The infinite products lecture was also pretty fantastic
That one had more interruptions because the original lesson plan basically had one computation at the beginning, so he was always to give examples of stuff
It's not necessarilly the identity. But Erdös proved that a multiplicative (not necessarilly totally) strictly increasing function $\Bbb N \to \Bbb R$ is a power law, and that if you force $\Bbb N \to \Bbb N$ you even have an integer power law
I could swear there was a question on multilinear algebra the last couple of weeks that has now reappeared in disguise, but I can't find the original. I actually thought about it, but someone else beat me to a good answer.
Ah. I just remembered. It was posed as a diff geo question about maps preserving $d$-dimensional volume for $d<\dim M$.
This question is closely related to Eigenvalues of binomial matrix, but is a bit more general.
I am trying to determine the eigenvalues and eigenvectors of the following matrix:
$$M_{ij} = 4^{-j}\binom{2j}{i}$$
where it is understood that the binomial coefficient $\binom{m}{k}$ is zero if $k<0...
Now $H_{N(n+1)}-H_{Nn} = \sum_{k=Nn+1}^{N(n+1)} {1\over k} \ge {N\over N(n+1)} = {1\over n+1}$ so you can never have equality, thus you get strict inequality @AkivaWeinberger (sorry for intensive ping)
Interestingly, @EricSilva, the OP hasn't mumbled one word at me regarding my sketch of an argument on that $\int \tau/\kappa\,ds$ thing. Plus, never responded to my point about needing closed curves.
You haven't even studied much algebra, have you, @Astyx?
This question is closely related to Eigenvalues of binomial matrix, but is a bit more general.
I am trying to determine the eigenvalues and eigenvectors of the following matrix:
$$M_{ij} = 4^{-j}\binom{2j}{i}$$
where it is understood that the binomial coefficient $\binom{m}{k}$ is zero if $k<0...
