Wikipedia talk:WikiProject Mathematics

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Merger of Unitary operator and Unitary transformation.

We have these two articles: Unitary operator, Unitary transformation. Should they get merged? Michael Hardy (talk) 21:04, 2 May 2024 (UTC)

I always thought the map C² → C³ given by sending (u, v) to (u, v, 0) would be an example of a unitary operator, with 'unitary' referring just to the preservation of a hermitian inner product. The notion suggested here is what I would call "unitary isomorphism." Have I been using the term in an unusual way? Gumshoe2 (talk) 18:59, 3 May 2024 (UTC)

The text is incorrect. A bounded linear operator U : H → H on a Hilbert space H that satisfies U*U = UU* = I need not be surjective unless H is finite dimensional. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 19:03, 9 May 2024 (UTC)

No, this statement is correct. (Since $x=Ix=UU^{*}x=U(U^{*}x)$.) Tito Omburo (talk) 20:26, 9 May 2024 (UTC)

The content of Unitary transformation which describes isomorphism between Hilbert spaces should be migrated to a section on Hilbert space, which currently has no discussion of morphisms of Hilbert spaces (in fact the word "isomorphism" is used only 8 times and never in the context of describing the natural notion of isomorphism of Hilbert spaces!). The rest of it is just a copy of content in Unitary operator which is a much more commonly discussed notion.

Should also point out there will be quite a bit of overlap with Unitary matrix and indeed Unitary group but I think unitary operators (i.e. automorphisms of infinite-dimensional Hilbert spaces, especially function spaces) are studied in their own right as a primary topic that is distinct enough in flavour and techniques (and of course in level of difficulty for the average reader to understand) that it makes sense to keep Unitary matrix/operator/group as 3 separate pages. Tazerenix (talk) 05:35, 10 May 2024 (UTC)

This seems sensible to me. Tito Omburo (talk) 09:16, 10 May 2024 (UTC)

request: birkhoff universality theorem

i began investigating my my belief that optimal function estimation of a 'target function', using purely geometric properties (i.e. properties of the curve wrt to the unit interval) was cyclical.

as a preface i want to emphasise to my peer group my understanding that using very finite-valued integers (i.e. in the hundreds or thousands) to describe cardinality on the real interval is asking to get slaughtered, but i need to lay some groundwork for my request. anyways!

i observed behaviour where the 'optimal function' estimating the target would exist in cycles. that is, define A > B > C \in \mathbb{Z}_+ as the cardinality of the set of uniformly-spaced points on the domain for which we have values of target function f. i.e. we have f(a) for all a \in A, etc.

whilst possible for B = A+1, i often found there was a 'gap' between A, B and C. again, ANYWAYS:

assuming the vertical line test is enforced, it seems that the 'optimal function' for a 'target function' can be (easily) estimated via composition of functions from, say, the space of square-integrable functions.

this lead me to the work of Joel Shapiro, which seems to point to Garrett Birkhoff's 1929 paper which, as i understand it, is considered the "birkhoff universality theorem".

don't you guys think we need a page for this? it seems kind of important in the era of function approximation and the ensuing evaluation of its optimality, for which the cyclical nature of the composition of functions are incredibly relevant.

https://math.osu.edu/sites/math.osu.edu/files/Birkhoff.pdf

George D Birkhoff. Démonstration d’un théoreme élémentaire sur les fonctions entières. C. R. Acad. Sci. Paris, 189(14):473– 475, 1929.

pinging the wikipedia math legend @D.Lazard: to see if this meets the WP notability.

toodles, my dear PEER GROUP 162.157.84.254 (talk) 22:53, 11 May 2024 (UTC)

Aaron Naber and Robin Pemantle

If anyone would like a suggestion on new wikipages to write, Aaron Naber and Robin Pemantle are mathematicians recently elected to the National Academy of Sciences (NAS, AMS). With this qualification there should be no issue on notability. Gumshoe2 (talk) 15:03, 12 May 2024 (UTC)

In the same spirit there's Vladimir Sverak, recently elected to the American Academy of Arts and Sciences (AMS news). Gumshoe2 (talk) 22:39, 14 May 2024 (UTC)

Biographical notability

I just posted the following, to Wikipedia talk:WikiProject Physics. Cross-post here, because WPM has exactly the same problem.

The physics project template counts the number of articles ranked by importance, and quality. Here: Wikipedia:WikiProject Physics/Quality Control There are currently 700+ articles with unassesed priority (marked "???"). Clicking through, almost all of these are biographies. I suspect that no one particularly wants to tackle this, because of the unpleasantness of tagging someone's biography as "unimportant". That, plus the true difficulty of actually assigning a relative ranking -- you have to be very cross-disciplinary to be able to assess such comparisons. And that's just within physics, never mind something like "my biologist is more important than your physicist" or god help us, "our TV anchor is more notable than your physicist". Thus, I'm wondering if perhaps there might be better to avoid this issue entirely? I'm thinking of allowing the template to have an "importance=biographical" value. Or maybe there is some better way to do this?

Please discuss there.

BTW, the WPM assessment is here: Wikipedia:WikiProject Mathematics/Assessment and clicking through to the server shows almost all unassessed articles are biographies, with a sprinkling of societies, journals and awards. 67.198.37.16 (talk) 00:18, 16 May 2024 (UTC)

Adjoint functor theorem, axiom of choice and anafunctor

I noticed that in the Formal criteria for adjoint functors it says that "for simplicity ignoring the set-theoretic issues". Does this refer to axiom of choice? Also, it seems that axiom of choice can be avoided by introducing a concept called anafunctor. It would be great if you could give me some advice or help with the draft (Draft:Anafunctor). SilverMatsu (talk) 05:16, 17 May 2024 (UTC)

I think it is not appropriate to ignore set theoretic issues in the statement of the theorem, because one of the conditions is essentially a set theoretic smallness condition already (it holds trivially in small categories for instance). MacLane states the theorem for (small-)complete categories with small hom sets. As far as I am aware, the proof uses choice. I don't know about anafunctors. Tito Omburo (talk) 10:52, 17 May 2024 (UTC)

Thank you for your advice. I think so, too. I think the theorem (SAFT) requires an axiom of choice. By the way, I'm thinking about whether to add Category: Axiom of choice to a new draft. --SilverMatsu (talk) 17:36, 21 May 2024 (UTC)

Also, Roberts (2011) says that, the etymology of anafunctor is an analogy of the biological terms anaphase/prophase. By the way, wiktionary has an wikt:anafunctor, and wikipedia has a profunctor. --SilverMatsu (talk) 15:46, 23 May 2024 (UTC)

Thanks for sharing. That's an interesting remark. Tito Omburo (talk) 15:56, 23 May 2024 (UTC)

Requested move at Talk:One half#Requested move 17 May 2024

There is a requested move discussion at Talk:One half#Requested move 17 May 2024 that may be of interest to members of this WikiProject. Remsense诉 13:18, 17 May 2024 (UTC)

Expectile

I have created a somewhat stubby new article titled Expectile.

• It could use further work.
• Its uses in mathematical economics could possibly be mentioned. I don't know enough about those to do that.
• Three articles link to it: Quantile, Expected value, and Risk measure. Possibly other links should be there.

Michael Hardy (talk) 17:32, 17 May 2024 (UTC)

Edit war in Tournament (graph theory)???

Just across the article via contributions, watchlist, or whatever it is, the article Tournament (graph theory) apparently has some kind of edit war (I suppose) between User:David Eppstein and User:Closed Limelike Curves. I have no clue about graph theory, but probably need some discussion per WP:BRD. Dedhert.Jr (talk) 15:31, 18 May 2024 (UTC)

CLC has now three times changed the lead to a wrong definition involving complete directed graphs. Tournaments are not complete directed graphs. Complete directed graphs have edges in BOTH directions between each pair of vertices. Tournaments have an edge in exactly ONE direction between each pair of vertices. They are orientations of complete UNDIRECTED graphs. The undirected part is important. CLC should be reverted a third time, at least. I reverted twice but more eyes would help. —David Eppstein (talk) 16:04, 18 May 2024 (UTC)

@David Eppstein Not to mention, the article has problems with citations, and more importantly, why does the article even put the theorem box in the first place? Will take care of these problems as much as I can. Let me know if someone has a different idea.

But seriously, for verifiability that tournaments are not the complete directed graphs, is it possible to expand the article, pointing it out alongside the supported sources, avoiding confusion or misinterpretation? Another problem here is the lead may already give some WP:TECHNICAL, and it seems that CLC relates this terminology to the round-robin tournament, from which I could not see anything about them instead of the list of see also section in the edit source. Dedhert.Jr (talk) 16:12, 18 May 2024 (UTC)

Hi David—very sorry if my last edit was unclear, my intention wasn't to start an edit war. The last time I edited this, however, I described tournaments as "Oriented complete graphs", which I believe to be correct. (I don't see any difference between "oriented complete graphs" and "orientation of a complete graph"—the term "oriented complete graphs" means you start with a complete graph, then orient it.)

I believe most people would understand the term "complete oriented graph" refers to a tournament by slight abuse of terminology (the meaning is clear because an oriented graph can't be complete, so it must mean "as complete as possible"). My citation of the Mathematica wiki shows the wiki using the term "complete oriented graphs".

If you think "Orientation of a complete graph" would be more technically correct language, I think that's reasonable, but I'd prefer if you edited that term directly rather than reverting the edit as a whole. –Sincerely, A Lime 17:16, 18 May 2024 (UTC)

Re your "I described tournaments as "Oriented complete graphs", which I believe to be correct": maybe you can argue that this is correct in a pedantic WP:TECHNICAL sense, if one understands the technical word "oriented" to mean adding directions to the edges of an undirected graph and "complete graph" to mean "complete undirected graph". However, it is also confusing, misleading, and totally inappropriate for the lead sentence of an article. When we talk about directed graphs, the natural interpretation of "complete graph" would be a complete directed graph, and casual readers are unlikely to notice the distinction between oriented and directed. These are not complete directed graphs. —David Eppstein (talk) 18:51, 18 May 2024 (UTC)

PS also please stop putting CS1-formatted citations into their own separate templates. This article uses CS2 (the citation template, not the cite templates) with short footnotes. When you put a citation into a template, rather than leaving it in the main text of an article, and then make a short footnote to it, it will always generate a harv linking error (look at the hidden categories). In addition, this violates WP:CITEVAR. —David Eppstein (talk) 18:55, 18 May 2024 (UTC)

In this case David Eppstein's edit is certainly better since it is clearer. However, Closed Limelike Curves' proposed definition as "oriented complete graph" seems to be identical, at least according to the lead sentence of Orientation (graph theory). The sentence "A tournament is an orientation of a complete graph" also appears on that page. If this is actually in error, presumably because of wiki conventions on graph theory language, perhaps that page needs to be changed. Gumshoe2 (talk) 19:51, 18 May 2024 (UTC)

Again, a definition that can be argued to be technically correct, if one uses the precise technical meanings of each term, can still be seriously misleading, if an un-expert reading of those terms would likely lead readers to a wrong understanding. We should aim for understanding, not merely technical correctness. —David Eppstein (talk) 20:11, 18 May 2024 (UTC)

0.999... still at Featured Article review

I think it's at a point where only some tidying remains, but I'm not sure when I'll have time to do that tidying. XOR'easter (talk) 01:26, 24 May 2024 (UTC)