That example "Comedy" sounds like the most soulless synthetic example of "music" ever. Like the music analogue of a PR-speak article of a big corpo generated by an LLM. It's still technically impressive but it's 0/10, absolutely empty of soul.
All AI generated art feels somewhat masturbatory to me, but AI generated music is the most baffling of them all. It's like what if someone thought Muzak had too much soul.
I’ve always been curious about differentials and how to build a rigorous theory of what the fuck dx, dy, dy/dx, etc. are. For example, if you study Tao’s Analysis and Analysis 2, you will not see anything at all about differentials, and I think that maybe you won’t see the dy/dx notation at all. So, can anyone recommend a textbook about differentials?
Depends what you're looking for. Full Frontal Calculus[0], Intuitive Infinitesimal Calculus[1], and Elementary Calculus[2] are all textbooks on the calculus sequence using an infinitesimal pov. The basic approach is to extend the Real numbers to include infinitesimals (greater than zero but smaller than every positive real number) and transfinites (greater than every positive real number), collectively called the Hyperreals.
If you're looking for a more formal approach, ie the infinitesimal analogue to the usual real analysis, it's called nonstandard analysis and you could probably start with the original, eponymous book written by the creator, Abraham Robinson, for which I unfortunately don't have a link.
If this stuff interests you btw I would also check out Knuth's book on surreal numbers[3], which I believe, in some sense, are the fullest possible extension of what we think of as numbers? But it's been a while since I read into those.
If we're going the hyperreal route, I quite like Goldblatt's GTM Lectures on the Hyperreals. You have to augment it with a paper or two if you want to work with other nonstandard objects, but when I was doing my graduate work it was the resource I kept going back to for clarity.
This looks like the most in-depth resource on the topic that I've seen so far; thanks for adding it! One of the reasons that I'm partial to the hyperreals is because it's such a natural thing, in the context of mathematical history, to extend the number system when that system isn't expressive enough to solve the problems we want to solve. The limit-based approach seems clumsy in comparison.
That same idea applies very nicely to other ideas of math too. Take Konig's Lemma (every infinite, locally finite, connected graph has an infinite path) as an example. The nonstandard proof goes something like:
1. For every natural, you can find a path of that length.
2. Therefore (nonstandard chicanery), for every hypernatural you can find a hyperpath of that hyperlength. Pick one for some infinite hypernatural.
3. Restricting that hyperpath to the original graph yields the infinite normal path you were looking for.
Whole problems melt away entirely as soon as you don't have to worry about clumsy "limit-based" approaches.
In representation theory we reduce problems of algebra to problems of linear algebra. E.g. the standard example is to find representations of groups, this way we can represent group operations as matrix operations. We do this because (1) linear algebra is mathematically very well-understood, (2) in terms of applications, linear algebra is computationally fast, faster than implementing the group with code manually (at least, in general).
In the OP post, author reduces quiver (which is a particular kind of algebra) to linear algebra. Once this is done, the intention is to solve problems of quivers in the language of linear algebra.
>however, according to Reddit, there’s a number of people who want to be left alone and can be irritated if you interrupted their workout to talk.
My suggestion would be not to read social advice on public websites on the internet, especially on Reddit, because per public internet, everything is not okay/forbidden, everyone should mind their own business, choose the safest and the most inoffensive action in every possible social situation. Public places such as reddit are full of terminally online socially awkward people who are very unrepresentative of people in real life. Also, there are incentives to recommend the safest course possible because then you won't get downvoted by haters. I don't even say to take advice there with a grain of salt. I say it's probably better not to read such stuff because your brain might subconciously internalize that people think like this even though actully, in real physical world, it works differently.
I heard about this game many many times due to software developers showcasing it as an example of a good libre videogame. However, I don't know a single person who played it and I have never seen anyone recommending it for its gameplay.
It is a relatively simple formula that is very combat heavy with extremely simple economy. The campaigns are excellent though and as long as the true randomness of attacks/defense doesn't drive you crazy it is a lot of fun. Very challenging and has real strategic and tactical depth as well as pretty well balanced.
I personally never did multiplayer but last I checked the multiplayer community was pretty healthy.
i played it, its fine, its a solid game. Easily can lose several hours in a session and probably played over 40 in total. Its enjoyable to play through due to the upgrading mechanics and wanting to see all the potential evolutions. That said, I'm not always a huge fan of the level design as you're often encouraged to play into negative fights (e.g. the timing for meeting the enemy aligns with their daytime bonuses) which forces you to play a bit more defensively than I'd like.
>The real reason Valve are being the "good guys" at the moment
Ok, but this “at the moment” has lasted at least since 2011. Basically my whole adult life Valve gas been a pretty great company delivering value and not being annoying.
For me that's how it works unless I already have a Firefox window opened. If I have multiple windows of different profiles already open (which happens all the time) it may add it to the wrong one, so I sometimes need to drag and drop the tab between the windows.
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