Ciro Santilli @cirosantilli 40

Capoeira music is amazing. And some Brazilian pop adaptations of it have also been awesome.

The Fourier series of an $L^2$ function (i.e. the function generated from the infinite sum of weighted sines) converges to the function pointwise almost everywhere.

The theorem also seems to hold (maybe trivially given the transform result) for the Fourier series (TODO if trivially, why trivially).

Only proved in 1966, and known to be a hard result without any known simple proof.

This theorem of course implies that Fourier basis is complete for $L^2$, as it explicitly constructs a decomposition into the Fourier basis for every single function.

TODO vs Riesz-Fischer theorem. Is this just a stronger pointwise result, while Riesz-Fischer is about norms only?

One of the many fourier inversion theorems.

Some interesting analysis by Parth Shukla twitter.com/pparth | www.linkedin.com/in/parth-shukla-59583b20/:Apparently most of the routers were Chinese. No surprise there.

visualizing the Riemann hypothesis and analytic continuation by 3Blue1Brown (2016) is a good quick visual non-mathematical introduction is to it.

The key question is: how can this continuation be unique since we are defining the function outside of its original domain?

The answer is: due to the identity theorem.

The second protein to have its structure determined, after myoglobin, by X-ray crystallography, in 1965.

Breaks up peptidoglycan present in the bacterial cell wall, which is thicker in Gram-positive bacteria, which is what this enzyme seems to target.

Part of the inate immune system.

It is present on basically everything that mammals and birds excrete, and it kills bacteria, both of which are reasons why it was discovered relatively early on.