Timeline for Why is the Gaussian so pervasive in mathematics?
Current License: CC BY-SA 4.0
35 events
| when toggle format | what | by | license | comment | |
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| Apr 18 at 3:25 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Sign error corrected
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| Aug 27, 2023 at 11:10 | comment | added | Tom Copeland | On matched pairings, the Gaussian, and the CLT: Terence Tao's central limit theorem, Double Factorials (!!) and the Moment Method (youtube.com/watch?v=oPQ4mNcqY7k&t=665s) by Nica. | |
| Feb 15, 2022 at 22:10 | comment | added | Tom Copeland | @MichaelHardy, I've done that for more important equations in other writings. Feel free to edit to your tastes as long as it doesn't change the content. I think I've invested quite a lot of time and effort on specifics in this answer to be forgiven for not going the whole ten yards on the formatting (28 taps vs. 4), and other unfinished notes beckon. | |
| Feb 15, 2022 at 21:42 | comment | added | Michael Hardy | @TomCopeland : If you just google "latex symbols" you can see at once how to write $$\langle t-\langle t\rangle\rangle^2 $$ instead of $$<t-<t>>^2. $$ | |
| Feb 15, 2022 at 19:09 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Intro / summary of assocations added
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| Feb 15, 2022 at 12:50 | comment | added | Tom Copeland | Ghys and Ranicki, in "Signatures in algebra, topology and dynamics", use the expression 'quadratic form' 51 times whereas 'Gaussian' is used only once in 'Gaussian elimination'. Functions, $f(z)$, are extremely important, of course, in mathematics also, but that doesn't explain the interpretation and utility of $e^{f(z)}$. | |
| May 31, 2021 at 22:06 | comment | added | Tom Copeland | "Gaussian processes and Feynman diagrams" by William G. Faris, related to the general $Z = e^F$ (OEIX A036040) enumerating number of disconnected objects formed from connected ones, in this case a single graph with one edge with the exponential generating function $F = x^2/2$. The cumulant expansion formula A127671 (cf. A263634 for relation to the raising op for Appell polynomials) gives the inverse relation--connected from disconnected--mentioned in the answer. | |
| May 31, 2021 at 18:14 | comment | added | Tom Copeland | More on the associations among the Heisenberg-Weyl algebra, matchings on simplices, and Gaussian distributions soon to come at oeis.org/A344678. | |
| May 30, 2021 at 18:18 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Introduced a combinatorial perspective
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| May 23, 2021 at 17:21 | comment | added | Tom Copeland | For more on the heat/diffusion equation, small oscillations, and quadratic relations, see Arnold's "Lectures on Partial Differential Equations." | |
| May 23, 2021 at 2:08 | comment | added | Tom Copeland | For a discussion of integral transforms with general quadratic equations in an exponential, see "On self-reciprocal functions under a class of integral transforms" by Kurt Bernardo Wolf. | |
| May 19, 2021 at 1:48 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Corrected variable
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| May 17, 2021 at 19:54 | comment | added | Tom Copeland | See also "On Riemann's zeta function" by Bump and Ng (eudml.org/doc/173728) and the closely related "Binomial Polynomials Mimicking Riemann’s Zeta Function" by Coffey and Lettington (arxiv.org/pdf/1703.09251.pdf). | |
| May 15, 2021 at 0:30 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Added generalization
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| May 15, 2021 at 0:23 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Added generalization
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| Apr 9, 2021 at 3:13 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Misplaced t removed.
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| Apr 8, 2021 at 13:02 | comment | added | Tom Copeland | Related: Gauss-Markov theorem. | |
| Apr 1, 2021 at 11:35 | comment | added | Tom Copeland | See also the Gauss-Weierstrass operator/transform en.m.wikipedia.org/wiki/Weierstrass_transform | |
| Mar 16, 2021 at 23:01 | comment | added | Tom Copeland | Some researchers are exploring extended Heisenberg algebras and categories, such as Khovanov in "Heisenberg algebra and a graphical calculus" (arxiv.org/abs/1009.3295) and Brunden, Savage, and Webster (arxiv.org/abs/2007.01642). For more Hermite integrals and uniqueness of the Hermite polynomials as the only orthogonal Appell family, see Anshelevich see-math.math.tamu.edu/~manshel/papers/Hermite-integrals.pdf. | |
| Mar 14, 2021 at 20:57 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Changed to more accessible link (with additional interesting articles by Hersh).
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| Mar 1, 2021 at 21:26 | comment | added | Tom Copeland | See also "Adelic harmonic oscillator" by Branko Dragovicha arxiv.org/abs/hep-th/0404160 | |
| Feb 28, 2021 at 19:15 | comment | added | Tom Copeland | On relation to the Heisenberg uncertainty relation, see this Oxford lecture m.youtube.com/watch?v=cz0-HcAtU5U | |
| Feb 26, 2021 at 7:32 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Corrected variable and a link
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| Feb 25, 2021 at 19:13 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Revamped
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| Feb 14, 2021 at 17:36 | comment | added | Tom Copeland | In another MO-Q Nik Weaver alludes to "Quantum Mechanics in Rigged Hilbert Space Language" by Rafael de la Madrid Modino--a good intro to the Hermite polynomials and the quantum harmonic oscillator and various spaces associated with them. | |
| Feb 12, 2021 at 1:01 | comment | added | Tom Copeland | @AbdelmalekAbdesselam, link to your notes? I found "Hermite expansions of some tempered distributions" by Chihara et al., which refs Simon and also Kagawa. Expansion of tempered distributions works as well for the Laguerre polynomials (see mathoverflow.net/questions/382735/…) and should also for similar orthogonal polynomial sequences. | |
| Feb 12, 2021 at 0:40 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Added the classic Cartier paper
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| Feb 12, 2021 at 0:17 | comment | added | Tom Copeland | @AbdelmalekAbdesselam, thanks. Bet you can find the germ of the idea in the convolutional soln. to the Cauchy problem mentioned above as you can for the Schwartz distributions in 19th century classic potential theory as sketched in mathoverflow.net/questions/127601/…. Sato runs with this. Would be interesting to find a good monograph comparing, contrasting the theories and elucidating the historical connections. | |
| Feb 11, 2021 at 23:43 | comment | added | Abdelmalek Abdesselam | BTW, Treves in his book on distributions does not mention this. Other authors credit Reed-Simon, but this was well known to Schwartz and Grothendieck (mentioned in Schwartz's book on distributions and A.G. in his thesis in Memoirs of AMS). I don't know who first discovered the result. Wiener? | |
| Feb 11, 2021 at 23:37 | comment | added | Abdelmalek Abdesselam | Don't know about nascent delta. For the Schauder basis, I have some notes from a course I taught. Otherwise there is aip.scitation.org/doi/abs/10.1063/… but not well written, it's also in Reed Simon Vol 1, not better, most of the proof punted in the exercises. Simon's new book on analysis is much better. | |
| Feb 11, 2021 at 23:29 | comment | added | Tom Copeland | @AbdelmalekAbdesselam, refs? Connected to the use of the Gaussian as a nascent Dirac delta and the Cauchy problem, similar to Sato's theory of hyperfunctions? | |
| Feb 11, 2021 at 22:49 | comment | added | Abdelmalek Abdesselam | Also good to know: the Hermite functions give a Schauder basis for the Schwartz space $\mathscr{S}(\mathbb{R})$ and therefore, and more importantly, also for the space of tempered distributions $\mathscr{S}'(\mathbb{R})$. They essentially trivialize lots (but not all) of functional analytic questions about these spaces. | |
| Feb 11, 2021 at 20:18 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Provided links, another ref to CM systems
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| Feb 8, 2021 at 21:29 | history | edited | Tom Copeland | CC BY-SA 4.0 |
Defined D
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| Feb 8, 2021 at 2:27 | history | answered | Tom Copeland | CC BY-SA 4.0 |