Timeline for On $e^{\pi\sqrt{4\cdot163}}$ and unusual connections
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Jan 23, 2015 at 5:48 | vote | accept | Tito Piezas III | ||
| Jan 14, 2015 at 2:28 | answer | added | S. Carnahan♦ | timeline score: 5 | |
| Jan 7, 2015 at 19:10 | comment | added | Tito Piezas III | @AaronBergman: I've clarified it in my answer/comment below. mathoverflow.net/a/193394/12905 | |
| Jan 7, 2015 at 19:08 | answer | added | Tito Piezas III | timeline score: 3 | |
| Jan 7, 2015 at 13:43 | comment | added | Aaron Bergman | Can you clarify what is the question that you want answered? Is it the mathematical derivation of the approximation? Do you want something beyond what is on the subsequent pages of Witten's paper? Or do you want to understand the physics behind the relation? If so, do you have a specific question beyond or about what's in the paper? | |
| Jan 7, 2015 at 1:32 | comment | added | José Figueroa-O'Farrill | @TitoPiezasIII: I agree. My point is simply that it was not claimed that the k=1 result was a good approximation. | |
| Jan 6, 2015 at 23:19 | comment | added | Tito Piezas III | @JoséFigueroa-O'Farrill: And to continue that quote in p. 32, "...Agreement improves rapidly if one increases k. For example, at k = 4..." While the paper did not said it was good, the rest of the quote implies it was just a first approximation, in a family of progressively better approximations. (As the post shows.) | |
| Jan 6, 2015 at 22:26 | answer | added | j.c. | timeline score: 2 | |
| Jan 6, 2015 at 20:47 | comment | added | José Figueroa-O'Farrill | @GerryMyerson: You are right: it's not a good approximation and, in fact, the original paper does not say it is. Quoting from the paper (p. 32): "It is illuminating to compare the number $196883$ to the Bekenstein-Hawking formula. An exact quantum degeneracy of $196883$ corresponds to an entropy of $\log 196883 \approx 12.19$. By contrast, the Bekenstein- Hawking entropy at $k = 1$ and $L_0 = 1$ is $4π \approx 12.57$. We should not expect perfect agreement, because the Bekenstein-Hawking formula is derived in a semiclassical approximation which is valid for large $k$." | |
| Jan 6, 2015 at 19:09 | comment | added | jjcale | for Bekenstein-Hawking entropy see en.wikipedia.org/wiki/Black_hole_thermodynamics | |
| Jan 6, 2015 at 18:52 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
OEIS link.
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| Jan 6, 2015 at 18:01 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Missed one number.
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| Jan 6, 2015 at 16:45 | comment | added | Tito Piezas III | @GerryMyerson: They define a function they call $Z_k(q)$. That was just $k=1$. To quote Witten's paper in page 32, "...agreement improves rapidly if one increases $k$." For $k=4$, they used the first coefficient of the q-expansion as, $$\log(81026609428)\approx 25.12,\quad 8\pi \approx. 25.13$$ | |
| Jan 6, 2015 at 15:05 | comment | added | Gerry Myerson | 12.19 is approximately 12.56? I'm not sure why such a crummy approximation would be mentioned anywhere. | |
| Jan 6, 2015 at 7:07 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Small typo, title.
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| Jan 6, 2015 at 6:22 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Tag; details
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| Jan 6, 2015 at 6:09 | history | asked | Tito Piezas III | CC BY-SA 3.0 |