User serge r. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T13:46:09Z http://mathoverflow.net/feeds/user/3770 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81353/interdefinability-of-two-expansions-of-the-real-field/81358#81358 Answer by Serge R. for Interdefinability of two expansions of the Real Field Serge R. 2011-11-19T17:02:13Z 2011-11-19T17:02:13Z <p>I am not sure I see the difficulty. </p> <p>Let's assume $f'>0$ on $I$. The function $s$ that sends $x$ to $\frac{f'(x)\cos(f(x))}{|f'(x)\cos(f(x))|}$ if $f'(x)\cos(f(x))\neq 0$ and to $0$ else is definable in the first structure and, as you noted, $\cos(f(x))=s(x)|cos(f(x))|$ is therefore also definable. </p> <p>Or did I miss something ?</p> http://mathoverflow.net/questions/78467/thom-polynomial-for-contact-algebraic-structures/78845#78845 Answer by Serge R. for Thom polynomial for contact algebraic structures Serge R. 2011-10-22T20:02:50Z 2011-10-22T20:02:50Z <p>I think that the answer to 1) is positive. $f$ could even be chosen not to depend on $g$ and the distribution would not need to be taken contact. This would be a consequence of the fact that the complex field has a so-called <em>strongly minimal theory</em> and that the tangency condition is closed.</p> http://mathoverflow.net/questions/29300/whats-wrong-with-the-surreals/30167#30167 Answer by Serge R. for What's wrong with the surreals? Serge R. 2010-07-01T11:04:58Z 2010-07-01T11:04:58Z <p>This is not a major issue but there was a remark made in the master thesis that can be found at the following address <a href="http://www.mamane.lu/concoq/" rel="nofollow">http://www.mamane.lu/concoq/</a> that there is a small gap in the proof of the transitivity of the ordre relation in the original book of Conway. See the report, page 49-53.</p> http://mathoverflow.net/questions/23259/thom-first-isotopy-lemma-in-o-minimal-structures/23263#23263 Answer by Serge R. for Thom first isotopy lemma in o-minimal structures Serge R. 2010-05-02T16:45:59Z 2010-05-02T16:45:59Z <p>Shiota sent this preprint on the arXiv not long ago : <a href="http://arxiv.org/abs/1002.1508" rel="nofollow">http://arxiv.org/abs/1002.1508</a></p> http://mathoverflow.net/questions/78467/thom-polynomial-for-contact-algebraic-structures/78845#78845 Comment by Serge R. Serge R. 2011-10-24T07:50:21Z 2011-10-24T07:50:21Z The proof would be similar to jvp's: as the parameters that describe the distribution and the curve vary, the set of points on the curve at which the distribution is tangent gives a family of sets each of dimension 1 or 0. When the dimension is 1, we get all the curve to be tangent. Else we get a finite collection of points; but by strong minimality, there should be an uniform bound for the number of these finitely many points. http://mathoverflow.net/questions/78467/thom-polynomial-for-contact-algebraic-structures/78845#78845 Comment by Serge R. Serge R. 2011-10-24T07:46:33Z 2011-10-24T07:46:33Z I don't have my books now but Poizat &quot;Cours de th&#233;orie des mod&#232;les&quot; should contain some about it (and is translated in many languages, some translations being freely available on the web). Marker's &quot;Model theory: an introduction&quot; should do too (chapter 3 and 8 ?). http://mathoverflow.net/questions/23259/thom-first-isotopy-lemma-in-o-minimal-structures/23263#23263 Comment by Serge R. Serge R. 2010-05-02T22:56:20Z 2010-05-02T22:56:20Z (Also a limitation of the results in Shiota's 97 book is that they &quot;only&quot; deals with expansions of the field of reals and not with general o-minimal expansions of a real closed fields.) http://mathoverflow.net/questions/23259/thom-first-isotopy-lemma-in-o-minimal-structures/23263#23263 Comment by Serge R. Serge R. 2010-05-02T17:21:44Z 2010-05-02T17:21:44Z It was indeed not a very sharp reference. More relevant could be the appendix (Theorem A.5.) of ### Computing o-minimal topological invariants using differential topology Author(s): Ya'acov Peterzil; Sergei Starchenko Journal: Trans. Amer. Math. Soc. 359 (2007), 1375-1401. ### (it states the result for a proper function, not a general proper mapping.)