If the figures / occasionally risqué cartoons scattered throughout are any indication, this book is going to be incredible.

c.f. figures 1.3, 9.2, but most especially the unlisted figure on page 146 in the introduction to part III

If the figures / occasionally risqué cartoons scattered throughout are any indication, this book is going to be incredible.

c.f. figures 1.3, 9.2, but most especially the unlisted figure on page 146 in the introduction to part III

As I’m writing in my journal this morning, I freshly mint the word “empathodepressed” to described depression due to being around and empathizing/mirroring someone else who’s depressed. And of course I expected OS X to tell me that my spelling didn’t produce a known word—and, well, it did do that—but I didn’t expect it to produce a suggested correction.

Most of all, though, I didn’t expect the suggested correction to be:

empathodepresbsed

just used

unsafePerformIO :: IO a -> a

for the first time, and it did not disappoint

Speaking of quantum information, here’s a quick proof of the no-cloning theorem if you don’t know it; it’s a simple proof that requires only rudimentary quantum mechanics knowledge.

Suppose you had a quantum state cloner, call it C, which could produce the following effect: C(ψ⊗e) = ψ⊗ψ, where the initial state e—which plays the role of a blank sheet of paper in a photocopier—is yours to choose, but your copier must work on any arbitrary unknown input ψ.

Then consider two different possible inputs ψ and χ. The initial states of the copier have inner product 〈ψ⊗e | χ⊗e〉. But since, in quantum mechanics, time evolution is unitary, C must be unitary and this inner product is equal to 〈ψ⊗e |UU| χ⊗e〉. But this is just 〈ψψ | χχ〉 by definition, which is the square of what we started in (choosing e to be normalized). But then ψ and χ are either orthogonal or identical; in any case, they aren’t two arbitrary states, so the generic copier C must not exist.

PS: If you just time-reverse this process—and unitary evolution is of course time-reversable—then you get the no-deleting theorem.

Lifting the successor function through as a pullback on the Fourier components g : Z → C of a function f : S1 → C actually induces a global phase twist in the image of f by mapping f(k) → f(k)e2πik. Imagining the constant function f = (λ x . 1) is particularly pleasing, visually.

(And the image of f under this transformation of course remains globally continuous, because the phase is just unity at k=0,1; going the other way, lifting the (+α) function into the Fourier domain of a complex valued function on the integers, of course has no such restrictions.)

These calculations are much more complicated and require an enormous amount of computer time, because a very large number of [points] (more than 5000) must be taken to achieve convergence… [our] calculations are thus very fast and convenient, and indeed we were able to perform them on a PC microcomputer.

– some scientist in 1988 (emphasis mine)

http://thesummerofmark.tumblr.com/post/93894004721/thesummerofmark-my-favorite-proof-that-the

thesummerofmark:

thesummerofmark:

my favorite proof that the fundamental group of the circle is the additive group of integers is the one where you imagine something going around in a circle, in either direction, arbitrarily many times

Err, sorry—I definitely meant higher homotopy groups, not higher spheres. I agree that the fundamental group for higher spheres isn’t so challenging, especially if you’re practiced at it.