Modular exponentiation by Ciro Santilli 37 Updated 2025-10-14
Can be calculated efficiently with the Extended Euclidean algorithm.
The beauty of this algorithm is that because exponentiation grows really fast, there is no hope that we can ever learn all the digits of an exponential, as there is simply not enough time or memory for that. Therefore, a natural sub-question is if we can know some part of that number, and knowing the smallest digits is the most natural version of that question.
X-ray crystallography by Ciro Santilli 37 Updated 2025-07-16
One of its main applications is to determine the 3D structure of proteins.
Sometimes you are not able to crystallize the proteins however, and the method cannot be used.
Crystallizing is not simple because:
  • you need a considerable amount of the protein
  • sometimes it only crystallizes if you add some extra small chemical that stabilizes it
Cryogenic electron microscopy can sometimes determine the structures of proteins that failed crystallization.
Dojo by Ciro Santilli 37 Updated 2025-07-16
"Dojo" is the japanese version of the word that unfortunately came to dominate in the West, the original is of course Chinese Dao4 chang3 (道場) which means:
  • dao4 (道): Taoism, the Enlightned Path to something
  • chang3 (場): suffix indicating "a place where you do something"
Ciro Santilli would like to fully understand the statements and motivations of each the problems!
Easy to understand the motivation:
Hard to understand the motivation!
Alexander Grothendieck by Ciro Santilli 37 Updated 2025-07-16
This dude looks like a God. Ciro Santilli does not understand his stuff, but just based on the names of his theories, e.g. "Yoga of anabelian algebraic geometry", and on his eccentric lifestyle, it is obvious that he was in fact a God.
Linear function by Ciro Santilli 37 Updated 2025-07-16
The term is not very clear, as it could either mean:
Riemann integral by Ciro Santilli 37 Updated 2025-07-16
The easy and less generic integral. The harder one is the Lebesgue integral.
Don't set goals for your students.
Ask students what they want to do, and help them achieve that goal.
If they don't know what to do, give suggestions of interesting things they could do.
Once they have a goal, just help them learn everything that is needed to achieve that goal
If they don't have a goal, any attempt to learn is a total and complete waste of time.
This is because the universe of potentially useful things that can be learnt is infinite, and no human can ever learn everything.
The only solution, is to try and learn only what seems necessary to reach your goal, and just try to reach your goal instead.
This approach is called backward design.
"Graduating" and "getting a diploma" are not valid goals, because they are useless. A goal has to be either an amazing specific technological or artistic development.
If you give a course in a classroom, you reach 10 people (the others were sleeping).
If you make a perfect course online, and answer questions online, you reach 10 thousand.
Not doing things online is a waste of time.
You are a highly trained professional, and your time is extremely valuable.
Even if it takes twice as long to create the material than giving course, you are still more efficient by a factor of 500.
It is as if there were 500 little copies of you working full time. It is a superpower.

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