{"?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-GOF06BLV/c34b1adc-7a0b-4ae6-bcd1-d003c9c536c3/PDF","dcterms:extent":"2045 KB"},{"@rdf:about":"http://www.dlib.si/stream/URN:NBN:SI:doc-GOF06BLV/085bc344-8d59-4558-a173-08b385fd561b/TEXT","dcterms:extent":"77 KB"}],"edm:TimeSpan":{"@rdf:about":"2008-2025","edm:begin":{"@xml:lang":"en","#text":"2008"},"edm:end":{"@xml:lang":"en","#text":"2025"}},"edm:ProvidedCHO":{"@rdf:about":"URN:NBN:SI:doc-GOF06BLV","dcterms:isPartOf":[{"@rdf:resource":"https://www.dlib.si/details/URN:NBN:SI:spr-UP1WMFAR"},{"@xml:lang":"sl","#text":"Ars mathematica contemporanea"}],"dcterms:issued":"2020","dc:creator":["Domokos, Gábor","Kovács, Flórián","Lángi, Zsolt","Regős, Krisztina","Varga, Péter T."],"dc:format":[{"@xml:lang":"sl","#text":"številka:1"},{"@xml:lang":"sl","#text":"letnik:19"},{"@xml:lang":"sl","#text":"str. 95-124"}],"dc:identifier":["ISSN:1855-3974","COBISSID_HOST:43350787","URN:URN:NBN:SI:doc-GOF06BLV"],"dc:language":"en","dc:publisher":{"@xml:lang":"sl","#text":"Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije"},"dc:subject":[{"@xml:lang":"en","#text":"f-vector"},{"@xml:lang":"sl","#text":"f-vektor"},{"@xml:lang":"en","#text":"monostatic polyhedron"},{"@xml:lang":"sl","#text":"monostatični polieder"},{"@xml:lang":"sl","#text":"polieder"},{"@xml:lang":"en","#text":"polyhedron"},{"@xml:lang":"en","#text":"static equilibrium"},{"@xml:lang":"sl","#text":"statično ravnovesje"}],"dcterms:temporal":{"@rdf:resource":"2008-2025"},"dc:title":{"@xml:lang":"sl","#text":"Balancing polyhedra|"},"dc:description":[{"@xml:lang":"sl","#text":"We define the mechanical complexity ?$C(P)$? of a 3-dimensional convex polyhedron ?$P$?, interpreted as a homogeneous solid, as the difference between the total number of its faces, edges and vertices and of its static equilibria; and the mechanical complexity ?$C(S;U)$? of primary equilibrium classes ?$(S;U)^E$? with ?$S$? stable and ?$U$? unstable equilibria as the infimum of the mechanical complexity of all polyhedra in that class. We prove that the mechanical complexity of a class ?$(S;U)^E$? with ?$S; U>1$? is the minimum of ?$2(f+v-S-U)$? over all polyhedral pairs ?$(f; v)$?, where a pair of integers is called a polyhedral pair if there is a convex polyhedron with f faces and v vertices. In particular, we prove that the mechanical complexity of a class ?$(S;U)^E$? is zero if and only if there exists a convex polyhedron with ?$S$? faces and ?$U$? vertices. We also give asymptotically sharp bounds for the mechanical complexity of the monostatic classes ?$(1;U)^E$? and ?$(S; 1)^E$?, and offer a complexity-dependent prize for the complexity of the Gömböc-class ?$(1; 1)^E$?"},{"@xml:lang":"sl","#text":"Definiramo mehansko kompleksnost ?$C(P)$? 3-dimenzionalnega konveksnega poliedra ?$P$?, interpretiranega kot homogeno telo, kot razliko med skupnim številom njegovih lic, povezav in točk ter številom njegovih statičnih ravnovesij; definiramo tudi mehansko kompleksnost ?$C(S;U)$? primarnih ravnovesnostnih razredov ?$(S;U)^E$? s ?$S$? stabilnimi in ?$U$? nestabilnimi ravnovesji kot natančno spodnjo mejo mehanske kompleksnosti vseh poliedrov v tem razredu. Dokažemo, da je mehanska kompleksnost razreda ?$(S;U)^E$? pri pogoju ?$S; U>1$? minimum izraza ?$2(f+v-S-U)$? po vseh poliedrskih parih ?$(f; v)$?, pri čemer se par celih števil imenuje poliedrski par, če obstaja konveksen polieder z ?$f$? lici in ?$v$? točkami. Posebej, dokažemo, da je mehanska kompleksnost razreda ?$(S;U)^E$? enaka nič natanko tedaj, ko obstaja konveksen polieder s ?$S$? lici in ?$U$? točkami. Predstavimo tudi asimptotsko ostre meje za mehansko kompleksnost monostatičnih razredov ?$(1;U)^E$? in ?$(S; 1)^E$?, ter ponudimo od kompleksnosti odvisno nagrado za določitev kompleksnosti Gömböcovega razreda ?$(1; 1)^E$?"}],"edm:type":"TEXT","dc:type":[{"@xml:lang":"sl","#text":"znanstveno časopisje"},{"@xml:lang":"en","#text":"journals"},{"@rdf:resource":"http://www.wikidata.org/entity/Q361785"}]},"ore:Aggregation":{"@rdf:about":"http://www.dlib.si/?URN=URN:NBN:SI:doc-GOF06BLV","edm:aggregatedCHO":{"@rdf:resource":"URN:NBN:SI:doc-GOF06BLV"},"edm:isShownBy":{"@rdf:resource":"http://www.dlib.si/stream/URN:NBN:SI:doc-GOF06BLV/c34b1adc-7a0b-4ae6-bcd1-d003c9c536c3/PDF"},"edm:rights":{"@rdf:resource":"http://creativecommons.org/licenses/by/4.0/"},"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 na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije"},"edm:object":{"@rdf:resource":"http://www.dlib.si/streamdb/URN:NBN:SI:doc-GOF06BLV/maxi/edm"},"edm:isShownAt":{"@rdf:resource":"http://www.dlib.si/details/URN:NBN:SI:doc-GOF06BLV"}}}}