Timeline for Maximal minimum for a sum of two (or more) cosines
Current License: CC BY-SA 3.0
6 events
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Jun 13, 2015 at 4:14 | vote | accept | Daniel Soudry | ||
| Jun 12, 2015 at 16:53 | answer | added | Dror Speiser | timeline score: 4 | |
| Jun 12, 2015 at 14:42 | comment | added | Daniel Soudry | Indeed, irrational numbers are out, as a result of the Kronecker's theorem (the second one here en.wikipedia.org/wiki/Kronecker's_theorem). However, I don't understand from your comment how to rule out non-integer rationals (e.g., note that $r=1.5$ is not bad). Also, note that in my parameter scan, $r=0:dr:10$, and I check $x\in[0,\frac{2}{dr}\pi]$. | |
| Jun 12, 2015 at 13:57 | comment | added | Aaron Meyerowitz | If $r$ is not an integer then it does not suffice to check just $0 \le x \le 2\pi.$ For example, with $r=2/19$ and $x=29\pi$ we have $rx=2\pi+\pi+\pi/19$ so $\cos(x)+\cos(rx)$ is nearly $-2$ at that $x$ and has a minimum even slightly smaller near by. I think these considersations could help you rule out irrational $r$ (find a close approximation of the form $x \approx \frac{m}{n}$ with $m,n$ odd then take $x=n\pi$ to bound the minimum from above.) Then non-integer rational numbers (note $2/19 \approx 3/29$.) Also, odd integers are clearly out. Leaving $r=2,4,6,8,\cdots$ and $3$ characters. | |
| Jun 12, 2015 at 10:39 | history | asked | Daniel Soudry | CC BY-SA 3.0 |