{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/22842"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/22842","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"The field of reals with Gevrey functions is model complete and o-minimal","abstract":"Infinitely representable functions are introduced, and it is proved that the expansion $\\IR\\sb{\\cal G}$ of the field of reals by a certain subfamily of all infinitely representable functions (namely, the family of so-called Gevrey functions) is model complete and o-minimal as well as polynomially bounded. The function $\\phi$ given on (1, $\\infty$) by$$\\log\\Gamma(x)=\\left(x-{1\\over2}\\right)\\log x-x+{1\\over2}\\log(2\\pi)+\\phi(x)$$is definable in $\\IR\\sb{\\cal G},$ as are all functions whose Taylor series expansion at the origin is multisummable in the direction $\\IR\\sp+$.","abstract_html":"Infinitely representable functions are introduced, and it is proved that the expansion $\\IR\\sb{\\cal G}$ of the field of reals by a certain subfamily of all infinitely representable functions (namely, the family of so-called Gevrey functions) is model complete and o-minimal as well as polynomially bounded. The function $\\phi$ given on (1, $\\infty$) by$<span class=\"etd-inline-math\">\\log\\Gamma(x)=\\left(x-{1\\over2}\\right)\\log x-x+{1\\over2}\\log(2&pi;)+\\phi(x)</span>$is definable in $\\IR\\sb{\\cal G},$ as are all functions whose Taylor series expansion at the origin is multisummable in the direction $\\IR\\sp+$.","abstract_has_math":true,"creators":["Speissegger, Patrick Urs"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["van den Dries, Lou"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:53:18Z","date_published":"2011-05-07T13:53:18Z","updated_at":"2026-07-22T22:25:20Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1996 Speissegger, Patrick Urs"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["9780591200003","AAI9712443","(UMI)AAI9712443"],"render_values":[{"text":"9780591200003","href":null,"code":true},{"text":"AAI9712443","href":null,"code":true},{"text":"(UMI)AAI9712443","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/22842","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["van den Dries, Lou"]},{"key":"dc:creator","label":"Author","values":["Speissegger, Patrick Urs"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:53:18Z","10000-01-01","1996"]},{"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":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1996 Speissegger, Patrick Urs"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["9780591200003","AAI9712443","(UMI)AAI9712443","http://hdl.handle.net/2142/22842"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Infinitely representable functions are introduced, and it is proved that the expansion $\\IR\\sb{\\cal G}$ of the field of reals by a certain subfamily of all infinitely representable functions (namely, the family of so-called Gevrey functions) is model complete and o-minimal as well as polynomially bounded. The function $\\phi$ given on (1, $\\infty$) by$$\\log\\Gamma(x)=\\left(x-{1\\over2}\\right)\\log x-x+{1\\over2}\\log(2\\pi)+\\phi(x)$$is definable in $\\IR\\sb{\\cal G},$ as are all functions whose Taylor series expansion at the origin is multisummable in the direction $\\IR\\sp+$.","The class of all infinitely representable functions is shown to be the same as the class of all finite sums of functions satisfying Gevrey estimates on disks (rather than on sectors of large enough opening). This leads to a generalization of the theory of multisummability in the direction $\\IR\\sp+$. An example of an infinitely representable function whose Taylor series is not multisummable in the direction $\\IR\\sp+$ is constructed.","Made available in DSpace on 2011-05-07T13:53:18Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9712443.pdf: 5129492 bytes, checksum: fb23db93eb185c9a03f891044180780d (MD5) Previous issue date: 1996","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T15:00:23Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:28:34-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["The field of reals with Gevrey functions is model complete and o-minimal"]}]}],"canonical_facts":{"dc:contributor":["van den Dries, Lou"],"dc:creator":["Speissegger, Patrick Urs"],"dc:date":["2011-05-07T13:53:18Z","10000-01-01","1996"],"dc:description":["Infinitely representable functions are introduced, and it is proved that the expansion $\\IR\\sb{\\cal G}$ of the field of reals by a certain subfamily of all infinitely representable functions (namely, the family of so-called Gevrey functions) is model complete and o-minimal as well as polynomially bounded. The function $\\phi$ given on (1, $\\infty$) by$$\\log\\Gamma(x)=\\left(x-{1\\over2}\\right)\\log x-x+{1\\over2}\\log(2\\pi)+\\phi(x)$$is definable in $\\IR\\sb{\\cal G},$ as are all functions whose Taylor series expansion at the origin is multisummable in the direction $\\IR\\sp+$.","The class of all infinitely representable functions is shown to be the same as the class of all finite sums of functions satisfying Gevrey estimates on disks (rather than on sectors of large enough opening). This leads to a generalization of the theory of multisummability in the direction $\\IR\\sp+$. An example of an infinitely representable function whose Taylor series is not multisummable in the direction $\\IR\\sp+$ is constructed.","Made available in DSpace on 2011-05-07T13:53:18Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9712443.pdf: 5129492 bytes, checksum: fb23db93eb185c9a03f891044180780d (MD5) Previous issue date: 1996","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T15:00:23Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:28:34-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["9780591200003","AAI9712443","(UMI)AAI9712443","http://hdl.handle.net/2142/22842"],"dc:language":["eng"],"dc:rights":["Copyright 1996 Speissegger, Patrick Urs"],"dc:subject":["Mathematics"],"dc:title":["The field of reals with Gevrey functions is model complete and o-minimal"],"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":"2026-07-22T22:25:20Z"}