most recent 30 from http://mathoverflow.net 2013-05-22T22:43:05Z http://mathoverflow.net/feeds/user/8528 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/35746/inaccessible-cardinals-and-andrew-wiless-proof Cosmonut 2010-08-16T10:41:32Z 2011-08-19T01:37:26Z

<p>In a recent issue of New Scientist (16 Aug 2010), I was surprised to read that a part of Wiles' proof of Taniyama-Shimura conjecture relies on inaccessible cardinals.</p> <p>Here's the link: <a href="http://www.newscientist.com/article/mg20727731.300-to-infinity-and-beyond-the-struggle-to-save-arithmetic.html?page=2" rel="nofollow">http://www.newscientist.com/article/mg20727731.300-to-infinity-and-beyond-the-struggle-to-save-arithmetic.html?page=2</a></p> <p>Here's the relevant bit from the article: "Large cardinals have been studied by logicians for a century, but their intangibility means they have seldom featured in mainstream mathematics. A notable exception is the most celebrated result of recent years, the proof of Fermat's last theorem by the British mathematician Andrew Wiles in 1994............... To complete his proof, Wiles assumed the existence of a type of large cardinal known as an inaccessible cardinal, technically overstepping the bounds of conventional arithmetic"</p> <p>Is this true ? If so, could someone please outline how they are used ?</p>

http://mathoverflow.net/questions/38795/borel-sets-on-rn Cosmonut 2010-09-15T09:34:23Z 2011-08-17T10:19:29Z

<p>Define the Borel sigma-algebra on R^n as the smallest sigma-algebra containing all n-rectangles (a1, b1) x...x (an, bn).</p> <p>Is it true that the Borel sigma algebra contains all sets of the form A1 x...x An, where each Ai is some Borel set in R ?</p> <p>I thought this would be trivially true, but I had a lot of trouble trying to prove it, and I'm not even sure its true anymore.</p> <p>If this is a well-known result, could you please refer me to a text where it has been (dis)proved ?</p>

http://mathoverflow.net/questions/39928/wiener-process-related-counterexample Cosmonut 2010-09-25T05:22:53Z 2010-11-09T05:13:42Z

<p>The Wiener process is defined by the three properties: 1. $W(0) = 0$, 2. $W(t)$ is almost surely continuous, and 3. $W(t)$ has independent increments with $W(t) - W(s) \sim N(0, t-s)$ (for $0 ≤ s &lt; t$).</p> <p>What would be an example of a process which satisfies 1) and 3), but not 2) ?</p> <p>I am going to teach an introductory class on Brownian motion at advanced undergrad level. Just wanted to make sure that all the conditions are mutually independent.</p>

http://mathoverflow.net/questions/38659/total-energy-of-the-universe Cosmonut 2010-09-14T07:55:08Z 2010-09-14T16:07:58Z

<p>In popular science books and articles, I keep running into the claim that the total energy of the Universe is zero, "because the positive energy of matter is cancelled out by the negative energy of the gravitational field".</p> <p>But I can't find anything concrete to substantiate this claim. As a first check, I did a calculation to compute the gravitational potential energy of a sphere of uniform density of radius R using Newton's Laws and threw in E=mc^2 for energy of the sphere, and it was by no means obvious that the answer is zero !</p> <p>So, my questions:</p> <p>1) What is the basis for the claim - does one require General Relativity, or can one get it from Newtonian gravity ?</p> <p>2) What conditions do you require in the model, in order for this to work ?</p> <p>3) Could someone please refer me to a good paper about this ?</p>

http://mathoverflow.net/questions/37610/demonstrating-that-rigour-is-important/38031#38031 Cosmonut 2010-09-08T05:04:01Z 2010-09-08T05:04:01Z

<p>Tim Gowers wrote:</p> <h2>But I was, and am, more interested in good examples of cases where a proof of a statement that was widely believed to be true and was true gave us much more than just a certificate of truth.</h2> <p>How about Stokes' Theorem ?</p> <p>The two-dimensional version involving line and surface integrals is "proved" in most physics textbooks using a neat little picture dividing up the surface into little rectangles and shrinking them to zero.</p> <p>Similarly, the Divergence Theorem related volume and surface integrals is demonstrated with very intuitive ideas about liquid flowing out of tiny cubes.</p> <p>But to prove these rigorously requires developing the theory of differential forms whose consequences go way beyond the original theorems</p>

http://mathoverflow.net/questions/2358/most-harmful-heuristic/3585#3585 Cosmonut 2010-10-03T03:39:36Z 2010-10-03T03:39:36Z

I second, third and fourth that, Darsh. That particular definition of tensor set back my understanding of differential geometry by at least a year.

http://mathoverflow.net/questions/38659/total-energy-of-the-universe/38690#38690 Cosmonut 2010-09-25T18:16:09Z 2010-09-25T18:16:09Z

@Willie: Yes, I am aware that quantum gravity is not solved. But I got intrigued when Sean Carroll mentioned the &quot;zero energy&quot; result the Cosmic Variance blog, giving the impression that it was an established consequences of classical general relativity. Learning the actual physics is definitely on the cards - but so much to do, so little time...

http://mathoverflow.net/questions/39928/wiener-process-related-counterexample/39935#39935 Cosmonut 2010-09-25T17:58:06Z 2010-09-25T17:58:06Z

Hello Byron and Reda, thanks for replying. Just one thing which is confusing me a little. Reda says, &quot;Byron exhibited another version by changing the process on a set of measure zero&quot;. Intuitively, what I understand of Byron's construction is that every path is being broken &quot;at a different point in time&quot;. Thus, all the paths are discontinuous, but the joint distributions of the random variables W(t) remain unchanged. Is that what you mean as well ?

http://mathoverflow.net/questions/38795/borel-sets-on-rn/38799#38799 Cosmonut 2010-09-16T08:44:08Z 2010-09-16T08:44:08Z

Ok, done. Proved Step 1, with the standard trick of: - Consider all sets of the form A1 x R^(n-1) which belong to Borel sets of R^n, where A1 is a set in R - Showed that was a sigma algebra - Since (a, b) x R^(n-1) is in Borel sets of R^n, A1 can any Borel set of R. Thanks for the help and the addendum.

http://mathoverflow.net/questions/38795/borel-sets-on-rn/38799#38799 Cosmonut 2010-09-15T11:15:59Z 2010-09-15T11:15:59Z

No, this wasn't a homework problem. I've been wondering whether B(R^n) should be defined as the smallest sigma algebra containing all rectangles, or the smallest sigma algebra containing all products of Borel sets in R. I was trying to prove that these are equivalent, and getting worried that I couldn't and I was getting stuck trying to do step 1 as you suggested. Ok, just need to try harder then...

http://mathoverflow.net/questions/38659/total-energy-of-the-universe/38690#38690 Cosmonut 2010-09-14T17:11:49Z 2010-09-14T17:11:49Z

Curiouser and curiouser ! The claim that the total energy of the Universe is zero is usually used to make the grander claim that &quot;hence, the Universe could spontaneously arise from Nothing without violating conservation of energy&quot;. But Shing-Yau result suggests that the zero energy claim and its corollary is nonsense !! Thanks. Let me post this on some physics blogs and see what they have to say.

http://mathoverflow.net/questions/38659/total-energy-of-the-universe Cosmonut 2010-09-14T08:02:12Z 2010-09-14T08:02:12Z

I've been searching the astronomy and physics blogs, but they all seem to state the result without further backup. Plus, the conditions under which this holds are nowhere clarified. It made me think this may be a question for mathematical physicists.

http://mathoverflow.net/questions/35746/inaccessible-cardinals-and-andrew-wiless-proof/35749#35749 Cosmonut 2010-08-16T11:56:59Z 2010-08-16T11:56:59Z

Thanks Pete and others. This is very interesting.