Timeline for Decomposition of a semialgebraic set into connected components
Current License: CC BY-SA 4.0
21 events
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| Oct 31, 2018 at 13:05 | history | edited | kirma | CC BY-SA 4.0 |
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| Jul 5, 2016 at 17:17 | comment | added | kirma | @MichaelE2 This is long overdue, but regarding your comment on closures... An anecdote from Finnish television years ago: an established F1 commentator: "black smoke comes from nozzles behind his car!" - to which an old F1 driver in the studio laconically comments: "in professional circles we call them 'exhaust pipes.'" :} I was feeling a bit like that guy. :) | |
| May 14, 2015 at 12:05 | history | edited | kirma | CC BY-SA 3.0 |
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| May 13, 2015 at 11:28 | vote | accept | Artem Malykh | ||
| May 12, 2015 at 5:09 | history | edited | kirma | CC BY-SA 3.0 |
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| May 12, 2015 at 3:48 | comment | added | Michael E2 | It could be why I gave up on this sort of thing in the past. :) | |
| May 12, 2015 at 3:47 | comment | added | kirma |
@MichaelE2 I'm not surprised of this, but you definitely are not going to get separated connected components for some cyclic (of couse, non-semialgebraic) functions with plain old Reduce without tricks.
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| May 12, 2015 at 3:45 | comment | added | Michael E2 |
I guess I don't know for sure. I thought Reduce[.., {x, y}] produced a CAD with respect to {x, y}: "For polynomial systems, Reduce uses cylindrical algebraic decomposition for real domains...." -- tutorial/SomeNotesOnInternalImplementation
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| May 12, 2015 at 3:39 | comment | added | kirma |
@MichaelE2 Unless I'm mistaken, the benefit of CylindricalDecomposition in comparison to Reduce is that individual cells it produces are guaranteed to be connected. In the case of Reduce, one "cell" (or for instance, DNF item) may present multiple disconnected components, and there's no trivial mechanical method of transforming them to connected cells like in CAD. Correct me if I'm wrong...
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| May 12, 2015 at 3:33 | history | edited | kirma | CC BY-SA 3.0 |
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| May 11, 2015 at 21:06 | history | edited | kirma | CC BY-SA 3.0 |
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| May 11, 2015 at 20:46 | history | edited | kirma | CC BY-SA 3.0 |
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| May 11, 2015 at 17:40 | history | edited | kirma | CC BY-SA 3.0 |
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| May 11, 2015 at 17:21 | history | edited | kirma | CC BY-SA 3.0 |
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| May 11, 2015 at 17:17 | comment | added | Michael E2 |
Yes, that's what I meant, and yes I know it goes beyond the literal scope of the question. It also sometimes will go beyond the capabilities of Reduce, which can be used instead of CylindricalDecomposition.
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| May 11, 2015 at 17:16 | comment | added | Michael E2 | In mathematics, we call your "expanded" components their closures. | |
| May 11, 2015 at 17:16 | comment | added | kirma | @MichaelE2 Do you mean transcendental functions in definitions of the set? That would be obviously outside the domain of semialgebraic sets which are at least somewhat easier to operate computationally... | |
| May 11, 2015 at 17:13 | history | edited | kirma | CC BY-SA 3.0 |
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| May 11, 2015 at 17:11 | comment | added | Michael E2 | If you really want to stress it out, try a transcendental component! :) (+1) -- I've often wanted to do this. | |
| May 11, 2015 at 17:10 | history | edited | kirma | CC BY-SA 3.0 |
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| May 11, 2015 at 17:00 | history | answered | kirma | CC BY-SA 3.0 |