. We prove that 1-dimensional persistence of Rips and selective Rips complexes are isomorphic up to reparametrization"},{"@xml:lang":"sl","#text":"V disertaciji vpeljemo nov simplicialni kompleks imenovan selektivni Ripsov kompleks. Ta kompleks je posplošitev Vietoris-Ripsovega kompleksa, kjer uporabimo zaporedje radijev namesto enega radija. Prvotna verzija selektivnega Ripsovega kompleksa je bila vpeljana z namenom, da zazna več geometrijskih lastnosti kot pripadajoči Ripsov kompleks. S proučevanjem lastnosti selektivnega Ripsovega kompleksa posplošimo že znano teorijo Vietoris-Ripsovih kompleksov. Dokažemo pripadajoč izrek o stabilnosti za selektivne Ripsove komplekse. Najpomembnejši doprinos te disertacije so različni rekonstrukcijski rezultati do homotopskega tipa z uporabo selektivnega Ripsovega kompleksa. Najprej pokažemo, da je selektivni Ripsov kompleks sklenjene Riemannove mnogoterosti ?$X$? homotopsko ekvivalenten prostoru ?$X$? za primerno majhne parametre. Ker je Vietoris-Ripsov kompleks poseben primer selektivnega Ripsovega kompleksa, s tem podamo nov dokaz Hausmannovega rekonstrukcijskega rezultata. Nadalje pokažemo končna rekonstrukcijska rezultata z uporabo selektivnega Ripsovega kompleksa in intrinzičnega Čechovega kompleksa: če je metričen prostor ?$S$? dovolj blizu sklenjeni Riemannovi mnogoterosti ?$X$? glede na Gromov-Hausdorffovo razdaljo, tedaj je selektivni Ripsov kompleks (oz. intrinzičen Čechov kompleks) na ?$S$? homotopsko ekvivalenten celotnemu prostoru ?$X$?. Kot poseben primer s tem podamo nov dokaz Latschevega rekonstrukcijskega rezultata. Nazadnje klasificiramo eno-dimenzionalno vztrajnost geodetskih prostorov porojeno s selektivnimi Ripsovimi kompleksi. Pokažemo, da sta 1-dimenzionalni vztrajnosti Vietoris-Ripsovih in selektivnih Ripsovih kompleksov do reparametirizacije izomorfni"}],"edm:type":"TEXT","dc:type":[{"@xml:lang":"sl","#text":"visokošolska dela"},{"@xml:lang":"en","#text":"theses and dissertations"},{"@rdf:resource":"http://www.wikidata.org/entity/Q1266946"}]},"ore:Aggregation":{"@rdf:about":"http://www.dlib.si/?URN=URN:NBN:SI:DOC-8MW6VJ50","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:DOC-8MW6VJ50"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:DOC-8MW6VJ50/b28eb6936f760--f-90bfc-92c505a4cf57b/PDF"},"edm:rights":{"@rdf:resource":"http://rightsstatements.org/vocab/InC/1.0/"},"edm:provider":"Slovenian National E-content Aggregator","edm:intermediateProvider":{"@xml:lang":"en","#text":"National and University Library of Slovenia"},"edm:dataProvider":{"@xml:lang":"sl","#text":"Univerza v Ljubljani, Fakulteta za matematiko in fiziko"},"edm:object":{"@rdf:resource":"http://www.dlib.si/streamdb/URN:NBN:SI:DOC-8MW6VJ50/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:DOC-8MW6VJ50"}}}}