/sci/ - Science & Math

>>15853575
Real quick, given the action of some group on a variety, what is the induced action on the regular functions? In particular, I'm interested in torus actions.

>mfw grad student TA and grading undergrad HW and exams.

Lol what a fucking abortion of a /mg/. The old one hasn't even hit the bump limit yet.

>>15854778
you're an abortion

>>15854890
What theoretical IQ would be needed to realistically do this?

The more I think about Axiom of choice, the less sense it makes to me.
1. Let D be a set of computable numbers.
2. According to AoC, we can somehow keep picking numbers from R \ D
It would be even weirder if instead of set of computable numbers we could've taken a set of definable numbers, but I'm not sure whether that's a legitimate set (the notion of first-order definability cannot be expressed as a formula in the language of set theory, so there's that).
I don't think I've disproved mathematics, but at this point I'm genuinely confused why anyone ever thought that it AoC (outside of countable sets) was intuitive in any way. It certainly doesn't seem empirical or applicable to real world in any way.

>>15855760
You are braindead.

>>15855766
Ok.

>>15855729
biggest obstacle would probably be motivation and discipline given youre not a midwit

>>15855760
>why anyone ever thought that it AoC (outside of countable sets) was intuitive in any way.
They didn't. That's part of why it was controversial for years.

>>15855773
I mean the problem is that the AoC IS intuitive, it just has extremely unintuitive consequences. No one would be arguing about it if the original axiom didn't seem obviously true but given a collection of sets, you would think that of course you can select an element from each one. Countable vs. uncountable doesn't make a difference until you start really inspecting it.

>>15855760
i would just say let infinity equal zero (hegel's "speculative judgements") and loop the end behavior back to the origin (asymptoto-axiomatically). godspeed pal.

I came across this integral by playing around with the Desmos graphic calculator. Can some kind of closed form be solved for this?

>>15855729
I have a theoretical IQ of 200.

>>15856180
that sounds like your problem
chaos pendulums are easy to set up

>>15856621
I think you meant to reply to someone else

>>15855760
You don't need AC to do that, just pick any noncomputable number x and consider x, x+1, x+2...
>>15856180
Why does being countable make a difference? Never understood that philosophical argument

>>15856699
You just don't get all the sketchy behavior with countable sets. They're much tamer and don't lead to counterintuitive mathematics. The well-ordering theorem, for example, is perfectly reasonable when applied to countable sets, but seems questionable for uncountable sets. It's equivalent to the axiom of choice, though, and things like that make some people think we should restrict to countable choice only.

>>15855760
I don't get what you're not finding sensible about this. Is this a weird finitism troll?
>I'm genuinely confused why anyone ever thought that it AoC (outside of countable sets) was intuitive in any way
How is it not?
>>15856792
"I don't like that I don't understand it, therefore it should be gotten rid of"? wut

>the banach-tarski paradox is impossi-
https://www.youtube.com/watch?v=9IKwPoNhP4E

>>15856809
Well it's an axiom, not a theorem. Axioms are supposed to be those "obvious truths" that are too basic to be proven but people generally agree about. It leads to counterintuitive results, so some people don't believe we should accept it as an axiom. At least it's independent of ZF, so a lot of mathematics can be salvaged if you dispense with it, but most people basically regard it as too useful. Functional analysis, for example, would cease to exist without choice, and it seems to be worthwhile.

>>15856817
>Well it's an axiom, not a theorem. Axioms are supposed to be those "obvious truths" that are too basic to be proven but people generally agree about.
The hell is with that weird implication that popularity factors in somehow? An axiom need merely be a stated assumption and whether or not anyone else likes it doesn't matter. What matters is whether or not the assumption(s) work to accomplish whatever goal you have in mind. Does not matter one piss whether anyone else finds it counterintuitive the only thing that matters is whether or not it works.
>It leads to counterintuitive results, so some people don't believe we should accept it as an axiom.
Just because something, when analyzed, leads to things hard to understand implies nothing about whether we should or should not accept something. Reality is hard to understand for a lot of people and guess who wins that argument? You are still not making sense.
>a lot of mathematics can be salvaged if you dispense with it, but most people basically regard it as too useful
Jesse, what the fuck are you talking about? "salvaged"? Wtf?

these fags need the axiom of choice to pick one cock from each family to suck LMAO get fukt setqueers

>>15856826
Why are you being so hostile?

>>15856838
I'm not. I'm still waiting for you to start making sense though.

Or is it really just a weird ass finitism troll?

>>15856699
>pick any noncomputable number x
How would you do that without computing it?

>>15855729
no iq needed for math.

>>15857439
Just say "let x be uncomputable"

hiiii /mg/ i have returned after years only to find dumb arguments about iq and foundations.
i found some stuff about synthetic algebraic geometry online, and how people are formalizing it in cubical agda. i think that is really neat. i need to learn about topoi for it to make sense though because the notion of internal language to a topos is new to me. this also is motivating me to care about intuitionistic stuff more, since even if you are using classical logic externally, when you are working internally, you still need intuitionistic logic.
anyone else doing this kind of stuff? what is your current autistic fixation? are you advancing your research career by doing it or are you just a bored NEET like me?

>>15856816
the banach tarski paradox is clearly ridiculous and shows that set theory is rubbish. And if you want my opinion, it's not the axiom of choice that's rubbish, it's one of the other axioms of set theory. But still people say there's nothing wrong with it. That's how far mathematics has fallen. They see ridiculous things and gloss over them saying they are just incredible results.

>>15857821
>the banach tarski paradox is clearly ridiculous and shows that set theory is rubbish
What's rubbish about being able to represent any given series in a series of infinite series with finite transformations?
>That's how far mathematics has fallen.
So you can't make sense of it, therefore "mathematics has fallen"?

>>15857821
>it's one of the other axioms of set theory.
Which one? Come on, don't be shy. Show us where the bad ZFC touched you.

>>15853575
3rd year Math bachelors grad here. Lost a significant part of the childlike wonder for math i had in HS as college curriculum was too fast for me
Still love math but i feel very worn out by college and dread the academic system. I am a very very slow learner . Also college forced Physics subjects down my throat that i did not enjoy + was taught terribly
How do i reignite the joy for math in me again sci

>>15858027
>How do i reignite the joy for math in me again sci
study what you enjoy not what you're forced to learn to suit the system you're in. Ideally balance both so you're not homeless and jobless.

>>15858027
what are the comfy parts of math for you? like any results or general ideas that just give you happy chemicals? try to find a cozy idea that can be explained in elementary terms and then use that to motivate study in more abstract shit

>>15858037
I need the grades, my college rushes A LOT, teaching insane material in short amounts of time
I can see that high school teenager that loved math dying in me, feels extra terrible as I am surround by gifted people who just can blitz things fast
I know I can get these things I just need more time

How to maintain my sanity and love for the subject

>>15858038
>what are the comfy parts of math for you? like any results or general ideas that just give you happy chemicals?
Everything actually. I love studying math, my bone of contention is I am too slow in learning speed compared to my curriculum. I just cannot match the speed
Else I love math

>>15858044
Ot atleast, used to love math
I just feel tired nowadays
Very very tired

>>15858039
>How to maintain my sanity and love for the subject
Told you how. If you're struggling to keep up in classes for grades you need more study and prep time between semesters to maintain or study ahead. Easiest way to maintain a 4.0 is study the material of your next semesters over the summer and such in between. Might be a bit late for that but that's "the secret" if you want the easiest A's ever. Study as much as you can every single day, even if it's just one hour, and study preemptively for your next classes.

>>15858050
Got it, thanks

>>15858052
Also, is there any realistic way to increase learning speed in math

>>15858050
>pay money to learn material
>study on your own
studying the prerequisites rigorously would be a better use of time/resources, and then moving onto the next semester material if time allows

>>15858055
>studying the prerequisites rigorously would be a better use of time/resources, and then moving onto the next semester material if time allows
obviously in preemptively studying your planned next semester's material you are going to review prerequisites you are struggling with in the process.

Why do people like you exist?
>Durrrr but the thing entailed by what you said is better to do

>>15858054
>Also, is there any realistic way to increase learning speed in math
learning "speed" is a function of exposure, including things like spaced repetitions, and cognitive ability. Control what you can work around what you can't and deal with your limitations as they are.

Same way learning a language is fastest by proactively trying to use that language and proactively forcing more and more diverse exposure. The more you do with it the more you'll learn about it.

>>15858062
The way you state it makes it seem as though the student is working through the next semester's syllabus and just doing the entire thing (learning prereqs as they go). What I stated is different, instead, working through all of the prerequisites rigorously (independently of the next semester) and then once mastered, starting the next semester's content.

>>15858217
>The way you state it makes it seem as though the student is working through the next semester's syllabus and just doing the entire thing (learning prereqs as they go).
Yeah. That's what you do.
>What I stated is different, instead, working through all of the prerequisites rigorously (independently of the next semester) and then once mastered, starting the next semester's content.
Learning isn't a linear thing and exposure of the next topics with what you've just been learning means broader exposure. If that worked for you, great, but I'm the least studious person I've ever met and I just got A's doing it the way I suggested and with comparatively little studying.

Sure, part of that is probably my more general pursuit and proactive exposure, so my repetitions of past content are probably a lot more frequent than average. Abstractly, though, the more connections made the more reliable your remembering of the subject material, and you get more connections with broader context than you do repeating things you already feel you studied. Unless you meant something very different by the words "work through prerequisites rigorously" which most people would just get bored by.

>>15858258
>Unless you meant something very different by the words "work through prerequisites rigorously" which most people would just get bored by.
I mean working through another book for the subject. Assuming this is the subject the person is into, then this is a purposeful endeavor.

>>15858269
>I mean working through another book for the subject. Assuming this is the subject the person is into, then this is a purposeful endeavor.
Depending on the level and that, yeah, sure, that'd be part of just general proactive exposure in a way. I'd take that as part of what I said here >>15858066 and here >>15858037
Being that'd be part of "study what you enjoy on the side". What I wrote about studying for a grade is purely about maximizing the grade with the least effort for shit you don't care that much about so you stay sane.

>>15858276
yea that's true, grades are pretty overrated in the states though

>>15858292
Can be. I sort of take the position of "whatever potentially gets the most with the least" and you never know where pulling a 4.0 out your ass gets you in that door. Though in the USA it's a lot more whose ass you kiss and nepotism or "muh social networking" (just nepotism with extra steps) than anybody's real competence. Either way.

>>15855760
Without the AoC weird shit happens. Really weird shit.

>>15858301
If the goal is maximizing capital then perhaps, I think maximizing understanding in statistics for machine learning, programming and combinatorics (for those interview questions) would be the way to go (grades would be secondary to understanding and competency here.. may not even need a course with enough preparation)

>>15858329
It's a kind of tiered thing. Ideally you want to get past whatever the filter is, then the secondary sorting of the filter, then as you note the technical competency portion. Add to that being casually friendly while letting the interviewer do whatever his amount of preferred talking is bla bla human bullshit.

But yeah I do treat it like maximizing capital or rather "potential capital" whether in monetary or other terms. You've got one life to live and everybody's got their head up their ass so get what you can get out of it without driving yourself crazy because of that.

>>15858339
it's like one filter though (the first job) after that, ideally you have enough of a niche or domain expertise built up which takes precedence over all previous work (and at a certain point, you may not even need to put what degree you got, like just saying BS/BA and that's it)

If someone wants to get into venture capital, then maybe it's a perk (again for someone just starting out) at a certain point, it's about cash flow and growth.

And going back to academia, it seems research experience and academic connections are the two big filters for landing competitive phd positions..

>>15858347
Depends on what you want to do or are doing or how high you want to climb and stuff, or how aggressively rather. Only constant is really that who you know and who you can get to like you matters more than any actual competency if you ask me, though there's often some kind of "minimum bar" for that competency it's way lower than it should be.

/these and other opinions nobody asked for but hey there ya go

Guy's I'm math BS and I'm seriously freaking out about grad school.
I'm cutting it close with my graduation. I don't have a lot to talk about. Most of the high level courses I'm doing are this year. The only high level thing I can "brag" about it doing reals 1 and 2. And my 1.2 years of research.
I'm not a putnam genius, I have zero honors or awards.
It's not like I'm applying to princeton. I'm applying to 12 schools.
I'm going to fucking an hero if I don't get accepted to a PhD program.

>>15858364
I'm gonna be real, you don't have much of a chance. Doing an MS first could help a lot.

>>15857689
Oh hey, are you one of the million identical trans women NEETs into those things with an anime profile pic on Mastodon?

>>15858415
I don't want to pay for more education
I was told it was possible to do BS to PhD and most schools waive tuition for PhD students.

im learning systems of linear equations. how long until i can do integrals and derivatives

>>15858634
also: am i missing out by just using matplotlib instead of graphing by hand? i kept making mistakes due to bad graphing

>>15858639
You need calculus to graph well by hand.

how do you find out who is the department rapist when going to a new school? I'm too shy to ask people but i heard there is one guy i need to watch out for

>>15858771
It's a surprise, no reason to spoil it.

>>15857488
Then you aren't picking an element, are you? You simply postulate the existence of a set that contains it without constructing it. Which is a known effect of AoC (non-constructive proofs). I think the main appeal of >>15855760 example is an easy demonstration of that effect which is usually demonstrated with more complicated examples like theoretical well-ordering of the reals which exists according to AoC but is impossible to construct.
>>15858327
Can you give one example? Would be even better if it's specifically about strong AoC (as opposed to weaker versions like Axiom of countable choice).

>>15855760
>1. Let D be a set of computable numbers.
>2. According to AoC, we can somehow keep picking numbers from R \ D
Why would this be a problem? Uncomputable numbers aren't necessarily undefineable, you just can't get their decimal expansion with Turing machines.
Your point about undefineable numbers is more interesting, but I don't really think this is a problem. AoC lets you pick numbers, but you don't actually learn anything about these and they aren't specific numbers, so you aren't defining any specific undefineable number.

>>15858639
>hasn't started calculus
>am I missing out by not hand drawing?
Yes but only slightly. You should try sketching rough plots by hand before your computer tells you the answer.
You probably don't have to get good at technical drawing in the modern era, but you should at least be able to quickly visualize the broad strokes.
>why tho???
google "chunking"

>>15858887
>Then you aren't picking an element, are you?
NTA but it's functionally equivalent to having picked some element, and there's plenty of ways one can do this. Such as by intersection, exclusion, modus tollens, and fancier things with too many syllables.
>You simply postulate the existence of a set that contains it without constructing it
So what? If the properties of the set to which it belongs works then by transitivity the postulated element if it exists in the set also works. This is also known as the law of syllogism. You could not have picked a more famous and fundamental law to transgress upon if you tried.
>usually demonstrated with more complicated examples like theoretical well-ordering of the reals which exists according to AoC but is impossible to construct.
So what?

>>15858887
>Can you give one example?

The follow are possible without AoC

The reals are a countable union of countable sets
The reals can be a union of two disjoint sets, each of smaller cardinality than the reals
You can partition a set into disjoint non-empty parts such that the number of parts is greater than the number of elements of the set.

I'm getting a situation where the inverse of a matrix [math]\begin{pmatrix} a & b \\ b & c \end{pmatrix}[/math] is [math]\begin{pmatrix} c & -b \\ -b & a \end{pmatrix}[/math]. What is this called?

>>15858887
>AoC (non-constructive proofs).
There's plenty of nonconstructive stuff in ZF already. For example, you can prove there exists a function from Turing machines to booleans that is true when they halt and false otherwise.

>>15859489
*Turing machine-input pairs

>>15859154
>>The reals are a countable union of countable sets
That means you have a surjection from [math]\mathbb{N} \times \mathbb{N}[/math] to [math]\mathbb{R}[/math], and therefore a surjection from [math]\mathbb{N}[/math] to [math]\mathbb{R}[/math]. Then we just go down the list removing duplicates when we find them (this is nonconstructive because deciding if two reals are equal isn't computable, but noncomputable functions that do so exist in ZF), turning the surjection into a bijection. And we know that there's no bijection between [math]\mathbb{N}[/math] and [math]\mathbb{R}[/math]. What part of this requires the axiom of choice?

>>15859526
>(this is nonconstructive)

Yeah...no shit...that's why you can't do it without the axiom of choice.

>>15859536
Where does the famous proof that there are two irrationals x and y such that x^y is rational use the axiom of choice?

>>15859526
Apparently ZF + "the reals are a countable union of countable sets" has been shown consistent (presumably assuming ZF is consistent) so I've most certainly made a mistake somewhere, still trying to figure out where.

>>15859547
Whenever you're trying to construct a function and you think "I can't pick an exact x but such an x must exist so i'll just pick an arbitrary one" That's the axiom of choice at work.

>>15859561
No. You don't need the axiom of choice for [math]\lnot (\forall x P(x)) \rightarrow \exists x \lnot P(x)[/math].

>>15859568
Well, you don't need choice for finite sets... I don't think.

>>15859154
I look this up (mathoverflow answer says "T. Jech, The Axiom of Choice. This particular proof appears in Chapter 10.") and find much wonkier things.
>Theorem 10.1. There is a model of ZF which has an infinite set of real numbers without a countable subset.
WTF?

>>15859573
Both accepting the AoC and not accepting it lead to wonky results. But rejecting AoC gives you far stranger ones.

>>15859578
Not sure if that's a fair comparison. Accepting AoC implies the wonky results directly whereas not accepting AoC only means your theory is consistent with the wonky results (presumably, under the assumption ZF is consistent in the first place).

>>15859575
Wait, I read that wrong, for some reason I misread it as the real numbers themselves not having a countable subset, which would be obviously wrong. So still a bit wonky, but not nearly as wonky as I initially thought it was.

>>15859584
>only means your theory is consistent with the wonky results

True. But if you want to reason about infinite things you kinda need the Axiom of Choice. You can go with some weaker axioms I suppose but I don't think you can do analysis without at least dependent choice

>>15859589
I'm not sure exactly how far he got, but from what I understand Bishop managed a lot of analysis without even the excluded middle. But obviously some results will just stop holding when you do it that way, such as the monotone convergence theorem.

>>15856498
Wolfram says yes, in terms of higher order hypergeometric functions (so not really).

the axiom of choice was the greatest mistake mathematicians ever made
those capable of higher brain function can see that the axiom of determinacy is the way to enlightenment

What fun math games do you know of that actually use anything above arithmetic? Like that one function graphing game.

>>15859598
>Equivalence relation on R that is larger than R

Yuck. No thanks!

>>15859526
Okay, after some searching I found where the argument in my head (only sketched in that post) used the axiom of choice. Let S denote the family of countable sets that [math]\mathbb{R}[/math] is the union of. It's true that there's a surjection f from [math]\mathbb{N}[/math] to S, and for each element T of S, there's a surjection from [math]\mathbb{N}[/math] to T. In order to make these into a surjection from [math]\mathbb{N} \times \mathbb{N}[/math] to [math]\mathbb{R}[/math], we would need to choose for each [math]T \in S[/math] a surjection [math]g_T[/math] from [math]\mathbb{N}[/math] to T. This uses countable choice. Then [math]h(m, n) = g_{f(m)}(n)[/math] can be proven to be a surjection from [math]\mathbb{N} \times \mathbb{N}[/math] to [math]\mathbb{R}[/math].

>>15859753
Without the Axiom of Choice some sets have incomparable cardinality. Kind of wacky

>>15857859
axiom of pairing is retarded

>>15858541
uhhh i like to think i am a unique individual but yes I am trans and post on fedi.

>>15859593
Which is cute and all but like the weird finitists or hardcore constructionists the problem is when you design a system of logic arbitrarily denying other valid things, such as transitivity, you end up with contradictions due to having an incomplete or arbitrary set of rules. I'd argue it gets even worse because properties like transitivity are contingent on others, and negating transitivity necessarily negates those as well (as mentioned, modus tollens).

It's really hard to tell whether this weird crap about some "big problem" with AoC is an elaborate troll or people who really don't understand it at all. Either way I almost want to talk to some of these crazy people. To me they're like flat earthers and I'm left going "... but why?" Really is it a troll? Is it a joke? Or is someone like Wildberger really unaware of just how many contradictions and self-refutations his system creates? For that matter, would he care if they're formally pointed out? Has anyone done so?

>>15860293
Why would having an incomplete set of rules give you contradictions?

>>15860293
Also who has a problem with "transitivity"? Transitivity of what?

>>15860330
>Why would having an incomplete set of rules give you contradictions?
Sorry guess I did not specify enough. I meant when you negate or hold invalid other properties in order to maintain the arbitrarily restricted ruleset, and so incomplete (in my opinion) by design. So incomplete in that sense, not merely unstated or incomplete in a broader way or technical sense. I'm not sure how to word this better.

As stated, problem is declaring certain properties of logic, rules, inferences, etc, to be invalid, these tend to be contingent and by their negation will negate other properties. Usually when I read some kind of crank paper they completely fail to realize by negating something they're also negating its contingent parts. I'm not sure what about modus tollens is hard for people to understand but a lot of people don't seem to be able to get that negation also has entailment. Or maybe it just seems weird to apply negation entailment to axioms? but that should be intuitive, like negating the law of identity would thus entail allowing a=b or if explicitly negated require "a != a" entailing still other negations of other rules or properties.

I'm sorry for the wordiness, it just seems to basic and intuitive to me I don't know if the issue is my choice in phraseology or it's just so basic nobody thinks about it?

>>15860334
>Also who has a problem with "transitivity"? Transitivity of what?
Law of syllogism. >>15858962
It was an example for people who want to demand a construction of some specific element to be able to claim its existence. Demanding that entails negating the law of syllogism and that has further entailments they would not like.

It's like dominos. You knock one down there are others that will fall too. That's why you can't just cherrypick things.

>>15860347
>>15860356
There's a big difference between not using a particular axiom and asserting its negation.

>>15860387
>There's a big difference between not using a particular axiom and asserting its negation.
Yes, it depends exactly on how derived the axiom and what system of logic is being employed and to what end. In this context we're talking to the ends of simply rejecting the validity of a given line of inference or reasoning, and by definition that entails rejecting the validity of the structure used.

>>15855760
Think of the finite version first. Given n sets and taking an element from each you can build a new set. Then extend this rule to allow such procedure to happen for an unlimited amount of sets

>>15853575
Not sure if this is a stupid question, but:

If f : M-->N is a continuous map between connected compact manifolds, without boundary, and of the same dimension,
then is f surjective?

>>15860443
I get your intuition, but no. Take any constant map for example. Not sure about the answer when you additionally require that f is an immersion.

>>15860455
Oops, right. But yes maybe requiring f to also be an immersion would give the result?

>>15860443
>>15860455
Can't you continuously embed the torus into the double torus?

>>15860146
Why don't any of you have degrees? Are you all living on trust funds popping estrogen and coding Rust all day or what?

>>15860463
Thought about it, but again no. Take for example both M and N as the unit circle viewed as a subset of the complex numbers, and choose [math]f(z)=z^k[/math] for any integer [math]k\geq 2[/math].

>>15860493
>>15860470
Sorry, I'm retarded. That's an example for an immersion that's not an embedding. Let me think some more.

>>15860498
Don't know if it matters, you're definitely better at this than I am, but I think he's right if it's an immersion.
So,
Continuous, connected, compact, without boundary, Immersion, of same dimension.
Either I'm being really dumb today or that seems to bijection, so an injective surjective mapping? Am I forgetting something?

>>15860520
I think that you are right, but I'll think about how to actually prove it. Hope that I find a proof by tomorrow.

Will transitioning improve my odds of getting into grad school?

>>15860520
maybe cylinder and plane is a counterexample?
i forget the exact defns for geometry stuff

>>15860586
>muh grad school
you don't need to transition if you're already a faggot

>>15860620
The plane is not compact

>>15860586
Especially if you're poor or brown, you need to lean into the social justice hard in addition to transitioning.

>>15860533
You can easily find a proof on StackExchange, for those curious.

>>15860620
A plane is not compact. Neither is an infinite cylinder. A finite cylinder is compact but has a boundary.

>>15860470
I have a bachelor's in math and CS and I started a math PhD at a top school but I dropped out because of mental health. I had a high-paying job coding rust after that but got fired again for mental health. Now I'm a depressed NEET living with my disappointed parents but I still do math sometimes so I can think about something other than suicide. :)

>>15860667
NTA but I know the feeling.

>>15853700
Hey, are you still around? I'm not sure how TeX works on this site so apologies if this appears as plaintext.

If we're working over an algebraically closed field $k$. Let $G$ be a group and let $V$ be an affine variety over $k$. Given an action $(g,x) \mapsto g \cdot x$ of $G$ on $V$, we define the induced action $(g,f)\mapsto g \cdot f$ of $G$ on the coordinate ring $k[V]$ of $V$ (which is isomorphic to ring of regular functions on $V$) by defining how $g \cdot f$ is evaluated at a general point $x \in V$,
\[ (g \cdot f)(x) = f(g \cdot x), \]
where the action on the left hand side is the action of $G$ on $k[V]$ and the action on the right hand side is the action of $G$ on $V$.

Since you mention tori and you're working with affine varieties, I assume you're interested either in linear algebraic groups. I can get into this topic if you like, as my area is the representation theory of algebraic groups. The most accessible reading for this would be Chapter VI of Humphreys or Chapter 3 of Springer. If you really want to get into the representation theory, you could go for Jantzen, though it may be a bit dense depending on what level you're at.

If you can elaborate on your interest in tori, I can provide a summary. The most interesting aspects of them (for me) are their weights and roots of maximal tori. You can understand a lot of the representation theory of an algebraic group $G$ by understanding the representation theory of a maximal torus of $G$.

>>15857689
>anyone else doing this kind of stuff? what is your current autistic fixation? are you advancing your research career by doing it or are you just a bored NEET like me?
Learning higher level mathematics, logic, ideas, concepts? Specifically intuitionist logic?
as for me >>15860679 I'm torn between boredom and continuing to feel ever deeper alienation as I work on things, can't find people who share interest in what I'm working on, and then try to bridge gaps anyway and waste time trying to explain myself to people who don't, can't, or won't, get it.

You definitely don't have to be a NEET to be miserable. For my part I just can't find a group of people who are both competent and creative. If they're competent I tend to find a lot of hostility toward exploring new ideas, and if they're creative I tend to find they're incompetent with no interest in gaining any. This dynamic drives me nuts.

>>15860751
>Hey, are you still around? I'm not sure how TeX works on this site so apologies if this appears as plaintext.
upper left on the input reply box click "TEX". you open/close with "math" and "/math" in square brackets as shown in the TEX previewer.

>>15858579
I got into a PhD program out of a BA in math at a small liberal arts school, so you're fine. Also, if they're making you pay to attend don't do it. You should not only have tuition waived, but be getting paid a stipend.

>>15861021
Should I add smaller schools and christian schools to my list?

>>15860667
>>15857689
Do you have any idea why your 'type' of mathematician seems, most of the time, to be studying abstract algebra, logic, or category theory, and rarely something like probability, combinatorics, or dynamical systems?
Maybe it's just a stereotype but I'm curious nonetheless.

>>15861036
I'm not an expert on this for what it's worth, but I'm not sure that's necessary. You can still apply to R1 schools just broaden your horizons to slightly lower in the rankings than you may want. I'll also add, depending on your research area there may be schools with very strong research groups in that area that are not top 50. That's the situation I'm in. At a school that is very well known for a specific research group, even though as a whole we're ranked in the 70s I think. Obviously you should also talk to faculty at your current undergrad, but your prospects are probably not that poor

where does one even go online to meet and talk to other people interested in advanced or abstract mathematics that isn't homework, dead, or drowned in cranks?

Is the internet just dead or do I need to trawl fediverse like I'm going through the jedi archives?

>>15855729
None such IQ exists, too low <60 and you might not be able >60 and you start to realize that learning random fields of math is a waste of time, and you start focusing on a specialization and/or practical problems

>>15861242
Why not make a post on /soc/?

>>15861297
Is that a joke or a one question IQ test?

>>15861127
NTA but it's probably because the former lean a little more verbal and a little less geometric. The default "trans culture" is all about hermetic shit like speaking the world into being and the superiority of inner reality to outer reality. It's schizo in its basic nature. Of course the people who are least in touch with the physical world and their bodies are into the ultimate levels of generality and abstraction.
Transgenderism is transhumanism with weak sense of self and a porn addiction.

/sci/tards ITT claim an IQ of 180 yet can't RTFM for TeX and need scriptkiddy tier LaTeX (they still don't RTFM)

>>15861299
I don't know what to tell you except that it's not guaranteed to fail.

>>15861310
My tolerance for crazy is very low and has gotten quite a lot lower in the past few years. That tends to happen when you've repeatedly done things like you suggest there, in various formats, and only gotten "even more extreme and hostile crazy".

Not guaranteed to fail? Sure. Am I tired of firing magic missiles at the darkness hoping I hit something? Abso-fucking-lutely. I didn't come out the womb putting in no effort I got here after consistent efforts and constant failure. Just giving context as to why I'm not going to "go fish".

>>15861316
>I want to find someone to talk to about maths.
>Should I go ask on the one place here designed for it?
>No, they are *all* crazy, hostile, and not worth the time of a divine being such as myself.
>I will now make an abrasive post in /mg/, that should get my point across.

>>15861316
Double check you're not just a narcissist. Triple check if you're a tranny. They're often especially narcissistic.

>>15861127
NTA but as a guy who likes those first three things and dislikes the latter three, proofs from algebra / category theory have an elegance that proofs from the latter three don't. On the other hand all three of those are obviously more concretely useful than the first three, but I don't care. As for logic, the proofs aren't as nice but I like the idea of noticing similarities in a wide range of math fields (also makes category theory attractive)

>>15861316
>Am I tired of firing magic missiles at the darkness hoping I hit something? Abso-fucking-lutely
oh no the trauma of writing an internet post about something you're passionate about

Guys... I just found a pattern in primes

>>15861346
Not until you prove it you haven't. Only famous prodigies get to present conjecture as fact.

>>15861329
>>15861331
>>15861338
Imagine feeling this personally attacked by a 4chan post lmao

>>15861346
I think there is a theorem that states that any set of numbers with pattern of digits
xyx
xyyx
xyyyx
where x is 1, 3, 7, 9
etc, have primes gaps ln(N) for large N

>>15861398
What is that theorem called?

>>15861460
Incremental Gap Lemma

>>15860293
>Wildberger
What's the general opinion on his rational trigonometry?

Wildberger's Rational Trigonometry represents a groundbreaking departure from traditional trigonometric norms, introducing a paradigm shift that replaces sine and cosine functions with rational expressions. This alternative approach simplifies geometric reasoning, making complex proofs more accessible and reducing reliance on intricate trigonometric identities.

The reimagining of fundamental relationships, such as the Pythagorean theorem, in terms of quadrances and spreads not only enhances clarity but also lays the groundwork for a deeper understanding of geometric concepts. In the realm of triangles, Wildberger's rational laws provide concise and elegant solutions, replacing the conventional laws of sines and cosines.

The brilliance of Rational Trigonometry becomes evident in its treatment of circles. Wildberger's rational parametrization using spread and quadrance offers a powerful tool for geometric analysis and simplifies computations. This alternative representation proves to be versatile and insightful, challenging the conventional unit circle and its associated trigonometric functions.

Beyond its theoretical contributions, Rational Trigonometry holds promise for reshaping how we teach and learn this fundamental branch of mathematics. By prioritizing conceptual understanding and leveraging the computational advantages of rational functions, Wildberger's approach makes trigonometry more accessible and engaging.

In essence, Rational Trigonometry stands as a testament to the power of reimagining mathematical fundamentals. It strategically moves towards a more elegant, computationally efficient, and conceptually intuitive framework, inviting us to appreciate the beauty of mathematics in a new light.

>>15861498
>>15861516
So basically, if we swap to Wildberger's system we'll be able to fail a bunch more undergrads when they see limits for the first time and completely break down.
Sweet.

Somebody please help me google the following:
Suppose I have some functions that form a finite sequence, like so:
[math]f_1(x \; -> \; x'), f_2(x' \; -> \; x''), \dots f_n(x^{n-1}' \; -> x^{n}') [/math]
Is there a special name for the function [math]F[/math] that takes [math]x[/math] to the n-tuple [math] \{ x, x', x'', \dots x^{n}' \}[/math]?

>>15861831

Anyone know how to make this squiggly line in tikz?

>>15859641
EVE Online?

>>15853575
does anyone know where I can download Strangs Introduction to linear algebra (6th edition)? I can only find the 5th edition on libgen.rs.

>>15862149
I don't know about tikz but I would make that graph by adding an absolute value of sin to the original line equation.

Bros. I'm trying to do some black magic by approximating sums with integrals.
Obviously there is Euler-Maclaurin summation but I'm trying to do better.
The first improvement I got was to shift the integration interval so that the sample point is the midpoint. This gets the first 2 leading order terms correct.
Then I got creative and shifted the interval off the real line.
[math]Re[\int\limits_{{i \over \sqrt{12}}-{1 \over 2}}^{{i \over \sqrt{12}}+{1 \over 2}}f(x)dx] \approx f(0)[/math]
This gets the first 4 leading order terms correct.
Now I'm trying to generalize this to use an NxN matrix in the integration bounds.
My hope is to get the first 4N terms correct from just the integral.
I want something like:
[math]Re[tr[W\int\limits_{M-{1 \over 2}}^{M+{1 \over 2}}f(x)dx]] \approx f(0)[/math]
You can assume M and W are just diagonal.
You end up with a system of equations: Re[tr[WM^k]] = c(k) but it is a pain.

Second order logic is based.

>>15862449
based on what?

https://www.youtube.com/watch?v=XuaHmVGtzAE

what are the implications of this?

>>15861498
>What's the general opinion on his rational trigonometry?
General where? I don't think there's a poll anywhere.
General mathematicians? Have no clue who the crank is probably. If they do I would hope they're comfortable enough with logic and mathematical logic to realize the implications and mistakes in his reasoning.
General here? Trolls/cranks/contrarians love it for 'some reason'.

As for my opinion it seems to rely on fallacious reasoning in order to reject the "realness" of numbers one can infer but cannot give a finite sequence to. For example, declaring an infinite approximation of sqrt(2) "meaningless" by arbitrary standard of "not being computed" necessarily requires rejecting transitivity and all kinds of problems that therefore entails by its negation. To reject the results of valid logic with true premises (valid and sound) is to reject the validity of the form, and so his entire system of reasoning is self refuting by its hypocrisy or entailment of contradiction by rejecting valid form.

Why does he do this, then? Why throw out the baby with the bathwater or be a hypocrite over something so silly? He rarely brings it up but I did eventually figure out why. Example, https://www.youtube.com/watch?v=REeaT2mWj6Y
I think he reveals what's really going on in his conclusion here. Paraphrasing slightly, "God possibly has the potential of understanding this thing [sqrt(2)] but not we humans. We are not at the level of God. It is wrong for us to be so arrogant".

So it would appear he does it for the same reason every other otherwise intelligent person throws their brain in the garbage, because it's somehow tied into his belief in adult santa. I have sometimes attempted to figure out if he merely means this as metaphor, or is in fact some kind of deist, but I don't care enough to literally trawl everything the man has ever done in his life. Far as I know he is a "believer" and that's the real motive behind this nonsense.

let's say I have a weird, real-valued power series like pic related.
Very obviously this converges quite strongly to some function. But it also quite rapidly diverges outside of the RoC.
I want to see more of the underlying function. I know of analytic continuation but I've only seen it done on complex-valued functions. Not only that, I've only seen it done with functions we know beforehand so we can guarantee they're analytic and all the other checkmarks for analytic continuation.

If I'm trying to continue a power series on the real line, for a function that we can't even know is analytic, what am I to do?

I've got a math riddle.

If x/i=10, what's the value of x? (Note: x is a positive real number)

>>15863048
Try Fourier interpolation instead.

>>15863060
i is the roman numeral for one
[math]\frac{x}{1} = 10[/math]
[math]\Rightarrow x = 10[/math]
trivial, really

>>15863060
x=10i, by simple multiplication, but we want it in terms of a real positive number
squaring both sides, x^2=-100
now obviously we are working in GF(13), because it makes everything easier and all jokes are best written around 13, in which case -100 is congruent to 4
so we have x^2=4 mod 13, so x=2 or -2 mod 13.
Hence x=2 or x=11, QED

>>15863048
Pade approximation should do the trick.
It is blowing up because there is obviously some pole(s) on the unit circle.
Pade should be able to properly imitate the pole(s) and go a little beyond the unit circle (the long range behavior might not be accurate since it will eventually just reflect the order of the chosen approximation).

https://www.wolframalpha.com/input?i=Pade+24%2C+24+Sum%5B%28-1%29%5En+*+x%5Eprime%5Bn%5D+%2F+prime%5Bn%5D%2C+%7Bn%2C1%2C20%7D%5D

You can play with different orders to see where they agree and where they don't agree to get an idea of how far you can trust them.

order m,n will agree with your function up to x^(m+n+1)

>>15863066
Or more obviously: x is the roman numeral for 10.
>trivial
>overcomplicated it

Is it true that American maths major students encounter proofs only in the third year of their undergrad studies?

>>15864199
I always find that weird too.

>>15864199
Obviously, I can't speak for everyone, but my uni didn't have a formal "proofs" class, but locked a pretty sizable amount of the curriculum (topology, diffeq, number theory, and basically all non-elementary algebra) behind what was essentially a very, very proof-heavy set theory class, and it was pretty obvious that they were using that as their introductory proofs class.
The only way I could think of that you'd get to even your second year of undergrad without taking it is if you loaded up your first year with elementary algebra, calculus, and other shit you probably should have already taken in high school, considering your major of choice. No idea how you draw that out to your third year

>>15864199
>>15864317
actually, upon double-checking the catalogue, I have determined that all of the courses on the list that applied mathematics students are required to take have an alternative prereq to that one. No idea how proofs-oriented it is, but if the answer is "not very", this exact situation could have ended up happening there.
Ah, well. Disgusting as it may be, it's just another reason to spit on the applied side of things.

Isn't the Axiom of Choice just "you can make a set by picking an element each from any number of other sets"?
I lack basic training in set theory. Where does the madness begin?

15864199
You have to go back

@15864406
>You have to go back

State of this board: reddit fags are flooding threads with basedjacks

>>15864374
Weird phenomena like Banach-Tarski paradox, non-measurable sets
I don't actually have a problem with Banach-Tarski since it'd be impossible to implement it in the real world, but it's def counterintuitive

>>15864199
American math majors are allowed to take whatever they want (which is good).

>>15864199
I wished I'd had the option, or was forced to take a course of proofs earlier. I somehow got into grad school without ever having to do one and it was a shock.

Bros, im getting filtered by differential geometry. I do understand the concepts, but im unable to do the problems on my own, do you have any material that could help me?

>>15864769
I hope you’re reading docarmo…

post things part of the status quo in maths that you think are suspicious
for me, it has to be the well ordering of the reals

to this day I still have no idea what the fuck anglos mean by "calculus" college courses
is it supposed to be integration by parts? is it multivariate differentials? is it finding the day of Easter? because this is all supposed to be highschool stuff
guess I'll never know

>>15857689
this is off-topic but it's crazy how all the "ex-4chan" twitter/fediverse/discord lgbt nazi punks fuck off type people are so obviously recognizable in their written affect.
>hiiii /mg/ i have returned after years
this is an anonymous website. nobody knows you, and nobody gives a shit what you've been doing recently.
>what is your current autistic fixation?
realistic systems for cheap verification of outsourced computation.

>>15853575
I cant manage to prove this fact that seems so simple. Please help.

Let [math]H_1, H_1[/math] be separable Hilbert spaces and let [math]a_{i,j} \in L(H_j, H_i)[/math] for all [math]i,j \in \{1,2\}[/math]. That is, each [math]a_{i,j}[/math] is a bounded linear operator from [math]H_j[/math] to [math]H_i[/math].
Furthermore, suppose the block linear operator [math]A \in L(H_1 \times H_2)[/math] given by
[eqn]\begin{pmatrix} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end{pmatrix}[/eqn]
is invertible and positive definite, with [math]a_{1,1}[/math] and [math]a_{2,2}[/math] postive definite over [math]H_1[/math] and [math]H_2[/math], respectively.

I want to show that the bounded linear operator given by
[eqn]\begin{pmatrix} a_{1,1} & a_{1,2} \\ 0 & a_{2,2} \end{pmatrix}[/eqn]
Is also invertible.

I am so fucking tired bros.

>>15864969
Motherfucker, I meant the operator
[eqn]\begin{pmatrix}a_{1,1} & 0 \\ a_{2,1} & a_{2,2} \end{pmatrix}[/eqn]

So the lower triangular part of A.

Sorry, typing latex on my phone is hard.

>>15864969
>>15864977
I'm calling your lower triangular operator T.
Have you tried finding an operator S so that ST = TS = I?
That would show invertibility.

>>15865021
I know the inverse for a_{1,1} and a_{2,2}, a_{2,1} is not invertible in general.
Let me see if I can get something with that approach

>>15864917
>is it supposed to be integration by parts? is it multivariate differentials?
yes and yes
"calculus I", at least around here, is basically single-variable differentiation and integration
calculus II is multivariate stuff
both of these are offered in high schools but you can also take them in college if you're retarded
calculus III is vector calculus and is the first one you actually have to wait for university for

>>15864967
>this is off-topic but it's crazy how all the "ex-4chan" twitter/fediverse/discord lgbt nazi punks fuck off type people are so obviously recognizable in their written affect.
Hood rats trying to talk fancy.

>>15865021
Thanks this works, found it. Amazing. Thank you so much

>>15857689
>foundations
>dumb
foundations are the bedrock of all maths, and they should be challenged and challenged again, in order to strengthen them, and improve upon them, until finally there's an objectively built foundation for maths rooted in facts about this world that we all live in.

Is there a theory about divisibility of infinite sets? Was thinking about infinite dimensional wedge products, sign of permutation etc.

It seems easy to show the half open unit interval is even, the closed unilateral triangle is not divisible by 3 and so on. Is this applicable in infinite dimensional analysis?

>>15857689
Because of the way you write and because you said you're returning after years, I get the feeling I might know you from before. Were you in a /mg/ discord server?
>are you advancing your research career
I'm trying to learn a foreign language so i can attend a different university than I'm currently in. I hope that counts. Not really looking for a research career.

>>15853707
It's a pain trying to read their handwriting.
One time I just told the instructor if I can't read an answer, that's a zero for the question, and if the homework has cheating, the entire homework gets zero.

>>15861346
It's easy to prove the "not prime" ones, because those are just divisible by 11.
Also, I think you're guaranteed to break the "prime" pattern, since eventually one of them will be divisible by 11.

>>15865609
>>15861346
Sorry, I take back the second part.
Residue mod 11 is always -2.

Henry SEXerman

>>15865615
Is this the guy that appears on Numberphile

do you have cute math gf like her?

>>15862409
I've got c(0)=1, c(2k) = 2k*(1-2^(1-2k))*zeta(1-2k).
c is 0 for odd values.
Doing the order (2N-1, 2N) pade expansion for Sum[c(k)*x^k] will give Q(x)/P(x).
The roots of P(x) are conjugate pairs.
Pick N roots, r, of P (one from each conjugate pair)
The 1/r are the entries of M.
The corresponding weights in W are -Q(r)/[r*P'(r)].

How to solve this shit?
[math]\sin(2025x) - \tan(2024x) = \cos(2023x)[/math]

Why is math like this? I just wanted the answer for a simple maths problem, and the answer I got doesn't even fit on screen. It just goes on and on even after ridiculous 20 mouse scrolls.

If you're interested, this is the side length of a square that is put between two circles with radii a and b that are tangent to each other and a straight line, the bottom of the square being on top of that line and the top two vertices being on the circumferences of the circles.

>>15866495
Can you show your work? There has to be a nicer solution.

>>15866490
Let [math]y = e^{ix}[/math] then the equation is just
[eqn]\frac{y^{2025} - y^{-2025}}{2i} + i \frac{y^{2024} - y^{-2024}}{y^{2024} + y^{-2024}} = \frac{y^{2023} + y^{-2023}}{2}
[/eqn]
Multiply with [math]y^{6073} + y^{2025}[/math] and you're left with a polynomial equation for [math]y[/math].
Find all complex solutions of it (there can be at most 8098 of them and then remove those solutions for which [math]y^{2024} + y^{-2024} = 0[/math]. If you are only interested for real solution of the original equation also remove all solutions with [math]|y| \neq 1[/math].
For each of those solution you get infinitely many solutions for [math]x[/math] namely [math]x = - i \log(y) + 2 \pi k[/math] where [math]k \in \mathbb{Z}[/math].

>>15866664
Thanks, big brain.
But, Is it possible to solve this without using imaginary numbers? It supposed to be school level problem.

Is the derivative of ln(tan(2x)) =
[math]
\frac{2}{tan(2x)cos^2(2x)}
[/math] ?
Online calculator says no but I'm pretty sure it is? Each part using the chain rule should be:
[math]
\frac{1}{tan(2x))} * \frac{1}{cos^2(2x)} * 2
[/math]
Or where is the error?

>>15868077
Out of curiosity, what's the calculator saying it is?

>>15866551
I divided the two circles into two semi-circles in the same way as in picrelated and I got the equations for those such that I have the x-coordinate as a function of y-coordinate:

x = 2 sqrt(a b) - sqrt(2 b y - y^2) + s
and
x = sqrt(2*a*y-y^2)+s

Then I took the difference of those x-coordinates and made that equal to y (this means that the width of the rectangular shape is equal to its height, effectively creating a square):

(2 sqrt(a b) - sqrt(2 b y - y^2) + s) - (sqrt(2*a*y-y^2)+s) = y

Now if you solve the 'y' from this, it gives you the side of the square which is that ridiculous formula. Don't worry about the 's' variable that was just the x-coordinate of the center point of the left circle.

>>15868187
I used derivative-calculator.net, you can try it. But basically, after it finishes calculating, there is a button to check if your result is equivalent to the one it calculated and it says no for my answer for some reason.

>>15868469
I don't know if you mistyped something or other, but it seems pretty happy with your answer when I try it (as it should)

>>15868614
Damn, I retyped it two times but was told wrong both times. Guess I had a typo, thanks anon.

>>15861036
Realistically why wouldn't you apply to every school you possibly can to maximize your chances?

>Taking measure theoretic based probability theory graduate course
>Even with an undergraduate degree in math, I am essentially lost, blown away by the speed and complexity of the course.
>measures in hilbert spaces, locally compact abelian groups, etc

Holy shit, I am a stupid fucking animal and the only reason I am probably going to pass, is all the exams are take home week long assignments that still take me days to work through. I don't think I have solved a single thing without the extensive use of references, notes, and AI assisted topic direction. I don't deserve to pass this course. I'm only halfway through this masters, and am doing a masters because I'm too stupid for a PhD.
Any other subhumans here?
For actual smart math geniuses, thank you for being kind to me and allowing me to be in your vicinity while you do real math.

>>15870219
A measure theoretic probability course isn't supposed to have all that.

Stupid question (explain me. plz)
Why this constraint [math] X1, . . . , Xn ∈ L, c(X1 ∨ . . . ∨ Xn) = c(X1) + . . . + c(Xn).[/math] is true only for finite sets of mutually exclusive. How does it work in infinite sets?

>>15870223
Why wouldn't it? The course is just called "Probability theory" and the pre requisites were two previous courses on probability theory and stochastic processes at the graduate level, which required a full undergrad degree to start as the prereqs for those were real analysis, multivariable calculus, and linear algebra. We are using Parthasarathy's book.

>101.3% in class

Professor literally has to give me a score over 100% in order to avoid giving out too many failures.

>"Anon, how do you do it!"

Just study nonstop every day. Instead of videogames, I study. Instead of going out with friends, I study. Instead of dating, I study. Instead of masturbating, I study. Instead of getting more than 6 hours of sleep, I. STUDY.

>>15870559
That's nice anon. Just make sure not to burn out.
>I won't
That's what everyone says. It's very common.

>>15870559
> Instead of getting more than 6 hours of sleep, I. STUDY.
lol.

>>15866935
>school level
use periodicity - sin(x)=sin(x+2pi*k) for any whole number k

N < K => Powerset(N) < Powerset(K)

Is independent of ZFC

I can't do this anymore bros. I can't fucking do it. How the fuck is this not an axiom.

If I don't get into UC Berkeley I WILL kill myself

>>15871803
Reddit school.

>>15871822
If I don't get into UPenn I WILL kill myself

>>15871649
Source?

What is a matter of days? Does the sun have a ring around it?

The Sun has a ring around it. Is that release of something or a second stage? Is a higher power encircling the Sun. I know you think it's not a problem but it is that. You are esquelle, the tranny. Longer than 250 years for being so disgusting in the face of life.

>>15870219
Get a job. Relax.

>>15864199
What really? We had to take one starting sophomore year alongside linear algebra. We had a choice between intro to math logic (which was basically a course focused on Hammocks "Book of Proof") and Discrete Math (also proof focused).

I really liked that math logic course. I remember it being really well taught and opened me up to the rigor of math for the first time.

>>15871828
https://arxiv.org/pdf/1203.4026.pdf

>>15872409
x<y <=> 2^x < 2^y is already well known
This is actually just the definition of monotone increasing functions.
There is probably something to be said about invertible functions.
There's probably something to be said here about invertible functions in R, but I don't feel like thinking about it.
Maybe something like when f is invertible in R then |x| < |y| => |f(x)| < |f(y)|

>>15853575
Any book/course recommendations to improve your ability to handle complex equations without forgetting a carry etc. I can write and debug complex code and immediately realize what it should be doing and why it is wrong. Not so for equations. Working through a book of exercises one does not have the same write code -> run -> observe error -> repeat, feedback loop. So what can I use to improve this ability, I remember from University physics tutorials really improved that skill, so something similar with just a bunch of exercises and answers so I can immediately realise my mistake and go back.

Where in the proof of Lemma 1.4.8 is it used that [math]f[/math] and [math]g[/math] are monomorphisms?
This precondition has to show up somewhere, because the final pullback square (on page 38) establishes that f is indeed monic,but I don't see which step of the proof fails without it.

>>15872434
It's about cardinals, not functions.

>>15872409
Uohhhhhhhhh

>>15872805
Set Theory bros support cute and funny math. This is basic knowledge.

>>15872458
I did some research and found the book, "Berkeley Problems in Mathematics" which seems to be a perfect fit.

>>15870444
I mean a first course in measure theoretic probability isn't supposed to have all that.
>measures in hilbert spaces, locally compact abelian groups
This assumes you know functional analysis and graduate algebra. And if you are at this level, you probably should have figured out by now whether math is for you or not. I mean it's very uncommon for someone to struggle with math at such high level because everyone else has already been filtered out.
> Parthasarathy
RIP. I assume you mean his (God rest his soul) book on probability on metric spaces. That is a fairly advanced book. I don't understand how could you get to this level if you are not already excellent at math.

>>15871649
Just join finitechads bro. It's as limiting as ZFC propagandists make it seem. Otherwise you can just ignore set theory in it's entirety, as most most mathematicians do.

>>15871649
Because it rules out the existence of certain sets whose powersets are bijective with the powersets of larger sets, which might annoy maximalists.
It's implied by GCH if that makes you feel better.

>>15872458
The best thing to do is to type it out. Basically solve it first on paper then type it out via latex or word doc (it doesn't matter which). When you see the problem worked out in a clean and organized fashion, you'll catch mistakes very quickly.

With derivations, you can often do logic tests too.

>>15873468
>Set theorists when everyone agrees that if A is bigger than B, it means A has more stuff in it

>>15873531
It's more like "If A has more elements than B, it must also have more subsets than B" which is a lot less reasonable if you don't hold the view that all sets are constructible.

Are all discontinuous functions where discontinuities only happen at finite amount of specific points otherwise the function is continuous (think piecewise functions) always Riemann integrable?
if so is it correct to claim so because for each point the integral on the interval from that point to that point is 0?

>>15873752
No, consider
[eqn]f: \mathbb{R} \to \mathbb{R} \\
x \mapsto \begin{cases} 0 & x=0 \\ \frac{1}{x} & x \neq 0 \end{cases}[/eqn]

>>15873541
Constructibility is a fine axiom though.

Why are there no closed forms for numbers like arctan(2/5)?

>>15873541
>which is a lot less reasonable

I think set theorists have been living in their bizzarro world for so long they don't allow themselves to have intuitions about anything.

>>15872796
>WTF COULD REAL NUMBER FUNCTION POSSIBLE HAVE TO DO WITH REAL NUMBERS
Take your medicine

>>15873541
>It's more like "If A has more elements than B, it must also have more subsets than B" which is a lot less reasonable
What's unreasonable about it? It's immediate for finite sets. On infinite sets, by definition [math]| A | < | B | [/math] implies the existence of an injection [math] f : A \to B [/math]. In a sane framework, and injection between sets should induce an injective map of the power sets [math] \Tilde{f}: 2^A \to 2^B [/math].

>>15873995
>>15873995

No one mentioned real numbers schizo chan. This is about the cardinality of sets.

intuitionism will win in the end

>>15874012
but why can't there be a surjection from 2^A onto 2^B?

>Uni career day
>every single math and physics graduate there works as a software engineer/IT consultant

>>15873789
the problem is coming from the 1/x and not the zero tho.
if we add the condition that besides the discontinuity points, the function is both bounded and continuous, would all such function be riemann integrable on every bounded interval?

>>15874012
>It's immediate for finite sets.
Not a very good heuristic. In ZF alone, all finite sets trivially have choice functions and all finite zero-sum 2 player games of perfect information are determined, but if you extend both those results to infinite sets (through the Axiom of Choice and Axiom of Determinacy) you can find an inconsistency immediately. Also if you accept the Axiom of Choice, Vitali Sets are unmeasurable (which is never the case for finite sets), while if you accept the Axiom of Determinacy, Vitali Sets can be partitioned into strictly more non-empty partitions than they have elements.

>>15873752
Yes, in fact you could weak the continuity further.
A function in a closed and bounded interval is (properly) Riemann integrable if it is continuous on a closed interval except for a countable set.
You can weaken it even further by allowing discontinuities on a set of Lebesgue measure zero,

>>15873752
Yes, but not just finite points; countably many points. See Lebesgue-Vitali theorem.

>>15874012
>In a sane framework, and injection between sets should induce an injective map of the power sets
It does, this is provable from ZF. What's not provable is whether there can be also be a surjective and, by extension, a bijective map.

>>15874012
Heuristically you can show that [math]|A|<|B|[/math] implies something like [math]\aleph_0^k|P(A)|\leq|P(B)|[/math], but this isn't enough to show inequality with just ZF it seems.

>>15860356
You can keep syllogisms, you just need to drop excluded middle and double negation elimination, that leads to constructive logic, which is closely related to computability and used in a lot of proof assistants and total programming languages.

>>15874909
>>15875114
This result is called Easton's Theorem btw

fake math is pretty to look at

>>15875201
whereas real math looks entirely boring

Thinking about it now, I used to consider choice to be weird but Easton's Theorem convinces me powerset is the most egregious axiom in ZFC.

>>15853575
What do you guys think about these alleged mathematical miracles? Could one find these patterns in any book? Are they too precise to be the result of random chance?
https://www.youtube.com/playlist?list=PL14w8uiQqssd-dQYNRRg6BjyPovlRz-mA

>>15874909
Okay I see. One would think by naive intuition that a surjective morphism of the power sets would induce a surjective morphism of the sets, though I believe this may fail even for easy examples like a dense open subset of an ambient set.

>>15870559
>Instead of masturbating, I study.
Don't you go crazy?

>>15875275
The catch with numerology in general is that they always invent new rules for each case to get the results they want.
They aren't just converting the entire book into a a string of numbers (e.g. a=1, b=2, etc) then parsing it for patterns, it's always a new rule for each case. here you count Bs, there you multiply the number of Gs by 7 (duh, 7 is the magic number) and add 1, etc.
basically it's just using the basic arithmetic to get any number from any number

>>15874731
>Vitali Sets can be partitioned into strictly more non-empty partitions than they have elements
This is where I draw the line. I got over them being nonmeasurable (once I learned a bit more about measure theory) and am not bothered by PSA, but I do not accept that a set can be partitioned into more nonempty parts than it has elements. What is the justification for that?

>>15875470
This is just my intuition so take it with a grain of salt:
Think about what cardinality actually means outside the safety of Well-ordering/Choice. It's essentially a statement about what mappings exist, and you can't necessarily pick out any of the parts to check what's in them. So there might be a surjection from the elements to the parts that exists "in reality" (or it's at least reasonable to think it does), but not in the model, or some of the parts might really be empty "in reality" but which the model thinks are non-empty. Or maybe the partitioning of finite sets gives us a false intuition for what's really happening when one "partitions" such a poorly-behaved infinite structure.

In some ways it's not so different from something like the Banach-Tarski paradox, except applied to cardinality rather than measure.

>>15875525
That intuition seems reasonable, but I take issue with the idea nonetheless. A set is defined purely by its elements in ZF. If every set in the partition is nonempty, it contains at least one element. You therefore immediately get an induced surjection from the Vitali set to the partition. If any set in the partition is missed, then by definition it cannot contain an element of the Vitali set.

>>15875640
Anyway to be a little clearer, I appreciate that such a thing may well be provable in the given system of axioms. I'm saying that if it is, then I do not regard AoD as a reasonable axiom because I view that conclusion as obviously false.

>>15875470
>got over them being
>not bothered by
>do not accept
>justification
meaning what?

>>15875651
I was initially uncomfortable with the existence of non-measurable sets and therefore was unconvinced that AoC was a reasonable axiom, but learned more about measure theory and no longer regard their existence as a problem. Axioms are judged based on how apparently reasonable they are since they are by definition unprovable. If they lead to unreasonable conclusions, then they should be excluded. I view the notion that a set can be partitioned into more nonempty, disjoint subsets than it has elements as completely unreasonable, and therefore think the axiom used to construct such a thing is suspect. I am not an absolutist, if someone can explain why my naïve perspective is wrong then I can accept the axiom, but like I said, it seems to me that such a partition function would contradict basic ZF since a set being defined by its elements induces a map to such a partition which appears to be surjective.

>>15875666
>how apparently reasonable
It's just a definition.

>>15875667
What are you getting at?

how do i prove that Y will always equal β?

>>15875673
recall that the measure of an inscribed angle that subtends a given arc is half of that of the central angle subtending the same arc

>>15875724
i know that but how do i prove that?

>>15875769
or is this correct already?

>>15875769
>>15875773
I would use a few more intermediate steps, just to be sure
something like
>2x+a=180
>x+c+90=180, x=90-c
>180=2(90-c)+a=180-2c+a
>2c=a

>>15875769
id be careful with your diagrams here, the blue and green c angles are not necessarily the same
the most straightforward way to use the double angle theorem is by using it twice, comparing alpha and beta, then alpha and gamma

>>15875666
Out of curiosity, would you consider the existence of infinite dedekind-finite sets in the context of ZF unreasonable?

>>15876058
Yes, I would consider that unreasonable. The definition of finite/infinite I know is that a set is finite iff it has a bijection with some natural number (viewed as a von Neumann ordinal). A set is infinite wrt that definition iff it is Dedekind-infinite, the two are equivalent, so "infinite Dedekind-finite" is a contradiction.

>>15876068
That's fair. I asked because your view of Vitali sets under AD reminded me of Skolem claiming that even though there are models that believe they contain a Dedekind-finite infinite set, they're wrong in an absolute sense.
Imo the strongest case to be made for Determinacy is that it's known to hold in L(R), assuming the consistency of certain large cardinals.

I did it. I applied for a job. Met all the education and experience requirements. A mathematician position for the department of energy. I have no idea what id even do at the job, I just want to stay in my math safe space.

>Axiom of Determinacy
I sleep
>Axiom of Global Choice
Real Shit

what is the bridge between number theory and analysis?

>>15876375
you mean like analytic number theory?

>>15876377
the actual step involved, not the subject.

I saw this problem on the internet and I thought it was interesting. In terms of the sides OA, AB, BC, CO and alpha, what is the angle beta?

>>15865123
>both of these are offered in high schools but you can also take them in college if you're retarded
I was literally the only freshman in my Calc III course at my college and I could only even get into that course because a weird staffing/scheduling issue at my high school caused everyone in my grade at the most advanced level of math to have to skip a year of Precalc. I was in my school's first ever Calc II equivalent class. They admittedly started an IB program that would let students intentionally skip a year to get in to future classes while I was there, but AP and other non-IB students wouldn't be able to get in going forward unless they were transferring in from other districts with different math schedules.

I've no fucking idea why you think it's normal to take Calc I and II in high school. You might be able to, but it's not exactly some universal experience.

>>15875670
I don't understand this fully myself, but it can't be wrong because it's just a definition. I mean, couldn't you just find something that acts like how this is defined?

For different axioms it could fit different physical things or something. A switch can either be up or down, but I could also have something that is programmed to have 3 values, up, down, and both. So in this case, the switch is both off and on. But for some things it is not like the value can be yes and no.

>>15875670
I mean, obviously if you just program what ever it is you're talking about, you have a system that acts like this definition.

>>15875470
Personally I don't accept that there can be a well-ordering of the reals.

>>15870559
You'd probably score even higher if you slept a little more.

>>15870699
I have winter break coming up and then my final semester in spring. I'm practically done already.

>>15875384
masturbation time is part of my sleep time. I stay awake during the day by chugging coffee or Celsius.


>>15877106
Explain

Any recommendations to learn Non-parametric stats? Ideally a textbook with complete, rigorous proofs?

>>15876386
I would argue that the crucial step is this.
Let [math] A(n) [/math] be an arithmetic function. You may define functions such as:
[math] F(x) = \sum_{k \leq x} A(k) [/math]

Whose behavior is, of course, entirely determined by the original arithmetic function, but is now an integrable function as it is continuous almost everywhere. As such, you can now apply the tricks of calculus. As a simple example, when you prove the prime number theorem regarding the prime-counting function [math] \pi [/math] you are actually not talking about an arithmetic function [math] \pi(n)[/math]. You are talking about the real function [math] \pi(x) [/math] defined in the way above, but such that [math] A(n) [/math] is the prime identity function.

>>15877163
Wasserman is a great book, but he skips a couple proofs so you have to do some hunting for them.

I feel so smug knowing constructivism will eventually take Over and y'all motherfuckers are wasting your time dealing with "how many angels can dance on the head of a pin" type questions which are the natural consequence of falling for the formalism set theory meme.

>>15877294
Are you agreeing with the posters saying that axioms need to be justified, or what? Explain

>>15877297
I don't care about anything they say, its completely meaningless to argue about it in the moment you subscribe to formalism, because then you night as well just play the language game that better represents your fefes. If you want your mathematical statements to have a minimum of meaning, you should simply become a constructivist.

/mg/, why do some normies claim to be good at math despite the fact that they struggle with basic arithmetic? These individuals sometimes cite remedial courses taken in high school and college as evidence of their "aptitude". It's quite sad.

>>15877350
They're a bunch Dunning–Kruger silly geese. Don't pay any attention to them, anon. Just focus on improving your own understanding of math which is far more rewarding.

>>15877386
bunch of*
Now I feel like a silly goose.

>>15877287
This is the first recommendation I got, with precisely the same warning.

Could you elaborate a little on how much he skips? Are we talking about "this proof is left to the reader" sprinkled towards the book for the trivial results, or are we talking about entire foundational theorems being left without proof?

And for the proofs that are there, are they completely rigorous with measure theory? Or are the proofs simplified for a non-math audience?

>>15875769
What shape do you get from arg((B-z)/(C-z)) = const?
(B-z)/(C-z) = k*(B*-z*)/(C*-z*) where k is on the unit circle.
Letting w = (B-z)/(C-z) gives:
w = kw* which is a line through the origin in the w plane.
Doing a mobius transformation to go back to the z plane gives a circle through B and C.

>>15877294
How can it take over when it requires that we throw a very large chunk of all modern results and doesn't provide the tools necessary to even come close to catching up?

>>15877485
*throw out

fuck math

>>15877531
unironically speaking, is there any sense to make out of surreal numbers or are they truly just made up by the utterly deranged

>>15877536
ive got no idea, idek what a surreal number is

>>15877350
Forms during high school as you say, when they have teachers that pick favorites. I had a 73 in Algebra II and a 55 in precalculus, yet scored a decent 640 in the math section of the SAT, and currently have a 3.8GPA as a junior undergrad in mathematics.
What I'm trying to say is, from my experience, high school is a social club and teachers are losers that want to be part of the cool kids and will bully nerds in their own way. I also received a 70 in keyboarding, and at the time I was top 1% on typeracer.

>>15877569
>decent 640
why do you say "decent" when that's like 98th percentile of nationally representative

>>15877350
For fun, lets make two student profiles. Typical HighGPA/LowSAT and LowGPA/HighSAT.

type 1 is a female with strong GPA, AP and honors classes, both of which are just bullshit GPA boosters are they're on a 5.0 scale. participates in class and submits a ton of busywork. Teachers dote on her for being a brownnoser and submissive when it comes to following current trends or opinions of her superiors. But in contrast to her GPA, her standardized test scores are abysmal, and this will be written off as "just a little bit of test anxiety!" as you can't curve standardized tests by making 40% of the grade be participation.

Type 2 is typically male, all standard college prep classes. The female guidance councilor doesn't find it appropriate to allow him to enroll in honors or advance placement courses.. Not all homework is turned in as its found to be repetitive and can be solved by inspection, so points get docked for not showing "work"(basic arithmetic). This student excels in standardized testing and is used as an example as to why standardized testing doesn't work as he's clearly just an imbecile that needs to learn his place, as he's [pick 1 or more: incel/loser/slacker/racist/disobedient/disrespectful/etc etc]


The normie you're talking about is type 1. I bet you were imagining a female when typing your post. Nothing against women of course, they're just falling where the overarching system is guiding them, and its that larger system that is creating this phenomenon among other issues. It could have quite as easily gone the other way.

>>15877485
Aren't you supposed to be dead, Hilbert? The world has evolved since then, and it turns out you don't need to throw out alot of things, as was shown by Bishop in his very celebrated book Foundations for Constructive Analysis. Alot of strides have been made in constructive mathematics in the past few decades, it just seems you are unaware of them, and you can bet alot more advancements will be made, it might go unnoticed for a while but I ber my mother whomst I love very much it will. Plus, even you have to admit there's alot of fantasy trash that doesn't matter anyways! I know you have wasted alot of time learning the old ways, but please have some dignity and don't fall for the sunken cost meme.

>>15877536
Read the games part of On Numbers and Games. Surreals and their addition, subtraction, and ordering weren't just made up; they arose out of studying game theory.

the sci sticky puts precalculus as the first thing to learn/refresh my memory on, i thought algebra was first no? and for brainlets like me who forgot everything they learned in school maybe even arithmetic.

>>15877792
Precalculus is simply a combination of algebra, trigonometry, and basic analytical geometry on the coordinate plane. Only the latter will be the real refresher and only when it comes to identifying the functional representation of basic conic sections or something.

Just breezing through Lang's basic mathematics would be enough for a refresher. If you intend to refresh for the purpose of tackling Differential/integral calculus, Its probably insufficient, and I'd recommend just hammering hundreds of problems in Stewarts precalculus instead.

Have any of y'all moved laterally into a math program? Or gotten a master's in mathematics w/o a bachelors in math?

What do I do if I get rejected from every PhD program again?

>>15878371
You get a job

>>15878439
Guess I'll learn to code

>>15878330
Universities post the requirements for their program on the program pages. Your question depends on country, university, among other factors. It doesn't seem you have a genuine interest in math if you're only now going for it and switching to a masters. Its not normal to do a masters in math unless you're playing catchup towards a PhD, and many schools simply don't offer a masters in the US, its bachelors or PhD only.

>>15877425
He skips pretty much every proof that isn't a few lines, or needs measure theory (which he doesn't use). That's not to say the book is easy; the theorems are all there. The topics he picks are broad enough to be worth your time.
Still, if you want the full autistic rigour of empirical processes and all the complications they bring with them, you can look at the book by van der Vaart and Wellner. I'm not sure that the additional insight you get from studying this topic for nonparametric stats is worth the amount of labour required for extracting it, but you can judge that yourself.

>>15877792
Arithmetic should be first, which is why I recommend everyone start with Serre, A Course in Arithmetic.

>>15878330
Yes to your last question, but this is in Europe where it's somewhat common.

>>15879257