User z norwood - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T20:18:59Z http://mathoverflow.net/feeds/user/11445 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/50343/what-would-you-want-to-see-at-the-museum-of-mathematics/63658#63658 Answer by Z Norwood for What would you want to see at the Museum of Mathematics? Z Norwood 2011-05-01T22:59:03Z 2011-05-01T22:59:03Z <p>What about some large-number phenomena? This seems to be something the general public would appreciate and could relate to the "Computers in Modern Mathematics" booth others have suggested.</p> <p>What I have in mind is not really Ackerman function/Graham's number business (which I don't think I could wrap my head around any more easily at a museum), but facts that involve small-ish large numbers. For instance:</p> <blockquote> <p>The smallest positive integer $n$ for which $n$ divides $2^n-3$ is $4,700,063,447$.</p> </blockquote> <p>There are many other great examples (though not all interesting or accessible to non-mathematicians) in answers to <a href="http://mathoverflow.net/questions/15444/the-phenomena-of-eventual-counterexamples" rel="nofollow">this MO question</a>. It also might be nice to see comparisons of smallest counterexamples like this to 'real-world' numbers like the population of China (~$1.34$ billion), or the number of cells in the human body (~$10^{14}$), or the number of elementary particles in the observable universe (~$10^{80(\pm10?)}$). </p> <p>To me, the goal of such an exhibit should be (1) to provide a few examples (like the one above) illustrating the importance of <strong>proof</strong> over <strong>verification of the first $10^{10}$ cases</strong>, and (2) to help museum-goers conceptualize the small-ish large numbers that come up in analyzing real-world phenomena.</p> http://mathoverflow.net/questions/52893/is-it-possible-to-construct-without-choice-even-a-non-finitely-generated-grou Is it possible to construct (without choice, even?) a non-finitely-generated group with no proper non-finitely-generated subgroup? Z Norwood 2011-01-23T01:01:32Z 2011-01-24T20:15:40Z <p>Is there a non-finitely-generated group each of whose proper subgroups is finitely generated? If so, what form of choice (if any) is required to construct such a group?</p> http://mathoverflow.net/questions/49349/simplest-example-of-an-infinite-p-group-with-trivial-center Simple(st) example of an infinite $p$-group with trivial center Z Norwood 2010-12-14T04:27:47Z 2010-12-18T14:04:44Z <p>The only examples I have encountered of infinite $p$-groups with trivial center employ non-elementary methods in their construction. For instance, Example 9.2.5 of Scott's <em>Group Theory</em> is a perfectly satisfactory example, but it requires the wreath product (which, though an invaluable group-theoretic tool, is not what I consider an "elementary method").</p> <p>Does anyone know of an example (of an infinite $p$-group with trivial center) that can be constructed and proven to have the claimed properties in a way that is friendly to, say, students of a first course in group theory? Perhaps a large product of finite groups or an easy-to-describe matrix group?</p> <p>(I also welcome arguments for the nonexistence of such an example!)</p> http://mathoverflow.net/questions/95611/books-on-logic-for-someone-aiming-to-go-to-grad-school-in-the-field Comment by Z Norwood Z Norwood 2012-04-30T23:24:30Z 2012-04-30T23:24:30Z I second Andreas's recommendation for Kunen's book. Marker's book on model theory is good, but beware of the errors:<a href="http://homepages.math.uic.edu/~marker/mt-errors.html" rel="nofollow">homepages.math.uic.edu/~marker/mt-errors.html</a>. http://mathoverflow.net/questions/80805/dimension-of-vector-space-without-the-axiom-of-choice Comment by Z Norwood Z Norwood 2011-11-13T16:54:31Z 2011-11-13T16:54:31Z @Asaf: Thanks! I didn't know that. I can't read German, so I have no idea what L&#228;uchli is doing in that paper. Just thought the OP might want the original source. http://mathoverflow.net/questions/20882/most-unintuitive-application-of-the-axiom-of-choice/20886#20886 Comment by Z Norwood Z Norwood 2011-11-13T16:46:05Z 2011-11-13T16:46:05Z I don't claim to have much intuition about what a well-ordering of $\mathbb{R}$ would look like (whatever that means), but the existence (even in the absence of AC) of a whole range of well-orderable uncountable sets makes it fairly believable that there's a bijection from $\mathbb{R}$ to one of them. Banach–Tarski, on the other hand, offers me no intuition even while I stare at its proof. http://mathoverflow.net/questions/80805/dimension-of-vector-space-without-the-axiom-of-choice Comment by Z Norwood Z Norwood 2011-11-13T16:27:37Z 2011-11-13T16:27:37Z I believe Jech offers an outline of the proof as a hint to the exercise Asaf mentioned. Also, L&#228;uchli's original paper is: L&#228;uchli, H. <i>Auswahlaxiom in der Algebra.</i> (German) Comment. Math. Helv. 37 (1963). MR here: <a href="http://www.ams.org/mathscinet-getitem?mr=143705" rel="nofollow">ams.org/mathscinet-getitem?mr=143705</a>. http://mathoverflow.net/questions/5329/requires-axiom-of-choice-vs-explicitly-constructible/9532#9532 Comment by Z Norwood Z Norwood 2011-10-14T13:09:58Z 2011-10-14T13:09:58Z You can find Feferman's paper here: matwbn.icm.edu.pl/tresc.php?wyd=1&amp;tom=56. The relevant result is Theorem 4.9, I believe. (A minor addition to Ashutosh's answer: It's consistent with ZFC+GCH (not just ZFC) that there is no definable well-ordering of ℝ.) http://mathoverflow.net/questions/30511/ebook-readers-for-mathematics Comment by Z Norwood Z Norwood 2011-04-21T23:51:31Z 2011-04-21T23:51:31Z Does anyone know if there is an organized effort to develop (La)TeX packages/classes specifically designed to create ereader-friendly PDFs? http://mathoverflow.net/questions/30511/ebook-readers-for-mathematics Comment by Z Norwood Z Norwood 2011-04-21T23:40:37Z 2011-04-21T23:40:37Z @Joseph: You do not incur a charge, provided you transfer the PDF to the Kindle via the USB cord (just open up the Finder and drag the PDF to the documents folder in your Kindle). I think Amazon does charge you to transfer the file wirelessly. http://mathoverflow.net/questions/58659/group-theory-isomorphic-subgroup Comment by Z Norwood Z Norwood 2011-03-16T16:55:37Z 2011-03-16T16:55:37Z This question is probably more appropriate for math.stackexchange.com. See <a href="http://mathoverflow.net/faq#whatnot" rel="nofollow">mathoverflow.net/faq#whatnot</a>. (Your example will have to be an infinite group, of course. Consider the least complicated infinite group you can think of and one of its proper subgroups.) http://mathoverflow.net/questions/54775/what-is-the-shortest-ph-d-thesis/54793#54793 Comment by Z Norwood Z Norwood 2011-02-08T18:06:41Z 2011-02-08T18:06:41Z There is an English translation here: <a href="http://arxiv.org/PS_cache/arxiv/pdf/0803/0803.3787v2.pdf" rel="nofollow">arxiv.org/PS_cache/arxiv/pdf/0803/0803.3787v2.pdf</a> That document is 17 pages (including title page, etc.). http://mathoverflow.net/questions/54775/what-is-the-shortest-ph-d-thesis/54796#54796 Comment by Z Norwood Z Norwood 2011-02-08T18:02:37Z 2011-02-08T18:02:37Z Totaro's 1989 thesis is titled &quot;K-theory and algebraic cycles&quot; and, according to ProQuest, is 20 pages. If your university library subscribes to ProQuest, you can see a PDF preview of the thesis by searching for &quot;Totaro, Burt&quot; in the Dissertations and Theses database. http://mathoverflow.net/questions/11591/suggestions-for-a-good-measure-theory-book/20665#20665 Comment by Z Norwood Z Norwood 2011-01-25T04:12:29Z 2011-01-25T04:12:29Z @Andrew L: I don't think Rudin (resp. Halmos) wrote <i>Real and Complex</i> (resp. <i>Measure Theory</i>) as bedtime reading for beginners, but during its four decades in print it has proven to be an important resource for mature students with a serious interest in analysis and (as Bill mentions) as a reference for analysts. I don't think it's productive to dismiss a standard (and valuable) text, even though it almost certainly isn't the ideal one for this stage of the OP's analysis education. http://mathoverflow.net/questions/52893/is-it-possible-to-construct-without-choice-even-a-non-finitely-generated-grou/52914#52914 Comment by Z Norwood Z Norwood 2011-01-24T22:37:08Z 2011-01-24T22:37:08Z @KConrad: I agree with Peter that the $\mathbb{Z}[1/p]/\mathbb{Z}$ construction is more concrete, but thank you for the alternative. Maybe thinking about this group from a different perspective will help me grasp its weirdness. http://mathoverflow.net/questions/52893/is-it-possible-to-construct-without-choice-even-a-non-finitely-generated-grou/52914#52914 Comment by Z Norwood Z Norwood 2011-01-24T22:35:21Z 2011-01-24T22:35:21Z @Peter: You're quite right. I'm not a group theorist, so I'm still a bit surprised to see that a group with (what is to me) such an unlikely property is so concrete. (I admit I didn't even expect an explicit construction.) http://mathoverflow.net/questions/52893/is-it-possible-to-construct-without-choice-even-a-non-finitely-generated-grou/52914#52914 Comment by Z Norwood Z Norwood 2011-01-24T04:26:32Z 2011-01-24T04:26:32Z Excellent example and explanation---thanks! It's still not clear to me to what extent choice is necessary for this example, but I'll look through the argument more closely later. (I was expecting a less clear-cut construction, I suppose.) http://mathoverflow.net/questions/49349/simplest-example-of-an-infinite-p-group-with-trivial-center/49352#49352 Comment by Z Norwood Z Norwood 2010-12-14T05:21:06Z 2010-12-14T05:21:06Z Brilliant! I was hoping some sort of matrix fiddling like this would do the trick. I guess I'm not sure how much more elementary this is than the example I gave, but at least the group can be described in more elementary language. This example is also cleverer, I think. Thanks!