Wikipedia talk:WikiProject Mathematics

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about complex analytic variety

I would like some advice on redirects and article names. see Talk:Complex analytic variety#Create a redirect to this article. thanks ! --SilverMatsu (talk) 01:17, 22 December 2021 (UTC)

Edit war started

Some help would be helpful for the edit-war that is starting at Inner product space. Pinging Mgkrupa. D.Lazard (talk) 15:01, 22 December 2021 (UTC)

@D.Lazard: it takes two to tango. If there are 4 reverts within a 24 hour period, that might lead to a report at WP:EWN, but not here. The edits to the article inner product space seem like cosmetic and harmless format changes (<math>, </math>, latex format vs. more primitive mathematical coding). Possibly it might be surprising that the complex conjugate of a Hilbert space is not mentioned in the article. [For a (complex) inner product space, its dual space is naturally a Hilbert space (with a canonical conjugate structure in the complex case).] Isn't it more usual to use the coding

$\int _{a}^{b}f(t){\overline {g(t)}}\,dt$

with dt instead of dt, which seems clunky? Mathsci (talk) 16:05, 22 December 2021 (UTC)

On the last point, apparently this is a known difference of conventions. The traditional usage in the States (and maybe Britain, not sure) is to use italic d, but in France and maybe some other places they prefer to put it in roman text, sometimes bold. I think the idea is to save italics for variables. Sometimes they go as far as to render e in roman, on the grounds that it's a constant rather than a variable. --Trovatore (talk) 18:00, 22 December 2021 (UTC)

@Trovatore: I thought that was more of a mathematicians-v.-physicists divide rather than a geographical difference. Michael Hardy (talk) 20:09, 22 December 2021 (UTC)

Could be. My memory of the discussions is hazy. --Trovatore (talk) 20:37, 22 December 2021 (UTC)

I don't think so. In my maths degree I have seen both $\mathrm {d} x$ and $dx$. (usually those that are very fussy about their presentation tending to use the former, many others using the latter) --George AKA Caliburn · (Talk · Contribs · CentralAuth · Log) 19:29, 24 December 2021 (UTC)

Looking at my hard copy of Laurent Schwartz's 1966 classic "Théorie des distributions", $dx_{1}$, etc, is the standard style adopted. The Notes for proofs in the article are unintelligible seas of red and blue, of little use for readers of wikipedia: Inner_product_space#Notes The French article Espace préhilbertien is fine. Mathsci (talk) 18:48, 22 December 2021 (UTC)

Hi, I'm the other editor and this 09:38, 22 Dec version of the article is what I would like to commit. After this 20:48, 21 Dec edit was partially reverted (resulting in this 21:18, 21 Dec edit that D.Lazard said was "clearer"), I changed my now-reverted 20:48, 21 Dec edit to be more similar to that of D.Lazard's 21:18, 21 Dec edit (resulting in what I consider to be an improvement) and I also made some changes that I hoped would remedy some of his concerns. Long story short, the result of my changes was this 09:38, 22 Dec edit (which I'd like to commit) that was fully reverted (resulting in the latest version of the article). Is D.Lazard right that my desired version of the article is flawed enough that it should not replace the current version of the article? Thanks. Mgkrupa 01:11, 23 December 2021 (UTC)

The eighth reference seems like a wiki, can it be used as a wikipedia reference like PlanetMath ?--SilverMatsu (talk) 13:29, 26 December 2021 (UTC)

BrilliantMath is a low-quality source. It was added (along with a bunch of other mediocre-to-poor sources) in this edit by Miaumee -- I think any of the sources added in that edit could/should be removed. --JBL (talk) 14:35, 26 December 2021 (UTC)

Thank you for your reply. A search of BrilliantMath on wikipedia found that it was used in the section of references in over 20 articles. Would you(we) like to create a new section for this? But today it takes longer than usual to load a wikipedia article.--SilverMatsu (talk) 15:39, 26 December 2021 (UTC)

Reference removal is complete in the Inner product space.--SilverMatsu (talk) 16:06, 26 December 2021 (UTC)

More uniformization articles?

We have an article titled Simultaneous uniformization theorem, which begins as follows:

In mathematics, the simultaneous uniformization theorem, proved by Bers (1960), states that it is possible to simultaneously uniformize two different Riemann surfaces of the same genus using a quasi-Fuchsian group of the first kind.

One of the links was put there by me today: [[uniformization theorem|uniformize]]. At Uniformization theorem we see this:

In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere.

The first quote above says "of the same genus", meaning this isn't only about simply connected surfaces, and yet the second quote deals only with simply connected surfaces. Thus the story of "uniformization" in this sense of the word is incomplete in this set of articles. Is there another article that deals with uniformization of surfaces that are not simply connected? If not, should we create one? Michael Hardy (talk) 20:42, 22 December 2021 (UTC)

There are several generalisations of the uniformization theorem, mostly dating back to Paul Koebe. The article planar Riemann surfaces surveys some of these generalisations (including circle packing). The Riemann mapping theorem also covers this material from a slightly different point of view using normal families. (Jänich wrote a pocket-sized Springer-Lehrbuch on it.) Mathsci (talk) 21:13, 22 December 2021 (UTC)

These (uniformisation and simultaneous uniformisation) are two completely different theorems; if anything, the classical uniformisation theorem is a pre-requisite for Bers' simultaneous uniformisation theorem. The latter is a parametrisation of a certain family of representations of surface groups into the group of isometries of hyperbolic 3-space; the statement amounts as saying that given a surface of genus at least 2, and any pair of points in its Teichmüller space, there exists an action of its fundamental group on hyperbolic 3-space whose domain of discontnuity on its boundary (the Riemann sphere) is a pair of discs, with actions corresponding to the points in Teichmüller space. This takes for granted the fact that Riemann surfaces of genus at least 2 are quotients of the disc, which is essentially the content of the classical uniformisation theorem. jraimbau (talk) 23:06, 22 December 2021 (UTC)

Articles on "differential calculus" and "integral calculus"

For whatever probably silly reason, I happened to be looking at old discussions in talk:calculus and I came across one in which the querent asks why we have an article called differential calculus but none called integral calculus (the latter is a redirect to integral).

D.Lazard asserted that "differential calculus" was much more used than "integral calculus", which seems unlikely to me. (D., do you want to elaborate on this claim?)

As I see it, "differential calculus" and "integral calculus" are not so much areas of mathematics, as they are units in a course of study. In the former, you teach students what a derivative is, how to compute it, and what it's used for/how to use it. In the latter, you do the same thing for integrals. There might be a case for writing about these separate portions of a course, from a math-education point of view. Otherwise it would make sense to me to merge "differential calculus" into derivative.

Thoughts? --Trovatore (talk) 19:07, 27 December 2021 (UTC)

My assertion (summarized by Trovatore) was a feeling. It results probably from my French culture. In fact, the French equivalent to "calculus" is "calcul différentiel et intégral". This long phrase is generally abbreviated into "calcul différentiel", and this is probably the origin of my feeling.

The strong relation between these two subjects makes artificial to distinguish them. For example, the fundamental theorem of calculus belongs to both calculi and says essentially that they are equivalent.

So, without reading again the articles, my first suggestion would be to merge Differential calculus partly to Calculus and partly to Differentiable function and/or Derivative, to redirect both Differential calculus and Integral calculus to Calculus, and editing Calculus for making clear that calculus is an abbreviation of "differential and integral calculus", or is an abbreviation of both "differential calculus" and "integral calculus" (the choice depends on sources that can be found). D.Lazard (talk) 20:06, 27 December 2021 (UTC)

Hmm, if we're going to get historical, I think "calculus" is an abbreviation of "the infinitesimal calculus".

As a practical matter, differential calculus has quite a bit of content, and may be useful to a certain contingent of readers as a separate article. I would just like to figure out its aboutness and make it clear, and probably avoid giving the implication that it's a separate area of study. I don't see any reason a parallel integral calculus article wouldn't be just as useful and for the same reasons, but I have no enthusiasm for working on it myself. --Trovatore (talk) 20:56, 27 December 2021 (UTC)

Well spotted by Trovatore. I’m still thinking about it. The problem isn’t confined to the two articles about differentiation because the process of integration is also covered by two similar articles: Integral and Antiderivative. Some merging is looking attractive. Dolphin (t) 06:58, 28 December 2021 (UTC)

To be fair, there is an articulable difference between integrals and antiderivatives, though it isn't one you'd explain to a first-semester calculus student. It might be reasonable to pitch antiderivative at a slightly higher level rather than merging it, perhaps comparable to Riemann integral (I haven't checked that article to see how it's written). --Trovatore (talk) 07:39, 28 December 2021 (UTC)

though it isn't one you'd explain to a first-semester calculus student !!! Understanding the ideas of integration and antidifferentation separately is an essential part of a good first course in calculus. --JBL (talk) 01:13, 29 December 2021 (UTC)

I'm glad to see someone explicitly distinguishing between topics in mathematics and units in a course of study. I've encountered actual mathematicians who, in some contexts at least, are somewhat challenge on that point. Michael Hardy (talk) 19:40, 28 December 2021 (UTC)

Analytic number theory expert needed

https://en.wikipedia.org/w/index.php?title=Prime-counting_function&oldid=1033755551

On July 30th, 2021, Vitamindeth misunderstood the derivation of the explicit formula for the prime-counting function and made some major edits. In particular, he wrote that the equality

$\sum _{m=1}^{\infty }\operatorname {R} (x^{-2m})={\tfrac {1}{\log x}}-{\tfrac {1}{\pi }}\arctan {\tfrac {\pi }{\log x}}$

does not hold (see the article for the details about Riemann's R-function), while it actually follows from Riesel&Göhl.

On December 23rd I noticed that flaw and made the corresponding fixes, but A1E6 completely reverted them. To avoid WP:WAR I started discussion on the Talk page, but A1E6 turned out to be intractable. I found the source with the expansion for $\mathrm {R} (e^{-t})$ directly leading to the equality questioned, but A1E6 decided that's not WP:CALC but WP:OR. In my opinion, article presupposes the reader's sufficient competence to understand how that equality follows from the sources cited. I would like to revert A1E6's recent edits to this version: https://en.wikipedia.org/w/index.php?title=Prime-counting_function&oldid=1062149402

Please take a look and help us to resolve this dispute. Droog Andrey (talk) 09:05, 28 December 2021 (UTC)

Please cite the page and line in which the equality was proven in Riesel and Göhl. vitamindeth

The equality

$\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\left(-{\frac {n}{2\log x}}+\int \limits _{x^{1/n}}^{\infty }{\frac {dt}{t(t^{2}-1)\log t}}\right)={\frac {1}{\pi }}\arctan {\frac {\pi }{\log x}}$

comes straightly from the expression (32) in paper by Riesel & Göhl.

Note that the sum $\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\sum _{\rho }\mathrm {li} (x^{\rho /n})$ does not converge because

$\sum \limits _{\rho }\mathrm {Ei} ({\rho }z)={\frac {1}{2z}}+{\frac {3}{2}}(\gamma +\log z)-\log {\sqrt {4\pi }}+{\frac {5}{6}}z+O[z^{2}]$ for $0<z<\log 2$, where $\gamma$ is Euler's constant.

Nevertheless, we still write

$\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\sum _{\rho }\mathrm {li} (x^{\rho /n})=\sum _{\rho }\mathrm {R} (x^{\rho })$

in analytical sense since

$\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}={\frac {1}{\zeta (s)}}$

and actually converging sum is

$\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\left(\sum _{\rho }\mathrm {li} (x^{\rho /n})-{\frac {n}{2\log x}}\right)=\sum _{\rho }\mathrm {R} (x^{\rho })+{\frac {1}{\log x}}$.

Summarizing this, we have

$\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\Pi _{0}{\bigl (}x^{1/n}{\bigr )}=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\left(\mathrm {li} (x^{1/n})-\sum _{\rho }\mathrm {li} (x^{\rho /n})-\log 2+\int _{x^{1/n}}^{\infty }{\frac {dt}{t\left(t^{2}-1\right)\log t}}\right)=\mathrm {R} (x)-\sum _{\rho }\mathrm {R} (x^{\rho })-{\frac {1}{\log {x}}}+{\frac {1}{\pi }}\arctan {\frac {\pi }{\log {x}}}$

Droog Andrey (talk) 17:53, 28 December 2021 (UTC)

@Droog Andrey: So, you did not cite the page and line in which the equality was proven in Riesel and Göhl. By the way, (32) in Riesel and Göhl is the following:

$\sum _{n=1}^{N}{\frac {\mu (n)}{n}}\left(\int _{x^{1/n}}^{\infty }{\frac {dt}{(t^{2}-1)t\log t}}-\log 2\right)={\frac {1}{2\log x}}\sum _{n=1}^{N}\mu (n)+{\frac {1}{\pi }}\arctan {\frac {\pi }{\log x}}+\epsilon (x,N)$

where $\epsilon \to 0$ as $N\to \infty$. A1E6 (talk) 18:56, 28 December 2021 (UTC)

Another way to write it is

$\sum _{n=1}^{N}{\frac {\mu (n)}{n}}\left(-{\frac {n}{2\log x}}+\int _{x^{1/n}}^{\infty }{\frac {dt}{(t^{2}-1)t\log t}}-\log 2\right)={\frac {1}{\pi }}\arctan {\frac {\pi }{\log x}}+\epsilon (x,N)$

and since $\epsilon \to 0$ and $\sum _{n=1}^{N}{\frac {\mu (n)}{n}}\to 0$ as $N\to \infty$,

$\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\left(-{\frac {n}{2\log x}}+\int \limits _{x^{1/n}}^{\infty }{\frac {dt}{t(t^{2}-1)\log t}}\right)={\frac {1}{\pi }}\arctan {\frac {\pi }{\log x}}$

immediately follows. Droog Andrey (talk) 20:55, 28 December 2021 (UTC)

@Droog Andrey: Yes, but here

$\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\left(\sum _{\rho }\mathrm {li} (x^{\rho /n})-{\frac {n}{2\log x}}\right)=\sum _{\rho }\mathrm {R} (x^{\rho })+{\frac {1}{\log x}}$

your ${\textstyle \sum _{n=1}^{\infty }\mu (n)=-2}$ was used. And ${\textstyle \sum _{n=1}^{\infty }\mu (n)=-2}$ is nowhere in Riesel and Göhl. There's not even any mention of zeta regularization in Riesel and Göhl. A1E6 (talk) 21:10, 28 December 2021 (UTC)

Good for you to agree about arctan. Now let's switch to ${\textstyle {\frac {1}{\log x}}}$.

Prime-counting_function states that $\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\sum _{\rho }\mathrm {li} (x^{\rho /n})=\sum _{\rho }\mathrm {R} (x^{\rho })$ by Möbius inversion.

Is it clear for you? Droog Andrey (talk) 21:25, 28 December 2021 (UTC)

The following equalities are in the prime-counting function article:

$\pi _{0}(x)=\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\Pi _{0}{\bigl (}x^{1/n}{\bigr )},$

$\pi _{0}(x)=\operatorname {R} (x)-\sum _{\rho }\operatorname {R} (x^{\rho })-\sum _{m}\operatorname {R} (x^{-2m}),$

$\sum _{\rho }\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\mathrm {li} (x^{\rho /n})=\sum _{\rho }\mathrm {R} (x^{\rho }).$

Your

$\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\sum _{\rho }\mathrm {li} (x^{\rho /n})=\sum _{\rho }\mathrm {R} (x^{\rho })$

is not in the article. And you're trying to prove

$\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\left(\sum _{\rho }\mathrm {li} (x^{\rho /n})-{\frac {n}{2\log x}}\right)=\sum _{\rho }\mathrm {R} (x^{\rho })+{\frac {1}{\log x}}.$

Notice the $-{\tfrac {n}{2\log x}}$ term. This leads you to use ${\textstyle \sum _{n=1}^{\infty }\mu (n)=-2}$ when you want to write

$\cdots =\sum _{\rho }\mathrm {R} (x^{\rho })+{\frac {1}{\log x}}.$ A1E6 (talk) 21:36, 28 December 2021 (UTC)

Are you sure that $\sum _{\rho }\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\mathrm {li} (x^{\rho /n})=\sum _{\rho }\mathrm {R} (x^{\rho })$ is the correct way of applying Möbius inversion?

If you dig up the details, you will find ${\textstyle \sum _{n=1}^{\infty }\mu (n)=-2}$ inside, because

$\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\sum _{\rho }\mathrm {li} (x^{\rho /n})$ diverges, while

$\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\left(\sum _{\rho }\mathrm {li} (x^{\rho /n})-{\frac {n}{2\log x}}\right)$ converges.

Droog Andrey (talk) 22:20, 28 December 2021 (UTC)

The convergence of

$\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\left(\sum _{\rho }\mathrm {li} (x^{\rho /n})-{\frac {n}{2\log x}}\right)$

does not prove ${\textstyle \sum _{n=1}^{\infty }\mu (n)=-2}$. Neither does the convergence of

$\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\left(\sum _{\rho }\mathrm {li} (x^{\rho /n})-{\frac {n}{2\log x}}\right)$

prove

$\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\left(\sum _{\rho }\mathrm {li} (x^{\rho /n})-{\frac {n}{2\log x}}\right)=\sum _{\rho }\mathrm {R} (x^{\rho })+{\frac {1}{\log x}}.$ A1E6 (talk) 22:29, 28 December 2021 (UTC)

The convergence of

$\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\left(\sum _{\rho }\mathrm {li} (x^{\rho /n})-{\frac {n}{2\log x}}\right)$

actually proves that

$\sum _{n=1}^{\infty }{\frac {\mu (n)}{n}}\left(\sum _{\rho }\mathrm {li} (x^{\rho /n})-{\frac {n}{2\log x}}\right)=\sum _{\rho }\mathrm {R} (x^{\rho })+{\frac {1}{\log x}}$

because generalized sum matches regular sum inside the convergence region. Droog Andrey (talk) 00:19, 29 December 2021 (UTC)

Two major mistakes: (1) The Möbius inversion of the oscillatory term converges. (2) The relation $\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{s}}}={\frac {1}{\zeta (s)}}$ only holds for s>1 if proven by the Euler product. You apply it for s=0. Vitamindeth

Vitamindeth, please sign your edits. (1) What is "oscillatory term"? (2) We are in the field of analytic, not arithmetic number theory. Droog Andrey (talk) 07:44, 29 December 2021 (UTC)

Your comments show how little you know about this topic. This discussion site is not a place to lecture someone in mathematics. Maybe someone with more time on their hands can explain it to you step by step, even though [User:A1E6|A1E6]] has already tried. Vitamindeth

Transcendental equation

Is this a meaningful concept, treated by reliable sources? I am skeptical. The fact that only one sentence is sourced is not encouraging. --JBL (talk) 01:20, 29 December 2021 (UTC)

• Aside from being dubiously sourced, the contents are also factually dubious. Of the three transcendental equations in the lead with allegedly no closed-form solution, the first and third do have closed-form solutions involving the Lambert-W function , which is not really that obscure. Reyk _YO! 01:27, 29 December 2021 (UTC)
• Even the two first sentences are factually dubious ("A transcendental equation is an equation containing a transcendental function of the variable(s) being solved for. Such equations often do not have closed-form solutions"): it is unclear whether ${\textstyle \exp({\frac {\log x}{2}})=5}$ is transcendental, as it simplifies easily to an algebraic function; the phrase "closed form solution" is meaningless without listing the accepted basic functions.
  It seems that there are no general theory nor significant results on such non-algebraic equations. So I suggest to nominate this article at AfD. D.Lazard (talk) 10:18, 29 December 2021 (UTC)
• I believe to remember that "transcendental equation" is used for an equation (over numeric domains) that is not (equivalent to) an algebraic equation. The large number of translation links indicates that the concept is widely known. The link de:Transzendente Gleichung gives 2-3 fairly reliable sources (in German); also the depicted Herschel book looks reliable, and interesting at first glance. So, I'd be in favor of fixing the flaws of this article and keeping it as a stub. In the long run, it could accumulate methods to solve particular kinds of nonalgebraic equations, which are useful to obtain "closed-form"/"analytic" solutions in special cases. - Jochen Burghardt (talk) 11:41, 29 December 2021 (UTC)