{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/50350"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/50350","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Extremal functions related to convexity and martingales","abstract":"We consider the problem of finding the extremal function in the class of real-valued biconvex functions satisfying a boundary condition on a product of the unit ball with itself, with a suitable norm in the plane. We want to maximize the biconvex function at a point in the domain where the second component is fixed and therefore we can consider the biconvex function as a convex function of the first component. We then find a representation for the convex function in terms of some functions of a suitable quotient of second order partial derivatives of the convex function, where these functions will satisfy certain conditions so that the biconvex function will have the given boundary values. From the quotient of second order partial derivatives of the convex function, we obtain a relation leading to the Hopf differential equation, whose solution involves a parameter function. With a given boundary function, we perform a variation of the parameter function by a small real-valued function. Then we find the change of the representation of the convex function. If the convex function is an extremal function, then the rate of change with respect to the variation made to the parameter function is zero. This will be the condition that we are looking for in an extremal function.","abstract_html":"We consider the problem of finding the extremal function in the class of real-valued biconvex functions satisfying a boundary condition on a product of the unit ball with itself, with a suitable norm in the plane. We want to maximize the biconvex function at a point in the domain where the second component is fixed and therefore we can consider the biconvex function as a convex function of the first component. We then find a representation for the convex function in terms of some functions of a suitable quotient of second order partial derivatives of the convex function, where these functions will satisfy certain conditions so that the biconvex function will have the given boundary values. From the quotient of second order partial derivatives of the convex function, we obtain a relation leading to the Hopf differential equation, whose solution involves a parameter function. With a given boundary function, we perform a variation of the parameter function by a small real-valued function. Then we find the change of the representation of the convex function. If the convex function is an extremal function, then the rate of change with respect to the variation made to the parameter function is zero. This will be the condition that we are looking for in an extremal function.","abstract_has_math":false,"creators":["Suwannaphichat, Sineenuch"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Hinkkanen, Aimo","Miles, Joseph B.","Loeb, Peter A.","Merenkov, Sergiy A."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-09-16T17:11:56Z","date_published":"2014-09-16T17:11:56Z","updated_at":"2026-07-22T22:25:40Z","subjects":["Extremal functions","Convex functions"],"languages":["en"],"rights":["Copyright 2014 Sineenuch Suwannaphichat"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/50350","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Hinkkanen, Aimo","Miles, Joseph B.","Loeb, Peter A.","Merenkov, Sergiy A."]},{"key":"dc:creator","label":"Author","values":["Suwannaphichat, Sineenuch"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-09-16T17:11:56Z","2016-09-22T20:59:30Z","2014-08","2014-09-16"]},{"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":["Extremal functions","Convex functions"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2014 Sineenuch Suwannaphichat"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/50350"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We consider the problem of finding the extremal function in the class of real-valued biconvex functions satisfying a boundary condition on a product of the unit ball with itself, with a suitable norm in the plane. We want to maximize the biconvex function at a point in the domain where the second component is fixed and therefore we can consider the biconvex function as a convex function of the first component. We then find a representation for the convex function in terms of some functions of a suitable quotient of second order partial derivatives of the convex function, where these functions will satisfy certain conditions so that the biconvex function will have the given boundary values. From the quotient of second order partial derivatives of the convex function, we obtain a relation leading to the Hopf differential equation, whose solution involves a parameter function. With a given boundary function, we perform a variation of the parameter function by a small real-valued function. Then we find the change of the representation of the convex function. If the convex function is an extremal function, then the rate of change with respect to the variation made to the parameter function is zero. This will be the condition that we are looking for in an extremal function.","Item withdrawn by Laura Spradlin (lspradl2@illinois.edu) on 2014-07-02T15:03:46Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Suwannaphichat_Sineenuch.pdf: 540281 bytes, checksum: 83b60b6adef81152b10b56797933a3be (MD5)","Made available in DSpace on 2014-09-16T17:11:56Z (GMT). No. of bitstreams: 2 Sineenuch_Suwannaphichat.pdf: 540276 bytes, checksum: 30910f8452149b9d20612c29a6314f5d (MD5) license.txt: 4074 bytes, checksum: 8e1020ca4452e9419375cd880af19db8 (MD5)","Embargo set by: Seth Robbins for item 50461 Lift date: 2016-09-16T17:13:01Z Reason: Author requested closed access (OA after 2yrs) in Vireo ETD system","Limited Restriction Lifted for Item 50461 on 2016-09-22T20:59:30Z."]},{"key":"dc:title","label":"Title","values":["Extremal functions related to convexity and martingales"]}]}],"canonical_facts":{"dc:contributor":["Hinkkanen, Aimo","Miles, Joseph B.","Loeb, Peter A.","Merenkov, Sergiy A."],"dc:creator":["Suwannaphichat, Sineenuch"],"dc:date":["2014-09-16T17:11:56Z","2016-09-22T20:59:30Z","2014-08","2014-09-16"],"dc:description":["We consider the problem of finding the extremal function in the class of real-valued biconvex functions satisfying a boundary condition on a product of the unit ball with itself, with a suitable norm in the plane. We want to maximize the biconvex function at a point in the domain where the second component is fixed and therefore we can consider the biconvex function as a convex function of the first component. We then find a representation for the convex function in terms of some functions of a suitable quotient of second order partial derivatives of the convex function, where these functions will satisfy certain conditions so that the biconvex function will have the given boundary values. From the quotient of second order partial derivatives of the convex function, we obtain a relation leading to the Hopf differential equation, whose solution involves a parameter function. With a given boundary function, we perform a variation of the parameter function by a small real-valued function. Then we find the change of the representation of the convex function. If the convex function is an extremal function, then the rate of change with respect to the variation made to the parameter function is zero. This will be the condition that we are looking for in an extremal function.","Item withdrawn by Laura Spradlin (lspradl2@illinois.edu) on 2014-07-02T15:03:46Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Suwannaphichat_Sineenuch.pdf: 540281 bytes, checksum: 83b60b6adef81152b10b56797933a3be (MD5)","Made available in DSpace on 2014-09-16T17:11:56Z (GMT). No. of bitstreams: 2 Sineenuch_Suwannaphichat.pdf: 540276 bytes, checksum: 30910f8452149b9d20612c29a6314f5d (MD5) license.txt: 4074 bytes, checksum: 8e1020ca4452e9419375cd880af19db8 (MD5)","Embargo set by: Seth Robbins for item 50461 Lift date: 2016-09-16T17:13:01Z Reason: Author requested closed access (OA after 2yrs) in Vireo ETD system","Limited Restriction Lifted for Item 50461 on 2016-09-22T20:59:30Z."],"dc:identifier":["http://hdl.handle.net/2142/50350"],"dc:language":["en"],"dc:rights":["Copyright 2014 Sineenuch Suwannaphichat"],"dc:subject":["Extremal functions","Convex functions"],"dc:title":["Extremal functions related to convexity and martingales"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:40Z"}