User leblanc - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T18:00:53Z http://mathoverflow.net/feeds/user/17173 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/121679#121679 Answer by LeBlanc for Trichotomies in mathematics LeBlanc 2013-02-13T06:47:18Z 2013-02-13T06:47:18Z <p>Every prime of a field either ramifies, splits or is inert in a field extension. </p> http://mathoverflow.net/questions/106400/is-every-countable-dedekind-domain-the-ring-of-integers-of-some-number-field Is every countable Dedekind domain the ring of integers of some number field? LeBlanc 2012-09-05T05:25:43Z 2012-11-01T21:39:29Z <p>Is every countable Dedekind domain the ring of integers of some number field? I tried googling different keywords, but did not find anything. Does anyone know of research in this area?</p> http://mathoverflow.net/questions/105147/measure-theory-treatment-geared-toward-the-riesz-representation-theorem/105149#105149 Answer by LeBlanc for Measure theory treatment geared toward the Riesz representation theorem LeBlanc 2012-08-21T09:51:27Z 2012-08-21T23:03:40Z <p>Rudin's <em>Real and Complex Analysis</em> proves the theorem for the following two cases where $X$ is a <strong>locally</strong> compact Hausdorff space. :</p> <ul> <li><p>For linear functionals on the space $C_c(X)$, the space of all continuous compactly supported functions. (Theorem 2.14)</p></li> <li><p>For linear functionals on the space $C_0(X)$, the space of all continuous functions vanishing at infinity. (Theorem 6.19)</p> <p>Since the second is the most general form of the theorem I know, surely this will suit your purposes?</p></li> </ul> http://mathoverflow.net/questions/104731/instances-where-an-existence-result-precedes-the-constructive-version/104746#104746 Answer by LeBlanc for Instances where an existence result precedes the constructive version LeBlanc 2012-08-15T06:07:18Z 2012-08-17T03:07:44Z <p>One example is indeed that of Transcendental numbers, as Yoav Kallus points out in the comments. </p> <p>Liouville showed in 1844 that numbers which do not satisfy a polynomial equation with integer coefficients exist, but he only gave an example in 1851, the famed Liouville constant, a celebrity among Transcendental numbers:</p> <p>$$\sum_{n=1}^\infty 10^{-n!}.$$ For more information, see <a href="http://www.jstor.org/stable/1988833" rel="nofollow">here</a> and <a href="http://www-mathdoc.ujf-grenoble.fr/JMPA/PDF/JMPA_1851_1_16_A5_0.pdf" rel="nofollow">here</a>. You might need JSTOR access to read the first.</p> <p>Cantor however, whose proof of the existence of Transcendental numbers follows directly from the uncountability of the Reals, only came up with his proof in 1874. Whether his proof of the Uncountability of the Reals is constructive or not is something people are still debating, so I will not comment on that. For more information on this, see this <a href="http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_proof#Is_Cantor.E2.80.99s_proof_of_the_existence_of_transcendentals_constructive_or_non-constructive.3F" rel="nofollow">Wikipedia Article</a></p> http://mathoverflow.net/questions/2533/homological-algebra-texts/2542#2542 Comment by LeBlanc LeBlanc 2012-09-23T03:34:41Z 2012-09-23T03:34:41Z @skupers Your link is broken. http://mathoverflow.net/questions/106400/is-every-countable-dedekind-domain-the-ring-of-integers-of-some-number-field/106401#106401 Comment by LeBlanc LeBlanc 2012-09-05T05:38:41Z 2012-09-05T05:38:41Z Ah, silly me. Thank you :) http://mathoverflow.net/questions/106171/the-probability-that-a-random-number-n-has-at-least-m-factors Comment by LeBlanc LeBlanc 2012-09-02T11:54:08Z 2012-09-02T11:54:08Z @Stefan I think the questioner is trying to ask something like: &quot;Given positive integer $N$, and numbers $L,M&lt; N$, if you pick a random number in the set $\{N+i,N−i|0\leq i&lt;L \}$ (Numbers &quot;around&quot; $N$) , what is the probability that it has at least $M$ factors? &quot;Or maybe not so symmetric sample around $N$. What if we pick a number out of the set of all numbers with the same number of digits as $N$ instead? http://mathoverflow.net/questions/105147/measure-theory-treatment-geared-toward-the-riesz-representation-theorem/105149#105149 Comment by LeBlanc LeBlanc 2012-08-21T22:55:46Z 2012-08-21T22:55:46Z @Yemon, How can that be? For example, if $X=\mathbb R$, $f(x)=e^{-x^2}$ is an element of $C_0(X)$ but not of $C_c(X)$. http://mathoverflow.net/questions/105147/measure-theory-treatment-geared-toward-the-riesz-representation-theorem/105149#105149 Comment by LeBlanc LeBlanc 2012-08-21T10:08:42Z 2012-08-21T10:08:42Z @Igor Please see edit. Rudin does treat a more general case. http://mathoverflow.net/questions/104731/instances-where-an-existence-result-precedes-the-constructive-version/104746#104746 Comment by LeBlanc LeBlanc 2012-08-17T03:08:25Z 2012-08-17T03:08:25Z @LeeMosher Thanks! Fixed it. http://mathoverflow.net/questions/96379/is-there-a-name-for-the-set-of-all-permutations-of-a-given-set/96383#96383 Comment by LeBlanc LeBlanc 2012-05-08T22:29:18Z 2012-05-08T22:29:18Z I suppose you're right. But,my real point was to say that $A^2$ may be used to denote it. My set theory professor often did. http://mathoverflow.net/questions/96379/is-there-a-name-for-the-set-of-all-permutations-of-a-given-set/96383#96383 Comment by LeBlanc LeBlanc 2012-05-08T22:21:12Z 2012-05-08T22:21:12Z Yes, that too. Please feel free to edit my answer to include this if you want to!