Quantum Jumps of Normal Polytopes

Bruns, Winfried; Gubeladze, Joseph; Michalek, Mateusz

Quantum Jumps of Normal Polytopes

Dateien

Zu diesem Dokument gibt es keine Dateien.

Datum

2016

Autor:innen

Bruns, Winfried

Gubeladze, Joseph

Michalek, Mateusz

Publikationstyp

Zeitschriftenartikel

Publikationsstatus

Published

Erschienen in

Discrete & Computational Geometry ; 56 (2016), 1. - S. 181-215. - Springer. - ISSN 0179-5376. - eISSN 1432-0444

Zusammenfassung

We introduce a partial order on the set of all normal polytopes in R^d. This poset NPol(d) is a natural discrete counterpart of the continuum of convex compact sets in R^d, ordered by inclusion, and exhibits a remarkably rich combinatorial structure. We derive various arithmetic bounds on elementary relations in NPol(d), called quantum jumps. The existence of extremal objects in NPol(d) is a challenge of number theoretical flavor, leading to interesting classes of normal polytopes: minimal, maximal, spherical. Minimal elements in NPol(5) have played a critical role in disproving various covering conjectures for normal polytopes in the 1990s. Here we report on the first examples of maximal elements in NPol(4) and NPol(5), found by a combination of the developed theory, random generation, and extensive computer search.

Fachgebiet (DDC)

510 Mathematik

Schlagwörter

Lattice polytope, Normal polytope, Maximal polytope, Quantum jump

Zitieren

ISO 690

BRUNS, Winfried, Joseph GUBELADZE, Mateusz MICHALEK, 2016. Quantum Jumps of Normal Polytopes. In: Discrete & Computational Geometry. Springer. 56(1), pp. 181-215. ISSN 0179-5376. eISSN 1432-0444. Available under: doi: 10.1007/s00454-016-9773-7

BibTex

@article{Bruns2016Quant-52322,
  year={2016},
  doi={10.1007/s00454-016-9773-7},
  title={Quantum Jumps of Normal Polytopes},
  number={1},
  volume={56},
  issn={0179-5376},
  journal={Discrete & Computational Geometry},
  pages={181--215},
  author={Bruns, Winfried and Gubeladze, Joseph and Michalek, Mateusz}
}

RDF

<rdf:RDF
    xmlns:dcterms="http://purl.org/dc/terms/"
    xmlns:dc="http://purl.org/dc/elements/1.1/"
    xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
    xmlns:bibo="http://purl.org/ontology/bibo/"
    xmlns:dspace="http://digital-repositories.org/ontologies/dspace/0.1.0#"
    xmlns:foaf="http://xmlns.com/foaf/0.1/"
    xmlns:void="http://rdfs.org/ns/void#"
    xmlns:xsd="http://www.w3.org/2001/XMLSchema#" > 
  <rdf:Description rdf:about="https://kops.uni-konstanz.de/server/rdf/resource/123456789/52322">
    <dc:date rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2021-01-08T12:21:34Z</dc:date>
    <dcterms:abstract xml:lang="eng">We introduce a partial order on the set of all normal polytopes in R&lt;sup&gt;d&lt;/sup&gt;. This poset NPol(d) is a natural discrete counterpart of the continuum of convex compact sets in R&lt;sup&gt;d&lt;/sup&gt;, ordered by inclusion, and exhibits a remarkably rich combinatorial structure. We derive various arithmetic bounds on elementary relations in NPol(d), called quantum jumps. The existence of extremal objects in NPol(d) is a challenge of number theoretical flavor, leading to interesting classes of normal polytopes: minimal, maximal, spherical. Minimal elements in NPol(5) have played a critical role in disproving various covering conjectures for normal polytopes in the 1990s. Here we report on the first examples of maximal elements in NPol(4) and NPol(5), found by a combination of the developed theory, random generation, and extensive computer search.</dcterms:abstract>
    <dc:language>eng</dc:language>
    <dcterms:issued>2016</dcterms:issued>
    <dc:rights>terms-of-use</dc:rights>
    <dcterms:title>Quantum Jumps of Normal Polytopes</dcterms:title>
    <dcterms:available rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2021-01-08T12:21:34Z</dcterms:available>
    <dc:contributor>Bruns, Winfried</dc:contributor>
    <dcterms:rights rdf:resource="https://rightsstatements.org/page/InC/1.0/"/>
    <dc:contributor>Michalek, Mateusz</dc:contributor>
    <dc:creator>Gubeladze, Joseph</dc:creator>
    <dc:creator>Michalek, Mateusz</dc:creator>
    <dspace:isPartOfCollection rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
    <foaf:homepage rdf:resource="http://localhost:8080/"/>
    <void:sparqlEndpoint rdf:resource="http://localhost/fuseki/dspace/sparql"/>
    <dcterms:isPartOf rdf:resource="https://kops.uni-konstanz.de/server/rdf/resource/123456789/39"/>
    <dc:contributor>Gubeladze, Joseph</dc:contributor>
    <bibo:uri rdf:resource="https://kops.uni-konstanz.de/handle/123456789/52322"/>
    <dc:creator>Bruns, Winfried</dc:creator>
  </rdf:Description>
</rdf:RDF>

Universitätsbibliographie

Nein

Begutachtet

Ja