Wikipedia talk:WikiProject Mathematics/Archive/2020/Sep

"Admits"

I'm having difficulty tracking down a good definition for the technical use of the word "admits" for use in the List of mathematical jargon article. (See Talk:List_of_mathematical_jargon#"Admits"). Can anyone provide one, preferably with a WP:RS to support it? -- The Anome (talk) 10:10, 20 August 2020 (UTC)

@The Anome: Although this is a well-used word in mathematical writing, it's not really a technical meaning of the word. Standard dictionaries give entries that are in line with the usage here: e.g., sense 4 in wikt:admit

(transitive) To be capable of; to permit. In this sense, "of" may be used after the verb, or may be omitted.

     the words do not admit such a construction.

–Deacon Vorbis (carbon • videos) 13:09, 20 August 2020 (UTC)

Agree. Paul August ☎ 14:20, 20 August 2020 (UTC)

This is a standard word but not actually a technical term, except possibly in some special cases. Michael Hardy (talk) 17:29, 2 September 2020 (UTC)

Mirror symmetry

Since the old discussion was archived, it seems like the consensus was to either redirect Mirror symmetry to Mirror symmetry (disambiguation) or just flat out delete the Mirror symmetry page so it doesn't have a redirect. Can an admin weight into this? Wundzer (talk) 20:16, 6 September 2020 (UTC)

Draft:Conservative system

For some totally stupid, utterly ill-thought-out reason, I decided to resolve some red-link issues at Talk:Markov odometer by filling in an article for conservative system (which used to be a wildly incorrect redirect to conserved quantity). Recall that a conservative system is that thing that is contrasted with a dissipative system, and that the Hopf decomposition theorem states that every dynamical system decomposes into a conservative part, and a dissipative part. In simplified terms, a system is conservative if and only if the Poincaré recurrence theorem applies. Hopf did his work in the 1930's, Poincare did his work in the 19th century, and I think the core ideas of a conservative system go back into the 18th century. You can type in "conservative dynamical system" into you favorite search engine and see the hits keep on coming. Of course, it has nothing to do with the current political miasma. Despite all this, some nice WPdian decided that the topic was non-notable and lacked a sufficient number of references and moved it to draft space. Here: Draft:Conservative system ... anyone care to, umm, do whatever it takes to do whatever it is that has to be done to move along the process that results in the publication of those kinds of articles that cover basic, remedial topics in mathematics and physics? 67.198.37.16 (talk) 17:54, 12 September 2020 (UTC)

• wow, very nice article, thanks for the efforts Walwal20 ^{talk ▾ contribs} 04:39, 13 September 2020 (UTC)

MathSciNet as a resource

Is mathscinet appropriate as a resource? I am aware that it might dip into "original research" and also that it may be bad because of the paywall/institutional access. Just as hypothetical examples (just in terms of the mathscinet use in and of itself) what would be the appropriateness of the following:

1. using the fact that Yoshikazu Giga has ten articles cited over 100 times on mathscinet as justification of notability for a wikipedia article

2. this kind of sentence in an article: "According to MathSciNet, Yau's article [ref] is the most widely cited differential geometry article of the 1970s."

3. this kind of sentence in an article: "According to MathSciNet, six of the ten most widely cited differential geometry articles in the 1970s were written by Yau."

4. this kind of sentence in an article: "According to MathSciNet, Serre's most widely cited article is [ref]."

Gumshoe2 (talk) 04:24, 13 September 2020 (UTC)

I think the first example is fine; original research and judgment are a basic part of AfD. But the others are OR, and not suited to an article. 2 and 3 look promotional as well, a further strike against them. --{{u|Mark viking}} {Talk} 04:49, 13 September 2020 (UTC)

Thank you for the feedback... I can definitely understand #3 as promotional or at least odd. I'm not sure I can understand the same about #2 though. For instance, what about the following example: giving on the differential geometry page the five most-cited differential geometry articles? This wouldn't seem to be promotional in any sense, and in terms of original research it seems to me to be about equivalent to #1. (As before I'm not suggesting this as an actual good addition for the page, just trying to understand some hypotheticals.) Gumshoe2 (talk) 05:02, 13 September 2020 (UTC)

I think citation counts on MathSciNet are too haphazard to be meaningful: the accuracy of their numbers depends too much on whether the work is in one of the central areas they index everything of versus something more peripheral where only some of the publications get indexed and many of the listed publications don't include references. In this respect they are worse than broader multi-disciplinary databases like Google Scholar where the subdiscipline doesn't affect the level of coverage. I think many mathematicians would take the position that citation counts, in general, are a bad way to measure or justify the importance of a mathematical result or of a mathematician, and should be avoided in principle. The numbers also vary significantly within mathematical subdisciplines, so numbers that might be very high for arithmetic geometry would be uninterestingly low for numerical differential equations, and this variation also means that when someone works in multiple subdisciplines (some of Giga's papers are more numerical, others more theoretical) you will get a skewed idea of which ones are important. Because they're dubiously meaningful, require a lot of subject-specific knowledge to interpret, time-varying, and sourceable only to search engines, I don't think that citation counts belong in articles at all. While I do regularly make arguments about notability based on citation counts in deletion discussions, they don't work well for mathematicians, because the citation patterns in many areas of mathematics lead to numbers that are much lower and less meaningful than in other nearby fields like computer science or physics. When a mathematician does have genuinely high citation counts (like Giga), there is generally also some better reason for justifying the article (like, in his case, being an AMS Fellow). So my answers to your questions are 1. "Sometimes, but you can usually do better", 2. "No", 3. "No", and 4. "No". On the other hand, MathSciNet and zbMATH are genuinely useful for the independent and reliably-published descriptions they provide of publications. The paywall isn't an issue; Wikipedia sources aren't required or expected to be free. —David Eppstein (talk) 05:19, 13 September 2020 (UTC)

Thanks very much for the thoughtful answer and for clarifying on the paywall. (As a side point, I think numerical pde are actually very well represented on mathscinet, much higher numbers than arithmetic geometry, and in the specific case of Giga I'm not aware of any major numerical work.) I am personally in complete agreement with you about the inherent meaningfulness of citation counts, at least that a low count absolutely cannot be taken as proof of insignificance (the converse might be a little more subtle and require some different analysis). I should say my main interest in the question is captured in your sentence

When a mathematician does have genuinely high citation counts (like Giga), there is generally also some better reason for justifying the article (like, in his case, being an AMS Fellow).

As you acknowledge, sometimes there isn't a better reason. In such a case it may nonetheless be clear to me (or someone else), as an expert in some specific fields, that some so-and-so and their work, unrecognized by significant memberships and awards, is very important. Sometimes this isn't reflected by citations, and I suppose I have to give up since individual expertise is encyclopedically irrelevant. But often it is reflected by citations. Should I still give up? (If it helps, one specific example I have in mind is Wan-Xiong Shi, who has in my view one of the approximately 5 or so most significant papers on Ricci flow. This is consistent with mathscinet citations but as far I know not by any kind of professional honors or recognition other than a few NSF grants) Gumshoe2 (talk) 06:03, 13 September 2020 (UTC)

To phrase it a bit more directly- I'd like to understand better the notability criteria for mathematicians who are only known for their research output and are not recognized by professional institutions. Gumshoe2 (talk) 06:08, 13 September 2020 (UTC)

The point about numerical pde was not so much its representation on mathscinet — that's a problem for a different subdiscipline — but that it is a subarea with higher citation counts than more-theoretical topics and so papers in it will appear more important regardless of whether they actually are. As for Wan-Xiong Shi, you can make a case for WP:PROF#C1 using Google Scholar citation counts, but in general the answer to what to do for people with good research output but inadequate recognition for it is: be patient. We are not here to provide that recognition ourselves, but to record it once others do. —David Eppstein (talk) 06:19, 13 September 2020 (UTC)

I see, I'd misunderstood your comment about numerical pde. I would argue that it is relevant that numerical pde is a significantly larger field than arithmetic geometry and that it actually has many more important papers, but I know this isn't the place for it... Anyhow, thanks for the clarification, much obliged. Gumshoe2 (talk) 06:34, 13 September 2020 (UTC)

Draft:Resonant interaction

I've created an article Draft:Resonant interaction and would like to request for a review, and/or for someone to move this to main article space. As this is primarily physics, I've also placed a request at WP:Physics for the same. I'm posting here, because one aspect, the three-wave equation, have seen intensive study, via inverse scattering methods, for the last 5-6 decades by a variety of prize-winning mathematicians. (They have a Lax pair, specifically, a 3x3 Lax pair). (Actually, I'm planning to split the three-wave resonant interaction into it's own article, the Draft:Three-wave equation. It's rather remarkable; the simplified case is solved by the Weirstrass elliptic functions and so has the Eisenstein series g_2 and g_3 as invariants. Apparently, the Diophantine equations show up too, so its rather number-theoretic-ish for something that shows up in non-linear optics, etc. Which is why I post here, and not just WP:Physics) 67.198.37.16 (talk) 21:43, 16 September 2020 (UTC)

Draft:Fake nodes

I came across this article while doing AFC review. I felt that this article will require some specialist review. --Salimfadhley (talk) 12:46, 12 September 2020 (UTC)

@Salimfadhley: I don't really do AfC and I tend to have a somewhat conservative view when it comes to notability, so feel free to take what I say with a grain of salt. But I think this is an easy sources-do-not-demonstrate-notability decline. The "introduced in 2020" with three sources, at least one author common to all of them, is a pretty big red flag. A quick look doesn't really find anything else out there about this, and it just feels like researcher(s) putting their work on Wikipedia. –Deacon Vorbis (carbon • videos) 13:06, 12 September 2020 (UTC)

I have left a comment on the draft that this draft cannot be accepted as an article because of the policy WP:PRIMARY. Since this concept has been introduced this year, it cannot be any reliable secondary source that discuss it. D.Lazard (talk) 13:18, 12 September 2020 (UTC)

@Deacon Vorbis: There are two authors common to all three sources and four authors common to sources no 1 and 3. --CiaPan (talk) 13:57, 12 September 2020 (UTC)

Thank you all for this swift scrutiny. --Salimfadhley (talk) 15:36, 12 September 2020 (UTC)

(Maybe this is already a settled matter) but the issue with the draft is WP:TOOSOON. I can’t think of any instance when a concept in mathematics introduced this recent can have a Wikipedia article. Mathematics develops quite slowly compared to other Internet-age stuff. —- Taku (talk) 03:42, 13 September 2020 (UTC)

Unrelated to fake nodes, but on the same topic of notability and primary sources, some people may want to look at Deformed Hermitian Yang–Mills equation. I added the notability tag in June, wasn't sure what else to do about it. The relevant physics articles (by Strominger, Witten etc) seem to be somewhat well known, but the article is mostly about recent math preprints, one of which is just listed as "in preparation". Gumshoe2 (talk) 03:58, 13 September 2020 (UTC)

I think this is a different case; the topic itself in this case seems notable. According to [1], there is a 2000 paper with over 100 citations that uses the term in the abstract, which suggests the notion is not too soon and is sufficiently well known. That article cites recent papers itself is not an issue and may be a good thing since recent papers are likely up to dates on the current research status. Also, the use of primary sources itself is not an issue if it does not lead to a biased treatment of the topic. (I don’t know enough about the topic to know if the current article is neutral or not). —- Taku (talk) 02:15, 17 September 2020 (UTC)

I think there are some things to say about the topic which could be notable. But even if we were to take as given that the topic itself is notable, the vast majority of the article is about articles from the last three years which have only appeared on arxiv (one of them not even released publicly in any form yet). That is what seems not too good to me Gumshoe2 (talk) 04:49, 17 September 2020 (UTC)

Draft:List of discoveries in mathematics

Hi all. I'm thinking of moving the draft Draft:List of discoveries in mathematics to mainspace. What do the others think? The topic itself seems reasonable except there is an obvious classic problem of: is mathematics inverted or discovered or both ? Since, in Wikipedia, we can't take a side. The draft should be probably named to Draft:List of discoveries and inventions in mathematics or something. Also, "timeline" instead of "list" may better reflect the current structure of the draft. (The draft itself looks incomplete but the development can still continue in mainspace) -- Taku (talk) 02:21, 17 September 2020 (UTC)

The scope of such a list seems ridiculously large. Wouldn’t basically every mathematics article on Wikipedia qualify for inclusion? The creation of any mathematical theory or object is a "mathematical discovery/invention". — MarkH₂₁^talk 02:54, 17 September 2020 (UTC)

In addition to the above, this was more of Diametakomisi's (aka whalestate) garbage (see Wikipedia:Sockpuppet investigations/Whalestate for some of the history. Most of his other junk has been G13ed by now, but apparently this one has been tweaked a bit by others. Seeing as how the clock's been reset, I'd suggest WP:MFD as mostly the creation of a sockpuppet with only minor edits in the mean time, combined with MarkH21's concerns about being way too broad/vague. –Deacon Vorbis (carbon • videos) 03:12, 17 September 2020 (UTC)

I concur with MarkH21 that the scope would be ridiculously large. I'd !vote to delete it at MfD. XOR'easter (talk) 15:05, 17 September 2020 (UTC)

I agree as well. As well as being hopelessly subjective and way too huge in scope, it is partly redundant to articles like History of mathematics and Timeline of mathematics. Reyk _YO! 15:20, 17 September 2020 (UTC)

I agree with the other responders. What the original poster is talking about is nothing less than a list of mathematics. That list is not useful. Mgnbar (talk) 00:30, 18 September 2020 (UTC)

Thank you all for the responses. It seems that Timeline of mathematics already covers the same topic (and is more complete). So I simply redirected the draft to it. To comment on the concern on the scope: it is ok and possible to have an article on a topic of large scope. In fact, timeline of mathematics does have quite a large scope. —- Taku (talk) 21:30, 18 September 2020 (UTC)

Not a great example though. That article makes it seem like 3/8 of the mathematical world in the 2010s was focused on calculating digits of pi, which unfortunately is probably what most of the world believes mathematicians do. — MarkH₂₁^talk 21:33, 18 September 2020 (UTC)

Well, the author of the article does seem to think the calculation of pi is on per with achievements like proving the fundamental lemma. Given the treatment in general mainstream media, I don’t know if that’s a fair view. —- Taku (talk) 21:41, 18 September 2020 (UTC)

Re-insertion of "nationality" column to the table at Fields Medal

The longstanding consensus at Fields Medal since 2014 was to remove the "nationality" column on the grounds of:

1. not providing particularly relevant information for a mathematics award,
2. reducing edit-warring and nonsense.

I had removed the column again after I realized that an IP had added the "nationality" column back into the table in 2019, and I opened Talk:Fields Medal/Archive 1#Removing the entire nationality column where a new editor is adamant on keeping the column. The discussion could use additional input from math editors here. Thanks. — MarkH₂₁^talk 06:40, 18 September 2020 (UTC)

On the basis of the new consensus there, I have nominated the closely related article List of countries by number of Fields Medalists for AfD at Wikipedia:Articles for deletion/List of countries by number of Fields Medalists (2nd nomination). — MarkH₂₁^talk 20:52, 18 September 2020 (UTC)

While we're on the topic, I think most of Ber31's edits of 14 September are unhelpful, but I wasn't sure if reverting is worth it. What are your thoughts? Brirush (talk) 02:37, 19 September 2020 (UTC)

@Brirush: I just took a look at those edits (here is the diff), and I agree that they aren't particularly helpful but I also don't feel particularly strongly for most of it. The note about winners mostly having PhDs in mathematics probably isn't an important fact for the article, at the very least not lead-worthy. — MarkH₂₁^talk 02:56, 19 September 2020 (UTC)

Thanks, that's how I generally felt and why I didn't revert them. I wanted confirmation though.Brirush (talk) 03:23, 19 September 2020 (UTC)

Ultragraph C*-algebra

Could someone with sufficient nous have a look at Ultragraph C*-algebra? To me that is just highly structured barbed wire, and I can't tell whether it's sufficiently distinct from C*-algebra to merit a separate article, or if it should be merged. In any case I suspect it would have to be a lot less technical; it currently looks like the deep end of an advanced text book. --Elmidae (talk · contribs) 04:02, 6 September 2020 (UTC)

These are examples of C*-algebras that generalise Cuntz algebras and the more general Cuntz-Krieger algebras due to Joachim Cuntz and Wolfgang Krieger. There is no accessible introduction to Cuntz-Krieder algebras: the two foundational papers on "topological Markov chains" were published in 1980 in Crelle's Journal and Invent. Math. The general theory is part of C* dynamical systems and the classification of C*-algebras via K-theory (cf George A. Elliott, Mikael Rørdam, Eberhard Kirchberg, et al). Looking at the article, however, it is quite surprising that it has been transferred to wikipedia. The article graph C*-algebra does not seem very readable. The article k-graph C*-algebra is also quite technical. Paul Muhly, Mikael Rørdam, Ruy Exel, Alex Kumjian and Iain Raeburn seem to have been connected with some parts of the theory. (Paul Muhly's interests are quite broad; they include operator theory.) Mathsci (talk) 14:07, 6 September 2020 (UTC)

Upon reading it, it seems like they are a special (distinguished) class of C*-algebras people study. I agree the article could be improved upon, but don't think deleting it is the correct path. Furthermore, if you went to the article's author's page, you would see they are open to suggestions for improvement. I am going to suggestion some improvements they could make so it can be more accessible to a wider wikipedia audience. Wundzer (talk) 20:21, 6 September 2020 (UTC)

It might be worth splitting out the first half into it's own article, the ultragraph. I'm kind-of surprised that's a redlink, since these kinds of structures are not uncommon in machine learning and knowledge representation and deep learning, they're right up there with hypergraphs and generalized hypergraphs aka metagraphs (whoa, more red-links?). Dettaching from the C* aspects might allow it to mature on it's own, without tension with the more abstract nature of operator algebras. Just an idea. (For example, the DPLL algorithm is "just" a way of walking an ultragraph, and pruning away all ultragraph trees, leaving only a densely connected interior. The interior is then solved by exhaustion, and then the remainder is reconnected and solved trivially. Foundational for the Boolean SAT problem.). (FWIW, there is a version of the ultragraph that has m incoming edges (instead of one) and n outgoing edges (same as the ultragraph); it is common in linguistics and in proof theory/theorem-provers (e.g. in natural deduction, incoming is what goes in, outgoing is what goes out; theorem-proving is "just the simple matter" finding a suitable collection of these to form a closed graph), and, in graph software, it's nicer than ordinary graph representations because it has better CPU-cache locality, uses less RAM in many cases, and is easier to traverse with a recursive algorithm. I don't know that it has a commonly accepted name, however. Generalized ultragraph, it would now seem?) 67.198.37.16 (talk) 01:21, 13 September 2020 (UTC)

@67.198.37.16: Hey! This is awesome you know so much about ultragraphs. Would you like to write up the ultragraph page and include all the intuition/applications to computer science you've mentioned. The wiki community would greatly benefit from these additions. Wundzer (talk) 01:04, 24 September 2020 (UTC)

Draft:Three-wave equation

I'd like to request a review of Draft:Three-wave equation and a move to main-article namespace, as appropriate. It is an applied-mathematics/physics article mostly, used in fluid mechanics, plasma physics, electronics and the like, where this is seen. Mathematically interesting because the uniform solutions are given by Lax pairs and are classified by modular invariants. 67.198.37.16 (talk) 14:41, 21 September 2020 (UTC)

Thanks! 67.198.37.16 (talk) 23:48, 24 September 2020 (UTC)

Mandelbrot set

I've tried to do a little work on Mandelbrot set — making notation consistent, streamlining the parts that are supposed to be summaries, etc. Naturally, the article is filled with pretty pictures, perhaps to an overwhelming extent. In particular, the "3D Images" section seems a bit over the top (and maybe OR-ish). Thoughts on how to improve the page would be most welcome. XOR'easter (talk) 02:59, 24 September 2020 (UTC)

Overall I think the following images can be safely removed or brought to a collapsible "Gallery" section:

Extended content
this fractal has its own article

as they are mostly redundant with other images, mainly those in section "Image gallery of a zoom sequence" which is very clean and organized (props to whoever did it), or they are not referenced from the text, or they have marginal importance to the main subject of the article, I believe.

Section "3D images of Mandelbrot and Julia sets" seems to be original "research". While very interesting, I don't think it belongs to wikipedia, so here's my support if you want to delete it... another option is to shorten the explanation text so that it doesn't look and feel like original research, and bundle the images together in a collapsible gallery. Best, Walwal20 ^{talk ▾ contribs} 03:30, 24 September 2020 (UTC)

Thanks. I've removed all but a couple of the images listed above and cut the "3D images" section. I'd have no objection to restoring a shortened and re-bundled version of it if more sources turn up. The topic naturally lends itself to nice pictures, but we need some analysis showing why any particular way of making pictures is significant. XOR'easter (talk) 19:04, 24 September 2020 (UTC)

Here are some ideas for "improvement" with "improvement" in scare quotes:

• Almost all the pictures show a smooth shading, but there is no explanation on how to get the smooth shading. The answer is one can approximate a "continuous" iteration count as $\nu =n+{\frac {1}{2}}\log \log |z_{n}|$ or something like that (I'm working from memory), for the $z_{n}$ being the last one to leave.
• The orbits of the points inside settle down to an invariant measure. What is it? Do the orbits converge to a stable attractor? (they do) What about the unstable, chaotic orbits? What fraction of the interior "C" lead to unstable, forever-ergodic orbits, vs what fraction are in the basic of an attractor? What is the shape of the neutral manifold? (oops redlink, its a synonyn for center manifold) i.e. the set that divides the basin of the stable attractor from the ergodic/hyperbolic attractor? There are many pictures online... The only problem being that "by popular convention", when people say "the Mandelbrot set" they really mean "that algo that draws the exterior in a colorful way". So talking about the interior is perceived by some as off-topic. BTW, the basins of attraction, the measure itself spread outside the M-set.
• Above is talking about the Normally hyperbolic invariant manifold see also stable manifold for an intro. to the concept, which should cross-pollinate with attractor but currently doesn't. Ditto the Normally hyperbolic invariant manifold is written as if the article lives in a vacuum, and fails to mention any of the other articles talking about related ideas.
• One can ask what happens if one differentiates w.r.t. c. There are some interesting results on the second derivative. If I recall, it has countable point-like singularities, and the pictures are pretty, too.
• The "external rays" should be explained. One solves the 2D Laplace equation for boundary conditions 0 at infinity, 1 on the boundary of the M-set. The circle at infinity, the phase, runs zero-2pi and the lines of constant phase are the rays. There was an important paper back at the time that explained the period doubling of rays, e.g. showing how to find rays that pinch off bulbs, etc. That paper does not seem to be cited (after my quick skim). Its nice because its "mathematical" yet relatively easy to understand, starts building a bridge to ergodic theory.

Yes, its a chore to do this but you asked :-) 67.198.37.16 (talk) 00:54, 25 September 2020 (UTC)

Prowling -- Center manifold is OK-ish, Stable manifold is barely more than a stub and should really be moved to Stable and unstable manifolds because it treats both. And appearing out of thin air: Lagrangian coherent structure is 100% pure awesomeness! What I'm saying is that the Center and the stable/unstable articles could be/should be made equally awesome, and that (b) there is a discrete-time variant of these which applies to the M-set and perhaps the M-set article would be a good way to introduce these concepts... 67.198.37.16 (talk) 02:34, 25 September 2020 (UTC)

Fractal measure page?

I was reading fractal-related the articles such as Hausdorff measure and fractal derivative, as well as doing some reading outside of wikipedia on these topics. In this reading, I noticed that many outside sources tend to lean towards a generic "fractal measure" rather than specifically mentioning Hausdorff measure. Indeed, other fractal measures such as packing measures do exist. However, the hausdorff measure is clearly the most used and most notable one, to the extent that I'd be surprised if most other fractal measures would be considered notable at all. So my question is: should fractal measure be created as a redirect to hausdorff measure, being the most significant of the fractal measures? Or does it warrant a small page of its own describing what makes a fractal measure? Is it notable enough on its own? Thanks, _{Integral Python} ^{click here to argue with me} 17:00, 29 September 2020 (UTC)

This is a case for a WP:DABCONCEPT. --Izno (talk) 17:22, 29 September 2020 (UTC)

Thanks! I've gone ahead and made Draft:Fractal measure, we'll see how it goes. _{Integral Python} ^{click here to argue with me} 22:17, 29 September 2020 (UTC)

I think this is good content for its own page but I think "fractal measure" may not be the appropriate primary framing. "Caratheodory construction" may be more appropriate. A standard reference is section 2.10 of Federer's book "Geometric measure theory". See also the page metric outer measure; the section "Construction of metric outer measures" is essentially the same but seems to me like a strange inclusion there; maybe it should be merged with your draft page. Gumshoe2 (talk) 05:31, 30 September 2020 (UTC)

Currently, Draft:Fractal measure never actually says what a "fractal measure" is. Instead, it just says that there are two examples (and we've got long articles on both examples.) Tell us what it is, and give at least one more example that is not one of the conventional two.67.198.37.16 (talk) 19:27, 30 September 2020 (UTC)

That is why I suggest "Caratheodory construction" as a good alternative. As far as I know, "fractal measure" has no particular meaning. Gumshoe2 (talk) 19:32, 30 September 2020 (UTC)