Wikipedia talk:WikiProject Mathematics/Archive/2020/Aug

In the article titled Centroid, we find this:

The centroid of a non-self-intersecting closed polygon defined by n vertices (x₀,y₀), (x₁,y₁), ..., (x_n−1,y_n−1) is the point (C_x, C_y),^[1] where

${\displaystyle C_{\mathrm {x} }={\frac {1}{6A}}\sum _{i=0}^{n-1}(x_{i}+x_{i+1})(x_{i}\ y_{i+1}-x_{i+1}\ y_{i}),}$ and

${\displaystyle C_{\mathrm {y} }={\frac {1}{6A}}\sum _{i=0}^{n-1}(y_{i}+y_{i+1})(x_{i}\ y_{i+1}-x_{i+1}\ y_{i}),}$

and where A is the polygon's signed area,^[1] as described by the shoelace formula:

${\displaystyle A={\frac {1}{2}}\sum _{i=0}^{n-1}(x_{i}\ y_{i+1}-x_{i+1}\ y_{i}).}$

In these formulae, the vertices are assumed to be numbered in order of their occurrence along the polygon's perimeter; furthermore, the vertex ( x_n, y_n ) is assumed to be the same as ( x₀, y₀ ), meaning ⁠${\displaystyle i+1}$⁠ on the last case must loop around to ⁠${\displaystyle i=0}$⁠. (If the points are numbered in clockwise order, the area A, computed as above, will be negative; however, the centroid coordinates will be correct even in this case.)

References

1. [1]

   Bourke (1997)

• Bourke, Paul (July 1997). "Calculating the area and centroid of a polygon".

Did I clumsily miss something, or should the second formula be as follows?

${\displaystyle C_{\mathrm {y} }={\frac {1}{6A}}\sum _{i=0}^{n-1}(y_{i}+y_{i+1})(y_{i}\ x_{i+1}-y_{i+1}\ x_{i})}$

Michael Hardy (talk) 22:23, 5 August 2020 (UTC)

Think the second product factor in ${\displaystyle C_{y}}$ just has to match what is in ${\displaystyle A}$. Deriving the formulas probably yield what you suggested, but with a negative ${\displaystyle A}$, and cancelling the negatives give the original formulas, and save us from defining ${\displaystyle A_{x},A_{y}}$. I'm just guessing, though. Walwal20 ^{talk ▾ contribs} 22:59, 5 August 2020 (UTC)

(ec) @Michael Hardy: To calculate the centroid we decompose the figure into oriented triangles between points ${\displaystyle O}$, ${\displaystyle P_{i}}$ and ${\displaystyle P_{i+1}}$. The centroid of the ⁠${\displaystyle i}$⁠-th triangle is ${\displaystyle {\frac {O+P_{i}+P_{i+1}}{3}}}$, hence the term ${\displaystyle 0+x_{i}+x_{i+1}}$ for ${\displaystyle C_{x}}$ and similar for ${\displaystyle C_{y}}$. The weight of the triangle's centroid in the sum is the triangle's area (signed!) which is a half of the cross-product of the ⁠${\displaystyle i}$⁠-th and ⁠${\displaystyle (i+1)}$⁠-st vectors - hence the same second paren both for x-es and y-s. The common factors 1/3 and 1/2 are moved outside the sum and make the 1/6 there.

If you swap the terms in the second parenthese, you'll effectively change the paren's sign and get a negated value for ${\displaystyle C_{y}}$ as a result. --CiaPan (talk) 23:07, 5 August 2020 (UTC), edited 07:11, 6 August 2020 (UTC) and 10:59, 6 August 2020 (UTC)

@Michael Hardy: You might just test it yourself: take a triangle with vertices (0,0), (0,1) and (1,0) and calculate its centroid. There will be just one non-zero term in each of there sums, and the hardest part of calculation will be a division (1/6):(1/2).
:) CiaPan (talk) 12:49, 6 August 2020 (UTC)

@CiaPan: and @Walwal: : Thank you for your attention to this. Michael Hardy (talk) 21:10, 10 August 2020 (UTC)

• Abstract cell complex (edit | talk | history | protect | delete | links | watch | logs | views)

I just saw this article, the vast majority of which was written (with some broken English) by a single user who appears to be the author of a book/application that the article is serving as a bit of a coatrack for. As far as I can tell, the basic topic is probably at least borderline notable enough for a separate article (the Klette article cited has a description and some further references), but the digression into the research of Kovalevsky is probably too much. But I'd certainly welcome a second opinion on this one. Thanks, –Deacon Vorbis (carbon • videos) 14:09, 11 August 2020 (UTC)

If anyone has experienced blurry math formulas in Chrome recently, you may be interested in a thread I've just started: Wikipedia:Village pump (technical)#Change to Chrome rendering of PNGs causing blurry formulas. –Deacon Vorbis (carbon • videos) 17:36, 12 August 2020 (UTC)

I would appreciate some experts on differential geometry to weigh in over here. Mathsci (talk · contribs) and I have both been editing the wiki page in the last few days. I find many of his edits to be written confusingly; according to my understanding of the material, his comments on the talk page do not address my complaints. Conversely he seems to think I am making too many major edits to his material. I have no interest in an argument and would like to defer to community consensus one way or another, but for the moment it's just me and him. Gumshoe2 (talk) 22:28, 10 August 2020 (UTC)

A lot of this article was edited by me in 2008. In the past I have edited a lot of mathematical content on wikipedia, starting in 2006. The concern with the current article has been to patch missing sections at the beginning of the article. That involved adding definitions from standard textbooks. Later, lecture notes of Nigel Hitchin suggested that it might be a good idea to have some material on covariant derivatives. I started by editing using Hitchin's notes and then Boothby's book on differential geometry. While editing, I had included Brioschi's proof of Gauss' Theorema Egregium (from Dirk Struik's book). Gumshoe2 then moved the material I edited on Brioschi's proof to another section. He also completely rewrote the section on covariant derivatives. From my point of view it is currently not properly sourced. That could probably be remedied using Thorpe's book on Elementary topics in differential geometry. The standard sources for this material are Helgason's book on Differential Geometry and do Carmo's book on Riemannian geometry. I like that material, but it is probably at the level of graduate students.

However, the aim in this article has always for it to be accessible to a wide readership. So covariant derivatives are probably out of the scope of the article, unless carefully presented (finding the sources).

I have twice given courses on the Atiyah-Singer index theorem in the UK: part of that required the theory of Riemannian covariant derivatives. It's on public record that I have had a stroke, which has had some consequences. Mathsci (talk) 23:35, 10 August 2020 (UTC)

I you have health problems that have no consequence on your work as Wikipedian, this is not worth to mentioning them. It they have consequences, it is to you to decide whether they allow you to conveniently edit Wikipedia, and if they not, to decide to stop editing Wikipedia. So, in any cases, it is not worth to mention here your health problems. D.Lazard (talk) 09:29, 11 August 2020 (UTC)

By the way, Mathsci has opened a WP:ANI discussion, entitled "D.Lazard and Differential geometry of surfaces", in which it is discussed whether WP:BOOMERANG must apply. D.Lazard (talk) 09:29, 11 August 2020 (UTC)

The usual five pillar always apply: articles require WP:RS when creating content. I am currently reading John A. Thorpe's book, "Elementary topics on differential geometry." I looked at your editing history. The article system of polynomial equations was created by you in 2010. That content described your own research. Mathsci (talk) 11:48, 11 August 2020 (UTC)

In my mind, each of the sentences in the previous post by Mathsci contains pretty much no relevant information to the discussion at hand. @Mathsci: if there is an issue here, it would be helpful to precisely (and concisely) state what is going on and refrain from adding unrelated bits of information. Jakob.scholbach (talk) 07:14, 12 August 2020 (UTC)

No problem. For smooth surfaces ${\displaystyle S}$, the standard formalism requires (tangent) vector fields ${\displaystyle X}$ viewed as derivations of ${\displaystyle C^{\infty }(S)}$. The notion of affine connections is introduced through the covariant derivative ${\displaystyle \nabla _{X}}$ which yields a vector field ${\displaystyle \nabla _{X}Y}$ from two vector fields ${\displaystyle X}$ and ${\displaystyle Y}$. The Riemannian connection of Levi-Civita is easy to construct directly using a projection operator, assuming an isometric embedding in Euclidean space. Its properties of compatibility with the metric and symmetry follow from the construction or can be proved directly as part of the fundamental theorem of Riemannian geometry. The curvature operator is defined as ${\displaystyle R(X,Y)=[\nabla _{X},\nabla _{Y}]-\nabla _{[X,Y]}}$. The assignment ${\displaystyle (X,Y,Z,W)\mapsto (R(X,Y)Z,W)}$ is ${\displaystyle C^{\infty }(S)}$-linear in all of its four variables. Elementary algebra shows that, if ${\displaystyle v_{1}}$ and ${\displaystyle v_{2}}$ are linear independent tangent vectors at a point in ${\displaystyle S}$, the quantity ${\displaystyle (R(v_{1},v_{2})v_{2},v_{1}))/((v_{1},v_{1})(v_{2},v_{2})-(v_{1},v_{2})^{2})}$ is independent of the choice of basis. It is the Gaussian curvature at that point. As a corollary, any local isometry of surfaces preserves Gaussian curvature.

This is standard material that generalizes to any even dimension; there are lots of sources (including Thorpe's book). The material can be presented in a coordinate-free way (as here), with Christoffel symbols (classical) or both (perhaps the best option). Mathsci (talk) 09:06, 12 August 2020 (UTC)

I fail to see how any of this (in the post right above) relates to an alleged misbehavior of D.Lazard. Jakob.scholbach (talk) 09:21, 12 August 2020 (UTC)

The question for the noticeboard was not about D.Lazard. It was about the section in the article on covariant derivatives. My post above yours specifically addressed that topic. The area of the article has always been studied by undergraduates. The aim of the article was to present the material for a general readership. The first diff of Gumshoe2 was about a technical detail with covariant derivatives, which can be resolved (using my previous post, specialized to the material of Hitchin, Thorpe et al). Where there might be a disagreement is with inline citations: for a general readership it seems they really have to be supplied. Mathsci (talk) 10:49, 12 August 2020 (UTC)

This is a notice that I started Draft:Calculus on Euclidean space. As I see, there are two issues: (1) how to cover calculus topics in some systematic and centralized manners as opposed to having separate articles and (2) how much background materials should be in differential geometry of surfaces. I am with Mathsci that the issue (1) is indeed an issue; thus, I have started that draft page by copying the section on functions in 2 variables. Mathsci has said above: "This is standard material that generalizes to any even dimension." I want to believe that draft page (eventually a mainspace article) can cover such standard materials. As for (2), I don't have much opinion other than one I made at the talkpage of the diff-geo article. -- Taku (talk) 00:11, 17 August 2020 (UTC)

I need to mention that there is Draft:Analysis of vector-valued curves, which has a section closely related and thus I am thinking of moving that section to Draft:Calculus on Euclidean space. -- Taku (talk) 00:52, 17 August 2020 (UTC)

Taku, thans for your replies. I agree with you about the topics you've raised. As far as calculus on Euclidean space is concerned, I think everything generalises to any dimension. Hörmander takes a little care about C^k issues. On the talk page of your calculus draft, I have mentioned that mixing elementary material on calculus with more advanced material, such as differential topology (smooth manifolds, differential forms, etc) seems unrealistic. Calculus is quite elementary. Understanding differential forms, orientablity, etc, is more advanced. Also for smooth surfaces, there's no need to invoke the general theory of manifolds, submanifolds, Stokes's theorem, etc. Mathsci (talk) 13:44, 17 August 2020 (UTC)

The disagreement over two-dimensional calculus was an independent problem some people had with Mathsci at the Differential geometry of surfaces page, not related to this one. I've made a comment on the talk page to your draft Gumshoe2 (talk) 09:29, 17 August 2020 (UTC)

Yes, but I do believe Mathsci has a valid point that some background stuff seems missing or not well-organized in a single place in Wikipedia, that is needed to read differential geometry of surfaces. He often refers to materials in existing literature, and, since Wikipedia is supposed to be *a summary of established math*, we do indeed need to have some presentation of those materials. The question is "how" not "if" (the draft like this one I started should be one approach). I will be responding to points specific to the draft in the talkpage of the draft. -- Taku (talk) 03:57, 18 August 2020 (UTC)

An IP editor has been editing Kenneth Kunen as though he has died. Is that true? I don't find any news story about it on a brief search. --Trovatore (talk) 18:53, 15 August 2020 (UTC)

I don't know but it's obviously not acceptable to put it into the article without a source, so I've reverted the IP. (I've also left a message on the IP's talk page asking if they have a source.) --JBL (talk) 19:11, 15 August 2020 (UTC)

I concur; I was about to revert it myself but you got there first. Thanks for taking care of it and for leaving a note at the IP's talk page. XOR'easter (talk) 19:43, 15 August 2020 (UTC)

Answer appears to be yes, sadly. Article now has a source for it. —David Eppstein (talk) 17:13, 18 August 2020 (UTC)

I have created an article titled Expected mean squares. I think the plural may be justified by the fact that nobody (as far as I know?) ever has occasion to think about just one expected mean square, in view of their use in forming F-tests. Michael Hardy (talk) 00:05, 20 August 2020 (UTC)

There is currently a discussion at Wikipedia:Administrators' noticeboard regarding an issue with Mathsci (talk · contribs), Gumshoe2 (talk · contribs), D.Lazard (talk · contribs) and the articles on Symmetry of second derivatives, Differential geometry of surfaces. At the moment I'd like this to focus more on a possible process to follow rather than the actual dispute. --Salix alba (talk): 16:03, 22 August 2020 (UTC)

Hi,

I migrated the material on Mirror symmetry conjecture to Mirror symmetry (mathematics). Will someone remove the first article since the second's title is more in line with wikipedia standards? Here's the analogy: Mirror symmetry is to Mirror symmetry (mathematics as Gauge theory is to Gauge theory (mathematics). Wundzer (talk) 16:35, 18 August 2020 (UTC)

Unfortunately, by simply moving the content over, you have destroyed the edit history. The preferred way to make such a change is described at Wikipedia:Moving a page. You can do it yourself. :) Mgnbar (talk) 16:46, 18 August 2020 (UTC)

Actually, now that you've created Mirror symmetry (mathematics), you won't be able to move Mirror symmetry conjecture. You'll have to create a request at Wikipedia:Move requests. Mgnbar (talk) 16:51, 18 August 2020 (UTC)

What does it mean that the second title is "more in line with wikipedia standards"? Generally article titles on Wikipedia use parenthetical disambiguation when more natural disambiguation is not available. I have marked the new article for speedy deletion; after it is removed, we can start over with appropriate process, including perhaps a discussion of whether retitling is appropriate at all. --JBL (talk) 16:54, 18 August 2020 (UTC)

(edit conflict):It appear that you have copied the article instead of doing a move (see WP:MOVE). Moreover the title you suggest is definitively not in line with WP standards, in particular with WP:LEAST, as, for most mathematicians, "mirror symmetry" is synonymous with reflection symmetry. So, I'll redirect the new article, and its talk page to Reflection symmetry. If you want to change an article title, please follow the procedure described at WP:Move request. D.Lazard (talk) 16:58, 18 August 2020 (UTC)

I think that's largely incorrect. As far as professional mathematicians are concerned, mirror symmetry typically refers to the results from string theorists in the 90's and the related programs which came out of it. Here's a list of resources by professional mathematicians alive today who use Mirror symmetry to refer to this area: http://www.claymath.org/library/monographs/cmim01c.pdf , https://ncatlab.org/nlab/show/mirror+symmetry , https://mathoverflow.net/questions/tagged/mirror-symmetry , https://arxiv.org/search/advanced?advanced=&terms-0-operator=AND&terms-0-term=mirror+symmetry&terms-0-field=all&classification-mathematics=y&classification-physics=y&classification-physics_archives=math-ph&classification-include_cross_list=include&date-filter_by=all_dates&date-year=&date-from_date=&date-to_date=&date-date_type=submitted_date&abstracts=show&size=50&order=-announced_date_first Wundzer (talk) 17:17, 18 August 2020 (UTC)

How can I recover the edits I made on Mirror symmetry (mathematics)??? I'm very frustrated these changes were deleted: I want to put them on the Mirror symmetry conjecture page... Wundzer (talk) 17:11, 18 August 2020 (UTC)

Is this page about the same topic as Homological_mirror_symmetry??? --JBL (talk) 17:12, 18 August 2020 (UTC)

No! Homological mirror symmetry is a separate formulation of mirror symmetry. These are not interchangeable and mean different things. Wundzer (talk) 17:14, 18 August 2020 (UTC)

There's some discussion on the talk page about the topic and the expository approach. (It's not 100% clear to me that it warrants its own page, but I'm not an expert on the topic.) It'd be nice to have some more commenters. Gumshoe2 (talk) 17:26, 18 August 2020 (UTC)

Let me explain the reasoning: Mirror symmetry generally refers to the more analytic and combinatorial part of the subject. It studies things like Quantum cohomology, Gromov-witten invariants, and Picard-Fuchs differential equations. On the other hand, homological mirror symmetry is more focused on studying a (partly) conjectural equivalence of categories, the derived category of coherent sheaves and the Fukaya category. This area uses tools like A-infinity categories, Hochschild homology, and derived geometry. I think separating out the subjects is warranted, but the Homological mirror symmetry page is in serious need of a clean-up: it reads like a pop-science article but contains sparse content. This could be re-written to include some of the formulations and discuss some of the proofs, like HMS on the elliptic curve, Seidel's proof for K3 surfaces using Vanishing cycles, and Sheridan's article on HMS for hypersurfaces. Wundzer (talk) 17:52, 18 August 2020 (UTC)

Just be careful to make sure your content is as accessible as possible. I believe the ideal wiki article is quite different from an ideal review article. Gumshoe2 (talk) 18:12, 18 August 2020 (UTC)

I agree with making the material accessible and that's the main reason why I care about writing these articles. I think writing in the survey/review format is good for a baseline model because it is easy to expand upon and link to relevant material. Moreover, it organizes the material in a top-down fashion which (I think) can help make it a great wiki article. I do try my best to make content accessible, but will make sure to keep a keener eye for linking relevant material. Wundzer (talk) 20:03, 18 August 2020 (UTC)

• Filters in topology (edit | talk | history | protect | delete | links | watch | logs | views)

This article was just created by Mgkrupa. We already have an article at Filter (mathematics), which (aside from basic definitions) is mostly focused on their use in topology. I'm a little skeptical that this should be spun out into a separate article, especially so because what's there now falls very squarely on the wrong side of WP:NOTTEXTBOOK. I'm trying to limit my activity here these days, but I'd certainly welcome others' opinions on this. Thanks, –Deacon Vorbis (carbon • videos) 00:30, 20 August 2020 (UTC)

On a side note, it makes extensive use of special Unicode characters, like (U+212C ℬ SCRIPT CAPITAL B) for example, which I'm under the impression should generally be avoided due to rendering support. –Deacon Vorbis (carbon • videos) 00:41, 20 August 2020 (UTC)

My general impression of Mgkrupa's contributions is that they tend to be very WP:TECHNICAL. For subjects that really are technical, this may be ok, but it tends to spill over onto other subjects where the technicality is an unnecessary obstacle (for instance, I had to revert several times Mgkrupa trying to rework the definitions in convex hull to focus on topological vector spaces rather than Euclidean spaces). —David Eppstein (talk) 00:53, 20 August 2020 (UTC)

I have had a view that the focus of the use in topology in Filter (mathematics) is a *problem*. A "filter" is after all a general concept. A filter (in particular an ultrafilter) is also used in abstract algebra in the form of ultraproduct and also is a useful concept in the theory of Boolean algebras (if I recall, please correct me if you know this stuff). It seems natural to have a separate article on the use of a filter in topology in an independent article (in that case, it is essentially equivalent to net but should probably be separated from that). -- Taku (talk) 01:11, 20 August 2020 (UTC)

This would require an expert (not me) but, for example, we might want to mention a result of G.M.Bergman - Ehud Hrushovski, Linear ultrafilters or some other similar results. In any case, the point is that a filter is a general concept. -- Taku (talk) 01:28, 20 August 2020 (UTC)

Another related matter: ultrafilter is a separate article from "filter (mathematics)"; which is not necessarily bad but if we can separate the topology stuff from "filter (mathematics)", then there will be more real estate to merge the ultrafilter article to the filter article. —- Taku (talk) 02:32, 22 August 2020 (UTC)

I don't really think ultrafilters should be merged into filters. There's plenty to say about ultrafilters as a topic on their own. Also the problems studied with regard to ultrafilters tend to have a bit of a different character from those that come up using filters in general. --Trovatore (talk) 22:43, 25 August 2020 (UTC)

Yes, "ultrafilters" certainly makes sense as an independent topic; that does not mean it needs a separate article. For example, "embedding" discusses both embeddings of a manifold and a category; even though there is *plenty to say* about embedding of a manifold (e.g., Whitney embedding theorem). I suppose the problem is that currently filter (mathematics) is somehow underdeveloped than ultrafilter; the latter mentions a filter (in particular an ultrafilter) on a Boolean algebra while the former has nothing on the use in a Boolean algebra. It would be helpful to the readers including non-specialists (like myself) if there is a single article to read on filters as well as ultrafilters; a spin-off of ultrafilter is needed only after that single article becomes too long. It's possible that filter (mathematics) ends up discussing mostly ultrafilters; I don't think that would be a problem. -- Taku (talk) 00:04, 26 August 2020 (UTC)

No, I really don't agree. I feel very strongly that ultrafilters should retain a dedicated article. Ultrafilters come up largely in the context of ultraproducts. They're mostly a technique for getting elementary embeddings. A typical ultrafilter is generally not definable. That's not true at all for general filters, which can easily be definable, and are studied in lots of other contexts. --Trovatore (talk) 00:37, 26 August 2020 (UTC)

I agree with Trovatore. JRSpriggs (talk) 04:12, 26 August 2020 (UTC)