Which limit to take as a key applied math decision - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T05:51:35Z http://mathoverflow.net/feeds/question/118160 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118160/which-limit-to-take-as-a-key-applied-math-decision Which limit to take as a key applied math decision R Hahn 2013-01-05T23:11:53Z 2013-01-06T20:17:00Z <p>The Borel-Kolmogorov paradox refers to situations where non-uniqueness in the notion of conditioning on a set of measure zero leads to apparent contradictions. As a formal matter, one requires instead to condition on "the" generating sigma algebra, which vanquishes non-uniqueness by fiat. For a technical explanations see <a href="http://econpapers.repec.org/article/eeejmvana/v_3a27_3ay_3a1988_3ai_3a2_3ap_3a434-446.htm" rel="nofollow">this paper</a>. Billingsley's measure theory book has a nice treatment as well. </p> <blockquote> <p>I am looking for examples where a formal non-uniqueness was resolved by applied considerations which suggested a natural "tie-breaker".</p> </blockquote> <p>A simple example from <a href="http://www.jstor.org/discover/10.2307/2685937?uid=3739656&amp;uid=2129&amp;uid=2&amp;uid=70&amp;uid=4&amp;uid=3739256&amp;sid=21101539327171" rel="nofollow">this paper</a> illustrates the idea. Let $X$ and $Y$ be independent standard normal random variables. What is the conditional distribution of $X$ given that you are on the (measure-zero) line where $X = Y$? The answer will vary depending on if you condition on $Z_1 = 0$ where $Z_1 \equiv X - Y$ or $Z_2 = 1$ where $Z_2 \equiv X/Y$, to give just two of an infinite number of examples. So in a given situation which $Z$ is the "right" one to use?</p> <p>My question is not about the Borel paradox or modeling random phenomena per se. </p> <blockquote> <p>I am interested broadly in hearing about situations where </p> <ul> <li>we have a mathematically well defined condition ($x = y$ as above)</li> <li>we want to study some applied model when that condition is satisfied</li> <li>the conclusions we reach will differ depending on the way we approach (as taking a limit) that condition formally</li> </ul> <p>Finally I am interested in how this ambiguity is resolved by "physical" considerations.</p> </blockquote> <p>I would make the problem sharper if I could, but the reason I want examples is precisely to help focus my thinking. I find it intriguing that it is not enough to have a well defined condition and a well defined model, one must also justify (by way of interpretation) which limit to take!</p> <p>I anticipate there are many examples from physics of which I am unaware and perhaps some from the literature on finite elements for solving PDEs.</p> <p>(Apologies for the pay-wall links.)</p>