Thread 16683009

[math]\mathfrak{mg}[/math]
Jiggery-pokery edition
Talk math

Previous thread: >>16657148

>>16683009 (OP)
Can interiors and exteriors of Jordan curves be uniquely defined? At least for smooth curves?

>>16683134
For smooth, yes. The exterior goes to infinity.

>>16683304
Can you link a proof?

>>16683009 (OP)
Any good recommendations on Geometric Measure theory (and potentially its usage within probability theory)?

I'm an EE (though I did a math minor in undergrad and have done a real analysis and measure course through the math dept. in grad), and my research is starting to involve probability on Riemannian manifolds. I haven't found very much that actually deals with probability densities with respect to measures that are not Lebesgue volume or Dirac/counting measures. It seems like there should be a decent amount of literature on probability when the dominating measure is a (finite) Hausdorff surface measure on a Riemannian manifold but I really haven't found anything useful yet.

>>16683009 (OP)
Precalculus - Stewart
Calculus 1/2/3 - Stewart
Differential Equations - Zill
Linear Algebra - Anton

-Signals and Systems - Laithi
-Electrodynamics - Griffith
Real Analysis 1/2/ - Tao, Rudin
Abstract Linear Algebra - Axler
Statistics - Casella

Rate this progression, im starting precalc today to go back to college

>>16683410
This ordering of books skips set theory entirely. Add in a book on methods of proof and basic set theory after the first linear algebra book. "Book of Proof" by Hammack is fine for this. Also, "Signals and Systems" is typically an engineering course. It is usually an elective course for undergrad math majors, not all need to take it. You can drop this book, or read it after a stats book if you like.

That said though, the precalculus, calculus, linear algebra, and differential will take you at least a year to get through if you are a top student. Most people take 2 years. I would say don't really worry about any of those books. You will probably learn if you actually hate math and don't want to study it all in the next two years. This isn't meant as an insult, but rather the acknowledgement that most people don't actually like the topic even after learning it. If you don't like your classes, just shift to something you can enjoy.

>>16683484
NTA, but Lathi's book is one of the best introductory "calculus based" Fourier analysis books out there. If you're a practitioner/applied math person who is interested in using spectral analysis techniques for solving real problems, it's a great approach to learning.

It's not an accident that many of the most important machine learning and optimization theory researchers came from the signal processing world.

>>16683410
If you're planning to be an EE focusing on RF or signal processing, then it's a pretty good starting list. Personally, I think a good textbook on linear systems/control theory would also be helpful for those goals. Nise's Control book is really great as an undergrad friendly all-around linear systems/control book that covers the important topics within all three major viewpoints in the control world (LTI-transfer functions, State space/state variable feedback, signal flow graphs). Modern control theory by Brogan is a great transition book between an undergrad level to an early graduate level of understanding control and linear systems problems.

How does one learn to cast off one's ego in mathematics? What are your personal coping strategies when encountering those who are obviously far more talented? Do you take solace in the fact that, despite individual differences in ability, we are all united in the search for beauty and truth? I genuinely struggle with this.

>>16683525
I don't doubt the quality of that book, it is the same one I used when I was a student. I also don't doubt its utility for math students. Laithi was the book I used to prepare for PDEs and it was perfectly fine. I'm just not certain that I would place it so early in the sequence of materials. I am also not certain that all undergrads need to be exposed to it. Definitely worth reading, but optional in the same sense that a cryptography or advanced abstract algebra class is optional.

>>16683410
I think Axler is actually more accessible than you might think. It would suffice as a first course desu if you have a slightly stronger intuition with set theory and proofs beforehand. In my opinion, learning set theory and doing proofs only takes a few weeks at most to be decent enough to handle Axler. In fact, Axler skips a lot of things that are more abstract.

>>16683542
That's totally fair. I actually ended up in a sub-discipline of signal processing (I'm this guy >>16683324 if you have any good Geometric measure or probability theory using GMT recs), so I'm probably quite biased towards it being more useful. I will say, having studied linear systems and control during my 3rd year of undergrad made my senior year complex analysis course a lot easier (aside from the habit of using j for the imaginary unit vector). I was definitely more used to working with functions of complex variables than my math major classmates.

>>16683534
For me, it helped to stop attaching my sense of self to the idea that I am intelligent. Then I stopped believing I am intelligent, which helped me work more effectively.

Am I retarded to think of division as repeatedly adding a number smaller than one?

10/2 = 10(1/2) = 1/2 + 1/2…. = 5

>>16683484
You're right I completely forgot to include a proof book, I had chosen Hammack also when comparing him to velleman and cummings just by skimming some of the pages, but i didn't like how hammack wouldn't sometimes explicitly define terms, for example he didn't give a name to AND,OR,NOT etc, he just said what they do, instead of explicitly grouping them together and calling them logical operators, so I got another logic book from suppes to use alongside it, so I will read those two before starting analysis.

I took calculus 1 in highschool but failed it, mainly because my algebra was shit, so I feel like it shouldn't take too much time to catch back up, I just need to finish calc 1/2 to be able to test into admission for calc 3 at my university, im not taking cal 1/2 at university because I want to save a bit on tuition.

I included those signals and elctro books because I'm going to finish my EE degree, i want to study analysis for my own understanding.

>>16683547
I'll check it out first maybe, I'm not required to take linear algebra in my EE degree but I heard studying it before hand can make some of the EE courses easier to understand but idk. I'm more focused on learning the computation stuff first so I can atleast start taking my classes and finish my degree but I am interested in the more abstract foundational stuff also for my understanding.

would you say picrel is about 2/3rds of a math undergrad degree?

>>16683589
Wait, your EE program doesn't require you to do a linear algebra course? What on Earth? Linear algebra is one of the few math disciplines you really need to have down to do EE at most schools.

>>16683590
No, absolutely not. It spends way too much time on "fundamentals" that are usually devoted to a single semester's course or assumed to have been learned in high school.

Most math undergrad degrees start at calculus. Your flowchart ends on the first semester of most math degrees.

>>16683599
They require Cal 1/2/3, Diff EQ, Statistics, and "Math for Engineers" which has the following description:

"This course covers the essentials of matrix theory, graph theory, numerical methods, and introduction to proofs needed for majors in Electrical and Computer Engineering. Topics include Gauss-Jordan elimination, matrix algebra, determinants, graphs, trees, root-finding algorithms, numerical differentiation, numerical integration, numerical matrix methods, propositional and predicate logic, and formal logic proofs."

Which I guess looking at that now it sounds like linear algebra or atleast the important stuff, i dont know why they dont just call it that

>>16683605
Yeah, so that math for engineers course seems like it covers the important topics within an applied linear algebra course. It probably would also be helpful to learn the basics of vector space mappings, but that might be thrown into there and just not explicitly described.

That makes a lot more sense than just not having a linear algebra requirement at all.

>https://www.youtube.com/watch?v=uE6q-dxjrlA&list=PLJHszsWbB6hoOo_wMb0b6T44KM_ABZtBs&index=22

I've been researching spinors for the last 19 days and I can confirm that they are the final redpill.
The series introducing you to spinors concludes with an introduction to quantum mechanics.

The most fucked up thing that I've ever discovered in mathematics is the fact that spinors exist and the rest of your life is spent integrating what exactly that means.

What’s the most advanced math that still has “practical” application? Like stuff that grad students go over but still has uses in some industries and you would have to know

>>16683556
Nope. That's actually a neat analogue of multiplication being repeated addition of numbers bigger or equal to one (integers, that is).

>>16683695
yeah thats what got me thinking about it, if subtraction is adding a negative number, multiplying is repeated addition, then i was thinking about how division could be thought about in terms of addition

>>16683656
Spinors?

Seriously, someone could correct me, but it could be.

Underpinning geometry, spherical geometry specifically, space, quantum physics and the natural world, abstract algebra, transformations, symmetries / group theory and probably something about associativity and commutativity are "spinors".
The way that electrons work are modeled by spinors and the only reason that anything has mass is because of electrons. It's a fascinating theory but then it also seems to have its roots in about 5 other absolutely fundamental and / or mindbending areas of mathematics.

Watch that 23 part series on what a spinor is if you're curious.

>>16683410
>Abstract Linear Algebra - Axler
What's this? I thought that the only LA book Axler did was Linear Algebra Done Right.

>>16683615
>they are the final redpill
the first redpill
If you venture off the paved path you might find them in more illuminating contexts.

>>16683589
I'm the anon who mentioned a proof book. For EE, the list of texts you provided are perfectly reasonable. I would also add one book on complex analysis. Also, if any one of those books doesn't quite mesh well don't hesitate to change to a different author. Essentially all of the Springer series in Mathematics have high quality textbooks. Peek around for the authors who explain a topic in a way you like the best. This is an essential part of trying to learn faster than your course curricula, imo.

>>16683967
Any recs for complex analysis? obviously far away but i figured i would add one to the list, i heard rudin is the generic one to pair with baby rudin on real analysis

>>16683982
Complex Variables and Applications by Brown and Churchill is the book I used in undergrad some years ago. I think it is considered an extremely common intro text. The authors have published several editions, updating it over time. If you look it up on your search engine of choice, you should be able to find a pdf of it hosted by multiple universities pretty easily. I'm not going to claim it is the best, but its good enough for a first time with the topic.

If pi is irrational that means that the number you get for the circumference of a circle using the formula C = 2 pi r is always an approximation right? So technically the answer should always be ≈ x right

>>16684032
I guess what I'm really asking is, due to the relationship between the circumference and radius of a circle involving pi, an irrational number, does that necessitate that at least one of the value (either circumference or radius) must be irrational as well?

>>16684033
yes, because pi cannot be written as a fraction. If both values are fractions, that's a contradiction

>>16684037
I dunno if he knows about irrational numbers and how they're basically just an arbitrary byproduct of a number system with the number 10 as a base.

>>16684032
Try thinking about "irrational numbers" more in terms of fractions.

>>16684039
Irrational numbers have nothing to do with base 10. They are the definitionally the complement of rationals with respect to the reals. The name literally means "not rational." The rationals are all numbers which can be written as an ordered pair of two integers, with one of those being nonzero. This is pair is usually called a fraction or ratio. The important part is that both are integers. It has nothing to numeric representation such as base. Episode 9 of this series on formal construction of numbers explains this:
https://youtube.com/playlist?list=PLBh2i93oe2quR7WUsPoIlzzEvFDAnEkhR&si=X_iSUe7ngmr0zMvW
Don't mislead people without references.

>>16684044
You can't "express" something if your number system doesn't allow it.

You're probably right, and I'm just gonna give you this one, but no where in this is proof that something can be expressed rationally.

Show me the formula or proof that 3.747234235 is rational or rational because it can or can't be expressed by a ratio of two integers.

Also, this is a genuine question, but what is the point of defining something as exactly two integers instead of a chain of ratios? (3 / 7 / 43) if the integers can be infinitely large?
Because it feels like you can produce any number by pulling two numbers out of your ass and creating a ratio out of them. I don't know how.

>be me
>27 years old
>weeks from delivering my dissertation
>I applied to 56 post-doc positions
>no acceptances
>not sure what I’ll be doing in a few months
>advisor asks me “what else have you been working on? Do you have a backup plan?”
>i say no
>he says “yikes”

>>16684067
I will respond as plainly as possible. I was never intending to have an argument, I was simply stating how the rationals and irrationals are defined and pointing out how it doesn't have to do with the choice of a base. I apologize if I came off as flippant, I was assuming you knew more about the topic, but the question you asked afterward makes it clear that you also have your own questions about numbers. I will answer some.

>You can't "express" something if your number system doesn't allow it.
It is possible to produce extensions of a number system. In the real numbers, this is done by something called a completeness property. The overall idea is pretty abstract, but one way to explain it is to create a sequence of numbers which can be thought of as an approximation, of a number that does not yet exist, then defining algebraic operations on these approximations. See this on Wikipedia: https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers#Cauchy_completeness

>Show me the formula or proof that 3.747234235 is rational or rational because it can or can't be expressed by a ratio of two integers.
That is a finite sequence of digits. It is rational. In particular, it is equal to the ratio 3747234235/1000000000

>Also, this is a genuine question, but what is the point of defining something as exactly two integers instead of a chain of ratios?
If you can express the chain of ratios in a way which provides for consistent and unambiguous algebraic rules (this is called a well-defined representation) then from a mathematical point of view, there is no problem. I am not sure if you are aware of this, but what you are talking about sounds like something called continued fractions: https://en.wikipedia.org/wiki/Continued_fraction

>>16684082
That's rough. If you are in the USA, do you think it is due to uncertainty in the current university climate? If you are not in the USA, what's your subfield and publication count?

>>16684088
I am in the USA.
It was 11 years ago btw.
Main reason is that there really just aren't many positions in most fields.
>what's your subfield and publication count
It was Applied Analysis and 2 at the time.

There was some bad timing, it was still the tail end of the great recession, and I finished my 2 year dissertation project just as people stopped caring about that topic and everyone rebranded themselves an ML researcher.

>>16684171
did you get a $300K job as a quant?

>>16684175
NLP researcher, I only make $250k a year but there's better work/life balance

>>16683808
Axler is a Jew using students’ penchant to overvalue free textbooks to self-promote his low quality textbooks

>>16684410
It is ok to admit that you don't understand how indexing and sum notation works, anon. Don't take out your anger at how you had to repeat linear algebra twice on free textbooks.

Where can i learn mathematics required for AI?

>>16684446
most of the mathematics involved is just basic linear algebra

>>16684446
>mathematics required for AI
unless you're doing theory, it's more software engineering than "math"

I really hate shapes and geometry. My brain starts to freak out. Its pretty interesting to me.

>>16683802
>the only reason that anything has mass is because of electrons
Not to nitpick, but the “reason” “anything” has mass is because of the Higgs mechanism. I put quotation marks around these two because by “anything” I mean the Standard Model fields. Most of a proton’s mass comes from the QCD potential, not quark mass.

So you can understand mass in two separate ways: as the quadratic Casimir of a unitary translation group representation (a quantum field) and as an invariant associated with the stress-energy tensor. The Higgs mechanism is needed because massive representations are not gauge-covariant. It can be understood as a solder form on the gauge bundle. This has nothing to do with mass in the latter sense of stress-energy tensor, as that one is defined via the metric on the manifold.

Is this accurate?
It looks like it to me, but I'd prefer to ask you bros..

Can anyone explain, logically from first principles why exactly two negatives multiplied equals a positive? Every example I see is either circular or tries to give a “real world” intuitive answer that isn’t satisfying. I’ve just accepted it for so long but I don’t know why

>>16684644
It's a natural consequence if you want the rational (or real) numbers to be an ordered ring.

>>16684661
i dont know what an ordered ring is

>>16684644
a + b = b + a
a(b+c) = (a)(b) + (a)(c) = ab + ac
a + (b + c) = (a + b) + c = a + b + c

0 = 0(a) = (0)a
(0 + a) = a
0 = (-a + a)

> e.g.
> 0 = (a + b)
> --> (0 + -b) = (a + b) + -b = a + (b + -b) = a + 0
> --> -b = a
> --> -a = b

0 = (a)(-b + b) = a(-b) + ab
0 = (-a + a)(b) = (-a)b + ab
--> a(-b) = (-a)b = -(ab)

0 = (-a + a)(-b + b) = (-a)(-b) + (-a)b + a(-b) + ab
--> 0 + ab = (-a)(-b) + (-a)b + a(-b) + ab + ab
--> ab = (-a)(-b)

>>16684680
i forgot about right-distribution but wtv, i gave left-distribute

>>16684674
Then, look it up retard.

>>16684680
oh also,

0 = (-a + a)(-b) = (-a)(-b) + a(-b)
= (-a)(-b) + -(a)(b)
--> ab = (-a)(-b)

>>16684688
i did, i didnt understand it

>>16684693
Not my problem.

>>16683009 (OP)
What books does "Introduction to Smooth Manifolds" by John Lee serve as a prerequisite to? Is this book worth reading if you want to go on to study fiber bundles and gauge theory? Apparently it doesn't cover banach or frechet manifolds so would it be better to just read a book that starts with such greater generality?

I ask this because it seems to be a very highly regarded book.

>>16684696
i didnt say it was, i just asked if you could explain it simply and logically, if you can't do that, THAT is a you problem

>>16683009 (OP)
what the hell is the thetaF mentioned on the top left part of the pic?

>>16684721
You don’t need Frechet and Banach manifolds for vanilla diff geo unless you’re doing calculus of variations or something like geometric quantization.

As per your question on what it can serve as a prerequisite to, the aforementioned things, gauge theory mentioned by (You), symplectic geometry, hardcore Lie theory (with all the root system and Dynkin diagram business), and possibly things like homotopy and cohomology in algebraic topology as many such constructions use diff geo tools.

>>16683009 (OP)
how do i do math n sheeeeeit? I am a nerd that never fully bloomed. I have always wanted to be a really smart nerd!

>>16684723
I explained it simply and logically. You are just malding because of low iq.

where da womanifolds at?

>>16684032
>>16684033
You should view the formula as a definition of π.

π is defined as the ratio of a circles circumference to its diameter.

>>16683304
>For smooth, yes.
Interesting, what can go wrong for non-smooth curves?

>>16684992
n=4

>>16685000
I deleted the post you're replying to, but here's a full answer:
https://math.stackexchange.com/questions/373708/when-is-the-group-of-units-in-mathbbz-n-cyclic

>>16684975
Unless you can use the JCT in your proof it is very hard to prove that no points in the interior can go to infinity like points on the exterior can when the boundary is something silly like radial weierstrass, even though it is obvious that they can't, it is just bullshit to try to prove.

>>16684847
you didn’t though, you just made a statement, a statement isn’t an explanation

>>16685065
If your curve is expressed in polar coordinates as [math] r = f(\theta) [/math]
where [math]f[/math] is positive, continuous, and periodic with period [math]2\pi[/math],
then it is easy to prove there is a bounded interior consisting of the points [math](r,\theta)[/math] with [math] r<f(\theta)[/math],
and an unbounded exterior consisting of the points [math](r,\theta)[/math] with [math] r>f(\theta)[/math].
You don't need JCT for this, and [math]f[/math] can be as un-smooth as you like, as long as it's continuous (and periodic and positive).

Abstract algebra question (field theory):

Let F be a field. If [math] f(t),g(t) \in F[t] [/math] are irreducible in [math]F[t] [/math] and are non-associated (i.e. are not nonzero constant multiples of each other),

then is [math] (f(x),g(y)) [/math] a maximal ideal in [math] F[x,y] [/math]?

Is math just applied philosophy in some sense?

Logic is the fundamental underpinning of both, in each philosophical field you take some axioms and construct from there, same in math, it’s just that math has some perfect abstracted correspondence with the real world.

I always loved philsophy subjects like epistemology, ethics, etc, and now that I am getting back into math for school, and learning how well constructed math is and reading Tao’s introduction paragraph in analysis 1, thinking about it like this has made me enjoy it much more.

>>16683324
Most measure theoretic results hold for general topological spaces and general measures, so what more do you need to do other than plug those in

>>16685189
I don't understand what you mean by "applied philoosphy". Both philosophy and math use logic, but are concerned with entirely different things.
>it’s just that math has some perfect abstracted correspondence with the real world
Math has all kinds of constructions not found in the "real" phenomenological world. Mathematicians can talk about irreducible representations of the Lie algebra [math]\mathfrak{e}_8[/math] in 30380 dimensions or the Alexandrov long line.

>>16685157
For anyone wondering, a counterexample is [math] F = \mathbb{Q} [/math], [math] f(t) = t^2 - 3 [/math], [math] g(t) = t^4 - 3 [/math].

>>16684573
yeah it is

>>16685677
thanks

>>16683534
>What are your personal coping strategies when encountering those who are obviously far more talented?
4chan's general coping strategy with that is to, in any order:
>accuse the talented person of being egotistic
>rally around some overrated celebrity
>aggressively promote mediocrity as a virtue
>seek solace in various forms of collectivism
>find a novel use for the word "tranny"
et al.
now you too can cope like a proper anon

Anons do you need a gold at the imo to be make it in maths academia?

I dont agree with calculas
Sry but getting infinetly closer to something is not the same as actually getting there, they're all wrong

>>16689054
Lisa Sauermann, a mathematician specializing in combinatorics, won a silver medal at the IMO.
The competition isn't the end-all for academic success, really.

Is ai capable of checking proofs yet or will it just hype me up? I think I'm right about stuff but I don't have anyone to check my work. I'm not the brightest and can sometimes make minor errors and not notice.

wolframalpha.com won't draw
1 square,
1 oblique rectangle,
6 small circles, and
2 large circles.

>>16689539
We clearly see muh AI make dogshit mistakes and hallucinate, but boomers are so involved in the bubble that they genuinely believe it will be capable of accurately translating mathematical papers into second-order logic strings and then check for consistency. At this point muh AI doesn’t even mean anything. Normalfags have the talent to make everything semantically meaningless (see quantum computing, crypto, etc.)

>>16689539
>Is ai capable of checking proofs yet or will it just hype me up?
No even close yet
>I'm not the brightest and can sometimes make minor errors and not notice
Try writing your proof in Lean. It will force to formalize and explain every little detail leaving no room for mistakes.

>>16689554
>wolframalpha.com won't draw [...]
nevermind

https://www.wolframalpha.com/input?i=Flatten%5B%7BAbs%5Bx+-+y%5D+%2B+Abs%5Bx+%2B+y%5D+%3D%3D+100%2C+Abs%5Bx+-+y%5D+%2B+Abs%5B0.5+x+-+y%5D+%3D%3D+25%2C+Table%5B%28x+-+8+k%29%5E2+%2B+%28y+-+6+k%29%5E2+%3D%3D+10%5E2%2C+%7Bk%2C+-5%2C+5%2C+2%7D%5D%2C+%28x+-+25%29%5E2+%2B+%28y+%2B+25%29%5E2+%3D%3D+25%5E2%2C+%28x+%2B+25%29%5E2+%2B+%28y+-+25%29%5E2+%3D%3D+25%5E2%7D%5D

>>16689679
>mistakes and hallucinate
It's kind of scary that high schools and below want to use AI as 'tutors', wonder how much fake shit people will learn.

IT'S UP
https://www.youtube.com/watch?v=aHUQ9347zlo

>>16685581
The measure theoretic results are useful in the sense that if I can demonstrate a (sigma-finite) dominating measure on a space, then I know a Radon-Nikodym density with respect to this measure will exist. It doesn't help me with actually producing this dominating measure, or defining proper Radon-Nikodym densities with respect to said dominating measure.

I need more than an existence/uniqueness proof, and actually need a mechanism to produce a viable density form or path to integration.

>>16691079
>that boomer zoom stare
kek

>>16683009 (OP)
Learning math is hard. I will learn the vocabulary first. I have a math book I learned the term Fact Family. Do you know what a Fact Family is?

>>16691674
Non-constructivist bros...

>>16689706
>Lean
Is that really a good way to check textbook problems? Does that actually work?

>>16685156
This requires the interior be star-shaped, which is not true for all jordan curves (try one tracing out the boundary of the letter C, for example).
So then your problem becomes finding a set of coordinates under which a given interior is star-shaped, which is the same as showing every interior is homeomorphic to a disk...
which is what we're trying to do anyway.

>>16691920
It's not really a constructivism thing, so much as my purpose is to actually use this "theory of integration on surfaces" to perform integration on a surface.

The existence and uniqueness proofs are good enough to know that it can be done, but the paper I'm working on requires that I actually have a path towards defining and evaluating a specific probability measure on a Riemannian surface embedded in R5.

What's the best book for learning mathematical argumentation?
I am taking more advanced courses and I can't follow a single thing

>>16692068
If the Jordan curve is expressed as [math] r = f(\theta) [/math] in polar coordinates, where f is continuous and positive and periodic with period 2pi, as was described in the post you're replying to, then obviously its interior is star-shaped.

is this right?

how do I cope with the fact that donuts are coffee cups and guitars and chairs. nobody will listen to me about it they just say I'm crazy

>>16693195
"That's what I said: Sodium Chloride."

>>16693195
You didn’t specify the equivalence class, buddy.

>>16693157
It's wrong.
You claimed that if [math]N = \lfloor \frac{1}{\varepsilon} \rfloor[/math] then [math] \frac{1}{\varepsilon} < N[/math] but this is wrong for some values of [math]\varepsilon [/math] like [math]1 [/math] or [math]\frac{1}{2} [/math]. You need a slightly bigger [math]N[/math] to get this strict inquality.

>>16693157
along with what the other guy said, did you even use the archimidean property? Why not?

>>16689429
I agree but it may be helpful to know that what actually happens is we squeeze the [math] \varepsilon \delta[/math] interval infinitely [math]\mid[/math] we leave the field R entirely and drop into an extension

this is a form of quantization, and yes, it's a discontinuity

Man, nah fuck this math shit yall niggas made this shit hard on purpose fuck yall

Just graduated college with a degree in math in minor in cs from uchicago. Decent math gpa abysmal overall gpa (3.7/3.1). Did a lot of phd classes here, and a lot of phd theory ml classes. Only one REU and no real experience.

Interested in operator algebras and rmt. Currently reading alon spencer probabilistic method and murphy c star algebras.

the question is, what should i do with my life? currently got no job/internship/grad school lined up. Planning on reading over the summer and applying to as many jobs in ai, swe, robotics, fintech, and quant as possible. gonna apply to math/theory cs/theory ml phds in the fall. Gonna also try and do quant recruiting for next summer.

if u were me, what would you do?

>>16694053
i also like pdes and computational geomtry/combinatorial optimization stuff i forgot to mention

>>16694053
>if u were me, what would you do?
That's an absurd hypothetical. I am not you.

>>16684082
Anon, don’t give up. At least you’ll have your PhD. You’re ahead of me and I haven’t given up. I’m 27 couldn’t even finish a math undergrad and am going to a post bacc now to try and do that and go for a masters and then we’ll see how things go. You’re talking like you’re a failure but you’re so far ahead of the vast majority of the population. What was your area of research? Are there any possible industry applications? Don’t give up!!!!

hey /mg/, just curious if there's any foundations on studying parametrized (and thus oriented) closed curves with self intersections and sharp extrema? Sort of topological stuff or something to relate back to harmonic analysis? I've already studied C* algebras and know about the continuous functional calculus before thats brought up, and I'm aware of enough knot theory to know I don't want to chase anymore knot theory

>>16694571
>with self intersections and sharp extrema
What I’m going to propose is quite specialized, but algebraic geometry studies algebraic curves (among other varieties) and how locus points indicate algebraic multiplicites. It’s quite similar in spirit to how one can analyze non-essential singularities in complex analysis via contour integrals and Laurent series. Algebraic geometry has a nice connection to differential geometry via the Serre-Swan theorem. So you can likely research your analytic problem using abstract algebra.

>>16694590
oh nice! that would be surprisingly well-suitef as I'm just a group theorist dipping my toes into some funny curves I found recently, thanks anon I'll see if any of the common wiki pages provide a good reference to chase for this result.

Plz anons, recommend good resources for mastering combinatorics and probability, right from scratch. It can be videos lectures, books, etc., I just wish to get an intuitive hang of the basics and advanced topics(undergrad level) and possess the confidence & ability to tackle hard problems. I have about 30-40 days of free time btw, something which I totally wish to dedicate for this. Btw, I only know basic coordinate geometry(eqns of straight lines, circles, conics, 3d lines & planes). basic high school algebra like quadratic eqns, arithematic-geometric progressions, binomial theorem and basic calculus.
I did try studying basic permutations on my own using a standard math textbook followed by high schools in my cunt, but it's dry, boring and seems to have little to no explanations & reasoning as to why shit just werks, which is understandable as permutations and combinations is just one of the 15 or so many chapters in that book

>>16695263
fuck off panjeet

>>16695263
For combinatorics start with basically any Discrete Math book that gets posted here a lot. Like How To Prove It. Book of Proof. Just read the sections on "counting". If you wanna really dive into it more check out A Walk Through Combinatorics.

Probability i don't have any suggestions but there's probably no shortage of good options. Most important is to just start and not get caught up on finding the perfect resource. The ideas are the same no matter who or what explains it to you. There's no rule saying you have to read one book or one lecture in a straight line. In fact I find it more productive to use several resources at once. If one book describes it in a way you can't understand try checking that section in another similar one

>>16683009 (OP)
Is a second course in analysis required? I finished reading and doing all the problems in Abbott's Understanding Analysis. Do I need a second course in analysis such as going through Pugh's Analysis textbook?

Also, I was thinking of going through Stein's complex analysis textbook, is it possible for me to do all the problems correctly in that book with just my knowledge from Abbott or do I need a prior exposure to complex analysis to do all the problems in Stein's textbook.

Because I was thinking of going through complex analysis by Bak then complex analysis by Stein incase Stein requires prior knowledge since it's classified as a graduate school level introduction to complex analysis while Bak is classified as a undergraduate introduction to complex analysis.

Thank you for reading this.

Saw this somewhere else and it bothers me that I could have answered it back in college but not now. What terms should I look up to relearn how to solve this? I'd check my old calc 3 notes but I haven't been able to find those

>>16696411
It depends on what you're trying to do. If you'd take a look at Abbot's analysis book and Royden, you'd see they have essentially no material overlap. In fact, most Measure oriented analysis books assume you've already done an "undergraduate" analysis course.

If you want to understand research level analysis, learning the basics of measure and functional analysis is absolutely required. You'd really want to do something like Royden or Folland or Rudin or something. If you aren't planning on being an analyst or applied mathematician (I've yet to run into an applied mathematics problem which measure and functional analysis don't help), you might not need it.

Do you suppose the space of meromorphic functions could form a nice algebraic structure given equivalence classes of functions with equal residues?

>>16696634
Thought about it, addition is no good because there are no inverses. Multiplication is highly non-trivial

>>16696613
So there is no point in going through real analysis by ross...? So instead ill try and do real analysis by pugh instead.

>>16696634
meromorphic functions already form a field

>>16696677
Yeah but I’m more interested in an algebra of residues

>>16696411
Abbott is not an analysis book. It's a calculus book written like an analysis book. It doesn't define trigonometric functions for instance and its treatment of integration is very basic: it does not talk about Stieltjes integration or polar coordinates or surface integration, etc. For that matter, it doesn't say anything about multivariable analysis, not even complex numbers. A lot of this can be compensated for by Rudin. However, I'll suggest you pick up Folland's Real Analysis. It's a graduate analysis book, which is a completely different topic, but if I am not wrong, it does derive most of the things missing from Abbott except for trigonometric functions and multivariable differentiation which you can consult from the relevant chapter of Rudin.

Pugh is a pretty sloppy book in my opinion, and I don't suggest it.

Could some anon give me a qrd for dummies on the whole sheaf, stalk, germ business? I encounter it all the time, but whenever I look at formal definitions, it just looks like mumbo-jumbo to me. I know a fair amount of topology and category theory.

>>16696795
What about the real analysis book by Stein? Would you reccomend that book after Abbott?

>>16696839
I don't know. It's very slow from what I have seen, which is why I have avoided it. The entire series except for the last one covers everything Folland does in a single book. Of course, if you want a gentler treatment go for it. It has pretty good reviews.

>>16696411
This is what you need. You need to unlearn all the false mathematics being taught in the standard curriculum.

>>16683009 (OP)
I want to download MATHLAB to see if I can start to understand how concepts and symbolic expressions work, among other things.
Because I don't know how to solve
>∫(x2 + cos(x)) dx from x=0 to x=π/2
or
>limₓa (2x + b)
or
>limₓc f(x)
But I read that to download MATHLAB I need an account, and after that a license.
I have neither.
Could you help me? Is it something that will get me in trouble If I download through someone or torrent?

>>16697053
Matlab is for pussies. Do math on paper or in your head. Learn how to find antiderivatives with a textbook

Let A be a positive-measure subset of R^n. (Using the Lebesgue measure)

Does the closure of A necessarily contain a (nonempty) open set?

>>16697350
fat Cantor set

>>16697355
Thanks anon

Does [math]f(\alpha) = (\cosh(\alpha),\sinh(\alpha))[/math] have an inverse?

>>16697636
[math] f( \alpha ) = ( \cosh ( \alpha ), \sinh ( \alpha) )[/math]

>>16697665
Uhh, that's not an inverse

>>16697636
(x,y) -> (ln(x+y))

What’s a good book for a more rigorous approach to tensors

>>16697958
Intro to Smooth Manifolds by Lee. That’s if you want to do tensor fields. Chapter 12 specifically. But tensors from an abstract point of view are just multilinear maps and tensor algebra is just a free algebra with vector spaces acting as generating sets. For that, something like Artin’s Algebra suffices. He covers free groups and polynomial rings (which are free objects in the category of groups and commutative rings respectively), so you can easily generalize to tensor algebras. Algebras have “ring like” behavior with ideals etc.

>>16697636
>>16697665
Using the horizontal line test, the given parametric function is injective, therefore you will have an inverse that is a valid function.

The yellow line in the picture is the original parametric function, the blue line is its inverse found by switching the hyperbolic functions around.

>>16697979
I think I have Lee on my shelf already so I’ll look at it, thanks. I have a solid background of rings and some category theory.

>>16697053
Matlab is great for a lot of things, but if you're using it for basic derivatives and limits that tells me that you should spend some time learning your basic calculus fundamentals.

Also, afaik it's not super easy to torrent/crack. Julia has a symbolic mathematics module that supposedly is fairly competent and is free. You could give that a shot.

I wrote a new blogpost about the sign of a permutation, providing a much simpler proof that it's well defined than in my previous blog post
https://radioclubjp.github.io/math/2025/06/14/signature-sign.html

Have any of you encountered the proof before? Would be interested to know.

My class is following Milne's notes on Galois theory but I cannot follow it /I find it too fast paced and compact
Can sci suggest a better Galois Theory book that has same content approximately as Milne but better paced with more exercises and explanations


Here is link for Milne's notes
https://www.jmilne.org/math/CourseNotes/FT.pdf

So I finished my undergrad degree in math a few months ago and I'd like to take some time this summer to prepare for grad school. Do any of you guys have suggestions for what I should work on?

I have a pretty decent understanding of which subjects I'm best and worst at and which subjects I'd like to go into for research. The question is should I take time and go through the material I'm weaker at (Algebra, Combinatorics, discrete stuff), or should I try and deepen my knowledge in my favorite subjects (Differential Geometry, Complex Analysis)?

Why can I say that [math]E(e^{tX}|Y)[/math] is the MGF of [math]X|Y[/math]?

>>16699757
I assume you have some kind of qualifier exam before your first semester. Prepare for that. Grad courses will deeper your understanding, don’t worry. You’ll be in for a ride.

>>16699874
I'm not sure exactly what you're asking for, but intuitively if you "know Y", you know the distribution of X, hence also that of [math]e^{tX}[/math], which gives you the MGF. The conditional notation "[math]\mid Y[/math]" is just an accounting of this knowledge.
Formally, with a little bit of regularity the law of X|Y is described by a kernel [math]K[/math] (satisfying some regularity assumptions) such that [math]K(y,A)=\mathbb P(X \in A \mid Y = y)[/math] which is well defined since [math]\mathbb E(1_{X\in A} \mid \sigma(Y))[/math] is actually a function of [math]Y(\omega)[/math] as opposed to simply of [math]\omega[/math] by Doob-Dynkin.
Then you can integrate functions [math]f:\mathbb R \to \mathbb R[/math] by [math]Kf(y)=\int_{\mathbb R}f(x)K(y,dx)=\int_{\mathbb R}f(x)\mathbb P(X\in dx \mid Y = y)=\mathbb E(f(X)|Y=y)[/math], all of which is just notation, and the point is that the act of conditioning on [math]Y[/math] is "embedded" in the measure you're integrating against.

How do we cope with the fact that the smartest man in the world is woke?

>>16699998
This is a requirement in the US academia. He may as well be a shadow chud, but he’s smart enough to not spill his ‘ghetti.
t. chud who put the pronoun bs on his resume

>>16685157
yes if F is algebraically closed (;
>>16696804
>I know a fair amount of topology and category theory.
I'm sure you're able to parse the definitions then. Anyways here's the typical example. Let X be your favorite topological space. Our sheaf F is a rule that assigns to each open subset U of X the set F(U) of continuous functions U->R. Here are the defining features of our sheaf:
For every inclusion V -> U of open sets, there is a corresponding map F(U)->F(V) given by restricting functions U->R to functions V->R.
If an open set U is covered by some U_i's, then a function f in F(U) is uniquely determined by its restriction to each U_i.
On the other hand if you have a function f_i on each U_i, and the functions agree on the overlaps (f_i and f_j have the same restriction on the intersection of U_i and U_j) then you can glue them to obtain a continuous function f on U.
The above features make F a sheaf.
If x is a point of your topological space, and f is a function defined on some neighborhood of x, then the germ of f at x is what f "looks like" "near" x. The stalk of F at x is the collection of all germs of all functions defined on any neighborhood of x.

Are there infinitely many Pythagorean triples (a,b,c) for which a and c are both prime?

>>16683009 (OP)
Somehow I feel safer about the future of Homo sapiens sapiens.

Does anyone know what book picrel is from?

>>16700183
This reduces to the trivial problem of showing there are infinitly many primes [math]p[/math] such that [math]\frac{p^2 + 1}{2}[/math] is also prime.
Then you can take Phythagorean triples
[eqn] (a,b,c) = \left( p , \frac{p^2 - 1}{2} , \frac{p^2 + 1}{2} \right) [/eqn]

>>16700200
Nevermind, after some more prodding cheepy cheepy found it was from Probability Theory and Combinatorial Optimization by J Michael Steele.

>>16700208
Oh lol then I guess my next question is, how do we show there are infinitely many primes p such that (p^2 + 1)/2 is also prime?

>>16700039
>Our sheaf F is a rule that assigns to each open subset U of X the set F(U)
Here’s where I immediately get lost. What’s a “rule”? A functor?

I get the intuition behind it, but every explanation I’ve seen is either this informal “rule” stuff or full-blown abstract nonsense.

>>16700217
yes a contravariant functor from the category of open sets of X to the category of sets - where an inclusion of open sets U->V gets sent to the restriction map F(U)->F(V). Such functors are called presheaves. Presheaves where you can glue in a unique way are sheaves

>>16700212
he might be pulling your leg because that's an open problem similar to the twin prime conjecture

>>16700266
>he might be pulling your leg because that's an open problem similar to the twin prime conjecture
Lol didn't realize, thanks for letting me know.

Then, is there any way to show >>16700183 without having to use >>16700212 ?

>>16700238
Great, that makes it much clearer! Thanks!

Wish there was some kind of iqlet math academia where your job is basically like a phd math guy, but you research easier stuff and teach easier courses, like you only need a masters in math to do it.

>>16700565
It’s called a hs teacher, anon.

It's springer summer sale. All books 25% off. Why the fuck did nobody tell me and where can I find more such good deals?

Is there an isomorphism from the space of 2x2 matrices in [math]\mathbb{C}^2[/math] to 4x4 matrices in [math]\mathbb{R}^4[/math]?
>>16700403
You are free to try but I think it might be similarly difficult

>>16701168
wait I figured out my confusion was about complex-linearity vs real-linearity. That's why [math]\hom(\mathbb{C}^2,\mathbb{C}^2)[/math] is only 8 real dimensions instead of 16, because there are real-linear transformations that aren't linear over complex scalar multiplication

What does the coset space [math] \mathrm{GL}_+(2,\mathbb{R}) / \mathrm{GL}(1,\mathbb{C}) [/math] look like topologically? This should be some 2-manifold, but I'm not sure how to determine what it is.

>>16701417
To add, my guess is it should be the open upper half-plane of [math] \mathbb{R}^2 [/math]. My reasoning is, given a positively-oriented real frame of [math] \mathbb{R}^2 [/math], we an use rotations and dilations (which are complex-linear) to rescale the first vector in the frame to (1,0), then the second vector can be anything in the open upper half plane. However, I'm not sure if this is correct, or how to make this rigorous

I’m like this close to solving Collatz, stay posted

>>16701417
I think it's all matrices spanned by [math]\begin{pmatrix} a&0\\0&1\end{pmatrix}[/math] for [math]a>0[/math] and [math]\begin{pmatrix}1&0\\b&1\end{pmatrix}[/math] for [math]b\in\mathbb{R}[/math]. My reasoning is that each complex number represents a scaling and a rotation (embedded in [math]\text{GL}_+(2,\mathbb{R})[/math] as the matrices [math]\begin{pmatrix}a&b\\-b&a\end{pmatrix}[/math]), so the remaining linear transformations are shear transformations that do not cause reflections (i.e. negative determinant).
You can do the work from here

>>16701521
rather, I should say [math]\begin{pmatrix}a&0\\0&\frac{1}{a}\end{pmatrix}[/math] to be more precise

>>16701526
[math]a[/math] is still positive only because multiplying by -1 is the same as -I which is a complex number representation, but the transformation to go from b to -b in the second matrix is not a complex representation.

Dilbert space

>>16699998
I don't have an issue with the pronouns thing.
My issue is with "non-binary" people.
Being trans is an actual mental disorder, I'm fine with them transitioning (as long as they eventually pass physically and vocally), and don't mind calling them he or she.

"Non-binary" is literally just attention-whoring.
I'm not gonna call you "they/them" you fucking child.

Intersectionality is category theory for progressives

>>16702004
>implying category faggots aren’t troons already

>>16702381
was grothendieck a closet tranny?

>>16702968
>was an anarchist Jew a tranny?
Is water wet?

Is [math] \mathrm{SO}(2n,\mathbb{R}) \cap \mathrm{GL}(n,\mathbb{C}) [/math] connected?

I know [math] \mathrm{SO}(2n,\mathbb{R}),\mathrm{GL}(n,\mathbb{C}) [/math] individually are connected. But certainly in general the intersection of two connected Lie subgroup (of a connected Lie group) need not be connected.

>>16703513
I think it’s isomorphic to U(n)

>>16703600
Ah I see, thanks anon

I'm starting a masters degree in Applied and Computational Mathematics in the fall. I am excited. I am 35 years old lol.

That is all.

>>16703929
Congrats! I also did grad school a bit later (started my PhD at 28, and now am in candidacy 4 years later). I think it's never too late to do a masters or PhD if you can get back into the mindset of being a student.

>>16693990
That sounds like bullshit
I think chatgpt is confused or is being pedantic
>teehee you didn't say WHICH cartesian coordinate system
>3 orthogonal basis vectors are not a basis for R^4 lmfao

>>16703994
If the vectors are 4 dimensional, you need 4 of them. The dimension is baked into the vectors. The word basis is very precise.

>>16703937
Honestly I love it. My life has never felt better. I don't stress out about the same small bullshit I used to, it kept my mind so pre-occupied and constantly took my focus off school. I see the young 20 year olds going through it now.

I have been taking 4th year classes and some graduate level classes to get back in the swing of things (and to get a few credits off the MSc) and I have never ever done so well in my entire life. I just aced, literally 100%, my graduate probability midterm.

I don't even care if this MSc leads to a job. I'm just so thankful I get to even do this. Working 9-5 in a job I hated was a billion times worse than being a student studying what I love.

>>16684644
When you look into math you stop thinking of math as being about "numbers" and start thinking about it in more abstract contexts.

You can multiply arrows and cubes and a bunch of other things including higher dimensional numbers. (I think) a tensor product is basically a tool that generalizes the concept of multiplying anything.

The term "inverse" is a better descriptor of what a negative number is. Regardless of what it's operating on, an inverse just undoes said operation, and an inverse of an inverse is just the thing before it was inverted.

>>16704074
>(I think) a tensor product is basically a tool that generalizes the concept of multiplying anything.
That's absolutely correct. It's the core of what the determinant is, and not the "multilinear map" explanation.

Hey fags.

I studied math and after 5 years forgot many nuances. I wanna refresh my memory and learn new shit.

I'm looking for a puzzle book ranging all possible math topics (calculus abstract algebra topology geometry probability whatsnew), from easy to very complex puzzles.

Any recommendations?

>>16703996
Yes obviously
But what chatGPT is saying there makes no sense

bump

How can I prove the equivalence between these two constructions of the differential?
1) [math](\mathrm{d}f)_p: X \mapsto (f\circ c)'(0)[/math] where X is a derivation and c is a curve s.t. c'(0) = X and c(0) = p;
2)[math](\mathrm{d}f)_p:X \mapsto X(f)[/math] where [math]X\in T_pM[/math]

>>16704625
This should be a basic undergrad-level proof anon. Show that both are finite-dimensional vector spaces. Then recall if there's any theorem in linear algebra that says whether or not there's mayhaps some convenient isomorphism?

>>16704632
I'm not him, but these are just two functionals acting on [math] T_p M [/math]. There's no need to construct an "isomorphism", we just need to show they both act the same on each [math] X \in T_p M [/math]

>>16704637
functionals are vector spaces. You’re thinking like an analysis fag. I’m thinking like an algebra fag. There’s a non-canonical isomorphism between fin dim vector spaces over the same field if they have the same dimension. You don’t have to prove it every single time.

>>16704639
Both of the definitions in >>16704625 already live in the dual space [math] T_p M ^* [/math]. What are you going to do, prove [math] T_p M ^* [/math] is isomorphic to itself? Lol

>>16704644
The two definitions aren’t equivalent and it’s anon’s question on how to show canonical equivalence. What are you on about?
>We can define it this way
>but we can define it the other way
>Claim: these definitions are equivalent
>(You): of course they are dude! They’re the same space!
Not by the initial assumption. One is using the collection of curves definition, the other is using the derivation definition.

>>16704648
You don't seem to understand what I'm saying. In >>16704625 we have defined two functionals [math] \alpha,\beta \in T_p M^* [/math]. Our goal is to prove that [math]\alpha = \beta[/math].

>>16704651
and, to be clear, you can't show [math] \alpha = \beta [/math] by just "finding an isomorphism."

>>16704651
He's not even talking about the cotangent space, anon.

Jesus Christ is the Son of God, Who died for our sins and rose from the dead to give us the free gift of eternal life. He also promises to heal your body. You can ask Jesus for His gifts.

>>16704657
He's talking about two linear functionals on [math] T_p M [/math], hence two elements of the linear dual [math] (T_p M)^* [/math]. Whether you call it the cotangent space yet is up to you.

>>16704662
Yes, but what I'm trying to say here is that in the first definition, he's using the collection-of-curves definition of T_pM, while in the second definition, he's using a derivation-based definition. The anon who posted it should have explained it better, but what's he's asking is not even showing that these two are equivalent as T^\ast_p M, but rather just showing that the definitions of T_pM are equivalent.

>>16704677
Guy who asked the question here. I know the two definitions are equivalent. For every derivation [math]X[/math] at a point [math]p[/math], I can find a curve [math]\gamma[/math] with [math]\gamma(0) = p[/math] s.t. [math]\gamma'(0) = X[/math].
With this relation between the two definitions, what I want to know is how are the two definitions of the differential equivalent

>>16704677
I agree that two definitions of [math] T_p M [/math] are being used.

I assume the "original" definition is the derivation on functions (at a point) definition.
Then, we can get the "curve" definition as follows: given a smooth curve [math] c(t) [/math] with [math] c(0) = p [/math], we can define [math] c'(0) \in T_p M [/math] as a derivation on functions at p, by [math] (c'(0))(f) := (f \circ c)'(0) [/math].
Of course, with this, >>16704625 is almost tautologically true.

If [math] F \rightarrow E \rightarrow B [/math] is a fiber bundle with F,E,B all topological manifolds (without boundary), and F and E are connected and orientable, then is the base space B orientable?

>>16704762
Nvm, I think I figured out a counterexample: the normal bundle of the Mobius strip in R^3

>>16704766
Are the fiber bundles definitely orientable in that case?

>>16704771
Yes: if I'm not mistaken, the normal bundle on the Mobius strip in [math] R^3 [/math] has fiber [math] \mathbb{R} [/math] and total space homeomorphic to an open solid torus, both of which are connected and orientable

>>16704766
Topology exercise:
Show if [math] F \rightarrow E \rightarrow B [/math] is a fiber bundle, and F,E,B are all topological manifolds,
and F and E are connected and orientable,
and B is simply-connected,
then B is orientable.

Topology exercise:
Show if [math] F \rightarrow E \rightarrow B [/math] is a fiber bundle,
and F,E,B are topological manifolds,
and F,E are both connected and orientable,
and B is simply-connected,
then B is orientable.

Note: I had to use the Serre spectral sequence to show this, but I'm wondering if anyone has a proof without using spectral sequences.

>>16704798
>>16704762
All simply connected manifolds are orientable. This is elementary.

*starts sweating*

How do I prove that the zero morphism is both a left and right annihilator: for all morphisms f we have f0 = 0 and 0f = 0?

>>16704945
How do you define the zero morphism

>>16704946
The unique morphism obtained by composing the morphisms to and from the zero object.

>>16704947
f0 and 0f both factor through the zero object and hence must be 0

>>16704955
Oh, thx. That's very obvious now, but I guess it's just processing the definitions.

Can someone tell me what an equivalence relation mathematically is? Say you start with a set of objects S, and then you define an equivalence relation, so now maybe there are some distinct objects A and B in your set such that A ~ B. Maybe we can do some interesting math, like create some two-variable function F where F(A,C) = F(B,C) if A ~ B.

But what IS the equivalence relation? Is it a partition of the original set S? So A~B means A and B are in the same subset of S?
In the same vein, what is an order relation? Are we dividing up S into subsets like above? Then saying that A < B if the subset A is in is contained in the subset B is in? Or A = B if they're in the same subset?

Like the rule for the relation are simple to understand, but wtf IS it? We're classifying objects within a set, so what else could the relation be if not another set?

>>16705006
Also, would this mean that an equivalence class is also just a set?

>>16705006
>Can someone tell me what an equivalence relation mathematically is?
A relation that satisfies all the same properties as equality.
>Is it a partition of the original set S? So A~B means A and B are in the same subset of S?
Yes, partition of the original set is one way to think about it. You can also think about it as a function f on the original set such that a~b if and only if f(a)=f(b). All of these are equivalent ways of thinking about equivalence classes.

>>16705006
An order is just a way to arrange the elements. By design a set doesn't have any order and there is neither first nor last element. To arrange them you need an extra relation called an order. You place them somehow and order them how you like, for example the list (3,2,1) orders the set {1,2,3}.
>Like the rule for the relation are simple to understand, but wtf IS it? We're classifying objects within a set, so what else could the relation be if not another set?
As for ontology of relations, usually a relation on a set X is conceptualized as a subset of XxX(that is, a a set whose elements belong in XxX), where XxX is the cartesian product of X with X, which is the set of all pairs (a,b) for a,b in X. Any subset is a valid relation. Some relations are functions, others are orders, others are equivalence relations.
For an example, Consider X = {1,2}. Then XxX = {(1,1), (1,2), (2,1), (2,2)} has four elements, hence it has 2^4 = 16 relations, i.e. subsets.
An example of a relation is the empty relation {}. Another one is {(1,1), (2,2)}, which represents the identity function as well as an equivalence relation induced by equality.

>>16705006
Just think of it as a systematic way to partition a set.

>>16705016
>>16705021
thx

>>16705016
>>16705021
>>16705054
thx

>>16704870
I see why simply-connected smooth manifolds are orientable. Is the same true for topological manifolds?

>>16705091
Orientability has to do with top forms, which only use the local homeomorphism property. I don’t see why topological manifolds would be different.

>>16705099
What do you mean by a “top form” on a topological manifold?

>>16705115
Top forms and volume forms can be defined on arbitrary fin dim vector spaces. There’s no assumption of differentiability at all. It’s just a fully antisymmetrized Hom(V,R) tensor of highest non-vanishing rank. Think about how a volume of a parallelepiped can be expressed in terms of vectors.

>>16705132
What vector bundle are you using on a topological manifold without smooth structure?

>>16705091
Yes.
>>16705132
I don't understand. Where do you get vectors from? Topological manifolds don't have tangent spaces.

>>16705142
You can define a bundle on any topological space. The only requirement is local trivilization. And in the case of the tangent bundle, it should trivialize to [math]M\times\mathbb{R}^{\dim M}[/math]. You should think about these basic definitions more and whether each one requires smooth structure. For smooth manifolds, the additional step is endowing the tangent bundle with a smooth atlas induced from the smooth atlas on the base manifold, but this is obviously something you only do for smooth manifolds to make the projection map smooth.

Fun fact, back when I attended Cambridge undergrad lectures on differential geometry the lecturer explained to us a way to define orientation on general topological manifolds. I was confused by the explanation and saw immediate flaws with it but didn't say anything. Next lecture he came and apologized, saying that was a completely wrong explanation and that didn't work at all, and that the real explanation for orientation on topological manifolds was too complex for us to understand.

>>16705143
>Topological manifolds don't have tangent spaces.
Every neighborhood is locally homeomorphic to R^n. You can do a bunch of sheaf-stalk-germ business to define a tangent space at a point without an atlas.

>>16705144
I don't think topological manifolds have a tangent bundle anon.

>>16705150
Can you walk me through the definition? Because that doesn't sound possible to me.

>>16705151
>>16705152
An atlas is literally just a sheaf with some additional gluing requirements (transition maps have to be k-times differentiable). Just drop the differentiability requirements.
https://ncatlab.org/nlab/show/Serre-Swan+theorem

>>16705154
Sheaf of what? What space do you assign to the full manifold? What about the union of two domains for two different charts?

>>16705165
Every topological manifold is, well, a topological space. So it has a topology on it by definition and an “atlas” on a topological manifold is the usual sheaf of open sets. The morphisms are continuous maps, yata-yata.

>>16705143
Thanks anon, could you tell me what book that is?

>>16705172
Spanier, Algebraic Topology

why should we care about a function when it spins clearly bullshit tales like this

>>16705500
What's bullshit about that?

Category Theory is the litmus test.
It's it. This is the thing that determines whether you really "get" math.

You can throw all the bullshit terminology like "fibrous bundle" you want at me, tell me how much you understand this field that tries to go one level of abstraction deeper where all the symbols and all the meaning you give to them get deconstructed and get shown to be less concrete than you thought they were.

Tell me how much you understand Category Theory.
If you can't tell me very much about it and insist that you can't because you haven't read an encyclopedia on it, then I'm afraid you're just a parrot that's spent enough time looking at nlab.

>>16705500
https://www.youtube.com/watch?v=beakj767uG4
feel free to screech like someone that hasn't watched the video or read the paper would
https://arxiv.org/abs/2401.10981

>>16705540
Are you ok, anon?

>>16705559
You do not understand.

It is okay.

>>16705576
What the hell do you want me to understand? You forgot your meds again, honey.

>>16705590
don't mind him, category theory tends to do that to some, sad display, but they tend to get better, although a bit fuzzy on what the hell they where doing during one of their episodes

>>16705626
>dude you’re not thinking about it right. It’s alshyally a (0,infintity) groupoid with an essentially surjective full functor to an exact protomodular category enriched with a daggered frame lattice
>Ok? You’re just regurgitating definitions in your autistic language.
>but dude abstraction lmao

sorry honey, /mg/ is an algroid thread, category theory is cool if and only if it was created by an algebraic geometer

A Fourier series question: if I have a continuous, [math] 2 \pi [/math]-periodic real valued function [math] f [/math] of a real variable, whose trigonometric Fourier coefficients are nonnegative, and are monotonically decreasing, does its trigonometric Fourier series converge to [math] f [/math] in every real point?

Using Fejer's theorem, the Fourier series will converge to [math] f [/math] in the sense of Cesaro, so if the series converges regularly, then it must also be to [math] f [/math]. How would I get regular convergence from the seemingly light conditions on the function? It seems like a pretty strong result, but I'm finding anything on it online. Am I on the right track here?

>>16705540
>Treating empty sets as instantiable objects inside any symbolic manifold.
You forgot to post a clown alongside your joke of a post.

>>16705166
Morphisms are irreducibly triadic, bud.

>>16705882
don't run, bitch >>16706429

How can I get better at doing simple arithmetic in my head without practising?

brainlet here. Why is Goldbach conjecture so hard? What is infinite complexity in context of such problems?

>>16706556
because it’s a random thing pulled out of someone’s ass that never shows up anywhere else. That’s the “difficulty” of 95% of number theory problems. Most people don’t care and would rather just do something more productive like study elliptic curves or Fredholm operators.

noob here. if i want to get into creating algorithms that can translate one language to another instantly, how important is math? How much and what kind of math should I learn?

>>16706745
Brother that problem is already solved using large language models.
So ML math I guess.

>>16706749
no, actually, i want to translate animal speech to human speech, particularly from dolphins and orcas

I'm an art school dropout, I know nothing about math. I'm gonna start learning calculus soon, but after that, im not too sure...

>>16706666
nice quads
what about a method for solving such problems, it might be quite useful.

Does anyone know any books that teach differential geometry needed for (functional) analysis?
I followed an undergraduate course on it, but whenever I read "cotangent bundle" and vector bundle sections my brain just turns off and I have no clue what they're talking about.

I'm not super interested in diff. geometry itself, but more in the language that it uses. That's why I am not just following the thread guide.

>>16706772
Think of how Fermat’s Theorem got solved. A bunch of people were doing completely unrelated stuff and then Wiles cobbled it all together Todd Howard style to prove it. I’m not dissing the guy, he’s a genius, but I don’t know any other field of mathematics that does things this way. The Poincare Theorem had a clear attack strategy behind it with Hamilton’s work on Ricci flow; Perelman’s mastery was in tackling singular behavior via surgery. Completely different vibes.

>>16706776
If you don't want to bother with diff geo, you just need the basic definitions from topology.

A topological fiber bundle is a triplet [math](E,\pi,B,F)[/math], where E, B, and F are topological spaces pi is a cts surjective map from E to B. We call E the total space, B the base space, and F the fiber. For this to be a topological bundle as opposed to just a bundle, you need the additional condition called local trivialization: for all [math]p\in B[/math] there exists a neighborhood U and a homeomorphism [math]\phi:\pi^{-1}(U)\rightarrow U\times F[/math] such that
[eqn]\pi = \phi \circ p_U[/eqn]
where p_U is the product projection onto U (this is a lifting property in the language of category theory). If you're doing things with smooth manifolds, you make the maps smooth, diffeomorphic, etc, but the idea remains the same.

In plain English, the fiber bundle is the following. You have a base space B to every point of which you "attach" some other space F to make the total space E. This is in a way a generalization of the topological product. For example, the Möbius strip cannot be expressed as a product of "simpler" spaces, but it can be express as the total space of a fiber bundle. If you know a bit of group theory, a fiber bundle to a topological product is what a semidirect product is to a direct product in some sense.

Now for the common definitions in diff geo. A vector bundle is where you attach a (topological) vector space at every point. A tangent bundle is you attaching the tangent space to every point of the manifold (which is the base space). The cotangent bundle is you attaching the dual space of the tangent space at every point. Hope that helps!

>>16706926
*quadruplet
started typing with a generic bundle in mind, but those don't show up in diff geo to my knowledge

>>16683534
>>16683551
Can confirm that the answer is to just accept you're a dumbfuck retard. You don't "overcome" or "cast off" the ego so much as you stop resisting it being crushed. The struggle is you struggling with your own grandiosity in the face of contradicting evidence.
Once you give up on your false self-image you'll feel much better and do better work. People will probably like you more, too.

>>16706926
>>16706927
Thanks. However, I don't think this covers everything needed from diff. geo right. That's why I'm looking for possible books that have a more "language" oriented approach than an actual field of study. Any textbooks that are more "intuitive" would also be good I guess.

>>16705817
The periodic extension of [math]f(x)=\pi-x[/math] from 0 to [math]2\pi[/math] has fourier sine coefficients [math]b_n=2/n[/math] but it has jump discontinuities at each [math]2\pi k[/math]

>>16707171
Oh I missed that f has to be continuous. If f is continuous then the fourier series converges at every point, doesn’t matter if it’s monotone or not

>>16707150
>However, I don't think this covers everything needed from diff. geo right
Absolutely.
>That's why I'm looking for possible books that have a more "language" oriented approach than an actual field of study.
ChatGPT, unironically. If you just want to know the terms rather than study the thing, ChatGPT is enough.

Realistically, how much mathematical maturity should I have as an undergrad to undertake functional analysis?

>>16707208
Depends on what you're trying to do. At a full graduate level, you should be quite comfortable with the graduate level real analysis topics you'd find in a measure theoretic real analysis book (e.g., Folland or Axler).

Always struggled with math and never felt confident with it. Any good sites for practicing and perfecting basic every day math or courses something like that?

>>16707312
Unironically Khan Academy. It's got a bunch of exercises and gives instant feedback.
If you feel like you want a book and to cover highschool exhaustively, the Haese Math IB books work fine.

>>16707282
So measure theory is basically a prerequisite for functional analysis? I'd be taking measure theory together with functional analysis actually

>>16707336
>So measure theory is basically a prerequisite for functional analysis?
Lebesgue integration hinges on measure theory. If your space has some additional structure, you need a measure compatible with that structure for integrals to be invariant quantities. For example, locally compact topological groups have the (left and right) Haar measure on them. You then want to define differentiation on these abstract spaces to have reflect the usual notions from real analysis: differentiability implies continuity, some notion of the fundamental theorem of calculus on that space, etc etc. These require you to define derivatives via measures. This is all covered in Papa Rudin.

>>16703929
Neat. I would like to study full time again, and slam some college age pussy for the first time

What is so "freeing" about a module that is free? Why not basisable or baseable or basable or baseble?

>>16707547
They're "free" of relationships among their generators.
This is the same reasoning that applies to basically any structure called "free"

>>16707547
I think you mean based, anon.

>I want to do pure mathematics for my master's degree.
>The professor tells me that there is a good scholarship to a leading university in Europe.
>I assumed it was for pure mathematics, I applied for the scholarship without reading the program in full.
>I am accepted into the master's program.
>The master's program is for applied mathematics.
>All my undergraduate degree was in pure mathematics, basically no programming or statistics.

Should I accept it? Even people who did a lot of applied math told me that the program is incredibly difficult, and I feel like I have zero chance, since all my background is in theoretical stuff. I haven't touched a programming language in like 4 years.

>>16707569
Sounds more like hikikomori or incels. Should've called em loner or incelibate module. Free module sounds like they have the right to bear arms.

>>16707611
If it's a master's, but you chug through it, but then you're sorta killing both your academic career (you won't manage a PhD if you don't like applied) and your industry career (a master's in math basically lands you hs teacher job at some private school at most).

Is there an example of a continuous map [math] f : X \rightarrow Y [/math] between some topological spaces X,Y, such that:
X is connected and Y is Hausdorff, and
f is continuous and surjective, but is not a topological quotient map?

>>16707749
Nvm I figured out an example. But it's a good exercise if you haven't seen this before

>>16707749
>>16707762
Here's a harder version though, and I'm not sure what the answer is:

Is there an example of a continuous map [math] f:X\rightarrow Y [/math] between some topological spaces X,Y, such that:
X is connected, and Y is *contractible* and Hausdorff, and
f is continuous and surjective, but is not a topological quotient map?

>>16707762
Here's a harder version though:

Is there an example of a continuous map [math] f : X\rightarrow Y [/math] between some topological spaces X,Y, such that:
X is connected and Y is *contractible* and Hausdorff, and
f is continuous and surjective but is not a topological quotient map?

Pretty sure this is a simple question
For the top pic, tangent vectors of [math] \mathbb{R} [/math] are supposed to be objects like [math] a \tfrac{\partial}{\partial t} \in T_p\mathbb{R} [/math], and tangent vectors of [math] M [/math] are supposed to be like [math] \sum b^i \tfrac{\partial}{\partial x^i} \in T_pM [/math]. But "canonical" time derivative, the author says, is defined to be the pushforward differential applied to the tangent vector "1", so [math] \dot\gamma(t) := D\gamma(1) [/math]. This is confusing because to me it'd make more sense for it to be [math] \dot\gamma(t) := D\gamma( \tfrac{\partial}{\partial t} ) = \sum \tfrac{d\gamma^i}{dt} \tfrac{\partial}{\partial x^i} [/math], cause at least now it's working on an actual tangent vector. Now, the "obvious" reason might be that he's just referring the tangent vector 1 means the coordinate 1 for basis [math] \tfrac{\partial}{\partial t} [/math], as there's an objious bijection there, but the guy never mentions a damn thing about this, nor has he ever done it before!

Similarly, for the lower pic, normally is should be that [math] Df(X)(g) = X(g \circ f) [/math], where [math] g : \mathbb{R} \rightarrow \mathbb{R} [/math]. To get close to what he wrote, it would work when [math] g(x) = x [/math], so something like [math] Df(X)(1) = X(1 \circ f) [/math], but except for the problem mentioned above, the author never talks about this intention or wtv they actually mean.

Outside of these two situations, he's been fairly straightforward about the notation. Is he really using "shorthand" for the 1-vector in the top and the 1-function (identity) in the bottom?

thx

>>16708400
Oh, also for the second one, it looks like that in this specific situation only, we can move the g to the outside, and the equation works in the sense that normally, [math] df(X)(g) = X(f) * \tfrac{\partial g}{\partial t} [/math] or [math] df(X) = X(f) * \tfrac{\partial}{\partial t} [/math]. So AGAIN, is the dude actually just shortening the vector [math] \tfrac{\partial}{\partial t} [/math] to be "1"? I don't understand why he's doing this other than pointing out the bijection

>>16708400
I agree the notation "1" shouldn't be used for the tangent vector, if you're treating tangent vectors as differential operators / derivations. It should just be called [math] \partial / \partial t [/math], where t is understood to be the "standard" coordinate on [math] \mathbb{R} [/math]

>>16683324
I think the Federer book is usually the go-to. I haven’t read it but apparently there’s a bit of a learning curve associated with it. Kinda sounds like fun ngl

>>16683534
>How does one learn to cast off one's ego in mathematics?
I guess the “””strategy””” that has always worked for me is to simply treat mathematics (and anything else i approach) the way a child does when first encountering a new toy/object. It’s not about ego at all, it’s about curiosity and satisfaction of learning and’s discovery being its own reward. This is what makes truly great Mathematicians/Scientists/Engineers. Ego and petty competition is for people who are mentally stuck in High school and feel the need to look cool in front of each other, even if it comes at the cost of their own growth and success. Fuck what other people think. They will never achieve anything and they are to be ignored

Yang Mills or Navier Stokes?

>>16709496
What the fuck are you on about? They’re entirely different equations.

>>16683009 (OP)
Any books/lecture notes/resources you bros can recommend on automata theory?

If [math]\varphi:\mathbb{R}^n\to U\subset M[/math] is a parametrization on a manifold [math]M[/math], and [math]\omega[/math] is a differential form, how should I interpret [math]\varphi^*\omega[/math]?

>>16710149
As the pullback of the differential form to R^n? Desu, that’s just “coordinatizing” the form. So a formal way of writing
A = A_mu dx^mu

>>16710155
So it'd be, basically [math]\sum_I(\omega_I\circ\varphi)\varphi^*\mathrm{d}x^I[/math]? But I'm unsure on how to see the pull back of [math]\mathrm{d}x^I[/math]

>>16710173
>the pullback of dx^I
That lives in the chart, anon. You pull back things in the codomain, not the domain. You have a point on the manifold that lives in the neighborhood U, you have a differential form evaluated at that point and when you pull it back to R^n, you essentially assign some coordinate system to the whole thing.

>>16709993
second this

last bump

>>>/adv/33295090
>I'm the fuckin booogeyman over at /math/.
Is this one of yours?

>>16711080
he be wildin

Bros, I really need your help.
Long story short, I am very bad at making a decision. But I have two possibilities to do my masters.
1) First, a uni in Europe, but I don't really like the program as it includes a lot of computer science stuff. There is a possibility to get into PDEs but it is incredibly competitive.
2) A uni in Brazil. I really like the program here as it is mostly pure mathematics (algebra, topology and analysis).
The European university is Paris Saclay, which has an incredible reputation. All my professors tell me that it is the right choice, as it is easier to find a job if you get a degree from that university.
On the other hand, the institute in Brazil is IMPA, which has great prestige in Latin America, but it is nowhere near the level of Paris-Saclay.
Does the country of the university matter when it comes to getting a job?
Does the ranking matter? In other words, is the Shanghai ranking really the determining factor in getting a place in academia?

If [math] f\in C^\infty(S \subseteq \mathbb{R}^n \rightarrow \mathbb{R}) = F [/math] is a smooth function from set S to R, what is the dual to the set F? Would it be the set of all all [math] \displaystyle g = \sum_{i=1}^{\dim S} \sum_{j=1}^{\dim S} \sum_{n=0}^{\infty} \int dx^i \delta_i(x^i - p^i) \hspace{1pt} a_n^j \left( \tfrac{\partial}{\partial x^j} \right)^n \in G [/math] with coordinates a_n^j? If not, what is F*, and what would G be called if anything?

If Collatz conjecture can be implemented with cellular automaton and according to Rice's Theorem cellular automata are undecidable, does it mean that Collatz is undecidable as well?

>>16711367
Well if you go to IMPA and do well I assume you could be in academia there if that is what you want in pure maths. On the other hand in terms of jobs of course CompSci is better especially from a prestigious uni.

You could do a PhD in Europe afterwards in a topic more intresting to you but pure maths would probably be out of reach. Just my two cents

New
>>16712226
>>16712226
>>16712226

>>16689752
Omg, how pretty!
Who could have drawn something this pretty?

>>16709993
Don't know, but if you are going for the Ullman--Hopcroft book, make sure you get an edition without the jeet. He removed a lot of things which were—I assume—too difficult for IIT students.