{"id":{"repo_id":"the-open-u","oai_identifier":"oai:oro.open.ac.uk:24046"},"canonical_url":"https://search.dev.ndltd.org/etd/the-open-u/oai:oro.open.ac.uk:24046","repository":{"repo_id":"the-open-u","name":"The Open University","base_url":"https://oro.open.ac.uk/cgi/oai2"},"display":{"title":"Combinatorial aspects of root lattices and words","abstract":"This thesis is concerned with two topics that are of interest for the theory of aperiodic order. In the first part, the similar sublattices and coincidence site lattices of the root lattice A4 are analysed by means of a particular quaternion algebra. Dirichlet series generating functions are derived, which count the number of similar sublattices, respectively coincidence site lattices, of each index. In the second part, several strategies to derive upper and lower bounds for the entropy of certain sets of powerfree words are presented. In particular, Kolpakov's arguments for the derivation of lower bounds for the entropy of powerfree words are generalised. For several explicit sets we derive very good upper and lower bounds for their entropy. Notably, Kolpakov's lower bounds for the entropy of ternary squarefree, binary cubefree and ternary minimally repetitive words are confirmed exactly.","abstract_html":"This thesis is concerned with two topics that are of interest for the theory of aperiodic order. In the first part, the similar sublattices and coincidence site lattices of the root lattice A4 are analysed by means of a particular quaternion algebra. Dirichlet series generating functions are derived, which count the number of similar sublattices, respectively coincidence site lattices, of each index. In the second part, several strategies to derive upper and lower bounds for the entropy of certain sets of powerfree words are presented. In particular, Kolpakov's arguments for the derivation of lower bounds for the entropy of powerfree words are generalised. For several explicit sets we derive very good upper and lower bounds for their entropy. Notably, Kolpakov's lower bounds for the entropy of ternary squarefree, binary cubefree and ternary minimally repetitive words are confirmed exactly.","abstract_has_math":false,"creators":["Heuer, Manuela"],"institution":"The Open University","degree_name":"phd","degree_level":"doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010-10","date_published":"2010-10","updated_at":"2026-07-24T05:01:55Z","subjects":[],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.sponsor","label":"Sponsor","values":["Engineering and Physical Sciences Research Council [grant number EP/D058465/1]"]},{"key":"dc:creator","label":"Author","values":["Heuer, Manuela"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2010-10-03"]},{"key":"dc:date.issued","label":"Date","values":["2010-10"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["ARRAY(0x7ffb403be800)"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["The Open University"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://oro.open.ac.uk/24046/"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["phd"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://oro.open.ac.uk/24046/1/thesis.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This thesis is concerned with two topics that are of interest for the theory of aperiodic order. In the first part, the similar sublattices and coincidence site lattices of the root lattice A4 are analysed by means of a particular quaternion algebra. Dirichlet series generating functions are derived, which count the number of similar sublattices, respectively coincidence site lattices, of each index. In the second part, several strategies to derive upper and lower bounds for the entropy of certain sets of powerfree words are presented. In particular, Kolpakov's arguments for the derivation of lower bounds for the entropy of powerfree words are generalised. For several explicit sets we derive very good upper and lower bounds for their entropy. Notably, Kolpakov's lower bounds for the entropy of ternary squarefree, binary cubefree and ternary minimally repetitive words are confirmed exactly."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Combinatorial aspects of root lattices and words"]}]}],"canonical_facts":{"dc:contributor.sponsor":["Engineering and Physical Sciences Research Council [grant number EP/D058465/1]"],"dc:creator":["Heuer, Manuela"],"dc:date":["2010-10-03"],"dc:date.issued":["2010-10"],"dc:description.abstract":["This thesis is concerned with two topics that are of interest for the theory of aperiodic order. In the first part, the similar sublattices and coincidence site lattices of the root lattice A4 are analysed by means of a particular quaternion algebra. Dirichlet series generating functions are derived, which count the number of similar sublattices, respectively coincidence site lattices, of each index. In the second part, several strategies to derive upper and lower bounds for the entropy of certain sets of powerfree words are presented. In particular, Kolpakov's arguments for the derivation of lower bounds for the entropy of powerfree words are generalised. For several explicit sets we derive very good upper and lower bounds for their entropy. Notably, Kolpakov's lower bounds for the entropy of ternary squarefree, binary cubefree and ternary minimally repetitive words are confirmed exactly."],"dc:format":["application/pdf"],"dc:identifier.uri":["https://oro.open.ac.uk/24046/1/thesis.pdf"],"dc:language":["en"],"dc:publisher.department":["ARRAY(0x7ffb403be800)"],"dc:publisher.institution":["The Open University"],"dc:relation.isreferencedby":["https://oro.open.ac.uk/24046/"],"dc:title":["Combinatorial aspects of root lattices and words"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["doctoral"],"dc:type.qualificationname":["phd"]},"updated_at":"2026-07-24T05:01:55Z"}