if they use an LLM to make the suggestions then it’s possible it ends up suggesting websites that don’t even exist. or it could accidentally suggest a malware website, or make a typo, etc.
this could be dangerous if they aren’t very careful
maybe the billionaires?
maybe it’s being developed in the arctic circle and there really are only four nights per year
Anyone can pay $150 to become a dues-paying member and rub elbows with the court’s nine justices at events like the dinner where Windsor spoke with Alito. (Tickets for the dinner were an extra $500.)
this is all it took for him to admit this stuff? anybody with 650$ could have walked in and asked him a couple prodding questions? these guys really arent even trying to hide it anymore
it was over as soon they casted kevin hart
mmmm cookies and cream
go for it
applied mathematics can get very messy: it requires performing a bunch of computations, optimizing the crap out of things, and solving tons of equations. you have to deal with actual numbers (the horror), and you have to worry about rounding errors and stuff like that.
whereas in theoretical math, it’s just playing. you don’t need to find “exact solutions”, you just need to show that one exists. or you can show a solution doesn’t exist. sometimes you can even prove that it’s impossible to know if a solution exists, and that’s fine too. theoretical math is focused more on stuff like “what if we could formalize the concept of infinity plus one?”, or “how can we sidestep Russel’s paradox?”, or “can we turn a sphere inside out?”, or The Hairy Ball Theorem, or The Ham Sandwich Theorem, or The Snake Lemma.
if you want to read more about what pure math is like, i strongly recommend reading A Mathematician’s Lament by Paul Lockhart. it is extremely readable (no math background required), and i thought it was pretty entertaining too.
some do, some don’t
Infinite-dimensional vector spaces also show up in another context: functional analysis.
If you stretch your imagination a bit, then you can think of vectors as functions. A (real) n-dimensional vector is a list of numbers (v1, v2, …, vn), which can be thought of as a function {1, 2, …, n} → ℝ, where k ∊ {1, …, n} gets sent to vk. So, an n-dimensional (real) vector space is a collection of functions {1, 2, …, n} -> ℝ, where you can add two functions together and multiply functions by a real number.
Under this interpretation, the idea of “infinite-dimensional” vector spaces becomes much more reasonable (in my opinion anyway), since it’s not too hard to imagine that there are situations where you want to look at functions with an infinite domain. For example, you can think of an infinite sequence of numbers as a function with infinite domain. (i.e., an infinite sequence (v1, v2, …) is a function ℕ → ℝ, where k ∊ ℕ gets sent to vk.)
and this idea works for both “countable” and “uncountable” “vectors”. i.e., you can use this framework to study a vector space where each “vector” is a function f: ℝ → ℝ. why would you want do this? because in this setting, integration and differentiation are linear maps. (e.g., if f, g: ℝ → ℝ are “vectors”, then D(f + g) = Df + Dg, and ∫*(f+g) = ∫f + ∫g, where D denotes taking the derivative.)
The default allocation for Recall on a device with 256 GB will be 25 GB, which can store approximately 3 months of snapshots.
this comes out to about 2 GB / week. it’s honestly terrifying they could be generating 2 GB of activity data for just a weeks worth of computer use. it’s both a privacy nightmare and an optimization nightmare
i forgot for a second that the winters and summers get flipped in the southern hemisphere
it will only be the strongest material in the universe until it gets boiled. trust me on this one
if they invent some new kind of fucked up math to do it then there could be far reaching consequences
“shittitest alchemist currently alive” has got to be one of the most challenging titles to hold onto for any serious length of time
you can always add an empty room without changing the total number of rooms, so there should be plenty of room for sisyphus and his boulder at the hotel
you got off easy. some of us have been trying for minutes
i think this is a fairly reasonable gut reaction to first hearing about the “unnatural” numbers, especially considering the ways they’re (typically) presented at first. it seems like kids tend to be introduced to the negative numbers by people saying things like “hey we can talk about numbers that are less 0, heres how you do arithmetic on them, be sure to remember all these rules”. and when presented like that, it just seems like a bunch of new arbitrary rules that need to be memorized, for seemingly no reason.
i think there would be a lot less resistance if it was explained in a more narrative way that explained why the new numbers are useful and worth learning about. e.g.,
i think the approach above makes the addition of these new types of numbers seem a lot more reasonable, because it justifies the creation of all the various types of numbers by basically saying “there weren’t enough numbers in the last number system we were using, and that made it a lot harder to do certain things”
they won’t even turn off the ads if you pay them. what a joke
edit: oops i just saw that these are the “free benefits”
i feel the same way but i make an exception for the live album “stop making sense” by the talking heads. specifically the songs “crosseyed and painless” and “once in a lifetime” sound so much more energetic on the live album. depending on my mood, i sometimes pick the live versions over the studio recordings. they feel like different songs to me