Timeline for Maximal (minimal) value of $S=x_1^2x_2+x_2^2x_3+\cdots+x_{n-1}^2x_n+x_n^2x_1$ on condition that $x_1^2+x_2^2+\cdots+x_n^2=1$ [closed]
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Mar 2, 2021 at 21:51 | review | Reopen votes | |||
| Mar 3, 2021 at 0:09 | |||||
| Feb 13, 2021 at 14:59 | history | closed |
abx user44191 David Handelman skupers Max Alekseyev |
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| Feb 10, 2021 at 15:39 | comment | added | Will Sawin | Is your question whether $x_1= \dots = x_n$ is a local maximum, local minimum, or neither, or is your question what the true global maximum and minimum are? For the second one, I will add that the symmetry $(x_1,\dots, x_n) \to (-x_1,\dots, -x_n)$ shows the set of values of $S$ is symmetric under negation and so the minimum is minus the maximum. | |
| Feb 9, 2021 at 12:02 | comment | added | user64494 | MSE is a proper forum for such questions. | |
| Feb 8, 2021 at 17:20 | answer | added | Michael Rozenberg | timeline score: 6 | |
| Feb 8, 2021 at 16:11 | comment | added | Zach Teitler | Bordered Hessian | |
| Feb 8, 2021 at 15:58 | history | edited | user64494 | CC BY-SA 4.0 |
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| Feb 8, 2021 at 15:27 | comment | added | user44143 | Proving that the equal $x$'s are the extreme $x$'s reduces to proving the inequality $$(x_1^2+\cdots+x_n^2)^3-n(x_1^2x_2+\cdots+x_n^2x_1)^2 \ge 0$$ | |
| Feb 8, 2021 at 15:22 | answer | added | user64494 | timeline score: 2 | |
| Feb 8, 2021 at 15:03 | comment | added | Liuyang Guo | @Hhan It's true the point $x1=...=xn$ is neither the maximal nor minimal, but I'm not sure if it's a local maxium or minimum. | |
| Feb 8, 2021 at 15:01 | history | edited | Liuyang Guo | CC BY-SA 4.0 |
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| Feb 8, 2021 at 14:11 | comment | added | Henry | @Hhan I do not understand your maximal example, and it seems possible to have $S$ negative | |
| Feb 8, 2021 at 13:48 | comment | added | Hhan | The point $x_1=...=x_n$ is neither the maximal nor minimal (probably for all $n$). For even $n$, choose $x_{2k}=0$ will give $S=0$ and $x_1=1/2\sqrt 2, x_2=1/\sqrt 2$ may give the maximal. | |
| Feb 8, 2021 at 13:18 | review | Close votes | |||
| Feb 13, 2021 at 15:01 | |||||
| Feb 8, 2021 at 12:39 | history | asked | Liuyang Guo | CC BY-SA 4.0 |