Timeline for Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order derivational operators
Current License: CC BY-SA 3.0
20 events
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| Aug 10, 2015 at 9:14 | history | edited | Raziel | CC BY-SA 3.0 |
Added explicit example on the sphere.
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| Aug 10, 2015 at 7:29 | history | edited | Raziel | CC BY-SA 3.0 |
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| Aug 10, 2015 at 7:28 | comment | added | Raziel | Correct, that's sufficient. So it seems the problem it's even easier (at least with this relaxed condition on the ''sum of squares''). I will edit my post. | |
| Aug 10, 2015 at 2:29 | comment | added | Robert Bryant | @Raziel: Actually, the equation $\sum_{i=1}^N\mathrm{div}(X_i)X_i=0$ is at most $n$ equations, not $N>n$ equations, since you are only trying to get the vanishing of a single vector field, not setting $\mathrm{div}(X_i) = 0$ for $i=1,\ldots,N$. | |
| Aug 9, 2015 at 9:30 | history | edited | Raziel | CC BY-SA 3.0 |
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| Aug 9, 2015 at 9:23 | history | edited | Raziel | CC BY-SA 3.0 |
Fixed typos, improved notation for consistency
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| Aug 9, 2015 at 9:22 | comment | added | Raziel | As I explained the formula is valid for N>n under the conditions on the $W_I$ that I described. This indeed put constraints on the $W_I$. For example all $\|W_I\| \leq 1$ (and at least some of them is strictly $<1$ as soon as $N > n$. You can test it explicitly with the example on the $2$-sphere that I suggested. As a matter of fact, it's a matter of linear algebra: you can write a positive definite matrix in $n$-dimension as $\sum_{I=1}^N w_I w_I^*$ for an arbitrarily large number of vectors $w_I \in \mathbb{R}^n$. | |
| Aug 9, 2015 at 3:53 | comment | added | Ali Taghavi | @Raziel Thanks again for your answer. You wrote the above formula works for N>n. Am I mistaken to think the formula is sensitive to orthonotmality?So not valid for N>n? | |
| Aug 8, 2015 at 8:41 | history | edited | Raziel | CC BY-SA 3.0 |
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| Aug 7, 2015 at 8:27 | history | edited | Raziel | CC BY-SA 3.0 |
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| Aug 7, 2015 at 8:12 | history | edited | Raziel | CC BY-SA 3.0 |
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| Aug 7, 2015 at 8:07 | history | edited | Raziel | CC BY-SA 3.0 |
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| Aug 7, 2015 at 8:01 | history | edited | Raziel | CC BY-SA 3.0 |
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| Aug 7, 2015 at 7:48 | history | edited | Raziel | CC BY-SA 3.0 |
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| Aug 7, 2015 at 7:35 | history | edited | Raziel | CC BY-SA 3.0 |
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| Aug 6, 2015 at 18:03 | history | edited | Raziel | CC BY-SA 3.0 |
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| Aug 6, 2015 at 9:43 | history | edited | Raziel | CC BY-SA 3.0 |
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| Aug 6, 2015 at 9:35 | comment | added | Raziel | Thanks for the comment. Indeed I believe the best I can do is build such a frame where the fields are divergence-free at a given point $x_0$. The requirement on a full neighborhood leads to a system of PDEs. The integrability conditions give then the local obstruction. | |
| Aug 6, 2015 at 9:25 | comment | added | Jean Van Schaftingen | Can you explain how can you construct in general a local frame of orthogonal and divergence-free vector fields? | |
| Aug 5, 2015 at 7:29 | history | answered | Raziel | CC BY-SA 3.0 |