Timeline for How to compute the asymptotic of a summation which involves binomial coefficients?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Jun 16, 2020 at 10:04 | comment | added | Jianrong Li | @Johannes Trost, thank you very much for your help last time. If $v_1, v_2$ have large overlap, how do you compute asymptotic of ${|v_1| - O(\min(|v_1|, |v_2|)^v) \choose x_1 - d}$. Is it possible to show that $f(v_1, v_2)<1$ when $|v_1|, |v_2|$ are large and $v_1, v_2$ have large overlap? | |
| Aug 5, 2018 at 19:02 | history | edited | Jianrong Li | CC BY-SA 4.0 |
added 36 characters in body
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| Aug 5, 2018 at 18:44 | history | edited | Jianrong Li | CC BY-SA 4.0 |
added 2 characters in body
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| Aug 5, 2018 at 15:36 | comment | added | Jianrong Li | @Johannes Trost, thank you very much. I only need to show that $f(v_1,v_2)$ is strictly smaller than 1 when $|v_1|, |v_2|$ are large enough. | |
| Aug 5, 2018 at 14:35 | history | edited | Jianrong Li | CC BY-SA 4.0 |
deleted 30 characters in body
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| S Aug 5, 2018 at 14:31 | history | suggested | Johannes Trost | CC BY-SA 4.0 |
Added missing set-of-natural-numbers and some absolute brackets
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| Aug 5, 2018 at 14:30 | review | Suggested edits | |||
| S Aug 5, 2018 at 14:31 | |||||
| Aug 5, 2018 at 14:20 | comment | added | Johannes Trost | The asymptotics will be very different depending on the size of $|v_{1} v_{2}|$. If the vectors have a small overlap, $|v_{1} v_{2}|\ll \min(v_{1}, v_{2})$, then one might neglect $|v_{1} v_{2}|$. If the vectors have large overlap, then $|v_{1} v_{2}| = O(\min(v_{1}, v_{2}))$ and should not be neglected. Or let it be $|v_{1} v_{2}|= O(\min(v_{1}, v_{2})^\nu)$ with some real exponent $\nu$, presumably smaller than one. And these are only some examples. The results could be very different in all these cases. Before I start working, it would be helpful, if many possibilities could be excluded. | |
| Aug 5, 2018 at 14:16 | history | edited | Brendan McKay | CC BY-SA 4.0 |
make abs value brackets bigger
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| Aug 5, 2018 at 13:40 | comment | added | Jianrong Li | @JohannesTrost, I am looking for something like ${2n \choose n} \sim \frac{4^{n}}{\sqrt{n \pi}}$ ($n \to \infty$). | |
| Aug 5, 2018 at 13:30 | comment | added | Johannes Trost | So you are not looking for an asymptotic expansion, but rather an efficient (approximate) algorithm for large $|v_{1}|$, $|v_{2}|$ ? | |
| Aug 5, 2018 at 13:06 | comment | added | Jianrong Li | @Johannes Trost, thank you very much. The vectors $v_1, v_2$ are given. So $v_1 v_2$ is known and $|v_1 v_2|$ is known. | |
| Aug 5, 2018 at 12:42 | comment | added | Johannes Trost | Have you any further information on the behaviour of $|v_{1} v_{2}|$ ? As it stands it could be anything in $\left [ 0, \min(|v_{1}|,|v_{2}|) \right]$. Maybe $|v_{1} v_{2}|= O(\min(|v_{1}|, |v_{2}|) )$ ? | |
| Aug 5, 2018 at 12:23 | history | asked | Jianrong Li | CC BY-SA 4.0 |