Hmm, this seems wrong but fixable. Namely, exp(A) is close to (I+A/n)^n, so raising both sides of det(exp(A))=exp(tr(A)) to the power of 1/n gives something like what we want. Still a bit too algebraic though, I wonder if we can do better.
Interesting, can you give a simple geometric explanation?
Yup, determinant is how much the volume stretches. And trace is how much the vectors stay pointing in the same direction (average dot product of v and Av). This explains why trace of 90 degree rotation in 2D space is zero, why trace of projection onto a subspace is the dimension of that subspace, and so on.
Very cool that you're thinking about this. I've been in a bit of funk since the news about Cogent, Lumen and LINX. It's good to hear that not everyone in the West subscribes to "bolt the door from outside".
Right now there's indeed an exodus of young qualified people from Russia. The easiest path goes to countries that are visa-free for Russians, like Armenia or Argentina.
Ukrainians have wanted to join the EU for years, it was one of the main points of the Euromaidan. Most in the EU were lukewarm to it, but now because of the war there are huge pro-Ukraine demonstrations in every European capital.
If everyone acts rationally, the result will be Ukraine growing closer to the EU, Russia becoming more isolated, and no WW3. But Russia isn't acting rationally, I'm losing count of distinct stupid things it has done since Feb 21. Extrapolating that stupidity into the future makes me think that WW3 is quite possible.
Can you describe what changed / what made you start feeling that the problem is solvable / what your new attack is, in short?
I think the acoustic has a better sound, but the electric one has more groove.
Sethares' theory is very nice: we don't hear "these two frequencies have a simple ratio", we hear "their overtones align". But I'm not sure it is the whole story.
If you play a bunch of sine waves in ratios 1:2:3:4:5, it will sound to you like a single note. That perceptual fusion cannot be based on aligning overtones, because sine waves don't have overtones. Moreover, if you play 2:3:4:5, your mind will sometimes supply the missing 1, that's known as "missing fundamental". And if you play some sine waves slightly shifted from 1:2:3:4:5, you'll notice the inharmonicity (at least, I do). So we must have some facility to notice simple ratios, not based on overtone alignment. So our perception of chords probably uses this facility too, not only overtone alignment.