/mg/ math general - Empirical Mathematics Edition

[Anonymous No.9549094]

Empirical Mathematics Edition [TQFT, String Theory, AQFT over exotic spacetimes, etc.]

NO Grothendieck or Serre.

What have you been studying, /mg/?

[Anonymous No.9549118]

reported

[Anonymous No.9549123]

>>9549094
where the fuck do i start with number theory? elementary, analytic or algebraic? getting sick of the attacks on /mg/

[Anonymous No.9549126]

>>9549094
>What have you been studying, /mg/?
I've been studying French to start reading Grothendieck's ETQFT.

[Anonymous No.9549140]

undergrad math major here. I can understand how to "do problems" but I'm having trouble with what it all means. I can't help but feel that there's always more out there that I can't grasp, and that I'm only scratching the surface of the surface of math. Anyone else relate?

[Anonymous No.9549208]

In Spivak's Calculus, chapter 1 problem 4 (v-viii), perhaps even more, how does the author expect you to prove inequalities such as:

>(v)[math] x^2-2x+2 > 0 [/math]
>(vi) [math] x^2+x+1 > 2 [/math]
>(vii) [math] x^2 -x + 10 >16 [/math]

I understand these are "completing the square" and quadratic equation problems, but how am I supposed to derives this myself given only the properties in pic related?

The only solutions I've seen involve so much creativity it seems infeasible:
>https://math.stackexchange.com/questions/1878298/spivak-calculus-chapter-1-question-4-6

Here is the top answer from said link, regarding question vi (I'm praying to god the tex works out):
[math]
x^2+x+1&>2 & \text{Given}\\
x^2+x+1+0&>2+0 & \text{By Addition}\\
x^2+x+1+0&>2 & \text{By P2}\\
x^2+x+0+1&>2 & \text{By P4}\\
x^2+x+\left( \frac{1}{2} \right)^2+(-1)\left( \frac{1}{2} \right)^2+1 &>2 & \text{By P3}\\
\left(x+\frac{1}{2}\right)\left(x+\frac{1}{2}\right)+(-1)\left( \frac{1}{2} \right)^2+1 &>2 & \text{By P9}\\
\left(x+\frac{1}{2}\right)\left(x+\frac{1}{2}\right)+ (-1)\left( \frac{1}{4} \right) + 1 &> 2 & \text{By Multiplication}\\
\left(x+\frac{1}{2}\right)\left(x+\frac{1}{2}\right) &> \left( \frac{5}{4} \right) & \text{By Addition, P3, and P2}\\
[/math]

In this example, it seems infeasible to be expected to see the possibility of [math]-1(\frac{1}{2})^2 + \frac{1}{2}[/math]

I understand these questions are intended to be difficult, but I'd like to leave no stone unturned throughout this book.

[Anonymous No.9549215]

>>9549208
Tex did not work out, here's a lazy screenshot:


The answer ends up being [math] x > \frac{-1 + sqr{5}}{2} /math] or [math] x < \frac{-1 + sqr{5}}{2} /math]

[Anonymous No.9549221]

>>9549215
>[math] x > \frac{-1 + sqrt{5}}{2} /math] or [math] x < \frac{-1 + sqrt{5}}{2} [/math]

[Anonymous No.9549226]

which fucking thread is the real one. fucking autists

[Anonymous No.9549231]

>>9549208
>(vii) x2−x+10>16

This should've been
>[math] x^2 +x + 1 > 0 [/math]

[Anonymous No.9549236]

>>9549226
well this one is the only one with math so far

[Anonymous No.9549280]

>>9549221
>>9549215
>>9549231
>>9549208
Nevermind, I'm getting it. I never really understood what completing the square really was, but finding a geometric example was insanely helpful. Turns out the 1/2^2 I complained about wasn't expected to be found by an arbitrary stroke of luck, but comes straight out as a result of completing the square

[Anonymous No.9549326]

When solving an equation or inequality, at any given point you have plenty of options, each of which may lead to a new assortment of options.

Are there any heuristics involved to determine which options are generally more productive?

One example might be above, the inequality [math] x^2 + x + 1 > 2 [/math], and it's partial solution, here: >>9549215.

My first reaction upon trying this problem was to simply manipulate it into something like this:
>[math] x^2 + x > 1 [/math]
>[math] x(x+ 1) > 1, x \neq -1 [/math]
And from here, it seems I've manipulated into a much more troublesome problem than what it was when I started. Often it seems the slickest solution is the path down a binary-tree-like structure, and I often find myself lost amid the branches, unsure which node is most promising. Do these kinds of issues lessen with experience? Clearly, a more experienced mathematician would instantly recognize that the above was a square looking to be completed, so it that the basis of my inadequacies? Of course, practice always helps, but I'd still be any related material or heuristics for improving my problem solving of this nature,

[Anonymous No.9549341]

>>9549326
>the slickest solution is the path down a binary-tree-like structure
I meant to say it's a select set of the possible paths down said structure, hopefully the analogy and intended meaning was clear.

[Anonymous No.9549421]

>>9549123
I think analytic is the best place to start. Algebraic number theory tends to use rather heavy machinery to do anything interesting, and elementary number theory is either stupid toy problems or special cases of respectable theorems done in ad hoc ways.
Analytic number theory lets you get to pretty cool stuff (like the PNT or Dirichlet's theorem) pretty quickly with only some undergrad tools.
Just don't go too deep or you'll become an unsalvageable epsilon-muncher

[Anonymous No.9549442]

>>9549421
>Analytic number theory lets you get to pretty cool stuff (like the PNT or Dirichlet's theorem) pretty quickly with only some undergrad tools.
So in other words one shouldn't waste any time on it?

[Anonymous No.9549450]

>>9549421
Thanks, I was leaning towards analytic anyway since Apostol wrote a book on it. Do I need to visit elementary at all if I've already worked through a couple discrete math/combinatorics/proof text that covered (what I assume to be) similar problems? Or will I understand elementary as I work through analytic/algebraic? And need I visit algebraic after analyitc?

[Anonymous No.9549493]

>[math] x^2 - 2x + 2 > 0 [/math]
My solution
>[math] (x-1)^2 > -1} [/math]
Because the exponent is odd, the left hand side of the equation will always be positive, and so any value in the set of reals will work for x.

But the solution here, https://etoix.wordpress.com/category/calculus-by-spivak/page/2/ , has a different answer. How did they arrive at that? Pic related. Ultimately we both had the same final answers, all reals, but I'm curious as to how I may have arrived at the same answer differently. My method for arriving at this answer was a straightforward completion of the square. It looks as though this person used a similar method, but I don't understand how they could've done anything differently.

[Anonymous No.9549527]

>>9549493
They essentially did the same thing as you but took the square root of both sides.

[Anonymous No.9549540]

>>9549450
You should know your elementary number theory (congruences, bezout, chinese lemma, prime factorization, valuations, etc.) since it is the only one that will prove useful whatever math you end up doing. Moreover, it is obviously a prerequisite for any other type of number theory and it's really easy to learn.
And, even if you end up doing analytic stuff, you might need to know some of the algebraic/geometric theory in some areas (eg automorphic forms)

[Anonymous No.9549629]

>>9549280
No one cares.

[Anonymous No.9549655]

>>9549493
Note that the function f(x) = x^2 - 2x + 2 is continuous and f(0) = 2. If, at any point c, f(c) <= 0, then by the intermediate value theorem we must have another point d with f(d) = 0. But there is no such point, since f(x) = 0 implies x is in C \ R. Thus there is no point c with f(c) <= 0.

[Anonymous No.9549678]

>>9549493
>>9549655

Jesus Christ what are all these autistic solutions? Just do [math] x^2 - 2x + 2 = (x-1)^2 + 1 [/math] which is obviously always positive.

[Anonymous No.9549745]

Is this proof right?

Theorem: The set [math]V[/math] of n-tuples of orthonormal vectors of [math]\mathbb{R}^{n+k}[/math] is a closed subspace in [math]\underbrace{S^{n+k-1}\times...\times S^{n+k-1}}_{n \;\text{times}}[/math].

Proof: The functions[eqn] \Psi_{ij}:S^{n+k-1}\times...\times S^{n+k-1}\to \mathbb{R}[/eqn] defined by [math]\Psi_{ij}(x_1,\ldots,x_i,\ldots,x_j,\ldots,x_n)= x_i\cdot x_j[/math] are continuous, for [math]i,j\in\{1,...,n\}[/math] and [math]i\neq j[/math], with [math]x_i\cdot x_j[/math] the standard inner product. Hence, the set[math] \bigcup_{i}\bigcap_{j}\Psi_{ij}^{-1}(0)[/math] is closed and it is precisely the set [math]V[/math]. Q.E.D.


The main worry I have is that the set defined from [math]\Psi_{ij}[/math] isn't actually [math]V[/math].

[Anonymous No.9549748]

>>9549527
Can you expand on this? I never took the square root. Here's my step by step process:
>[math] x^2 -2x +2 > 0 [/math]
Complete the square, of the form [math] ax^2 + bx + c [/math]
>[math] [x^2 -2x + 1] - 1 + 2 > 0 [/math]
Factor and rearrange
>[math] (x-1)(x-1) > -1 [/math]
And thus
>[math] (x-1)^2 > -1 [/math]
And it is clear that this will always be positive, for all reals.

>>9549655
Thanks. However I was really trying to prove it by only working within the axioms given.

>>9549629
I didn't want someone to waste their time explaining it, faggot. Doing otherwise would discourteous and selfish.

>>9549678
That is the solution I came to, as stated in my original post. I'm asking how the other person arrived at their answer.

[Anonymous No.9549751]

>>9549748
>faggot
Why the homophobia?

[Anonymous No.9549763]

>>9549748
They used the quadratic equation, which is the same as completing the square and then taking the root. They proved the original statement by using the fact that [math]f(x) = x^2 - 2x + 2[/math] is concave up and that [math]f(x) = 0[/math] has no real solutions therefore [math]f(x) > 0[/math]. Basically taking an analytic approach over an algebraic approach.

[Anonymous No.9549767]

>>9549751
why the faggotry?

[Anonymous No.9549768]

How would you convince someone of the joys of mathematics? This particular person comes from a standard, run of the mill USA background: little exposure to trig, no exposure to any maths other than that of a tedious computational nature (and the accompanying lack of rigor), and, given the circumstances, this person carries an appropriate disdainful attitude towards the subject.

I was thinking Stillwell's "Elements of Mathematics" would be a good fit, and maybe "What is Mathematics?" by Courant. I don't want sources that 'teach' or 'drill' necessarily - really I want the opposite. Things that just might initiate an interest. Any videos, resources, etc would all be welcome.

>pic semi-related

[Anonymous No.9549771]

>>9549748
>>9549763
To clarify, the quadratic equation was used to come up with [math]f(x) = 0[/math] having no real solutions.

[Anonymous No.9549776]

>>9549763
It all makes sense now, thank you.

[Anonymous No.9549779]

>>9549745
just define [math]\Psi \colon V \to \text{Mat}_{n+k}(\mathbb{R})[/math] sending each [math](n+k)[/math]-tuple to the Gram matrix. then [math]V = \Psi^{-1}(I)[/math].

[Anonymous No.9549782]

>>9549779
But is what i did right or not?

[Anonymous No.9549787]

>>9549782
I don't know, you tell me

[Anonymous No.9549824]

>>9549768
Give him a #1 Olympiad problem, an hour, and promise a blowjob as a reward if he gets it right.

[Anonymous No.9549836]

>>9549824
He's too far away for a bj in a reasonable amount of time. Plus he's not gay, and I don't think he's into incest.

[Anonymous No.9549844]

>>9549540
Does Apostol's Analytic Number Theory cover elementary as well?

[Anonymous No.9549849]

>>9549836
He sounds like a fag.

[Anonymous No.9549854]

>>9549849
Would you recommend this then?

[Anonymous No.9549861]

>>9549844
Not the guy you are talking to but Apostol is my bible. Apostol is a weird beast. In his first chapter, he presents the basics of divisibility theory (divisors, gcd, factorization, etc.) and just that. This is technically all you need for the next 3 chapters, which are 100% calculus. Then on the 5th chapter, he goes back to elementary number theory, this time for congruences. He covers everything you need to know about congruences. After this he goes back to calculus, but he actually sprinkles a ton of really heavy algebra (at least for an undergrad) aswell so be ready for that. Then finally after getting ass-raped by algebra and calculus, again in chapter 9 Papa Apostol goes back to elementary number theory. This time for quadratic reciprocity (quick hacks to solve quadratic congruences). Then he moves on to primitive roots, which is also elementary number theory. This is the last he will touch of elementary number theory, and it is, in my opinion, everything that counts as elementary number theory (except for diophantine equations, but every now and then he covers a little bit of that). Even stuff like primitive roots is slightly pushing it in the direction of algebra, but primitive roots were in my elem. number theory course so it counts.

So Apostol will teach you elementary number theory but I can tell you this much: He covers those topics as if you were a 500 IQ grad-student who needs no explanation for anything, or if you already know everything but just need a quick refresher in preparation for his problems (which count as statutory rape in my state). He mentions in his introduction that his book is indeed for grad students and very advanced undergraduate students. If you are looking to be able to solve his problems, you will end up severely depressed (unless you have 500 IQ) as you will barely be able to solve his first problem, and very likely be considering suicide by his #10.

[Anonymous No.9549868]

>>9549861
You sold me. I'm not that anon but I am downloading the book right now.

[Anonymous No.9549881]

>>9549868
That's ok. If you are an advanced student then definitely go for it, it is a really fun ride. I should mention that while some of his problems are standard olympiad-level stuff, some others are full of research problems. In chapter 2 there is a problem that is literally a result that was published back in like the 1900's. And other problems are so weird you will probably need math software to crunch some numbers in order to find out what the deal is with the problem.

After chapter 2 I personally had to re-read the various chapters multiple times before I could actually understand Apostol enough to tackle his problems. If you are going to push through I recommend having advanced knowledge of:

>Inequalities (at the analysis level, not petty high school shit)
>Integration
>Linear Algebra
>Algebra
>Combinatorics
>Creativity

For when you give up, there is a blog online with all the solutions to all of his problems. Have fun.

[Anonymous No.9549883]

>>9549881
not that guy but
>solutions to all the problems exist
thanks i can just read the book and pretend like i wouldve gotten the answers by myself eventually after looking at the solutions

thanks now i have 500iq

[Anonymous No.9549884]

>>9549883
Reading a solution only gives +0.001 IQ. Coming up with a solution gives +0.1 IQ.

[Anonymous No.9549886]

>>9549861
this is top quality sci right here

thanks for the recommendation my dude

[Anonymous No.9549891]

>>9549861
thanks anon, 'preciate it. have you read any text in algebraic number theory that you might endorse?

[Anonymous No.9549906]

>>9549891
No, I haven't touched anything in that direction except for the quick rundown Apostol gives in his book.

[Anonymous No.9549921]

>>9549123
>where the fuck do i start with number theory? elementary, analytic or algebraic? getting sick of the attacks on /mg/
Ireland and Rosen

[Anonymous No.9550196]

Theres a lot of arithmetic, algebra and geometry books. Can you recommend any computation, combinatorics and logic books?

[Anonymous No.9550200]

>/mg/ - Book Recomendations & /sqt/

[Anonymous No.9550205]

>>9550200
More like
>/mg/ - /b/ but for math

[Anonymous No.9550225]

>>9549123
Classical then whichever camp you like more (protip analytical is the bestest)

[Anonymous No.9550228]

>>9550196
http://4chan-science.wikia.com/wiki/Mathematics#Primers_in_Combinatorics_and_Graph_Theory
http://4chan-science.wikia.com/wiki/Mathematics#Proofs_and_Mathematical_Reasoning
http://4chan-science.wikia.com/wiki/Mathematics#Introductory_Logic
http://4chan-science.wikia.com/wiki/Computer_Science_and_Engineering#Automata.2C_Computability_Theory.2C_and_Complexity_Theory

[Anonymous No.9550234]

>>9549094
cute marisa

[Anonymous No.9550252]

>>9550228
don't do that

[Anonymous No.9550255]

>>9550225
Ok. I'm probably going with Ireland and Rosen then Apostal, based off what I've read here and elsewhere. Thanks, I'll come back for some algebraic down the road

[Anonymous No.9550338]

>>9550255
>Ok. I'm probably going with Ireland and Rosen
Make sure you get the most recent edition, it's been heavily updated

[Anonymous No.9550357]

I have Kaczynski's dissertation. I am not going into its substance as such, but I am studying its layout. Pic related is a diagonalization argument figure presented on page 27.

The body of the text is 75 pages with constant lemma/theorem proving and occasional comments. It's straight, terse business throughout, basically.

25 lemmas and 10 theorems are proven, so the business averages a result proven every two pages or so.

[Anonymous No.9550378]

>>9550357
Why did he show the diagonalization argument? Are these his own personal proofs or is he just repeating other people's shit?

[Anonymous No.9550398]

Am I doomed because I cant get Lang's Algebra?
Im still babby tier starting 2nd semester now and I gotta do a seminar about the first like 30 pages.
I understood it pretty well up to factor groups but then he started listing theorems with literally nothing more than bare bone proof sketches.
He doesnt write functions in his graphs, he doesnt write down Isomorphisms, how the fuck am I supposed to understand this.
Now he introduced subset towers and Im pretty much lost.
I cant develop any intuition for how factor groups and normal subgroups behave. I just dont know what's "allowed" and not with the notation.

[Anonymous No.9550399]

>>9550398
Lang is a meme. Try a different algebra book.

[Anonymous No.9550400]

>>9550378

In the context of the dissertaion, Kaczynski uses a form of diagonalization argument in the service of a particular lemma, which has to do with complex analysis, topology, and geometry. That much is perfectly clear. Admittedly, I haven't checked details because those are over my head, but I have enough education to get the general thrust. Your statement, "THE diagonalization argument" suggests to me that you think that that K is just aping Cantor https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument , when I instead simply wanted to post a fun pic from the dissertation for some color.

The thing itself also contains several results which are really his, as stated, and also contains several things which are slight modifications or else fully-cited repetitions of previous results. "Repeating other people's shit", when done properly and with the right context, is also known as "academic research", but you don't know that.

[Anonymous No.9550403]

>>9550378
His main work was in complex analysis. You can trivially think of a bijection from R^2 to C (a+bi is equivalent to (a,b)). I suppose the diagonalization argument was to then form a bijection from R to R^2 (you can akshually algebraically prove a bijection from R to any R^n, but this diagonalization argument gives us a neat visualization for the specific case of R -> R^2).

[Anonymous No.9550448]

>>9550399
what is that supposed to mean

[Anonymous No.9550453]

>>9550448
>what is that supposed to mean
It means you should read something else.

[Anonymous No.9550467]

>>9549140
That feeling comes and goes, just keep going.

[Anonymous No.9550529]

>>9549787
It was wrong, I forgot that I needed a union over all the possible [math]x[/math] in my space and that wouldnt have been necessarily closed. Thanks bud

[Anonymous No.9550550]

>>9550398
If you've never done any algebra it would be stupid to learn from Lang. Try Dummit Foote

[Anonymous No.9550568]

>>9550453
>>9550550
ok thanks for the advise.
I just got the book because it was listed as reference material.

[Anonymous No.9551229]

From Spivak's Callculus 1.1.5:
>[math] (x^2 + X^{n-1}y +...+x^2 y^{n-2} + y^{n-1}x) - (yx^{n-1} - y^2x^{n-2} -...- xy^{n-1} - y^n) [/math]

Which readily simplifies to:
>[math] x^n + x^2y^{n-2} - x^{n-2}y^2 + y^n [/math]

But the answer is supposed to be [math] x^n + y^n [/math], and I'm failing to see how to simplify further. Is it something to do with the expansion in the ellipses? I'm having the same problem this guy is: https://math.stackexchange.com/questions/580397/spivak-calculus-chapter-1-problem-1v


It seems the top answer by 'oks' simply continued the expansion for one more step, is this the solution?

[Anonymous No.9551246]

>>9551229
The original problem is detailed in the link provided, please check that so I don't have to type tex for 15 minutes. It should be very easy to tell, I'm just too skeptical to assume that and move on without confirmation.

My original attempt was done identically to oks', except I didn't add [math] x^{n-3}y^2 [/math] and such because as you can see in the OP it wasn't there. Is the only honest solution to go one more step down the expansion like oks did?

[Anonymous No.9551250]

>>9551229
>Which readily simplifies to:
>>xn+x2yn−2−xn−2y2+yn
That's not true.

[Anonymous No.9551252]

>>9551246
>In the OP it wasn't there.
But it is.

[Anonymous No.9551264]

>>9551252
It was in the expansion, is that what you're saying? And I was supposed to poke around in there?

[Anonymous No.9551293]

>>9551252
>>9551264
I mean there's obviously a symmetry there between the terms and cancellations that I understand would reduce to x^n - y^n, but how do 'prove' that that? What if the problem wasn't set up so that the next term didn't fix everything? It seems the top answer on the link just added one more explicit term and then everything canceled smoothly, but perhaps in a future situation it may not conveniently be the next term in line that evens everything out.

No summation notation allowed because that hasn't been introduced yet. Excuse my blabbing, just trying to make sure there aren't mistakes in my thoughts because I feel there are.

[Anonymous No.9551454]

What is mathematical maturity, and how does one gain it?

[Anonymous No.9551465]

Reminder that string theory is not physics.

[Anonymous No.9551553]

>>9551454
Basically, do that opposite of this guy
>>9551465

[Anonymous No.9551556]

>>9549094
>studying
Been out of school for about 4 years.
Trying to grind out math from basic algebra all way back to calc.
So far I'm up to college algebra. Shit is going to take me some time.

[Anonymous No.9551584]

>>9551454
it's a measure of the ease of which you can work through, digest and retain new and/or difficult text.

you have to work through, digest and retain a lot of new and rigorous text

[Anonymous No.9551588]

Nice job anon, It's worth it, keep it up.

[Anonymous No.9551599]

>>9551584
>>9551454
also, see:
>https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/
https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/

basically it's a measure of the length and intensity that your nose has been at the mathematical grindstone

[Anonymous No.9551613]

Do we have a definition of n-category that allows us to compute coherence laws for any n?

[Anonymous No.9551673]

>>9551599
Thanks, so I just gain mathematical maturity by just doing it.

[Anonymous No.9551709]

>>9551673
basically, but you can't be doing it mindlessly and expecting solid results. while drilling is great, you should also emphasize looking for patterns and how the pieces connect and you should always be working toward gaining deeper insight:

>http://calnewport.com/blog/2008/11/14/how-to-ace-calculus-the-art-of-doing-well-in-technical-courses/

[Anonymous No.9551713]

>>9551673
>>9551709

and I'd definitely try to read text that demand a more mathematically mature reader. Compare Spivak to Stewart in their expositions on Calculus and you'll see what I mean.

[Anonymous No.9551758]

>>9551613
It's still an open problem for n = 1.

[Anonymous No.9551985]

>>9551758
...why

[Anonymous No.9552039]

>>9551985
It's hard. Cubical TT is getting there but still unable to fully generalize to HoTT - which is only for n = 1 from my understanding. Unless I'm not getting what you mean by "compute coherence laws".

I also don't really know what I'm talking about, I'm just a filthy type theorist and category theory is mostly incomprehensible for me. So take what I say with a lot of salt.

[Anonymous No.9552046]

>>9552039
ok...I thought you were trolling lol. There are no coherence laws for 1-categories. I'm talking about things like the pentagon identity. It's extremely hard to work out what they are supposed to be in higher dimensions, but I thought HoTT or something might help with that. Unless I'm mistaken we only know up to n = 3 or 4.

https://ncatlab.org/nlab/show/coherence+law

[Anonymous No.9552073]

>>9549126
>Geothendieck
So sick of this meme. Etale orbifolds can be topologized without reference to topos.

[Anonymous No.9552109]

>>9552046
My bad, I was thinking of [math](\infty, n)[/math]-categories. I'm definitely not clear on exactly the differences between all the different higher categories. HoTT handles higher dimensions as just higher homotopies, which are all equivalences. Or in cubical TT, hypercubes. I've seen the pentagon law done for n = 2, and I imagine one could theoretically pick any n and do it for that. But I'm not sure how you would go about doing it in one shot for any n, if it's possible in HoTT at all.

[Anonymous No.9552121]

>>9549094
>What have you been studying, /mg/?

Lately, synthetic geometry and the physics of spacetime

[Anonymous No.9552159]

>Prove that the sequence 10^-n converges to 0
>Prove that sqrt(1+(sqrt(1+... = golden ratio
>Prove that the bounded sequence [math]x_n[/math] such that [math](2 - x_n)(x_n+1) = 1[/math] converges to 1
>Prove that the sequence a_n = (1/n^2)([x] + [2x] + ... [nx]) converges to x/2

What
the
fuck

I'm going to start a math major in like a week and these are some of the problems the last weeks of intro to calculus. I'm really good at self-studying, I've studied a lot of undergraduate stuff these months and I've managed to master a lot of exercises from books (Spivak, Herstein, Enderton)

But I feel like a fucking brainlet whenever I check sequence related stuff. I can do convergence tests and proofs related to infinite series but I can't prove a fucking sequence convergence. Heck, I'm even able to prove some functions' limits through epsilon-delta

So, any help? What's a good book that'll help me prove sequence theorems/exercises? Rudin is too topologic, Spivak is too simple, Stewart doesn't even feature proofs.

also... am I panicking too soon for not being able to do these problems?

[Anonymous No.9552169]

>>9552159
the third one should have a [math]x_{n+1}[/math] as the second factor

[Anonymous No.9552199]

>>9552159
>am I panicking too soon for not being able to do these problems?

yes. that's what your math major is for.

[Anonymous No.9552281]

>Doing my assignment in the library today
>Guy next to me stares at my problem sheet for a solid minute then looks at me like some kind of freak

Does a math career guarantee no social life?

[Anonymous No.9552284]

I've read (cover to cover) texts in precalc, calc, discrete maths/combinatorics, linalg and analysis yet I still get stumped ju stupid shit like quadratic equations, factoring, trig, algebra and a long list of similar issues. How do I repair these deficiencies?

[Anonymous No.9552290]

>>9552281
stop humble bragging faggot

[Anonymous No.9552293]

>>9552281
>Guy
Did you fuck?

[Anonymous No.9552294]

>>9552284
do the exercises

[Anonymous No.9552306]

>>9552294
I have, as should go without saying. Replies like yours bug me deep down because I have to solve a captcha to say something that should go without saying. How about you either recommend something given my criteria or just call me a brainlet? I'd rather be insulted than solve a captcha for this nonsense

[Anonymous No.9552317]

>>9552281
>things that never happened

[Anonymous No.9552340]

It's like with a language. You can read the book, learn the vocab and grammar, do well on the test, but if you're not using it constantly it fades to dust.

[Anonymous No.9552344]

>>9552340
>>9552306
my bad meant to (you) you

[Anonymous No.9552357]

>>9552340
I do use it constantly. I mean I don't use the particular problems I have constantly, but maybe I should start incorporating some drills everyday with problems that stress my issues rather than just relearning it after I fuck up then proceeding to work through the text. Obvious in hindsigiggt, but this still leaves the question of where I will find a text resource with the kind of problem repository I desire

[Anonymous No.9552379]

>>9552357
Are you focusing on concepts or problem solving? Exercises are only good to entrench or illustrate the concept.

[Anonymous No.9552392]

>>9551553
>Basically, do that opposite of this guy
I'm not a "guy".

[Anonymous No.9552396]

>>9552290
>faggot
Why the homophobia?

[Anonymous No.9552422]

>>9552379
Both, they aren't mutually exclusive. Though it's not as if I'm preparing for Olympiad's or anything. This struck me as a silly question, I hope I'm not talking to a Stewart's PreCalculus know it all.

[Anonymous No.9552428]

>>9552396
Why the faggotry and stale, unfunny meme?

[Anonymous No.9552433]

>>9552428
>Why the faggotry and stale, unfunny meme?
Please do not reply to our posts.

[Anonymous No.9552440]

>>9552422
Yes I did Stewart first year when I was in science and it's a piece of shit with 0 rigor. I know that now, but I didn't know that then. I'm not sure what your background is which is partly why I asked. But even with proper textbooks one can lose sight of why one does the exercises, or you do the exercise feel good about it and don't fully think of the implications.

[Anonymous No.9552509]

>>9552073
Subhuman avatarfag physishits are not welcome here. Refer to >>>/toy/physical/

[Anonymous No.9552531]

>>9552039
>a filthy type theorist
>category theory is mostly incomprehensible for me
So what you're saying is that you aren't even a type theorist?

[Anonymous No.9552661]

>>9552440
Ok, but this still doesn't answer my questions.

[Anonymous No.9552725]

>>9552661
Your questions are low mathematical maturity trash and are not worth answering.

[Anonymous No.9553144]

>>9552357
do more exercises

[Anonymous No.9553214]

>>9552531
He meant that he's a CSfag.

[Anonymous No.9553304]

https://www.math.u-psud.fr/~limic/som/stateofmath.html

Does /mg/ have the so-called "capacity for rigor"?

[Anonymous No.9553315]

>>9552531
Mostly it's higher category theory I'm ignorant of, I can work at the basic level with functors and monads and cartesian or monoidal categories and stuff like that.

[Anonymous No.9553938]

>>9552725
Not really, it's not my mathematical maturity that's the problem. I can digest rigorous texts and concepts fine, but it's the tedium of computation and manipulative tricks that are often my downfall. Many here probably have the same issues, I know we're not all olympiad finalist.

[Anonymous No.9553955]

I was never taught mathematics while growing up (long story). I'm in my early twenties and I currently work in IT. Recently I've been dabbling in programming as part of my day job and I've enjoyed it a great deal.

I've decided I'd like to educate myself. When I say that I wasn't taught math, I really mean it. I'm slow to do simple mental arithmetic, and I don't even know where to begin with percentages, ratios etc.

Is Khan Academy really the best place for somebody in my situation? I've heard a lot of negative press about it, but I don't know what else I should do. I can't find anything online about studying below the level of American "high school" mathematics. Does anybody have any advice?

[Anonymous No.9553967]

>>9553304
tl;dr

[Anonymous No.9554068]

>>9553955
Im not sure what negatives the say from Khan academy, but I would totally recommend it. Would also recommend getting your ged, and typically community colleges have remedial math classes. Here on the east coast, my CC has to get people to learn simple addition, and they usually seem to succeed.
But no matter what, the best way to learn math is practice.

[Anonymous No.9554077]

>>9554068
Thanks for replying anon. I'll press ahead with Khan academy. I'm not in the USA, so I can't take the second part of your advice.

[Anonymous No.9554143]

>>9553304
>putting your personal diary online
>not giving the reader any explanation or motivation up front to read it

[Anonymous No.9554199]

>>9553967
>https://www.math.u-psud.fr/~limic/som/stateofmath.html
Thanks to grade inflation, affirmative action, copypasting LaTeX code, &c. people can graduate with a degree in math and be a complete brainlet.

[Anonymous No.9554336]

>>9554199
I already knew that, especially for women

[Anonymous No.9554341]

>>9554199
Man, I'm only in linear algebra right now, but damn some if the people here make me feel like a genius. I'm pretty stupid myself, so how do these people function in reality?

[Anonymous No.9554352]

>>9554199
Quote from someone in grad school at my uni, "Anyone can get a math phd, you should see some of the idiots here it's unbelievable".

[Anonymous No.9554409]

>>9553967
>>9554199
and it's also written by a glorified statistician.

[Anonymous No.9554418]

>>9553304
>my ""research field"" is probability theory.
Not interested in what a non-mathematician has to say on "the State of Mathematics".

[Anonymous No.9554419]

Does the Finnish girl still post on these? I liked her topology posts.

[Anonymous No.9554431]

>>9554419
I'll go ask her in the cute girls topology department.

[Anonymous No.9554520]

The answer to pic related is that it's a "straightforward check". I'm completely lost as to what I'm supposed to do.

[Anonymous No.9554524]

>>9554520
Do you know what it means for a number to satisfy an equation?

[Anonymous No.9554532]

>>9554520
Substitute those numbers for x.

[Anonymous No.9554538]

>>9554520
Plug and chug boi.

[Anonymous No.9554541]

High school senior, thinking about majoring in math but I don't want to code.

Am I fucked if I'm not a minority or a superstar already? Wouldn't mind professorship

[Anonymous No.9554558]

>>9554532
Thanks. Can't believe I didn't think of that kek.

>>9554524
I know what it means to satisfy for x, fuck lol

[Anonymous No.9554585]

>>9554558
Don't worry, questions will be phrased more rigorously as you study more advanced topics. It's just that when questions are as simple as >>9554520, too much rigor makes it trivial to answer.

[Anonymous No.9554710]

>>9554541
Bruh that math shit isn't gonna go far if you aren't making models and computations, both needing coded programs to get anything done in a reasonable amount of time.
But you can learn.

[Anonymous No.9554717]

>>9554541
>>9554710
>>>/g/hetto/

[Anonymous No.9554858]

>>9554710
Sorry, meant to mean I don't want to code for a living

[Anonymous No.9554862]

I am looking for recommendations for short but hard books on early undergrad topics.

To explain what I mean, I will be taking the Putnam this year but I am already at a point in my undergrad career where I have already taken all of the low-level undergrad topics, and now I only have classes on topics that go beyond the scope of the Putnam (topology and the deeper abstract algebra stuff, etc). This means that I already know linear algebra and calculus, so I just want a book on those topics that very quickly goes through the core results and then presents me with hard problems in those topics.

To give a good example of what I mean, there is a book called "Putnam and Beyond" which has a chapter on each core topic seen in the Putnam exam, and in each chapter it presents the core results and then a bunch of truly hard problems. The problem is that as each main topic only has a chapter, the theory sections are too short. This means that for the topics that I know very well, these short explanations serve me nicely, but on the topics that I am not that good at the short explanations leave me completely unprepared for the mass shooting of a problem section that follows. That is why I am instead looking for a whole book on each topic that is roughly in that spirit.

The two main topics I am looking a book for are calculus and linear algebra, as these are the cornerstone of Putnam problem-solving. But I appreciate books on other topics like geometry and probability if someone has them.

Another example of what I mean is "Equations and Inequalities, Elementary Problems and Theorems in Algebra and Number Theory". This book does what I want, but for advanced high-school tier topics instead. I, unfortunately, can't find similar books for calculus and linear algebra.

[Anonymous No.9554881]

>>9554862
Oh, by the way, a "short" book is roughly like a 200 page book. I don't mind if its longer, but longer books are typically so because they are geared towards non-experts.

[Anonymous No.9554882]

>>9554862
go read about topological signal processing and tell me if there's anything worth value there and get back to me

[Anonymous No.9554885]

>>9554882
I am sure topological signal processing won't appear in a competition.

[Anonymous No.9554890]

>>9554419
>girl

[Anonymous No.9554958]

How would I identify what kind of equation this is and solve it? I'm not sure if it's because it's midnight or if I'm just a sped, but I seriously can't do this.

[Anonymous No.9554973]

Every differential geometry textbook I've encountered uses different notations...

[Anonymous No.9554985]

>>9552159
this anon here

Seriously, what is a good boon on this? Or at least how can I prove #3

[Anonymous No.9555207]

>>9554862
>>9554862
Problems and Theorems in Analysis
Problems and Theorems in Linear Algebra
Selected Problems in Real Analysis

[Anonymous No.9555210]

>>9554973
yeah. you have to develop your own notation and translate, seriously that's the only way.

[Anonymous No.9555214]

This equation does not belong to any common kind of differential equation, but I will teach you how to solve it.

First, you should have in mind a list of all the equations you know how to solve. First order linear equations, linear equations with constant coefficients, exact equations, etc. Then you should scan your equation, in your case your mental scan reveals that it doesn't belong to any case.

When it doesn't belong to any case, that means you need to discover a way to reduce your equation to a different equation you do know how to solve. Here, you just need to get creative. In your equation both [math] y^3 [/math] and [math] y^2 y' [/math] appear. But [math] (y^3)' = 3y^2y' [/math] which implies that if you let [math] u = y^3 [/math] the equation becomes:

[eqn] x^3 + u - \frac{x}{3} u' = 0 [/eqn]

This is now a first order linear differential equation, which you can solve trivially by finding the integrating factor.

[Anonymous No.9555229]

>>9552159
Well there are a number of things you have to know that you will probably learn in due time. That being said, a few tips:
. Know your usual functions and their limits (or lack thereof), eg: cos, sin, polynomials, exp, log, and all their products and quotients (not so easy).
. Prove (and remember) the following result: let f be any function having a limit L at a point x_0 (which may be a real, or +/- infinity) and (u_n) a sequence converging to x_0. Then the sequence (f(u_n)) converges to L
. Learn your classical identities: sum of geometric progressions, arithmetic progressions, sum of squares and cubes, things about binomials etc.
. Most importantly (because this is the most applicable when you aren't doing "rigged" problems), learn to bound tightly whatever it is that you are estimating.

[Anonymous No.9555241]

>>9549768
Lockhart's Lament

[Anonymous No.9555316]

>>9554419
Occasionally. I'm basically just watching the faggotry unfold. You are entertaining, but too much below me in the universal hierarchy of Thulean hyperdoctrines for me to actually interact with you.

[Anonymous No.9555715]

>>9555316
Do you still have an eating disorder?

[Anonymous No.9555839]

So I'm doing some vector stuff as you can see, and I'm trying to find the angle between two vertices. I worked out the dot product first, then I worked out the magnitude of the two vectors and did dot product / magnitude * magnitude. I ended up getting 0.10 for A in this problem set.

My problem is, the answer sheet I'm looking at has "≈1.4716 radians" as the answer.

Can anyone see where I went wrong? I literally can't figure it out, I feel like they've made an error on the result sheet.

[Anonymous No.9555841]

>>9555839
You need to take arccos of 0.1, which gives the correct angle.

[Anonymous No.9555843]

>>9554431
Sorry but we don't take appointments here.

[Anonymous No.9555845]

>>9555841
Oh cheers cunt, I was doing cos, which is the 0.10. I would've been stuck on that for hours, you're a legend.

[Anonymous No.9555870]

>>9554985
>>9552159
Consider the bounded constant sequence with [math]\frac{1+\sqrt5}{2}[/math] as every term. Then [math](2-\frac{1+\sqrt5}{2})(\frac{1+\sqrt5}{2}+1)=1[/math]. And it is constant so it doesn't converge to 1.

We have been trolled anons

[Anonymous No.9555871]

>>9554958
*What kind of differential equation
I'm dumb but I'm not that dumb

[Anonymous No.9555909]

>>9555841
>implying you can't constructively prove the law of excluded middle

[Anonymous No.9555931]

>>9555909

[Anonymous No.9555948]

>>9555931
Linear logic my nigga.

[Anonymous No.9555950]

>>9555909
Go ahead and do it then

[Anonymous No.9556001]

>>9555950
http://homepages.inf.ed.ac.uk/wadler/papers/dual/dual.pdf
Pages 6-7

[Anonymous No.9556083]

>>9555715
I guess some people could say that. Why do you ask?

[Anonymous No.9556157]

>>9554958
lol mate this is for you >>9555214

[Anonymous No.9556234]

Did anyone here ever do Budapest Semesters in Mathematics, and if so would you recommend it? I'm looking to study abroad and my counselor said I should be eligible. I'd also be interested in any other stories of studying math abroad.

[Anonymous No.9556245]

>>9556234
What year are you in? https://www.mccme.ru/mathinmoscow/ is a good one

[Anonymous No.9556271]

>>9556245
I'm a freshman looking to travel next spring, by then I'll have taken at least intro classes in most fields (algebra, diff. geo, analysis, PDE, etc.). I've seen that Moscow one, but Budapest as a city just seems a more interesting place to visit. Did you do it?

[Anonymous No.9556475]

>>9555843
Those are the usual false rumors.

[Anonymous No.9556494]

Why should I read an algebraic topology book? What's the pay-off?

Is there a fundamental theorem of algebraic topology?

What can I do with it that I can't do with linear algebra and calculus?

Sell it to me, /sci/

[Anonymous No.9556502]

>>9556494
>Why should I read an algebraic topology book?
Please don't. We don't need your kind shitting it up.

[Anonymous No.9556507]

>>9556271
>Did you do it?
I only did Moscow, but I know two people who did both Budapest and Moscow (you can even do them back to back if you want) and they both thought the Moscow program was more difficult. I'm not sure what they think about the actual cities

[Anonymous No.9556523]

>>9556494
All of algebraic topology can be done using only basic engineering machinery such as the deep "Fundamental Theorem of Calculus".

[Anonymous No.9556536]

>>9556494
>Is there a fundamental theorem of algebraic topology?
Probably Brown Representability

[Anonymous No.9556545]

>>9556523

Ok, but is there any examples of a problem where it's difficult/impossible to solve using linear algebra and calculus, but can be using methods or theorems taught in a typical algebraic topology text?

I.e. motivation?

Asking honestly, I have a hard time reading something if it just seems like definitions.

[Anonymous No.9556576]

>>9556545
> is there any examples of a problem where it's difficult/impossible to solve using linear algebra and calculus, but can be using methods or theorems taught in a typical algebraic topology text?
Nope, even problems in algebraic topology can usually be solved by using trivial geometric insights from calculus and linear algebra.
For example, some French guy showed that [math]\pi_z(S^n)[/math] for [math]z \in \mathbb{C}[/math] is not not finitely generated as a [math]\mathbb{Q}^{\flat}[/math]-vector space using only the geometrical interpretation of matrix multiplication.
>I have a hard time reading something if it just seems like definitions.
I agree. I think definitions should mostly be dealt away with as well.

[Anonymous No.9556581]

>>9556576
>For example, some French guy showed that πz(Sn) for z∈C is not not finitely generated as a Q-vector space using only the geometrical interpretation of matrix multiplication.
Source?

[Anonymous No.9556585]

>>9556545
Homological methods simply many proofs.

Brouwer Fixed Point theorem, Borsuk-Ulam theorem, Hairy Ball Theorem, Jordan Curve Theorem, etc.

>>9556576
this is nonsense

[Anonymous No.9556591]

>>9556581
You should be better prepared before you ask such questions.
>>9556585
How is a trivial fact "nonsense"?

[Anonymous No.9556610]

i'm currently an engineering science major whom'st is thinking of switching to maths or double majoring
i have an appointment with my advisor next week
how do i convince him i'm smart enough for maths?

i actually hate maths but if i can suffer through a few years of uni and get a fancy piece of paper with my name on it then i'll do it

[Anonymous No.9556612]

>>9556610
>>>/sci/sqt/

[Anonymous No.9556616]

>>9555931
>>>/r/eddit/

[Anonymous No.9556631]

>>9556536

Cool, though I don't understand what this is saying. Are wedge sums at all related to exterior products?

>>9556585
>Brouwer Fixed Point theorem
>Hairy Ball Theorem

I know these two, and they seem pretty intuitive and can at least be explained with geometry. Do you think there any additional insights gained from using the methodology of in algebraic topology to rigorously prove them? (Not that I even know what that would look like.)

[Anonymous No.9556658]

>>9556631
>Are wedge sums at all related to exterior products?
No. The exterior product is an algebraic construction, an alternating tensor product.

The wedge sum is just gluing two space togethers along a basepoint.

>I know these two, and they seem pretty intuitive and can at least be explained with geometry. Do you think there any additional insights gained from using the methodology of in algebraic topology to rigorously prove them?

Yes the give you some topological insight of course. But mainly the homological proofs are just simpler than their counterparts. The exception being proofs via generalized Stokes theorem, as those are very closely related via deRham Cohomology.

[Anonymous No.9556659]

>>9556585

Also, I guess I'm wondering if there is any additional 'machinery' there is for calculating things.

Linear algebra has the matrix, calculus the derivative, ect.

[Anonymous No.9556665]

>>9556658
>The exception being proofs via generalized Stokes theorem, as those are very closely related via deRham Cohomology.

Is this the proof saying the integral of a boundary of a p-form is the same as integrating the enclosed volume of the (p-1)-form?

I haven't read the proof but I've read it in a text for explaining to physicists. I thought that was really cool.

[Anonymous No.9556666]

>>9556659
(co)homology groups are the main things you calculate in algebraic topology.

Also homotopy groups, which have a more intuitive definition but are much much harder to actually compute.

[Anonymous No.9556676]

>>9556665
Here is the proof for Brouwer via stokes.

https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem#A_proof_using_Stokes's_theorem

[Anonymous No.9556688]

>>9556665
Look it up in Bott-Tu. It's actually very simple and more or less boils down to integration by parts (step 0 would be to understand why it generalizes integration by part).

[Anonymous No.9556919]

>>9556083
I liked your blog posting. It was inspiring to read how you had your mental problems but you still kept doing topology. What's your BMI now babe? Have you studied anything cool lately?

[Anonymous No.9556942]

>>9556919
Where can I obtain access to her blog?

[Anonymous No.9556958]

>>9556475
Not false. Just what the club vice president and I decided.

[Anonymous No.9556975]

>>9556942
These threads used to be that

[Anonymous No.9557029]

>>9556958
That false flag operation was recently called off by the club vice president herself. Check the mailing list.

[Anonymous No.9557047]

>>9557029
The fact that she made decisions without my approval means she will soon be relieved of her duties.
The position of vice president is now open anon, would you like to try and apply?

[Anonymous No.9557137]

>>9556545
The best one is the proof of the fundamental theorem of algebra using the fundamental group

[Anonymous No.9557157]

Why do /mg/ study math?

[Anonymous No.9557192]

how do i convince other fellow physicists to have a nice anime posting general like this?
im jealous

[Anonymous No.9557205]

>>9557192
Start one. There had been like 2 or 3 successful ones before they were ruined by autists from here.

[Anonymous No.9557212]

>>9557157
Cause it's the fucking best

[Anonymous No.9557226]

>>9557192
>physics
Such garbage is not welcome here. Take it to some other shithole.
>>9557205
>before they were ruined by autists from here.
You mean similar to how your dog-eating kind shits up these threads?

[Anonymous No.9557236]

>>9549094
>What have you been studying
literally my dick

[Anonymous No.9557253]

>>9557226
>>physics
Who are you quoting?

[Anonymous No.9557276]

>>9557205
>Start one.
me?

pls more 2hu science :3

[Anonymous No.9557282]

>>9557276
>me?
Sure, why not?
>pls more 2hu science :3
:333333

[Anonymous No.9557284]

>>9557192
>>9557205
>>9557276
>>9557282
Refer to >>>/toy/. That's where the physics threads are.

[Anonymous No.9557621]

Is anyone still working on comparing definitions of n-category?

https://ncatlab.org/nlab/show/n-category

[Anonymous No.9557640]

>>9557621
People are probably more interested in (∞,n)-categories

[Anonymous No.9557977]

thank god for anonymity. i need help with a literal elementary-school-tier problem (i never went to school growing up)

Solve for z
42 = -7(z - 3)

I got this shit wrong, and when I checked the answer, it talked you through the steps but started with dividing both sides by -7. What gives? I thought you had to do the things in parentheses first? shouldn't it have become 45 = -7z?

[Anonymous No.9557995]

>>9557977
you can't do the thing in parenthesis because you have the unknown there, instead you just ignored the parenthesis and took the -3 away to the other side
42 = -7(z - 3); 42 = -7z + 21; z = -3

[Anonymous No.9558001]

>>9557995
thanks, i get it now. i must have missed a video that explained that. the khan academy ui is messy

[Anonymous No.9558033]

>tfw have to memorise definitions, theorems and proofs for a degree in econ
Just fuck me in the ass and call me princess

[Anonymous No.9558131]

For which n does 2n+1 divide 2^n + 1?

[Anonymous No.9558139]

>>9558131
n=0, 1, 2, 5, 6, 9...

[Anonymous No.9558152]

>>9558139

Are there infinitely many solutions?

[Anonymous No.9558174]

>>9558152
Probably

[Anonymous No.9558273]

Is this just a meme? It looks pretty comprehensive but I feel like a couple of those books might be superfluous. Also, I hear that Spivak is pretty difficult. Is there a place for a first year calculus text like Thomas' Calculus & Analytic Geometry?

[Anonymous No.9558284]

>>9558273
>Is this just a meme?
Yes,

[Anonymous No.9558318]

>>9558273
It's a poor meme at that. Just read Velleman and skip to Spivak, he's not that hard.
If you need 2 set theory books, 3 proof books, and 4 books on basic math for calculus you should just give up desu.

[Anonymous No.9558355]

>>9556919
>What's your BMI now babe?
Don't objectify me. It's probably somewhere between 17 and 18.
>Have you studied anything cool lately?
I haven't had much time to study anything lately. I do have a homotopy book by Baues next to my bed, though. I'll continue reading it in a few days.

[Anonymous No.9558363]

>>9556919
>blog posting
link?

[Anonymous No.9558376]

>>9558363
I may have been a bit hyperactive and posted a lot of unrelated stuff to these threads a few years ago. That's what zê is referring to. I won't do that again. I have bettered my ways, and there is nothing xõ can do to make me do it again!

[Anonymous No.9558383]

>>9558376
FEMALE
HUMAN
DAUGHTER

[Anonymous No.9558425]

ok category theory >not a he's

https://en.wikipedia.org/wiki/Exterior_algebra#Universal_property

Is it fair to say that the exterior algebra is the free alternating algebra?

[Anonymous No.9558652]

I will never understand how the anime posters in this thread are the ones who know more math than anyone else. Everyone else is jerking off about basic math while those motherfuckers are always talking big league shit. I'm done.

[Anonymous No.9558655]

>>9558425
>ok category theory >not a he's
I'm not a "not a he."
>Is it fair to say that the exterior algebra is the free alternating algebra?
Only for nonpointed vector spaces.

[Anonymous No.9558661]

>>9558655
What do you mean by non-pointed vector spaces

[Anonymous No.9558669]

>>9558652
Probably they're the type of people that read on fringe fields like category theory that have fancy words and just spew incoherent babble that they don't really understand

[Anonymous No.9558672]

>>9558669
>fringe
>category theory

i've been bamboozled, haven't i?

[Anonymous No.9558674]

>>9558661
>non-pointed vector space
An object in the category of bimodules over a nonpointed ring.

[Anonymous No.9558683]

>>9558672
Studying category theory is like studying set theory. Everyone uses the basic facts, maybe once in a while you need to go a little deeper, but you have to be autistic to study it for the sake of studying it.

[Anonymous No.9558693]

>>9558683
>you have to be autistic to study [math] for the sake of studying it.
Quite. That's why we study string theory and TQFT in /mg/, subjects which are empirically founded and deal with reality, not abstract nonsense.

[Anonymous No.9558700]

>>9558683
I study it for type theory.

[Anonymous No.9558721]

>>9558672
Category theory is irrelevant to most of mathematics

[Anonymous No.9558777]

>>9551556
You should instead start with calculus, discrete math or linear algebra and fill in the gaps as you go.

[Anonymous No.9558783]

>>9558273
This is one of the worst progressions I've ever heard of.

Just open up Spivak. If the proofs are hard, read Book of Proof or How to Prove It.

[Anonymous No.9558878]

there are 2 coins

first coin has a 50/50 chance of landing on either side

second coin has a 60% chance of landing on head

how many times do you have to flip both coins for a 95% probability of more heads on the second coin than the first coin?

you should be able to solve this

[Anonymous No.9558897]

>>9558878
Do your own homework.

[Anonymous No.9558958]

>>9558318
>>9558783
I made this guide. I see the same criticism levied against it every time; that it's too much preparation to learn calculus. The guide isn't intended to just get you ready for calculus. If that's your goal you can literally just read any pre-calc book and then go into Spivak/Apostol.

The set theory you get in book of proof will be more than enough for calculus. I put Enderton in there because there should be a book dedicated to set theory in the guide. The reason I call it a "foundational approach", is that it takes a highly formal and axiomatic road to calculus. That's why I put Foundations of Analysis BEFORE Basic Mathematics. That's why the guide starts with a book on first-order logic.

The guide ends with calculus because after that point you can branch out pretty much wherever you want, not because it's some sort of "end goal". The guide is made by, and for, people who enjoy studying foundational issues.

[Anonymous No.9558978]

>>9558273
>Lang
list disregarded

[Anonymous No.9558981]

>>9558958
>I made this guide. I see the same criticism levied against it every time; that it's too much preparation to learn calculus. The guide isn't intended to just get you ready for calculus. If that's your goal you can literally just read any pre-calc book and then go into Spivak/Apostol.
>The set theory you get in book of proof will be more than enough for calculus. I put Enderton in there because there should be a book dedicated to set theory in the guide. The reason I call it a "foundational approach", is that it takes a highly formal and axiomatic road to calculus. That's why I put Foundations of Analysis BEFORE Basic Mathematics. That's why the guide starts with a book on first-order logic.
>The guide ends with calculus because after that point you can branch out pretty much wherever you want, not because it's some sort of "end goal". The guide is made by, and for, people who enjoy studying foundational issues.
t. non-mathematician

[Anonymous No.9558997]

>>9558273
>>9558958
>>9558981
This guide is better

[Anonymous No.9558999]

>>9558997
>This guide is better
The following is the most up to date and rigorous.

High School:
• Euclidean geometry, complex numbers, scalar multiplication, Cauchy-Bunyakovskii inequality. Introduction to quantum mechanics (Kostrikin-Manin). Groups of transformations of a plane and space. Derivation of trigonometric identities. Geometry on the upper half-plane (Lobachevsky). Properties of inversion. The action of fractional-linear transformations.
• Rings, fields. Linear algebra, finite groups, Galois theory. Proof of Abel's theorem. Basis, rank, determinants, classical Lie groups. Dedekind cuts. Construction of real and complex numbers. Definition of the tensor product of vector spaces.
• Set theory. Zorn's lemma. Completely ordered sets. Cauchy-Hamel basis. Cantor-Bernstein theorem.
• Metric spaces. Set-theoretic topology (definition of continuous mappings, compactness, proper mappings). Definition of compactness in terms of convergent sequences for spaces with a countable base. Homotopy, fundamental group, homotopy equivalence.
• p-adic numbers, Ostrovsky's theorem, multiplication and division of p-adic numbers by hand.
• Differentiation, integration, Newton-Leibniz formula. Delta-epsilon formalism.

[Anonymous No.9559000]

>>9558997
I don't agree.

[Anonymous No.9559001]

>>9558999
Freshman:
• Analysis in R^n. Differential of a mapping. Contraction mapping lemma. Implicit function theorem. The Riemann-Lebesgue integral. ("Analysis" by Laurent Schwartz, "Analysis" by Zorich, "Theorems and Problems in Functional Analysis" by Kirillov-Gvishiani)
• Hilbert spaces, Banach spaces (definition). The existence of a basis in a Hilbert space. Continuous and discontinuous linear operators. Continuity criteria. Examples of compact operators. ("Analysis" by Laurent Schwartz, "Analysis" by Zorich, "Theorems and Problems in Functional Analysis" by Kirillov-Gvishiani)
• Smooth manifolds, submersions, immersions, Sard's theorem. The partition of unity. Differential topology (Milnor-Wallace). Transversality. Degree of mapping as a topological invariant.
• Differential forms, the de Rham operator, the Stokes theorem, the Maxwell equation of the electromagnetic field. The Gauss-Ostrogradsky theorem as a particular example.
• Complex analysis of one variable (according to the book of Henri Cartan or the first volume of Shabat). Contour integrals, Cauchy's formula, Riemann's theorem on mappings from any simply-connected subset C to a circle, the extension theorem, Little Picard Theorem. Multivalued functions (for example, the logarithm).
• The theory of categories, definition, functors, equivalences, adjoint functors (Mac Lane, Categories for the working mathematician, Gelfand-Manin, first chapter).
• Groups and Lie algebras. Lie groups. Lie algebras as their linearizations. Universal enveloping algebra, Poincaré-Birkhoff-Witt theorem. Free Lie algebras. The Campbell-Hausdorff series and the construction of a Lie group by its algebra (yellow Serre, first half).

[Anonymous No.9559002]

>>9559001
Sophomore:
• Algebraic topology (Fuchs-Fomenko). Cohomology (simplicial, singular, de Rham), their equivalence, Poincaré duality, homotopy groups. Dimension. Fibrations (in the sense of Serre), spectral sequences (Mishchenko, "Vector bundles ...").
• Computation of the cohomology of classical Lie groups and projective spaces.
• Vector bundles, connectivity, Gauss-Bonnet formula, Euler, Chern, Pontryagin, Stiefel-Whitney classes. Multiplicativity of Chern characteristic. Classifying spaces ("Characteristic Classes", Milnor and Stasheff).
• Differential geometry. The Levi-Civita connection, curvature, algebraic and differential identities of Bianchi. Killing fields. Gaussian curvature of a two-dimensional Riemannian manifold. Cellular decomposition of loop space in terms of geodesics. The Morse theory on loop space (Milnor's Morse Theory and Arthur Besse's Einstein Manifolds). Principal bundles and connections on them.
• Commutative algebra (Atiyah-MacDonald). Noetherian rings, Krull dimension, Nakayama lemma, adic completion, integrally closed, discrete valuation rings. Flat modules, local criterion of flatness.
• The Beginning of Algebraic Geometry. (The first chapter of Hartshorne or Shafarevich or green Mumford). Affine varieties, projective varieties, projective morphisms, the image of a projective variety is projective (via resultants). Sheaves. Zariski topology. Algebraic manifold as a ringed space. Hilbert's Nullstellensatz. Spectrum of a ring.
• Introduction to homological algebra. Ext, Tor groups for modules over a ring, resolvents, projective and injective modules (Atiyah-MacDonald). Construction of injective modules. Grothendieck Duality (from the book Springer Lecture Notes in Math, Grothendieck Duality, numbers 21 and 40).
• Number theory; Local and global fields, discriminant, norm, group of ideal classes (blue book of Cassels and Frohlich).

[Anonymous No.9559003]

>>9559001
>this meam again

[Anonymous No.9559004]

>>9559002
Sophomore (cont):
• Reductive groups, root systems, representations of semisimple groups, weights, Killing form. Groups generated by reflections, their classification. Cohomology of Lie algebras. Computing cohomology in terms of invariant forms. Singular cohomology of a compact Lie group and the cohomology of its algebra. Invariants of classical Lie groups. (Yellow Serre, the second half, Hermann Weyl, "The Classical Groups: Their Invariants and Representations"). Constructions of special Lie groups. Hopf algebras. Quantum groups (definition).

Junior:
• K-theory as a cohomology functor, Bott periodicity, Clifford algebras. Spinors (Atiyah's book "K-Theory" or AS Mishchenko "Vector bundles and their applications"). Spectra. Eilenberg-MacLane Spaces. Infinite loop spaces (according to the book of Switzer or the yellow book of Adams or Adams "Lectures on generalized cohomology", 1972).
• Differential operators, pseudodifferential operators, symbol, elliptic operators. Properties of the Laplace operator. Self-adjoint operators with discrete spectrum. The Green's operator and applications to the Hodge theory on Riemannian manifolds. Quantum mechanics. (R. Wells's book on analysis or Mishchenko "Vector bundles and their application").
• The index formula (Atiyah-Bott-Patodi, Mishchenko), the Riemann-Roch formula. The zeta function of an operator with a discrete spectrum and its asymptotics.
• Homological algebra (Gel'fand-Manin, all chapters except the last chapter). Cohomology of sheaves, derived categories, triangulated categories, derived functor, spectral sequence of a double complex. The composition of triangulated functors and the corresponding spectral sequence. Verdier's duality. The formalism of the six functors and the perverse sheaves.

[Anonymous No.9559006]

>>9559004
Junior (cont):
• Algebraic geometry of schemes, schemes over a ring, projective spectra, derivatives of a function, Serre duality, coherent sheaves, base change. Proper and separable schemes, a valuation criterion for properness and separability (Hartshorne). Functors, representability, moduli spaces. Direct and inverse images of sheaves, higher direct images. With proper mapping, higher direct images are coherent.
• Cohomological methods in algebraic geometry, semicontinuity of cohomology, Zariski's connectedness theorem, Stein factorization.
• Kähler manifolds, Lefschetz's theorem, Hodge theory, Kodaira's relations, properties of the Laplace operator (chapter zero of Griffiths-Harris, is clearly presented in the book by André Weil, "Kähler manifolds"). Hermitian bundles. Line bundles and their curvature. Line bundles with positive curvature. Kodaira-Nakano's theorem on the vanishing of cohomology (Griffiths-Harris).
• Holonomy, the Ambrose-Singer theorem, special holonomies, the classification of holonomies, Calabi-Yau manifolds, Hyperkähler manifolds, the Calabi-Yau theorem.
• Spinors on manifolds, Dirac operator, Ricci curvature, Weizenbeck-Lichnerovich formula, Bochner's theorem. Bogomolov's theorem on the decomposition of manifolds with zero canonical class (Arthur Besse, "Einstein varieties").
• Tate cohomology and class field theory (Cassels-Fröhlich, blue book). Calculation of the quotient group of a Galois group of a number field by the commutator. The Brauer Group and its applications.
• Ergodic theory. Ergodicity of billiards.
• Complex curves, pseudoconformal mappings, Teichmüller spaces, Ahlfors-Bers theory (according to Ahlfors's thin book).

[Anonymous No.9559007]

>>9559006
Senior:
• Rational and profinite homotopy type. The nerve of the etale covering of the cellular space is homotopically equivalent to its profinite type. Topological definition of etale cohomology. Action of the Galois group on the profinite homotopy type (Sullivan, "Geometric topology").
• Etale cohomology in algebraic geometry, comparison functor, Henselian rings, geometric points. Base change. Any smooth manifold over a field locally in the etale topology is isomorphic to A^n. The etale fundamental group (Milne, Danilov's review from VINITI and SGA 4 1/2, Deligne's first article).
• Elliptic curves, j-invariant, automorphic forms, Taniyama-Weil conjecture and its applications to number theory (Fermat's theorem).
• Rational homotopies (according to the last chapter of Gel'fand-Manin's book or Griffiths-Morgan-Long-Sullivan's article). Massey operations and rational homotopy type. Vanishing Massey operations on a Kahler manifold.
• Chevalley groups, their generators and relations (according to Steinberg's book). Calculation of the group K_2 from the field (Milnor, Algebraic K-Theory).
• Quillen's algebraic K-theory, BGL^+ and Q-construction (Suslin's review in the 25th volume of VINITI, Quillen's lectures - Lecture Notes in Math. 341).
• Complex analytic manifolds, coherent sheaves, Oka's coherence theorem, Hilbert's nullstellensatz for ideals in a sheaf of holomorphic functions. Noetherian ring of germs of holomorphic functions, Weierstrass's theorem on division, Weierstrass's preparation theorem. The Branched Cover Theorem. The Grauert-Remmert theorem (the image of a compact analytic space under a holomorphic morphism is analytic). Hartogs' theorem on the extension of an analytic function. The multidimensional Cauchy formula and its applications (the uniform limit of holomorphic functions is holomorphic).

[Anonymous No.9559010]

>>9559007
Specialist: (Fifth year of College):
• The Kodaira-Spencer theory. Deformations of the manifold and solutions of the Maurer-Cartan equation. Maurer-Cartan solvability and Massey operations on the DG-Lie algebra of the cohomology of vector fields. The moduli spaces and their finite dimensionality (see Kontsevich's lectures, or Kodaira's collected works). Bogomolov-Tian-Todorov theorem on deformations of Calabi-Yau.
• Symplectic reduction. The momentum map. The Kempf-Ness theorem.
• Deformations of coherent sheaves and fiber bundles in algebraic geometry. Geometric theory of invariants. The moduli space of bundles on a curve. Stability. The compactifications of Uhlenbeck, Gieseker and Maruyama. The geometric theory of invariants is symplectic reduction (the third edition of Mumford's Geometric Invariant Theory, applications of Francis Kirwan).
• Instantons in four-dimensional geometry. Donaldson's theory. Donaldson's Invariants. Instantons on Kähler surfaces.
• Geometry of complex surfaces. Classification of Kodaira, Kähler and non-Kähler surfaces, Hilbert scheme of points on a surface. The criterion of Castelnuovo-Enriques, the Riemann-Roch formula, the Bogomolov-Miyaoka-Yau inequality. Relations between the numerical invariants of the surface. Elliptic surfaces, Kummer surface, surfaces of type K3 and Enriques.
• Elements of the Mori program: the Kawamata-Viehweg vanishing theorem, theorems on base point freeness, Mori's Cone Theorem (Clemens-Kollar-Mori, "Higher dimensional complex geometry" plus the not translated Kollar-Mori and Kawamata-Matsuki-Masuda).
• Stable bundles as instantons. Yang-Mills equation on a Kahler manifold. The Donaldson-Uhlenbeck-Yau theorem on Yang-Mills metrics on a stable bundle. Its interpretation in terms of symplectic reduction. Stable bundles and instantons on hyper-Kähler manifolds; An explicit solution of the Maurer-Cartan equation in terms of the Green operator.

[Anonymous No.9559014]

>>9559007
Specialist (cont):
• Pseudoholomorphic curves on a symplectic manifold. Gromov-Witten invariants. Quantum cohomology. Mirror hypothesis and its interpretation. The structure of the symplectomorphism group (according to the article of Kontsevich-Manin, Polterovich's book "Symplectic geometry", the green book on pseudoholomorphic curves and lecture notes by McDuff and Salamon)
• Complex spinors, the Seiberg-Witten equation, Seiberg-Witten invariants. Why the Seiberg-Witten invariants are equal to the Gromov-Witten invariants.
• Hyperkähler reduction. Flat bundles and the Yang-Mills equation. Hyperkähler structure on the moduli space of flat bundles (Hitchin-Simpson).
• Mixed Hodge structures. Mixed Hodge structures on the cohomology of an algebraic variety. Mixed Hodge structures on the Maltsev completion of the fundamental group. Variations of mixed Hodge structures. The nilpotent orbit theorem. The SL(2)-orbit theorem. Closed and vanishing cycles. The exact sequence of Clemens-Schmid (Griffiths red book "Transcendental methods in algebraic geometry").
• Non-Abelian Hodge theory. Variations of Hodge structures as fixed points of C^*-actions on the moduli space of Higgs bundles (Simpson's thesis).
• Weil conjectures and their proof. l-adic sheaves, perverse sheaves, Frobenius automorphism, weights, the purity theorem (Beilinson, Bernstein, Deligne, plus Deligne, Weil conjectures II)
• The quantitative algebraic topology of Gromov, (Gromov "Metric structures for Riemannian and non-Riemannian spaces"). Gromov-Hausdorff metric, the precompactness of a set of metric spaces, hyperbolic manifolds and hyperbolic groups, harmonic mappings into hyperbolic spaces, the proof of Mostow's rigidity theorem (two compact Kählerian manifolds covered by the same symmetric space X of negative curvature are isometric if their fundamental groups are isomorphic, and dim X> 1).
• Varieties of general type, Kobayashi and Bergman metrics, analytic rigidity (Siu)

[Anonymous No.9559078]

What are some good resources that generate random practice problems in selected topics? Something not quite Olympiad level, but just a general tool for excersizing my elementary skills (arithemtic, algebra, geometry, computation, combinatorics, logic, probability and calculus).

[Anonymous No.9559254]

How many times is it acceptable to use "in particular" in a math paper? I have used it 3 times in a 10 line proof.

[Anonymous No.9559275]

>>9558958
>That's why I put Foundations of Analysis BEFORE Basic Mathematics
kek, either you're actually retarded or a masterful baitman.

The chart has good books in it (for the most part) on an individual basis, but there is little productive synergy or cohesiveness, the order sucks, and a mathematician who spent a year or two doing this chart would be miles and miles behind a mathematician who just did Spivak supplemented with a proof book if necessary, and then spent the extra 11-23 months learning tons of other great math.

>>9558997
Equally shit, way to much prep for a James Stewart book, absolutely retarded. Everything before Stewart can be cut and replaced with "see Khan Academy as needed", or something similar. Proofs books only need to be an option, not exactly something that should be required reading cover to cover, as a real math text with force you to learn proofs anyhow.

Starting the "real math" with set theory is also retarded if you've never dealt with higher maths. This is why you don't learn set theory before you learn to count. A dedicated text for set theory is not necessary, especially in conjunction with a proof book just before it. Many, many books that are not about set theory explicitly contain reviews of it, so set theory is best kept as an option.


The real patricians guide is to just open up Spivak and power through it, using auxiliaries and supplements as necessary. Alternatively, Concrete Mathematics is likely just as good of a starting point, as is any renowned linear algebra book.

[Anonymous No.9559410]

I just won a Fields Medal.

[Anonymous No.9559414]

>>9558669
>>9558683
>the grapes are sour

[Anonymous No.9559416]

>>9559275
>>9559000
>>9559003
>>9558318
>>9558981
>>9558284
>>9558783
>>9558978
I'd like it if /sci/ could stop dismissing these guides and stating why they're wrong instead of making a new and good one that we can post so we don't get shitty meme ones like this: >>9558273
I'll give you guys a day to make a full list ranging from Highschool to undergrad (just up to analysis) that we can circle around without some faggot dismissing it and turning the thread into debates complaining about a picture

[Anonymous No.9559417]

>>9559410
gz

[Anonymous No.9559422]

>>9559416
but reading lists are for brainlets and /sci/ is nothing but brainlets so why would that ever change? just go read anything and enjoy it, it just doesn't matter.

[Anonymous No.9559423]

>>9559416
>faggot
Why the homophobia?

[Anonymous No.9559426]

>>9559416
>I'd like it if /sci/ could stop dismissing these guides and stating why they're wrong instead of making a new and good one that we can post so we don't get shitty meme ones like this: >>9558273
>I'll give you guys a day to make a full list ranging from Highschool to undergrad (just up to analysis) that we can circle around without some faggot dismissing it and turning the thread into debates complaining about a picture
see >>9558999
>>9559001
>>9559004
>>9559006
>>9559007
>>9559010
>>9559014

[Anonymous No.9559446]

>>9549094
>NO Grothendieck or Serre.
Fuck you: https://arxiv.org/pdf/1802.10425.pdf

[Anonymous No.9559450]

>>9559422
You're contradicting yourself if you dismiss lists like >>9558273 yet suggest reading whatever they want, if they go for a list, they probably want to learn those things

There would be a lot less brainlets if we gave them a good list on reading materials so we don't get threads debating how good a book is and confused high schoolers coming in here getting meme list turning the thread into just a shitshow bashing on a list cause it sucks

>>9559426
An original list not made by a man who masturbates to russian action figures please, try again next time

[Anonymous No.9559469]

>>9558683
>Studying category theory is like studying set theory.
Clearly you have studied neither. And while you had enough intelligence to make the correct decision for "set theory", it seems like your brain power wasn't enough when it comes to the theory of cats. It's a common issue among the not so bright ones so I don't really blame you for this.

[Anonymous No.9559482]

Ibn Al-Haytham's Completion of the Conics.

It's probably one of the smartest things I've read in mathematics thus far. It DOES have some errors in it, as notated by the translator, but between Ibn Al-Haytham's flawless execution of the Analytical-Synthetical mode and Maimonides comments/amendments, this piece of mathematical literature is just downright fascinating to read.

It has me going "WOAHHHH" "OH MY GOOOOOOOD" sometimes haha. I find myself drop-jawed looking at the mental process that I've just observed, as sometimes I do.

[Anonymous No.9559513]

>>9549094
Power series differential equations and i need help with a question:
(x^2+1)y''+2xy'=0

[Anonymous No.9559535]

>>9559416
I already gave you one. Start with Spivak's or Apostol's 'Calculus' (followed by Schlomo's 'Advanced Calculus' or Spivak's 'Calulus on Manifold's or Apostol's 'Calculus, Vol.2'), Knuth's 'Concrete Mathematics' or Hoffman and Kunze's 'Linear Algebra' and refer to auxiliary and supplementary material like Book of Proof, Khan Academy, Paul's Online Math Notes, etc as needed to fill in the gaps. If you're totally at a loss, backtrack to something like Axler's 'Precalculus' and/or Lovasz's 'Discrete Mathematics'. Don't waste time with 500 meme books to learn calculus, like this board often suggest.

After that, do analysis (since that's what you requested) via Rudin, or for a more gentle introduction, Tao. But if you've read any of the canonical works above, you're ready to handle anything and can read about whatever you want rather than reading another 5 books on intro set theory.

[Anonymous No.9559542]

>>9559535
>Rudin
Rudin is a meme.

[Anonymous No.9559545]

>>9559535
Additionally, for each of those materials I've listed there's a plethora of lecture notes, video series, MOOCs, etc to accompany the material. Stop wanking and start doing maths, faggot.

[Anonymous No.9559550]

>>9559542
Rudin is a canonical work in analysis, and if you're going to self learn there's an advantage to staying with the canon. But of course, canon = meme.

Do you have any real criticisms of the exposition, or are you just upset that your too mathematically immature to understand Rudin?

[Anonymous No.9559552]

>>9558683
>Studying category theory is like studying set theory.

[Anonymous No.9559565]

Can someone help me understand this?

[math]S_5 [/math] is a symmetric group. In 1 and 2, I see the maps because 1 maps to 3, 3 to 4 and 4 to 1, but I don't understand how this forms a composition function 2.

[Anonymous No.9559569]

>>9559565
Me again, just to say I understand how they arrived here https://en.wikipedia.org/wiki/Symmetric_group#Elements , but I only understand the syntax, not the underlying concept of what's actually happening. I don't understand at all how this is a composition of functions, only that the syntax represents two mappings (in ex2, 1 maps to 3, which maps to 4, which maps to 1, and then there's a separate correlation between 2 mapping to 5, but what's actually happening and being represented is unclear to me).

[Anonymous No.9559577]

>>9559569
>I don't understand at all how this is a composition of functions
Cycles are bijective functions. If the only action the cycle takes is swapping 2 and 5, then for any element outside those two, (2,5) doesn't move that element, i.e. it maps it to itself. This is the definition of a cycle.

If you break a function into a bunch of disjoint cycles, there's no harm done in daisy-chaining them all together because only one of them at a time can ever do anything, and the rest will be identities.

[Anonymous No.9559611]

>>9559535
>>9559545
This isn't really enough. Your list is not as in-depth (why hoffman & kunze, why knuth?) as those meme guides like >>9558273 and badly structured (cluster of backtracking). Also it isn't a picure as I requested, your post will be forgotten after this thread dies while high schoolers will continue to flood this board and people will continue to post meme lists.
Get to work and make a good list anon! Or forever let shitty lists roam and stop complaining

[Anonymous No.9559639]

I'm looking to work through a differential geometry book over the summer, before I go to graduate school. I've asked a few peers and they recommended either Lee's Introduction to Smooth Manifolds or Tu's Introduction to Manifolds. What is the difference between these, and what would be better? I will have finished a course on algebraic topology covering singular homology and cohomology (chapters 2 and 3 in Hatcher), and an algebra sequence covering most of Dummit and Foote, in case that makes any difference. I think that in general, I learn best when a textbook is closer to a sequence of exercises that guides you through a topic, rather than lots of long exposition.

[Anonymous No.9559651]

>>9549208
Bro x^2 is only less than 2x in [0,1], so in [-1,1] we have that x^2 -2x has a range of [-1,3], and outside of [-1,1] we have x^2 -2x >=0. So just add 2 and you get a range thats always above 0

[Anonymous No.9559675]

>>9559577
Ok, I think I'm beginning to understand (just watched a lecture on symmetric groups too). But pic related is still confusing to me.

So I'd begin with the innermost function, (1364)
>1 2 3 4 5 6
>3 2 6 1 5 4

What is the procedure for applying the outside function correctly though? I would (45) would mean
>1 2 3 4 5 6
>3 2 6 5 4 4

Since I just remapped 4 to 5, and 5 to 4. But that clearly isn't right.

[Anonymous No.9559682]

>>9559611
>your post will be forgotten
I don't care if you continue to suck at math.

>not as in depth
You mean it doesn't waste time piddling around with 100 prerequisite books? Hoffman and Kunze and Knuth because they are unrivaled canon in their respective fields.

You will always be backtracking and growing your mathematical roots, it's part of the process, not "cluster".

>>9559639
I would look into Do Camo's and Spivak's respective works in the fields. Also, for fun, maybe look at "Functional Differential Geometry by Sussman if you like to program. I haven't read these though so I don't know.

[Anonymous No.9559699]

Who the hell makes crap like this and sticks it on Wikipedia

[Anonymous No.9559708]

>>9559682
>if you like to program.
Post disregarded. >>>/g/hetto

[Anonymous No.9559725]

>>9559708
obviously I'm not big on programming, didn't you notice that unclosed quotation mark?

[Anonymous No.9559726]

>>9559675
nevermind, got it. thank you math irc guy

[Anonymous No.9559840]

>>9559699
I like it

[Anonymous No.9559882]

new thread when

[Anonymous No.9559900]

>>9559882
Given some of the last threads, we should just die off.

[Anonymous No.9559943]

do you guys smoke weed

and be honest how many hours a day/week/month do you study? I see lots of larpers saying they are autist and "study all day", yet they find time to pounce on an opportunity to humblebrag in every relevant thread

[Anonymous No.9559980]

>>9559943
>do you guys smoke weed
No. If I am going to poison myself with drugs might as well be ones that increase my productivity, not worsen it.

[Anonymous No.9560071]

>>9559980
why does weed worsen your productivity? tip: you're a brainlet

[Anonymous No.9560080]

>>9560071
Weedfags are so obnoxious. Nobody gives a shit that you smoke.

[Anonymous No.9560277]

>>9559682
>I don't care if you continue to suck at math.
What's this supposed to mean?
>You mean it doesn't waste time piddling around with 100 prerequisite books?
Not at all, extremely dumb assumption. I meant you didn't justify your books well enough as why they're the best choice.
>You will always be backtracking and growing your mathematical roots, it's part of the process, not "cluster".
True, can't argue with that, but the list isn't supposed to be a cluster, you could rearrange stuff and get shit in correct order

[Anonymous No.9560380]

>>9559639
I'd recommend Tu, mostly because I can't stand how Lee's books are formatted (And Tu's is great)

[Anonymous No.9560463]

this shit needs to be a threadly reminder
>>9552509
>Subhuman avatarfag physishits are not welcome here. Refer to >>>/toy/physical/

[Anonymous No.9560494]

Threadly reminder to work with physicists.

[Anonymous No.9560497]

>>9560494
>physishits
See >>>/toy/.

[Anonymous No.9560516]

>mg - math homework help, math literature recommendations, and whining about physics

[Anonymous No.9560518]

>>9559545
>faggot
Why the homophobia?

[Anonymous No.9560545]

>>9560516
>he didn't mention anime shitposting and not being hes

[Anonymous No.9560552]

>>9560545
>he

[Anonymous No.9560585]

>>9549854
Ye

[Anonymous No.9561361]

>>9560277
>What's this supposed to mean
That I don't care if you continue to suck at math, because you're waiting on a meme chart and other people to research for you. In the meantime I'm going to continue studying along the patrician path,

>Not at all, extremely dumb assumption. I meant you didn't justify your books well enough as why they're the best choice.
>extremely dumb assumption
'extremely dumb' phrasing and hyperbole. You said it wasn't as "in depth", if you meant it wasn't descriptive enough you should've said so - depth implies a quantity, given the context I assumed books. Further, DYOR.


>>9560277
True, can't argue with that, but the list isn't supposed to be a cluster, you could rearrange stuff and get shit in correct order
Are you unable to rearrange things mentally? This will be your first exercise.

[Anonymous No.9561393]

>>9561361
>In the meantime I'm going to continue studying along the patrician path,
cringe

[Anonymous No.9561745]

>>9561361
>That I don't care if you continue to suck at math, because you're waiting on a meme chart and other people to research for you. In the meantime I'm going to continue studying along the patrician path
cringe af to read, if you don't care about people being bad at math don't bash on their meme charts and provide a list of books you'd recommend instead.
>'extremely dumb' phrasing and hyperbole.
cringe as hell again, don't read into things too much and take them too literal, you make yourself appear retarded
>You said it wasn't as "in depth", if you meant it wasn't descriptive enough you should've said so - depth implies a quantity, given the context I assumed books. Further, DYOR.
sigh
>Are you unable to rearrange things mentally? This will be your first exercise.
bait attempt of a mentally challenged /pol/ user, I'll give you an imaginary applause for your efforts

[Anonymous No.9561777]

>>9561745
>because you're waiting at other people to do research for you
>lol if ur gonna say that just make me a list of books

[Anonymous No.9561819]

>>9561777
>lol if ur gonna say that just make me a list of books
Should've read as "and yet* provide a list of books you'd recommend instead" since you provided it anyways here >>9559535
I'm surprised you're so retarded I had to clarify it for you

[Anonymous No.9561823]

>>9561822

New

[Anonymous No.9562210]

>>9561745
>cringe af to read, if you don't care about people being bad at math don't bash on their meme charts and provide a list of books you'd recommend instead.
>cringe af
If you didn't observe the irony in "patrician path", improve your reading comprehension. The cringe was intentional. Additionally, I've given you my recommendations already.

>cringe as hell again, don't read into things too much and take them too literal, you make yourself appear retarded
Says the one waiting indefinitely for recommendations I've already delivered. Even more retarded is that a simple series of searches via google will give you more than enough leads.

>sigh
nice arguement, brainlet.

>bait attempt of a mentally challenged /pol/ user, I'll give you an imaginary applause for your efforts
I've yet to say anything that would imply any politic alignment, nor have I discussed politics at all, other than perhaps a subtle endorsement of a socially capitalistic philosophy in that if you wanna git gud, you should put in the work.

You have failed your first exercise.


>>9561819
This doesn't even make sense, you are truly beyond help. Just start fucking studying man, 'aint nothin to it but to do it'.