Mathematics all-in-one cheat-sheet (2013) [pdf] (ourway.keybase.pub)

Fixed^2

e^iπ+1=0 from

e^ix=sinx + icosx with x=π

φ_T[f] = ∫ dt p(t) exp(-2πift) = ⟨ exp(-2πifT) ⟩

If two variables X ~ r and Y ~ s are independent then you can prove [from ⟨f(X) g(Y)⟩ = ⟨f(X)⟩ ⟨g(Y)⟩ or X+Y ~ q where q(z) = ∫ dx r(x) s(z — x)] that their sum has a characteristic function

φ_{X+Y} = φ_X + φ_Y

And therefore the “sample mean” M of n IID variables is itself a random variable with characteristic function

φ_M[f] = ( φ[f/n] )^n.

So we find that

log φ_M[f] = n log φ[f/n] ≈ 0 + i a f – b f²/n + O(f³/n²).

These terms [a, b] from expanding the log of the characteristic function constitute the cumulant expansion and for large n the other terms shrink to zero, so that the characteristic function is to first order in 1/n a Gaussian.

The characteristic function was a Fourier transform of a PDF, so an inverse Fourier transform gets it back:

p(t) = ∫ df φ_T[f] exp(2πift)

But the Fourier transform of a Gaussian is just a Gaussian.

    90%+ => A (4.0)
    80%+ => B (3.0)
    70%+ => C (2.0)
    60%+ => D or F depending (1.0 or 0.0)

At my university the letter grades were

    A+  99.00 - 100.00%
    A   93.00 -  98.99%
    A-  90.00 -  92.99%
    B+  87.00 -  89.99%
    B   83.00 -  86.99%
    B-  80.00 -  82.99%
    C+  77.00 -  79.99%
    C   73.00 -  76.99%

It's even in the formal title.

Four years later in banking, I discovered the entire industry does the same, but in Excel. Enjoy your studies!

Well, Fermat made lots of similar claims wrt. other propositions, and for most of them the proof was found easily, or perhaps they were refuted altogether and shown to be wrong. FLT gets its name because it was a very rare case of a claim that just couldn't be solved, one way or the other. In fact, it seems that Fermat himself may have realized at some point that what he thought of as a proof he had, was in fact wrong - and dropped his claim altogether as a consequence. Which would then explain why it was only found as a margin note in a textbook. It's fascinating because it's such a simple claim to state, and yet the proof is incredibly complex. To be sure, logicians can predict that such cases will occur, in the abstract; it's a bit like having hard-to-solve instances of the SAT. But it's still nice to have such a natural example!

I wasn't sure if the author was making an intentional callout to Fermat's lost marvelous proof he couldn't fit in the margin. But indeed, looks like he was, since he doesn't seem to have really mentioned the length of any other proofs.