most recent 30 from http://mathoverflow.net 2013-06-20T09:04:25Z http://mathoverflow.net/feeds/user/19334 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110327/what-time-does-it-take-for-irrational-rotations-to-hit-an-interval Antoine Levitt 2012-10-22T12:50:28Z 2012-10-29T03:04:33Z

<p>Hi,</p> <p>Consider $\theta_n = (\theta_0 + n \theta) \mod 1$, $\theta$ being an irrational number, and $\theta_0$ an uniform random variable in $(0,1)$. Is there any estimates for the time it will take this process to hit $(0,\alpha)$ ? From the ergodic theorem I know that, if I denote $N(n)$ the number of times $\theta_n \in (0,\alpha)$, then $N(n)/n \to \alpha$. What I want to know is how much time it will take for this limit to be attained.</p> <p>Another way of framing this question is : is there any "central limit theorem" (or weakening thereof ; I'm mainly interested in guaranteed bounds for $P(N\geq 1)$) for ergodic processes? From what I've read, there is no general answer to this for a generic ergodic process and function f. There are some results that depend on $f$ being smooth, which it isn't here.</p> <p>The same question was asked on <a href="http://mathoverflow.net/questions/4411/quantitative-versions-of-ergodic-theorem" rel="nofollow">http://mathoverflow.net/questions/4411/quantitative-versions-of-ergodic-theorem</a>, but I haven't found anything there that relates to my question.</p>

http://mathoverflow.net/questions/81172/algorithm-for-the-smallest-algebraic-eigenvalues-of-a-symmetric-sparse-matrix Antoine Levitt 2011-11-17T13:26:14Z 2012-06-15T05:22:00Z

<p>Hi,</p> <p>I'm looking for a way to get the negative eigenspace of a large (sparse) symmetric matrix. This matrix is basically a discretized version of the operator $-\Delta + V$, $V$ negative, on some domain $[-L,L]$ with Dirichlet BC, so its spectrum consists of a few negative eigenvalues (which I want to find), and a lot of positive ones (whose distribution is roughly known). </p> <p>The way I currently do it is to use the shift-invert mode of ARPACK (so Lanczos), with a negative shift and 'LM' mode (lowest magnitude). This requires me to choose a good shift: too large a shift might miss negative eigenvalues, and too small a shift leads to slow convergence. The 'LA' mode (lowest algebraic) is just not an option, it's too slow/imprecise.</p> <p>Is there any better method out there?</p>

http://mathoverflow.net/questions/90440/extrapolation-on-the-p-norm-sphere-using-the-exponential-map Antoine Levitt 2012-03-07T10:06:02Z 2012-03-07T10:35:25Z

<p>Hi,</p> <p>In order to follow a branch of solutions to an implicit equation on the manifold $M = \lbrace x \in \mathbb R^n, \|x\|_p = 1 \rbrace $, I'm interested in the following problem. Given two points $a$ and $b$ in $M$, is there a natural way to extrapolate them to a third point $c$, as in the flat case $c = 2b - a$ ? </p> <p>In particular, I'm trying the following strategy. Find v such that $\exp_b(-v) = a$, and compute $c$ as $\exp_b(v) = a$, where $\exp_b$ is the exponential map on $T_bM$. This should be a well-posed problem for $b$ and $a$ close enough, but my knowledge of differential geometry is very limited, and I do not know if it possible to solve this problem in closed form for general $p$. I expect it only makes sense for $p \geq 2$, but I'd be happy to be proved wrong. I have solved the case $p=2$ using the formula from <a href="http://www.math.duke.edu/~bryant/267/Day12Exercises.pdf" rel="nofollow">http://www.math.duke.edu/~bryant/267/Day12Exercises.pdf</a> (it reduces to simple geometry)</p> <p>If the inverse problem (logarithm map) is too hard to solve, I'm still interested in the direct problem (exponential map), because if I have an explicit derivative of the curve available, I can use that as $v$. Of course, a trivial strategy is to extrapolate linearly the curve and then project back, but I feel that's clumsy.</p>

http://mathoverflow.net/questions/82582/integrating-a-differential-functional-equation/82607#82607 Antoine Levitt 2011-12-04T09:35:13Z 2011-12-04T09:35:13Z

<p>Huh, that's one strange equation. Where does it come from? Are you sure there's not a simple form? (in particular, one you can differentiate to give a simple differential equation) Anyway, look up litterature for integro-differential equation. Or just bruteforce it with an Euler method and piecewise constant integration, if you don't need high precision.</p>

http://mathoverflow.net/questions/75925/do-you-know-this-form-of-an-uncertainty-principle/81462#81462 Antoine Levitt 2011-11-20T22:20:44Z 2011-11-20T22:20:44Z

<p>I find the neatest "standard" uncertainty principle is the one with commutators, see e.g. <a href="http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/GenUncertPrinciple.htm" rel="nofollow">http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/GenUncertPrinciple.htm</a>. I think that readily gives both your inequalities.</p>

http://mathoverflow.net/questions/81259/on-the-periods-in-the-periodic-table-or-why-is-a-noble-gas-stable/81272#81272 Antoine Levitt 2011-11-18T19:01:29Z 2011-11-18T19:01:29Z

<p>The question is probably not appropriate for MO, but it's interesting nevertheless. I think (but have no proper reference nor physical knowledge for this) the reason for these different shells is that, loosely speaking, the inner electrons screen the potential, such that the outer electrons see an effective potential of charge Z - Ninner. This shifts the eigenvalues, and accordingly the order in which the shells are filled. This is probably discussed in quantum chemistry books, although I haven't found anything satisfactory yet.</p> <p>The appropriate theory to discuss this is Hartree-Fock.</p>

http://mathoverflow.net/questions/81250/functional-minimization-when-is-this-heuristic-rigorous/81254#81254 Antoine Levitt 2011-11-18T15:14:15Z 2011-11-18T15:14:15Z

<p>This is pretty easy : find out an appropriate Banach space for f (the one that makes your integrals well-defined; usually, some kind of Sobolev space), prove that h is C^1 (with respect to differentiation in Banach spaces), and then standard arguments apply. Keywords you might want to look up are: differentiation in Banach spaces, weak solutions, Sobolev spaces.</p> <p>For instance, if $h(f) = \int f^2 + f'^2$, then $h$ is $C^1$ as a functional in $H^1$, and the solution is a (weak) solution to $f'' + f = 0$</p>

http://mathoverflow.net/questions/80146/is-the-mendeleev-table-explained-in-quantum-mechanics/81210#81210 Antoine Levitt 2011-11-17T23:16:52Z 2011-11-17T23:16:52Z

<p>I'm arriving after the war, but this is an interesting question, so I'm going to write up what I understand about it.</p> <p>First of all, for a comprehensive mathematical understanding of the periodic table, you have to settle on a model. The relevant one here is quantum mechanics (for large atoms, relativistic effects start to become important, and that's a whole mess). It's entirely axiomatic, and requires no further tweaking. Then you basically have to solve an eigenvalue on a space of functions of $6N$ coordinates (ignoring spin). That gives you a "mathematical explanation" of the table, in the sense that knowledge of the solution $\psi(x_1,x_2,\dots,x_N)$ is all there is to know about the static structure of an atom. Notice that in this formulation, all electrons are tied together inside one big wavefunctions, so an "electronic state" has no meaning. Mendeleev table is not even compatible with this formulation.</p> <p>Of course, solving the full eigenproblem is not possible, so all you can do is mess around with approximations. A simplistic but illuminating approximation is to completely neglect electron repulsion. Great simplification occurs, and it turns out one can speak of "electronic states". Non-trivial behaviour occurs because of the Pauli exclusion principle. This is known as the "Aufbau" principle: one builds atoms by successively adding electrons. The first electron gets itself into the lowest energy shell, then the second one gets into the same state, but with opposite spin. The third begins to fill the second shell (which has three spaces, times two because of spin), and so on. This is the basic idea behind the table, and provides a clue as to why it is organised the way it is. So this might be the theory you're looking for. It's explicitely solvable, and only requires the theory of the hydrogenoid atoms.</p> <p>Of course, because of the approximations, the quantitative results are all wrong, but the organisation is still there. Except for larger elements, where the Mendeleev table is, from what I understand, an ad-hoc hack. You can improve the approximation using ideas like "screening", and this leads to the Hartree-Fock method, which still preserves the notion of shells.</p> <p>Hope that helps. Then again, if you're looking for a completely logical approach to physics that'll readily explain real life, you're bound to be disappointed. Even simple theories such as the quantum mechanics of atoms are too hard to be solved exactly, which is why we have to compromise and make approximations.</p>

http://mathoverflow.net/questions/110327/what-time-does-it-take-for-irrational-rotations-to-hit-an-interval/110958#110958 Antoine Levitt 2012-10-30T08:01:19Z 2012-10-30T08:01:19Z

That's amazing. I'm accepting that as an answer, thanks! The reason I'm interested in this is that I've got some numbers from a simulation that I'm trying to explain. This is exactly what I was looking for. I'll try and fit this Cauchy distribution, and see where that takes me.

http://mathoverflow.net/questions/110327/what-time-does-it-take-for-irrational-rotations-to-hit-an-interval Antoine Levitt 2012-10-22T17:31:05Z 2012-10-22T17:31:05Z

Yes, but this is to get strong bounds, ie bounds on $N(n)$ uniform in $\theta_0$. I'm still hoping for faster rates on hitting times averages.

http://mathoverflow.net/questions/110327/what-time-does-it-take-for-irrational-rotations-to-hit-an-interval Antoine Levitt 2012-10-22T15:08:37Z 2012-10-22T15:08:37Z

Oh, of course, sorry, how stupid of me. I'm a bit lost in this maze of theorems, but that does imply an upper bound on the hitting time, independent on $\theta_0$. The downside is that this bound depends on the diophantine approximation of $\alpha$. Even in what seems to be the most favorable case of $D_N = O(log N / N)$, I get lower bounds which are solutions of $\alpha n = log n$, and so grow (a bit) faster than $1/\alpha$. I was hoping for hitting times on the order of $1/\alpha$, but hey, that's life. Maybe other methods can do better though. Thanks!

http://mathoverflow.net/questions/110327/what-time-does-it-take-for-irrational-rotations-to-hit-an-interval/110329#110329 Antoine Levitt 2012-10-22T14:20:30Z 2012-10-22T14:20:30Z

Thanks a lot! The question seems harder than I thought! The limit $\alpha \to 0$ is precisely the case I'm interested in. If I understand correctly the paper, it says that when $\varepsilon \to 0$, one can find subsets of nonzero measure whose hitting time arbitrarily exceeds the expected return time $1/\varepsilon_n$. That's unsettling, but fair enough. What about the case $\varepsilon$ fixed? I did not know about Kac's Lemma, is it invalid outside $J_\varepsilon$? Ie is $\int_{S^1} \tau(\omega) d\omega \neq 1/\varepsilon? If so, what's it equal to? This would be an answer to my question.

http://mathoverflow.net/questions/110327/what-time-does-it-take-for-irrational-rotations-to-hit-an-interval Antoine Levitt 2012-10-22T13:47:09Z 2012-10-22T13:47:09Z

Thanks! This is very interesting. If I understand correctly, the strategy is to use the Koksma–Hlawka inequality. This fails in my case because $f$ is an indicator function, which is not BV.

http://mathoverflow.net/questions/81172/algorithm-for-the-smallest-algebraic-eigenvalues-of-a-symmetric-sparse-matrix Antoine Levitt 2012-10-19T22:00:44Z 2012-10-19T22:00:44Z

Unfortunately, no. I just used a well-chosen shift.

http://mathoverflow.net/questions/83418/asymptotics-of-fourier-coefficients-of-power-type-functions Antoine Levitt 2011-12-14T16:08:33Z 2011-12-14T16:08:33Z

There's a lot of sites like that, usually used for pasting code. I didn't actually know about mathurl, it's pretty neat!

http://mathoverflow.net/questions/83418/asymptotics-of-fourier-coefficients-of-power-type-functions Antoine Levitt 2011-12-14T16:02:28Z 2011-12-14T16:02:28Z

Actually, this asymptotic behaviour is quite easy to prove using a simple change of variable. It doesn't give the full asymptotics and constants though, see Igor Rivin's answer below for that.

http://mathoverflow.net/questions/83418/asymptotics-of-fourier-coefficients-of-power-type-functions Antoine Levitt 2011-12-14T15:50:13Z 2011-12-14T15:50:13Z

Just did some crude numerics on that for fun, result is that $\hat f(n) \approx n^{-(1 - alpha)}$. Code (python/numpy) is at <a href="http://pastebin.com/rL5QNMnv" rel="nofollow">pastebin.com/rL5QNMnv</a> if you want to take a look. I'm interested in the proof.

http://mathoverflow.net/questions/83182/on-the-convergence-of-a-special-fixed-point-iteration Antoine Levitt 2011-12-11T19:41:15Z 2011-12-11T19:41:15Z

In general, the way to prove such statement is with the Banach fixed point theorem. Have you tried it?

http://mathoverflow.net/questions/83019/fourier-transform-of-a-differential-operator Antoine Levitt 2011-12-09T11:55:22Z 2011-12-09T11:55:22Z

Also, your operator looks, modulo rotation in the x-y plane, like a fractional laplacian, and there's quite a bit of litterature on that (also lookup sobolev space with non-integer exponent on wikipedia).

http://mathoverflow.net/questions/83019/fourier-transform-of-a-differential-operator Antoine Levitt 2011-12-09T11:08:12Z 2011-12-09T11:08:12Z

This is not a multiplication, but an operator application. When you differentiate a function, you can either differentiate it in real space, or multiply by $i\omega$ its Fourier transform. Same applies here.

http://mathoverflow.net/questions/83019/fourier-transform-of-a-differential-operator Antoine Levitt 2011-12-09T10:26:02Z 2011-12-09T10:26:02Z

Making sense of this requires the theory of tempered distributions. Assuming everything makes sense, the Fourier transform of the constant function is the dirac distribution, so your guess is correct.

http://mathoverflow.net/questions/82739/approximation-algorithmic-considerations Antoine Levitt 2011-12-06T06:36:12Z 2011-12-06T06:36:12Z

I didn't understand what you wrote about the singularity. Assume $a = 0$, $f(x) = 1/x + x + o(x)$ at 0. Then, define $g(x) = f(x) - 1/x$. $g$ isn't singular anymore, so you can interpolate it, and reconstruct f from it. I ask about the cost because I'm not convinced it's worth the bother to go higher up in the approximation scheme. Have you tried a simple linear interpolation? Why is it unsuitable? You haven't answered the question of your end goal.

http://mathoverflow.net/questions/82739/approximation-algorithmic-considerations Antoine Levitt 2011-12-05T22:46:03Z 2011-12-05T22:46:03Z

Just a thought - instead of splitting your interval to use your asymptotic expansion, can't you interpolate f - f_a - f_b, where f_a and f_b are your expansions at the endpoints? That way you get a non-singular problem. In the end I guess it depends what your end goal is - do you want to integrate f? Solve a BVP that depends on f ? If your f is simple enough, I'd suggest a trivial linear interpolation, based on evaluation of f at each point of a grid. About your last remark, presumably, it isn't cheaper to compute $f^(n)(x_i)$ than $f(x_i)$, is it?