Wikipedia talk:WikiProject Mathematics/Archive/2022/Sep

Classification of torsion-free abelian groups

This rather vaguely written popularization says it was recently proved that the classification of torsion-free abelian groups is a "hard" problem. Should our article titled Torsion-free abelian group mention the classification problem?

I have never thought about torsion-free abelian groups, so in reading this, my first thought was "Isn't every torsion-free abelian group a direct product of infinite cyclic groups?", but then I thought: The rationals with addition are a torsion-free abelian group, and so is the group of binary rationals with addition (i.e. rationals whose denominator is a power of 2). This raises another question: Even if the answer to the first question above is "no", should we add a diversity of examples to the article? Michael Hardy (talk) 02:35, 31 August 2022 (UTC)

If I remember, the classification of torsion-free abelian groups is very difficult and it is a field itself. By Googling, at least I found this [1]. The Wikipedia article certainly doesn’t do the justice and it’s one of those that require expertise to have an adequate treatment. —- Taku (talk) 06:33, 31 August 2022 (UTC)

Paolini and Shelah's result seems too technical to state properly in this article as there seems to be no articles on wikipedia covering the relevant background on model theory and descriptive set theory in sufficient depth. It seems reasonable to add more non-finitely generated examples and a few more elementary definitions, as the article could include some of Baer's theory (https://zbmath.org/?q=an%3A63.0074.02), in particular the classification of groups of rank 1 (subgroups of the additive group of the rationals). jraimbau (talk) 07:15, 31 August 2022 (UTC)

There is actually the article torsion-free abelian group of rank 1, which covers the classification in rank 1. I am not too sure if it should be a separate article; as far as I understand it doesn't generalize to higher ranks -- Taku (talk) 07:55, 31 August 2022 (UTC)

I think this should be merged into the torsion-free abelian group article, since as noted in this discussion it suffers from lack of content rather than overabundance thereof. Maybe some of the content of the article on Rank of an abelian group#Groups_of_higher_rank could also be mentioned in this article. jraimbau (talk) 08:05, 31 August 2022 (UTC)

The classification result is for finitely generated torsion-free Abelian groups. In general one expects more complicated behaviour. It is similar to (more or less the same as) trying to understand how operators behave as finite-dimensional matrices (where the Jordan decomposition completely answers the question) vs infinite dimensions (the entire field of functional analysis). Tazerenix (talk) 09:35, 31 August 2022 (UTC)

This is not about a classification result but a classification problem. I'm not a specialist but it seems that people think that the classification problem for countable torsion-free abelian groups is intractable beyond rank-1 groups, and so have taken to the approach of trying to quantify its "difficulty" using descriptive set theory. If i understand it correctly the theorem of Shelah and Paolini that started this discussion states that in this sense the problem for torsion free abelian group is hardest possible among classification problems for structures on countable sets.

I don't find your analogy with finite dimension/functional analysis to be very convincing, is it backed by an actual result? jraimbau (talk) 12:28, 31 August 2022 (UTC)

I was merely replying to the sentence "Isn't every torsion-free abelian group a direct product of infinite cyclic groups?". My comment is just about the proof of the classification for finitely generated Z-modules using the existence of Jordan decompositions. I agree the article should mention the classification problem beyond the finitely generated case. Tazerenix (talk) 17:48, 31 August 2022 (UTC)

I took a swing at editing the article following this conversation (my thanks to the participants). I did include a small paragraph on what prompted it---the Paolini--Shelah preprint---but it is not very good and anybody with working knowledge of the relevant fields would be welcome to improve on it (i find that very interesting but i don't have the time to delve into it now). Maybe writing an account of the Friedman--Stanley paper somewhere on wikipedia would also be useful (maybe it's already here, didn't have the courage to look for it). jraimbau (talk) 13:26, 2 September 2022 (UTC)

Harmonic function vs. Laplace's equation

There is at present one wiki page for harmonic function and one for Laplace's equation. I take these two topics to have purely grammatical difference (a harmonic function is defined as a solution of Laplace's equation), and no mathematical differences. (There are very likely some generalizations or modified contexts with a difference, probably for instance in graph theory or functions valued in metric spaces, but that is not presently relevant to either page - generalizations present on the harmonic function page all satisfy generalizations of Laplace equation.)

So I propose the two pages should be merged. Any thoughts? If agreed, the obvious followup question is whether the page should be called "harmonic function" or "Laplace's equation"? Both are extremely fundamental and widespread vocabulary.

Relatedly, there is also some content split between Laplace's equation and Laplacian, and I would also argue some material presently in the former page (such as fundamental solution) is actually about the Laplacian, and not Laplace's equation. (A differential operator has a fundamental solution, a PDE doesn't have a fundamental solution - although admittedly it is common to misspeak in that way.) Gumshoe2 (talk) 19:27, 7 September 2022 (UTC)

I would say all the content of the Laplace's equation page needs to be found somewhere on the wikipedia, even not on that page. Right now the situation is there are three separate pages which roughly cover:

• the properties of the operator at Laplace operator including generalisations
• the properties of the equation at Laplace's equation. This should include the common set ups of boundary value problems, descriptions of the equation in different coordinates (things which are likely to be incredibly useful and commonly looked for for visitors of that page)
• properties of solutions to the equation at Harmonic function including generalisations.

I think I believe these three topics are different enough that they should either have 3 separate pages, or combined into a single page. In particular I think the content on Laplace's equation is important enough to the average visitor that it doesn't make sense to bury it in either harmonic function or Laplace operator without making it abundantly clear where you would find it. If I had to choose it would be absorbed into Laplace operator almost entirely. Tazerenix (talk) 22:51, 7 September 2022 (UTC)

I agree that essentially all material on Laplace's equation page is significant and would be looked for by someone going to that page (the Schwarzschild section is an exception - it seems wrongly stated and is of dubious significance) - or moved elsewhere and, as you say, clearly linked to. But I think all that material, with possible exception of "boundary conditions" section, would be equally looked for by someone going to the harmonic function page.

For completeness, here is virtually all material on Laplace equation page (except "boundary conditions" section), minimally rephrased so as to be a fundamental aspect of harmonic functions: that the real and imaginary parts of a holomorphic function are harmonic, that any harmonic function is analytic and hence has locally has a conjugate harmonic function, that harmonic functions have a particular type of Fourier series expansion, that the Cauchy-Riemann equations arise in fluid flow and hence that the derivatives of a harmonic function appear as the velocity field, that harmonic functions describe electrostatic configurations, that the only rotationally symmetric harmonic functions are $c_{1}\log |x|+c_{2}$ and $c_{1}|x|^{2-n}+c_{2}$, that any harmonic functions on any region is represented by a certain kind of convolution of a certain region-dependent function with the boundary values, that harmonic functions can be expanded by spherical harmonics, and that harmonic functions describe gravitational vacuum in classical field theory.)

And even the content of "boundary conditions" section is equally fundamental as a statement directly about harmonic function, i.e. a harmonic function on any compact region is uniquely determined by its values along the boundary, and the choices of boundary values parametrize the harmonic functions on interior. It is just a little verbally clunkier to describe this way. (It can also be phrased as a bijection between function space of harmonic functions and function space of boundary values, although this is a little nonstandard and so not good for wiki.)

Conversely, if I were looking for information about Laplace equation, I would want the information on harmonic function page.

By contrast I think the scope of Laplace operator/Laplacian is rightfully broader, as it belongs equally well to Poisson equation, heat equation, Schrödinger equation, wave equation, etc. Gumshoe2 (talk) 00:56, 8 September 2022 (UTC)

So, I don't have a strong opinion on the matter but one thing that comes to mind: we have separate articles for holomorphic function and Cauchy–Riemann equations. If we were to merge harmonic func and Laplace equations into one article, then it seems logical that they too should be in one article. Maybe they should. It's essentially the question of what style editors (really math editors) would like to adopt (I am not sure what style is good). -- Taku (talk) 07:16, 9 September 2022 (UTC)

From a physics POV, identifying a PDE and its set of solutions as the same thing doesn't make a lot of sense--the PDE, and its symmetries, are the fundamental objects in a physical model of say, electric fields. How to calculate solutions, whether analytically or numerically, is a conceptually separate topic. But if you all wanted to go the unification route, I think you would need to consider the potential theory article as well. --{{u|Mark viking}} {Talk} 17:34, 9 September 2022 (UTC)

I essentially agree with what you say in and of itself, but it does not seem to describe the actual difference between these two pages, even in the form they are currently written. But thank you for pointing to potential theory page. So my issue is if I want to add material to wiki about harmonic functions, I would have no clue which of these three pages to add it to.

For instance, consider the Cheng–Yau estimate for harmonic functions. Its proof and reasoning is entirely a manipulation of the equation itself and the key point is about its symmetry of commutation with the derivative, so perhaps it should go to Laplace equation; however the property itself is very purely about solutions of the equation, so maybe instead to harmonic functions; however in Davies' book (a standard ref), it is the primary topic of a section called "potential theory".

I think this example is characteristic and there is no natural/obvious division between these pages. And if there is to be a division then it should be made clear to readers and editors on each of the three pages. Gumshoe2 (talk) 19:02, 9 September 2022 (UTC)

A note on the relationship between holomorphic functions and Cauchy–Riemann equations is the Looman–Menchoff theorem. However, currently the redirect target of holomorphic is a holomorphic function, but I think the Cauchy–Riemann equation might be a better target. --SilverMatsu (talk) 03:53, 11 September 2022 (UTC)

As a general rule I think it’s just fine to keep such subjects separate, even if they are largely overlapping. –jacobolus (t) 04:04, 11 September 2022 (UTC)

I agree; I also think that as a reader it's helpful when each article in such a grouping links the others in fairly prominent ways. --JBL (talk) 18:08, 11 September 2022 (UTC)

There is no way we are going to eliminate all overlaps in Wikipedia articles, and an overlapping style is even encouraged by some Wikipedia editing guidelines (notably Wikipedia:Summary style). I don't have a strong opinion or knowledgeable point of view on this specific case. But it is an obvious principle that when overlaps occur the articles should point to each other. —David Eppstein (talk) 20:23, 11 September 2022 (UTC)

I agree that overlap is ok, and in many cases desirable. My problem here is that three different pages (harmonic function, Laplace equation, and potential theory) are about exactly the same topic, to the extent that they are (to my eyes) indistinguishable, something like having one page for "mathematical constant e" and another page for "Euler's number". (caveat: I do have expertise on the mathematics involved here, although embarrassingly "potential theory" is new to my vocabulary!)

But I can see that I have a minority point of view. So in terms of adding content I will just try to select which of the three pages already has content with the closest fit. But the minimum to ask for is that each page very clearly link to the other two, perhaps via a note at the top, above the main text. I'm not sure specifically of the most appropriate format. Gumshoe2 (talk) 20:54, 11 September 2022 (UTC)

Fair enough, but we do have e (mathematical constant), Natural logarithm, Exponential function, List of representations of e, Proof that e is irrational, List of logarithmic identities, Characterizations of the exponential function, Exponential growth, Exponential decay, Matrix exponential, Logarithm of a matrix, Exponential map (Lie theory), Derivative of the exponential map, Exponential map (Riemannian geometry), Exponential map (discrete dynamical systems), History of logarithms, Slide rule, Logarithmic scale, Logarithmic derivative, Logarithmic differentiation .... (see List of exponential topics for a longer list). There is significant overlap involved in many cases here. Some questions: can these separate articles be extended in different directions? Would either article be improved by cutting down an extended section into a simple summary and including a link to a fuller treatment at the other page? –jacobolus (t) 00:42, 12 September 2022 (UTC)

As an example, I think Potential theory could include more discussion of its applications to approximation theory that would not necessarily be appropriate to include in harmonic function. Cf. literature search for «"potential theory" approximation». –jacobolus (t) 00:45, 12 September 2022 (UTC)

Well, again, my issue is with synonymous topics (Laplace equation and harmonic function), not overlapping topics. None of those topics (aside from the first itself) are synonymous with Euler number itself, or reasonably interpretable as such.

Anyway, on inspection, I think that the opening sentence of potential theory wikipage (copied from planetmath) is incorrect, and so perhaps potential theory should be excluded from consideration here. See for instance short discussion in example 9.13 in Renardy & Rogers "An introduction to PDE". Classical potential theory seems to be about newtonian potential; if this is the case, then harmonic functions are certainly very relevant to potential theory but far from the whole story. Poisson equation (with Laplace equation as only a special case) would be a closer point of reference; however it seems that modern potential theory or nonlinear potential theory contains an even broader section of equations than Poisson. See e.g. section 2.6 of Morrey's "Multiple integrals in the calculus of variations" in dealing with what it calls "generalized potential theory" (here standard potential theory, in section 2.5, is indeed about Newtonian potential, and the relevant PDE is indeed Poisson equation, not Laplace equation/harmonic function) Gumshoe2 (talk) 21:24, 12 September 2022 (UTC)

Numerical methods for PDE

I would like to propose that WikiProject Mathematics take a more deliberate approach to talking about numerical methods for PDE. Especially for non-linear PDE, I suspect that many people who search for e.g. Burgers' equation, the Monge-Ampere equation, or the Korteweg–De Vries equation are interested in numerical methods. However, the vast majority of articles on PDE only address analytical aspects, like well-posedness, etc. (None of the articles for the aforementioned PDE examples speak at all about numerical methods, as of 2022-9-12.) I'm only a casual Wikipedia editor, so I'm not sure how to implement this administratively, but I strongly believe it would be a very useful feature of WikiProject Mathematics.

I'm also not sure what would be the best way to summarize numerical methods for a given PDE, but I can propose some ideas:

• If a survey article exists in the literature, simply linking to it would be most useful.
• If there are only a few methods proven to converge, then it seems reasonable to mention them directly. E.g. "Such and such researcher has provided a wide-stencil method with proven convergence properties."
• If there are many methods proposed in the literature, but no available survey article, then it is usually possible to extract general features of the known solution methods from e.g. the introductions to papers on the subject. In such a case, it would be nice to recapitulate such general features, along with a few links to example literature. E.g. "Many methods proceed by converting the elliptic equation into a time dependent parabolic equation and solving the latter by finite difference schemes [refs]."

--171.64.108.74 (talk) 03:27, 13 September 2022 (UTC)

What is stopping you from adding that material to the relevant Wikipedia pages (besides the obvious: research and writing takes a lot of work)? Edit: For anyone who likes clicking links more than typing into search boxes, relevant pages here might include Nonlinear partial differential equation, pages linked from List of nonlinear partial differential equations including Burgers' equation, Monge–Ampère equation, Korteweg–De Vries equation, and perhaps other pages linked from List of partial differential equation topics. –jacobolus (t) 04:48, 13 September 2022 (UTC)

It is partly "the obvious" which is stopping me, and partly the lack of precedent. I would expect that I am a typical user of these articles on non-linear PDE; I have subject matter knowledge on some of them, but I rarely make edits which are not incremental. My main purpose in raising this issue here is to alert more experienced editors--people who are invested in WikiProject Mathematics--that there is a big gap between what is offered and what is needed for these articles. If there were any sort of established template for how to write about numerical methods for a given PDE, I would probably contribute to those articles for which I have knowledge. I was hoping that this would be an OK place to alert the community that this is desired change, and to start a discussion on how to go about it. I am not qualified to implement this myself. I can add knowledge, but I do not wish to invent the paradigm for writing about numerical methods for PDE all by myself. --171.64.108.74 (talk) 19:35, 13 September 2022 (UTC)

If you did a careful survey, I imagine you’d find that the majority of the content of articles in Wikipedia about every technical field are written by (more or less) amateurs such as hobbyists and students. There are surely some excellent professional mathematicians around here but most tenured professors are busy with other projects. In an ideal world perhaps every world-class expert would write (or at least review) the article(s) about their specific topic of study as a public service, considering that the Wiki page is likely to have at least an order of magnitude more readers than any of their papers. But since that doesn't happen, even relative novices shouldn’t hesitate to make "non-incremental" edits if they think they are warranted, without worrying too much about "precedent". The additions are (hopefully) still helpful compared to nothing. –jacobolus (t) 00:50, 14 September 2022 (UTC)

Is this perhaps a proposal to create *separate* articles for numerical methods in PDE? I noticed we have Numerical methods for ordinary differential equations, while we don't have a corresponding article for PDE. That sounds quite reasonable. (We have an article titled Numerical Methods for Partial Differential Equations, but that article is about a specific journal.) -- Taku (talk) 11:26, 13 September 2022 (UTC)

Sorry we have Numerical methods for partial differential equations, which is about numerical methods in PDE. So maybe that article covers what is to cover? -- Taku (talk) 11:52, 13 September 2022 (UTC)

No, this is definitely not a proposal for an article on numerical methods for PDE. The difficulty is that numerical methods for non-linear PDE are very much tailored to the specific details of the PDE. This is precisely why it is very important for the numerical methods to be discussed on the same page as the PDE itself. What I would like is (1) community guidance (or precedent) on how to write a "Numerical Methods" section for a given non-linear PDE article, and (2) for this to be considered a priority by the community for PDE articles. --171.64.108.74 (talk) 19:35, 13 September 2022 (UTC)

Generally this is a glaring missing feature of the analysis articles on the wikiproject (there seems to be quite a bias towards pure methods). I think adding sections to the relevant pages even with one or two sentences of content and a banner indicating a need to expand would be appropriate. Tazerenix (talk) 22:50, 13 September 2022 (UTC)

I wholeheartedly support this. It would be an excellent start. 171.64.108.74 (talk) 00:50, 14 September 2022 (UTC)

Although I agree, as far as I know there are not any very active editors here who are knowledgeable about numerical analysis of PDE. And there is danger of doing more harm than good when people try to add content about material they don't understand very well, even if following good references and guidelines. With this in mind, I think Tazerenix's suggestion is good. Gumshoe2 (talk) 02:11, 14 September 2022 (UTC)

Geometric examples in composition function

I had read this section, but it was confusing me until now. As a matter of fact, I don't know that there were other examples of geometry in composition function. Dedhert.Jr (talk) 05:35, 14 September 2022 (UTC)

I have reverted the (recent addition) of this section. D.Lazard (talk) 09:34, 14 September 2022 (UTC)

@D.Lazard Thanks a lot. I was trying to restore those edits, but then I was afraid that my edit become considered disruptive. In this case, I had to cancel it and then discuss this problem here. Dedhert.Jr (talk) 11:47, 14 September 2022 (UTC)

My eyes are bleeding from the style violations by Arthur Baelde at Rotational symmetry, Wallpaper group#What this page calls pattern, and Multiview orthographic projection. 171.66.135.140 (talk) 12:29, 14 September 2022 (UTC)

I removed the addition at Rotational symmetry which, in addition to being impenetrable, was entirely off-topic. 100.36.106.199 (talk) 12:40, 14 September 2022 (UTC)

Also at Multiview orthographic projection, for the same reasons. Someone else will have to consider Wallpaper group. 100.36.106.199 (talk) 12:43, 14 September 2022 (UTC)

Someone should do something about the caption on the remaining geometric image at Function composition, too, it's a real formatting nightmare. 100.36.106.199 (talk) 12:47, 14 September 2022 (UTC)

Yet again "weierstrass substitution"

Sigh. Continuing on the discussion/disputes from prior years (Cf. WikiProject math talk archives from 2013, 2020, 2022), we again have disagreement at Tangent half-angle substitution about how to present Weierstrass’s (lack of) involvement. Does anyone else want to weigh in at Talk:Tangent half-angle substitution#Name? (Also see Talk:Tangent half-angle substitution#Common name for context.) –jacobolus (t) 20:48, 21 September 2022 (UTC)

So far XOR'easter and I have weighed in, but I do think this dispute would benefit from broader input. JBL (talk) 00:58, 23 September 2022 (UTC)

Fractal articles

Hello WPM. Could someone with some expertise on fractal analysis please have a look recent edits to analysis on fractals, and the new article fractal calculus? There appears to be significant overlap, and a conflict of interest by the new article's creator. The older analysis article currently opens with "Analysis on fractals or fractal calculus..."

Should the two articles be merged? At the least, the newer article needs cleanup for essay tone, but I lack the subject knowledge for a good rewrite. Thanks for any help with this. Storchy (talk) 08:36, 20 September 2022 (UTC)

It seems like the same editor added a bunch of stuff to analysis on fractals back in 2020. Here's what it looked like before he weighed in. The "seminal" 2009 article does have a lot of citations but (at the risk of sounding elitist) none in journals I've heard of. It's also only applicable to fractal subsets of the real line. Given the cited earlier books by Kigami and Robert Strichartz, overfocusing on the 2009 article would seem like an due weight issue.

That said, I also lack subject knowledge. We might not even have any active Wikipedians with the right subject knowledge. I can try to put it on my to-do list and read some survey articles, but I also need to cut back on my Wikipedia editing... —platypeanArchcow (talk) 16:34, 20 September 2022 (UTC)

The fractal calculus article is a mess and appears unsuitable for wikipedia in any case. I'm not sure whether the contents should be incorporated into the analysis on fractals article, and i don't have anything substantial to add to what @PlatypeanArchcow: wrote above, lacking as he does the relevant expertise.

I'm going to make some proposals to advance the conversation: that the fractal calculus be AfDed and the legitimate article about analysis on fractals be reverted to its state previous to the edits by Golmankhaneh, perhaps adding a short sentence mentioning the development of a "fractal calculus" by Gangal--Parvate. It could also be useful to add some content from Strichatz's 2006 survey that is cited in the article. jraimbau (talk) 06:21, 22 September 2022 (UTC)

I believe that analysis on fractals needs a {{stub}} template and fractal calculus needs a definition of nomenclature, e.g., $S_{F}^{\alpha }(x)$. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 11:13, 22 September 2022 (UTC)

Fractal calculus hypes up the work of cranks (Nottale, Hameroff) and is full of word salad (e.g., In this model, time deepens into timelessness as energy folds back on itself in repeating cycles building matter, building time). Recommend killing with fire. XOR'easter (talk) 21:16, 25 September 2022 (UTC)

That's very poetic! JBL (talk) 23:36, 25 September 2022 (UTC)

"In this model, time arises from the enfolding of non-time and the erasure of points that do not become matter, as in the construction of the Cantor set. Hold on, I'm taking a hit..."

Oh, and it's copyvio. XOR'easter (talk) 01:42, 26 September 2022 (UTC)

Thank you for the work! i restored the old version of analysis on fractals. jraimbau (talk) 13:43, 26 September 2022 (UTC)

Thanks everyone for your help with this. Storchy (talk) 05:25, 29 September 2022 (UTC)

Working on a draft: Ordinal Priority Approach

Hello everyone,

I hope you are having a great time. I'm working on this draft right now. Please feel free to help me to improve it. Thank you so much for your great help in advance. Scholartop (talk) 16:00, 29 September 2022 (UTC)

I'd like to read a simple introductiory example there, if possible. - Jochen Burghardt (talk) 19:04, 29 September 2022 (UTC)

This sounds like it is about business management rather than mathematics. JRSpriggs (talk) 21:14, 29 September 2022 (UTC)

Thank you for your comment. Actually, I agree with you. I will remove it from this category and will discuss on the new one. Scholartop (talk) 21:19, 29 September 2022 (UTC)

Scholartop This does not seem to be a mathematics article. It makes use of mathematics, but it seems to be more of an applied science thing. Not sure how to categorize it (statistics?, engineering, management science?) , but I don't think it should be categorized as a mathematics article per se. PatrickR2 (talk) 21:15, 29 September 2022 (UTC)

Thank you. I will remove it from this category. You are completely right. Scholartop (talk) 21:20, 29 September 2022 (UTC)