Timeline for Show that these vectors are linearly independent almost surely
Current License: CC BY-SA 4.0
28 events
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| Apr 3, 2022 at 0:03 | comment | added | Gerry Myerson | Follow-up question: mathoverflow.net/questions/375654/… | |
| Nov 4, 2020 at 17:18 | vote | accept | FeedbackLooper | ||
| S Nov 4, 2020 at 15:04 | history | bounty ended | FeedbackLooper | ||
| S Nov 4, 2020 at 15:04 | history | notice removed | FeedbackLooper | ||
| Oct 30, 2020 at 15:48 | answer | added | RaphaelB4 | timeline score: 1 | |
| Oct 29, 2020 at 17:51 | history | edited | FeedbackLooper | CC BY-SA 4.0 |
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| Oct 29, 2020 at 9:25 | history | edited | FeedbackLooper | CC BY-SA 4.0 |
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| Oct 29, 2020 at 7:57 | history | edited | FeedbackLooper | CC BY-SA 4.0 |
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| Oct 29, 2020 at 7:52 | comment | added | FeedbackLooper | @Qfwfq Yes, the notion of genericity is in $\epsilon$, not on $E$. And yes, ideally I would like to show that given generic $\varepsilon$, every point in $E(\epsilon)$ is regular. However, as I stated in the "more context" part, if one cannot show that the whole $E(\epsilon)$ is regular, it would be desirable to show at least that non-regular points are sufficiently rare, such to ensure that a certain optimum $x^*$ (in the sense of a cost $x^TA_0x$) is regular. Hope this motivation is clear. And again, any suggestion/advice would be helpful. Thanks! | |
| Oct 29, 2020 at 7:51 | comment | added | FeedbackLooper | @Keba Because my idea was to perturb the matrices using "random noise". Hence, I would need almost any perturbation to lead to regular values. However, If you have ideas on what single perturbation could lead to regular values I would like to hear it. Thanks for your time! | |
| Oct 29, 2020 at 1:18 | comment | added | Qfwfq | Let $P(\epsilon)$ the datum of the perturbed matrices and $E(\epsilon)$ the corresponding set. Is the condition that you require the following: For "generic" $\epsilon$ every point of $E(\epsilon)$ is regular; or did I misunderstand the quantifiers? In particular, you need a measure (or a notion of genericity) on the space of $\epsilon$'s, not on $E$, correct? | |
| Oct 29, 2020 at 1:11 | comment | added | Keba | Not sure whether I understand your main goal but why does it not suffice to find, for each $\varepsilon >0$, a single pertubation leading to regular values only? | |
| Oct 29, 2020 at 1:02 | comment | added | Qfwfq | Just a remark: $P_i x$ is the gradient of $x^T P_i x -1$. So the condition of linear independence of the $P_ix$, at $x\in E$, is that the $E_i$ cut $E$ transversally. | |
| Oct 28, 2020 at 21:39 | history | edited | FeedbackLooper | CC BY-SA 4.0 |
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| Oct 28, 2020 at 21:31 | history | edited | FeedbackLooper | CC BY-SA 4.0 |
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| Oct 28, 2020 at 21:30 | comment | added | FeedbackLooper | What more information do you think we should add (or what am I missing?) to define a meassure on E? Isnt the meassure that matters implied in the probability distribution of the matrices? | |
| Oct 28, 2020 at 21:28 | comment | added | FeedbackLooper | Sure. I missed that. Surely $m<n$. Thanks. | |
| Oct 28, 2020 at 21:20 | comment | added | Bill Bradley | Should we assume $m<n$? And I guess we need a measure on $E$ to talk about the probability, so can we choose anything "reasonable"? | |
| Oct 28, 2020 at 18:14 | history | edited | FeedbackLooper | CC BY-SA 4.0 |
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| Oct 28, 2020 at 14:51 | history | edited | FeedbackLooper | CC BY-SA 4.0 |
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| S Oct 28, 2020 at 14:50 | history | bounty started | FeedbackLooper | ||
| S Oct 28, 2020 at 14:50 | history | notice added | FeedbackLooper | Draw attention | |
| Oct 27, 2020 at 17:52 | history | edited | FeedbackLooper | CC BY-SA 4.0 |
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| Oct 26, 2020 at 17:02 | history | edited | FeedbackLooper | CC BY-SA 4.0 |
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| S Oct 26, 2020 at 16:35 | history | suggested | mlk |
It's related to measure theory and geometric, however its not related to geometric measure theory...
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| Oct 26, 2020 at 16:00 | review | Suggested edits | |||
| S Oct 26, 2020 at 16:35 | |||||
| Oct 26, 2020 at 10:20 | history | edited | LSpice | CC BY-SA 4.0 |
Proofreading
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| Oct 26, 2020 at 10:16 | history | asked | FeedbackLooper | CC BY-SA 4.0 |