{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/8337"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/8337","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Rigidity and invariance properties of certain geometric frameworks","abstract":"Given a degenerate (n + 1)-simplex in a n-dimensional Euclidean space Rn, which is embedded in a (n + 1)-dimensional Euclidean space Rn+l. We allow all its vertices to have continuous motion in the space, either in Rn+l or restricted in Rn. For a given k, based on certain rules, we separate all its k-faces into 2 groups. During the motion, we give the following restriction: the volume of the k-faces in the 1st group can not increase (these faces are called \"k-cables\"); the volume of the k-faces in the 2nd group can not decrease (\"k-struts\"). We will prove that, under more conditions, all the volumes of the k-faces will be preserved for any sufficiently small motion. We also partially generalize the above result to spherical space Sn and hyperbolic space Hn.","abstract_html":"Given a degenerate (n + 1)-simplex in a n-dimensional Euclidean space Rn, which is embedded in a (n + 1)-dimensional Euclidean space Rn+l. We allow all its vertices to have continuous motion in the space, either in Rn+l or restricted in Rn. For a given k, based on certain rules, we separate all its k-faces into 2 groups. During the motion, we give the following restriction: the volume of the k-faces in the 1st group can not increase (these faces are called "k-cables"); the volume of the k-faces in the 2nd group can not decrease ("k-struts"). We will prove that, under more conditions, all the volumes of the k-faces will be preserved for any sufficiently small motion. We also partially generalize the above result to spherical space Sn and hyperbolic space Hn.","abstract_has_math":false,"creators":["Zhang, Lizhao, 1973-"],"institution":"Massachusetts Institute of Technology","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. Dept. of Mathematics.","school":null,"contributors":[],"advisors":["Daniel J. Kleitman."],"committee_chairs":[],"committee_members":[],"year":2002,"date_issued":"2002","date_published":"2002","updated_at":"2026-07-22T22:21:19Z","subjects":["Mathematics."],"languages":["eng"],"rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"rights_urls":["http://dspace.mit.edu/handle/1721.1/7582"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1721.1/8337","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Daniel J. Kleitman."]},{"key":"dc:contributor.department","label":"Department","values":["Massachusetts Institute of Technology. Dept. of Mathematics."]},{"key":"dc:contributor.other","label":"Dc Contributor Other","values":["Massachusetts Institute of Technology. Dept. of Mathematics."]},{"key":"dc:creator","label":"Author","values":["Zhang, Lizhao, 1973-"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2005-08-23T19:17:41Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2005-08-23T19:17:41Z"]},{"key":"dc:date.issued","label":"Date","values":["2002"]},{"key":"dc:publisher","label":"Institution","values":["Massachusetts Institute of Technology"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics."]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://dspace.mit.edu/handle/1721.1/7582"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1721.1/8337"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002.","Includes bibliographical references (leaves 59-60)."]},{"key":"dc:description.abstract","label":"Abstract","values":["Given a degenerate (n + 1)-simplex in a n-dimensional Euclidean space Rn, which is embedded in a (n + 1)-dimensional Euclidean space Rn+l. We allow all its vertices to have continuous motion in the space, either in Rn+l or restricted in Rn. For a given k, based on certain rules, we separate all its k-faces into 2 groups. During the motion, we give the following restriction: the volume of the k-faces in the 1st group can not increase (these faces are called \"k-cables\"); the volume of the k-faces in the 2nd group can not decrease (\"k-struts\"). We will prove that, under more conditions, all the volumes of the k-faces will be preserved for any sufficiently small motion. We also partially generalize the above result to spherical space Sn and hyperbolic space Hn."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Rigidity and invariance properties of certain geometric frameworks"]}]}],"canonical_facts":{"dc:contributor.advisor":["Daniel J. Kleitman."],"dc:contributor.department":["Massachusetts Institute of Technology. Dept. of Mathematics."],"dc:contributor.other":["Massachusetts Institute of Technology. Dept. of Mathematics."],"dc:creator":["Zhang, Lizhao, 1973-"],"dc:date.accessioned":["2005-08-23T19:17:41Z"],"dc:date.available":["2005-08-23T19:17:41Z"],"dc:date.issued":["2002"],"dc:description":["Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2002.","Includes bibliographical references (leaves 59-60)."],"dc:description.abstract":["Given a degenerate (n + 1)-simplex in a n-dimensional Euclidean space Rn, which is embedded in a (n + 1)-dimensional Euclidean space Rn+l. We allow all its vertices to have continuous motion in the space, either in Rn+l or restricted in Rn. For a given k, based on certain rules, we separate all its k-faces into 2 groups. During the motion, we give the following restriction: the volume of the k-faces in the 1st group can not increase (these faces are called \"k-cables\"); the volume of the k-faces in the 2nd group can not decrease (\"k-struts\"). We will prove that, under more conditions, all the volumes of the k-faces will be preserved for any sufficiently small motion. We also partially generalize the above result to spherical space Sn and hyperbolic space Hn."],"dc:description.degree":["Ph.D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/1721.1/8337"],"dc:language.iso":["eng"],"dc:publisher":["Massachusetts Institute of Technology"],"dc:rights":["M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission."],"dc:rights.uri":["http://dspace.mit.edu/handle/1721.1/7582"],"dc:subject":["Mathematics."],"dc:title":["Rigidity and invariance properties of certain geometric frameworks"],"dc:type":["Thesis"]},"updated_at":"2026-07-22T22:21:19Z"}