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Visit this link of Original Article
What does this mean?
Quantum makes everything Qooler, even Qalculus
My guess this is some q-analog stuff.
There's a "q-analog" to a lot of functions from combinatorics, which are continuous, and the regular ones are the limiting case as q goes to 1.
It's also related to "quantum" groups, and possibly quantum mechanics.
Measuring a Quantum-equation (x²-2x-3=0) falls into one of its solutions: |Ψ> = 0.707..| -1> + 0.707..|3>
🥸
To simplify the calculations at 10:53, we can notice that the quantum factorial is also recursive like the “normal” factorial, where [n]!_q = (1-q^n)/(1-q) * [n-1]!_q instead of multiplying everything from scratch.
Michael, thank you for making the "click bait" comment up front! We all know how the YouTube algorithm works, but it's so refreshing to see a presenter call it out so candidly! 👍
One easy way to count the inversions is to go through the numbers in the 1st, 2nd,…,nth place, counting how many numbers before them are larger than the number in the current place.
for example, for 4 2 1 3:
4 –> 0 larger
2 –> 1 larger
1 –> 2 larger
3 –> 1 larger
———————
total: 4 inversions
This also proves the formula for the max # of inversions, n*(n-1)/2, as follows: you can at max have 1 inverted (larger) element before the 2nd, 2 before the 3rd, … (n-1) before the n-th, thus in total:
1+2+…+(n-1) = n*(n-1)/2 inversions.
How do we know the # of inversions is unique?
Whats the use of it? Some examples please
One thing I truly appreciate about these videos (beyond the interesting mathematics) is the legible consistent handwriting and the organization of spacing, coloring, and symbols. The consistent speech patterns, delivery, and general organization of thoughts are clear as well. It's not very often I cannot follow something.
You baited me and you threw it right in my face the first 10 seconds. Love it.
Oh this relates closley to the number of n-cycles in a permutation. Good ol' exponential families 🥲 （；＿；）
I love q-analogs.
If it counts the „number of swaps“ then it could also be used to determine the sign in the sum of a determinant, which is exactly the number of swaps counted and positive if it is even and negative if it is odd, right? Or how could we relate this to matrices, I mean a determinant is closely related to permutations and in my first semester linear algebra course we were given the definition with the permutation formular.
i'm loving the new clickbait michael penn era
You don't need any clickbait for me. I watch your videos every day before I go to sleep without any exceptions.
In general, polynomials are quite useful for counting choices.
Nice! On my combinatorics course I've learned a lot of finite and infinite formal power series and what sequences they generate. This is a very interesting topic!
Is there an advantage to writing the q-factorial using the 'sum of finite geometric series' formula for this discussion? In this explanation, Michael is only using the polynomial expansion anyway, so (for this application, at least) why not just use that as the definition?
You must be one of the Q from the Star Trek Q Continum planet😂