Timeline for Microwaving Cubes
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
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| May 11, 2010 at 19:25 | vote | accept | Mark Bell | ||
| May 11, 2010 at 14:32 | comment | added | Sergei Ivanov | Another example is any periodic function. I don't think there is a sensible "if and only if" condition with this formulation. If you require something stronger - e.g. a cube of any size must be cookable within any cubical region of size 100 times that of the cube - then being harmonic is necessary (at least if $f$ is $C^2$). | |
| May 11, 2010 at 13:42 | comment | added | Willie Wong | Uh... of course not? You just need f to be harmonic (heck, even constant) on an open set that can strictly contain, say, the ball of radius 1. Or if $f$ has compact support, then modification of $f$ outside the set $supp(f) + B_1$ will not change this property. For any $f$ with a solution $\gamma$, you can of course modify $f$ outside the image of $I$ under the transport by $\gamma$ and still have a new function with the requisite property. Since the property you are looking for is strongly local, any global characterization of the function can be made as bad as possible. | |
| May 11, 2010 at 12:32 | comment | added | Mark Bell | Yes, $f$ being harmonic (or having compact support) is certainly sufficient however is it necessary? | |
| May 11, 2010 at 8:48 | comment | added | Willie Wong | Remark: the harmonic function construction obviously won't work in the 1 dimensional case, since SO(1) does not act transitively on "S^0". But the compact support construction works well for 1 dimension. | |
| May 10, 2010 at 22:13 | history | edited | Sergei Ivanov | CC BY-SA 2.5 |
simplified the final remark
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| May 10, 2010 at 21:58 | history | edited | Sergei Ivanov | CC BY-SA 2.5 |
added 10 characters in body
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| May 10, 2010 at 21:20 | history | answered | Sergei Ivanov | CC BY-SA 2.5 |