{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/45075"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/45075","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"A Kruskal-Katona theorem for cubical complexes","abstract":"The optimal number of faces in cubical complexes which lie in cubes refers to the maximum number of faces that can be constructed from a certain number of faces of lower dimension, or the minimum number of faces necessary to construct a certain number of faces of higher dimension. If <i>m</i> is the number of faces of <i>r</i> in a cubical complex, and if s > r(s < r), then the maximum(minimum) number of faces of dimension s that the complex can have is m<sub>(s/r)</sub> +. (m-m<sub>(r/r)</sub>)<sup>(s/r)</sup>, in terms of upper and lower semipowers. The corresponding formula for simplicial complexes, proved independently by J. B. Kruskal and G. A. Katona, is m<sup>(s/r)</sup>. A proof of the formula for cubical complexes is given in this paper, of which a flawed version appears in a paper by Bernt Lindstrijm. The n-tuples which satisfy the optimaiity conditions for cubical complexes which lie in cubes correspond bijectively with f-vectors of cubical complexes.","abstract_html":"The optimal number of faces in cubical complexes which lie in cubes refers to the maximum number of faces that can be constructed from a certain number of faces of lower dimension, or the minimum number of faces necessary to construct a certain number of faces of higher dimension. If &lt;i&gt;m&lt;/i&gt; is the number of faces of &lt;i&gt;r&lt;/i&gt; in a cubical complex, and if s &gt; r(s &lt; r), then the maximum(minimum) number of faces of dimension s that the complex can have is m&lt;sub&gt;(s/r)&lt;/sub&gt; +. (m-m&lt;sub&gt;(r/r)&lt;/sub&gt;)&lt;sup&gt;(s/r)&lt;/sup&gt;, in terms of upper and lower semipowers. The corresponding formula for simplicial complexes, proved independently by J. B. Kruskal and G. A. Katona, is m&lt;sup&gt;(s/r)&lt;/sup&gt;. A proof of the formula for cubical complexes is given in this paper, of which a flawed version appears in a paper by Bernt Lindstrijm. The n-tuples which satisfy the optimaiity conditions for cubical complexes which lie in cubes correspond bijectively with f-vectors of cubical complexes.","abstract_has_math":false,"creators":["Ellis, Robert B."],"institution":"Virginia Tech","degree_name":"Master of Science","degree_level":"masters","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Day, Martin V."],"committee_members":["Brown, Ezra A.","Haskell, Peter E."],"year":1996,"date_issued":"1996-06-06","date_published":"1996-06-06","updated_at":"2026-07-22T22:19:59Z","subjects":["cubical","simplicial","comples","Lindstrom","Kruskal"],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-10072005-094842"],"render_values":[{"text":"etd-10072005-094842","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/45075","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Day, Martin V."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Brown, Ezra A.","Haskell, Peter E."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Ellis, Robert B."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T21:47:06Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T21:47:06Z","2005-10-07"]},{"key":"dc:date.issued","label":"Date","values":["1996-06-06"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["cubical","simplicial","comples","Lindstrom","Kruskal"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-10072005-094842"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/45075"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The optimal number of faces in cubical complexes which lie in cubes refers to the maximum number of faces that can be constructed from a certain number of faces of lower dimension, or the minimum number of faces necessary to construct a certain number of faces of higher dimension. If <i>m</i> is the number of faces of <i>r</i> in a cubical complex, and if s > r(s < r), then the maximum(minimum) number of faces of dimension s that the complex can have is m<sub>(s/r)</sub> +. (m-m<sub>(r/r)</sub>)<sup>(s/r)</sup>, in terms of upper and lower semipowers. The corresponding formula for simplicial complexes, proved independently by J. B. Kruskal and G. A. Katona, is m<sup>(s/r)</sup>. A proof of the formula for cubical complexes is given in this paper, of which a flawed version appears in a paper by Bernt Lindstrijm. The n-tuples which satisfy the optimaiity conditions for cubical complexes which lie in cubes correspond bijectively with f-vectors of cubical complexes."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Master of Science"]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["BTD"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["A Kruskal-Katona theorem for cubical complexes"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Day, Martin V."],"dc:contributor.committeemember":["Brown, Ezra A.","Haskell, Peter E."],"dc:contributor.department":["Mathematics"],"dc:creator":["Ellis, Robert B."],"dc:date.accessioned":["2014-03-14T21:47:06Z"],"dc:date.available":["2014-03-14T21:47:06Z","2005-10-07"],"dc:date.issued":["1996-06-06"],"dc:description.abstract":["The optimal number of faces in cubical complexes which lie in cubes refers to the maximum number of faces that can be constructed from a certain number of faces of lower dimension, or the minimum number of faces necessary to construct a certain number of faces of higher dimension. If <i>m</i> is the number of faces of <i>r</i> in a cubical complex, and if s > r(s < r), then the maximum(minimum) number of faces of dimension s that the complex can have is m<sub>(s/r)</sub> +. (m-m<sub>(r/r)</sub>)<sup>(s/r)</sup>, in terms of upper and lower semipowers. The corresponding formula for simplicial complexes, proved independently by J. B. Kruskal and G. A. Katona, is m<sup>(s/r)</sup>. A proof of the formula for cubical complexes is given in this paper, of which a flawed version appears in a paper by Bernt Lindstrijm. The n-tuples which satisfy the optimaiity conditions for cubical complexes which lie in cubes correspond bijectively with f-vectors of cubical complexes."],"dc:description.degree":["Master of Science"],"dc:format.medium":["BTD"],"dc:format.mimetype":["application/pdf"],"dc:identifier.other":["etd-10072005-094842"],"dc:identifier.uri":["http://hdl.handle.net/10919/45075"],"dc:language.iso":["en"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["cubical","simplicial","comples","Lindstrom","Kruskal"],"dc:title":["A Kruskal-Katona theorem for cubical complexes"],"dc:type":["Thesis"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["masters"],"thesis:degree_name":["Master of Science"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:59Z"}