Talk:univalent

The following discussion has been moved from the page [[user talk:msh210]].

This discussion is no longer live and is left here as an archive. Please do not modify this conversation, but feel free to discuss its conclusions.

---

univalent

Could you add the math sense? I can't tell from w:Univalent function whether it's "(of a holomorphic function on an open subset of the complex plane) Injective", or "(of a function on an open subset of the complex plane) Holomorphic and injective", or "(of a function) Holomorphic, injective, and having an open subset of the complex plane as its domain", or maybe something else entirely.

Thanks in advance!
—Ruakh_TALK 08:17, 27 May 2014 (UTC)

And what is the difference between the three? Because it does not seem to be a mathematical one. — Keφr 09:24, 27 May 2014 (UTC)

The difference is between presupposition and entailment. For example, can someone say “We seek to prove that this function is not univalent. We will do this by proving the stronger statement that the function is not holomorphic”? Or does the mere fact of saying “We seek to prove that this function is not univalent” already presuppose that the function is holomorphic?
Note that this is not necessarily a firm distinction, and it's possible that more than one of these definitions would work (though one may still be more natural than the others).
—Ruakh_TALK 15:15, 27 May 2014 (UTC)

I think I got it. So I guess the question is whether saying things like "this function is injective but not univalent, because it is not holomorphic" makes so-called sense. And I think it sounds rather silly. When someone states they want to disprove that a function is univalent, then I guess it would be the case that they want to disprove that it is injective, while its holomorphicity and the domain over which it is considered have already been established. But of course, context could show otherwise. However, functions can be considered holomorphic over subsets of their domains, and I could expect univalent could be used similarly. (I have never actually seen this term used at all however, despite having taken a complex analysis course.) — Keφr 16:54, 27 May 2014 (UTC)

I last looked at complex analysis when I took the qual course in, um, 2000–01 or '01–'02, and don't remember. I note that WP has, in its first sentence, your first possibility, but you've probably seen that already. I don't have time now, but will look at bgc when I do, Template:soplink.​—msh210℠ 21:25, 27 May 2014 (UTC)

Thanks. It's not urgent; I just noticed the gap when I came across the Wikipedia article while trying to determine if the word had a linguistics sense “having only one meaning”. (And funnily enough, I guess that technically means I've looked at complex analysis more recently than you. But not as would count: I dropped the course within the drop/add period. It's the closest I came to taking a math class in college. :-P   ) —Ruakh_TALK 23:38, 27 May 2014 (UTC)

Unusual choice for the only math class to take in college. Usually it has prereqs like, I don't know, integral calculus at the least; though I guess integral calculus can be taken AP also. (Adverbial AP, I guess.  :-) ) My previous "I don't have time now, but will look at bgc…" stands.​—msh210℠ 03:36, 28 May 2014 (UTC)

I was lucky enough to be able to take several years of college math during high school. (For some unfathomable reason, my parents and sister were willing to take turns driving me from high school to the local university, where my parents teach, almost every day. In retrospect I don't think I appreciated it enough. Mental note: I should thank them. And my school system paid a good chunk of the tuition, which I know that many schools don't do.) I got up to the first-year-graduate-student level in some areas (topology, real analysis), though obviously with some undergraduate-level gaps (modern algebra, for example, was never offered at a time slot I could make). So when I actually went off to college (for engineering degrees), all my math requirements were covered by transfer credit — including whatever the prerequisites for complex analysis were, had I chosen to actually take it. —Ruakh_TALK 07:04, 28 May 2014 (UTC)

Okay, a few data points.

[1] seems clearly to be including "holomorphic" in the definition of univalent. (Thurston does, too.)

[2] uses analytic and univalent (note analytic=holomorphic for complex function), but I don't know that that means univalent doesn't include "analytic" in its definition: maybe it's a redundancy like meaning "not only analytic but univalent". But I tend to think that author means "analytic" is not in the definition of univalent. Similarly, [3] and [4] seem to use univalent to mean only "injective".

That would make two senses. I would rather see them listed as one if there's a way to do so: perhaps with our friend the semicolon. Any thoughts? (Anyone?) (Meanwhile I've not edited the definition, though I have made some other, more minor edits to the entry.)​—msh210℠ 04:06, 28 May 2014 (UTC)

Oh, and I don't think "having an open subset of ℂ as its domain" is part of the definition, though I'm not sure why I say so.​—msh210℠ 04:10, 28 May 2014 (UTC)

Thanks for your research. Re: "That would make two senses. I would rather see them listed as one": Me, too. —Ruakh_TALK 07:06, 28 May 2014 (UTC)

In that case, I propose we change the current definition "(complex analysis) Of a holomorphic function: injective in a given open subset of its domain" to "(complex analysis) Holomorphic and injective" with a usage note "The term univalent is often used of holomorphic functions to mean 'injective', but also of functions generally to mean 'holomorphic and injective'". (That's per the above discussion, except that I also removed "in a given open subset of its domain" because uses I saw say things like univalent on D or univalent (i.e. everywhere in its domain), so "in a given open subset of its domain" is implied by context — I think.)

Actually, now I'm thinking we should split it into two definition lines. I mean, if there's need for a usage note to say "It means '…' and '…'", doesn't that mean it has two senses?​​—msh210℠ 04:36, 29 May 2014 (UTC)

It's up to you. I feel like it's pretty O.K. to write "(of an X Y) Z" even if it's sometimes used more like "(of a Y) X and Z"; the difference is minor, all things considered. Since it's a precise term of art, I thought that one definition might be unambiguously correct and the other one unambiguously wrong, in which case I'd obviously have preferred the correct one, but since that's not the case — meh. :-P   —Ruakh_TALK 06:35, 29 May 2014 (UTC)

I added "in a given open subset of its domain" precisely to note the usual implied context. But whatever. By the way, do we have a standardised way of adding "of a" labels? {{label}} does not seem quite fitting here. — Keφr 06:50, 29 May 2014 (UTC)

We use {{context}}/{{label}}. There are even context ex-templates {{of a}} and {{of an}} for ease of adding such context labels. (Those were more useful before we required explicitly calling context: then one could write {{of a|bird}}. But they're still useful.)​—msh210℠ 03:31, 30 May 2014 (UTC)

Yes, but say {{label|en|of a|plant}} renders as (of a botany) because of our aliases setup. Never mind categorisation, which may or may not be desired here. I think "of a" labels should be separated from topical/lexical context labels. — Keφr 07:49, 30 May 2014 (UTC)

Yes, there've always been some redirects like that. One can deal with it, e.g. by using |of a|foo| when the categorization is desired and |of a foo| when it's not. You may be right this should be adjusted; I suggest you bring your proposal to the BP.​—msh210℠ 08:45, 30 May 2014 (UTC)

I've just adjusted the entry per this discussion. See what you think and (natch) please re-edit as desired, user:Ruakh, user:Kephir, all.​—msh210℠ 04:27, 6 June 2014 (UTC)

Thanks!   You know how you can write something like “a ± b ∓ c” to mean “either a + b − c or a − b + c”? It's too bad there's no corresponding way to indicate that the two parenthesized bits in your def are similarly complementary. But even without that, I find your presentation very appealing. —Ruakh_TALK 05:19, 6 June 2014 (UTC)