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:
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 |U†U| 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.
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.
Negative radiation forces act opposite to the direction of propagation, or net momentum, of a beam but have previously been challenging to definitively demonstrate. We report an experimental acoustic tractor beam generated by an ultrasonic array operating on macroscopic targets (>1 cm) to demonstrate the negative radiation forces and to map out regimes over which they dominate, which we compare to simulations. The result and the geometrically simple configuration show that the effect is due to nonconservative forces, produced by redirection of a momentum flux from the angled sides of a target and not by conservative forces from a potential energy gradient. Use of a simple acoustic setup provides an easily understood illustration of the negative radiation pressure concept for tractor beams and demonstrates continuous attraction towards the source, against a net momentum flux in the system.