mmmm cookies and cream

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, …, v_n), which can be thought of as a function {1, 2, …, n} → ℝ, where k ∊ {1, …, n} gets sent to v_k. 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 v_k.)

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.)

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

the standard (set theoretic) construction of the natural numbers starts with 0 (the empty set) and then builds up the other numbers from there. so to me it seems “natural” to include it in the set of natural numbers.

what if you just attach a second magnet to the car so that it pulls the first magnet forwards?

what were they doing for all that time

watching oppenheimer felt like watching a 3 hour compilation of trailers for oppenheimer. everything was so dramatic, there was no downtime, and they always had some kind of music playing. it felt like every scene wasn’t allowed to last more than 5 minutes

the teacher is the one doing the real lab experiment here

but can he grind on the topologists sine curve:

under an appropriate level of abstraction you can make lots of types of cylinders. in topology you can define a cylinder of a (topological space) X to just be X × [0,1]. this kind of definition comes up pretty frequently, and is used to create mapping cylinders, which i suppose are another type of cylinder.

the “normal” kind of cylinder is then just (circle) × [0,1], or (filled in circle) × [0,1], depending on whether you want it to look like an empty paper towel roll or a (full) can of beans

what if it’s not euclidean but only locally euclidean?

completely agree. and it’s even more insidious when you take into account how he’s spent the past 6 years bragging about how he has a salary of $0 because he’s “only working for the betterment of humanity” or some nonsense like that.

Board chair Robyn Denholm wrote in a letter included in the regulatory filing: “Elon has not been paid for any of his work for Tesla for the past six years… That strikes us, and the many stockholders from whom we already have heard, as fundamentally unfair.”

Musk’s compensation for 2023 was $0, the filing showed, as the billionaire does not take a salary from the company and is compensated through stock options.

it’s so unfair that elon hasnt gotten a single pay check and has instead had to settle for making billions off of his stock options. think of all the mega yachts and social media companies he could’ve bought if only he had been paid a salary.

i think this is a really clean explanation of why (-3) * (-3) should equal 9. i wanted to point out that with a little more work, it’s possible to see why (-3) * (-3) must equal 9. and this is basically a consequence of the distributive law:

0  = 0 * (-3)
   = (3 + -3) * (-3)
   = 3 * (-3) + (-3) * (-3)
   = -9 + (-3) * (-3).

the first equality uses 0 * anything = 0. the second equality uses (3 + -3) = 0. the third equality uses the distribute law, and the fourth equality uses 3 * (-3) = -9, which was shown in the previous comment.

so, by adding 9 to both sides, we get:

9 = 9 - 9 + (-3) * (-3).

in other words, 9 = (-3) * (-3). this basically says that if we want the distribute law to be true, then we need to have (-3) * (-3) = 9.

it’s also worth mentioning that this is a specific instance of a proof that shows (-a) * (-b) = a * b is true for arbitrary rings. (a ring is basically a fancy name for a structure with addition and distribute multiplication.) so, any time you want to have any kind of multiplication that satisfies the distribute law, you need (-a) * (-b) = a * b.

in particular, (-A) * (-B) = A * B is also true when A and B are matrices. and you can prove this using the same argument that was used above.

one time i had to call a company during regular 9-5 business hours to cancel a subscription after starting a free trial.

that experience was so horrible ive since sworn off free trials altogether. nowadays, if i need a free trial to use an app or website for a couple days, then i will simply not use that app or website.

Everybody knows what free speech means.

i really dont think so.

free speech is a pretty complicated thing and i feel like many people dont have a solid grasp on it. i think a good number of people think they know what free speech means because they know “it only applies to what the government can do to you”, but there’s quite a bit more to it than that. like how to deal with hate speech, threats, misinformation, disinformation, etc.

and this is directly related to the problems twitter is facing: elon musk started out by saying hes a “free speech absolutist”, but twitter has been slowly rediscovering why “free speech absolutism” doesnt work. and you can see those discoveries in real time with twitter reintroducing moderation policies (among other things)

is apa the one that makes you cite things in parentheses? because that one gets a thumbs down from me