Timeline for Why is the Gaussian so pervasive in mathematics?
Current License: CC BY-SA 2.5
29 events
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| May 16 at 13:49 | history | edited | Martin Sleziak |
added the tag (gaussian)
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| Feb 15, 2022 at 20:28 | comment | added | Michael Hardy | Probably answers to this question could be the content of a long book. | |
| May 15, 2021 at 14:35 | comment | added | jjcale | To "The gaussian appears as the fixed point to the Fourier transform" : This is nothing special, the eigenspaces of the fourier transform have infinte degeneracy. | |
| Apr 9, 2021 at 3:44 | comment | added | A rural reader | Simple diffusion models ... | |
| Feb 25, 2021 at 19:45 | comment | added | Ryan Budney | Do we really want to use the word "pervasive"? Dictionaries often struggle to find examples of it being used in a positive light. The answers seem to indicate that the Gaussian is not all over mathematics, but contained to some fairly specific types of mathematics. | |
| Feb 8, 2021 at 2:27 | answer | added | Tom Copeland | timeline score: 5 | |
| Nov 14, 2012 at 9:17 | answer | added | jbc | timeline score: 16 | |
| Sep 29, 2010 at 17:16 | comment | added | Vectornaut | @RBega: It's true that when the position and momentum observables are represented by the operators f(x) -> xf(x) and f(x) -> -ihf'(x) acting on L^2(R), the Gaussian shows up in the energy eigenvectors of the QHO. However, this rep isn't special! Pick some other separable Hilbert space of functions---for example, one of the "weighted Bergman spaces" defined in Sec. 3 of "Holomorphic Methods in Analysis & Mathematical Physics," by Hall---and unitary equivalence gives you a rep of the position and momentum operators on this space. In general, the QHO energy eigenvectors won't involve Gaussians. | |
| Sep 29, 2010 at 15:08 | answer | added | Alex Bloemendal | timeline score: 8 | |
| Sep 29, 2010 at 11:47 | answer | added | S. Carnahan♦ | timeline score: 13 | |
| Sep 28, 2010 at 21:29 | history | edited | Bjørn Kjos-Hanssen |
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| Sep 28, 2010 at 21:10 | vote | accept | Randy Qian | ||
| Sep 28, 2010 at 20:39 | answer | added | Terry Tao | timeline score: 62 | |
| Sep 28, 2010 at 20:32 | vote | accept | Randy Qian | ||
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| Sep 28, 2010 at 20:28 | vote | accept | Randy Qian | ||
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| Sep 28, 2010 at 20:25 | vote | accept | Randy Qian | ||
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| Sep 28, 2010 at 20:21 | comment | added | Randy Qian | Makes sense, thanks for that explantation about spheres. A thought: there seems to be some similarity between the formulation of the heat equation as the limit of squares with heat flow through its edges proportional to the difference of temperature and the dropping balls through pegs by relating balls to heat flow. Maybe instead of thinking of the ball as going one way or another when meeting a peg, we can consider its expectation of going either way which would be closer to the heat flow model. | |
| Sep 28, 2010 at 19:34 | comment | added | Mark Meckes | The factoring that appears in that volume calculation happens precisely because the standard normal distribution has independent components; the reason you can relate the normal distribution to the sphere is its rotation invariance. So that use of the Gaussian is an application of my favorite property which I mentioned above. As for the CLT and the heat equation, there certainly are deep relationships, but I don't know about "obvious" or "less heavy handed". | |
| Sep 28, 2010 at 19:28 | comment | added | Randy Qian | Thanks for all the comments and nice examples. Another one of my very favorite uses of the Gaussian is to find the volume of the n-sphere planetmath.org/… but its appearance here seems to have more to do with the calculation got "factored" than some especially deep fact about spheres. Is there some obvious relation between the CLT and the heat equation? (I know that the Fourier transform gives 2 matching ODEs in those two scenarios, but is there some less heavy handed explanation?) | |
| Sep 28, 2010 at 19:13 | answer | added | Kevin O'Bryant | timeline score: 13 | |
| Sep 28, 2010 at 17:15 | comment | added | Rbega | Another place the Gaussian shows up is in the quantum harmonic oscillator (i.e. the eigenfunctions are the product of Hermite polynomials times the Gaussian). | |
| Sep 28, 2010 at 13:01 | comment | added | Mark Meckes | The central limit theorem and the heat equation make no mention of the gaussian in their basic setups; it turns out to be the answer to very natural questions. So I think those two (on the surface completely unrelated) examples show that this phenomenon is definitely not just an artifact of techniques. In particular, there are other methods of proving the CLT or solving the heat equation besides Fourier transforms. | |
| Sep 28, 2010 at 12:11 | comment | added | Mark Meckes | There is a book "The Normal Distribution: Characterizations with Applications" by Bryc which could be taken as a large collection of probabilistic examples to add to your list. One of my favorites: the normal distribution in $\mathbb{R}^n$ is the unique (up to scaling) rotation-invariant probability measure with independent components. | |
| Sep 28, 2010 at 12:01 | comment | added | Paul Siegel | I just want to point out that the appearance of the Gaussian in Atiyah-Singer isn't so mysterious. In the McKean-Singer formula - $Index(D) = Tr_s (e^{-tD^2})$ - the Gaussian can be replaced with any smooth function which rapidly decays at infinity. The Gaussian is used because $e^{-tD^2}$ is the solution operator for a very well understood differential equation, namely the heat equation, and thus we can look up the relevant asymptotic analysis in old PDE textbooks. | |
| Sep 28, 2010 at 9:01 | answer | added | Bjørn Kjos-Hanssen | timeline score: 21 | |
| Sep 28, 2010 at 7:39 | comment | added | Suvrit | I guess, the gaussian is as "natural" as euclidean spaces---this remark is in light of the bijections between divergences and exponential families as discussed in: ideal.ece.utexas.edu/pdfs/126.pdf (see Section 3); I would, however, love to learn of deeper connections. | |
| Sep 28, 2010 at 5:35 | comment | added | Steve Huntsman | I think the appearances you mention can be sensibly grouped in two broad classes: Fourier-analytic (including the CLT, see e.g. terrytao.wordpress.com/2010/01/05/…) and heat-kernelish (incl. Atiyah-Singer). Perhaps microlocal analysis is the best bridge between these. | |
| Sep 28, 2010 at 5:13 | history | edited | Randy Qian | CC BY-SA 2.5 |
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| Sep 28, 2010 at 5:06 | history | asked | Randy Qian | CC BY-SA 2.5 |