Timeline for Regularity of the reparametrization map between curves [closed]
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30 events
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| Apr 21, 2017 at 10:49 | comment | added | Todd Trimble | I have rolled back to a previous revision. Please note that vandalism of a question (by erasing virtually all its content) is considered a site violation. | |
| Apr 21, 2017 at 10:48 | history | rollback | Todd Trimble |
Rollback to Revision 13
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| Apr 21, 2017 at 9:11 | history | closed |
Michael Albanese Franz Lemmermeyer R.P. Friedrich Knop Stefan Kohl♦ |
Not suitable for this site | |
| Apr 21, 2017 at 3:34 | review | Close votes | |||
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| Apr 21, 2017 at 3:08 | review | Low quality posts | |||
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| Apr 8, 2017 at 16:03 | history | edited | Romeo | CC BY-SA 3.0 |
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| Apr 5, 2017 at 21:12 | history | edited | Romeo | CC BY-SA 3.0 |
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| Apr 5, 2017 at 9:46 | history | edited | Romeo | CC BY-SA 3.0 |
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| Apr 4, 2017 at 7:09 | history | edited | Romeo | CC BY-SA 3.0 |
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| Apr 3, 2017 at 19:58 | comment | added | Romeo | For sure there is a natural reparameterization of the $\gamma_\epsilon$ but to be honest I do not see how this can be related to the Borel regularity of the reparametrization operator. In a sense, if you want, we can directly consider the curve in $\mathbb R^{d+1}$ defined by $(h, \gamma)$. But how can this be used to prove the Borel regularity of $\mathcal R$? | |
| Apr 3, 2017 at 19:54 | comment | added | Rbega | My thought was that there is a natural reparameterization of the $\gamma_\epsilon$ and this might be helpful. | |
| Apr 3, 2017 at 19:43 | comment | added | Romeo | @Rbega Thanks for your comments! Yes, exactly, that was a typo, I fixed it. Not precise applications in mind, it is a question which arose several times and I believe there must be somewhere in the literature a general result of this kind. I am not sure of getting your hint: what is the point in considering the curve you suggest (and using arc-length there)? Thanks. | |
| Apr 3, 2017 at 19:40 | history | edited | Romeo | CC BY-SA 3.0 |
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| Apr 3, 2017 at 19:38 | comment | added | Rbega | Do you have a specific application of this sort of result in mind? A useful trick (that may be completely unrelated to what you are interested in) is to consider $\gamma_\epsilon(t)=(\gamma(t), \epsilon t)$ as a map in $Lip([0,1, \mathbb{R}^{d+1})$. For instance, the curves $\gamma_\epsilon$ can be parameterized by arclength in a natural way and one can study what happens as $\epsilon\to 0$ (which recovers $\gamma$ in a sense). | |
| Apr 3, 2017 at 19:26 | comment | added | Rbega | Shouldn't the target space of $R$ be maps from $[0,T]$ to $\mathbb{R}^d$? Do you want to restrict the domain of $R$ to $B$? | |
| Apr 3, 2017 at 18:50 | history | edited | Romeo | CC BY-SA 3.0 |
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| Apr 2, 2017 at 15:40 | comment | added | Romeo | @PietroMajer Thanks again for your kind reply! Exactly, that was also an idea I considered but I got stuck because: 1. I am not completely sure of having an argument to prove that (inverses) of strictly monotone reparameterizations induce Borel maps between curves: do you have any references for this? 2. I had not a clear idea of how the perturbations converge (pointwise?); 3. I thought there has to be a general (well-known) argument behind (thus I came here to ask). Thanks for your valuable comments. | |
| Apr 2, 2017 at 15:32 | comment | added | Pietro Majer | Ok, thank you. Then I would first consider $\gamma\mapsto h_\gamma + {1\over n}\mathrm{id}$, since I guess $s_\gamma$ can be obtained as the limit of $(h_\gamma + {1\over n}\mathrm{id})^{-1}$ for $n\to\infty$... | |
| Apr 2, 2017 at 14:19 | history | edited | Romeo | CC BY-SA 3.0 |
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| Apr 2, 2017 at 14:18 | comment | added | Romeo | Thanks for the useful comment. Yes, I have been imprecise. Let me add that I am assuming the association $\gamma \mapsto h_{\gamma}$ is Borel (between the space of Lipschitz curves in $\mathbb R^d$ and the Lipschitz maps in $\mathbb R$). Say now that I define $s_\gamma$ to be $s_\gamma(r) = \inf \{t: h_\gamma(t)>r \}$ (this should be the left inverse of $h_\gamma$). Is the corresponding association $\gamma \mapsto s_\gamma$ now at least Borel? Thanks again. | |
| Apr 2, 2017 at 14:06 | comment | added | Pietro Majer | Not quite clear to me: in these assumptions, $B\ni \gamma\mapsto s_\gamma$ could be any map taking $\gamma$ to some $C^1$ diffeomorphism $s_\gamma$ in any crazy irregular way, possibly having nothing to see with $\gamma$, since (1) would be automatically satisfied. Say that the function $s_\gamma$ is always either $\arctan$ or $\exp$, according whether $\gamma$ is in some ineffable set $C\subset B$ or not. Why should $R$ be any nicer than $\gamma\mapsto s_\gamma$? | |
| S Apr 2, 2017 at 13:25 | history | suggested | Henry.L |
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| Apr 2, 2017 at 13:07 | review | Suggested edits | |||
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| Apr 2, 2017 at 13:06 | answer | added | Henry.L | timeline score: 0 | |
| Apr 2, 2017 at 12:56 | history | edited | Romeo | CC BY-SA 3.0 |
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| Apr 2, 2017 at 7:45 | history | edited | Romeo |
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| Apr 1, 2017 at 22:51 | history | edited | Romeo |
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| Apr 1, 2017 at 18:03 | history | edited | Romeo |
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| Apr 1, 2017 at 12:57 | history | edited | Romeo |
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| Apr 1, 2017 at 11:41 | history | asked | Romeo | CC BY-SA 3.0 |