For other dimensions, this is an open question; it seems unlikely to be true in general. For some dimensions the densest known irregular packing is denser than the densest known regular packing.
> For some dimensions the densest known irregular packing is denser than the densest known regular packing.
I thought that was one of the important results from the paper, the most efficient packing for all dimensions is symmetrical again and this increase was significant enough it seems unlikely that existing non-symmetrical methods will be able to beat it.
Perhaps so. If you hunt you might be able to find a new summary table somewhere (I didn't find one in a very brief skim around). My impression was this new work was more about high-dimensional cases than necessary a dramatic improvement for every low-dimensional example.
> His result has also revived a debate in the field about the nature of the optimal packing in arbitrarily high dimensions. For a while, mathematicians considered highly symmetric, lattice-based packings to be the best way to arrange spheres as densely as possible. But in 2023, a team found a packing that didn’t rely neatly on a repeating lattice; before Klartag’s result, it was the record to beat. Some mathematicians saw it as evidence that more disorder was needed in the search for an optimal sphere packing.
Clearly, this improvement doesn't apply to the few dimensions for which we already proven optimal packing, but the proof was general.
If you click through to the link from your quoted sentence, the 2023 paper "provided a new recipe for how to densely pack spheres in all arbitrarily high dimensions", and from the description the new paper also seems to address the same question.
This is different from asking about the best-known specific packing in n dimensions when n = 7, 10, 19, or whatever.
In a few of the specific low dimensional cases, my impression is that the best currently known packing is not regular, and I don't see any evidence that this has changed. Perhaps this new method will give people a way to beat those records, or perhaps not.
Not necessarily—in 3d there are uncountably many non-lattice packings. They all have the same density as the FCC lattice though. To construct these packings, shift horizontal layers of FCC horizontally with respect to each other.
It is conjectured that in higher dimensions, the densest packing is always non-lattice. The rationale being that there is just not enough symmetry in such spaces.
Well these new results (denser packings than before) are regular lattices which might suggest that the optimal packing could be a lattice. (Until the record is broken again by a irregular packing ;-)
If you read the ruling, training was considered fair use in part because Claude is not a book generation tool. Hence it was deemed transformative. Definitely not what Suno and Udio are doing.
I was "stuck" (someone dropped me off and someone else was supposed to pick me up, but they were late so..) on a highway/motorway in <country> (edit: I wrote it and then deleted it for privacy) some weeks back. The weather was great! Sun was shining, a cool 24 degrees, I was wearing my hoodie, it was windy. I got bored on waiting by the highway/motorway.
No village/coffee place anywhere near, so I decided to take a vertical small road and walk by a green field. And it was windy. And I could see the bushes and leaves from the trees swinging back and forth. And it was windy and very calming (to my soul) so I stood there gazing at the wonderful nature. And I was thinking, why the fuck do we live in cement boxes in cities? I could buy "a few sqm, build something with glass/brick/steel, no deep foundations, and smaller 25sqm "hut" as my office right next to the "house" and live next to a field and have a great life...
Anyway, my friend arrived, picked me up and we continued driving.
I was thinking that the cost of remote land/house/'office' would be 50% on the cost of a 100sqm flat in (most), with the pro of the calmness and the con of being alone in the middle of nowhere.
It makes intuitive sense and maybe is true in many places (I’m not in the US, for example), but some of the quietest flats I happened to have seen were in subsidized government housing or cost almost half of my rent. I simply did not luck out with one of those.
If you are in top N% then yes, you probably just do not have to worry about noise. However, at this point we are just arguing about the definition of “the poor”. If you mean anyone who lacks cash to forfeit the tenancy agreement like it was nothing, then yes.
The system while better is biased towards parties who can get the majority of individual constituencies based on geographic location. It relies on localized monocultures to get democracy for smaller parties. But that doesn't happen.
House of representatives is not designed to provide proportional representation based on aggregate % vote country wide. Senate is more aligned that way and it's reflected in the numbers, in AU:
> Dependency injection is just passing your dependencies in as constructor arguments rather than as hidden dependencies that the class itself creates and manages.
That's the best way to think of it fundamentally. But the main implication of that which is at some point something has to know how to resolve those dependencies - i.e. they can't just be constructed and then injected from magic land. So global cradles/resolvers/containers/injectors/providers (depending on your language and framework) are also typically part and parcel of DI, and that can have some big implications on the structure of your code that some people don't like. Also you can inject functions and methods not just constructors.
That's because those containers are convenient to use. If you don't like using them, you can configure the entire application statically from your program's entry point if you prefer.
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