Ciro Santilli @cirosantilli 37

TODO understand this stuff:

After something broke on the website due to SQLite vs PostgreSQL inconsistencies and took me a day to figure it out, I finally decided to update the test system so that OURBIGBOOK_POSTGRES=true npm test will run the tests on PostgreSQL.

Originally, these were being run only on SQLite, which is the major use case for OurBigBook CLI, which came before the website.

But the website runs on PostgreSQL, so it is fundamental to test things in PostgreSQL as well.

We know that superfluidity happens more easily in bosons, and so electrons joins in Cooper pairs to form bosons, making a superfluid of Cooper pairs!

Isn't that awesome!

Ultimate explanation: math.stackexchange.com/questions/776039/intuition-behind-normal-subgroups/3732426#3732426

Only normal subgroups can be used to form quotient groups: their key definition is that they plus their cosets form a group.

Intuition:

One key intuition is that "a normal subgroup is the kernel" of a group homomorphism, and the normal subgroup plus cosets are isomorphic to the image of the isomorphism, which is what the fundamental theorem on homomorphisms says.

Therefore "there aren't that many group homomorphism", and a normal subgroup it is a concrete and natural way to uniquely represent that homomorphism.

The best way to think about the, is to always think first: what is the homomorphism? And then work out everything else from there.

The busy beaver game consists in finding, for a given $n$, the turing machine with $n$ states that writes the largest possible number of 1's on a tape initially filled with 0's. In other words, computing the busy beaver function for a given $n$.

There are only finitely many Turing machines with $n$ states, so we are certain that there exists such a maximum. Computing the Busy beaver function for a given $n$ then comes down to solving the halting problem for every single machine with $n$ states.

Some variant definitions define it as the number of time steps taken by the machine instead. Wikipedia talks about their relationship, but no patience right now.

The Busy Beaver problem is cool because it puts the halting problem in a more precise numerical light, e.g.:

Bibliography:

Why. Why. Why is there no limit to how much I can help, but there is a limit to how many thanks I can get?

At most, limit it to a single answer to avoid highly publicized events, e.g. an answer being shared on Reddit. But across answers? It makes no sense.

The two ways main ways to overcome this limit are the 15 point answer accept reputation and bounties.

200 reputation per day works out 73k a year BTW.

Digits 0 to 9, white on black background:

Bibliography:

The inaugural that predicted the Josephson effect.

Published on Physics Letters, then a new journal, before they split into Physics Letters A and Physics Letters B. True Genius: The Life and Science of John Bardeen mentions that this choice was made rather than the more prestigious Physical Review Letters because they were not yet so confident about the results.

Paywalled by Elsevier as of 2023 at: www.sciencedirect.com/science/article/abs/pii/0031916362913690

Correspond to the angular part of Laplace's equation in spherical coordinates after using separation of variables as shown at: en.wikipedia.org/wiki/Spherical_harmonics#Laplace's_spherical_harmonics

This is the possibly infinite dimensional version of a Hermitian matrix, since linear operators are the possibly infinite dimensional version of matrices.

There's a catch though: now we don't have explicit matrix indices here however in general, the generalized definition is shown at: en.wikipedia.org/w/index.php?title=Hermitian_adjoint&oldid=1032475701#Definition_for_bounded_operators_between_Hilbert_spaces