Wikipedia:Reference desk/Archives/Mathematics/2023 October 8

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

October 8

Does every closed plane curve admit a Jordan subcurve?

This question is inspired by the September 29 question on closed plane curves admitting regular polygons. In particular, since it is known that all triangles can be inscribed in any arbitrary Jordan curve, it must be the case that this is true for all closed non-simple plane curves too if they admit simple subcurves. To formalize the question, let $\gamma :[0,1]\rightarrow \mathbb {R} ^{2}$ be a continuous function such that $\gamma (0)=\gamma (1)$. Does there exist some continuous and injective (except for the endpoints) $\gamma _{s}:[0,1]\rightarrow \mathbb {R} ^{2}$ such that $\gamma _{s}(0)=\gamma _{s}(1)$ and $Im(\gamma _{s})\subseteq Im(\gamma )$? GalacticShoe (talk) 01:21, 8 October 2023 (UTC)

It was pointed out to me that the answer, pretty trivially, is no; consider a closed plane curve whose image is an interval. However, some analogs of the problem still stand, and I will update if there are any developments there. GalacticShoe (talk) 18:31, 10 October 2023 (UTC)

So it turns out that in general, if $\gamma :I\rightarrow \mathbb {R} ^{n}$ is a nonconstant Euclidean curve, then there is a nonconstant simple subcurve $\gamma _{s}:I_{s}\rightarrow \mathbb {R} ^{n}$ such that $Im(\gamma _{s})\subseteq Im(\gamma )$. Moreover, if $I=[a,b]$ is closed, then $\gamma _{s}$ can be specified so that $\gamma _{s}(a)=\gamma (a),\gamma _{s}(b)=\gamma (b)$. This is because $Im(\gamma )\subseteq \mathbb {R} ^{n}$ is an arc-connected space, as a consequence of $\mathbb {R} ^{n}$ being $\Delta$-Hausdorff. GalacticShoe (talk) 04:35, 12 October 2023 (UTC)

39 is the thirteenth...

The article says:

The thirteenth Perrin number is 39, which comes after 17, 22, 29 (it is the sum of the first two mentioned).

I looked up the Perrin numbers in the article itself (NOT OEIS, but the Wikipedia article Perrin number) and 39 is the fourteenth number in the sequence; 29 is the thirteenth. Who is right?? Georgia guy (talk) 01:32, 8 October 2023 (UTC)

$P(13)=39$, but the sequence of Perrin numbers starts from $P(0)=3$, making $P(13)$ the fourteenth number in the sequence. Technically you're both right, it's just a bit of language ambiguity. GalacticShoe (talk) 01:38, 8 October 2023 (UTC)

I have corrected the article (not mentioned by the OP, but it is 39 (number)). I don't think it's ambiguous to say something is the fourteenth number in a sequence, even if the formula for the sequence traditionally indexes from zero. BTW, the statement in the article was added way back in 2006 by a now-blocked editor. CodeTalker (talk) 01:40, 8 October 2023 (UTC)

Honestly, I think it might be worth just leaving the number out of the description, instead opting to just say that 39 is a Perrin number. It's less ambiguous that way, and it's also the approach that all other pages for numbers that are Perrin numbers (except 10) take. GalacticShoe (talk) 01:59, 8 October 2023 (UTC)

I support GalacticShoe’s latest suggestion: “39 is a Perrin number.” Dolphin (t) 02:16, 8 October 2023 (UTC)

I'm fine with that. CodeTalker (talk) 03:23, 8 October 2023 (UTC)

Alrighty, I've gone ahead and taken the liberty to change the wording in both 39 (number) and 10. Thanks for the confirmation y'all. GalacticShoe (talk) 03:29, 8 October 2023 (UTC)

Integration

Hi, someone help please?

$\int \ln(x)\ln(1-x)\,dx$ over $[0,1]$ TheAlienMan2002 (talk) 12:38, 8 October 2023 (UTC)

Numerical integration gives ≈ 0.355065933152, which is too close to 2 − π² / 6 = 2 − ζ(2) to be a coincidence.  --Lambiam 17:01, 8 October 2023 (UTC)

There are several proofs here, some more valid than others.  --Lambiam 17:17, 8 October 2023 (UTC)

Like many other problems, it's a big help to know what the answer is supposed to be before you start; it's then just a matter of finding the steps in between. The last pair of entries in the Physics Forums page seems to be the most elementary. Basically, start with the observation that

$\int _{0}^{1}x^{i}\log x\,dx=-{\frac {1}{(i+1)^{2}}}.$

This is integration by parts, but taking limits because the integral is improper. Expand ln(1-x) as a series to get

$\int _{0}^{1}\ln(x)\ln(1-x)\,dx=-\int _{0}^{1}(x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}+\dots )\ln(x)\,dx.$

Blithely ignoring convergence issues (such as whether the series converges uniformly), integrate term by term to get

$\int _{0}^{1}\ln(x)\ln(1-x)\,dx={\frac {1}{1\cdot 2\cdot 2}}+{\frac {1}{2\cdot 3\cdot 3}}+{\frac {1}{3\cdot 4\cdot 4}}\dots .$

(This is similar to $-\int _{0}^{1}{\frac {\ln x}{1-x}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}\dots$ given in Basel problem#Integral representations.)

Note that

${\frac {1}{i(i+1)}}={\frac {1}{i}}-{\frac {1}{i+1}},$

and apply this twice to get that the series is equal to

${\frac {1}{1\cdot 2}}-{\frac {1}{2\cdot 2}}+{\frac {1}{2\cdot 3}}-{\frac {1}{3\cdot 3}}+{\frac {1}{3\cdot 4}}-{\frac {1}{4\cdot 4}}\dots$

$={\frac {1}{1\cdot 2}}+{\frac {1}{2\cdot 3}}+{\frac {1}{3\cdot 4}}\dots -{\frac {1}{2\cdot 2}}-{\frac {1}{3\cdot 3}}-{\frac {1}{4\cdot 4}}\dots$

$={\frac {1}{1}}-{\frac {1}{2}}+{\frac {1}{2}}-{\frac {1}{3}}+{\frac {1}{3}}-{\frac {1}{4}}\dots -{\big (}{\frac {\pi ^{2}}{6}}-1{\big )}$

$=2-{\frac {\pi ^{2}}{6}}.$ --RDBury (talk) 14:22, 9 October 2023 (UTC)

The integral is also related to the Beta function:

$\int _{0}^{1}\ln(x)\ln(1-x)\,dx=\left.{\frac {\partial ^{2}B(z_{1},z_{2})}{\partial z_{1}\partial z_{2}}}\right|_{z_{1}=z_{2}=1}$

Ruslik_Zero 20:44, 9 October 2023 (UTC)