Timeline for Can we perturb a map $\mathbb{R}^n \to \mathbb{R}^n$ to have distinct singular values?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 20, 2019 at 11:02 | vote | accept | Asaf Shachar | ||
| Nov 20, 2019 at 5:19 | comment | added | Dap | @AsafShachar: yes | |
| Nov 18, 2019 at 14:06 | comment | added | Asaf Shachar | Thank you. So the idea is to replace $\zeta$ with $\zeta+N\psi$, right? | |
| Nov 18, 2019 at 10:41 | history | edited | Dap | CC BY-SA 4.0 |
added 136 characters in body
|
| Nov 18, 2019 at 10:39 | comment | added | Dap | @AsafShachar: I've fixed the perturbation argument - I had got confused about what lives where. It's no trouble to fix mistakes of course; sorry for rushing | |
| Nov 18, 2019 at 10:35 | history | edited | Dap | CC BY-SA 4.0 |
added 51 characters in body
|
| Nov 11, 2019 at 12:47 | history | bounty ended | Asaf Shachar | ||
| Nov 11, 2019 at 12:44 | comment | added | Asaf Shachar | Thank you, really. I am sorry to trouble you again, but I have two more questions: (1) In the definition of $\phi:U\times X^c\to \mathbb R^{2\times 2}$, should $\zeta(x)$ be replaced by $df(x)$? (otherwise this does not compile, and I think that the replacement gives you what you want). (2) Can you please elaborate on the "pushing the isolated points out of the unit ball" - via composing with a diffeomorphism $\phi:\mathbb{R}^2 \to \mathbb{R}^2$? What is the exact action you are doing? replacing $\zeta$ with $\zeta \circ \phi$, or taking the pullback or pushforward of $\zeta$ using $\phi$? | |
| Nov 11, 2019 at 8:39 | history | edited | Dap | CC BY-SA 4.0 |
added 30 characters in body
|
| Nov 11, 2019 at 7:15 | history | edited | Dap | CC BY-SA 4.0 |
added 43 characters in body
|
| Nov 11, 2019 at 7:03 | comment | added | Dap | I'm using Hodge decomposition on $\mathbb R^n$ for $L^2$ forms. This is in "Geometric function theory and non-linear analysis" 10.6, but that is proving something stronger ($L^p$ decomposition for all $1<p<\infty$) - the $L^2$ decomposition is easy in Fourier space but I don't have a more direct reference. | |
| Nov 11, 2019 at 6:38 | history | edited | Dap | CC BY-SA 4.0 |
added 24 characters in body
|
| Nov 11, 2019 at 6:37 | comment | added | Dap | @AsafShachar: I've added some more detail. I only know of Hodge decomposition for compact manifolds and for $\mathbb R^n,$ and I thought the latter would be easier. I am using a smoothness result for the components of the Hodge decomposition, which I've added an argument for. (There was a mistake before, where I only used $\bar f$ to be $C^\infty\cap L^2,$ where I really want all derivatives to be $L^2.$) I don't use the rank assumption on $df$ - it's easy to ensure by perturbation anyway. | |
| Nov 11, 2019 at 6:32 | history | edited | Dap | CC BY-SA 4.0 |
more detail
|
| Nov 9, 2019 at 20:17 | history | edited | Dap | CC BY-SA 4.0 |
deleted 11 characters in body
|
| Nov 9, 2019 at 19:31 | history | answered | Dap | CC BY-SA 4.0 |