To comment something then, the symmetric group (bijections) is generated by permutations of two elements, in the braid group you can braid the left whisker on top of the right one, or below. If you permute twice, you do nothing, but you can twist hair one, two, three, ... any times. that is the contrast in the generators and relations of both groups. If you go to the wiki page of topological quantum computer, the photo expresses a unitary representation of a braid group element. Schrödinger evolution in discrete time is given by unitary matrices (one of the Stone theorems). Now look at the pic, and imagine threads from input 1 to output 1, input 2 to output 2, etc, in adition to the colored threads of the pic. With the extra threads as gluing spec (topological identification) you get a single colorful closed curve (or several ones). The chapter is going to talk about how this is a link/knot. So you get an algebraic understanding from a structural object as the symmetric group is, opening what these closed curves are. People in loop quantum gravity could have had their fingers on that kind of page. There is an accesible description of the Jones polynomial. If you bookmark this, next time the pros juggle the name before you you have a place to go to avoid showing "that face".
I am more optimistic, good blogs will continue being there, and crap ones are no new invention or menace, be it LLM slop or Markov chain SEO babble content of 10 years ago.
There is an economy here that the effort investment is paid with a reward of quality. Many teens feel the discomfort of being locked into platform attention farms and are stoic about it. They deserve the opportunity of being told other options exist.
Are you a teen or you speak on behalf of the whole world? My impression of said demographic is totally different, though. And I would not speak for them just like that, as they be very diverse depending on where you find them.
As I understand, the technology was protected by a patent help by guys at Leptonica and it exprided. There is a crude project for encoding images to jbig2 at https://github.com/agl/jbig2enc. I am sharing my personal scripts here [1] (windows) that wrap that for end to end djvu to pdf for scanned texts using jbig2 compressed images in the pdf instead of jpeg. This combines decent compression with pdf handiness. djvu still compresses better but pdfs can be got under twice the side, that sounds no impressive, but many common available pipelines produce sizes x3, x4 and worse, a particular offender those using ghostscript pdfwriter. The sripts have worked months locally but are given "as is" without testing, with zero support, you deal with python dependencies and having jbig2 and djvu-libre tools in the path. Beyond image compression tech, they support OCR-layer (cut/pasteability), bookmark and page label migration from djvu to pdf info.
Interesting, he also talks about quantum computing (a first?): p. 191, "We now go on to consider how such a computer can also be built using the laws of quantum mechanics. We are going to write a Hamiltonian, for a system of interacting parts, which will behave in the same way as a large system in serving as a universal computer."
p. 196: "In general, in quantum mechanics, the outgoing state at time t is
eⁱᴴᵗ Ψᵢₙ where Ψᵢₙ is the input state, for a system with Hamiltonian H. To try to find, for a given special time t, the Hamiltonian which will produce M = eⁱᴴᵗ when M is such a product of non-commuting matrices, from some simple property of the matrices themselves, appears to be very difficult.
We realize, however, that at any particular time, if we expand eⁱᴴᵗ out (as 1 + iHt − H²t²⁄2 + …) we'll find the operator H operating an innumerable arbitrary number of times — once, twice, three times, and so forth — and the total state is generated by a superposition of these possibilities. This suggests that we can solve this problem of the composition of these A’s in the following way..."
Feynman is indeed often quoted among the first people to propose the idea of a quantum computer! This talk he gave in ‘81 is among the earliest discussion of why a quantum universe requires a quantum computer to be simulated [1]:
> Can a quantum system be probabilisticaUy simulated by a classical (probabilistic, I'd assume) universal computer? In other words, a computer which will give the same probabilities as the quantum system
does. If you take the computer to be the classical kind I've described so far, (not the quantum kind described in the last section) and there're no changes in any laws, and there's no hocus-pocus, the answer is certainly, No! This is called the hidden-variable problem: it is impossible to represent the results of quantum mechanics with a classical universal device.
Another unique lecture is a 1959 one [2] about the potential of nanotechnology (not even a real thing back then). He speaks of directly manipulating atoms and building angstrom-scale engines and microscope with a highly unusual perspective, extremely fascinating for anyone curious about these things and the historical perspective. Even for Feynman’s standards, this was a unique mix of topics and terminology. For context, the structure of DNA has been discovered about 5 years prior, and the first instruments capable of atomic imaging and manipulation are from at least the 80’s.
If you’re captivated by this last one as I was, I can also recommend Greg Bear’s novel “Blood Music”. It doesn’t explore the nanotechnology side much, but the main hook is biological cells as computers. Gets very crazy from there on.
If you're into atomic physics and getting a feel for the intricate structure of the basic processes, the best find I had recently is this MIT course by Wolfgang Ketterle. The first lecture is an informal overview, and he gives vivid and detailed descriptions of the phenomena they can create and control now, like why we see different kinds of thing happening at very low temperatures: the atoms are moving past each other so slowly that it gives their wavefunctions time to overlap and interact, using intersecting lasers to create arrays of dimples in the electromagnetic field to draw in and hold single atoms, this kind of thing. It gives a more tangible insight into the quantum aspects of matter that can otherwise seem inscrutable
The quote is not suggesting a quantum computer can’t be simulated classically, it can in fact, just slowly, by keeping track of the quantum state where n qubits is 2^n complex amplitudes.
It relates more to the Bell results, that there doesn’t exist a hidden variable system that’s equivalent to QM.
“There’s plenty of space at the bottom” only really took off in popularity decades later. Feynman’s accomplishments are undeniable, Nobel prize and all, but his celebrity status is given by other aspects of his personality. No Feynman equivalent I can think of is alive today. Perhaps Geoffrey Hinton and his views on the risk of AGI? He’s far from the only one of course.
I loved to build backing tracks for guitar in Band-in-a-box, just from the chord progression and some settings. Leveraged little effort to interesting results. And the idea of a DSL is super. But I dunno how would you stand comparisons with audio rendered by pro DAW software loaded with a production quality sound library such as Hollywood Strings or similar if you render the audio yourself.
Ha, I was just playing with making a simple pad in webaudio and it evolved into a progression-playing backing track tool (vanilla html/js/css page). It would appear there are a lot of us in the Venn intersection of programmer/guitarist/practice time alone enjoyers.
Baez wrote some ideas in [1], one I'm liking connects Lorentz group in dimensions 3,4,6 and 10 with the modular group SL(2,Z) that is at a crossroads of several hardcore math themes. For Lie algebras:
sl(2, R) ≅ so(2,1)
sl(2, C) ≅ so(3,1)
sl(2, H) ≅ so(5,1)
sl(2, O) ≅ so(9,1)
Dirac equation is the C case, the other cases have their uses.
I skimmed the chapter on operators (7) and always liked that way of thinking (plug things like the derivative operation D into things that expect numbers instead, and see what happens). So plugging into 1/x and getting integrals. Dattoli and Tom Copeland do serious stuff starting from that kind of considerations that go way beyond cocktail party tricks.
reply