Wikipedia talk:WikiProject Mathematics/Archive/2023/Aug

Original proof of Gödel's completeness theorem

Some additional eyes would be generally appreciated at Original proof of Gödel's completeness theorem. There is some longer standing confusion about the argument itself, which is not helped by the fact that the article draws on none of the available secondary sources that are available on the topic. Felix QW (talk) 16:42, 25 July 2023 (UTC)

A lot of the mathematical notation is not well done. I've begun cleaning it up. Michael Hardy (talk) 21:00, 4 August 2023 (UTC)

Contrast these:

(it is thus of degree k-1)

(it is thus of degree k − 1)

I changed the former to the latter. Notice that (1) I expunged the bolding, (2) the k is italicized and the digit 1 is not (consistently with TeX and LaTeX style), (3) spacing before and after the minus sign (This is different in cases where a minus or a plus sign is used as a unary operator rather than as a binary operator. In those cases no such space is added. This is also consistent with what TeX and LaTeX do.) (4) the minus sign is not just a stubby little hyphen. Michael Hardy (talk) 21:07, 4 August 2023 (UTC)

Contrast these:

∀x₁∀x₂...∀x_k ∃y₁∃y₂...∃y_m φ(x₁...x_k, y₁...y_m)

∀x₁ ∀x₂ ... ∀x_k  ∃y₁ ∃y₂ ... ∃y_m φ(x₁ ... x_k, y₁ ... y_m)

Note that under WP:MOSMATH in this context one does not italicize digits or parentheses or other punctuation or things like cos, exp, det, max, sup, log, etc., but one italicizes variables. Note also that some space has been added between characters where appropriate. (Similarly, in the foregoing sentence I did not write "Notealsothatsomespacehasbeenaddedbetweencharacterswhereappropriate.") Michael Hardy (talk) 21:15, 4 August 2023 (UTC)

signed area

I just created a stub page for signed area which previously redirected to integral. This is a widely used concept which is not well described by integral and should have its own page (though it should probably eventually also be summarized in a section of area), so I created a stub for now with a basic definition.

I don't precisely know when I'll get around to expanding it, so other folks should feel free to jump in with any improvements or sources. –jacobolus (t) 18:15, 20 July 2023 (UTC)

You've just made me realize I've never actually properly considered what the area of a Möbius strip is or are there a couple of different reasonable areas :-) NadVolum (talk) 15:45, 22 July 2023 (UTC)

A Riemannian metric on any smooth manifold, oriented or not, defines a density, which can be integrated just like differential forms. This gives an unambiguous meaning to something like the volume of a Möbius strip with a particular Riemannian metric. (This is the general form of Riemannian volume, which in many textbooks is only dealt with for oriented manifolds.) Gumshoe2 (talk) 23:26, 25 July 2023 (UTC)

I've made it more general, but it still needs a lot of work. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 19:21, 25 July 2023 (UTC)

I've deleted the material you added, since the 2-form you wrote down has positive integral over every region; it's an unsigned area. A better differential-geometric generality might instead be based on interpreting the signed area 'under' the graph of y = f(x) as the integral of the 1-form y dx on R², taken over the graph curve. Gumshoe2 (talk) 23:23, 25 July 2023 (UTC)

Since circular angle is bounded there is no need for negative angles. On the other hand, hyperbolic angle and difference of slopes are doubly unbounded so they require a zero and orientation + or −. To give these three concepts a unified foundation, signed area is appropriate. See signed measure#Usage. Incorporation here of signed area as a fundamental tool for foundations is long overdue.Rgdboer (talk) 23:29, 25 July 2023 (UTC)

• In illustrations in Wikipedia articles the typesetting is generally of poor quality and cannot be edited without recreating the illustration. And in this one we see something that, although not abominable, is substandard and by reasonable standards incorrect:
  $${\begin{aligned}{\text{right: }}&\int _{a}^{b}f(x)\,dx\\[8pt]{\text{wrong (as it appears in the article): }}&\int _{a}^{b}f(x)dx\end{aligned}}$$
  Michael Hardy (talk) 16:35, 5 August 2023 (UTC)

  This figure is mediocre, but missing a thin space before the dx is the least of its problems. Feel free to make a new figure, and please include enough contrast between labels and background colors. Aside: For all your concern about typography, what's the deal with the big matrices inline in the text? It completely wrecks the leading (inter-line spacing) in body copy. –jacobolus (t) 18:41, 5 August 2023 (UTC)

• Oriented area was a redirect to Vector area. I've changed it to a disambiguation page also pointing to Signed area. Michael Hardy (talk) 16:51, 5 August 2023 (UTC)

  Vector area is a weird and confusing name. This redundant article should probably be merged into bivector. Or should at the least clarify that given any bivector in Euclidean space, the "vector area" is its dual (same magnitude but a complementary and perpendicular subspace of Euclidean 3-space, which can be found by multiplying [using the geometric product] by a unit pseudoscalar), but that bivector is a more general concept which generalizes to arbitrary dimension and works in the affine setting, which has traditionally (in Hamilton's quaternion system and Gibbs/Heaviside's vector analysis) been represented by a vector because these formalisms don't have any concept of bivectors. –jacobolus (t) 18:42, 5 August 2023 (UTC)

• One place where I forgot to mention that a particular area should be viewed as a signed area is on page 4 of this document. Referring to the illustration on that page, if μ is in the convex hull of the chord and the arc, then the area of the triangle is negative; otherwise positive. Michael Hardy (talk) 16:56, 5 August 2023 (UTC)
• Should we have a page titled Oriented volume? And perhaps incorporate this page into it and redirect Signed area to that? Michael Hardy (talk) 17:03, 5 August 2023 (UTC)

  There is a huge amount that could be written about signed area per se. I don't think it needs to be merged anywhere. –jacobolus (t) 18:54, 5 August 2023 (UTC)

Hypercomputation

The article Hypercomputation needs attention. It starts to become a confusing read at the second sentence. XOR'easter (talk) 21:58, 4 August 2023 (UTC)

In this version the lead was fine. --JBL (talk) 23:13, 4 August 2023 (UTC)

April 1 edits last year seem to be the culprit. --JBL (talk) 23:14, 4 August 2023 (UTC)

I have excised the problematic sentences; I haven't checked beyond the lead for other similar problems. --JBL (talk) 23:17, 4 August 2023 (UTC)

Will anyone be shocked to learn that the only other edits by that IP address are to Hava Siegelmann? --JBL (talk) 23:18, 4 August 2023 (UTC)

It also does not shock me to learn that someone into bio-inspired AI might also be into hypercomputation. —David Eppstein (talk) 02:14, 5 August 2023 (UTC)

Thanks for the excisions. The page is still tagged for various things; I'm guessing that this is a topic where plenty of offbeat ideas have been aired, and anything that anybody thought sounded cool eventually got added to Wikipedia. XOR'easter (talk) 22:30, 5 August 2023 (UTC)

Two math articles at WP:FAC

Two articles with substantial mathematical content are currently under consideration to become featured articles: they are Quine–Putnam indispensability argument and Affine symmetric group. (Currently, fewer than 10 articles on mathematics (as opposed to, say, mathematicians) are featured articles.) As part of the FAC process, reviews are required by members of the community. If you might be interested in reviewing one of these articles, you can follow the instructions at Wikipedia:Featured article candidates#Commenting, etc. --JBL (talk) 17:52, 3 July 2023 (UTC)

If you would like/be willing to contribute reviews of either of these articles, it would be good to do so sooner rather than later! You can comment at Wikipedia:Featured article candidates/Affine symmetric group/archive1 and Wikipedia:Featured article candidates/Quine–Putnam indispensability argument/archive1. --JBL (talk) 18:53, 24 July 2023 (UTC)

Another featured article candidate is Logic, which relates to mathematics, although it mainly deals with philosophy. See Wikipedia:Featured article candidates/Logic/archive1 for someone who interests in reviewing the article. Dedhert.Jr (talk) 10:47, 26 July 2023 (UTC)

This should be a reminder to keep Wikipedia:WikiProject Mathematics/Article alerts on your watchlist. Then you get watchlist notices of everything ongoing! Headbomb {t · c · p · b} 11:41, 26 July 2023 (UTC)

I see that Quine–Putnam indispensability argument has been promoted. Affine symmetric group looks close. XOR'easter (talk) 22:32, 5 August 2023 (UTC)

is there an article about degeneracy to an infinite/infinitesimal approximation?

I've been working on articles about spherical geometry, cartography/geodesy, etc., and one common observation/tool used, especially in practical applications, is that in the infinitesimal limit the sphere (or any other 2-dimensional Riemannian manifold for that matter) looks locally Euclidean, so that tools of affine or Euclidean geometry can be used without too much error for small areas on the sphere, or e.g. a flat map of some part of the world can be treated as a faithful enough representation. Many theorems in Euclidean geometry are degenerate limiting cases of analogous theorems in spherical or hyperbolic geometry. This is a common theme more generally, with e.g. Taylor polynomials often used as function approximations for very small arguments in numerical computations. Right now the titles much greater than and much less than redirect to Inequality (mathematics), which (a) doesn't do a very good job explaining what these mean or why they are used, and (b) seems like the wrong home for these infinite or infinitesimal approximations. We also have Degeneracy (mathematics) and Limiting case (mathematics), but these both seem much broader than what I am talking about (and are also not very good articles). We also have Small-angle approximation which is one specific example but a bit narrower in scope than I am imagining. There is also Asymptotic analysis but this seems like a distinct (albeit related) topic.

Do we have an article matching this idea? If not, would it be a good idea to add one? (I would imagine e.g. redirecting much greater than and much less than there.) What would a good title be, and can anyone think of good canonical sources? –jacobolus (t) 23:58, 5 August 2023 (UTC)

Wikipedia:Notability (academic journals) has an RFC

Wikipedia:Notability (academic journals), which is within the scope of this WikiProject, has an RFC for possible consensus. A discussion is taking place. If you would like to participate in the discussion, you are invited to add your comments on the discussion page. Thank you. Tercer (talk) 13:57, 6 August 2023 (UTC)

Proofs in math articles

Is there any general guidance about whether a proof should be given for a result stated in an article? Thanks. ByVarying (talk) 00:08, 3 August 2023 (UTC)

This is up to Wikipedians' discretion, but personally I like to see a proof or proof sketch if the article is about a theorem directly, especially if the proof is not too complicated. –jacobolus (t) 01:35, 3 August 2023 (UTC)

We've discussed this before; check the archives. My preference is to include proofs when they are separately noteworthy (as in, multiple sources have discussed that specific proof) or when the proof itself would help readers understand the topic, but not when they are just routine manipulation or don't have a proper source. —David Eppstein (talk) 01:51, 3 August 2023 (UTC)

A reference to a proof in a reliable source is always preferable. Occasionally the idea of a proof can be mentioned if it helps to understand the topic, as David Eppstein mentioned. But giving a proof for every single result in an article is not recommended and is counterproductive, as it makes the whole article hard to read and buries the overall content in minute technical details. Remember, the purpose of Wikipedia is not to "teach" a subject, but to summarize mathematical knowledge as it is present in reliable sources. PatrickR2 (talk) 03:11, 3 August 2023 (UTC)

A proof for every result in a broad article (say calculus or ellipse or group) turns it from an encyclopedia article into something more like a mathematical monograph or textbook, ballooning the content without necessarily much value to readers looking for an overview. But in the case that the result itself is the direct subject of the article, then having a proof (or sometimes several proofs) can help to explain why the result is true, why it is interesting, what connection it has to other fields, etc. For some arbitrary examples of famous relatively accessible theorems, see Pythagorean theorem, quadratic reciprocity, Chinese remainder theorem, fundamental theorem of calculus, square root of 2#Proofs of irrationality, central limit theorem, Sylow theorems, Brouwer fixed-point theorem. –jacobolus (t) 05:50, 3 August 2023 (UTC)

We are both in complete agreement. PatrickR2 (talk) 18:39, 3 August 2023 (UTC)

I think that nearly everyone would agree, as a general but non-absolute rule, that it's bad to give complete details of a proof. I agree with jacobolus that on a page for a specific theorem it's good to give, at the least, an indication of what's involved in the proof. It's easy to give more details than appropriate (and I've certainly been guilty of it!). Gumshoe2 (talk) 08:12, 3 August 2023 (UTC)

@Gumshoe2: That depends very much on the context. If the article is titled Proof that pi is irrational then the proof needs to be there. Michael Hardy (talk) 18:26, 3 August 2023 (UTC)

Agreed, that's why I say it's a non-absolute rule. There are some obvious exceptions. Gumshoe2 (talk) 18:33, 3 August 2023 (UTC)

I think that it's good to give proofs of important and notable theorems (for example, the functional equation of the Riemann zeta function). You can also make them collapsible so that they don't take up space. A1E6 (talk) 12:21, 3 August 2023 (UTC)

I'd leave out the 'important' and only have ones which are notable in themselves as proofs. And no need for the proof to be a watertight proof - what it should show is why the proof is notable. Sometime it can be worthwhile to include very simple proofs which make the article hang together without too much clicking - one or two lines. Textbooks or papers can be referred to anything more. What I've left out there is things like the multiple proofs of root 2 being irrational which I fully approve of including but haven't rationalized yet ;-) NadVolum (talk) 18:53, 3 August 2023 (UTC)

+1. Lots of important theorems have unenlightening or uncondensable proofs (the four color theorem comes to mind immediately). A good encyclopedic treatment might summarize the key ideas of the proof. --JBL (talk) 18:59, 3 August 2023 (UTC)

Of course; if a proof is too long, only summarizing its key points is appropriate. A1E6 (talk) 17:35, 4 August 2023 (UTC)

I did my best to summarize what I take to be the general attitude about this in my "So, you've decided to write about physics and/or mathematics on Wikipedia" advice page. XOR'easter (talk) 17:43, 7 August 2023 (UTC)

That's nicely put. --JBL (talk) 19:55, 7 August 2023 (UTC)

I definitely agree that important theorems, closely related to the subject of the article, generally deserve a proof or proof sketch in the text. The objective should be to show why the result is true, if possible, and to give an overview of the proof. If the proof is simple enough, a complete proof may be appropriate; in general probably proof sketches are better. - CRGreathouse (t | c) 13:51, 7 August 2023 (UTC)

Should I include a proof about the example of non-surjective epimorphism in the category of rings ? If we don't include the proof, this example might be better added to epimorphism. --SilverMatsu (talk) 02:41, 6 August 2023 (UTC)

This example is already (without proof) in Homomorphism#Epimorphism.

By the way, in this article there are proofs that, for "most" algebraic structures, "monomorphism=injection" and "epimorphism=surjection". This is a case where proofs are needed for verifiability, because, as far as I know, sources for the general cases are generally much more technical (general category theory) than the level of the article, and therefore not accessible for the common reader. D.Lazard (talk) 14:27, 6 August 2023 (UTC)

It seems that users who insert proofs into articles without references are often the ones that make the most mistakes. Something can look correct but be deeply misleading. (For instance, an editor might argue that it's common knowledge that the antiderivative of 1/x is ln|x|+C, but that ignores the fact that the two branches can have different constants). I agree with others that proofs should only be included if references are given; otherwise it is the literal definition of original research. Brirush (talk) 14:31, 6 August 2023 (UTC)

Personally, I like to see proofs when they are unusually elegant, or when they shed light on other topics. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 09:41, 8 August 2023 (UTC)

Too closely following text

On the page Projective tensor product, two sections, "Seminormed spaces" and "Examples" follow a book by Trèves on topological vector spaces very closely, basically all but verbatim. I also removed a lot from the page that I thought trivial or not notable, and these sections could very well fall under that too. I'm wondering what to do about these sections if anyone could give them a look. ByVarying (talk) 19:05, 9 August 2023 (UTC)

Oops, I forgot that the "Examples" section is cited to Schaefer and Wolff's book on topological vector spaces (of which I only have access to the older edition) but I have the same concern. ByVarying (talk) 19:45, 9 August 2023 (UTC)

@Mgkrupa: this appears to be your doing. Care to explain yourself? —David Eppstein (talk) 19:19, 9 August 2023 (UTC)

You said:

"follow a book by Trèves on topological vector spaces very closely, basically all but verbatim."

I just compared my edits with Trèves pp. 434-438 and it is not a verbatim copy.

Recall the definition of "verbatim":

verbatim (not comparable)

1. Word for word; in exactly the same words as were used originally. [quotations]

I have copied his speech verbatim, so this is exactly what he said, word for word.

Unlike my edit, the definition above was copied verbatim.

I did use much - but clearly not all (e.g. where's $\theta$?) - of the notation found in Trèves. (Is this not allowed for notation?) But please tell me what notation I should use for the tensor product of $p$ and $q$ if not $p\otimes q$? Ditto for the tensor product of sets?

And if you ignore all the content from Trèves that I didn't include and also ignore the parts that are out of order (e.g. I defined tensor products of seminorms before describing neighborhoods of the origin whereas Trèves presented neighborhoods first), then I did follow Trèves's presentation (as can be concluded from my citations to him) but nowhere did I plagiarize. Mgkrupa 02:34, 10 August 2023 (UTC)

I said "all but verbatim," not "verbatim." The point is that taking large portions of the source text and changing notation, phrasing, and order only just enough that it's not verbatim, which looks like what happened here, does not comply with policy. See WP:PARAPHRASE, which says among other things, Close paraphrasing ... when extensive (with or without in-text attribution) may also violate Wikipedia's copyright policy. ByVarying | talk 03:06, 10 August 2023 (UTC)

Thank you for clarifying that by "all but verbatim" you meant "closely paraphrasing". I just read through Wikipedia:Close paraphrasing in its entirety (thank you for the link) and although I feel that my edits are consistent with this policy (and I can argue this), I'd prefer to take the path of least resistance and instead work with you to edit the article in a way that resolves your concerns.

And since you are (presumably) better acquainted with the Wikipedia:Close paraphrasing policy than I am, perhaps you'd be willing to help me understand your concerns by discussing in detail some specific sentences that you think are problematic. I realize (based on your original post) that you think that every sentence in the section on seminormed spaces violates WP:PARAPHRASE but my paraphrasing of Prop 43.1 (for instance) is so substantially different from Trèves's statement on p. 435 that I do not see how WP:LIMITED doesn't apply since there's only so many ways you can change certain equations/definitions/facts before you start to violate some other Wikipedia policy. (For example, Trèves's definition of $p\otimes q$ that I gave in the article is clearly just the Minkowski functional of the set and that's the wording I would use, but I can't define it that way since then I'd be accused of violating WP:SYNTHESIS or WP:ORIGINAL RESEARCH).

The article's talk page (Talk:Projective tensor product) is the appropriate place to do discuss this futhure. I will start making changes once I can dedicate enough time to this, but you should please feel free to change the article as you wish (including the sentences that you consider problematic). Mgkrupa 18:13, 10 August 2023 (UTC)

Ok, thank you for being open to discussing this and I'll respond there. ByVarying | talk 18:27, 10 August 2023 (UTC)

Hilbert space pre-GAR

Hilbert space is currently designated a Good article. The GA criteria were recently tightened to require more inline citations. It would be nice if we could address this before going through the whole rigmarole of a Good Article Review. Somebody with the right textbook on their shelf could probably go through and knock out most if not all of the required footnoting in a single pass. See the Talk page for details. XOR'easter (talk) 17:51, 7 August 2023 (UTC)

And on the topic of Hilbert space, should the category be Category:Hilbert spaces or Category:Hilbert space? It was moved from the latter to the former a few months ago [1][2], seemingly without any notifications or discussion anywhere, though I might have missed something. I'd say that it should be in the singular, since it contains articles that pertain to the theme of Hilbert space but aren't about examples of Hilbert spaces. XOR'easter (talk) 00:24, 12 August 2023 (UTC)

I agree. The plural would be correct if we had lots of articles on individual Hilbert spaces, but for a category about the theory of these spaces, it should be singular. —David Eppstein (talk) 02:03, 12 August 2023 (UTC)

Affine symmetric group

Affine symmetric group is at FAC and would benefit from someone familiar with this subject inputting on the sourcing. See the bottom of the page here. Thanks. Gog the Mild (talk) 18:03, 17 August 2023 (UTC)

To be explicit about what is requested: in the section Source review, Jo-Jo Eumerus has marked six footnotes as needing to be checked by a mathematician as to whether the material to which they are attached is appropriately sourced (the footnotes support the article, the content of the sources is correctly represented in the article, the sources are reliable, etc.). Any help with this would be greatly appreciated. --JBL (talk) 20:40, 20 August 2023 (UTC)

Smith–Volterra–Cantor set

I have a concern about whether the name "Smith–Volterra–Cantor set" is in fact standard. I posted a note at talk:Smith–Volterra–Cantor set#Naming, but it's a low-traffic article and I don't really know whether anyone will notice it. Anyone who can contribute, it would be appreciated. --Trovatore (talk) 20:24, 21 August 2023 (UTC)

Lead image in Tree (set theory)

I am having a dispute with an editor of Tree (set theory) who insists on adding a lead image that depicts a graph-theoretic tree, not a set-theoretic tree (the difference is that in graph-theoretic trees every node except the root has a parent, and following parent links will eventually reach the root, but in set-theoretic trees the ancestors of any node can have arbitrary well-orderings in which not all elements have a parent). Additional informed opinions would be very welcome at Talk:Tree (set theory) and in editing the article (or maybe finding or creating a more appropriate lead illustration). —David Eppstein (talk) 04:32, 27 August 2023 (UTC)

Italics on Pentagramma mirificum

Should the title of this article be italicized? I forgot if every Latin term should be so, according to the guidelines.--ReyHahn (talk) 14:51, 25 August 2023 (UTC)

According to WP:ITALICTITLE, yes: Use italics when italics would be necessary in running text; for example, ... foreign phrases are italicized both in ordinary text and in article titles. —David Eppstein (talk) 04:36, 27 August 2023 (UTC)

It appears that it is used in italics in the article so I will carry out the italicization of the title. Be free to revert otherwise.--ReyHahn (talk) 12:29, 28 August 2023 (UTC)

Hurwitz space

I've promoted this from AFC. Its seems genuine and well referenced. I'm not a mathematician so I looked up some references to confirm its existance and to determine if there was an article on Wikipedia on the subject, but doesn't appear so. scope_creep^Talk 06:05, 31 August 2023 (UTC)

It's definitely a real thing; I am not an expert in this subject, but I am inclined to agree that it is not duplicative of existing content (at least that I could find). Thanks for promoting it. --JBL (talk) 17:22, 31 August 2023 (UTC)

Emmy Noether

Algebraically inclined editors may wish to look at Talk:Emmy Noether#WP:URFA/2020, where there is discussion of whether Emmy Noether meets current featured article standards (and if not, what must be improved). — Bilorv (talk) 17:36, 28 August 2023 (UTC)

So I don't have anything to add to that as a discussion per se but are there particular tasks that if someone did them it would be helpful for ensuring that the article stays at FA? --JBL (talk) 17:37, 31 August 2023 (UTC)

"Background on abstract algebra and begriffliche Mathematik (conceptual mathematics)" has only one reference, and the following subsection has none. The book containing the van der Waerden quote isn't a very good reference for the rest of the subsection; a textbook on abstract algebra would be more helpful there. XOR'easter (talk) 18:53, 31 August 2023 (UTC)

Where is the right place to feature request support for LaTeX macros/packages?

Wikipedia's Math/LaTeX syntax doesn't support the \rlap, \llap, \clap macros. I'd like to request feature support for these macros (perhaps in mathtools package? I'm not sure). These macros are incredibly useful to pad/align equations and terms, especially when the equations are in a column in a table and can't rely on a single align block. For instance, compare the table at Inverse trigonometric functions#Solutions to elementary trigonometric equations with its layout/markup prior to my edits. I had to use a lot of \;\,\qquad -style spacing. But instead, I could have used a more "semantic" spacing à la the following, if \rlap were available (ignore the align block; I used that in an external Latex editor to help me get the alignment):

\begin{align*}
\theta &= \rlap{(-1)^k}\phantom{(-1)^k\;}\rlap{\arcsin(y)}\phantom{arccos(x)} + \rlap{}\phantom{2\,}\pi k \\
\theta &= \rlap{(-1)^k}\phantom{(-1)^k\;}\rlap{\operatorname{arccsc}(r)}\phantom{arccos(x)} + \rlap{}\phantom{2\,}\pi k \\
\theta &= \rlap{\pm}\phantom{(-1)^k\;}\rlap{\arccos(x)}\phantom{arccos(x)} + \rlap{2}\phantom{2\,}\pi k \\
\theta &= \rlap{\pm}\phantom{(-1)^k\;}\rlap{\operatorname{arcsec}(r)}\phantom{arccos(x)} + \rlap{2}\phantom{2\,}\pi k \\
\theta &= \rlap{}\phantom{(-1)^k\;}\rlap{\arctan(s)}\phantom{arccos(x)} + \rlap{}\phantom{2\,}\pi k \\
\theta &= \rlap{}\phantom{(-1)^k\;}\rlap{\operatorname{arccot}(r)}\phantom{arccos(x)} + \rlap{}\phantom{2\,}\pi k \\
\end{align*}

Where do I lodge a feature request? Or should I lobby for it here first, and get community support for it? What's the process for Latex package addition? Thanks.  — sbb (talk) 17:53, 31 August 2023 (UTC)

You would want to use the Phabricator bug tracking system using the math tag. See https://phabricator.wikimedia.org/project/view/130/ There are other feature requests for extended maths syntax there. Don't expect a quick resolution, though.--Salix alba (talk): 18:13, 31 August 2023 (UTC)

Thank you. I completely understand not to expect a quick resolution. But it doesn't hurt to try to get reasonable requests in the queue! =)  — sbb (talk) 20:02, 31 August 2023 (UTC)

If you just care about this specific case, you could do something like:

${\begin{alignedat}{4}\theta ={}&&(-1)^{k}\,&\arcsin y&&+{\phantom {2}}\pi k,\\\theta ={}&&(-1)^{k}\,&\operatorname {arccsc} r&&+{\phantom {2}}\pi k,\\\theta ={}&&\pm \,&\arccos x&&+2\pi k,\\\theta ={}&&\pm \,&\operatorname {arcsec} r&&+2\pi k,\\\theta ={}&&&\arctan s&&+{\phantom {2}}\pi k,\\\theta ={}&&&\operatorname {arccot} r&&+{\phantom {2}}\pi k,\\\end{alignedat}}$

–jacobolus (t) 18:37, 31 August 2023 (UTC)

Thanks for that suggestion! Yeah, I always get confused about alignat positioning. I know each & demarcates an alternating left- or right-aligned column, but it still always confuses me.

But that doesn't work at all in cases where the equations are on separate rows in tables, which is what motivated my original question. Here's a reduced form of the table in the article I linked to:

Solution ​(for some $k\in \mathbb {Z}$)
$\theta =(-1)^{k}\arcsin(y)+{\phantom {2\,}}\pi k$
$\theta =(-1)^{k}\operatorname {arccsc} (r)+{\phantom {2\,}}\pi k$
$\theta ={\phantom {\quad \,\,}}\pm \arccos(x)+2\,\pi k$
$\theta ={\phantom {\quad \,\,}}\pm \,\operatorname {arcsec} (r)+2\,\pi k$
$\theta ={\phantom {(-1)\,\,}}\arctan(s)+{\phantom {2\,}}\pi k$
$\theta ={\phantom {(-1)\,\,\,}}\operatorname {arccot} (r)+{\phantom {2\,}}\pi k$

 — sbb (talk) 19:59, 31 August 2023 (UTC)