{"id":{"repo_id":"rice","oai_identifier":"oai:repository.rice.edu:1911/20643"},"canonical_url":"https://search.dev.ndltd.org/etd/rice/oai:repository.rice.edu:1911/20643","repository":{"repo_id":"rice","name":"Rice University","base_url":"https://repository.rice.edu/server/oai/request"},"display":{"title":"Convergence of Gibbs measures and the behavior of shrinking tubular neighborhoods of fractals and algebraic sets","abstract":"Annealing is a physical process that motivates our definition of a Gibbs measure, which is a certain probability measure on Euclidean space. In this paper we examine a sequence of Gibbs measures characterized by the distance function. In Chapter 2 we conclude that the sequence of measures converge to a Hausdorff probability measure equally distributed along self-similar fractals with Hutchinson's Open Set Condition. In Chapter 3 we study spaces of concentric circles (which we call targets) in the plane, and examine how the sequence of probability measures distributes over the targets. By varying the number of targets and the size of the circles, we find probability measures that divide their mass between different point masses and spaces. Finally, in Chapter 4 we conclude that the measure will distribute evenly over the highest-dimensional strata of any semi-algebraic set.","abstract_html":"Annealing is a physical process that motivates our definition of a Gibbs measure, which is a certain probability measure on Euclidean space. In this paper we examine a sequence of Gibbs measures characterized by the distance function. In Chapter 2 we conclude that the sequence of measures converge to a Hausdorff probability measure equally distributed along self-similar fractals with Hutchinson's Open Set Condition. In Chapter 3 we study spaces of concentric circles (which we call targets) in the plane, and examine how the sequence of probability measures distributes over the targets. By varying the number of targets and the size of the circles, we find probability measures that divide their mass between different point masses and spaces. Finally, in Chapter 4 we conclude that the measure will distribute evenly over the highest-dimensional strata of any semi-algebraic set.","abstract_has_math":false,"creators":["Samansky, Eric Michael"],"institution":"Rice University","degree_name":"Doctor of Philosophy","degree_level":"Doctoral","degree_discipline":"Natural Sciences","degree_department":null,"school":null,"contributors":[],"advisors":["Hardt, Robert M."],"committee_chairs":[],"committee_members":[],"year":2007,"date_issued":"2007","date_published":"2007","updated_at":"2026-07-24T04:10:24Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/1911/20643","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Hardt, Robert M."]},{"key":"dc:creator","label":"Author","values":["Samansky, Eric Michael"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2009-06-03T21:08:13Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2009-06-03T21:08:13Z"]},{"key":"dc:date.issued","label":"Date","values":["2007"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Natural Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Rice University"]}]},{"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":["Copyright is held by the author, unless otherwise indicated. 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In Chapter 3 we study spaces of concentric circles (which we call targets) in the plane, and examine how the sequence of probability measures distributes over the targets. By varying the number of targets and the size of the circles, we find probability measures that divide their mass between different point masses and spaces. Finally, in Chapter 4 we conclude that the measure will distribute evenly over the highest-dimensional strata of any semi-algebraic set."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Convergence of Gibbs measures and the behavior of shrinking tubular neighborhoods of fractals and algebraic sets"]}]}],"canonical_facts":{"dc:contributor.advisor":["Hardt, Robert M."],"dc:creator":["Samansky, Eric Michael"],"dc:date.accessioned":["2009-06-03T21:08:13Z"],"dc:date.available":["2009-06-03T21:08:13Z"],"dc:date.issued":["2007"],"dc:description.abstract":["Annealing is a physical process that motivates our definition of a Gibbs measure, which is a certain probability measure on Euclidean space. In this paper we examine a sequence of Gibbs measures characterized by the distance function. In Chapter 2 we conclude that the sequence of measures converge to a Hausdorff probability measure equally distributed along self-similar fractals with Hutchinson's Open Set Condition. In Chapter 3 we study spaces of concentric circles (which we call targets) in the plane, and examine how the sequence of probability measures distributes over the targets. By varying the number of targets and the size of the circles, we find probability measures that divide their mass between different point masses and spaces. Finally, in Chapter 4 we conclude that the measure will distribute evenly over the highest-dimensional strata of any semi-algebraic set."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["https://hdl.handle.net/1911/20643"],"dc:language.iso":["eng"],"dc:rights":["Copyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder."],"dc:subject":["Mathematics"],"dc:title":["Convergence of Gibbs measures and the behavior of shrinking tubular neighborhoods of fractals and algebraic sets"],"dc:type":["Thesis"],"thesis:degree_discipline":["Natural Sciences"],"thesis:degree_level":["Doctoral"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["Rice University"]},"updated_at":"2026-07-24T04:10:24Z"}