Ciro Santilli @cirosantilli 35

In this solution of the Schrödinger equation, by the uncertainty principle, position is completely unknown (the particle could be anywhere in space), and momentum (and therefore, energy) is perfectly known.

The plane wave function appears for example in the solution of the Schrödinger equation for a free one dimensional particle. This makes sense, because when solving with the time-independent Schrödinger equation, we do separation of variable on fixed energy levels explicitly, and the plane wave solutions are exactly fixed energy level ones.

Cute synonym for second brain, sample usage: beepb00p.xyz/exobrain/#cntnt

They are pioneers in making superconducting magnets, physicist from the university taking obsolete equipment from the uni to his garage and making a startup kind of situation. This was particularly notable for this time and place.

They became a major supplier for Magnetic resonance imaging applications.

Bibliography: discuss.bbchallenge.org/t/decider-cyclers/33

Example: bbchallenge.org/279081.

These are very simple, they just check for exact state repetitions, which obviously imply that they will run forever.

Unfortunately, cyclers may need to run throun an initial setup phase before reaching the initial cycle point, which is not very elegant.

Also, we have no way of knowing the initial setup length of the actual cycle length, so we just need an arbitrary cutoff value.

And unfortunatly, this can lead to misses, e.g. Skelet machine #1, a 5 state machine, has a (translated) cycle that starts at around 50-200M styeps, and takes 8 trillion steps to repeat.

Bibliography: discuss.bbchallenge.org/t/decider-translated-cyclers/34

Like a cycler, but the cycle starts at an offset.

To see infinity, we check that if the machine only goes left N squares until reaching the repetition, then repetition must only be N squares long.

Described at: www.sligocki.com/2022/06/10/ctl.html

The following things come to mind when you look into research in this area, especially the search for BB(5) which was hard but doable:

Bibliography:

If you can reduce a mathematical problem to the Halting problem of a specific turing machine, as in the case of a few machines of the Busy beaver scale, then using Turing machine deciders could serve as a method of automated theorem proving.

That feels like it could be an elegant proof method, as you reduce your problem to one of the most well studied representations that exists: a Turing machine.

However it also appears that certain problems cannot be reduced to a halting problem... OMG life sucks (or is awesome?): Section "Turing machine that halts if and only if Collatz conjecture is false".

ECDH has smaller keys. youtu.be/gAtBM06xwaw?t=634 mentions some interesting downsides:

Stopped 2019 apparently. Shame. We need something to be upstreamed!

bbchallenge.org/story#what-is-known-about-bb lists some (all?) cool examples,

wiki.bbchallenge.org/wiki/Cryptids contains a larger list. In June 2024 it was discovered that BB(6) is hard.

Turing machine acceleration refers to using high level understanding of specific properties of specific Turing machines to be able to simulate them much fatser than naively running the simulation as usual.

Acceleration allows one to use simulation to find infinite loops that might be very long, and would not be otherwise spotted without acceleration.

This is for example the case of www.sligocki.com/2023/03/13/skelet-1-infinite.html proof of Skelet machine #1.

Best busy beaver machine known since 1989 as of 2023, before a full proof of all 5 state machines had been carried out.

Entry on The Busy Beaver Challenge: bbchallenge.org/1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA

Paper extracted to HTML by Heiner Marxen: turbotm.de/~heiner/BB/mabu90.html