Readit News> 'Additive' here means that if u(K1) is defined as the unknotting number of the knot K1, and u(K1#K2) the unknotting number of the knots K1 and K2 joined together, then u(K1#K2) ≤ u(K1) + u(K2).
Kinda like the triangle inequality[1] of knots?
I recall the triangle inequality was useful for several cases in Uni, if so I guess I can see it might be a similarity useful inequality in knot theory.
I incorrectly had a ≤ instead of =, my apologies.
awanderingmind commented on The wall confronting large language models arxiv.org/abs/2507.19703... · Posted by u/PaulHoule
There is a lot of focus in the comments on the authors' credentials and, apparently, their writing style. It is a pity, because I think their discussion of scaling is interesting, even if comparing LLMs to grid-based differential equation solvers might be unconventional (I haven't convinced myself whether the analogy is entirely apt/valid yet, but it could conceivably be).
Maybe the article is dumbing it down too much, but the conclusion seems unsurprising. Why shouldn't a single unknotting do double-duty in some cases?
It feels akin to the classic trick of joining a tetrahedron to a square pyramid: 4 faces + 5 faces == 5 faces total!
According to the actual paper (https://arxiv.org/pdf/2506.24088), it has been an open conjecture since at least 1977. The quote:
> Unknotting number has long been conjectured to be additive under connected sum; this conjecture is implicit in the work of Wendt, in one of the first systematic studies of unknotting number [37]. It is unclear when and where this was first explicitly stated; most references to it call it an ‘old conjecture’. It can be found in the problem list of Gordon [13] from 1977 and in Kirby’s list [16].
'Additive' here means that if u(K1) is defined as the unknotting number of the knot K1, and u(K1#K2) the unknotting number of the knots K1 and K2 joined together, then u(K1#K2) = u(K1) + u(K2). It is this that has (assuming the paper is correct) been proven false. A deceptively simple property!
edit: I initially incorrectly had a ≤ sign instead of =
This looks like it'll be useful for CAPTCHA purposes.
According to the researchers, “the triggers are not contextual so humans ignore them when instructed to solve the problem”—but AIs do not.
Not all humans, unfortunately: https://en.wikipedia.org/wiki/Age_of_the_captain
Cool example in that link, thanks!
Cool study. I really want to believe the results, but the effect on life extension is so large (see figure 2B) that I find it hard to. Maybe there was some uncontrolled confounding factor? It is noted in the 'Methods' section that 'Researchers were not blinded to group allocation [...]', which is unfortunate.
awanderingmind commented on The Collapse of the FDA nytimes.com/2025/07/08/ma... · Posted by u/littlexsparkee
i know this is a very political thing now but i've had friends (smart phd people who work industry) very annoyed at the fda for many years, and maybe this collapse is good!
the fda started with a noble mission but they've been getting heavy handed. or better cliched - slow handed with getting things certified.
you can solve this one or two ways: drop regulation or increase staffing.
so many institutions have unnecessary fluff, tremendous red tape (why do i need environmental review to stick a shed in my backyard??), our modern lives have too much regulation.
let's hope for the best.
the old system is holding back drugs.. there should have been more ozempics, more breakthroughs had the fda not been so slow. companies have a strong incentive not release bad drugs now.. lawyers are not cheap and law firms know money can be made.. it's not the 1930s anymore.. (okay it's still the 1930s in certain places of the world, that's a criticism)
typing this out hoping to convince any regulation reduction is good reduction, i thought of a third fda option: the fda let's everyone go hog wild initially but looks at the top consumed products and checks them for safety and efficacy each year.
This perspective is addressed in the article... TLDR; that doesn't seem to be where this is going.
This is a rare type of article - a concrete analysis of different approaches to programming (that are arguably themselves reflections of different cognitive styles), that outlines the shortcomings of one approach in a specific domain, without generalising too much.
Production of electricity like production of steel makes most economical sense at scale. When the economic policy fails so hard it has to resort to backyard anything you know where it's going.