{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/97758"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/97758","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Some problems in polynomial interpolation and topological complexity","abstract":"This thesis is comprised of two projects in applied computational mathematics. In Chapter 1, we discuss the geometry and combinatorics of geometrically characterized sets. These are finite sets of n+d choose n points in R^d which impose independent conditions on polynomials of degree n, and which have Lagrange polynomials of a special form. These sets were introduced by Chung and Yao in a 1977 paper in the SIAM Journal of Numerical Analysis in the context of polynomial interpolation. There are several conjectures on the nature and geometric structure of these sets. We investigate the geometry and combinatorics of GC sets for d at least 2, and prove they are closely related to simplicial complexes which are Cohen-Macaulay and have a Cohen-Macaulay dual. In Chapter 2, we will discuss the motion planning problem in complex hyperplane arrangement complements. The difficulty of constructing a minimally discontinuous motion planning algorithm for a topological space X is measured by an integer invariant of X called topological complexity or TC(X). Yuzvinsky developed a combinatorial criterion for hyperplane arrangement complements which guarantees that their topological complexity is as large as possible. Applying this criterion in the special case when the arrangement is graphic, we simplify the criterion to an inequality on the edge density of the graph which is closely related to the inequality in the arboricity theorem of Nash-Williams.","abstract_html":"This thesis is comprised of two projects in applied computational mathematics. In Chapter 1, we discuss the geometry and combinatorics of geometrically characterized sets. These are finite sets of n+d choose n points in R^d which impose independent conditions on polynomials of degree n, and which have Lagrange polynomials of a special form. These sets were introduced by Chung and Yao in a 1977 paper in the SIAM Journal of Numerical Analysis in the context of polynomial interpolation. There are several conjectures on the nature and geometric structure of these sets. We investigate the geometry and combinatorics of GC sets for d at least 2, and prove they are closely related to simplicial complexes which are Cohen-Macaulay and have a Cohen-Macaulay dual. In Chapter 2, we will discuss the motion planning problem in complex hyperplane arrangement complements. The difficulty of constructing a minimally discontinuous motion planning algorithm for a topological space X is measured by an integer invariant of X called topological complexity or TC(X). Yuzvinsky developed a combinatorial criterion for hyperplane arrangement complements which guarantees that their topological complexity is as large as possible. Applying this criterion in the special case when the arrangement is graphic, we simplify the criterion to an inequality on the edge density of the graph which is closely related to the inequality in the arboricity theorem of Nash-Williams.","abstract_has_math":false,"creators":["Fieldsteel, Nathan Mulvey"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Schenck, Hal","Nevins, Tom","Baryshnikov, Yuliy","Hirani, Anil"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017-08-10T20:33:17Z","date_published":"2017-08-10T20:33:17Z","updated_at":"2026-07-22T22:24:34Z","subjects":["Approximation theory","Polynomial interpolation","Motion planning"],"languages":["en"],"rights":["Copyright 2017 Nathan Fieldsteel"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/97758","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Schenck, Hal","Nevins, Tom","Baryshnikov, Yuliy","Hirani, Anil"]},{"key":"dc:creator","label":"Author","values":["Fieldsteel, Nathan Mulvey"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2017-08-10T20:33:17Z","2019-08-11T09:15:39Z","2017-04-21","2017-05"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Approximation theory","Polynomial interpolation","Motion planning"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2017 Nathan Fieldsteel"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/97758"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This thesis is comprised of two projects in applied computational mathematics. In Chapter 1, we discuss the geometry and combinatorics of geometrically characterized sets. These are finite sets of n+d choose n points in R^d which impose independent conditions on polynomials of degree n, and which have Lagrange polynomials of a special form. These sets were introduced by Chung and Yao in a 1977 paper in the SIAM Journal of Numerical Analysis in the context of polynomial interpolation. There are several conjectures on the nature and geometric structure of these sets. We investigate the geometry and combinatorics of GC sets for d at least 2, and prove they are closely related to simplicial complexes which are Cohen-Macaulay and have a Cohen-Macaulay dual. In Chapter 2, we will discuss the motion planning problem in complex hyperplane arrangement complements. The difficulty of constructing a minimally discontinuous motion planning algorithm for a topological space X is measured by an integer invariant of X called topological complexity or TC(X). Yuzvinsky developed a combinatorial criterion for hyperplane arrangement complements which guarantees that their topological complexity is as large as possible. Applying this criterion in the special case when the arrangement is graphic, we simplify the criterion to an inequality on the edge density of the graph which is closely related to the inequality in the arboricity theorem of Nash-Williams.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2019-05-01","The student, Nathan Fieldsteel, accepted the attached license on 2017-04-20 at 19:04.","The student, Nathan Fieldsteel, submitted this Dissertation for approval on 2017-04-20 at 19:16.","This Dissertation was approved for publication on 2017-04-21 at 09:14.","DSpace SAF Submission Ingestion Package generated from Vireo submission #10948 on 2017-08-10 at 15:06:39","Made available in DSpace on 2017-08-10T20:33:17Z (GMT). No. of bitstreams: 2 FIELDSTEEL-DISSERTATION-2017.pdf: 511198 bytes, checksum: c9ab8e733b1b77d2e68f4b3331c278d7 (MD5) LICENSE.txt: 4214 bytes, checksum: f8b6cc51e88916bba385469854c3b8a1 (MD5) Previous issue date: 2017-04-21","Embargo set by: Colleen Fallaw for item 102811 Lift date: 2019-08-10T21:27:21Z Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","U of I Only Restriction Lifted for Item 102811 on 2019-08-11T09:15:39Z."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Some problems in polynomial interpolation and topological complexity"]}]}],"canonical_facts":{"dc:contributor":["Schenck, Hal","Nevins, Tom","Baryshnikov, Yuliy","Hirani, Anil"],"dc:creator":["Fieldsteel, Nathan Mulvey"],"dc:date":["2017-08-10T20:33:17Z","2019-08-11T09:15:39Z","2017-04-21","2017-05"],"dc:description":["This thesis is comprised of two projects in applied computational mathematics. In Chapter 1, we discuss the geometry and combinatorics of geometrically characterized sets. These are finite sets of n+d choose n points in R^d which impose independent conditions on polynomials of degree n, and which have Lagrange polynomials of a special form. These sets were introduced by Chung and Yao in a 1977 paper in the SIAM Journal of Numerical Analysis in the context of polynomial interpolation. There are several conjectures on the nature and geometric structure of these sets. We investigate the geometry and combinatorics of GC sets for d at least 2, and prove they are closely related to simplicial complexes which are Cohen-Macaulay and have a Cohen-Macaulay dual. In Chapter 2, we will discuss the motion planning problem in complex hyperplane arrangement complements. The difficulty of constructing a minimally discontinuous motion planning algorithm for a topological space X is measured by an integer invariant of X called topological complexity or TC(X). Yuzvinsky developed a combinatorial criterion for hyperplane arrangement complements which guarantees that their topological complexity is as large as possible. Applying this criterion in the special case when the arrangement is graphic, we simplify the criterion to an inequality on the edge density of the graph which is closely related to the inequality in the arboricity theorem of Nash-Williams.","Submission published under a 24 month embargo labeled 'U of I Access', the embargo will last until 2019-05-01","The student, Nathan Fieldsteel, accepted the attached license on 2017-04-20 at 19:04.","The student, Nathan Fieldsteel, submitted this Dissertation for approval on 2017-04-20 at 19:16.","This Dissertation was approved for publication on 2017-04-21 at 09:14.","DSpace SAF Submission Ingestion Package generated from Vireo submission #10948 on 2017-08-10 at 15:06:39","Made available in DSpace on 2017-08-10T20:33:17Z (GMT). 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