NYTimes competing with NYTimesPitchBot for funnier headlines, I see. What a bizarre and awesome piece of science. I like crystals for the miracle of uncountable numbers of atoms transferring symmetry from the smallest scale to the visible scale.
It's mentioned in the article that the chimpanzees only relinquished the crystals in exchange for many bananas, so it seems they're more into crystals...
This is just a general pattern: applied mathematicians are often using things pure mathematicians haven't proved to be true yet. The examples are widespread for the generalized Riemann hypothesis. There are statements we aren't sure about, but there's also a lot that we are sure about but not sure about the proof of.
The bull case is that everyone losing their jobs will accelerate and bring about the socialist revolution, giving us universal basic income and universal healthcare.
Lee taught Intro to Topological Manifolds for one quarter, and then the next two quarters where Intro to Smooth Manifolds. Then Riemannian, then vector bundles, and then complex manifolds.
Yesterday there was an article about how the ear works more like a Gabor transform or a wavelet transform than a Fourier transform, both of which are Short Time Fourier Transforms, so yes!
A function (which is an isomorphism) from complex numbers a+bi to matrices is a+bi |-> [[a,-b],[b,a]] where the matrix is listed by rows. So i is sent to the matrix R with a 0 in the top left, 1 in the bottom left, 0 in the bottom right and a -1 in the top right. R is a 90 degree rotation, you can check that it sends the unit vector [1,0] on the x-axis to [0,1], and the unit vector [0,1] on the y-axis to [-1,0].
I literally saw the sin one yesterday, so I'll rewrite it!!
Let e be an infinitesimal.
We write st(x) for the function dropping the infinitesimal part of a number.
Then f'(x) = st(1/e (f(x+e)-f(x)))
Now use the angle sum identity and cos(e) = 1 - e^2, sin(e) = e. I don't know how to justify these values other than the power series identities for sin and cos...
Here "like" means similarly in magnitude, not direction. If I could predict the future etc.