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?
God, Ciro Santilli respects this guy.
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.
Bulerias by Camaron de La Isla and Paco de Lucía
. Source. TODO date and location. Rondeña by Camaron de La Isla and Paco de Lucía
. Source. TODO date and location Online forums that lock threads after some time by 
Ciro Santilli 35 Updated 2025-03-28 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-03-28 +Created 1970-01-01And 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.
How to make an LM386 audio amplifier circuit by Afrotechmods (2017)
Source. Builds the circuit on a breadboard from minimal components, including one discrete transistor. Then plays music from phone through headset cables into a speaker.This technique has managed to determine protein 3D structures for proteins that people were not able to crystallize for X-ray crystallography.
It is said however that cryoEM is even fiddlier than X-ray crystallography, so it is mostly attempted if crystallization attempts fail.
By looking at Figure 1. "A cryoEM image", you can easily understand the basics of cryoEM.
We just put a gazillion copies of our molecule of interest in a solution, and then image all of them in the frozen water.
Each one of them appears in the image in a random rotated view, so given enough of those point of view images, we can deduce the entire 3D structure of the molecule.
Ciro Santilli once watched a talk by Richard Henderson about cryoEM circa 2020, where he mentioned that he witnessed some students in the 1980's going to Germany, and coming into contact with early cryoEM. And when they came back, they just told their principal investigator: "I'm going to drop my PhD theme and focus exclusively on cryoEM". That's how hot the cryo thing was! So cool.
Basically the same remarks as for university, just 10 times more useless, see also: Section "Motivation".
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