There are two cases:
- (topological) manifolds
- differential manifolds
Questions: are all compact manifolds / differential manifolds homotopic / diffeomorphic to the sphere in that dimension?
- Original problem posed, for topological manifolds.AKA: classification of compact 3-manifolds. The result turned out to be even simpler than compact 2-manifolds: there is only one, and it is equal to the 3-sphere.
- for differential manifolds:Counter examples are called exotic spheres.Totally unpredictable count table:is an open problem, there could even be infinitely many. Again, why are things more complicated in lower dimensions??
Take the element and apply it to itself. Then again. And so on.
In the case of a finite group, you have to eventually reach the identity element again sooner or later, giving you the order of an element of a group.
The continuous analogue for the cycle of a group are the one parameter subgroups. In the continuous case, you sometimes reach identity again and to around infinitely many times (which always happens in the finite case), but sometimes you don't.
One of the first formal proof systems. This is actually understandable!
This is Ciro Santilli-2020 definition of the foundation of mathematics (and the only one he had any patience to study at all).
TODO what are its limitations? Why were other systems created?
Watching www.youtube.com/watch?v=-SbZZPX-y9g in 2022, who was one of his inspirations, made Ciro miss his guitar so much... one day, maybe, one day.
And of course, 4chan just takes that to a whole new level, usually closing on the same day, and then getting deleted within a week. Why would anyone contribute non-illegal content to that king of system?!
Ridiculous, so when new information comes out, we just duplicate all the old comments on a new thread again?
Remember, Ciro Santilli is the Necromancer God.
They do seem to have been very innovative, and have had a very good work culture. They also had a huge impact on the Silicon Valley startup scene.
Some products they are known for:
- oscilloscopes
- Atomic clocks, notably highly portable ones, see e.g. Video "Inside the HP 5061A Cesium Clock by CuriousMarc (2020)"
- pocket calculator
Main implementations: the same as electronic switches: vacuum tubes in the past, and transistors in the second half of the 20th century.
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