User gregory putzel - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T19:14:51Z http://mathoverflow.net/feeds/user/3291 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22940/integral-involving-exponential-of-fractional-power Integral involving exponential of fractional power Gregory Putzel 2010-04-29T05:06:16Z 2010-12-04T19:48:54Z <p>Can anything be said about the Fourier integral</p> <p>$\int_{-\infty}^{\infty} \exp\left[ika - (\gamma + ik)^{2/3}\right]dk$</p> <p>where $a > 0$ and $\gamma > 0$?</p> <p>Can it be related to some special function? It appears in the physics application described <A HREF="http://mathoverflow.net/questions/22281/can-i-relate-the-l1-norm-of-a-function-to-its-fourier-expansion" rel="nofollow">in this MO question</A>.</p> http://mathoverflow.net/questions/22281/can-i-relate-the-l1-norm-of-a-function-to-its-fourier-expansion Can I relate the L1 norm of a function to its Fourier expansion? Gregory Putzel 2010-04-23T00:28:23Z 2010-04-26T15:41:07Z <p>I would like to express the integral of the absolute value of a real-valued function $f$ (over a finite interval) in terms of the Fourier coefficients of $f$. Failing that, I would like to know of any constraints or statistical correlations (in a sense explained in the motivation) relating these quantities.</p> <p>Motivation: This comes from a biophysics application, but is perhaps best explained as follows. If a rubber band of tension $t$ is stretched along the $x$ axis from $0$ to $L$, then it is easy to calculate the thermal fluctuations of its arc-length by letting $z(x)$ be the (small) deviation from the $x$ axis, and then writing the energy (arc-length times tension) in terms of the Fourier coefficients of $z(x)$. The Boltzmann weight turns out to be a Gaussian since in the limit of small deviations the arc-length becomes a sum of squares of the Fourier coefficients. My problem is more complicated: We have two rubber bands stretched over the same interval, with deviations $z_{1}(x)$ and $z_{2}(x)$. The energy includes not only the stretching of the rubber bands, but also a term proportional to the (positive) area enclosed between them, which is</p> <p>$\int_{0}^{L}|z_{1}(x) - z_{2}(x)|dx$</p> <p>Hence my question. So it would be nice to know how this area can be related to the Fourier coefficients of $z_{1}$ and $z_{2}$ or perhaps just to the arc-lengths of the rubber bands. By "statistical correlations" I am referring to the Boltzmann probability distribution with energy equal to the stretching energy plus the area-energy.</p> <p>Edit: Specifics on the Boltzmann probability distribution, more motivation.</p> <p>The state of the system is the pair of functions $z_{1}(x)$ and $z_{2}(x)$ describing the deviation of the two rubber bands from the x axis. Let's say it's the set of pairs of functions defined on [0,L] and that these functions are identified with a finite number of Fourier coefficients - I am a physicist and would like to avoid nasty functions or mathematically honest discussions of path integrals.</p> <p>The probability of occurrence of a state (z_{1}, z_{2}) is (before normalization)</p> <p>$\exp(-\beta E\left[z_{1},z_{2}\right])$</p> <p>where $E\left[z_{1},z_{2}\right]$ is the energy of the system, which in this case is the functional</p> <p>$E\left[z_{1},z_{2}\right] = \frac{t}{2}\int_{0}^{L}\left[(\frac{dz_{1}}{dx})^{2}+(\frac{dz_{2}}{dx})^{2}\right]dx + \kappa \int_{0}^{L}|z_{1}(x)-z_{2}(x)|dx$</p> <p>Where the tension t and the "surface tension" $\kappa$ are just numbers; set them equal to 1 if you wish. The first integral is the energy cost of stretching the rubber bands (in a linearized regime) and the second is the strange term proportional to the area enclosed between them. Without the second term, it is easy to diagonalize this functional in terms of the Fourier series of the two functions. That is why I was interested in writing the second term in terms of Fourier coefficients. That may be too much to ask, but perhaps it is still possible to calculate some quantities such as the statistical average of $z_1(x)^{2}$ - that is the kind of thing I ultimately want to know.</p> <p>I realize this is an unnatural-looking problem, so I will just mention that it's not really about rubber bands, but rather about fluctuating interfaces which occur in lipid bilayers with coexisting phases. There are two phase boundaries (one for each monolayer) with their respective "tensions" but there is also a term proportional to the area between them.</p> http://mathoverflow.net/questions/25054/different-ways-of-thinking-about-the-derivative Comment by Gregory Putzel Gregory Putzel 2012-08-02T02:49:51Z 2012-08-02T02:49:51Z Trivial point: In (3), one of the deltas should be an epsilon. http://mathoverflow.net/questions/50968/what-do-americans-mean-by-calculus Comment by Gregory Putzel Gregory Putzel 2011-01-02T23:45:07Z 2011-01-02T23:45:07Z We mean differential and integral analysis. As far as I know, everyone considers it to be part of mathematics. http://mathoverflow.net/questions/48671/examples-of-non-rigorous-but-efficient-mathematical-methods-in-physics Comment by Gregory Putzel Gregory Putzel 2010-12-09T03:53:01Z 2010-12-09T03:53:01Z How about: the use of the renormalization 'group' (including perturbative renormalization group calculations using the epsilon expansion), in particular in the study of self-avoiding walks. It seems that very much is &quot;known&quot; by physicists which is not yet proven by mathematicians. I would love to see a complete discussion about this from both sides. What is the mathematical perspective on these methods? I'm not qualified to make an answer out of this. http://mathoverflow.net/questions/23881/voyager-2-relative-velocity Comment by Gregory Putzel Gregory Putzel 2010-05-07T18:33:05Z 2010-05-07T18:33:05Z 25 km/sec is not anywhere near 8% of the speed of light. It is more like 0.008% http://mathoverflow.net/questions/22281/can-i-relate-the-l1-norm-of-a-function-to-its-fourier-expansion/22511#22511 Comment by Gregory Putzel Gregory Putzel 2010-04-25T22:11:13Z 2010-04-25T22:11:13Z Oh I see, you just posted the sign-gordon question. Curious to see what people can say... http://mathoverflow.net/questions/22281/can-i-relate-the-l1-norm-of-a-function-to-its-fourier-expansion/22511#22511 Comment by Gregory Putzel Gregory Putzel 2010-04-25T22:09:46Z 2010-04-25T22:09:46Z Thanks for the comments and the SIGN-Gordon link. I am going to start working out the details using the Schrodinger eqn/ Airy function approach. Thinking about it now, it is very similar to what is commonly done in the mean-field theory of Gaussian polymers. What I'm looking for is the free energy of a Gaussian polymer in one dimension, in an external V-shaped potential. Yes, I'm pretty sure the V-potential is what I want, since upon integration along x it gives the area between the two interfaces. http://mathoverflow.net/questions/22281/can-i-relate-the-l1-norm-of-a-function-to-its-fourier-expansion/22367#22367 Comment by Gregory Putzel Gregory Putzel 2010-04-24T04:51:17Z 2010-04-24T04:51:17Z That last comment is about the statistical problem, by the way. http://mathoverflow.net/questions/22281/can-i-relate-the-l1-norm-of-a-function-to-its-fourier-expansion/22367#22367 Comment by Gregory Putzel Gregory Putzel 2010-04-24T04:35:55Z 2010-04-24T04:35:55Z I walked down the hall to where the high-energy physicists are and my friend pointed out that this is probably equivalent to the path-integral formulation of a single quantum mechanical particle in a V-shaped potential - which can be solved analytically in terms of Airy functions. I think he is exactly right and the averages can be worked out that way. So the specific question about L_1 norms is probably misguided (but prompted an interesting connection above), while the statistical problem which seemed more difficult to me might be surprisingly tractable analytically... http://mathoverflow.net/questions/22281/can-i-relate-the-l1-norm-of-a-function-to-its-fourier-expansion Comment by Gregory Putzel Gregory Putzel 2010-04-23T16:10:25Z 2010-04-23T16:10:25Z Yes, I'll add that to the question.