Timeline for When is "metric dimension" well defined?
Current License: CC BY-SA 3.0
18 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Aug 6, 2017 at 8:40 | comment | added | Benoît Kloeckner | Know that this confusion has been cleared, the question makes sense. But you have to realize that youe mistake, however honest, caused much time spent on needless work for you. | |
| Aug 6, 2017 at 3:08 | comment | added | Chill2Macht | @BenoîtKloeckner I find this comment to be unnecessarily insulting and not an entirely accurate description of the situation. I confused "cardinality of minimal metric generating set" with "metric generating set minimal with respect to cardinality" - the post has been edited to reflect this. | |
| Aug 6, 2017 at 2:33 | history | edited | Chill2Macht | CC BY-SA 3.0 |
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| Aug 5, 2017 at 21:48 | review | Close votes | |||
| Aug 6, 2017 at 9:24 | |||||
| Aug 5, 2017 at 21:31 | comment | added | Benoît Kloeckner | I vote to close as ""not research-level". Too much time was already consumed by a plain confusion on minimality, and enough information is available to answer the question in the Answers and comments. | |
| Aug 5, 2017 at 16:46 | history | edited | Chill2Macht | CC BY-SA 3.0 |
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| Aug 5, 2017 at 13:45 | comment | added | Aaron Dall | The line "A metric generating set $B$ is called a metric basis if it has minimal cardinality." seems to be causing some confusion that would be cleared up by changing it to "A metric generating set $B$ is called a metric basis if it has minimal cardinality among all metric generating sets." | |
| Aug 5, 2017 at 13:41 | answer | added | Aaron Dall | timeline score: 4 | |
| Aug 4, 2017 at 13:24 | answer | added | Aaron Dall | timeline score: 4 | |
| Aug 3, 2017 at 20:20 | comment | added | Chill2Macht | @BenoîtKloeckner Ultimately the issue is that I don't know all of the relevant definitions correctly. I mean the same minimality which occurs for basises in vector spaces or more general matroids (inclusion/cardinality/bijection -- I don't know which it is for sure). As for metric dimension possibly not being well-defined, see the examples in the accepted answer to my previous question. It is obviously more difficult for it to not be well-defined when the metric dimension is infinite, but when it is finite it seems possible to cook up counterexamples which are not extremely complicated. | |
| Aug 3, 2017 at 20:15 | comment | added | Chill2Macht | @BenoîtKloeckner If I'm being honest, I don't quite know what exactly cardinality means or what cardinals are. I thought that two sets have the same cardinality if and only if they belong to the same equivalence class/are isomorphic in the category Set, i.e. there exists a bijection between the two. Then finite sets with different numbers of elements would have different cardinalities. In the case of finite sets this should be the same thing as minimality with respect to inclusion, for infinite sets not necessarily since bijections with proper subsets are possible in the infinite case. | |
| Aug 3, 2017 at 20:10 | comment | added | Chill2Macht | @MattF. This seems like it might be a crucial observation -- while connected graphs aren't convex metric spaces, they are "almost" convex in the sense that the convexity condition is satisfied for any two vertices which are not neighbors. | |
| Aug 3, 2017 at 20:04 | comment | added | Benoît Kloeckner | I don't get how it could not be well-defined. Do you mean finite? Or in the definition, do you mean that a basis should be minimal with respect with inclusion? Defining as you do bases as having minimal cardinality implies that they all have the same cardinal! | |
| Aug 3, 2017 at 19:42 | comment | added | user44143 | Murphy imposes a requirement of convexity, so I wouldn't expect his definition to work well outside of convex spaces. The requirement means: for any x and y, there is a z with d(x,z) + d(z,y) = d(x,z). | |
| Aug 3, 2017 at 15:51 | answer | added | Wlod AA | timeline score: 1 | |
| Aug 3, 2017 at 0:37 | history | edited | Chill2Macht | CC BY-SA 3.0 |
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| Aug 2, 2017 at 23:59 | history | asked | Chill2Macht | CC BY-SA 3.0 |