• Hoàng Xuân Sính, Gr-catégories.

]]>I would guess that a rewrite rule like WW → W to remove repeated subwords […] allow us to reduce any element of the free idempotent monoid on n generators to a ‘normal form’

The difficulty is that we’re allowed to apply the rewrite rule backwards too, as W → WW. Sometimes you can simplify a word only after you make it longer with the backwards rule. There was actually a very relevant problem posed in the KöMaL contest: A. 234., posed in 2000. “https://www.komal.hu/verseny/2000-03/A.h.shtml” has the solution, “https://www.komal.hu/verseny/2000-03/mat.e.shtml” has the English specification. This asks you to prove that the monoid is finite by proving that you can rewrite any word to a word that is not longer than 8 letters. The solution page also notes that there are multiple essentially different cases when you do need to make the word longer at first. The simplest example that it gives is:

abacbcacb = abacbca(cb) → abacbca(cbcb) = ab(acbcacbc)b → ab(acbc)b = aba(cbcb) → aba(cb) = abacb

I don’t really understand how this monoid works either though.

]]>The moments of come from those of , and involves sums like for even $m$, and the moments of a uniform distribution between -1/2 and 1/2. Perhaps someone knows formulas for these.

The m’th derivative of (for ) can be found (I think!) by replacing of the by its derivatives , and those are made of Dirac delta functions at something like . The remaining part has to be evaluated at these points.

]]>Okay, I’m doing it now. Good luck!

]]>Ah, I see! Thanks for that note, that makes sense.

]]>Convolutions are definitely the key to understanding this problem! I like your line of thinking. But there’s a nuance here. For the original Borwein integrals

we are interested in how far falls. But for the *cosine* Borwein integrals

we are interested in how far falls. We need to reach for to start falling, but we need to reach for to start falling. All this is explained by Greg Egan here:

• Patterns that eventually fail, *Azimuth*, September 18, 2018.