{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T22:16:42Z","timestamp":1768342602063,"version":"3.49.0"},"reference-count":15,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":4394,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2002,3]]},"abstract":"<jats:p>In this paper, we continue investigations into the asymptotic behavior of solutions of differential equations over o-minimal structures.<\/jats:p><jats:p>Let \u211c be an expansion of the real field (\u211d, +, \u00b7).<\/jats:p><jats:p>A differentiable map<jats:italic>F<\/jats:italic>= (<jats:italic>F<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026,<jats:italic>F<\/jats:italic><jats:sub>1<\/jats:sub>): (<jats:italic>a, b<\/jats:italic>) \u2192 \u211d<jats:sup><jats:italic>i<\/jats:italic><\/jats:sup>is<jats:bold>\u211c-Pfaffian<\/jats:bold>if there exists<jats:italic>G<\/jats:italic>: \u211d<jats:sup>1+<jats:italic>l<\/jats:italic><\/jats:sup>\u2192 \u211d<jats:sup><jats:italic>l<\/jats:italic><\/jats:sup>definable in \u211c such that<jats:italic>F<\/jats:italic>\u2032(<jats:italic>t<\/jats:italic>) =<jats:italic>G<\/jats:italic>(<jats:italic>t, F<\/jats:italic>(<jats:italic>t<\/jats:italic>)) for all<jats:italic>t<\/jats:italic>\u2208 (<jats:italic>a, b<\/jats:italic>) and each component function<jats:italic>G<jats:sub>i<\/jats:sub><\/jats:italic>: \u211d<jats:sup>1+<jats:italic>l<\/jats:italic><\/jats:sup>\u2192 \u211d is independent of the last<jats:italic>l<\/jats:italic>\u2212<jats:italic>i<\/jats:italic>variables (<jats:italic>i<\/jats:italic>= 1, \u2026,<jats:italic>l<\/jats:italic>). If \u211c is o-minimal and<jats:italic>F<\/jats:italic>: (<jats:italic>a, b<\/jats:italic>) \u2192 \u211d<jats:sup><jats:italic>l<\/jats:italic><\/jats:sup>is \u211c-Pfaffian, then (\u211c,<jats:italic>F<\/jats:italic>) is o-minimal (Proposition 7). We say that<jats:italic>F<\/jats:italic>: \u211d \u2192 \u211d<jats:sup><jats:italic>l<\/jats:italic><\/jats:sup>is ultimately \u211c-Pfaffian if there exists<jats:italic>r<\/jats:italic>\u2208 \u211d such that the restriction<jats:italic>F<\/jats:italic>\u21be(<jats:italic>r<\/jats:italic>, \u221e) is \u211c-Pfaffian. (In general,<jats:bold>ultimately<\/jats:bold>abbreviates \u201cfor all sufficiently large positive arguments\u201d.)<\/jats:p><jats:p>The structure \u211c is<jats:bold>closed under asymptotic integration<\/jats:bold>if for each ultimately non-zero unary (that is, \u211d \u2192 \u211d) function<jats:italic>f<\/jats:italic>definable in \u211c there is an ultimately differentiable unary function<jats:italic>g<\/jats:italic>definable in \u211c such that lim<jats:sub><jats:italic>t<\/jats:italic>\u2192+\u221e<\/jats:sub>[<jats:italic>g<\/jats:italic>\u2032(<jats:italic>t<\/jats:italic>)\/<jats:italic>f<\/jats:italic>(<jats:italic>t<\/jats:italic>)] = 1- If \u211c is closed under asymptotic integration, then \u211c is o-minimal and defines<jats:italic>e<jats:sup>x<\/jats:sup><\/jats:italic>: \u211d \u2192 \u211d (Proposition 2).<\/jats:p><jats:p>Note that the above definitions make sense for expansions of arbitrary ordered fields.<\/jats:p>","DOI":"10.2178\/jsl\/1190150053","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T19:12:10Z","timestamp":1197573130000},"page":"438-448","source":"Crossref","is-referenced-by-count":5,"title":["Pfaffian differential equations over exponential o-minimal structures"],"prefix":"10.1017","volume":"67","author":[{"given":"Chris","family":"Miller","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Patrick","family":"Speissegger","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200010082_ref012","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511525919"},{"key":"S0022481200010082_ref011","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1093\/oso\/9780198538622.003.0007","volume-title":"Logic: From foundations to applications","author":"van den Dries","year":"1996"},{"key":"S0022481200010082_ref010","doi-asserted-by":"publisher","DOI":"10.1515\/crll.1999.508.189"},{"key":"S0022481200010082_ref009","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1987-0869411-2"},{"key":"S0022481200010082_ref007","first-page":"357","volume-title":"Logic colloquium 98","volume":"13","author":"Peterzil","year":"2000"},{"key":"S0022481200010082_ref006","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/28.1.7"},{"key":"S0022481200010082_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1994-1195484-5"},{"key":"S0022481200010082_ref002","volume-title":"Proceedings of the American Mathematical Society","author":"Lion"},{"key":"S0022481200010082_ref013","doi-asserted-by":"publisher","DOI":"10.2307\/2118545"},{"key":"S0022481200010082_ref004","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1093\/oso\/9780198538622.003.0016","volume-title":"Logic: From foundations to applications","author":"Miller","year":"1996"},{"key":"S0022481200010082_ref015","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-96-08416-1"},{"key":"S0022481200010082_ref001","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.1999.8128"},{"key":"S0022481200010082_ref014","doi-asserted-by":"publisher","DOI":"10.1112\/S0024610797005437"},{"key":"S0022481200010082_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-98-02288-0"},{"key":"S0022481200010082_ref008","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1983-0716843-5"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200010082","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,20]],"date-time":"2024-02-20T10:41:41Z","timestamp":1708425701000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200010082\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,3]]},"references-count":15,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2002,3]]}},"alternative-id":["S0022481200010082"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1190150053","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,3]]}}}