Ben Crowell
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 Nov 22 comment Location of Archimedes' grave in Syracuse (math/archaelogy trivia) This would have been perfect for hsm.SE. Nov 19 comment Probability distribution of the distance between two random, uniformly distributed points in the unit ball see mathoverflow.net/help/on-topic -- "MathOverflow's primary goal is for users to ask and answer research level math questions, the sorts of questions you come across when you're writing or reading articles or graduate level books." This would be more appropriate for math.stackexchange, and if you've figured out what you think the integral is, you should include that in your question. Nov 19 comment Probability distribution of the distance between two random, uniformly distributed points in the unit ball Maybe I'm misunderstanding something, but isn't this not really a research-level question? It would just be some integral, wouldn't it? Nov 10 comment Intermediate value for a vector-valued function I see. But if Jordan-Brouwer was proved in 1911, and this is the only deep issue involved, it still seems a little surprising that the Poincare-Miranda conjecture wasn't proved until 1940. Nov 10 comment Intermediate value for a vector-valued function @user78588: Hmm...OK, to be more explicit, how about this. If there is a continuous curve C connecting p to the origin, such that C does not intersect the image of the surface, then the sign is positive. (We don't need to appeal to an orientation of the image surface.) Nov 10 comment Intermediate value for a vector-valued function It surprises me that it seems to have taken 57 years between the conjecture and the proof. Is there something wrong with the following elementary argument? Take the boundary of the domain and continuously contract it onto the origin O. Let h be the signed distance from p to the nearest point on the image of the contracting surface. The sign indicates whether p is inside (-) or outside (+). Once the surface is fully contracted to the origin, h is either zero, in which case O is the solution, or positive. If the latter, then by the intermediate value theorem h had a zero somewhere. Oct 26 comment How much energy will be released in the explosion when one shoots a superelastic billiard ball into a collection of still superelastic billiard balls? @JosephVanName: OK, could you edit the question to clarify that? BTW, there is nothing about $\alpha>1$ that violates Newton's laws. Oct 25 comment How much energy will be released in the explosion when one shoots a superelastic billiard ball into a collection of still superelastic billiard balls? In the statement of the problem, is the rule $\gamma\rightarrow\gamma\alpha$ supposed to apply in some fixed and arbitrarily chosen frame of reference, or in the center of mass frame? If the former, then the motivation for the question seems weak, since it would violate Galilean relativity. Oct 9 comment Geometric meaning of the black hole horizon The fundamental reason that this has to be a coincidence is that event horizons are observer-dependent. A good example is an observer with constant proper acceleration in Minkowski space. Such an observer sees an event horizon, which clearly can't be connected to any intrinsic property of the spacetime, since the Riemann tensor vanishes identically. The horizon that we customarily talk about in the case of the Schwarzschild spacetime just happens to be the one seen by an observer at null infinity. Oct 8 comment Proof of asymptotic non-flatness Re the topology, I'm not 100% sure that I'm even right, but I also can't really tell from your comment what in particular it is that you are asking for explanation of. There definitely are topological properties implied by asymptotic flatness, though. See, e.g., Wald, p. 279, where he discusses the implied topology of scri+. Another way of proving that NUTty spacetimes aren't asymptotically flat may be that they don't have the asymptotic symmetries that are implied by asymptotic flatness. See Wald p. 284ff. The singularity stretches to i0 and would break rotational symmetry. Oct 8 comment Proof of asymptotic non-flatness Re your definition of asymptotic flatness, Wald describes on p. 282 an earlier, simpler version by Penrose, which sounds like it may be what you're using...? Re the definition of $\partial M$, see p. 276; this is the result of a nontrivial construction, not just the ordinary notion of the boundary of a point-set. Oct 7 comment Proof of asymptotic non-flatness One thing that I don't quite understand about your definition of asymptotic flatness, if it's intended to be a rigorous one rather than a schematic outline, is that you refer to $\partial M$, but normally M is going to be a manifold (not a manifold-with-boundary). As far as I know, the standard definition these days is the one in Wald, ch. 11, and it's much more complicated than the one you've stated. As an example, $\Omega$'s derivatives have to behave in a certain way at $i^0$, whereas different conditions hold at scri+ and scri-. Oct 7 comment Proof of asymptotic non-flatness I think the idea is that a spacetime with a NUT parameter always has one or more semi-infinite wire-like singularities (which I think can be either a curvature or a non-curvature singularity, depending on the specific example). Since this singularity extends to infinite distances, clearly the spacetime doesn't look like Minkowski space far away. As far as formal proof, isn't the topology just wrong for an asymptotically flat space? Since the singularity is like a line removed from the manifold, you should have non-contractible curves going around it. Oct 7 comment Are there any standard analysis facts that can be proven or arrived only by means of non-archimedean extensions of reals and non-standard analysis? This sounds to me like it might be overstating the reduction in complexity achieved with NSA. I think in most cases what is achieved is something like eliminating one layer of quantifiers. Oct 1 comment Why should we believe in the axiom of regularity? We could turn this around and ask whether there is any reason to disbelieve the axiom of regularity. A strong reason to disbelieve it would be if there were some part of mathematics (group theory, differential geometry, number theory, ...) in which we found that it was inconvenient to be restricted to the kind of sets that are well-founded. Sep 21 comment Why is the exterior algebra so ubiquitous? Grassmann's argument is really about orientation. The basic insight is that you can make geometry simpler (cut down on case splitting, etc.) by considering oriented quantities rather than unoriented ones. Sep 17 comment Dynamics of electrons on a sphere @NoamD.Elkies: non-unique solutions can happen but only when the system starts at a singularity or reaches a singularity I don't think this is true, since Norton's dome would seem to be a counterexample. It certainly doesn't have any singularity of the particular type you describe. Or did you have in mind some definition of singularity more general than the example you gave? If so, what is it -- a singularity in what function? The indeterminacy of Norton's dome is usually described not as an issue involving a singularity but as one involving a lack of Lipschitz continuity. Sep 17 comment Dynamics of electrons on a sphere This seems closely analogous to packing of hard spheres, where it is known that random packing is always worse than the optimal packing. So I think it's very unlikely that such a system achieves the global minimum in all cases. Sep 17 comment Dynamics of electrons on a sphere @NoamD.Elkies: The answer to Q1 must be Yes, because the differential equation has a unique solution so it must retain the initial symmetry. I don't think this really constitutes a proof, since there are counterexamples such as the famous Norton's dome: en.wikipedia.org/wiki/Norton%27s_dome . The problem is basically that solutions to the equations of motion in Newtonian mechanics may be nonunique, i.e., Newtonian mechanics is not actually deterministic in all cases. Sep 17 comment electron configuration on manifolds Another issue you might want to clarify is the dimensionality $n$ of M. If $n\ne3$, then retaining the form of Gauss's law is not consistent with a $-2$ exponent of the Coulomb force law (at short distances -- it doesn't make sense to talk about such a force law at long distances in a curved space). And are you imagining M as having only intrinsic structure, or as being embedded in a higher-dimensional space? If the latter, then Earnshaw's theorem doesn't apply.