The whole thing is unbelievable slop ! I think the whole article is llm-generated unfortunately (just reading the first paragraph I got an immediate smell).
Fair on all accounts! Surely, this could be made way more lively if I were in front of a blackboard waving my hands and drawing images, but alas, the medium is what it is :)
well, the statement is: is there a single operation, built from elementary operations, such that all _other_ elementary operations have finite representations.
this preprint answers that in the affirmative
otoh, (x, y) -> 1/(x-y) does not answer this question at all. you can argue that the preprint does so "via the infinite series in an operation" (which I have no idea what that means; surely if exp(x) qualifies then so must 1/(x-y) if we pick a monomial basis?) but ¯\_(ツ)_/¯
now, do I think that this is groundbreaking magical research (as I'm currently seeing on twitter) no... But it's neat!
I don't think this can do any of the "standard" constants or what we generally consider to be closed-form expressions, though ! (E.g., no e, pi, exp, log, etc.)
> True, but with such numbers you will normally not do anything else except computing an approximate value of them.
That's what I think people do with other numbers like "pi" at the end of the day, no? :)
> That is what I meant by "interesting", i.e. the necessity of using symbols of such numbers, obviously for use in symbolic computations, since in numeric computations you would never use the actual numbers, but only some approximations of them.
It's very much an encoding problem, I think. Though we probably, on aggregate, use "unnamed computable numbers" implicitly on the order of as much as we use "named computable numbers" the former just has way more of a "tail" of uses where the "encoding of the symbol" is, e.g., "here's the PDE you use to compute this number"!
(It gets a little weird since we're kind of not distinguishing between the approximation that can be used to construct said numbers to arbitrary precision vs the specific program instance that constructs one specific approximation, but the idea is mostly there.)
> few of the computable numbers that are not algebraic are interesting, the main exceptions being the numbers that are algebraic expressions containing "2*Pi" and/or "ln 2".
I don’t think this is true at all. For example: the solution to a generic PDE that has no closed form solution at some point of import is likely transcendental, not algebraic, but definitely computable. (Think, say, Navier-Stokes being used for weather predictions in some specific place.)
I don't want to put OP on blast here, but this is unfortunately just complete slop writing.
The points being made are fine, I think, but look, if it's faster for you to generate than it is for us to read, I think this qualifies as denial-of-service-lite.
Yes definitely a great extension would be to add a camera in the image plane (alternatively, defocusing the image slightly and using a photodiode would also be fun!)
Ooh, great question. Usually fractions (~1/20th?) of a turn for alignment, it’s hard to go below that since the mounts are so small and the springs don’t have the tension to keep it super stable. (This is plenty for such a “coarse” set up like a Michaelson but might not be up to par for more delicate ones. This can be improved very easily but it was enough for this experiment!) If you want to observe something on the outputs, you have to do something like exhale on one of the arms or put a soldering iron near one of them—merely touching one of the screws gives you indiscernible output, even if the mirrors are aligned.
Very interesting re: JWT, I will definitely take a peek, thanks !
I think getting an electron source and creating a robust-ish adjustable set up is v doable, but is definitely more of a Real Project(TM) than this silly little interferometer :)