Timeline for Smallest root of a degree 3 polynomial
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Sep 18 at 1:22 | comment | added | Iosif Pinelis | @Venus : Do you have a further response to this answer? | |
| Sep 16 at 13:11 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Sep 16 at 13:08 | comment | added | Iosif Pinelis | @Venus : $t_*$ is the only critical point of $p(t)$ that is $\le1/5$. I have not tried other $t_*$'s. | |
| Sep 16 at 5:29 | comment | added | Venus | @Iosif Pinelis Thank you for your answer. May I ask how you found $t_{*}$? And, are there other $t_{*}$'s that are significantly different and would work? Thanks. | |
| Sep 15 at 21:33 | comment | added | Iosif Pinelis | @AlekseiKulikov : Such a quest, with such a rather simple but apparently unremarkable function $g$, would look rather strange to me. | |
| Sep 15 at 19:35 | comment | added | Aleksei Kulikov | Yes, the new argument is ok, but do you have a nicer argument for $3$ being the only critical point of $g$, other than "just take the derivative and solve the equation, high-school style". | |
| Sep 15 at 19:26 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Sep 15 at 19:20 | comment | added | Iosif Pinelis | @AlekseiKulikov : I have added two details here. Please let me know if more details are needed. Also, of course, here we need $\min$ rather than $\max$. | |
| Sep 15 at 19:18 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Sep 15 at 18:57 | comment | added | Aleksei Kulikov | The new argument looks extremely condensed, even the logic is hard to follow, it would benefit from a few english words I think. Anyway, if I understood it right, in the middle you claim that $g(u) \ge g(3)$ for $0 \le u \le 3$ -- why is that true? (you also claim that $h(u) \ge \max(h(0), h(3/5))$ which is just straight up false, but I assume you've meant $\min$ there and not $\max$, which I managed to convince myself in) | |
| Sep 15 at 18:34 | comment | added | Iosif Pinelis | The proof is now much simpler and without using Mathematica. | |
| Sep 15 at 18:33 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Sep 15 at 18:23 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Sep 15 at 15:59 | comment | added | Iosif Pinelis | @AlekseiKulikov : You are quite right. I have further simplified the expressions for $t_*$ and $p(t_*)$ to reduce the total Mathematica execution time to under 0.5 sec. | |
| Sep 15 at 15:56 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Sep 15 at 14:49 | comment | added | Aleksei Kulikov | "Claims (1) and (2) can be verified algorithmically" -- technically, the initial claim can also be verified algorithmically (by writing all the conditions in terms of sines and cosines and using Tarski--Seidenberg). I think what you implicitly mean is that they "can be verified algorithmically before the heat death of the Universe". | |
| Sep 15 at 13:37 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |