Timeline for Chebyshev net in 3D
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Mar 27, 2012 at 2:54 | vote | accept | Anton Petrunin | ||
| Mar 27, 2012 at 2:54 | history | bounty ended | Anton Petrunin | ||
| Mar 19, 2012 at 12:25 | comment | added | Robert Bryant | @Deane: Your analysis of the conditions is pretty much the same as mine. What I required was that the $u_i$, which satisfy $u_1+\cdots+u_n=0$, must span a subspace of dimension $n{-}1$ (or else the `solution' $f$ can't be local diffeomorphism) and that the circumradius of the simplex they define must be less than $1$. If these conditions are satisfied, there will be exactly two solutions for $v$, and the corresponding $\partial_if$ will be linearly independent along $\mathbb{W}$. | |
| Mar 19, 2012 at 10:59 | comment | added | Deane Yang | Robert, yes you have the same uniqueness statement as for real analytic data. | |
| Mar 19, 2012 at 10:00 | history | edited | Deane Yang | CC BY-SA 3.0 |
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| Mar 19, 2012 at 7:34 | history | edited | Deane Yang | CC BY-SA 3.0 |
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| Mar 19, 2012 at 7:07 | history | edited | Deane Yang | CC BY-SA 3.0 |
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| Mar 18, 2012 at 23:29 | comment | added | Robert Bryant | @Deane: Excellent response! Very satisfying. Thanks for the reference and the analysis in dimension $n$. The full nonlinearity means that you actually have some choices (exactly $2$ when $n=3$ and the right inequalities are satisfied) as to how to specify the rest of the derivatives of $f$ along the hypersurface, but, once that choice is made the extension is unique in the analytic category. I'm assuming that you also have (local) uniqueness in the smooth category as well. Is that so? | |
| Mar 18, 2012 at 22:25 | history | edited | Deane Yang | CC BY-SA 3.0 |
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| Mar 18, 2012 at 22:07 | history | answered | Deane Yang | CC BY-SA 3.0 |