Timeline for What is the analog of "monotonic" for scalar functions on surfaces?
Current License: CC BY-SA 3.0
16 events
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| Jul 8, 2012 at 17:33 | comment | added | Alec Jacobson | BTW, we decided to just abuse the term monotonic for our paper: igl.ethz.ch/projects/monotonic | |
| Jun 1, 2012 at 2:56 | history | edited | François G. Dorais |
edited tags
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| Apr 22, 2012 at 19:41 | comment | added | Alec Jacobson | @BS: I like this but it's just as hard/long to say as "lacks local extrema" | |
| Apr 21, 2012 at 9:52 | comment | added | BS. | What about "harmonic for some conformal structure" ? | |
| Apr 21, 2012 at 9:41 | comment | added | Alec Jacobson | @Misha: I would like to use this word to say the following: Given the value of an unknown function $g(x,y)$ with $x=[0,1]$ or $y=[0,1]$ (that is, on the boundary of $(x,y) \in [0,1]$, we find values for the function $g(x,y)$ with $(x,y) \in (0,1)$ such that $g(x,y)$ is _________ there. Where ______ is the word that means "has no local extrema". | |
| Apr 21, 2012 at 9:37 | history | edited | Alec Jacobson | CC BY-SA 3.0 |
mathcal --> mathbb for reals
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| Apr 21, 2012 at 3:08 | comment | added | Misha | For instance, there are open maps ${\mathbb R}\to {\mathbb R}$ which are discontinuous everywhere (cf. Igor's suggestion). By the way, you can also use the class of submersions (this makes sense in both continuous and smooth setting). This is more restrictive than Igor's suggestion (except for functions of one variable), but behaves nicer in some ways. All in all, I think the answer depends on what are you going to do with this generalization. | |
| Apr 20, 2012 at 7:15 | comment | added | Alec Jacobson | I'm OK with only considering continuous functions, but ideally the word would be a catch all like monotonic is in 1d. | |
| Apr 20, 2012 at 2:58 | comment | added | Will Sawin | @Ricky: There is a good reason for that, I think. Any definition of monotonicity that a) depends only on topology and b) never says a function on a subset of the real line is monotonic when it isn't has that property. @Misha: I'll look it up. | |
| Apr 20, 2012 at 1:16 | comment | added | Anton Petrunin | There is a standard notion of "monotonic vector field", it is defined using the following inequality: $$\langle v_p-v_q,p-q\rangle>0.$$ | |
| Apr 20, 2012 at 0:31 | comment | added | Misha | @mangledorf: Do you consider only continuous functions? Otherwise, it's unlikely you will get a nice generalization. | |
| Apr 20, 2012 at 0:23 | comment | added | Misha | @Will: Your notion is very close to "cell-like" map. | |
| Apr 19, 2012 at 23:31 | comment | added | user5810 | Constant functions are monotone but are only open in trivial cases. $\:$ Will's suggested criterion would never hold for (non-empty) disconnected domains and contractible ranges. $\:$ (I don't have my own suggestion at this point.) $\;\;$ | |
| Apr 19, 2012 at 23:03 | comment | added | Igor Rivin | WHat about "open"? | |
| Apr 19, 2012 at 21:53 | comment | added | Will Sawin | "Inverse image of a contractible set is contractible" is what I have often thought the correct generalization of monotonicity to higher dimensions to be, and it fits your requested criteria. But I don't know a word for it. | |
| Apr 19, 2012 at 21:29 | history | asked | Alec Jacobson | CC BY-SA 3.0 |