DevelopingElk

A simpler way to see this is to first focus on the central region of the big square that's 0.5 units away from the edge. All squares centers must lie in this region. Break this region into a grid of 25 equally sized square tiles. Each tile is now small enough that the center of two squares can't lie within the same tile. This gives 25 choose 11 possibilities. The article instead broke this region into hexagons that were still small enough that the centers of two squares can't lie in the same hexagon. This let them use only 16 tiles, which vastly reduced the space of possibilities.

I'm working on a reproduction of the proof with some personal changes. The basic approach is the standard computer assisted "unavoidable set" approach. First, choose some regions small enough that two square's centers don't fit in the same region, the article used 16. Each region must contain or not contain a square, which is 16 choose 11 cases, about 2000. For each case you try and rule it out. You do this by identifying areas that must be covered by a square, and propagating this information. You can also use packing LPs like Stromquist did in 1989 to rule out more configurations. You then narrow in on the remaining cases and subdivide them more.

I think the only reason this wasn't done pre-AI was due to it not being a topic of serious focus. 1989's computers were too weak to handle all the cases. But all the basic ingredients were present in the Kepler conjecture proof. What AI did was lower the effort enough that amateurs who just liked square packings could perform and formally verify such a proof. I consider myself among such amateurs. So this isn't a case of AI stealing mathematicians proofs, or doing something superhuman, its a case of democratization. I am concerned about how AI is affecting math and how the AI companies are behaving, but this isn't the case to be worried about. The calculations for proving this arrangement optimal will always be too big to be checked by hand. However, I'm hoping to produce some nice visualizations of the packing LP or core overlap that rejects each configuration

They do work, but transformers work better on the accuracy/compute tradeoff curve. Transformers let far away positions in the sequence exchange info in 1 step, while a dilated CNN will need many layers. Additionally, Transformers can be fairly efficiently trained.

I do think if Transformers weren't invented you would still be able to train powerful language models, but they would take more ram and be slower so nobody does this.

CPS is a way of embedding imperative computation into an FP language. I think they built a mini compiler for their binary relation language, which is then Jitted by Julia.

Russian one time pads were frequently broken. How? They were inconvenient to distribute, so people reused them.

Symmetric cryptography is safer because one key can easily protect as much data as you need.

One of the data center's cooling loops broke.

I ran into this problem where I wanted to generate balanced trees for test cases, and I decided to crack it with math.

Let X = the number of remaining open parenthesis required to reach your target length, and Y = the number of remaining close parenthesis.

Let P(X) = X/(X+Y) * (1 + 1/(Y-X+1)). Generate an open parenthesis with probability P(X), and otherwise a close parenthesis.

This will efficiently generate an unbiased random plane tree. The reason this works has to do with counting the valid ways to complete the plane tree, using a calculation similar to the one in the article.

His claim is that we exp-minus-log cannot compute the root of an arbitrary quintic. If you consider the root of an arbitrary quintic "elementary" the exp-minus-log can't represent all elementary functions.

I think it really comes down to what set of functions you are calling "elementary".

According to the article “This work effectively rules out explanations of the Hubble tension that rely on a single overlooked error in local distance measurements". So any systemic errors would need to affect multiple measurement types.

Good? Bad? Doesn't matter as long as you had fun.

Have you tested this GCs performance? Sometimes a baby GC can be fast enough.

I'm fairly sure this was human written.

Rust's discussion boards has an idea of "keyword generics" for expressing some of these concepts. The idea is that a function can be generic over const, async or some other keyworded effect. I like this description. It shows the benefits without too much theory.

Yours might go a little less into the details, but its really clear and I like the diagrams and explanation around glitch hazards. Please do follow up on your tangents if you have time.

I have commented once or twice on articles being AI generated. I don't put them when I think the writer used AI to clean up some text. I added them when there are paragraphs of meaningless or incorrect content.

Formats, name collisions or back-button breakage are tangential to the content of the article. Being AI generated isn't. And it does add to the overall HN conversation by making it easier to focus on meaningful content and not AI generated text.

Basically, if the writer didn't do a good job checking and understanding the content we shouldn't bother to either.