@Zee A serious answer: Complex differentiable functions have really nice properties (they're infinitely differentiable, for example). Real differentiable functions, and their derivatives, can be horrible.
@arctictern `A set, two binary operations (+ and *) and the operations satisfy: Distributive Laws and * is associative. I have no clue how any of this applies here
@DemCodeLines You literally have a checklist of things to check to verify that it's a ring. For example, does addition defined in Z18 satisfy the associative property for example? It's like, if I define a giraffe to be an animal with properties A, B, C and D, then in order to determine if an animal is a giraffe, all I need to do is verify that it has properties A, B, C and D.
@DemCodeLines do you not know what Z18 is? if you don't know what it is, you should have led with that question, instead of asking how to prove it's a ring.
Anyway, I remember reading that a locally integrable function that is CR is actually holonorphic which is mind blowing, but I didn't know enough to read more
Is continuity and existence of partials not enough? Or do we want the real and imaginary parts to be differentiable functions of two variables as well?
> Looman pointed out that the function given by $f(z) = \exp(−z^{−4})$ for $z \ne 0$, $~f(0) = 0$ satisfies the Cauchy–Riemann equations everywhere but is not analytic, or even continuous, at $z = 0$.
Oh, and the partials need to exist everywhere (or at least in an open set around the point)
Existing at one point, and satisfying Cauchy–Riemann there, is not enough.
> The function given by $f(z) = z^5/|z|^4$ for $z \ne 0$, $~f(0) = 0 $is continuous everywhere and satisfies the Cauchy–Riemann equations at $z = 0$, but is not analytic at $z = 0$ (or anywhere else).
Another weird thing about that series which was pointed out to me, and which is apparently due to Mahler: It's transcendental for any algebraic value of $z$ (besides $z=0$).
It shows up at a curricular level very strongly. For example, we've got a version of first year calculus which is more computational (but even they have to learn some proofs)
Then we have a 2 quarter "Math for social science" sequence, and a 2 quarter "Math for physical science" sequence
But that's about it, as far as I know. Physics was not happy and went and did their own math classes, so the latter is really more for geophysical sciences and chemistry
Math majors just do Spivak/IBL -> Rudin -> Dummit and Foote
But it's not enough, I still hold that algorithms in finite groups should be a core class and that Peter May should teach the 160s @Araske
Though here they go pretty hard on the stuff, everyone does a year of graduate algebra which is a quarter of representation theory, a quarter of commutative algebra and algebraic geometry, and a quarter which is more generic topics. Usually Kato teaches it and does more algebraic geometry, plus algebraic number theory
But yeah I mean, I'd say it's generally good to say that everyone ought have a minimum of knowledge across various areas. It shouldn't be like, a 4 year ordeal that takes everyone to research level
@Avery @bwDraco well, I looked. Turns out @TheGreatDuck and @Dodsy have been trading flags here for a couple of days. Latest volley was courtesy of Dodsy, but TheGreatDuck still leads by a fair margin for raw numbers pointless flags.
Ah, that makes sense @Semi. Usually I think of the "general math topics" as the basics of real analysis, complex analysis, point-set topology, and algebra
...I definitely have not been making any flags to target anyone and I have no recollection of flagging dodsy or anyone else much lately. As I said, I've been largely outside of the site recently.
@Zee I'd say perhaps the way my school does it is reasonably good. The first year everyone does the core courses, which are a year of analysis (real, functional, complex), a year of algebra (rep theory, comm alg + alg geo, topics), and a year of geometry/topology (atop, difftop, diffgeo)
@Dodsy You misunderstand completely. Dodsy. If you would just talk somewhere private I think I might be to explain, but I'm not comfortable giving out such information here...
I'm being honest. I find that to be pushing the boundaries of what is considered allowed, but I did not find it bad enough to choose to flag it. Plus, I barely even talk to Balarka and yet they've been getting tons of flags. Why would I flag someone for no reason for not even no reason.
@Dodsy I don't flag people for personal reasons. At least get that through your skull. and I have definitely not been flagging as much as you claim. I only flag things that are really bad.
and that hasn't happened in a while
besides, like i said: I cannot approve flags
so who the heck is going around approving nonsense flags, hm?
here you go, @Dodsy and @TheGreatDuck. I'd like to see the math chatroom chat about something other than this dispute (unless there are mods who can weigh in).
I'd recommend that we don't have this fight here, migrate to another room if you want to keep going. For factual claims, proof can now be requested, and the discussion on intention and what's offensive can go on forever. I will suggest that we return to math (and also will not continue participating in this conversation)
@Dodsy and just to clarify. I really don't want to get into a large discussion. It's kind of embarrassing but... you're gravely misinterpreting what I'm saying and I suggest that if you have some other form of external communication (such as email) that we have a little chat. There's been some stuff going on in my online life I haven't told a lot of people. Simple Art knows.
@SimplyBeautifulArt I messaged you on discord. You'll know what this is about.
That TGD has been flagging things has been public suspicion for a long while, and that's what has been clarified by community mods. I don't think there's any point to beating a dead horse to further levels of death.
@Dodsy and I should be very clear dodsy that i am not at all mad at you. In fact, I thought we were having a nice discussion the other night. I was very positive about it. But finding out these flags have been going on just makes my blood boil. If what that mod said is true I will most certainly be dealing with some shit.
@Dodsy alrighty. and stop bringing me back into chat with stuff. I'm technically chat banned.
you want me to be perma-banned for evading a ban?
I was told to stay out.
@Dodsy and if you have an email, I would please ask that you email me at thegreatduckinh@gmail.com. I'll explain why I have been flagging everyone.
@BalarkaSen Just to clarify. The flags were primarily just legitimate flags at first. I guess when people starting making fun of the whole thing (see I just assumed they'd get deleted) I guess I just figured marking a few once in a blue moon to make people laugh wouldn't be a big deal. I guess after a while it just got out of hand. I honestly never had anything against you. It was legitimately just you happening to say something profane at the wrong time and the mods being way to trigger happy.
but still use the email. I think there's more things that need to be said of off stack exchange. Too private for this site. If I were to say them, we'd probably all get banned.
and I'm not joking
@Dodsy one thing I should heavily emphasize though Dodsy is that I am far more self-destructive than rude towards other people and if I somehow could with a dupe account, I'd probably go through my whole chat history and flag any bad language I see... just to shit on myself more. You don't realize that many of my favorite posts on here are gone because I either reported them or cast close and delete posts on them just to punish myself for spending time on here.
Very Secret Number Theory Study Group
Very Secret Number Theory Study Group
