User josh guffin - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T04:09:56Z http://mathoverflow.net/feeds/user/3912 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37408/undergraduate-probability-topics Undergraduate Probability Topics Josh Guffin 2010-09-01T18:10:21Z 2011-03-04T04:55:02Z <p>I am teaching undergraduate probability this semester, and I am looking for some suggestions about inspiring applications that could be reasonably covered over the course of two one-hour lectures or less. For example, here are two very cool topics I covered last time I taught this course:</p> <ul> <li>Search engines and Markov chains; <a href="http://www.ams.org/samplings/feature-column/fcarc-pagerank" rel="nofollow">AMS' description</a> and <a href="http://www.rose-hulman.edu/~bryan/google.html" rel="nofollow">The $25 billion eigenvector</a></li> <li><a href="http://terrytao.wordpress.com/2009/07/03/benfords-law-zipfs-law-and-the-pareto-distribution/" rel="nofollow">Benford's Law and the Pareto Distribution</a></li> </ul> <p>Thanks in advance!</p> http://mathoverflow.net/questions/35736/the-canonical-line-bundle-of-a-normal-variety/40226#40226 Answer by Josh Guffin for The canonical line bundle of a normal variety Josh Guffin 2010-09-27T22:26:07Z 2010-09-29T18:31:20Z <p>Your formula is not quite right for toric varieties. In particular, the sum is not over "representatives of the class group", but over a set of minimal generators for the free group on torus-invariant divisors. Such a set is furnished by the 1-cones in the fan. More precisely,</p> <blockquote> <p>Let$X_\Sigma$be the toric variety associated to a fan$\Sigma$, and assume that$X_\Sigma$has no torus factors. Then for each$\rho \in \Sigma(1)$, there is a torus invariant divisor$D_\rho$, and $$\mathcal O_{X_\Sigma}\Big(-\sum_{\rho} D_\rho\Big)\cong \omega_{X_\Sigma}$$</p> </blockquote> <p>This is Proposition 8.2.7 in Cox Little Schenck.</p> <p>An easy toric proof that no projective toric variety is Calabi-Yau is that, as you said, the minimal generators of the one-cones must lie in a hyperplane. The positive hull over such a set of generators is strongly convex, so that the support of the fan cannot be all of$N_{\mathbb R}$, and thus$X_\Sigma$is not complete and thus not projective.</p> <p>I believe that your question really concerns the existence of a natural set of generators for the "total coordinate ring" of an Calabi-Yau variety and if they obey a linear relation. The total coordinate ring is defined to be $$R=\bigoplus_{D\in Cl(X)} \Gamma(X,\mathcal O_X(D)).$$</p> <p>Here$Cl(X)$is the class group of$X$. See <a href="http://arxiv.org/abs/0801.3995" rel="nofollow">0801.3995</a> for more details.</p> <p>For toric varieties$X_\Sigma$, this is simply$\mathbb C[x_\rho | \rho \in \Sigma(1)]$. If this is indeed your question, it is probably too much to hope for, as$R$is not known to be finitely generated! It is known to be so for toric varieties (<a href="http://arxiv.org/abs/alg-geom/9210008" rel="nofollow">Cox</a>), and for varieties of Fano type (<a href="http://www.ams.org/journals/jams/2010-23-02/S0894-0347-09-00649-3/home.html" rel="nofollow">Birkar–Cascini–Hacon–McKernan</a>). Some other specifically constructed examples (<a href="http://www.iag.uni-hannover.de/~prendergast/Papers/Example.pdf" rel="nofollow">Prendergast-Smith</a>) are known, but a general characterization is not.</p> <p><strong>Edit:</strong> Updated the links and fixed the unclear notation - thanks Artie!</p> http://mathoverflow.net/questions/33716/cox-rings-of-toric-varieties-over-arbitrary-fields/40466#40466 Answer by Josh Guffin for Cox rings of toric varieties over arbitrary fields Josh Guffin 2010-09-29T13:20:10Z 2010-09-29T13:20:10Z <p><a href="http://www.sciencedirect.com/science?_ob=ArticleURL&amp;_udi=B6WH2-4M04JC5-4&amp;_user=489256&amp;_coverDate=02/15/2007&amp;_rdoc=1&amp;_fmt=high&amp;_orig=search&amp;_origin=search&amp;_sort=d&amp;_docanchor=&amp;view=c&amp;_acct=C000022721&amp;_version=1&amp;_urlVersion=0&amp;_userid=489256&amp;md5=f83e7b141483e665059b365a5dd7c507&amp;searchtype=a%20%22Parametrizations%20of%20toric%20varieties%20over%20any%20field%22" rel="nofollow">Parametrizations of toric varieties over any field</a>: also available via <a href="http://docs.google.com/viewer?a=v&amp;q=cache%3aQRVOFXwUFAQJ%3awww.math.uoi.gr/~athoma/papers1/Parametrizations.pdf+Parametrizations+of+toric+varieties+over+any+field&amp;hl=en&amp;gl=us&amp;pid=bl&amp;srcid=ADGEESiyPr_qirHV1DVlCl-vT-Q0z-uCQIchB87s67gPKcNVkx0PT508B0Q8_g5zfB3L7-OIg0WblbnJdsDxsXXlWcUcVtDCDa8oS5-kGVC2lk5MrcZjW_ylh64aiMzWm53dv1YdkARs&amp;sig=AHIEtbTHPwicWDs-Nw30IHbYmKL8sWd4fw" rel="nofollow">google</a></p> http://mathoverflow.net/questions/40454/practical-applications-of-eigenvalues-and-eigenvectors/40462#40462 Answer by Josh Guffin for practical applications of eigenvalues and eigenvectors Josh Guffin 2010-09-29T12:46:51Z 2010-09-29T12:46:51Z <p>I would comment on <a href="http://mathoverflow.net/questions/40454/practical-applications-of-eigenvalues-and-eigenvectors/40461#40461" rel="nofollow">Peitro's answer</a>, but I don't have enough reputation; for a marvelously-titled explanation of Google's Pagerank, see <a href="http://www.rose-hulman.edu/~bryan/googleFinalVersionFixed.pdf" rel="nofollow">The$25,000,000,000 Eigenvector</a>.</p> http://mathoverflow.net/questions/38856/jokes-in-the-sense-of-littlewood-examples/39303#39303 Answer by Josh Guffin for Jokes in the sense of Littlewood: examples? Josh Guffin 2010-09-19T14:11:38Z 2010-09-19T14:11:38Z <p>Another joke in the spirit of the chain rule; you solve separable differential equations by "multiplying by g(y)dx" $$g(y) dx \left(\frac{dy}{dx} = \frac{f(x)}{g(y)}\right) \Rightarrow g(y)dy = f(x) dx$$</p> <p>Then, there's nothing to be done but integrating to get rid of the dx and dy.</p> <p>I also like to point out to students who ask about cancellation in the chain rule that you can cancel there just like you can cancel the sixes and nines respectively in </p> <p>$$\frac {16}{64} = \frac 1 4 \qquad \text{and} \qquad \frac{19}{95}=\frac 1 5;$$ that is, carefully, and when it makes sense to do so.</p> http://mathoverflow.net/questions/44379/is-there-an-analogue-of-mathscinet-for-physics Comment by Josh Guffin Josh Guffin 2010-10-31T21:10:59Z 2010-10-31T21:10:59Z Not exactly the same, but for high-energy physics there is spires: <a href="http://www.slac.stanford.edu/spires/" rel="nofollow">slac.stanford.edu/spires</a> http://mathoverflow.net/questions/40672/constructing-affine-hypersurfaces-with-one-singularity/40677#40677 Comment by Josh Guffin Josh Guffin 2010-09-30T22:49:22Z 2010-09-30T22:49:22Z His condition that $S$ lies in a hyperplane is equivalent to the existence of a $\mathbb C^*$ action for which each monomial is of the same degree. The weights are given by the normal vector to $H$. For instance, in example 3, $(6,0,0).(1,2,3)=(1,1,1).(1,2,3)=\cdots=6$, so the exponents lie in the hyperplane $a + 2b + 3c = 6$. http://mathoverflow.net/questions/15292/why-cant-there-be-a-general-theory-of-nonlinear-pde Comment by Josh Guffin Josh Guffin 2010-09-29T18:26:50Z 2010-09-29T18:26:50Z This also brings to mind the preface (<a href="http://books.google.com/books?id=qlNJAYwmfTcC&amp;lpg=PR5&amp;ots=tphTO4rxEA&amp;dq=Preface%20to%20the%20%20of%20%22Lectures%20on%20Partial%20Differential%20Equations%22&amp;pg=PR5#v=onepage&amp;q=Preface%20to%20the%20%20of%20%22Lectures%20on%20Partial%20Differential%20Equations%22&amp;f=false" rel="nofollow">books.google.com/&hellip;</a>) from &quot;Lectures on Partial Differential Equations&quot; by Arnol'd. Unfortunately the google books version cuts out after the first page, and I can't find another English version online. You can find a Russian version by googling for &quot;Лекции об уравнениях с частными производными&quot;.