Timeline for Cylinders dividing $\mathbb{R}^{3}$
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| Aug 31, 2017 at 5:07 | history | edited | Martin Sleziak |
Removed deprecated (discrete-mathematics) tag - see the tag info: https://mathoverflow.net/tags/discrete-mathematics/info (if there are some other suitable tags, choose some of them instead)
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| S Aug 7, 2017 at 4:59 | history | suggested | Martin Sleziak |
removed deprecated (geometry) tag - see the tag info: http://mathoverflow.net/tags/geometry/info; if there are some other geometry-related tags which are suitable, please use some of them instead
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| Aug 7, 2017 at 4:25 | review | Suggested edits | |||
| S Aug 7, 2017 at 4:59 | |||||
| Sep 20, 2012 at 12:13 | vote | accept | Victor | ||
| Sep 20, 2012 at 2:00 | answer | added | Joseph O'Rourke | timeline score: 0 | |
| Sep 16, 2012 at 12:24 | history | edited | Victor | CC BY-SA 3.0 |
elminate sentence
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| Sep 16, 2012 at 12:06 | comment | added | Victor | Yes, the cylinders have top and bottom. That is, they are compact. More explicitely, they are copies of $D\times\{0\}\cup S^{1}\times [-3,3]\cup D\times\{1\}$, where $D$ denotes the unit 2-disk. | |
| Sep 16, 2012 at 0:42 | answer | added | Joseph O'Rourke | timeline score: 8 | |
| Sep 15, 2012 at 23:24 | answer | added | Gerhard Paseman | timeline score: 1 | |
| Sep 15, 2012 at 21:42 | comment | added | algori | Steven -- in the first version the top and bottom were misssing. VCF -- it is not too hard to give an explicit bound cubic in $n$, assuming there are no quadruple intersections. I don't know if this helps though. | |
| Sep 15, 2012 at 21:31 | comment | added | Steven Landsburg | algori: Standardly embedded $S^1\times [-3,3]$, together with top and bottom, has an inside and an outside. (At least that's how I interpreted "with top and bottom", though it would be good for the OP to make this clearer.) | |
| Sep 15, 2012 at 21:00 | history | edited | Victor | CC BY-SA 3.0 |
Typo correction
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| Sep 15, 2012 at 20:55 | comment | added | algori | VCF -- don't you in fact consider infinite (i.e., non-compact) round cylinders? Standardly embedded $S^1\times [-3,3]$ does not subdivide $\mathbb{R}^3$ at all. | |
| Sep 15, 2012 at 20:54 | comment | added | Ben McKay | Does the cylinder have a top and bottom on it? | |
| Sep 15, 2012 at 20:49 | comment | added | Victor | Thank you, I didn´t kow that. It makes sense to me since, for example, I know that if we consider $n$ planes instead of cylinders we get at most $(n^3+5n+6)/6$ regions. | |
| Sep 15, 2012 at 20:43 | comment | added | Joseph O'Rourke | This is likely not of interest to you, but general theorems from the theory of arrangements of surfaces indicate that, asymptotically, the number of regions is $O(n^3)$. | |
| Sep 15, 2012 at 20:38 | history | edited | Victor | CC BY-SA 3.0 |
tag added
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| Sep 15, 2012 at 20:31 | history | edited | Victor | CC BY-SA 3.0 |
sentence correction
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| Sep 15, 2012 at 20:05 | history | asked | Victor | CC BY-SA 3.0 |