User carl offner - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T03:01:05Z http://mathoverflow.net/feeds/user/6172 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78129/a-brouwer-fixed-point-theorem-on-finite-sets/78133#78133 Answer by Carl Offner for A Brouwer fixed point theorem on finite sets Carl Offner 2011-10-14T13:21:38Z 2011-10-14T13:21:38Z <p>Wouldn't the Tietze extension theorem (with range $X$) show that any permutation of $A$ extends to a function $f$?</p> http://mathoverflow.net/questions/52708/why-should-one-still-teach-riemann-integration/57625#57625 Answer by Carl Offner for Why should one still teach Riemann integration? Carl Offner 2011-03-07T02:02:21Z 2011-03-07T02:02:21Z <p>Although it's true that the Lebesgue integral is technically more complicated than the Riemann integral, I don't think that's really the point. In fact, the Riemann integral is pretty technical already, as far as most undergraduates are concerned. The real point is the "fundamental theorem of the calculus", which is at least conceptual and amenable to a really good plausibility argument in the case of the Riemann integral, but requires some genuinely deep analysis in the case of the Lebesgue integral.</p> <p>Don't get me wrong -- I really love integration theory. And I've never taught serious undergraduate analysis, so I don't have any actual experience in this area. I do have a funny story, however:</p> <p>I first learned "real analysis" from a course taught by Richard Brauer. Brauer was of course an algebraist, but at the time, he taught in consecutive years the basic graduate "real analysis", "complex analysis" and algebra courses. He was as close as I felt I would ever get to the great tradition of mathematicians who had a broad view of the field as a whole. Of course the book we used as a reference for measure theory and integration was Halmos. At one point, he remarked, "Halmos is a curious book. When you finish this book you will have a very good understanding of how to integrate the function 1. But you will have no idea of how to integrate the function $x$."</p> <p>Of course he was referring to the fact that Halmos doesn't even touch on the question of differentiation in Euclidean space, and so doesn't really deal with the "fundamental theorem".</p> http://mathoverflow.net/questions/28147/why-do-we-care-about-lp-spaces-besides-p-1-p-2-and-p-infinity/28176#28176 Answer by Carl Offner for Why do we care about L^p spaces besides p = 1, p = 2, and p = infinity? Carl Offner 2010-06-14T21:55:23Z 2010-06-14T21:55:23Z <p>I guess that $L^1$, $L^2$, and $L^\infty$ just seem natural because they are so intimately related to obvious everyday concepts -- sums, averages, maxima, root-mean-square (Hilbert space, ...). So I doubt that the occasional theorem that involves another $L^p$ space explicitly will ever convince anyone that any other $L^p$ space is equally significant.</p> <p>But let me just mention another one of them anyway: there's a marvelous theorem of Beurling that states that the family of functions of the form $f(x) = \sum_{k=1}^n a_k\rho(\theta_k/x)$---where $\rho(x) = x -\lfloor x\rfloor$ and $\sum a_k\theta_k = 0$---is dense in $L^p(0,1)$ (for some $p\in[1,\infty]$) iff the Riemann Zeta function has no zeros in the half-plane $\sigma > 1/p$.</p> http://mathoverflow.net/questions/25161/why-is-lebesgue-integration-taught-using-positive-and-negative-parts-of-functions/25193#25193 Answer by Carl Offner for Why is Lebesgue integration taught using positive and negative parts of functions? Carl Offner 2010-05-19T01:10:56Z 2010-05-19T01:10:56Z <p>It's really the difference between two kinds of completions:</p> <ol> <li><p>An order-theoretic completion. For this, it's easiest to start with non-negative functions, and have infinite values dealt with pretty naturally.</p></li> <li><p>A metric completion. For this, it's more natural to start with finite-valued signed simple functions.</p></li> </ol> <p>It's not exactly that simple -- historically, signed simple functions (well, actually, I think they used step functions) were used in an order-theoretic treatment by Riesz and Nagy. But I think this is a good way to look at the two ways of approaching this integral.</p> <p>And needless to say, these two approaches generalize in two different contexts. They are both interesting and illuminate somewhat different aspects of the Lebesgue integral, even on the real line. For instance, the order-theoretic approach leads quickly to results such as the monotone convergence and bounded convergence theorems, while the metric approach leads naturally to the topology of convergence in measure and completeness of the $L_p$ spaces.</p> http://mathoverflow.net/questions/74841/an-example-of-a-beautiful-proof-that-would-be-accessible-at-the-high-school-level/74865#74865 Comment by Carl Offner Carl Offner 2012-10-04T00:30:06Z 2012-10-04T00:30:06Z It's certainly a great proof -- certainly immediately convincing. For some reason, it always irritated me. I've thought on and off over the years why this is. The only thing I can think of is that, while it is constructive, it is also outrageously inefficient. Now at the time I learned this, no one that I knew was even talking in those terms, so maybe it's just some instinct that I had. I really wish I liked it more. G.H. Hardy did, right? You can't argue with that. http://mathoverflow.net/questions/74841/an-example-of-a-beautiful-proof-that-would-be-accessible-at-the-high-school-level/74864#74864 Comment by Carl Offner Carl Offner 2012-10-04T00:24:13Z 2012-10-04T00:24:13Z I guess it depends on how it's presented. In my experience, it mainly started out &quot;suppose (a/b)^2 = 2 and a/b is in lowest terms&quot;, and then derived a contradiction, showing that a/b was not in lowest terms. Invariably, students would respond, &quot;Well, then <i>put</i> it in lowest terms.&quot; It just wasn't convincing. I always felt that casting the argument in terms of prime factorization was more convincing, but to be honest, I'm not convinced that I saw a lot of students being astonished at this, as I had been. There's a certain amount of irreducible sophistication here. http://mathoverflow.net/questions/98992/connected-components-of-the-boundary-of-an-open-subset Comment by Carl Offner Carl Offner 2012-06-06T22:16:50Z 2012-06-06T22:16:50Z Suppose $f$ maps $x$ to the norm of $x$? (And you could make $f$ as smooth as you want, really.) http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/28630#28630 Comment by Carl Offner Carl Offner 2012-02-29T02:10:53Z 2012-02-29T02:10:53Z This linguistic wrinkle has evidently occurred to a number of people. To follow on Victor Protsak's comment, Kelley's General Topology has a humorous and not-very-serious exercise on &quot;door spaces&quot; -- such a space is a topological space in which every subset is either open or closed. http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/28637#28637 Comment by Carl Offner Carl Offner 2012-02-29T02:02:41Z 2012-02-29T02:02:41Z I don't think this question has anything to do with what Whorf was suggesting. (And for that matter, I'm under the impression that what people refer to as the &quot;Whorf hypothesis&quot; is arguably rather stronger than Whorf himself would have argued for.) In any case, what's at issue here is simple confusion of terminology. That happens in any language, in any field, and has nothing to do with the idea (quite likely wrong, or at least vastly overstated) that the inherent structure of a language has some determining effect on how its speakers view the world at large. http://mathoverflow.net/questions/74841/an-example-of-a-beautiful-proof-that-would-be-accessible-at-the-high-school-level/74879#74879 Comment by Carl Offner Carl Offner 2011-12-03T02:05:20Z 2011-12-03T02:05:20Z I haven't taught high school in many years, and I never tried teaching these things there. My impression however, is that the Halting Problem, although the proof is indeed elementary, is conceptually very sophisticated. I think it's one of those things that people have to mull over for years to really get an appreciation for. Maybe some teachers, with some classes, could do this. But I bet that most high school students might be able to follow &quot;every line of the proof&quot;, but they'd see it as just a trick. I'd love to be proved wrong. Maybe the same is true for Russell's paradox. http://mathoverflow.net/questions/78129/a-brouwer-fixed-point-theorem-on-finite-sets/78133#78133 Comment by Carl Offner Carl Offner 2011-10-14T13:25:58Z 2011-10-14T13:25:58Z Yes, that was my point. As originally stated, I don't believe what you are hoping for is true. But I see that you have edited your question in the meantime, and I'm not sure what you are asking any more. http://mathoverflow.net/questions/50343/what-would-you-want-to-see-at-the-museum-of-mathematics Comment by Carl Offner Carl Offner 2011-05-02T01:28:41Z 2011-05-02T01:28:41Z I agree about the Exploratorium. It's wonderful, and it's the only good Science Museum I've ever seen. Maybe there are others, but all the others I've been to (with the exception of &quot;special-purpose&quot; ones like the Museum of Natural History in NYC) are just embarrassing. I'll bet a good Museum of Mathematics could be constructed. But it would take some really good people -- people who were both experts in mathematics and excellent teachers -- to do it. And leave out the marketeers and &quot;let's make math fun&quot; types. http://mathoverflow.net/questions/37610/demonstrating-that-rigour-is-important/37644#37644 Comment by Carl Offner Carl Offner 2011-04-28T02:58:14Z 2011-04-28T02:58:14Z It seems to me that items 3-7 are regarded by most people as &quot;monsters&quot; and as such not really worthy of serious consideration. As for items 1 and 2, I think that not only have most people not heard of them, when they do hear of them, they regard them either as jokes or don't really get the point at all. So it doesn't seem to me that these are convincing arguments for most people. (They are, to be sure, convincing arguments for me.) To some extent, I'm sure this is something that can only be appreciated by some experience. I think for instance that the Pythagorean theorem is [out of space http://mathoverflow.net/questions/49670/history-of-irrationality-results Comment by Carl Offner Carl Offner 2011-01-14T00:08:57Z 2011-01-14T00:08:57Z @Chris: I can't resist; please don't be offended; you're in good company. But what does &quot;this is just semantics&quot; mean? Is there really anything else? I used to be a compiler writer, and believe me, compiler writers find that phrase just hilarious. (I suppose what you meant was that the two concepts your professor was distinguishing were really the same in different words. And I don't mean to sidetrack the discussion. My apologies.) http://mathoverflow.net/questions/51217/computer-science-for-mathematicians/51229#51229 Comment by Carl Offner Carl Offner 2011-01-06T22:28:58Z 2011-01-06T22:28:58Z @Igor: Well, that's a different matter. I was responding to your statement that Knuth's work had &quot;very little to do with computing&quot;. Just on the face of it, that's not true at all. I in fact teach an analysis of algorithms class. I use Cormen et al. And I also point out to my class that the very phrase &quot;analysis of algorithms&quot; was coined by Knuth in his preface to Volume 1. His work is fascinating, both for the wealth of information it contains, and also for the historical insights of someone who was &quot;there at the time.&quot; http://mathoverflow.net/questions/51217/computer-science-for-mathematicians/51229#51229 Comment by Carl Offner Carl Offner 2011-01-06T02:03:31Z 2011-01-06T02:03:31Z @Igor: Well, Knuth is where I learned about trees and stacks. Both of which I've used -- a lot -- in programming. Sure, since then other books have come along. But Knuth is where a lot of that sort of stuff was first coherently organized. (The MIX business, though, I think is pretty forgettable and ignorable.) http://mathoverflow.net/questions/28788/nontrivial-theorems-with-trivial-proofs/28797#28797 Comment by Carl Offner Carl Offner 2010-07-27T22:35:25Z 2010-07-27T22:35:25Z I think these kind of sayings have been with us for a while. Here's another, similar example, having nothing to do with category theory, and which I heard attributed to George Mackey: he was supposedly lecturing, wrote something down on the board, and said it was obvious. Then he stopped, evidently realizing he didn't know why. He left the room and came back 10 minutes (?) later, and said, &quot;It is obvious. But it's not obvious that it's obvious.&quot; Did this actually happen? I don't know. But I wouldn't be surprised if this kind of story goes back to the ancient Greeks. http://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical-p/20828#20828 Comment by Carl Offner Carl Offner 2010-05-31T13:46:30Z 2010-05-31T13:46:30Z Fair enough. If you get a chance, take a look at Fremlin's statement. I think you'll like it. http://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical-p/20828#20828 Comment by Carl Offner Carl Offner 2010-05-31T02:24:45Z 2010-05-31T02:24:45Z I'm not sure exactly what you mean by additional structure, because it seems to me that expectations are pretty immediate. But I think your general point is correct. A great quote illuminating that (which due to space limitations I can't quite copy here) is in Fremlin's Measure Theory treatise -- Vol.2, the second paragraph of the introduction to Chapter 27. I think it's well worth looking at, and also very well written